i_,,_,^,^ii,,,,i,,,^...._'..;''''- g, _' = _ WHIPPLE ARCH-TRUSS BRIDGE. WHIPPLE TRAPEZOIDAL BRIDGE. AN ELEMEN TARY AND PRACTICAL TREATISE ON AN ENLARGED AND IMPROVED EDITION OF THE AUTHOR'S ORIGINAL WORK, BY S. WHIPPLE, C. E ALBANY, N. Y., INVENTOR OF THE WHIPFLE BRIDGES, &C. NEW YORK: D. VAN NOSTRAND, PUB3LISHE23 Murray St., and 27 Warren St. 1872. Entered according to Act of Congress, By S. WHIPPLE, In the office of the Librarian of Congress at Washington. MIUNSELL, PRINTER. ALBANY. INTRODUCTION. Itis about thirty years since the Author's attention was especially directed to the subject of BRIDGE CONSTRUCTION; and, his Original Essays published in 1847, are believed to have aided considerably toward establishing the foundation upon which a knowledge of the principles involved, and the conditions required in the proper construction of TRUSS BRIDGES, has been built up, and carried to a high state of advancement. However that may be, the flattering terms in which his former labors in the premises have often been referred to, as well as the frequent applications for copies of his former publication, since the supply became exhausted, have prompted the issue of the present volume. This work inculcates the same development of GENERAL PRINCIPLES, and treats of essentially the same General Plans, Combinations, and proportions for bridge work, as were discussed and recommended in its humble predecessor; with such iv INTRODUCTION. additions and improvements as subsequent experience and observation have enabled the Author to introduce. The design has been to develop from Fundamental Principles, a system easy of comprehension, and such as to enable the attentive reader and student to judge understandingly for himself, as to the relative merits of different plans and combinations, and to adopt for use, such as may be most suitable for the cases he may have to deal with. It is hoped the work may prove an appropriate Text Book upon the subject treated of, for the Engineering Student, and a useful manual for the Practicing Engineer, and Bridge Builder. But as to this, the decision must be left to those into whose hands it may fall; and to that arbitrement, without further remark or explanation, it is respectfully submitted. CONTENTS. Page. Preliminary remarks, &c. - - 1 Two Panel Trusses.,- - - - 9 Three Panel Trusses 16 Five Panel Trusses -. 20 Seven Panel Trusses. The Arch Truss - - 31 Trapezoid Trusses with Verticals -- 46 Trapezoid Trusses without Verticals -- 54 Decussation of forces, &c. - 64 The Warren Girder - -- 69 The Finck Truss - - 73 Characteristics of the Arch - - - 74 Weight of Structure 77 Double Cancelated Trusses with Verticals - - - 80 Double Cancelated Trusses without Verticals - - 86 Decussation in Trusses with Verticals - - - 92 Deck Bridges - - - - 97 Ratio of length to depth of Truss - 100 Inclination of Diagonals 1- 104 Width of Panel - - - 09 Arch Bridges - - - 112 Construction of equilibrated Arches - - 114 Webbed Arches - - - - -117 Ordinates of equilibrated curves determined by calculation 119 Action of the web in webbed Arches - - - 125 Effects of heat upon Arches without Chords - - - 126 Bridge Materials, Wood and Iron compared - - 132 IRON BRIDGES. Strength of Iron - - - i42 Experiments on Cast Iron, &c. - - 146 Safe practical strain of Iron - - - - 51 Note giving examples of stress of Wrought Iron in several structures in use - - - - 153 to 155 Table of Negative Strength of Iron in pieces of various lengths and sections - - - - 159 vi CONTENTS. Page. TRANSVERSE strength of Iron - - - 161 to 171 ARCH TRUSS BRIDGES -- - 172 Iron Beams for Bridges - - - - - 182 Modes of Insertion of Iron Beams - - -184 The Link Chord - - - - - - 188 The Eye-bar Chord - -- - 192 Size of connecting Pins - -- 193 Riveted Plate Chord -- - 196 Trapezoidal Truss Bridges,-details - - - 200 Double Cancelated Bridges,- details - - - 205 Wrought Iron thrust members -- 214 Double Chord, a suggestion - - -224 Rivet-work Bridges - - - - - - 227 Sway-bracing - - - -- - 236 Comparison of Bridge Plans -- 241 Bollman Truss - - - -241 Finck Truss - - - 244 Post Truss - - - - - - - 246 Whipple Trapezoid - - - - 251 The Isometric (without verticals) - - - 253 The Arch Truss - 256 Synopsis of results of analyses - - - - 257 General Remarks - 258 COUNTER-BRACING,-value of - - 263 WOODEN BRIDGES -Strength of timber - - 274 Table of Negative resistance of timber - - - -276 Transverse strength of Wood - - - - - 277 Resistance to cleavage --- - 278 Connections of Tension pieces - 279 Connecting pins of Wood and Iron - 281 Splicing - - - - 285 Construction of Wooden Trusses - - - - 288 Two panel Wooden Truss Bridge - - - - 288 Three panel Wooden Truss Bridge - - - - 294 Four and Six panel Wooden Truss Bridge - - 296 The Howe Bridge - -302 Wooden Trapezoid without verticals - - - 306 MODULUS of strength of Trusses - - - 314 PRELIMINARIES. I. A bridge is a structure for sustaining the weight of carriages, animals, &c., during their transit over a stream, gulf or valley. Bridges are constructed of various plans and dimensions, according to the circumstances and objects requiring their erection; and it is the purpose of this work, after a few remarks upon the general nature and principles of bridges, to attempt some analyses and comparisons of the respective qualities and merits of various general plans, with a view of deducing practical results, as to a judicious and economical choice and application of materials in the construction of these useful and important structures. II. The force of gravity, on which the weight of bodies depends, acts in vertical lines, and consequently, a heavy body can only be prevented from falling to the earth, by a force equal and opposite to that with which gravity impels the body downward. This resisting force must not only act vertically upward, but the line of its action must pass through the centre of gravity of the body it sustains. All the forces in the world, acting parallel with, or perpendicular to, the vertical passing through its centre of gravity, could not prevent a 1 2 BRIDGE BUILDING. musket ball (concentrated to the point of its centre of gravity) from falling to the centre of the earth, unless it were a horizontal force capable of giving the ball a projection, such that the centrifugal tendency should equal or exceed gravity a kind of force which could never be made available toward preventing people from falling into the water in crossing rivers; consequently, having no application in bridge building. In fact, nothing but a continuous series of unyielding material particles, extending from an elevated body downward to the earth, can hold or sustain that body above the earth, by vertical and horizontal action alone, either separately, or in combination. III. Suppose a body, no matter how great or small, placed above the earth, with a deep void, or an inaccessible space beneath it. Attach as many cords to it as you please, strain them much or little - only horizontally - the body will fall, nevertheless. Thrust any number of rods, with whatever force you may, horizontally against it; still the body will fall. This is obvious from the fact that horizontal forces, acting at right angles with the direction of the force of gravity, have no more tendency to prevent, than to promote the fall of the body. Moreover, the space beneath being inaccessible, there is no foundation, or foot hold, upon which to rest a post or stud that may directly resist the action of gravity, while the lines of all other vertical forces or resistances, pass by the body without touching it. In the case here supposed, the body can only be prevented from falling by oblique forces; that is, by forces whose lines of action are neither exactly horizontal, nor exactly perpendicular. Attach two cords to the PRELIMINARIES. 3 body, draw upon them obliquely upward and outward, in opposite directions, or from opposite sides of the void, with a certain stress, and the body will be sustained in its position. Apply two rods to it obliquely upward, of a proper degree of stiffness, in the same vertical plane, and on opposite sides of the perpendicular, a certain thrust exerted upon those rods, will prevent the descent of the body. IV. Here, then, we have the elementary idea -the grand fundamental principle in bridge building. Whatever be the form of structure adopted, the elementary object to be accomplished is, to sustain a given weight in a given position, by a system of oblique forces, whose resultant shall pass through the centre of gravity of the body in a vertically upward direction, in circumstances where the weight can not be conveniently met by a simple force, in the same line with, and opposite to, that of gravity. For a more clear illustration of this elementary idea, let us suppose a a', Fig. 1, to represent the banks of a river, or the abutments of a bridge; and gg', the line of transit for carriages, &c.; and, let us further suppose a load of a certain weight, w, to have arrived at a point centrally between a a'. The simplest method of sustaining the weight is, perhaps, either to erect two oblique braces aw.. atw, or suspend two oblique chains or ties pw, p'w, from fixed supporting points a a', or p p. It is not necessary that the weight be at the angular point w, of the braces or chains, but it may be sustained by simple suspension at w' below, or simple support at w" above, and such obliquity may be given to the braces or chains as may be most economical; a consideration which will be taken into account hereafter. 4 BRIDGE BUILDING. V. Thus we see how a weight may be sustained centrally between the banks of a river, or the extremities of a bridge. But the structure must not only provide for the support of weight at this point, but also at every other point between a a', or g g; and it is obvious that the same plan and arrangement will apply as well at any other point as at the centre, with only the variation of making the braces or chains of unequal length. FiG. 1. J w" 7\ D I This, however, would require as many pairs of braces or chains as there were points between g g', a thing, of course, impracticable, since the oblique members would interfere with one another, and be confounded into a solid mass. We therefore resort to the transverse strength and stiffness of beams, -phenomena with which all have more or less acquaintance, and without digressing in this place to investigate their principles and causes, it will be assumed as a fact sustained by all experience, that, for sustaining weight between two supporting points upon nearly the same level, a simple beam affords the most convenient and economical means, until those points exceed a certain distance asunder, which distance will vary with circumstances; PRELIMINARIES. 5 but in bridge building, will seldom be less than 10 to 14 feet, where timber beams are employed. Hence, for bridges of a length of 12 to 14 feet, usually, nothing better can be employed than a structure supported by longitudinal beams, with their ends resting upon abutments or supports upon the sides of the stream. Of course, no reference is here had to stone or brick arches. For, though these are advantageously used for short spans, and in deep valleys, where the expense of constructing high abutments for supporting a lighter superstructure, would exceed or approximate to that of constructing the arch, it is the purpose of this work to speak only of those lighter structures, composed mostly of wood and iron, and supported by abutments and piers of stone, or by piles, or frames of wood. Having then adopted the use of beams for supporting weight upon short spaces, it is only necessary upon longer stretches, to provide support for a point once in 10 or 14 feet, by braces, &c., from the extremities; and for intermediate points, to depend on beams or joists extending from one to another of the principal points provided for as above.* VI. For a span of 20 or 30 feet, it would seem that no better plan could be devised, than to support a transverse beam midway between abutments, by two pairs of braces or suspension chains, proceeding from points at or over the abutments, one pair upon each side of the road-way; this transverse beam affording support for longitudinal beams or joists extending * It is susceptible of easy demonstration that the power of beams to sustain weight by lateral stiffness, forms no exception to the principle that oblique forces alone can sustain heavy bodies over inaccessible spaces. But this matter is deferred for the present. 6 BRIDGE BUILDING. therefrom to the abutments. When suspension chains are used, it may properly be called a suspension bridge. If braces be employed, it is usually termed a truss-bridge. HORIZONTAL ACTION OF OBLIQUE MEMBERS. VII. Before advancing further, it will be proper to refer to an important principle or fact which has not yet been taken into account, though a fact by no means of secondary interest. The sustaining of weight by oblique forces, gives rise to horizontal forces, for which it is necessary to provide counteraction and support, as well as for the weight of the structure and its load. The two equal and equally FIG. 2. inclined braces, ac and be, Fig. 2, in supporting the weight w eQ ~ at c, act in the directions of II ^^T \their respective lengths, each _/ -j. \ with a certain force, which is f 6....., b - equivalent to the combined a action of a vertical and a horizontal force, [Elementary Mechanics -Statics,] which may be called the vertical and horizontal constituents of the oblique force. These two constituent forces bear certain determinate relations to one another, and to the oblique force, depending upon the angle at which the oblique is inclined. INow, we know that the vertical constituent alone contributes to the sustaining of the weight, and consequently, must be just equal to the weight sustained, in this case equal to ~w. We know moreover, from the principles of statics, that three forces in equilibrio, must have their lines of action in the same plane, and PRELIMINARIES. 7 meeting at one point; and must be respectively proportional to the sides of a triangle formed by lines drawn parallel with the directions of the three forces; and that each of the three forces is equal and opposite to the resultant of the combined action of the other two. We have, then, at c, the weight I w, the oblique force in the line ac, and a third force, equal and opposite to the horizontal constituent of the oblique force in the line ac. Then, letting fall the vertical de, and drawing the horizontal ad, the sides of the triangle acd, are respectively parallel with the three forces in equilibrio at the point c. Hence, representing the vertical cd, by v, the horizontal ad, by A, and the oblique by o and calling the horizontal force x, and the oblique force, y, we have the following proportions: (1). w: x:: v: h, whence, x = — w (2). ~w:: v: o, whence, y - w~ But }w equals the weight sustained by the oblique ac. Therefore, from the two equations above deduced, we may enunciate the following important rule: The horizontal thrust of an oblique brace, equals the weight sustained, multiplied by the horizontal and divided by the vertical reach of the brace; and the direct thrust (in the direction of its length), equals the weight sustained multiplied by the length, and divided by the vertical reach of the brace. VIII. Now, it is obvious that the brace exerts the same action, both vertically and horizontally, at the lower, as at the upper end, though in the opposite directions; the brace being simply a medium for transmitting the action of weight from the upper to the 8 BRIDGE BUILDING. lower end of the brace. Hence, the weight sustained by the brace ac, exerts the same vertical pressure at the point a, as it would do if resting at that point, while the brace requires a horizontal resistance to prevent its sliding to the left, as would be the case if its foot simply rested upon a smooth level surface. This horizontal resistance may be provided by abutments of such form, weight, and anchorage in the earth, as to enable them to resist horizontally as well as vertically, or by a horizontal tie, in the line ab, connecting the feet of opposite braces. These two methods are both feasible to a certain extent, and in certain cases; and, both involve expense. Under particular circumstances, it may be a question whether the former should not be resorted to, wholly or partially. But for general practice, in the construction of bridges for heavy burthens, such as rail road bridges, and especially iron truss bridges, where expansion and contraction of materials produce considerable changes, it is undoubtedly best to provide means for withstanding the horizontal action of obliques, within the superstructure itself; and this principle will be adhered to in the discussions following. The preceding remarks and illustrations as to the action of braces, or thrust obliques, obviously apply in like manner to obliques acting by tension, with only the distinction, that in the latter case, the weight is applied at the lower, and its action transmitted to the upper end of the oblique, and the horizontal action (at the remote end), is inward, and toward the vertical through the weight, instead of outward; and consequently, must be counteracted by outward thrust, as by a rigid body between the points p p', Fig. 1, or by heavy towers, and anchorage capable of withstanding Two PANEL TRUSSES. 9 the inward tendency. Hence, in applying the rule before given, to tension obliques, and their vertical and horizontal constituents, the word pull should be substituted for the word thrust, wherever the latter occurs in said rule. TWO PANEL TRUSSES. IX. There are three forms of truss adaptable to bridges with a single central beam or cross bearer (which may be called two FIGd. B panel trusses), the general a characteristics of which, are respectively represented by Figures 3, 4 and a - c~~ 5 li5. Fig. 3 represents a b C pair of rafter braces, with feet connected by a horizontal tie, and with a vertical tie by which the beam is suspended at or near the horizontal tie, or the chord, as usually designated. For convenience of comparison, let bd = v= 1= vertical reach of oblique members in each figure. Also, let each chord equal 4v, = 4, and the half chord = 2 = h = horizontal reach of obliques in Figs. 3 and 4. Then ad, Fig. 3, equals Vh2+v2 = V5, and if the truss be loaded with a weight w, at the point b, bd will have a tension equal to w, and abc, [see rule at end of Sec. VII], a tension equal to 1w, ( = weight sustained by ad), multiplied by the horizontal, and divided by the vertical reach of ad; that is, equal to w^-, ==-w, -= w; while ad suffers compression from end to end, equal to d. But ad=V/5, and v=1. Whence wd =~wV5. 2 10 BRIDGE BUILDING. Now, as the cross-section of a piece, or member, exposed to tension (or to thrust, when pieces are similar in figure), should be as the stress, it follows that the weight of each such member should respectively, be as the stress sustained, multiplied by the length, + an additional amount taken up in forming connections; which latter, for purposes of comparing the general economy of different plans, may be neglected. X. Then, representing by M, the amount of material required to sustain a stress equal to w, with a length equal to bd, = 1, we have only to multiply the stress of a member in terms of w, by the length in terms of bd, or v, and change w to M, to obtain the amount of material required for the member in question, omitting the extra amount in the connections. Hence, the length of the vertical tie bd, being equal to 1, and having a stress equal to w, requires an amount of material equal to 1M. For the horizontal tie, or chord, length = 4, and stress (as seen above), w whence material = 4Mo This added to IM, required for the vertical, makes a total of 5M, for material exposed to tension in truss 3. The two thrust braces, as already seen, sustain compression equal to $wV5, which multiplied by length, /5, and w changed to M, give material = -I M, for each, or 5 M, for the two. XI. In the case of truss Fig. 4, the obliques manifestly sustain a weight = -w, by tension, giving stress = ~w V/5, which multiplied by length, = v5, gives jM = material for each, and 5M, for the two. The compression of ki, equals ~wxh = -wx2 = w, while the length= 4, whence, material = 4M; and, each end post sustain Two PANEL TRUSSES. 11 ing ow, with length =-I, the two require material =-M making the whole amount of thrust material = 5M. Thus we see that the two plans require each the same precise amount of material for sustaining both tension and thrust, upon TFTn. 4. the supposition that the S __: __ iif, i material is capable of sus"^^:taining the same stress to the square inch of cross* section, in the one plan as in the other. This is true as to tension material; but with regard to thrust material, the power of withstanding compression, varies with the ratio of length to diameter of pieces, as well as with the form of cross-section; and it will hereafter be seen, that in this respect, plan 3 has the advantage in having the compression sustained mostly by shorter pieces, unless hi be supported vertically and laterally by a stiff connection with the beam atf, which would increase the amount of material. XII. Plan Fig. 5 has three members (in, mp and mq) exposed to tension, and the remaining three exposed, to compression. With the same length and depth L. _.P of truss, and the same load = w, at m, and with obliques equally inclined 1~-T"'"" (at 45~), it is manifest that the vertical and horizontal reaches, each for each, is equal to 1, and the length, equal to V/2; while the weight sustained by each equals ~w. Hence, the action (of tension or compression), equals wV2, and the material equals 2/2 x V/2.M - IM; 12 BRIDGE BUILDINGa making for the four pieces, 2M for tension, and 2M for compression. The tie or chord in, suffers tension equal to the horizontal constituent of the thrust of ql, manifestly equal to the weight sustained by ql, or equal to l w. Therefore, the length being equal to 4, the material required in its construction, equals 2M. The remaining member pq (= 2) sustains compression equal to the combined horizontal constituents of the tension of mq, and the compression of ql, each of said constituents equal to ~w, making compression of pq, equal to w, and length being 2, material = 2M We have therefore, for this plan of truss, 4M, for thrust material, and 4M for tension material, which is - less than in case of Figs. 3 and 4. Consequently, this plan is decidedly more economical than either of the others, unless the compression material acts with better advantage in the latter than the former; that is, unless the thrust members in 3 and 4, have a greater power of resistance to the square inch of cross-section, than those in Fig. 5o XIII. As to this, both theory and experiment prove, as will be shown in a subsequent part of this work, that the long thrust members in bridge trusses, are liable to be broken by deflection, rather than by a crushing of the material; that in pieces with similar cross-sections, with the same ratio of length to diameter, the power of resistance to the square inch is the same. That, since the cross-section is as the square of the diameter, and the diameters (in similar pieces), as the lengths, the absolute powers of resistance (being as the cross sections), are as the squares of the lengths. Two PANEL TRUSSES. 13 Hence, if the compressive forces acting upon two pieces of different lengths, be to one another as the squares of the lengths of pieces respectively, and the diameters be as the lengths, the forces are as the crosssections, and proportional to the power of resistance in each case, and the material in the two pieces, acts with equal advantage, as far as regards cross-section, so that the products of stress into length of pieces, are the true exponents of amount of material required in the two pieces respectively. It follows, that, if on dividing the forces acting upon the pieces in question respectively, by the squares of the lengths, the quotient be the same in both cases, the two pieces have the same power of resistance to the square inch, and in general, the greater the value of such quotient, the greater the power per inch, and the'greater the economy, though not necessarily in the same precise ratio. XIV. Applying this rule to thrust members in plan Fig. 3, being the braces, the compressive force equals ~wV5, and square of length = 5. Hence the quotient 1 w 4/5 - 0.2236w. The piece ki Fig. 4, has length = 4 and compression= w, whence, force divided by square of length gives ew = 0.0625w. This shows the material to be capable of sustaining much more to the square inch in the former, than in the latter case, though it does not give the true ratio. On the other hand, ek and gi, with length =1, and stress = —w, give quotient = w- = 0.5w. Hence, with similar cross-sections, these parts have greater power to the inch than either of the former, but not enough to balance the inferiority of ki, as compared with ad and do, in Fig. 3. 14 BRIDGE BUILDINGo With regard to truss Fig. 5, ql and pn, suffer each compression equal to iwV/2, with square of length = 2, giving quotient = 4w/2, = 0.371w, while pq, has com.. pression = w, and square of length = 4, and quotient -= w = 0.25w. Hence it appears that this plan not only possesses a decided advantage in the less amount of action* upon materials, but also, a considerable advantage as to ability of compression, or thrust members, to withstand the forces to which they are exposed. XV. Still another modification for a truss to support a single beam, is formed by reversing Fig. 3, thus converting tension members into thrust members, and vice versa; the oblique members falling below, instead of rising above the grade, or road-way of the bridge. In this case, the long horizontal thrust member ac, is divided and supported in the centre, and its economy of action becomes the same as that of pq, in Fig. 5; and the truss gives the same exponents for both thrust and tension material as when in the position of Fig. 3. This arrangement affords no side protection, and is not always admissible, on account of interference with the necessary open space beneath. DEDUCTIONS. XVI. We seem to learn from what precedes, that ~ (1). Since all heavy bodies not in motion toward, or not approaching the centre of the earth (or receding from it under the influence of previous impulse), exert a pressure equal to their respective weights [viii],'- By the expression, amount of action, is meant, the sum of products of stresses into lengths of parts, or members. Two PANEL TRUSSES. 15 either directly or indirectly upon the earth; and, since, a body crossing abridge, having (as bridges are always supposed to have), avoid space underneath, preventing a direct pressure, it follows, that every such body exerts an indirect pressure at some point or points at greater or less horizontal distance from the body. (2). That the pressure of a body at a point or points not directly below it, can only take place through one or more intermediate bodies, or members, capable of exerting (by tension or thrust), one or more oblique forces upon the first named body, and it is the office of a bridge to furnish the medium of such horizontal transfer of pressure [iv]. (3. That a single oblique force can not alone prevent a heavy body from falling toward the earth (since two forces can only be in equilibrio when acting oppositely in the same line), and that each oblique force is equal to the combined action of a vertical and a horizontal constituent, of which the first alone is equal to the weight sustained and transferred by the oblique member, while the horizontal constituent, acting at both extremities of the oblique medium, must be counteracted by means outside of the oblique and the weight sustained by it; which means are usually to be supplied by other members of the structure [viii]. (4). The direct force exerted by an oblique member (in the direction of its length), is equal to the weight sustained, multiplied by the length, and divided by the vertical reach of the oblique, while the horizonta. constituent equals the weight sustained multiplied by the horizontal, and divided by the vertical reach of the oblique [vii]. (5). The amount of material required in a tension member, is as the stress multiplied by the length ot 16 BRIDGE BUILDING. the member [Ix] (disregarding extras in connections) and the same is true of thrust members of similar formed cross-sections, sustaining stress proportional to the square of the length of pieces respectively. (6). The respective stresses of two thrust members, divided by the squares of respective lengths, give quotients indicative of, though not proportional to, the relative efficiency of material in the two members, - the greater quotient showing the greater efficiency, or greater power of resistance to the square inch of crosssection [xIIi]. With these rules or principles in view, we may proceed advantageously with general analyses and comparisons of different plans, or systems of bridge trussing, adapted to different lengths of span. THREE PANEL TRUSSES. XVII. In structures exceeding 25 or 30 feet in length, the length of joists from the centre to the ends, would require cross-sections so great, to give them the requisite stiffness, that their weight and cost would become objectionable. It becomes expedient, then, in such cases, to provide support for more than one principal point, or transverse beam, or bearer. A superstructure from 30 to 40 feet long, may be constructed with two cross beams, supported by two trusses with two pairs of braces each, with the feet connected by a horizontal tie or chord, as seen in Fig. 6. The cross beams, may be at b b', or suspended at c and d, at equal horizontal distances from a a', and from one another; which latter position they will be re THREE PANEL TRUSSES. 17 garded as occupying in this instance. Or, the figure may be inverted, thus reversing the action of the several thrust and tension members. XVIII. Another, and a more common form of truss for two beams, is shown in Fig. 7. These may be called three panel trusses. FIG. 6. Fig. 7. To compare these two trusses, suppose the two to have the same length and depth, and to be loaded with uniform weights, w, at the two points c and d in each. Then, since we know from the principle of the lever, that each weight produces upon each abutment, a pressure inversely as its horizontal distance from them respectively, and that the pressure upon the two abutments is equal to the weight producing it, it follows that ab, Fig. 6, sustains 2w, and compression-= ~ bw. Hence making ab = D,... be =v, and ac = A, w. labecomes 2 -w, and multiplying this stress by length, = D, and 3 18 BRIDGE BUILDING. changing wto M*,we have for material in ab,...-1 Mxm Diagonals,... (3'+ 3 )M Diagonals,.. (6 + 6-v)M End braces,... (6 —+ 6)M Verticals,. 3,-v. 7.2 ~+6v Total,......, (36 +9v)Mi Total,...... (386 -+9qMV) Making I = v = 1, Comp. 45.714M. Ten. = 45.714M. Grand total,...................................... 91.428M. We have here a little over 3 per cent less action upon material, than in case of truss Fig. 12, with verticals, The difference is a little less than was shown in my original analysis, that being based on trusses loaded at the upper, and this, at the lower chord; the former giving a trifle more action for the truss with verticals, and a trifle less for the other. Moreover, the difference was made to show greater still, by assuming that deductions might be made on account of certain diagonals being liable to two kinds of action. For instance, it was supposed that a member formed to sustain a considerable tension stress, might also sustain a small compressive force without additional material (not at the same time, of course), which is undoubtedly the case, on certain occasions; especially in the use of wooden trusses. This would give still greater apparent advantage to truss 13, with regard to economy of material. XLVII. There is, however, another view as to the action of load upon truss Fig. 13, which may modify the results above shown to a small extent. TRAPEZOID WITHOUT VERTICALS. 61 If we strike out the diagonals cm and me, and also dl and if, all the determinate forces necessary to sustain uniform weights at the nodes of the lower chord, would be exerted by remaining members, although we have assigned to those members, each, the sustaining of weight equal to 2w" under the full load, and twice that weight under certain conditions of partial load; and it is quite certain that they are necessary to the stability of the truss when partially loaded. But with both halves loaded uniformly, the weight upon each half could be transferred to the nearest abutment, producing equal thrust in both directions upon the central portion of the upper, and equal tension in opposite directions upon the lower chord; whereas, with one-half loaded, there is no means by which the pressure due at the farther abutment could be transferred past the centre, without oblique members in the centre panel. Still, which mode of action takes place under the uniform load, when the diagonals are in place, is a matter involved in a degree of uncertainty. If the centre diagonals do not act, under the uniform load, then ek and fj must sustain each 7w", instead of 6w" for the former, and 9w" for the latter, as above estimated. Also, kg would sustain 7w" by thrust, and different results would be produced as to stresses of various parts of upper and lower chords. The maximum stress for ek anddn, and fornb kg, would be 7w"D instead of 6w"-, as found above, and would occur under the full, instead of the partial load. The tension of gj and ob, also, would be increased to 14w". The weight sustained by J', would be only 7w" under the full load, though liable to the same maximum weight of 9w", under a partial load. 62 BRIDGE 3BUILDING. For the lower chord, we should have the same coefficient, (21) of w"' to express the tension of ab and ig, 28 for be, 35 (a decrease), for cd, and 42 for de. For upper chord, the co-efficients of w"- would be 28 for on and kj, and 42 for the three middle sections; no action being imparted by diagonals at m and 1. XLVIII. This uncertainty of action has no place in trusses of an even number of panels, as in such cases, no transfer of the action of weight can be supposed to take place past the centre, under a uniform load, without involving the absurdity of supposing the same member to carry weight by tension and compression at the same time; except, however, that in case of diagonals crossing two panels, or having a horizontal reach equal to twice the space between nodes of the chords, there will be diagonals filling the same condition of crossing in the centre of the truss, both vertically and longitudinally, as in Fig. 13. We may obviate mostly, any mischief liable to result in cases of the kind under consideration, by estimating the stresses upon the several parts under both hypotheses, and taking for each member the highest estimate, which will mostly meet all contingencies. Estimating action upon truss 13 in this manner, we obtain the following representative expressions for material: Compression. Tension. 6 h' 4 h2 Chord,......... 26-X Chord,....... 30 X M 4 vM Diagonals,... (64h Diagonals..... (4 Diagonas... (6 +6)M End braces 6.... 6 + 6v)M Verticals,...... 4vM 6 h2( 2 Total, (367 --- + 10OV)M Total, (37 7 104V)> TRAPEZOIDAL TRUSS.- DECUSSATION. 6 Making h = v = 1. These expressions give, Compression material = 46.855M + Tension do, 47.714M = total 94,571M. This shows an aggregate amount of compression and tension action, identical with that of truss Fig. 12, [XLIII.] DECUSSATION AND NON-DECUSSATION. XLIX. The elasticity of materials affords a means of answering the question as to decussation of forces through diagonals crossing in the centre of the truss, vertically and longitudinally (as in Fig. 13), in specific cases. But the results will vary in trusses of different numbers of panels, and different inclinations of diagonals. Suppose the truss Fig. 13 to be so proportioned that the maximum stresses of the several parts and members, will produce change of length equal to E, multiplied by the lengths of parts respectively; the vertical ob, = -, being the unit of length. Then, the truss being uniformly and fully loaded, and the chords being under their maximum stress, the upper chord is contracted, and the lower one extended at a uniform degree; and, if the diagonals be unchanged in length, their vertical and horizontal reaches have not been changed by the change in length of chords. Hence, the distance between chords is not altered by change in their length. But the diagonals being under stress, by which some are extended and others contracted, according to the stress they are under, as compared with their maximum stresses respectively, the nodes of the chords are allowed to settle to positions below what they are brought to by the mere change in lengths of chords. 64 BRIDGE BUILDING. Hence, the panels are (generally) thrown into more or less obliquity of form, in consequence of inequality in length of diagonals in the same panel. But the centre panel can not assume obliquity, because any tendency of forces to change the length of one diagonal, is attended by a like tendency of equal forces to produce exactly the same change in the other; so that the vertical reaches of both must suffer the same change, if any, and both must be under tension or compression, according as the acting forces tend to bring the chords at the centre, nigher together or farther apart. Now, the forces produced by the load being all concentrated at the points o andj (Fig. 13), the point d is depressed with respect to o, by the extension of ob and nd, and by the compression of bn. Hence, assuming decussation to have place, giving tension to the diagonals dl and me, equal to what is due to a weight of 2w", ob is under maximum tension and gives depression equal to E, to the point b, - bn and nd are under X maximum stress, and give depression, each equal to E X D2*= (1.666h2 -- 1.666) E, for the two (D representing length of diagonals, =vh2+1). Then, adding 1E for effect of ob, we obtain (1.666h2 + 2.666) E, = depression of point d. The point m is depressed by extension of oc under a maximum stress, giving an amount equal to D2E (A2 + 1) E. Also, by compression of cm under one-half maximum stress, to the extent of ( h2 + 1) E. H ence, depression of point m = (lkh2 + 1-) E. This shows the point d to be depressed more than,, by(1.66612 + 2.666) Ei- (1.5k2 + 1.5)E = (0.166h2 + Let the diagonals bd and Bd, of two rectangular panels ac and Ac, Fig. 14 (c and d, being fixed points), be exposed to tension in proportion to their respective cross-sections, receiving each thereby, extension TRAPEZOIDAL TRUSS.- DECUSSATION. 65 1.166) E, and the spaces md and le to be increased to that extent; of course producing tension upon dl and me. Noow, by hypothesis, these diagonals are under the weight of 2w", giving half maximum stress, and requiring an increase of vertical reach, equal to (h2 + 2) E. If then, we give such a value to h, as will make the last co-efficient of E equal to the one above, it will show that the chords have receded just enough to give the assumed tension to dl and me, and the decussation is a demonstrated fact. To find the value of h, producingthisresult, make,.5h2 +.5 =.166h2 + 1,166, and we deduce.333h2.666; whence h = /2. But this requires too great an inclination of diagonals, and a less value of h, gives a space from d to m too great for the supposed tension of dl and me. Making h = 1 = v, we have increase of distance from d to m = 1.333E, requiring a weight of 2.666w", to stretch dl down to the point d. But as no weight or stress can be added to the 2w" assigned to dl and me, without afequal to hbe and B/E, respectively. This will cause the points b and B to drop to b/ and B/, in ab and AB produced. Join b e and BE. Then, the infinitesimal triangles bb/e and BB/E, right-angled at e and E, are essentially FIG. 14. similar, respectively to the triangles db/a and a dBA. Hence, the following relations: A d (1). b- e:: bde:: bdab, and (2). BB': BE:: Bd: ab. whence, bb/ X ab =- e X bd, and BB' X abl= BE X Bd, From these two equations we derive/ - (3). bb/ X ab: BB' X ab:: b'e X bd: B/E X Bd. But, by the law of elasticity B cG (4). b'e: BE::h bd: Bd; whence, Bt- (5). b/e X bd: 1E X BdE:: bd2: Bd2. Hence dividing the first ratio of proportion (3) by ab, and substituting for the last ratio of (3), its equivalent found in (5), we have, ()), bb': BB': bd': Bd2. Hence the depression due to the extension of a dia'cr.al retaining the same vertical reach, is as the stress (per square inch), sustained, multiplied by the square of the length of diagonal. 9 66 BRIDGE BUILDING. fecting all the 5 members contributing to the depression of the points d and m, and in all cases, so as to diminish the elongation of distance between d and m, it is reasonable to conclude that by assigning some', or thereabouts, of the extra weight of.666w" required on dl alone, it would, by affecting the whole 5 members, be sufficient to correct the error. Let us, then, assume that dl and me sustain 2.15w", instead of 2w" as by previous supposition.' This change requires reduction of weight upon dn, nb and bo, from 5w" to 4.85w" for the two former, and from 12w" to 11.85w" for bo. Also an increase of weight on me and co, to 2.15w" on me, 11.85 and to 9.15w" on co. Then bo sustains -2 -of the 12 11.85 maximum, and gives depression =-1- =.9875E 4.85 bn and nd, sustaining 4 — of the maximum, give depression for the two, equal to 3.233E, making 4.2205E = whole depression of point d. With regard to the point m, we have the maximum 2.15 stress on oc, giving depression equal to 2E, and _4 of the maximum on me, giving depression = 1.075E, making a total of 3.075E for the point m. Hence, the elongation of the space dim, equals (4.2205 - 3.075) E, - 1.145E, whereas 2.15w' upon dl, increases its vertical reach by only 1.075E. This shows that dl sustains still a little more than 2.15w". On the other hand if we assume a weight of 2.2w" on dl, we obtain the opposite result, showing that dl sustains less than 2.2w", and hence the actual amount must be between 2,15w' and 2.2w". We conclude then, that dl and me are not inactive under the full load of the truss, but on the contrary, they sustain, in this case, even more than the decussation theory assigns to them. We learn, moreover, that TRAPEZOIDAL TRUSS.- DECUSSATION. 67 the question is affected by the horizontal reach of the diagonal, or the value of h. And, since in this case, the point d being depressed by action upon 3 members, and the point m by action upon only 2, we have an elongation of space between d and m requiring more than the theoretical stress upon centre diagonals, it is natural to conclude, that, in case of a greater number of panels, nine, for instance, where 4 members contribute to the depression of the upper, and only three to that of the lower chord at the centre, the increase of distance between chords, would be less than that required to give the theoretical stress upon diagonals in the centre panel; and, such is found to be the case. In a nine panel truss on the plan ofFig. 13, the increase of distance between chords, due to the stresses assigned by the decussation theory, is only about one-third of what would be required to give the centre diagonals the stress assigned them. Hence in this case there is less decussation than the theory requires; and, one or two trials, by assigning different weights as sustained by centre diagonals, in the manner pointed out above, would enable a near approxination to the actual amount of decussation to be arrived at, in the case of the nine panel truss, or any other. Let us take one more view of this matter, by assuming no action by centre diagonals, under the full load. Then cm (Fig. 13), is also out of action, and oc alone, under' maximum stress, contributes to the depression of point m, giving depression equal to 2x7E, = 1.55E, (assuming h = v). The point d is depressed by the maximum change of two obliques, and one vertical, giving depression = 5E. Therefore the distance md, is increased by (5 - 1.55) E, = 3.45E. Hence dl and me must be elongated by 1.725 68 BRIDGE BUILDING. times the amount due to the maximum stress, in order to escape action. Suppose the member to be of wrought iron, proportioned to a maximum stress of 10,000I O to the square inch. Then, the extension due to 17250fo to the inch, is about 1'oW72 x length.00069 x length, and if length - 15 feet, the extension is equal to.00069x15, =.01035 ft. or, say i inch. Hence, in a 7 panel truss, as represented in Fig. 13, with h = v, if the diagonals in the middle panel be slack, by { inch in 15 feet of length, no decussation will take place, and the centre diagonals will be inactive, under the full load of the truss. If those members have less than that degree of slackness, they will be in action in such circumstances. It would be a very badly adjusted piece of work in which such a degree of slackness should occur, and we may fairly conclude, that the centre diagonals, in this class of trusses, are never entirely inactive. But the quantity E, is so very small, with any kind of material, and with any co-efficient that may affect it, in practice, that a slight inaccuracy of adjustment, may so change the practical form the theoretical results deduced by calculation, as to decussation, as to render the latter of no great practical reliability. Hence, after all, perhaps the most unexceptionable course, in this regard is, to follow the rule given before [xLVIII], of estimating stresses on both hypotheses, and taking the highest estimate for each part. Now, perhaps, this subject has been discussed at greater length than its practical importance demanded, considering the small percentage of error liable to occur in any case; but with regard to this, as well as to other matters, it is well to know, what may be known TRAPEZOIDAL TRUSS.-WARREN GIRDER. 69 without inconvenient or unreasonable effort at investigation. THE WARREN GIRDER. L. There is another form of truss operating upon the same principle as truss Fig. 13, in which one set of oblique members is left out, so that only one diagonal remains to each panel. The diagonals meet and connect with one another and with the chords, forming alternate nodes at the upper and lower chords. This truss, represented in Fig. 15, requires an even number of panels that the two half-trusses may be symmetrical. This is an extension of truss Fig. 5, with tension verticals for suspending floor beams from the upper nodes, when the travel-way is along the lower chord, and thrust verticals ascending from the lower nodes, in the case of what are technically called deck-bridges.* We compute the stresses of the members of this truss, by placing the figures 1, 2, 3, etc., over the diagram as in preceding cases, and from a second line or series of figures, by adding all those in the first series, as in case of truss Fig. 12, because each weight tends to act upon every diagonal. Each figure in the second line, is the co-efficient of w" in the expression of the greatest weight transferred to the right hand abutment, through the diagonal crossing the panel next on the right hand of the figure; and the action is tension or thrust, according as the diagonal ascends or descends toward the right. Thus, the Fig. 6 over o, indicates that 6w" is the greatest weight acting by thrust upon oe, while 10 over the point e, indicates 10w" as the maximum weight acting by tension upon em. These * The author built several small bridges upon this plan, to carry a rail road track over common highways, in 1849 or 1850, believed to have been the first application of this form of truss. T70 3gBRIDGE BUILDINGo figures only indicate weight transferred from left to right, and it is evident that the same weights in a reversed order, are transferred from right to left, through the same diagonals. Hence, a third series of figures under the second, composed of the same figures in a reversed order, shows the weights carried by the several diagonals from right to left. The figures in the third line, show the weights acting on diagonals next on the left of respective figures. It will be seen also, that the figures under odd numbers of the upper line, FIG. 15. 1 2 3 4 5 6 7 1 3 6 10 15 21 28 28 21 15 10 6 3 1 q p o n m, 1 k a b c d e f g i j indicate weights acting by thrust, and those under even numbers, by tension. The figures 6 and 15 over o, indicate 6w" acting on oe, and 15w", on oc, both by thrust. Again 3 and 21 under 2, indicate 3w" acting on co, and 21w" upon cq, both by tension. The figure 28 at the right and left, under 1 and 7, indicate 28w" acting by thrust upon aq and jk. Now if we add all the figures in the second and third lines standing under odd numbers of the upper line, we obtain the co-efficient of w" for the aggregate maximum weights acting by thrust upon oblique members, Twhile the sum of all the figures in like manner, under even numbers, forms the co-efficient of w" for the aggiregate maximum weights acting by tension upon obliques. The former gives 1Ov",=12.5w for compression, and the latter, 68w",=8.5w, for tension. Ilence, TRAPEZOIDAL TRUSS —WARREN GIRDER. 71 makinga h=v=l, we have as expressions for amount of thrust and tension action upon material in oblique members, 251M for thrust and 17M for tension. One half of the lower chord obviously sustains a stress of 28w", equal to horizontal thrust of the end braces, and the other half, 60O",= horizontal action of aq, qc and co (under full uniform load), at one end, and of corresponding diagonals at the other end, giving required material for chord equal to 44M. The compression of the upper chord, equals the horizontal thrust and pull of aq and qc,= 48w", for 3 of its length, with the addition of 12w" for horizontal thrust of co, and 4tv" for pull of oe, making 64w" for the two middle panels. Hence expression for material is 40M. The verticals obviously require tension material equal to 4M, and the aggregate for the truss, is, For Compression. For Tension. hod,................. 40 Chord,......... Ch......... 44M Obliques.............. 25M Diagonals.............. 17M Verticals,............. 4M Total,.................... 65M Total,..................... 65M A corresponding truss with 2 diagonals in each panel, on the plan of Fig. 13, shows the same expressions for materials, or amount of action of both kinds, item for item, and any advantage possessed by either plan, must depend essentially upon the more advantageous action of compression material. Truss Fig. 15, has fewer intermediate thrust diagonals, and greater concentration of weight upon them; which is favorable; while in the other, the diagonals crossing one another, are enabled to afford mutual support laterally, in certain modes of construction. 72 BRIDGE BUILDING. The upper chord in Fig. 15, acts at a decided disadvantage, in having no vertical support for a length of 2 panel widths, unless it be especially provided at additional expense. As a deck bridge, with struts, or posts at p, n, I, and lateral tying and bracing, the truss may answer an excellent purpose. But even in that case, it can scarcely be considered as preferable to the truss with a double system of diagonals. The Ohio river bridge at Louisville, Ky., has its long spans (about 400 ft.), constructed upon the plan of Fig. 15, and no plan which we have considered, shows a less amount of action upon material. These are believed to be the longest spans of Truss Bridging in the country. An eight-panel truss upon the plan of Fig. 12, gives the following expressions for amount of material. Compression. Tension. Chord,..................43 Chord,..................41i Ends,....................14M Diagonals,..............28 Verticals,............... 7M Verticals,............... 2 64M 71M This indicates a difference of nearly 4 per cent, as to amount of action upon material, in favor of the truss without vertical members, generally speaking; i. e. in which there is no regular transfer of action from one to another, between diagonal and vertical members, as in truss Fig. 12. This advantage is made still larger in certain modes of construction, by the circumstance that the same members, in trusses 13 and 15, may sometimes act by tenlsion and thrust, on different occasions, without any more material than would be required to act in one direction only. FINCK TRUSS. 73 LI. It may be proper in this place, to refer to still another form of trussing, which has enjoyed a degree of popular favor, and which differs somewhat from any we have hitherto considered. The plan is seen in outline, in Fig. 16. Each weight is sustained primarily by a pair of equally inclined tension members, and thereby transferred either to the king posts standing upon the abutments, or, to posts sustained by other pairs of equally inclined suspension rods of greater horizontal reach; which in turn, transfer a part to king posts, and another part to a post sustained by obliques of still greater reach, until finally, the whole remaining weight is brought to bear upon the abutments by a single pair of obliques, reaching from the centre to each abutment. FIG. 16. THE FINCK TRUSS. In Fig. 16, are represented four different lengths of obliques, in number, inversely as the respective horizontal reaches. The first setcontains 8 pieces reaching horizontally across one panel, and sustaining each -w. The next longer set, of four pieces, reach across two panels, and sustain each 1w; one-half applied directly, and the other, through posts and short diagonals. The third and longest set, contains but two pieces, reach across four panels, and sustain together 4t; of which 1w is applied directly, 1w through two short diagonals, and 2w through two intermediates. Now, as each set sustains the same aggregate weight, namely 4w, the material in each set, will 10 74 BRIDGE BUILDING. be represented by this weight multiplied by the square of the lengths respectively, and divided by v: and, making k = v = 1, the squares of respective lengths are 2, 5 and 17, which added together and multiplied by 4w, and w changed to m\, gives 96M=amount of material in tension obliques, the only tension members in the truss. The upper chord sustains compression equal to the horizontal pull of one oblique member of each class, obviously equal to 10w, with length = 8. Hence, required material equals 84M. End posts sustain together, 7w, centre post 3w, and the two at the quarters, one w each, in all 12w, and the representative for material is 12M; whence the total for thrust material is 96M, making a grand total of thrust and tension material= 192M. The 8 panels trapezoid with verticals, requires,... 135M Do GS 66 without verticals,......... 130M This comparison exhibits an amount of action in case of the first (Fig. 16), which, considering that it possesses no apparent advantage as to the efficient working of compression material, would seem to exclude it, practically, from the list of available plans of construction. DISTINCTIVE CHARACTERISTICS OF THE ARCH. LII. We have seen that all heavy bodies near the earth's surface (except when falling by gravity or ascending by previous impulse), exert a pressure upon the earth equal to their respective weights. We have also seen that the object of a bridge, in general, is, to sustain bodies over void spaces, by transferring the pressure exerted by them upon the earth, from the THE ARCH TRUSS. 75 points immediately beneath them, to points at greater or less horizontal distances therefrom. -Ve have, moreover, seen that this horizontal transfer of pressure can only be effected by oblique forces (neither exactly horizontal nor exactly vertical), and have discussed and compared, in a general way, various combinations of members, capable of effecting this horizontal transfer of pressure. But, without going into unnecessary recapitulation, we find two or three styles of trussing, possessing more or less distinctive features, which promise decidedly more economical and satisfactory results than any others; and, to make the properties and principles of action of the best and most promising plans as thoroughly understood as may be within the proposed limits of this work, will form a prominent object in the discussions of succeeding pages. The distinctive feature of the arch, as a sustaining structure, consists in the fact that all the oblique action required to sustain a uniformly distributed load, is exerted by a single member of constantly varying obliquity from centre to ends; each section sustaining all the weight between itself and the centre, or crown of the arch, and none of the weight from the section to the end; so that the weight sustained at any point, is as the horizontal distance of that point from the centre. Consequently (the arch being supposed in equilibrio under a uniform horizontal load), the horizontal thrust at all points must be the same, and the inclination of the tangent at any point should be such that the square of the sine, divided by the cosine of inclination (from the vertical), may give a constant quotient. For, regarding each indefinitely short section of the arch as a brace coinciding with the tangent at 76 BRIDGE BUILDING. the point of contact, its horizontal thrust equals the weight sustained, multiplied by the horizontal, and di. vided by the vertical reach of the brace. But the horizontal and vertical reaches are respectively as the sine and cosine of the angle made by the tangent with the vertical; that is, as ab and bd, Fig. 17, while the weight is also as the sine ab, FIG. 17. of the angle adb. Hence, the weight by the horizontal / g reach, isas ab, or as the square C of the sine of adb; and the 2 _______ constant horizontal thrust of f/ ^~ the arch at all points, is as aby ab2 ~b; or, as-a. Now this condition is answered by the parabola, in ab2 ab2 which be = cd bd, and l- -constant C, whence ab2= cb x constant 2C, which is the equation of the parabola. This quality of the arch truss, allowing nearly all of the compressive action to be concentrated upon almost the least possible length, and consequently, enabling the thrust material to work at better advantage than in plans where this action is more distributed, and acts upon a greater number and length of thrust members, enables it to maintain a more successful competition with other plans than we might be led to expect, in view of the greater amount of action upon materials in the arch truss, than what is shown in trusses with parallel chords. Hence, we should, not too hastily come to a conclusion unfavorable to the arch truss, on account of the apparent disadvantage it labors under, as to amount of action upon material. These apparent disadvantages are frequently overbalanced by advan WEIGHT OF STRUCTUREo 77 tages of a practical character, which can not readily be reduced to measurement and calculation. The preceding general comparisons are to be regarded only as approximations, and should not be taken as conclusive evidence of the superiority or otherwise, of any plan, except in case of very considerable difference in amount of action, with little or no probable advantage in regard to efficient action of material. EFFECTS OF WEIGHT OF STRUCTURE. LIII. In preceding analyses, and estimates of stresses upon the various members in bridge trusses, regard has only been had to the effects of movable load, which may be placed upon, or removed from the structure, producing more or less varying strains upon its several parts. But the materials composing the structure, evidently act in a similar manner with the movable load, in producing stress upon its members; the only difference being, that the weight of structure is constant, always exerting or tending to exert the same influence upon the members, instead of a varying action, such as that produced by the movable load. In order, therefore, to know the absolute stress to which any member is liable, and thereby to be able to give it the required strength and proportions, we have to add the stresses due to constant and occasional loads together. The weight of structure evidently acts upon the truss in the same manner as if it were concentrated at the nodes along the upper and lower chords, and of the arch, in case of the arch truss. And, since much the larger proportion of it acts at the points where the 78 BRIDGE BUILDING. movable load is applied, if we regard the whole as acting at those points, the results obtained as to stresses produced by it, will be sufficiently accurate for ordinary practice. Still, more closely approximating results may be obtained by assigning to both upper and lower nodes, their appropriate shares of weight sustained, as may easily be done when deemed expedient. If we divide the whole weight of superstructure supported by a single truss, by the number of panels, the quotient, which we may represent by w', will show the weight to be assumed for each supporting point, on account of structure; and the stresses produced by such weights, added to the maximum stresses of the several members, due to the movable load, will represent the true absolute stresses the respective members are liable to bear. Now, as far as relates to parts suffering their maximum stresses under the full load, such as chords, ariches, king braces, and verticals in the arch truss, as to their tension strain, we have only to substitute W, (=w+w'), in place of w, in expressions obtained for stresses due to movable load. In other cases, w and w' will have each its peculiar and appropriate co-efficient. The diagonals of the arch truss, are obviously not affected by weight of structure, as they are not so under full and uniform movable load. Moreover, the weight of structure acts in constant opposition to the compressive action of movable load upon verticals. Hence, in truss Fig. 11, where we find the varying movable load gives a maximum compression upon the longest, equal to 3tw, and upon the next shorter, equal to 2w", the weight of structure diminishes -those quantities to 3t~"-w', and 2zw"-w' respectively. Or, if we WEIGHT OF STRUCTURE. 79 would be more exact, we may add in both cases, the weight of a segment of the arch, which has no tendency to produce tension upon the verticals; or we may subtract only 2 or T of w'; thus, 3w"-w',n and 2zV" — w' may be taken to represent the compressive action upon the verticals in Fig. 11. LIV. In the case of truss Fig. 12, the only diagonals acting under uniform load, are oc, fj, nd and ek; the two latter sustaining, of weight of structure, lw', and the two former, 2w'. And, the maximum movable weight borne by those members, being [XL] 10w" and 15w", the absolute maximum will be 10"+/t' for nd and ek, and 15w"+2w' for oc andfj. Now, if we place the figure 1 under d and e, (Fig. 12 A), and the figure 2 under c and f, and so on, in case of agreater number ofpanels, to the foot of the last diagonal each way, inclining outward from the lower nodes, these figures are obviously, the co-efficients of w' to express the weights contributed by the material of the structure, to the stresses of diagonals extending upward and outward from the points to which the figures respectively refer. FiG. 12 A. 1 2 3 4 s 6 3 6 10 15 21 o n m I k j a, b c d e.f i 2 1 1 2 Again, we have seen [XL], that a certain condition of the movable load, tends to throw 1w" upon bn, and another condition of such oad, tends to throw 3w" upon 80 BRIDGE BUILDING. cm. But, since, as we now see, the weight of structure tends to throw a constant weight of 2w' upon oc, which is antagonistic to bn, the actual maximum weight upon bn, is lw"-2w', which will always be a negative quantity, in practice; whence bn must always be inactive, and may be dispensed with. The maximum weight upon cm, as modified by weight of structure, is in like manner reduced to 3w" -w', which will in practice, be either negative, or of quite small amount. Hence, we have the following rule: For the absolute maximum stresses of diagonals (in case of parallel-chord trusses with verticals), we add the effects of weight of structure to the maximum effects due to variable load, where both fall upon the same, and subtract the former, in cases where the two forces fall upon counter, or antagonistic diagonals. In case of parallel-chord trusses without verticals, we add the effects of constant and variable load upon each diagonal, when alike, i. e., when both tensile or both compressive, and subtract the former when the effects are alike. DOUBLE CANCELATED TRUSSES. LV. The use of chords in a truss being to sustain the horizontal action (whether of thrust or tension) of the oblique members, it follows that the aggregate stress of chords, is equal to the aggregate horizontal action of all the diagonals acting in either direction And, the horizontal action being obviously as the number and horizontal reach directly, and as the vertical reach inversely; also, the length of truss being as DOUBLE CANCELATED TRUSSES. 81 the number and horizontal reach of diagonals, while the vertical reach is as the depth of truss, it follows that the stress of chords is directly as the length and inversely as the depth of truss, other conditions being the same. Hence, if the depth of truss be so reduced as to make the ratio of length to depth indefinitely large, the stress and required material of chords, become indefinitely large. On the contrary, if the depth be indefinitely great, although the stress of chords be ever so small the length and required material for diagonals and verticals must be indefinitely large. It is manifest, then, that between these two extremes there is a practical optimum,- a certain ratio of length to depth of truss, which, though it may vary somewhat with circumstances, will give the best possible results as to economy of material in the truss. This matter will be taken into consideration hereafter, and is referred to here, to show the expediency of generally increasing the depth, with increase of lengths in the truss. Now, in trusses of considerable length, and, consequently, depth, it becomes expedient, in order to avoid too great a width of panel (horizontally),or an inclination of diagonals too steep for economy of material in those members, to extend them horizontally across two or more panels, or spaces between consecutive nodes of the chords. In such cases, the truss may be called double or treble cancelated, according as the diagonals cross two or three panels. LVI. To estimate the stresses of the members of double caneelated trusses with vertical m1embers, a slight modification of the process already described, [XL, &c.], is required, as follows: II 82 BRIDGE BUILDING. Having placed the numbers 1, 2, 38 &c., over the nodes of the upper chord, as seen in Fig. 18, place under each odd number, the sum of all the odd numbers in the first series, up to and including the one under which the sum is placed; and the same with respect to the even numbers. Then, the second series of figures may be used in precisely the same manner as that explained with reference to Fig. 12, to determine the weight sustained by, and the maximum stress produced upon, each diagonal and vertical, by equal weights upon all or any of the nodes of either chord. For example; supposing the truss to have tension diagonals and thrust verticals; take the diagonal havits lower end under 5 (upper series), and its upper end under 7. This diagonal may be represented by 5/7, while 5\7 may indicate its antagonist, and so of other diagonals. Then, as we see 9 (the sum of 1+3+5), in the second series, over the lower end of 5/7, and, as the diagram represents a truss of 16 panels, we know that the diagonal in question is liable to a maximum weight of,w, = 9w". This amount is to be diminished, of course, by the weight due from weight of structure to the counter diagonal. Again, the diagonal 9/11 sustains as amaximum from variable load, 25w"; which will require to be increased on account of weight of structure, since the latter, in this case, acts upon the main, and not upon the counter diagonal, as in case of 5/7. Now, to obtain the effects of weight of structure and uniform load, the truss having even panels, we place 1 under the centre node of the lower chord, because half of the weight w', which is supposed to 1e concentrated at that point, tends to act On each of the diagonals rising from that point. DOUBLE CANCELATED TRUSSES. 83 At the next node from the centre, each way, the figure 1 \^ ^is set, because, of the weights (W'), concentrated at those points, p-r Os. 0 each bears upon its nearest abutment (the truss being uniformly 3 ~-^^~ floaded), through the diagonals running upward and outward ~;tS "' 10from those points. If this be not so, each must transmit a part ^~ v^. of its amount past the centr-e, through the antagonistic dia^ \/ f l gonals 7/9 and 7\9, which is contrary to statical law. Then we put 12 21 381, etc., 06 > nunder alternate nodes from the rcentre, and 1, 2, 8, etc., under alternates beginning at the first _ _ _\__ \_ on each side of the centre; as shown in diagram Fig. 18. B e-L~ Q weights acting, or tending to -. /".. act, upon the diagonals running. /l. ~ upward and outward from these numbers respectively, arising /_. /_._. from weight of structure, and also, the co-efficients for (w w'), /.. /. to express the load tending to,/" act on diagonals, arising from both superstructure and movable weight, when the truss is "N^ ~fully loaded. For illustration; 84 BRIDGE BUILDINGO the diagonal 5/7 we have seen to be liable to a maximum stress of 9w" from variable load, and, as we have the figure 1 at the foot of 5\7, it shows that the weight due to the latter on account of structure is lw', which must be subtracted from 9w" to obtain the actual maximum to which 5/7 is liable; which is 9w" W'e If w' be equal to or greater than 9w", then 5/7 is subject to no action, and may be dispensed with. As to the advantage of introducing counter diagonals, merely for the purpose of stifening the truss, the results of my investigations will be given in a subsequent part of this work. The maximum weight sustained by any thrust upright, is manifestly equal to the greatest weight borne by either diagonal connected with it at the upper end, since any weight borne by 3/5, for instance, being transferred to the antagonist of 5\7, thereby diminishes by a like amount, the maximum action of the latter. Whence the upright at 5, can receive no more load from the two diagonals, than the maximum load of one, and this relation holds in general. The reason of adding alternate figures to form the second series over the diagram, will be obvious, when it is observed that there are two independent systems of uprights and diagonals; one of which includes the uprights under even numbers in the upper series, and the diagonals connecting therewith, and the other, the remaining uprights and diagonals. Now weight applied at the nodes of either of these systems, can only act upon members of that same system; that is, weight applied at nodes indicated by even numbers in the upper series can only act upon the first above named system of uprights and diagonals, and vice versa. DOUBLE CANCELATED TRUSSES. 85 The main end braces are acted upon by both systems; so that to obtain the weight sustained by them, we must add the numbers 56 and 64 (and corresponding numbers in other cases), making in this case 120w" equal to 71w. The uprights under 1 and 15, sustain each a tension equal to w, for variable load, and to w+w', for weight of variable load and superstructure together; which obviously gives their greatest strain. Having thus determined the greatest weights to which the several verticals and diagonal members are liable, we proceed as in former cases, to multiply those weights by lengths of diagonals, and divide the products by lengths of verticals, to obtain the stresses of diagonals; remembering to take into account the difference in length between those having a horizontal reach of only one panel, at and near the ends of the truss, and those that reach across two panels. The mode of estimating the stresses upon the different portions of the chords, depending upon the horizontal action of diagonals, has been sufficiently explained. It is only necessary to observe that the end braces produce compression upon the upper, and tension upon the lower chord, through their whole lengths, equal to I (w+w'), multiplied by the number of nodes of the lower chord, and that product multiplied by -. and that each pair of intermediate diagonals analogously situated with respect to the ends of the truss, whether acting by thrust or tension, produce tension and thrust in like manner, upon the portions of the lower and upper chords, between their points of connection with the chords. Thus is generated a progressive and determinate increase of action upon succeeding 86 BRIDGE BUILDINGo portions of the chords from the ends to the centre of the truss. In the case of a deck bridge, the weights sustained by thrust uprights, are respectively indicated by the figures over the diagram on the right hand half of the truss, prefixed to w", for movable load, and the figures under the diagram prefixed to w, for weight of structure, being the same weight which gives the maximum.stress to the diagonal running upward and outward from the foot of the upright. Tension verticals at the ends sustain no weight. TRUSSES WITHOUT VERTICALS. It will be seen upon a general view of the action of the different parts of a truss with parallel chords, that the diagonals (and verticals when used), form media through which weight acting upon the truss, is reflected back and forth between the upper and lower chords, until it comes finally to bear upon the abutment. A weight applied at one of the nodes of the lower chord, of course, cannot be sustained by the tension of that chord, which acts only in horizontal directions; but is suspended by a tension piece, whether oblique or vertical, from a node in the upper chord. But the upper chord acting also horizontally, cannot sustain the weight. Consequently, athrust member, either oblique or vertical, must meet the force at that point, to prevent the weight from pulling down the upper chord, and destroying the structure. Hence, we see, that in all the cases we have considered, of trusses with parallel chords, the weight, whether applied at the upper or lower chord, acts alter TRUSSES WITHOUT VERTICALS. 87 nately upon thrust and tension pieces, extending directly or obliquely from chord to chord. With reference to Fig. 18, we have regarded the weight as transferred from tension diagonals to thrust verticals, and the contrary. But if we conceive the verticals to be removed, except the endmost, we have only to insert a thrust brace from the abutment to the second node (or the first from the angle), of the upper chord, and to so form and connect the other diagonals as to enable them to act by either tension of thrust, and we have a truss capable of sustaining weights applied at all, or any of the nodes of the upper and lower chords, in the same manner as the truss with verticals, represented in Fig. 18. In this condition, the truss will act upon the principles discussed with reference to Fig. 13. For this modification of the truss, see Fig. 19. To estimate the strains upon the several parts of such a truss, due to weights w, w, etc., at the nodes of the lower chord; we may place the figures 1, 2, 3, etc., over the nodes of the upper chord, as was done in the case of Fig. 18. But, instead of adding alternate figures to form the second series, to be used as co-efficients of w", for expressing the weights sustained by diagonals, we add every fourth figure; because it is only the weights at every fourth node, that act upon the same set of diagonals. For instance; the weights at 1, 5, 9 and 13, act upon their peculiar set of 8 pieces (excluding the end braces, but including the tension vertical at 1), and none of the weights at the other nodes have any action upon those pieces; as is made obvious by an inspection of Fig. 19. Again, the weight at 2, 6, 10 and 14, have their peculiar and independent set; and so of those at 3, 7, 11 and 15, and those at 4, 8 and 12. Therefore, in form" 88 BRIDGE BUILDING. ing our second series of numbers, we place under each figure T" ~ of the first series, the sum of that figure, added to every 4th ean ~ >X > 3 figure preceding; that is, under 12, place the sum of 12, 8 and 4 C*" 0 y^ %'24. Under 5, the sum of 5+ 1 =6. The four first figures, CQ'!.x having no 4th preceding figures, are simply transferred, without ^ / T- > ~ addition or alteration. These numbers in the second ooco /O O ^ series, are the co-efficients of w' (=w divided by the number of 3o /\/ \ panels in the truss, being 16 in this case), to express the greatest. ar/,\ weights acting by tension on w *+'^ / ~ ~each diagonal having its lower end under the number used, and Ho ^ \/ ~the upper end under a higher /o \ 0number. Also the weight act~\/ / ing by thrust upon the diagonal meeting the former at the upper c~ e xs\ / chord. The last, or highest number, determines the weight CQn^ sustained by the tension vertical under the number, the vertical me:< \ >< / being a member of one of the four sets of alternate thrust and C"m Sx tension pieces connecting the two chords. -^,^, ^ J A third series of figures, formed by reversing the order of the second-placing the low TRUSSES WITHOUT VERTICALS. 89 est number of the third under the highest of the second series, and vice versa, prefixed as before to w", will show the weights sustained by thrust and tension of diagonals in the reversed order; i. e., whereas one series shows the amount of tension a particular diagonal is liable to, the reversed series shows the thrust the same piece must exert in a different condition of the load. Thus we ascertain, as in the case of truss Fig. 13 [XLV], that nearly all of the diagonals are exposed to two kinds of action, thrust and tension; and it is only the preponderance of the larger over the smaller of these forces, which has place when the truss is fully loaded, and it is only this preponderance which is to be used as co-efficient to (w+w') in estimating the stresses upon the different portions of the chords, and as co-efficient to w', in modifying the effects of the variable load upon diagonals, as affected by weight of structure.. But it is to be remembered that the numbers over the diagram are to be divided by the number of panels, before being used before w and w', in the expression of stresses of members. Thus, we have, as the effect of variable load upon the diagonal 2/4..., 2w"(=2w), as the 18 greatest weight acting by tension, and 8w, the greatest acting by thrust. Hence the weight upon this piece, due to weight of structure, is (18-2 )W/,=w'^ and it produces thrust or compression, because the thrust tendency is the greater. This weight (w'), added to 18 tw, the greatest effect of variable load shows the maximum weight which can act by thrust upon that diago18 nal, to be Lw+w' We have, also, for the greatest weight acting by tension as modified by weight of strucl 12 90 BRIDGRE BUILDING, ture, 2W-_' which is a negative quantity when w is less than 8w', as will usually be the case in practice; consequently that diagonal can seldom or never be exposed to the force of tension. 16 A Again _1(w+w'), (a and v representing horizontal and vertical reaches of the diagonal, as in previous discussions), is the amount contributed toward the maximum tension of the lower chord by the diagonal in question, not affecting, of course, that portion of the chord outside of the connection therewith, or a like portion at the opposite end. LVIII. It is to be remembered that the tension or thrust of a diagonal, is always equal to the weight sustained, multiplied by the length, and divided by the vertical reach of the diagonal. The method here under discussion for estimating stresses, seems to need no further illustration. But the question as to decussalion, affects the case of Fig. 19, as well as that of Fig. 13. The two sets of diagonals which meet' the upper and lower chords in the centre, have symmetrical halves on each side of the centre, and no action can pass the centre upon either, when they are uniformly loaded; whereas, the two sets to which 7/9 and 7\9 belong, have thehalf of one on either side of the centre, a counter part to the half of the other set on the opposite side; and the diagonals 7/9 and 7\9, will act or not, according as their opposite points of connection with upper and lower chords, are carried farther apart, or the contrary. Now, as the points 9 and 7, upper chord, are depressed by the change in one vertical and 3 diagonals, while the opposite points at the lower chord are depressed by the TRUSSES WITHOUT VERTICALSo 91 change in 3 diagonals only, we might naturally expect to find greater depression in the upper than the lower points; though this does not follow as a matter of necessity, since the less number of members, by being more nearly under a maximum stress, might give greater depression than the greater number, under less stress, as compared with their maximum. Now, the vertical at 1, being under maximum weight, gives depression = E; (adopting the notation used with reference to Fig. 13 [xLIx.]). The two diagonals 1/3 and 3\5 being under 2o maximum, give depression equal to - x4E (making h=v=1), = 3.81E; while the diagonal 5/7, under -4- maximum, gives depression = 0.4x2E. = 0.SE, making a total depression of point 7, upper chord, = 5.61E. Again, the diagonal 1\3, under maximum stress, gives depression = 2E, while 3/5 and 5\7, under {1 maxm u m stress, give depression = 4 X 4E, = 3.2E, making a total for the point 7, lower chord, equal to 5.2E, which is less by 0.41E, than the depression of the opposite point in the upper chord, whereas it should be greater by 0.8E, in order to give to 7\9 and 7/9, the tension assigned to them by the decussation theory. But we must not conclude from this fact, that there is no decussation in this case. For, if we assume that 7\9 is inactive under the full load, it follows that 5/7 is also inactive, and that 1/3 x 3\5 sustain only Imaximum stress, producing I E, = 3.05 E, which added to 1E for the vertical at 1, makes 4.05 E, = depression at point 7, upper chord; while the 3 diagonals contributing to depression of the opposite point in lower chord, are under maximum stress, producing depression = 6 Eo Rence, we see, that upon this hypothesis, the distance between these two points, measuring the vertical reach 92 BRIDGE BUILDING, of the diagonal, is increased by (6 - 4.05) E = 1.95 E. This can not be, without producing tension upon diagonals 7\9 and 7/9. Since then these members can not be entirely without action, and as previously shown, they can not have as much action as the decussation theory assigns to them, it follows, in this case, that they must act, but with less intensity than the theory assigns them. In this case, as well as in that of Fig. 13, the result would be changed somewhat, by taking into the calculation the weight of structure, which would change to a small extent, the relation between the maximum. stresses of diagonals, and the stresses they sustain under a full load. For the stress due to weight of structure, is constant, and that due to variable load, is greater, upon most of the diagonals, under certain conditions of a partial, than under a full load. Hence, while 5\7 sustains (under full load), only { maximum upon that part of the material provided for variable load, it sustains a full maximum upon the part provided to sustain weight of structure. It is easy enough to take these things all into account, in estimating the amount of decussation in special cases. Still, it is doubtful whether any better practical rule can be adopted, than the one previously given, [XLVIII]; namely, to estimate stresses upon both hypotheses, and take the highest estimate for each part. DECUSSATION IN TRUSSES WITH VERTICALS. LIX. In trusses of this class with odd panels, and diagonals crossing two panels, as in Fig. 20, it will be seen, on subjecting them to analysis, such as was explained with reference to Fig. 18 [xvI], that, while in trusses of even panels, the figures in the second line DECUSSATION, ETCo 93 over the diagram, indicate the maximum stresses of diagonals, and those under the diagram, the stresses under uniform load (which are generally less than the maximum under partial loads), in case of the truss with odd panels, the bottom figures show, for certain diagonals, greater stresses for the full, than the upper figures give as the maximum for partial loads. Thus, in Fig. 20, the number 16 over m, indicates 16w" (=G6w), as the maximum weight for ii, while the figure 2 under the point i, indicates that il sustains 2w (= 18w"), under the full load. It should be remarked here, that the figure 1 under the first two nodes on either side of the centre, and the figure 2 under the next, are thus FIG. 20. 1 2 2 4 5 6 7 8 1 2 4 6 9 12 16 20 s'P q p o n I a b c d e f g i j k 1 1 1 1 1 placed upon the assumption that all the weight on either side of the centre, is made to act on its nearest abutment. This would necessarily be so, if en and f were removed or relaxed. But, with those members in place, and properly adjusted, there may be a decussation of forces through them, whereby a portion of the weights at e and f, may be made to bear upon the more remote abutments. Now, as the maximum on en is 6w" and that of its antagonist only 4w" the latter is not sufficient to neutralize the former entirely, but leaves a balance of 2w" which may be transmitted through en to gl, as an offset for a like amount trans 94 BRIDGE BUILDING. mitted through fq to ds. If this be so, then fm and er do not sustain the full weight of 1w, but only 7w" which, being transmitted to il, makes, with the weight w (=9w"), applied directly at i, 16w", as indicated by the figures over the diagram, instead of 2w (= 18w"), as the figure 2 under the point i would indicate. Now, whether the two diagonals en and fq, being apparently, in a state of partial antagonism, do in whole or in part neutralize the tendency of each other to transmit weight past the centre each way under a uniform load of the truss, is not quite obvious, and it may be proper to estimate stresses under both hypotheses, and take the highest estimate for each part of the truss. It will be seen that il and cs are the only diagonals in Fig. 20, which show greater stress with a full than a partial load, upon the non-decussation hypothesis. But all the diagonals undergo different stresses, with the uniform load, as viewed under the different theories, and consequently, their effects upon the chords are different. The end brace as, sustains 4 (w+w') = 4WV substituting W for w+w'), under either theory, and the tension of ac equals 4w; (making h = ab, and v = bs). cs sustains 2W, or 16W, whence cd sustains either 6W or 57WT. Again, ds sustains W, or 12W, the former without, and the latter with decussation. This diagonal having a horizontal reach of 2h, adds 2W, or 24W- to tension of chord, making 8W-, or 81 WA, as the tension of de. For er, we have W without decussation, making a tension of 10W for ef; while with decussation, er sustains TW, from which we subtract AW, for opposite action of en, leaving W DECUSSATION, ETC. 95 giving horizontal pull =1I-W- to be added to 89WA making 91 W = tension of ef. Upon the non-decussation hypothesis, s r and m 1, of the upper chord sustain thrust equal to 8W- and the remainder of the chord, 1OWA. By the other hypothesis, sr and ml sustain 81 W f- rq and nm sustain 99 W, and the other 3 sections, 101 W_ LX. We may derive some more light upon this subject, by considering the conditions resulting from the elasticity of materials. Supposing the upper and lower chords to be so proportioned as to be uniformly contracted or extended under a uniform load of the truss, this does not require or imply any appreciable difference in lengths of diagonals. But the stress upon chords being produced by the action of diagonals, the latter, when, as here supposed, acting by tension, necessarily undergo extension, by which means, the panels (except the centre one), are changed from their original form of rectangles, to that of oblique trapezoids. For instance, the figure gj I n becomes longer diagonally from g to 1, than from n toj, whence the point g falls lower than it would do, if the diagonal suffered no change. Suppose then the truss to be fully loaded, and the diagonals il, gl and fm, to be each exposed to the same stress to the square inch of cross-section. In that case, il and gl suffer extension proportionally to their respective lengths, thereby causing depression of the points i and y respectively as the squares of those lengths. [See note in section xLIX.] Hence, the point g is de 96 BRIDGE BUILDING. pressed more than the pointj, by the extension of diagonals, by as much as the square of gl exceeds the square of ii, or as 8 to 5; assuming diagonals to incline at 45~. The panel gm, must therefore be oblique, and the distance gm, greater than hi. Again, the pointf suffering the same depression from the extension offm, as the point g suffers from that of gl, and a still further depression from the compression of mi, and the extension of i, it. follows that the panel fn must also be oblique, and the distance fn, greater than the distance og. Now, the obliquity of both of the panels gm and Jn, manifestly contributes to the excess of distance fm, over oi. On the contrary, the centre panel eo has no obliquity due to extension of diagonals, or compression of uprights; since there is no cause for obliquity in one direction, more than the other. It seems to follow that en, crossing one oblique panel, must undergo extension; but not so much as fm,which crosses two. Now, fm and en being equal in length, the weight sustained by each, is manifestly as the cross-section and extension combined; and as the former,fm, should be the larger in the ratio of 9 to 6, or as their maximum stresses; if we allow their extensions to be as 2 to 1, the greater for fm, the relative weights sustained would be as 18 to 6, or as 6 to 2. Our decussation theory gave their relative stresses as 7w" to 2w". This is not a wide discrepancy, seeing that the above computation is based in part upon a mere approximate data. We may conclude then, that in cases like the one under consideration, decussation does actually take place. Still it obviously depends upon conditions which are not of the most determinate character. For, if en and fq, be relaxed or removed, under a full load of the DECK BRIDGES. 97 truss, decussation can not take place, for the same obliquity of the two panels next to the centre one, which produces the tendency toward tension of en andfq, on the contrary, tends to relax do and gp, through which latter alone decussation could take place, in the absence of the former. On the other hand, if en and fq be sufficiently strong, they may be strained to such a pitch as to bear all the weights at e and f, and leave fm and er entirely inactive. Hence, there is an uncertainty as to the action of these diagonals, which may be best obviated by estimating stresses upon both theories, and taking the highest estimates; as recommended with reference to trusses without verticals, and as previously suggested with reference to the case in hand. In view of preceding facts and principles, it may be advisable to avoid the odd panel in trusses with verticals, when practicable without incurring more important disadvantages in other respects. DECK BRIDGES. LXI. Are those having the movable load applied at the nodes of the upper, instead of the lower chord, as generally assumed in preceding analyses. It will readily be seen, on a brief contemplation of Figures 12 and 13, for instance, that weights applied at the upper chord, act directly upon compression members, either erect or oblique, as the case may be; and are thence transferred to tension members at the lower chord; according to the general principle, that weight applied at the upper end of a member, always acts by compression, and that which is applied at the lower end, by tension. 13 98 BRIDGE BUILDING. In the case of truss Fig. 12, the action of tension diagonals is precisely the same, whether the weight be applied at the upper or lower chord. But the compression verticals, in the deck bridge, sustain as their maximum, the weights indicated by the figures immediately above them respectively, from the centre toward the right hand; and these weights, of course, are equal to those acting upon the diagonals respectively meeting the verticals at the lower chord; and consequently, greater than when the weight is applied at the lower chord. For illustration, in Fig. 12, as the truss of a deck bridge, the vertical f/ sustains 15w", the same as fj, whereas, in the case of a "' Through bridge" (with load applied at the lower chord), ft sustains only 0lw' communicated to it through ek. In the deck bridge also, the tension verticals bo and jy are essentially inactive, merely sustaining a small portion of the lower chord. The chords suffers the same stress in both through, and deck bridges. LXII. Load applied at the upper chord of truss Fig. 13, acts by thrust directly upon the diagonals meeting at the upper chord, and the maximum weight (from movable load), sustained by diagonals meeting at one of the upper nodes, are indicated by the two figures immediately over the node; the larger figure referring to the diagonal running toward the nearer abutment e. g., the numbers 4 and 6 over the point m, signify 6w" =_ greatest weight borne by mc, and 4tw" = the greatest borne by me. It is obvious also, that the maximum thrust of any diagonal, equals the maximum tension of the diagonal meeting the former at the lower chord; that is, maximum thrust of me, is equal to 6w/~, = maximum ten DECK BRIDGES. 99 sion of co. The maximum thrust of bn being equal to 9w"-, the maximum tension of bo, equals 9w" This is an extra weight thrown upon the point o, in consequence of the vertical bo, being turned out of its regular direction of a diagonal in the position of bp,* in order to throw its load upon oa, whereby op and pa are rendered unnecessary. The weight borne by oa, therefore, instead of being 12w", as indicated by the figure 12 at o, is 12w"+9w", -- 21w", = 3tvo The figure 1 over o, denotes the tendency of lw" to act upon oc, by thrust, by which tendency the tension of oc, under a full load of the truss, is reduced to 5WP-. LXIII. If Fig. 12 be assumed to represent a truss with tension verticals and thrust diagonals, the figures over the upper nodes, prefixed to w" indicate the weights tending to act by compression upon the diagonals descending toward the right from the nodes respectively; which weight is transferred to the vertical meeting the diagonal at the lower chord. This constitutes the maximum load of the vertical, in case of a deck bridge. Otherwise, the maximum stress of verticals is shown by the figures immediately over FIG. 13A. p3 o nz T6 I J j ca b c d e f g i' The point p, not seen in Fig. 13, is assumed to be at the intersec tion of a vertical line through the point a, with the upper chordproduced. The arrangement above alluded to, gives the truss a reclangular, instead of a trapezoidal form of outline, which involves no more action upon material,'though it increases the number of members in the truss. [See Fig. 13A.] 100 BRIDGE BlUILDrNG. them, prefixed to w", provided, that in this case, the maximum stress of a vertical can never be less than w. = the weight applied immediately at its lower end. RATIO OF LENGTH TO DEPTH OF TRuss. LXIV. Having explained and illustrated, it is looped intelligibly, methods by which may be computed the stresses of the various parts composing most of the combinations of members capable of being used in bridge trusses, with a view to giving to each part its due proportions, it may be proper to give attention to the general proportions of trusses, and such other considerations as may affect the efficient, and economical application of materials in bridge construction. The ratio of length to depth of truss is susceptible very great range, and it is obvious that some certain medium, in this respect, will generally give more advantageous results, than any considerable deviation toward either extreme. For, it will be observed, that in the expressions we have derived for the amount of action open chords, - appears as a factor; v representing the depth of truss, between centres of chords. Hence, the smaller the value of v, the greater the stresses of chords, so that when v=0, these stresses become infinite, and the chords require an infinite amount of material; in other words, the case is impossible. On the other hand, if v be infinitely great, though the stress of chords be reduced to nothing, the verticals and diagonals being infinitely long, and sustaining a definite weight, also require an infinite a-mount of material. lNow, between these two impracticable extremes where shall we look for the most advantageous ratio? LENGTH TO DEPTH OF TRUSS. 101 It can not be the arithmetical mean, for there is no such mean between v = 0, and ^ = infinity. Uncoubtedly, we shall be unable to do more than answer this question approximately; and that, only with reference to specific cases; for the ratio suitable for one length of span, and in one set of circumstances, will often be found quite unsuitable under different circumstances. We have seen that the material required in chords, is in general, inversely as the depth of truss, or as -. Also, that the material for verticals and diagonals, increases with increase in the value of v; though not in a determinate ratio. But assuming the latter classes of members, including the main end braces of the Trapezoidal truss, to increase in the ratio at which v increases, while the chords diminish at the same rate, we might reasonably assume, that the minimum amount of action upon materials would occur when the amount of action upon chords were just equal to that upon all other parts of the truss. By recurrence to the analysis of truss Fig. 12, [xLIII], we find amount of action upon chords, represented by 561-M, and that upon all other parts, by ( 16U + 22.57v) M. Here, h is equal to part of the length of truss, while v is variable; and, by making these two co-efficients of M equal, and deducing thence the value of v, we have the depth of a 7 panel truss in which the amount of action upon chords, equals that of all other parts. Thus, putting 56-t = 16 — + 22.57v, substracting 167, and multiplying byv, we have 40h2 = 22.57v2; whence v= v/( 4 ), =1.34h nearly. This gives length to depth of truss, as 5.2 to.o 102 BRIDGE BUILDINOo Again, referring to analysis of truss Fig. 10, we find action upon chords represented by 20- M, and action upon other parts, by (8- + 11.2v) M. To make these quantities equal, requires that v = 1.03h, and that the length of truss be equal to -5 times its depth, or nearly 5 to 1. From this last case, we may infer as a probability, that a ratio of length to depth as 5 to 1, is the most economical for a truss of 5 panels, other things the same. We know, moreover, that by making v = Th, in the same truss, we double the amount of action upon chords - making it equal to the aggregate upon all parts with the ratio of 5 to 1, while the action, and consequently, the material of the other parts is probably reduced one-half. Hence, a ratio of length to depth as 10 to 1, probably increases the aggregate amount of action by some 25 per cent, over what takes place with a ratio of 5 to 1. We may therefore unhesitatingly conclude, that whether the ratio of 5 to 1 be too small or not, the ratio of 10 to 1 is much too large. Referring again to the 7 panel truss, it appears above, that a ratio of 5.2 to 1 indicates the same amount of action upon chords, as on all other parts. But we can not with certainty infer that the absolute amount of action upon the truss, is less with v=1.34h, than with v=h; in which case length is to depth as 7 to 1. In fact, if we estimate the absolute amount of action, assuming these two values of v successively, we shall find no essential difference in the results. Hence, if other conditions were the same in both cases, it would follow that the ratios of 5.2 to 1 and 7 to 1 were equally favorable to economy, and that there is a better ratio still, between the two; probably, about 6 to 1, LENGTH TO DEPTH OF TRUSS. ]03 But the conditions are not the same in the two cases, aside from the different values of v. For, while with v-h, the diagonals incline at 450, in the other case, their inclination from the vertical is considerably less, being only about 37~. This, we shall see hereafter, is a less favorable inclination for diagonals acting by tension, than 45~; and, since the ratio of 5.2 to I shows an equality as to economy, with the ratio of 7 to 1, with the more favorable conditions on the side of the latter, it would seem at least, highly probable that the ratio of 5.2 to 1 is the more near approximation to the desired optimum. Now, after much thought and investigation, with some considerable experience in planning and constructing truss bridges, I can give no better practical rule as to the proper depth of a truss of a given length, than to adopt that ratio between 7 to 1 and 5 to 1, which will best accommodate the desired length of panel (or value of h), and afford the best, and most economical inclination of diagonals; matters to which attention will shortly be directed. It is not supposed, however, that these limits of ratio will not frequently be exceeded, particularly in the adoption of a greater ratio than 7 to 1. In case of the very long spans dared and achieved in this age of rail roads and locomotion, engineers may recoil from the towering altitudes of 50 or 60 feet depth of truss which some of the long spans now occasionally constructed would require, perhaps more in deference to European precedent, and from an instinct of conservatism, than from regard to economy, and a true appreciation of the real merits of the question. But for important bridges for heavy burthens, a ratio greatei 104 BRIDGE BUILDINGo than 8 to 1 can not be regarded as commendable, except in rare and peculiar circumstances. INCLINATION OF DIAGONALS. LXV. We have seen the absolute importance of oblique members in bridge trusses, and we have also seen the excellence, in point of theoretical economy, of the trapezoidal truss, with parallel chords connected by diagonal members, with or without verticals. Now, since there is an endless variety in the positions which a diagonal member may assume, it becomes an important question, what degree of inclination these members should have, to give the most economical and satisfactory results. The inclination may be increased till it reaches a horizontal position, or diminished till it becomes a vertical; when, in either case, the member ceases to be a diagonal, and becomes incapable of performing the office of effecting a horizontal transfer of vertical pressure. The greatest efficiency of material used in diagonals, is manifestly, when the weight sustained by a given quantity of material, multiplied by the horizontal reach, gives the largest product; and, when the member acts by tension, the weight capable of being sustained by a given amount of material, is as the crosssection directly, and as the rate of strain inversely. But the rate of strain, or stress produced by a given weight, is as the weight multiplied by the length of diagonal (D), and divided by the vertical (v), or as D~ while the cross-section is inversely as D, or as I Ienc the weight is as - =V Rence, the weight is as - - -, ^= INCLINATION OF DIAGONALS. 105 Now, representing the horizontal reach by h, the efficiency of the material must be as t equal to ~ 7Zo Then, making v, constant, and dividing by v, the expression becomes, -,-A still being proportional to the efficiency of material. Consequently, that value of h, which gives the largest value to the expression — + - will indicate the inclination at which the diagonal will act with the greatest efficiency. This value of h, is found by differentiating the function,-, (h being the variable), and putting the differential equal to 0: by which process we obtain: d ( ) (2h2 )d-21&2dhA0, whence canceling the denominator, v2dh + h2dh = 2h2hA, and h2dh = v2dh. Then, dividing by dh, and extracting the square root, we have h=v; thus showing that an inclination of 450 is the most advantageous for tension diagonals as far as relates to those members alone. THRUST DIAGONALS. LXVI. The efficiency of material in a thrust brace, is directly as the useful effect produced by the member, and inversely as the amount of material required in it. Now, the useful effect, as in the previous case, is as the weight sustained and the horizontal reach, while the amount of material, depends not only upon the stress and length, but also upon the ratio of length to diameter, which affects the power of resistance. Theoretically, the power of resistance is as the cube of the diameter (d), divided by the square of the length (=v2xh2), a rule which is not sustained by experience, c(Ronman), before a variable, or the function of a variable, denotes the differential of such variable or function. 14 106 BRIDGE BUILDING. except in case of long slim pieces which break by lateral deflection, under a comparatively small compressive force. We will, however, use the rule for the present occasion. The efficiency of the material then, will be as the power of resistance and the horizontal reach directly, and as the stress produced by a given weight, inversely; which stress is asVva+/2r. Whence we have cd1~ 2~/ d~'vh\ v - ^ ( (-+ I. )) proportional to the efficiency of mate ial in a thrust brace. Making d3v=l, the last expression becomes 7 _), and the value of h which gives (V2 h62)l the greatest value to this function, will indicate the inclination at which a thrust brace will act with the greatest efficiency, as it regards the brace alone. Differentiating, and putting the result equal to 0, we have: d ( h )dh @24+ h2) 2h (u2'-2)) L><2i=h_0; whence, (V2 + b2) 3 (V 24k2)3 multiplying by the denominator (v2+h2)3a we obtain dh (v2+h2)j - 3h (v2 +') 2)x 2hdh, and, dividing by /(v2+h2)dh, we have v2+h2 = 3hx2h = 32 whence, V2 3h2-h2 = 2h2, and by evolution, v = h2, and h =- 0.7072v. If we deduce the value of the expression (which is equal to the horizontal reach divided by the cube of the length of brace), putting h=v and h==v successively, we find the degree of efficiency less than the maximum, as above determined, by about 9 per cent in the former, and 8 per cent in the latter case; showing that considerable deviations may be made in the inclinations of thrust braces without much detriment to efficiency of material in braces, when required INCLINATION OF DIAGONALS, 107 by other considerations; which will often be the case, as will be seen hereafter. EFFECTS OF INCLINATION OF DIAGONALS UPON STRESS OF CHORDS AND VERTICALS. LXVII. The comparative effects of different posisitions of diagonals upon the chords, may be illustrated with reference to Fig. 21. It is manifest that a given weight w on the centre of this truss, will produce a vertical pressure equal to ~w at each of the points a and b, and that each oblique member between a and w, will sustain a weight equal to ~w; and will exert each a horizontal action upon the upper and lower chords, equal to 1 w. Hence, the stress of chords in the centre, will equal 21Wxn, in which n represents the number of oblique members between a and w, or between a and c. But n equals T whence wh = 1wh Xc 2 2 v y W —. FIG. 21. e w a d c b The term h having been eliminated from the last expression, it shows that the inclination of diagonals has no effect upon the stress of chords in the centre, produced by weight in the centre of the truss; and by similar reasoning it is shown that the same is true in relation to other parts of the chords, or to weight at any other points in the length of the truss; the only difference being that the shorter the panels, or the smaller 108 BRIDGE 3BUILDING the value of h, the shorter the intervals at which the increments in the stress of chords are added, and the less the magnitude of such increments, in the same proportion. Hence, in general, there is no difference in the stresses of chords, whether the diagonals have one inclination or another. With regard to the effect upon verticals, that part of their stress which they receive through diagonals, is equal to the weight sustained by those diagonals, and is the same for a given weight, whatever be their inclination. On the vertical we, the pressure is received directly from the weight. But on the next adjacent vertical, on either side, one-half of the same pressure is received through to the intervening diagonal, and transmitted to the next, and so on to the end. Consequently the aggregate action of verticals, produced by the weight w, is equal to tw + ~wn, taking n for the number of verticals receiving their stress through the medium of diagonals, and which is equal to the whole number less 3, when the number is odd, and the verticals act by thrust, as assumed in the case of Fig. 21. If the weight be applied at the lower chord, the whole action of verticals is communicated through diagonals, the latter acting by tension. Hence the aggregate action of verticals increases and diminishes with their number, and economy as regards those members, would require the diagonals to incline at a greater angle with the vertical than that which is most favorable as to the diagonals themselves. We have seen, however [LXVI] that by placing the diagonals at 450 when they act by thrust, we lose about 9 per cent in economy of those members, and we now learn that such an arrangement increases the economy in verticals to a considerable extent by diminishing WIDTH OF PANEL. 109 their number; the actual amount depending somewhat upon the number, and not deducible by a general rule. We shall not, however, err greatly in assuming, that with an inclination of 45~, for thrust diagonals in conjunction with tension verticals, the loss upon the former is quite made up by saving in the latter, and that a less inclination in this case, should be regarded as very questionable practice. In case of tension diagonals and vertical struts, a saving in material may undoubtedly be made by making the horizontal greater than the vertical reach of the diagonal, whenever such a course is found consistent with a proper regard to just proportions of the truss in other respects; such as width of panel, depth of truss, etc. THE WIDTH OF PANEL. LXVIII. Which we have represented in our formula by A, has only been hitherto considered as to its relations to v, representing the depth of truss. With regard to the best absolute value of h, the question is affected by the relative expense of floor joists, and the extra amount of material and labor in forming connections at the nodes of the chords; as well as, in some cases, the lengths of sections in the upper chord. The latter requires support laterally and vertically at intervals of moderate length, depending upon the absolute stress, which, other things the same, governs the cross-section. The upper chord usually, of whatever material, has a cross-section so large as to exclude all danger of breaking by lateral deflection, in sections of 10 to 14 feet; and, as there will seldom be occasion for exceeding these lengths in cancelated trusses, the increased 110 BRIDGE BUILDING. expense of joists for wide panels, and the expense of extra connections in narrow ones, are the principal considerations affecting the absolute value of h, as an element of economy. The transverse beams, supposed to be located at the nodes between adjacent panels, may, of course, be proportioned to the width of panel, so as to require essentially the same material in all cases. But the joists, or track stringers of rail road bridges, the depth being proportional to the length between supports, have a supporting power as their cross-sections; and since the load, at a given weight to the lineal foot, is directly as the length, it follows, that to support the same load per foot, as bridge joists are required to do, the cross-section should be as the length. The expense of joists and stringers, therefore, is directly as the width of panel.* On the contrary, the expense of connections will be as the number of panels, nearly, and consequently, inversely as their width, or inversely as the * The thickness of joist most economical for a short reach would be liable to buckle with greater length and depth. Hence joists require increase of thickness with increase of length and depth. The thickness should be as the depth, and the cross-section, as the square of the depth (cd). Upon this basis, the required material for joists, increases at a greater ratio than the increase in width of panels. The supporting power of a joist or beam of a given form of section, or a given ratio of depth to thickness, is as the cube of the depth directly, and the length (1) inversely; or, as -. If there be two joists of depths respectively as d and x, and lengths as I and Ml, their supporting powers P, P/, for load similarly applied, will be as-i to l. But the power should be as the load; in other words, as the length of joists. Hence we have the proportion, Xs3 s3 X3 --:,: ": l, whence, cnd =' and x == cld-. Now A is as the length of joists, and the depth, therefore, is as the - power of the lenth, and the cross-section, and consequently the required material, as the W power of the length. Hence, if m represent the material for joists with panels of a given width, the material for panels twice as wide, will be represented by m X 2-/ l= 16 -- m11.52. But this is rather anticipating the subject of lateral, or transverse strength of beams. WIDTH OF PANEL, 111 length of joists. Hence, if we could find the point where the cost of connections (consisting of extra material in the lappings of parts, connecting pins, screws and nuts, and enlarged sections at the ends of members, together with the extra labor in forming the connections), becomes equal to the whole cost of material in joists or stringers; that would seem to indicate the proper width of panel, or value of h, as far as depends upon these elements. But aside from the fact that our data upon this question are so few and so imperfect, that it would be mere charlatanism to attempt to reduce the matter to a mathematical formula, the occasions would be so rare which would admit of the application of such formula, without incurring disadvantages in other respects, such as improper inclination of diagonals, unsuitable ratio of length to depth of truss, &c., that no attempt will be here made to give any thing more definite upon this point, than to refer to the best precedents and practice of the times; which seem to confine the range of width of panel mostly within the limits of 8 and 15 feet. Within these limits, and seldom reaching either extreme, plans may be adapted to any of the ordinary lengths of span, by adopting the single or double cancelated trusses, Figs. 12 and 13, or 18 and 19, or the arch truss Fig. 11, (which unquestionably contain the essential principles and combinations of the best trusses in use), according to length of span, the purposes of the bridges respectively, or the taste and judgment of engineers and builders. 112 BRIDGE BUILDING. ARCH BRIDGES. LXIX. An arch bridge may be distinguished from an Arch Truss Bridge, by the fact that in the former, the bridge and its load are sustained by one or more arches without chords; and, consequently, requiring external means to withstand the horizontal thrust or action of the arches at either end; which means are afforded by heavy abutments and piers, in case of erect arches, and by towers and anchorage in the earth, in case of inverted, or suspension arches. It is not the purpose of this work to treat elaborately of either of these forms of bridging, as the author's experience and investigations have been mostly confined to truss bridge construction. But as some of the largest bridge enterprises and achievements of the age are designed upon the principles here referred to, a brief notice of the subject, and some of the conditions affecting the use of these classes of bridges, may be regarded as desirable in a work of this kind. Suspension, or inverted arch bridges of very great spans, have long been in use, both in this and foreign countries; and the capabilities of that system have been pretty thoroughly tested experimentally and practically. But bridges supported by erect metallic arches, have hitherto been confined to structures of moderate span. Within a few years, however, the magnificent enterprise of spanning the Mississippi at St. Louis by three noble stretches of about 500 feet each, supported each by four arched ribs of cast steel, has been undertaken and is understood to be in rapid process of execution. The interest naturally felt in the progress and final result of this grand enterprise, by students and practi ARCH BRIDGES. 113 tioners in the engineering profession, will perhaps aid in rendering the following brief, and somewhat superficial discussion acceptable. LXX. An erect arch subjected to the action of weight, or vertical pressure, is in a condition of unstable equilibrium; and can only stand while the weight is so distributed that all the forces acting at each point of its length, are in equilibrio. To illustrate this, we may assume the arch to be composed of short straight segments meeting and forming certain angles with one another, and the weights applied at the angular points. A weight at c, Fig. 22, for instance, acts vertically, and, if dc be produced till it meet the vertical drawn FIG. 22. 1V through b in m, then the triangle bcm has its sides respectively parallel with the directions of three forces acting at the point c; namely, the weight at the point c, the thrust of the segment be, and that of de. Hence, if these three forces be to one another as the sides of said triangle,-that is, if the weight (w): thrust of be: thrust of de:: bm: be: cm, then they are in equilibrio. If w be greater than is indicated by this proportion, the point c will be depressed, bed approaching nearer and nearer to a straight line, and becoming less and 15 114 BRIDGE BUILDING. less able to support the weight, and a collapse must result. If w be less than the above proportion indicates, it will be unable to withstand the upward tendency of the point c, due to the thrust of be and de (or, to the preponderance of the vertical thrust of be, over that of de), the point c will rise, the upward tendency becoming greater and greater, and the result will be a collapse, as before. The same reasoning, and the same inference, apply to any other angular point, as at c. It is, therefore, only in theory that such a thing as an equilibrated erect arch, can exist. The arch is here considered as a geometrical line without breadth or thickness. It is this property of instability, in the Erect Arch, that the diagonals in the Arch Truss, [Figs. 5 and 11] are designed to obviate, and to enable the arch to retain its form and stability under a variable load. LXXI. Still, in theory, an arch may be in equilibrio with any given distribution of load, whenever the points a, b, c, etc., are so situated that the sides of the triangle bcm, for instance, formed by a vertical with lines respectively coinciding or parallel with the two segments meeting at c, are proportional to the 3 forces acting at c, as above stated, and so at the other angular points of the figure. To construct an equilibrated arch adapted to a given distribution of load, consisting of determinate weights at given horizontal intervals between the extremities of the arch, we may proceed as follows: Draw a horizontal line representing the chord ak, and upon the vertical Cft, erected from its centre, take Cf equal to the required versed sine, or depth of the ARCH BRIDGES. 115 arch at the centre. Also, takeft= Cf, and erect verticals upon the chord, at all the points at which the load is applied, and join a and 1. Then, if the load be uniformly distributed (horizontally) upon the arch, we have seen, [LII], the arch should be a parabola, to which of course, at is tangent at the extremity, a. But, regarding ab as tangent to the curve at r, half way between a and b (horizontally), we seek the abscissa fs, which is to Cf:: rs2: a(l2. Then, taking the distance offu=lfs, au is tangent to the curve at r, and coincides with the first segment (ab) of the archo (These segments are supposed to be so short, that the tangent and curve may be regarded as essentially coinciding, for the length of a single segment). Now, the thrust of ab, is to the whole weight bearing at a, as ru to us; and, erecting the vertical al, such that al: ab:: weight at b: thrust of ab, and drawing the straight line lbc cutting the second vertical in c, we have be for the second segment of the arch, being in the line of b1, which represents the resultant of the two forces acting at b; namely the weight at b, and the thrust of ab, In like manner, take bm, representing the weight at c, and the straight line mcd, meeting the third vertical in d, gives cd as the third segment of the arch. Repeat the same operation for each of the succeeding segments de, ef, &c., till the arch is completed, and it is obvious that the forces acting at each of the several angular points b, c, d, &Co, are in equilibrio; and that the arch throughout is, theoretically, in a state of equilibrium. We may vary this process so as to secure greater accuracy of construction, in the following manner o 116 BRIDGE BUILDING. Producing lbc till it meets Ct in v, we see that abl and ubv are similar triangles, and al: uv: horizontal distance of I: horizontal distance of v, from the point 6. Hence, we may take the point v instead of the point 1, by which to establish the position of the line be, and thereby secure greater relative accuracy of measurement. So may we also take vv', or V' instead of bn, to determine the line ed. By this means we multiply the small spaces al, bi, &c., and diminish the amount of error in measurement, and if the angular points, or nodes be at uniform horizontal distances, the process is very simple. LXXII. We have assumed, in describing the arch a, b, c, d, &c., a uniform distribution of load, horizontally. But the general process is obviously the same for an unequal distribution, after locating the first segment ab; which we may do by first ascertaining the amount of bearing at a, due to the load of the arch. This will be to the whole load, as the distance of the centre of gravity of load from k, horizontally, to the whole chord ak. For instance, if the centre of gravity be half way between C and k, one quarter of the load bears at a. The weight bearing at a, whatever it be, may be represented by A; and supposing it to exert the same horizontal thrust at a as half the load (V), would do when uniformly distributed, we take u' in ft, so that i W: A:: uC: u'C* Then au' gives the direction of ab', and we proceed in the same manner * We may assume any amount of horizontal thrust, and the greater the assumed thrust, with a given load, the less will be the depth of the arch, and vice versa. It is proposed here to construct an equilibrated arch a, b, c, d, &c., of about the same rise at the crown, as the normal curve, a, b c, d, &c., has. ARCH BRIDGESo 117 as before using the weights given for the several nodes of the arch, to determine the points c' d', &c. These being connected by straight lines, we have an equilibrated arch adapted to the given distribution of load. LXXIII. But of course, this arch will not stand under any other disposition of the load. To obviate this difficulty, and to construct an arch which will stand under a variable load, without the chord and counter-bracing of the arch truss, the device has been adopted, of constructing the arch of such vertical width that all the equilibrated arches or curves, required by all possible distributions of load; may be embraced within the width of the arched rib. Then, if there be sufficient material to oppose and withstand the forces liable to act in the lines of said several equilibrated curves, complete vertical stability must result. The proper width, or depth of the arched rib, will depend upon the length and versed sine of the arch, as well as the amount and distribution of load; and the material will act most efficiently, when mostly disposed in the outer and inner edges, or members of the rib, and connected, either by a full, or an open web, to distribute the action between the outer and inner members, according as the resultant line of action approaches the one or the other of those members. The normal form of the arch should be such as to be in equilibrio under a uniform load,* and hence it will be parabolic, as to the movable load, and the weight of road-way, and catenarian, as to the weight C The method above explained, for describing an equilibrated arch, is applicable to all cases where the load, both constant and variable acting on the several parts of the arch, is known, whether it be the normal curve, adapted to a full load, or a distorted curve, suited to an irregular distribution of load. 118 BRIDGE BUILDING. of arches (as far as they are uniform in section), and should approach the one or the other form, according to the weight of arches, as compared with the other weight to be supported thereby. The distance between outer and inner members, or the width of web, reckoned from centre to centre of those members, should be such that no condition of unequal and partial load, could throw greater action at any point of either member, than the extreme uniform load would throw upon both. Let us suppose that the curve a, b, c, d, etc., be centrally between the two members and that dd', and hh' be the greatest vertical departures, inward and outward, of any equilibrated curve, from the normal curve a, b, C, etc. Let us further suppose that the thrust of the arch at the points d and h, be 3 as great under the load acting in the curve of greatest departure from the normal as the extreme uniform load produces at those points. Then, if the outer and inner members of the rib, be placed at a distance of six times the greatest departure of the distorted from the primary, or central curve, one member will be twice as far from the line of action (at the point of greatest departure), as the other, and the latter will sustain two-thirds of the action, equal to one-half the action of the full load, and the same as in the latter case. If the width of web be less than six times the greatest aberration of the distorted curve, the action, under the suppositions above, will be greater upon one member than that due to a full uniform load; a condition altogether to be avoided. A few trials at constructing curves adapted to assumed possible distributions of load, may determine ARCH BRIDGES. 119 satisfactorily what condition gives the curve of greatest distortion and the greatest departure from the normal; and the amount of action under that condition, can be readily calculated with sufficient nearness, whence the proper width of web may be deduced. LXXIV. The points of the equilibrated curve may be located by calculation, and perhaps with as much ease, and greater accuracy than by construction. Suppose Fig. 22 to have a vertical depth, CJ; equal to one of ten equal sections of the chord ak. Having found the length of fX, in the manner already explained [LXXI], it is known that for a uniform load at each angle, the vertical reaches of the several segments, begining at the centre, are as the odd numbers, 1, 3, 5, 7 and 9; and, if we conceive Cf to be divided into 25 equal parts (25 being the sum of these numbers), each of these parts will be equal to 0.04 Cf, or.04v; and this factor, multiplied by the numbers 1, 3, 5, &c., give the vertical reaches of the respective straight segments, which vertical reaches being substracted successively from v, and successive remainders, show the several verticals to be as follows: At the centre, f, vertical = Cf =v. At e, vertical = v -. 04v=.96v. At d, vertical= (.96-.12)v =.84v; at c, vertical = (.84-.2)v =.64v, and at b, vertical = (.64-.28), =.36v. This establishes the normal curve for uniform load. Now, supposing the weight of structure to be equal to lw at each of the angles of the arch, and also, that a movable load of a like weight, w, be acting at each of the five pointsf, g, h, i,j; the permanent weight of structure gives a bearing of 4.5w at a, and the movable weights at f, g, h, &c., give respectively.5w,.4w,.3w, 120 BRIDGE BUILDING..2w, and.lw, together, equal to 1.5w; making the whole bearing at a, equal to 6w, which is tth less than if the same weight were distributed uniformly. Then taking Cu' = Cu, and drawing the line au', (not shown in the diagram), we have the inclination of ab', the first segment of the required curve, which gives the same horizontal thrust at a, as the normal curve would exert under the same load uniformly distributed. We findfu (=fs), by the proportion. C: f:: Ca2 (- 5v2) sr2 (= 4.5v2: 25v2: 20.25v2:: v:.81v; and, reducing 1.81v (= Cu), by one-seventh, we obtain Cu' = 1.5514v. This length is to aC:: A (=6w),: horizontal thrust of ab'; that is (making v=l), 1.5514: 5:: 6w: 30 = 19.33w, = horizontal thrust ab. 1.5514 Now, if this thrust be represented by laC, = v=1, then w will be represented by a space equal to 93, -.05173, which is equal to the vertical departure (D), of b'c' from ab'u'. Knowing the value of this departure which, of course, is directly as v, and inversely as aC), we can locate the points c', d', e' and f, by their vertical distances from au', as follows: The vertical at b', is evidently equal to xl1.5514, =.31026; consequently, the vertical at c' = 2x.31026-.05173 =.56879. Vertical at d' = 3x.31026 —3x.05173 =.77559. Vertical e' = 4x.31026-6x.05173 =.93068, and the vertical at f', equals 5x.31026 —10x.05173 = 1.034, showing that the new curve crosses the normal, between e and f, and j' is abovef, but not shown. Then, if each of the segments b'c', c'd', &c., be produced to meet the indefinite vertical drawn through a, they will evidently cut that line at intervals of D, 2D, 3D and 4D together, equal to 10 D, =.5173. Then, ARCH BRIDGES. 121 the weight atf being equal to 2w, it follows that fg' makes twice the deflection from e'f that the latter makes from d'e', that is, equal to 2D in the horizontal distance of lv, or 1, or IOD (=.5173), in the distance aC, or 5. Hence, f'' produced, cuts the vertical at a, twice as high as e'f cuts it, or, at a point 1.0346 above a; being just as high as the pointf/; except a small difference resulting probably from omitted fractions. This shows that fqg' is horizontal, and tangent to the curve at its vertex. It follows that all the weight at f, and at the left of that point, is brought to bear at a, aud all that at gy' and on the right thereof, bears at k. This affords a check upon our work thus far, as we already knew that the bearing at a was equal to 6w, and we now see that this is made up of l1w at each of the four points b', c', d' e', and 2w at f'. Iff'y' were not horizontal the arch could not be in equilibrio under the assumed condition of load. Now, as we manifestly have for the 4 remaining segments, a vertical reach for each, as the weights they respectively sustain; i. e., equal respectively to 2D, 4D, 6D, and 8D; making 20D (= Cf'); altogether, we have only to subtract these quantities successively from Cf (=1.0346), to obtain the lengths of verticals at h', i', j'; as follows: 1.0346 -2 x.05173 -.93114 = vert. at h'.93114-4 x.05173 =.72422- " i'.72422-6 x.05173=.41384- 4 "4 j'.41384-8 x.05173 = 0 k The differences between these lengths of verticals, and those of the normal curve at the same points, show 16 122 BRIDGE BUILDINGo the aberrations vertically, of the distorted, from the normal curve, as below. Nor. Dist. Below. Above. bb'=.36-.31026==.04974 G cc' =.64-.56879=.07121 6 dd'=.84.77559=. 06441 ee' =96-.93068=.02952 6 Dist. Nor. if' =1.034 -1.00-.034 gg'=1.034 -.96-.07446 hh'=.93114=.84-.09114 ii' - e72422-.64=.08422 jj'=.41384-.36=.05384 LXXV. From this exhibit, we perceive that the greate est vertical aberration externally for the condition of load here assumed, is at hh', and equals.091v, and the greatest internally, at cc' (or a little to the right of these points in both cases), equal to.071v, traversing a zone equal in width to.162v. nearly j of the versed sine of the normal curve. Now, we have seen that the horizontal thrust of the arch for a gross load of 14w, equals 19.23w, with the assumed proportion of versed sine to span, as I to 10, whether upon the normal or the distorted curve; and, the thrust being evidently as the gross load, other things the same, it follows that, with the full gross load of 18w, or 2w at each angle, the thrust would be to 19.33w as 9 to 7. Hence the load, as above assumed, produces A, or 77 7 per cent of the maximum thrust under the full uniform load. The uniform load being supposed to act equally upon the outer and inner members of the rib, the action of 50 per cent is due to each; and, in order that neither ARcH BRIDGES. 123 member, at the nearest approach to the equilibrated curve, may be subjected to greater stress than under the greatest uniform load, the web should be so wide that (assuming the outward and inward aberrations to be each equal to the mean of.081v, and putting x= ~width of web), x: ~x+.081v: 77.7: 50. Whence 50z =38.88x+77.7x.081v; and x=.557v. But this value of x being equal to the distance verti, cally across the web between c and d, or between h and i, is greater than the distance square across, about in the ratio of distance from a to f, to the line a C, in this case asV26: 5. The actual width of web, therefore, is only.545v, still considerably more than half the versed sine Cf. The condition of load here supposed, may or may not be the one requiring the greatest distortion of the equilibrated curve. The case has been assumed to illustrate this discussion, as it seemed likely to be near the condition requiring the greatest width of web; and I leave this part of the subject, without attempting a more general and determinate solution of the question. LXXVI. The movable load has been taken as only equal to the weight of superstructure, upon the supposition that this style of bridging would seldom be adopted, except for very considerable lengths of span, where the weight of superstructure is relatively greater than in case of short spans. This doubl'e arch, as here under consideration, consisting of an outer and an inner curved member, connected by a web, in order to act most efficiently should be so adjusted that the outer and inner members may be subjected to equal action under a full maximum, 124 BRIDGE BUILDING. uniform load. Hence, the normal and equilibrated curves, representing the line of the resultant of forecs acting upon the arch, have been assumed as terminating at each end, at points centrally between the extremities of the outer and inner curved members. It might seem possible that the distorted curve adapted to the above assumed condition of load, might so fall as to recross the normal between the points of greatest departure and the ends, and thus diminish the extent of aberration, and the necessary width of web. If the curve a, b', c', etc., be turned upon its centre, by raising the end at a, by 2rds of the greatest departure, that is, by 2x.081v,=.054v, the aberration half way between a andf, where it is at or near its maximum point, would be reduced by.027v, and become.054v just the same as at the end. The other end would drop to the same extent, and would reduce the outward aberration in the same degree. This, of course, would be the least possible extent of aberration; and if we could rely upon the resultant stress following this curve in such a position, it would enable us to diminish the width of web to.364v. But there seems to be no obvious reason why we should assume the equilibrated curve to take the position just described, rather than one with the left end below a, and the other above k, thus increasing instead of diminishing the aberration. Hence, in the case of an arch ribbed bridge, liable to a movable load equal to the weight of structure, foot for foot, upon the whole or any part of its length, if the web of the ribs be less than 36-100th, of the versed sine (Cf Fig 22), certainly, and if less than 54-100ths probably, the material in the principal members is liable to greater strain in some parts, under a partial, than under the extreme load; ARCH BRIDGES. 125 which would be decidedly an unfavorable condition, with regard to economy. LXXVII. The operation of the web in distributing the action upon the outer and inner curved members of the rib, and transferring it from one to the other, may be understood by the diagram Fig. 23, exhibiting said curved members, connected by a web consisting of a simple system of diagonals, capable of acting by thrust or tension as may be required. The normal curve is represented parallel with, and midway between the curved members; and the equiliFiG. 23 brated curve is represented as crossing the normal nearf, meeting it again at a and k, at the ends; and having its greatest aberrations at c and h. It is manifest that the action of the outer member at i, is to that of the inner one atj, asjh to ih (inversely as their distances from the distorted curve), and that the action upon the outer diminishes, while that of the inner one increases each way from i and j, until the action upon the two becomes equal at the meeting of the curves at k, and at the crossing point nearf. Hence the diagonals leaning toward the point i must act by thrust, while those leaning fromj, act by tension. On the contrary at d, where the greatest compression is upon the inner member, and diminishes each way, the diagonals leaning from e, act by thrust, while those lean 126 BRIDGE BUILDING ing toward c, act by tension. The tension diagonals are represented by single, and the thrust diagonals, by double lines. But the action changes more or less with every change in the position of the load, and if the load were reversed upon the two halves of the arch, each diagonal here represented as acting by thrust, would then act by tension, and vice versa. Now, assuming that de = -ce, and that the action upon the inner member at this point equals twice that of the outer one, it follows, since the action should be. come equal upon the two at a, that 1 of the whole thrust of the rib must be transferred from the inner to the outer member between c and a, by the thrust and pull of diagonals, exerted in the direction of the normal curve; the action accumulating and increasing upon successive diagonals each way from c, and in like manner from h. The action of diagonals is still further affected by the transfer of the action of load, from the outer to the inner member; the load being first applied directly to the outer curved member. Hence it becomes a somewhat complicated problem to determine the maximum action of diagonals; especially as the complication becomes increased by taking into account the EFFECTS OF TEMPERATURE. LXXVIIT. The expansion and contraction of metallic arches without chords, the ends remaining fixed as to position and distance asunder, must obviously cause the intermediate portions to rise and fall with the increase and decrease of temperature. The outer and inner members, if parallel, being similar concentric arcs, will rise and fall, by the same EFFECTS OF TEMPERATUREB 127 changes of temperature, proportionally to their respective radii;* the outer one undergoing the greater vertical change, whence, it must follow that in warm weather the outer, and in cold weather the inner member sustains the greater relative compression, a result for which there appears to be no obvious remedy, except by balancing the end bearings upon pivots at a and k; which would allow the two curved members to adjust themselves to an equal action upon the two. Or, if the curves be formed upon the same radius, and of equal length, they would rise and fall alike, and the distance across the web vertically, would be the same at all parts of the arch. In this case, as in all others, of the arched rib, the depression of the arch, whether from reduction of temperature, or the action of load, would be attended by increased thrust action, or diminished tension action upon diagonals less inclined from the vertical position, and the reverse of action, upon those more inclined. The absolute rise or fall of an arch, resulting from a given change of temperature, may, without essential error, be regarded as proportional to the change in the length of a circular arch of the same span and depth (from chord to vertex), within the limits of change produced by temperature; and, may be found by the following process: Divide the square of the half chord by the depth of arch (v), add the divisor to the quotient, and half the sum equals the radius. Divide the half-chord by the radius, to get the natural sine of half the arc; find in the table of natural sines, the angle corresponding - The curves not being supposed to be circular arcs, it is not strictly correct to speak of their radii, but the meaning will be comprehended. 128 BRIDGE BUILDING. with the sine thus found, and double that angle, for the number of degrees in the arc. Multiply the number of degrees (reducing minutes and seconds to the decimal of a degree), by.01745329 (= length of a degree, radius being equal to 1), for the length of the are. Then, in the same manner, find the length of an arc upon the same chord, and with a depth (v') one or two per cent greater or less than v; and, the difference in length of arcs thus found, is to the difference between v and v', as the change in length of arch due to the given change of temperature, to the rise or fall of the arch, resulting from such change. By applying this rule to a specific case, we can the better appreciate the importance of the effects of change of temperature upon this species of arched ribs. If we assume an arch of 500 feet chord and 50 feet depth, =v, we find the length of arc to be 513.25 ft. The length of an arc of the same span, and a depth (v') = 51ft., is 513.715ft., the difference being 0.465ft. The expansion of steel for a change of 110~ Fahrenheit, is.0007271 x length (513.25), =.37318. We have, therefore,.465: Ift..37318:.8025ft. = rise or fall of the arch in the centre, resulting from a change of 110~. Regarding this rise in the centre as the abscissa of a parabola, and the half chord as the corresponding ordinate, the rise at any other point of the curve is equal to the difference between.8025, and the abscissa answering to the ordinate of the given point. Suppose the point be 10 feet from the end, and the ordinate, of course, 240ft., we have, 2502: 2402::.8025 -.7395 =abscissa for the given point; whence, the rise at that point, equals.8025-.7395 =.063ft. EFFECTS OF TEMPERATURE. 129 Let Figo 24 represent the end portion of the arch, abe the upper, and ge the lower member, ag and be the width of web,=12'. b, with a FEa. 24. horizontal reach of 10', equals if.'\,,e, 10.75'. Then, bg being re-::. " garded as a rectangle, the d10........'^ / diagonal ae= 16.Ift and the'/."- /E\,/.:: temperature being raisedl10, \... \ //.. the points b and c rise to b' and At,\ C~/,c', bb' being equal to the vertical rise multiplied by the cosine of the angle abd, i. e., \ equal to.062 x cosine abd. This angle is a little over 22~, and its cosine about.93 whence bb'=.058ft,=c'. Joining a with c', and drawing cf at right angles with ac, and ac' (as these lines are essentially parallel), we have cf,=cc'x sin. acb=cc' x — =.038ft,= the contraction required to take place in the length of the diagonal ac, to accommodate a change of 110~ in temperature. In the mean time the point e rises to e', the distance ee' being equal to.1147, so that c'e' is extended about the same as ac' is contracted; a change equal to what would be produced by a force of 70,000 lbs to the square inch of cross section. If the normal length of the diagonals be adjusted for a medium temperature, the change would be half the above amount each way, or equal to that produced by 35,000 lbs to the inch. Succeeding diagonals toward the centre would be affected in a similar manner, though in a less degree; and the consequence must be an accumulation of thrust or compression upon the inner member toward the centre, and the outer one toward the ends, upon a rise 17 130 BRIDGE BUILDINGo of temperature, and the reverse on a fall below the normal point. THE WIDTH OF WEB, LXXXIX. For an arch of 500 ft. chord, and 50 foot depth. We have seen that, with a load as assumed [LXXIV], with reference to Fig. 22, the aggregate aberration outward and inward, traverses a zone of.162v, equal in this case, to.162 x 50 = 8.1 ft. If the web, therefore, be 8.1 feet wide between centres of curved members, the equilibrated curve will reach the centre of said members at the points of greatest aberration, both ways, and the whole thrust at these points, will ifll upon a single member, producing as we have already seen, 77 -7 per cent of the amount of thrust due to a maximum uniform load; being over 55 per cent more stress under a partial than under a full load. Again, suppose the web to be 12 feet wide. The distorted curve would approach within two feet of the outer and inner curved members, throwing upon one member at one point, and upon the opposite member at another point, almost 30 per cent more action than what is produced by the full maximum load, It was shown moreover [Lxxv] that nothing short of.545v=.545x50 =27.25 feet width of web, could be relied on to give as small a stress upon the curved members in this particular case of a partial, as that produced by a full maximum load. This would be an inconvenient, and an expensive width of web, and probably a less width would be preferable, even with a greater occasional stress upon the curved members which might be enlarged in section in parts liable to the greater stress. But I shall not undertake at this time, to determine the exact optimum. BRIDGE MATERIALS. 131 Finally, considering the difficulty of securing the most efficient thrust action of the curved members of the arch, the serious disturbances as to the action of the diagonals composing the web system, occasioned by changes of the temperature, together with the extra weight and strength of piers and abutments to withstand the horizontal thrust of the arches, it seems reasonable to conclude that the erect metallic arch bridge will only be adopted under rare and peculiar circumstances; and that in such cases, the plans should be subjected to especial examination and investigation. Truss bridges possess the advantage of having all the forces in operation, except the vertical action of weight, and the opposite resistance of the end supports resisted by means of members contained within the structures themselves, and composed of materials of so nearly uniform expansibility by heat, that no important disturbance in the relations of the different members, can be produced by changes of temperature. Plans, also, may be so arranged as to secure a near approximation to uniform maximum stress upon all the parts; at least, to a much greater degree than seems practicable in the case of the arch without chords. 182 BRIDGE BUILDING. BRIDGE MATERIALS. LXXX. Having discussed the general principles and relative characters and merits of different plans and forms of bridge trusses, and their proper proportions, particular and general, the question as to the best materials for the purposes of bridge construction may properly be considered. We have seen that the materials of a bridge truss are principally subjected to two kinds of action, that of tension, and that of compression. Lateral, or transverse action should be avoided in the principal parts and members of the truss. It is obvious then, that those materials best calculated to resist these kinds of force respectively, should, when practicable without sacrifice of economy, be employed in the situations where those forces are respectively exerted. For instance, when the diagonals act by tension, the upper chord (or the arch, in case of the arch truss), and the verticals, should be composed of the material best adapted to the sustaining of a compressive force, while the lower chord and the diagonals, should be of the.best material for sustaining tension. Wood and iron are the only materials that have been employed in the construction of bridge superstructures to an extent worthy of notice; and it seems reasonable to conclude that on these we must place our dependence. Cast iron resists a greater compressive force than any other substance whose cost will admit of its being used as a building material. Steel has a greater power of resistance, but its cost precludes its employment as BRIDGE MATERIALS. 133 a material for building purposes.* Wrought iron resists compression nearly equally with cast iron. But its cost is twice as great, which gives the cast iron a decided advantage. On the other hand, wrought iron resists a tensile force nearly four times as well as cast iron, and 12 or 15 times as well as wood, bulk for bulk. Not only are these the strongest materials, but they are also the most durable. In fact, with proper precautions, they may be regarded as almost imperishable, It would seem then, that wrought iron for tension, and cast iron for compression, were the best materials that could be employed in building bridges. But wood, though greatly inferior in strength and durability, is much cheaper and lighter, so that, making up with quantity for want of strength, and by frequent renewals, its want of durability, it has hitherto been almost universally used in this country for bridge building; and, in the scarcity of means, and the unsettled state of things in a new country, where improvements are necessarily, to a great extent, of a temporary character, this is undoubtedly the most economical material for the purpose. But it is believed that the state of things has now assumed that degree of settled permanency in many parts of this country, and available means have accumulated to that extent which renders it consistent with true economy to give a character of greater permanence to our improvements; and, in the erection of important works, to have more reference to durability, even at the cost of a greater present outlay. In this view * This remark, made originally some twenty-five years ago, may require some modification at the present time, when steel is being employed extensively for rail way track, and in some important arch and suspension bridges; but not in truss bridges, to the writer's knowledge. 134 BRIDGE BUILDING. of the subject, it seems highly probable that one of the channels in which this tendency of things will develop itself, will be in the extensive employment of iron in the construction of important bridges. With this impression, I proceed to some general comparisons as to the relative cost and economy of wood and iron as materials for bridges. LXXXI. The power of cast iron to resist compression, equals some twenty times that of wood; consequently, it will only require one twentieth as much of the former to withstand a given force, provided it can be put into a form in which its liability to fiexure, and yielding laterally, is not greater than that of wood. This may be accomplished in part, by giving the iron a hollow form, so as to make the diameter of the pieces approximate to an equality with twenty times the same amount of wood, which must generally be used in a simple rectangular, or cylindrical form of section. Assuming, then, that a cubic foot of cast iron will do the same work as 15 cubic feet of wood (after making allowance for the necessarily smaller diameter of the iron), we can institute a comparison which would seem, upon the surface, to show the relative economy of the two materials. A cubic foot of cast iron, manufactured for the work will cost about $13.00. 15 cubic feet of woodin abridge will cost, say $6.00. Whence it appears that the cast iron is more than twice as expensive, in the first outlay, for sustaining a compressive force, as wood. Again a cubic foot of wrought iron in the work, say 450 lb at 7~cts.=$34.00. Wood is about ~ as strong as iron. But about onehalf of its fibres must be separated in order that the BRIDGE MATERIALS. 135 other half may be so connected in the structure, as to be available to their full strength, acting by tension. Hence, it will take some 30 feet to equal one of iron; for which it will cost, say $12; showing a difference of a little less than three to one; making the average for both kinds of iron, reckoning equal quantities of each, about 2.6 to 1. To offset against this, we have the superior durability of the iron, which, as before observed, may be regarded as imperishable; whereas, wood requires frequent renewals, at a cost each time, equal to the first outlay. Now, the first cost of the iron is sufficient to provide for the first cost of the wood, and nearly two renewals. Besides this, money, though an inanimate substance, is, nevertheless, in these usurious times, made to be exceedingly prolific; insomuch, that with good management, it is found to double itself once in ten or twelve years, according to the hardness offace in the lender, or of fortune in the borrower. Assuming 5 per cent per annum as the net income of money invested, the term of time in which the 1 6 dollars saved in the wooden structure, will require to produce one dollar for renewal, will show the time that wood ought to last, to be equal with iron in economy, One dollar and sixty cents at compound interest will yield, at 5 per cent, one dollar in a little less than ten years. Therefore, if an imperishable iron structure cost 2.6 times as much as one of wood, and the latter last but ten years, and money will net 5 per cent, compound interest, the two materials are nearly upon a par as to economy. Experience has shown that wooden bridges, unprotected by roofing and siding, seldom last with safety over eight years, or thereabouts; and, the more there 136 BRIDGE BUILDING. be expended to increase the durability, the less surplus capital will be left to be invested toward renewals. LXXXII. But the above comparison is too superficial and general to be entitled to a great deal of confidence, except, perhaps, as it regards the sustaining of a given weight by a simple post, or suspending it by a bar or rod of iron or wood. In the complicated assemblage of pieces forming the superstructure of a bridge, there are numerous other facts and considerations which materially vary the results. First, there is a difficulty in connecting pieces of timber in such a manner that every part may be proportioned to the strength required of it, to the same extent as can be done with iron. Second, it is frequently necessary to use considerable quantities of iron in bolts and fastenings for putting together a structure of wood requiring great stability. Third, wood soon loses a portion of its strength by partial decay, and consequently, requires additional strength in the beginning, that it may be safe for a time after decay has commenced. Hence, but little can be predicated upon the simple general comparison of wood and iron as to strength and cost, relative to the comparative economy of the two materials for bridge building. It is only by comparing the results of actual experience, or, where this has not been had, by comparing the results of detailed estimates, upon well matured plans, founded on well established principles, that a satisfactory conclusion can be arrived at. With regard to wooden bridges, much experience has been had, and the reasonable presumption is, that a good degree of economy has been attained in their construction. But the idea of building iron bridges in this BRIDGE MATERIALS. 137 country, is of recent date, and but little has been experimentally proved in relation, to their cost and qualities. LXXXIII. This much, however, my own experience has demonstrated. Having received Letters Patent for an " Iron Truss Bridge," upon the arch truss plan, and constructed two bridges thereon, over the Enlarged Erie Canal (of 72 and 80 feet spans), one of which has been in use for six years, it may be regarded as a demonstrated fact, that bridges may be sustained by iron trusses. It has also been shown that the cost of the above class of bridges, is only about 25 per cent more than the same class of bridges of wood, as heretofore built, under the most favorable circumstances, upon the Erie Canal. That the iron portion, constituting some three-fourths of the whole, as regards expense, in the iron bridge, gives fair promise of enduring for ages, while the wooden structure can only be relied on to last eight or ten years. Upon these facts, experimentally established, I found the following comparison: A common road bridge of 72ft. span (the usual length for the enlarged Erie Canal), will cost, with iron trusses: For 7,000 lbs. of cast iron at 3cts.,................ $210. 6,000 " " wrought iron, manufactured for the work, at 7cts.,.................... 420. " Timber, labor and painting,............... 230. "Superintendence and profit,............... 80. W hole first cost,.............................. $940. $175 will renew the perishable part once in 9 years, to produce which, at 5 per cent compound interest will require capital of,' 320. Total for a perpetual maintenance, $1,260. 18 138 BRIDGE BUILDING. With wooden trusses, fastened with iron for timber, labor, paint and profit, $550, 2,000 lbs. of iron fastenings,............... 150. Whole first cost,............................. $700. (Some have cost $1000, or $12,000, and taken 3 to 4 thousand pounds of iron).......... To renew $550 worth of perishable material once in 9 years, will require, at 5 per cent, compound interest,.................... $1,000. Total for perpetual maintenance,...... $1,700. The reason of the apparent difference between this result, and that arrived at from the general comparison of the cost, &c., of wood and iron, is, that the bridges here referred to, have been constructed with a very large amount of iron fastenings, and with large quantities of casing and painting for protection and appearance.'Were the comparison confined strictly to the expense of timber work, in the sustaining parts of the trusses, the result would be found not to differ so essentially from that of the general comparison. The above estimate of $700, for the first cost of a 72 foot wooden bridge, though considerably below the average cost of canal bridges of that description, is nevertheless believed to be greatly above the minimum for which bridges may be built, dispensing with the parts which are not essential to strength. It is probable that bridges may be built for $500, as about the minimum, of equal strength and convenience, and nearly the same durability, as those hitherto built upon the Erie Canal Enlargement at a cost of from 800 to 1,000 dollars. Upon this supposition, which may be regarded as an extreme case in favor of wood, the comparison will stand thus: BRIDGE MATERIALS, 139 First cost of wooden structure,....................o 500 Capital invested at 5 per cent to produce $500 once in 9 years for renewal,...................... 909 Total for perpetual maintenance,............... $1409 The same for iron structure, as above,......... 1260 Balance in favor of the iron bridge,............ 149 Finally, since theoretical calculation and general comparison show a probable advantage, for a long term of time, and experience, as far as it has gone, shows a decided advantage in favor of iron, it would seem very unwise to discard the latter, without at least a fair trial of its merits. If in the first essays at iron bridge building, the iron bridge has competed so successfully with wooden bridges, improved by the experience of ages, may not the most satisfactory results be anticipated from an equal degree of experience in the construction and use of iron bridges? LXXXIV. Presuming the affirmative to be the only rational answer to the above question, I have arranged the details of plans for carrying into practice the preceding principles and suggestions in the construction of rail road bridges of iron. I have also made careful detailed estimates of the expense of bridges of different dimensions and in different circumstances, some of the more general results of which I will here state. In proportioning the parts of a rail road bridge, I have assumed thatit may be exposed to a load of 2,000lbs. per foot run, for the whole, or any part of its length, in addition to its own weight; and in case of tension, have allowed one square inch cross section of wrought iron for every 10,000 lbs. of the maximum strain produced 140 BRIDGE BUILDING. upon every part by such weights, acting by dead pressure. In case of thrust, or crushing force, I have allowed one square inch cross section of cast iron, for every 12,000lbs. acting on pieces (mostly in the form of hollow cylinders), of a length equal to 18 diameters, and a greater amount of material, where the ratio of length to diameter is greater; always having regard to practicability, as well as theoretical proportions, in adjusting the dimensions of the part. My estimates, made upon these bases, have fully satisfied me that a bridge of 100 feet span, with track upon the top (with wooden cross-beams), will cost about $2,000, or $20 per foot, assuming the present prices of iron (1846), in ordinary circumstances. If the track pass near the bottom of the trusses, the expense will be increased by two or three dollars a foot. For a span of 140 feet, by a liberal detailed estimate I make, in round numbers, a cost of $4,000. For 70 feet, I estimate a cost of 9 to 10 hundred dollars, according to circumstances. Thus it will be seen that actual estimate makes the cost of a single stretch of any length, very nearly as the square of the length, as should be expected from the nature of the case. Hence, knowing the cost of a span of any given length, we readily deduce that of a span of any other length, in similar circumstances, with reliable certainty. Now, although my investigations have forced the conviction upon me, that where strong and durable bridges are required, iron should be preferred in their construction, still there is a multitude of cases where wooden structures should be preferred; especially in sections of country comparatively new, where timber is PRACTICAL DETAILSo 141 plenty and capital scarce; and where improvements must necessarily be of a more temporary character. With this view of the subject, I have given considerable attention to the details of wooden bridges; and, with a good deal of investigation and experiment, have arranged plans which are confidently believed to possess important advantages over the plans generally in use. The preceding few pages have been transcribed from the author's original and first essay upon bridge building; and are introduced here, not on account of any practical value they may possess in the present state of progress in the science of bridge construction. But they may possess some little interest as marking about the starting point of the construction and use of Iron Truss Bridges. If the estimates above exhibited, of the cost of iron bridges, appear small and inadequate, under the lights furnished by the experience of a quarter of a century, much allowance may be claimed on account of the change of times and circumstances within the period in question. And, when it is borne in mind that the author actually contracted for, and built iron railroad bridges of 40 and 50 feet span, for $10, and of 146 feet for $30 per foot, the estimates above given may not seem entirely preposterous, although much higher prices are obtained for bridges of like dimensions at the present day. PRACTICAL DETAILS. LXXXV. In preceding pages I have endeavored to give a short and comprehensive general view of the 142 BRIDGE BUILDING. subject, and to ascertain and point out the best general plans and proportions, for the main longitudinal trusses, or side frames of bridges, and the relative stresses of their several parts. The side trusses may be regarded as vastly the most important parts of the structure, and the strength and sufficiency of these being secured, there is much less difficulty in arranging the remaining parts, the forces to which they are exposed being much less than those acting upon the trusses. I propose now to enter more into details, and give such practical explanations and specifications as to the strength of materials, the methods of connecting the several parts or pieces, both in the main trusses, and other parts of the structure, illustrated by the necessary plans and diagrams, as, it is hoped, will enable the young engineer and practical builder to proceed with judgment and confidence in this important branch of the profession. IRON BRIDGES. STRENGTH OF IRON. LXXXVI. Iron has the power of resisting mechanical forces in several different ways. It may resist forces that tend to stretch it asunder, or forces which tend to compress and crush it; the former producing what is sometimes called a positive, and the latter, a negative strain. It may also be exposed to, and resist forces tending to produce rupture by extending one side of the piece, and compressing the opposite side; as where a bar of iron supported at the ends, is made to sustain a weight in the middle, which tends to stretch the IRON BRIDGOES. 143 lower, and compress the upper part. This is called a lateral, or transverse strain. Iron may likewise be acted upon by forces tending to force it asunder laterally, in the manner of the action of a pair of shears. This is called a shear strain; and though less important than either of the preceding cases, it will frequently have place in bridge work, partially at least, in the action of rivets, and connecting pins. With regard to the simple positive and negative strength of iron it is only necessary for me to state in this place, as the result of a multitude of experiments, that a bar of good wrought iron one inch square, will sustain a positive strain of about 60,000Tbs. on the average; and a negative strain, in pieces not exceeding about twice the least diameter, of 70 or 80 thousand pounds. But in both cases, the metal yields permanently with much less stress than the amounts here indicated; and hence, as well as for other considerations, it can never be safely exposed in practice, to more than a small proportion of these stresses, say from X to j. Cast iron resists apositive strain of 15,000 to 30,0001s. to the square inch, but usually, not over 18,000. But it is seldom relied on to sustain this kind of action especially in bridge work, wrought iron being much better adapted to the purpose. On rare occasions, it may perhaps safely be exposed to a strain of 3,000 to4,000lbs. to the square inch, but should not be used under tension strain, when wrought iron can be conveniently substituted. Cast iron, however, is capable of resisting a much greater negative strain than wrought iron; its power of resistance in this respect, being from 80,000 to 144 BRIDGE BUILDING. 140,000Os.; seldom less than 100.000 to the square inch, in pieces not exceeding in length, twice the least diameter. But in pieces of such dimensions as must frequently be employed in bridge work, fracture would take place by lateral deflection, under a much smaller force than what would crush the material. It is therefore necessary to take into account the length and diameter, as well as the cross-section, in order to determine the amount of compression which a piece of cast iron, or any other material may be relied on to sustain. LXXXVII. The cause of lateral deflection resulting from forces applied at the ends, and tending to crush a long piece in the direction of its length, is supposed to be a want of uniformity in the material, and a want of such an adjust of the forces that the line joining the centres of pressure at the two ends, may pass through the centre of resistance in all parts of the piece. These elements are liable to considerable variation, and can not be very closely estimated in any case. Therefore the absolute power of resistance for a piece of considerable length, can not be deduced by calculation from the simple positive and negative strength of the material, but resort must be had to direct experiment upon the subject; and, even wide discrepancies should naturally be expected in the results of experiment, unless the lengths of pieces experimented upon, be very considerable. In respect to pieces, however, having their lengths equal to twenty or more times their diameters, a somewhat remarkable degree of uniformity is found in their powers of negative resistance, and the following formula, deduced theoretically, though not fully sustained IRON BRIDGES. 145 by experiment, may be useful in determining approximately the relative powers for pieces of similar crosssections, but different dimensions. The power of resistance (R), is as the cube of the diameter (d), directly and as the square of the length (1), inversely, that is, a, is as 2. The reason of this formula may be illustrated with reference to Fig. 25, in which adb represents a post loaded at a, so as to bend it into a curve, of the half of which cd is the versed sine. It is obvious that in this condition, the convex.side of the post is exposed to tension (or at least, to less compression FIG. 25. than the other side), and the concave side to compression; also, that the effect of the load at a, toward breaking the post at d, is as the versed sine ed, which is as the square of ab. But the power of the post to resist rupture transversely, is manifestly d as the cross-section of the post (i. e., as the square of the diameter), multiplied by the diameter. Hence, the power is as the cube of the diameter. Now, the ability of the post to sustain the load at a, is directly as the power to resist rupture, just determined, and inversely as the mechanical advantage with which the load acts, above seen to be as the square of the length of the post. Hence, the formula. We shall see as we progress, the relation which this formula seems to bear to the results of experiment. The following list of experiments made by the author some 25 years ago, though few in number, and upon a somewhat diminutive scale, nevertheless, may afford some light as to the law governing the resisting power of cast iron in pieces of different lengths, as compared 19 146 BRIDGE BUILDING, with their diameters. It may at least enable us the better to appreciate the better lights since shed upon the subject. LXXXVIII. EXPERIMENTS UPON THE NEGATIVE STRENGTH OF OAST IRON, IN LONG PIECES. Ends, flat cones or pyramids. Inches. i Form "a I of on. Remarks. 6 section. 1 1 Cylinder 9. 0.16 990 1062 Broke — in. from centre. 2 " " " 978 990 Broke 4 in. from centre. 3 Square " 0.15 803 854 Deflected cornerwise, and flew out without breaking. 4 cc 914 938 Broke in half a minute not cornerwise, d inch from centre. 5 Cylinder -5 7.1 0.126 1417 1437 Broke in 3 seconds, 1 in. from centre. 6 " " 1377 1397 Broke 15 in. from centre. " " "; " "{ ~;Piece flattened by flask not shutting true, and had been straightened with the hammer where it broke. 7 " " 4.5 2580 2580 Broke in 1 minute into 4 pieces of nearly equal lengths. Piece of same as last experiment. 8 " " 4.5 3218 3218 Broke in 4 minute into 3 pieces in centre, and 1 in. from centre. 9 Square' 4.5 2813 2838 Broke in minute, g- in. from centre, deflected parallel with sides. From experiments 7 and 8, in the above table, it appears that cast iron will sustain at the extrene, in cylindrical pieces whose lengths equal about 14~ diameters, a negative strain of 41,000 to 51,0001s to the square inch, say an average of 46,000. Square bars, according to experiment 9, length equal to 18 diameters (or widths of side), will sustain about 45,0001s to the square inch. IhoN BRIDGES. 147 Now, a hollow cylinder of a thickness not exceeding about A of the diameter, according to calculation, has a stiffness transversely, about 50 per cent greater to the square inch than a solid square bar whose side equals in width the diameter of the cylinder. Hence, a hollow cylinder of a length equal to 18 times its diameter, should sustain a negative strain of 67,500 fbs. to the square inch. But it should be observed, however, that direct experiments upon the transverse strength of the pieces used in the experiments leading to the results and conclusions above stated, as to negative strength, showed themto possess uncommon strength transversely, even to from 30 to 50 per cent greater than the fair average transverse strength of cast iron; as will be seen hereafter. It is therefore not considered proper to estimate the strength of hollow cylinders of the proportions above stated at more than 45,000 or 46,0001ks. to the square inch. The hollow cylinder is undoubtedly the form best adapted to the sustaining of a negative strain, having equal stiffness in all directions. It is therefore highly desirable that the power of that form of pieces to resist compression, with different lengths, should be ascertained by a careful and extensive series of experiments. But until that shall have been done, and the results made known, I shall assume the above estimate upon the subject, as probably not very far from the truth; subject, however, to correction, whenever the facts and evidences shall be obtained, upon which the correction can be founded.* In the mean time, since we know not the exact ratio between the greatest safe practical stress, and the ab* Since the original writing of this paragraph (25 years ago), extentensive experiments and investigations have been made, in the direction 148 BRIDGE BUILDING. solute strength of iron, and therefore should in practice keep considerably within the limits of probable safety, it becomes a matter of less importance to know the exact absolute strength; though this, of course, is desirable. LXXXIX. Having decided upon a measure of strength for pieces of a given length, we may properly endeavor to ascertain the rate of variation for different lengths as compared with the diameters. It is seen in the table, [LXXXVIII] that two cylindrL cal pieces of 9 inches in length, bore the one 990, and the other 978tbs., giving a mean of 984 pounds. Now, by the formula d3 the same cylinders reduced to 4.5 inches, should sustain four times as much, or 89361bs. But, by experiments 7 and 8, we find that they bore only 2,580, and 3,218, a mean of 2,899 pounds. Whence it appears that, the diameter being the same, the strength diminishes faster than the length increases, but not so fast as the square of the length increases.; being about half way between the two.. In fact, if we examine the results of these experiments throughout, we find that the weights borne by pieces of like cross-sections, whether round or square, were very nearly the arithmetical mean between the results obtained by considering them to be inversely as the simple length, and as the square of the length, successively. For illustration; take experiments 1 and 5. If the piece 9 inches long bore 990 res., taking the strength here indicated, and ingenious and convenient form.ulm deduced upon the subject involved, which might perhaps, be profitably substituted for the writer's own crude deductions in this behalf. But, as previously remarked on other occasions, the latter may possess interest as affording a monument upon the line of the march of progress. IRON BRIDGES. 149 to be inversely as the length, we have this proportion 1 1.: -::990: 1,255. Then, taking the strength to be inversely as the square of the length, we have: 1: 50.41: 990: 1,591. Taking the mean of these results, we find (1,255 + 1,591), - 2 = 1423. This is the weight which, according to the rule, the piece in experiment 5 should have borne, and it varies only G6bs. (less than I of one per cent), from what it actually did bear. Again, take experiments 1 and 8; in which the lengths were as 2 to 1. Supposing the weights to be inversely as the lengths, and as the squares of the lengths successively, and taking the mean of the results,we have (1,980+3,960! -2=2,970, which is 248ks. less than the weight borne in experiment 8. But it is also 3901bs. greater than that borne in experiment 7, by a piece of similar form and dimensions, but an inferior specimen. It does not seem, therefore, that the rule is widely at fault. The same rule applied to experiments 4 and 9, lengths being also as 2 to 1, gives 2,784 lbs. as the bearing weight, and 2,814 as breaking weight for No. 9; the former varying 71lbs. and the latter 241bs. from the weights shown in the table. Now, if we observe that the one broke in a quarter of a minute, and the other endured half a minute, it is no extravagance to assume that if No. 9 had been loaded with 24ibs. less, it would have stood' of a minute longer, giving a result in precise accordance with the rule. From what precedes, it is believed that the following may be adopted as a safe practical rule for determining the power of resistance to compression, for pieces of similar cross-sections, after knowing from experiment, the power of a piece of given dimensions, and similar cross section. 150 BRIDGE BUILDING. DT D3 Rule: Make the power of resistance as -- and as -- sucL cessively, and take the mean of the results thus obtained, as the true result; D representing the diameter (or width of side, in square pieces), and L, the length of the piece. This rule will be probably apply without material error, to pieces of lengths from 15 to 40 times as great as their diameters, and perhaps for greater lengths; although, in bridge building, greater lengths will seldom be employed.* But, as the length is reduced to 8 or 10 diameters, or less, it is manifest that the power of resistance increases at a less rate than that given in the rule. For, we see by the table of experiments, that a square piece of a length equal to 18 diameters (experiment 9), bore at the rate of 45,000Tbs. to the square inch, which is nearly one-half of the average crushing weight of cast iron, and one-third that of the strongest iron. But according to the rule, a piece of half that length, or equal to 9 diameters, should sustain 135,0001s. which is about the maximum for cast iron; whereas, experiment shows that the power of resistance increases with reduction of length, down to about 2 diameters. It may, therefore, be recommended to apply the rule above given, to hollow cylindrical, and square pieces above 15, and to solid cylinders, above 12 diameters. From those lengths down to 2 diameters, it cannot lead to material error to estimate an increase of power proportionate to diminution of length, according to the differences between the weights, or resisting powers determined as above, for square pieces and hollow cylinders of 15, and solid cylinders of 12 diameters in length, and the absolute crushing * It is probable that for greater lengths than 40 diameters, the formola -^ alone, would be more nearly sustained than in case of smaller lengths. IRON BRIDGES ] 51 weight of the iron; that is, if a square piece whose length equals 15 diameters bear m pounds, and the crushing weight for pieces of 2 diameters be n pounds to obtain the resistance (R), of a piece of (15-a), diameters in length, take m + ~ (n m)=R. XC. It has already been remarked that in practice, materials should be exposed to much less strain than their absolute strength is capable of sustaining for a short time. This fact is universally recognized, and the reasons for it, are perhaps, sufficiently obvious; still it may be proper to mention a few of them in this place, First, there is a great want of uniformity in the quality and strength of materials of the same kind, and no degree of precaution can always guard against the employment of those containing defective portions possessing less than the average strength. Again, when materials are exposed to a strain, although it be but a small part of what they can ultimately bear, a change is produced in the arrangement of their particles, from which they are frequently unable fully to recover; and whence they generally become weakened, especially if they be repeatedly exposed to such process. Hence, it often happens that a piece is broken with a smaller strain, than it has previously borne without apparent injury. Now, there is no means of estimating exactly the allowance necessary to be made on account either of these facts, as well as, probably, many others. Consequently, we can not determine with certainty, how much of a given material may be relied on to sustain with safety a given force. We should therefore, incline toward the side of safety, the more strongly, in pro 152 BRIDGE BUILDING. portion as the consequences of a failure would be the more disastrous. The breaking of a bridge is liable, in most cases, to be a serious affair, involving hazard to life and limb, as well as destruction of property. Hence, they should be constructed of such strength asto render failure quite out of the range of probability, if not absolutely impossible. XCI. Good wrought iron bars, will not undergo permanent change of form under a tensile strain of less than from 20,000 to 30,000 pounds to the square inch and though they will not actually be torn asunder with a stress below 50 or 60 thousand, and often more, to the inch, any elongation would certainly be deleterious to the work containing them, even if not dangerous from liability to fracture. Hence, it is certainly not advisable to expose the material to a stress beyond the lowest limit of complete elasticity. In the original predecessor of this work, the traditional allowance of 15,000lbs. to the square inch, was adopted as the tensile stress to which wrought iron might safely be exposed, and beyond which it was deemed improper to rely upon it. No evidences or arguments since that time, have induced a change of opinion in this respect. But in the case of a bridge, there is variety and uncertainty as to the exact amount of load, as well as in relation to the limit of safe strain for the material; and while it seemed probable that the load of a single track rail road bridge would never exceed 2,000lbs. to the' lineal foot upon any part of its length, still, seeing that rail roads were comparatively a new institution, and iron bridges for rail roads almost unheard of, especially in this country, it was deemed wise, in recommending their introduction, to so adjust IRON i B IDG-ES 153 their proportions as to meet almost any possible contingencies. This could be accomplished either by assuming a greater possible load for the bridge, or a lower limit to the stress of materials with the smaller load, with the same ultimate result. Ancd, perhaps the former would have been the more consistent course, as avoiding the seeming absurdity of the assumption that iron could safely stand a strain of 15,000s. in a common bridge, but only 10,000b in a rail road bridge; and the no less seeming absurdity of assuming that the same material could stand 50 per cent more strain in na bridge composed partly of wood, than in one entirely constructed of iron. Now, instances in great numbers could be pointed out, of rail road bridges of wood and iron, where 2,000lbs. to the lineal foot would produce a, stress considerably exceeding 15,000 to the inch upon certain bolts of wrought iron.* * The author had occasion several years ago to refer to the following instances in corroboration of the statement above made, in this wise " The best evidence that exists as to the capacity of a material to bear a strain with safety, is derived from experience as to the strain it has been exposed to in works, and conditions similar to those in which it is proposed to employ it, and where it has by long usage, proved itself adequate to the labor required of it. If wrought iron, or1 example, has been used in railroad bridges for a great number of years, in numerous and repeated instances, where a given load, in addition to the weight of structure, would produce upon it a tension of 15,0001bs. to the square inch, and has withstood such usage without cases of failure not caused by manifest defects in the quality of material, or by casualties which such structures are not expected to be proof against; it may be fairly assumed to be reasonably safe and reliable in other railroad bridges where a similar gross load can not produce a greater stress; and much more so, where a like load can only produce a stress one-half, or two-thirds as great. Now, it is provided in the plan herewith presented, that a load of 2,000lbs. to the lineal foot upon each pair of rails, on the whole, or any part of the length of the bridge, can not produce upon any part of the wrought iron work in the trusses, a tension exceeding 10,OOO1bs. to the square inch; and, to show that such provision is eminently safe and liberal, I proceed to give some examples of what the same material is liable to with the same load in other structures, where long and severe usage has fully proved its sufficiency. 16 154 BRIDGE BUILDING. And yet, it was deemed expedient by the author of this work, in the outset of the introduction of iron rail road bridges, to provide that 2,000bs. to the foot upon each pair of tracks, should not give a stress exceeding 10,000b to the square inch upon any part of the wrought iron work, not from a conviction that the material was unsafe under a stress of15,0001bs. but to provide against the possible contingency of its being sometimes exposed to greater stress than that produced by a dead weight of 2,0001b. to the lineal foot. XCII. The use of cast iron to sustain a tensile strain, should undoubtedly be avoided, as a general To begin with an instance near at hand; the bridge from the island to the main shore on the Hudson River rail road at East Albany, has, in one of its stretches, trusses 48 feet long, in 8 panels. It is a double track bridge with three trusses, of which the middle one sustains onehalf of the two pairs of tracks, and of the loaas passing over them. The truss is composed of top and bottom chords, and thrust braces of timber, and vertical suspension bolts of wrought iron, in pairs; and it is at once obvious that 7 of the weight of the tracks and their loads (or, of the half bearing upon the centre truss), is concentrated on the two pairs of suspension rods located 6 feet from each end. [See diagram.] The weight of middle truss, and other parts of the structure sustained by it, probably exceeds 16,000 lbs., of which 8, or 14,000 lbs. bear FIG. 25A. upon the endmost suspension bolts. Add 2,000 lbs. per foot for 1 of one pair of tracks, or rails, and it makes 56,0001bs. upon the suspension bolts in question, with only one track loaded. These bolts are 4 in number, and 1t" in diameter; and, allowing — // to be cut away by screw thread, the aggregate net, available cross-section of the four, is equal to 4.43 square inches; whence the tension, with only one track loaded, is 12,641 lbs. to the square inch, and 22,120 lbs. to the inch with both tracks loaded. 2. The bridge leading into the freight house of the Boston rail road, at East Albany, is a " Howe bridge," and acts upon the same principle as the one just spoken of. It is a double track bridge with two trusses, having 8 panels of 10' 8", and is a heavy covered bridge. Allowing 64 tons for weight of superstructure, or 56,000 lbs. for the portion sustained by the endmost bolts of each truss, and 2,000 lbs per foot upon one track, of which -- at least, bears on one truss, giving IRON BRIDGES. 155 rule; and, if on certain occasions it should be liable to that kind of action to a small extent, the stress should probably not be allowed to exceed 3,000 to 4,00 pounds to the square inch. When exposed to compression, in pieces of such length as to break by lateral deflection, it is believed it may be safely loaded to one-third of its absolute capacity. If a long piece exposed to a negative strain have a defective part, it does not diminish its power of resistance to the same extent as when it acts by tension. The power of negative resistance being, in a measure, inversely as the deflection produced by a 100,000 lbs. on the end bolts, we have 156,0001bs. sustained by 6 bolts of 1-" diameter, containing 8.1 square inches, besides screw thread. This is a strain of 19,259 lbs. to the square inch with one track, and 25,432 lbs. with both tracks loaded with 2,000 lbs. to the lineal foot. 3. The East bridge over the creek in the south part of Troy, is a double track covered bridge with three trusses, having 8 panels of 12'8' each, or 88.66 ft sustained by the endmost suspension bolts. Say, of weight of structure bearing on end bolts of middle truss, 35,000 lbs. and of load upon one track 88,666, making 123,666 lbs. on 4 bolts of 1-" diameter and two of 1// diameter, having a net cross-section of about 7.65 square inches. Hence the stress must be 16,156 lbs. to the inch, with one track loaded, and 27,750 lbs., with 2,000 lbs. to the foot upon each track. 4. The West bridge over the same stream, a few rods below the last mentioned, has three trusses containing 9 panels of 10~ ft. each in length. It is a high truss bridge with roof and siding. For weight of superstructure on endmost bolts of middle truss, say 28,000 lbs. and for load on one track, 84,000, making 112,000 lbs. on 4 bolts of 1i-' containing a net section of 5.41 square inches, giving a tension of 20,702 lbs. to the inch for one track, and 36,229 lbs. for both tracks loaded with 2,000 lbs. to the lineal foot. 5. The bridge across the Erie canal near Canastota, on the N Y. C. R. R., is a double tack bridge with 2 trusses, which have 9 panels of 10 feet. If the superstructure be estimated to weigh 40 tons, it gives a little over 35,000 Ibs. on the end bolts of each truss. Add - of 80 tons for 2,000 lbs. per lineal foot upon one track, and it gives 141,666 lbs. on 4 bolts of 1~/ diameter, and 5.41 square inches of net cross-section equal to 26,173 lbs. to the inch, with one track, and 36,044 lbs. with both tracks loaded." All these cases are stated from personal examination by the author, except the last, which was reported to him from authority considered reliable. The cases were not selected, but taken as the most accessible, and conveniemt for the author's observation. And still, he can not help regarding them as qremarkable, and somewhat exceptional cases 156 BRIDGE BUILDING. given weight, and the deflection depending on the stiffness of the piece throughout its whole length, the power is manifestly only diminished as the amount of defect, multiplied by the ratio of length of the defective part, to the whole length; that is, if the piece be defective so as to lose one-fourth of its stiffness, for that part of its length to which is due one-tenth part of the deflection, the deflection will only be increased by x- = -, and the power of resistance is diminished in the same ratio; whereas the power of positive resistance would be diminished by 1. The effect of negative strain, moreover, is believed not to be so deleterious to the strength of iron, as that of positive, or tension strain; though I can refer to no particular facts or evidences in coroboration of the opinion. Upon the whole, I am inclined to estimate the power of cast iron to resist compression (as against th.e tension of wrought iron at 15,000lbs. to the inch), in pieces of lengths equal to 18 diameters, for hollow cylinders, at 15,0001is. for solid cylinders, at 8,000, and solid square pieces, at 10,000sis. to the square inch of cross-section There are other forms of section for cast iron members of bridges, which it will frequently be convenient and economical to employ where lateral stiffness, as well as longitudinal resistance is required, amongwhich may be named, the cruciform +, the T, and the H form. The former of these, with equal leaves, probably possesses about the same resistance to the square inch, as a solid square which will just contain the figure. For, though it is not so stiff to resist a simple lateral force dicgonally of the including square, as parallel with its sides, and would be broken by tearing asunder the flange, or leaf upon the convex side, still when under NEGATIVE STRENGTH OF IRON. 157 longitudinal compression, the tension upon that leaf would be somewhat relieved. The T and H section will usually be employed where greater stiffness is required in particular directions, and if proportioned withjudgment, will usually possess about the same power to the inch, as the including solid square, or parallelopiped. XCIII. Having determined (approximately, at least), the safe strain for pieces of a certain length, and the ratio of variation in power, depending upon change of length, we readily deduce the safe strain for pieces of similar action, with any given dimensions. The following table, exhibiting the negative power of resistance to the square inch of cross-section, for hollow and solid cast iron cylinders, and solid square pieces (under which class may be included the + T and H formed sections, under proper conditions), calculated for length of from 2 to 60 diameters, is intended to show the safe practical rate of strain for the material, being about one-third of its absolute strength, in columns headed ~, and one-fourth of the absolute, in those headed I; the former to be used against wrought iron at 15,000, and the latter, where wrought iron is estimated to sustain 10,000lbs. to the square inch. This is the author's original table, slightly modified with the addition of two columns showing corresponding weights at ~ and I of the absolute strength, as calculated by "Gordon's formula," deduced from Hodgkinson's, experiments upon cast iron hollow pillars; which is regarded as the best authority upon the subject at the present day. Also, two corresponding columns for wrought iron hollow pillars, according to the same authority. 158 BRIDGE BUILDING. The Gordon formulhe are: for cast iron, S = 80,000O. + (1 +.0025-i2), for wrought iron, S = 36,000bs. ~ (I +.00033-), S representing absolute strength per square inch of section, I, the length, and d, the diameter of column, both referring to the same unit of length. Or making d = 1, we have =1 2 The table of negative resistances, presents a scale of numbers so adjusted as to touch at certain points established by experiment, and running in consistent gradations from one to another of such points. The columns for cast iron hollow cylinders, are the only ones referring to the same class of pieces, and exhibiting the difference in results, arising from difference in the mode of calculation. The Gordon formula is supposed to give results agreeing with those of experiment, for lengths included within the range em braced by the experiments from which the formula was deduced. Within that range, those results may be presumed to be more reliable (being founded on trials of the same kind of pieces as those to which they refer), than those in the author's original table, based upon trials of solid cylinders and parallelopipeds. Taking the 4th and 6th columns, it will be seen that the numbers agree at some point between the lengths of 18 and 20 diameters; the numbers above that point, being the larger in column 6, while, below that point, they are larger in column 4, down to about 50 diameters, where they come together and cross again, and those in 6, are thenceforward the larger. But the differences are small, for the range of lengths principally employed in bridge work. Power of negative resistance to the square inch, in pounds. Hollow Pillars, by Gordon's Formulae. Of cast iron, as by the author's original table. Length in Sqvaare, and Le-noth in diameters, VBWrought iron. Gast iron. IHoJow Cylinders. Solid cylinders. Squarec,, and Length in H sections. diameters. 1 1 1 1 1 1 3A 3 4 3 17 3 4 3 2 11,984 8,988 26,400 10,800 33,333 25,000 83,333 25,000 33,333 25,000 2 4 11,936 8,952 25,641 19,231 31,132 23,349 29,666 22,250 30,277 22,710 4 H 6 11,859 8,894 24,464 18,348 28,932 21.699 26,000 19,500 27,223 20,419 6 8 11,751 8,813 22,988 17,241 26,731 20.048 22,333 16,750 24,169 18,128 8 10 11,616 8,712 21,333 16,000 24,531 18,398 18,666 14,000 21,115 15,837 10 12 11,455 8,591 19,608 14,706 22,330 16,748 15,000 11,250 18,061 13,547 12 H 14 11,271 8,453 17,897 13,423 20,130 15,098 11,755 8,817 15,007 11,256 14 ~ 16 11,065 8,298 16,260 12,195 17,930 13,448 9,562 7,172[ 11,953 8,965 16 d 18 10,841 8,131 14,733 11,050 15,000 11,250 8,000 6,000 10,000 7,500 18 20 10,600 [7,950 13,333 10,000 12,825 9,619 6,840 5,130 8,550 6,413 20 o 22 10,347 7,760 12,066 9,049 11,156 8,367 5,950 4,463 7,438 5,579 22 24 10,083 7,564 10,927 8,195 9,844 7,383 5,250 3,938 6,562 4,922 24 26 9,811 7,358 9,913 7,435 8,787 6,590 4,688 3,515 5,858 4,398 26 28 9,533 7,150 9,009 6,757 7,921 5,941 4,224 3,168 5,280 3,939 28 30 9,254 6,942 8,205 6,154 7,200 5,400 3,840 2,880 4,800 3,600 30 o 32 8,969 6,727 7,490 5,618 6,592 4,944 3,515 2,636/I 4,395 3,297 32? 34 8,686 6,515 6,855 5,141 6,072 4,554 3,238 2,429 4,048 3,036 34 36 8,405 6,304 6,289 4,717 5,625 4,219 3,000 2,250 3,750 2,813 36 38 8,127 6,095 5,784 4,338 5,235 3,924 2,792 2,094 3,490 2,618 38 40 7,853 5,890 5,333 4,000 4,893 3,669 2,610 1,958 3,262 2,447 40 45 7,193 5,395 4,397 3,298 4,200 3,150 2,240 1,680 2,800 2,100 45 50 6,575 4,931 3,678 2,759 3,672 2,754 1,952 1,464 2,448 1,836 50 55 6,005 4,504 3,113 2,335 3,257 2,443 1,737 1,303 2,171 1,628 55 60 5,484 4,113 2,666 2,000 2,925 2,194 1,560 1,170 1,950 1,463 60 ~~~~~~~~~~~. _-~ 160 BRIDGE BUILDING. One obvious reason of the more rapid increase of numbers in the 6th column, for lengths under 15 or 16 diameters, is, that in the latter, the crushing weight for the iron is assumed at 100,00ls. to the square inch, whereas, by the Gordon formula it is limited at 80,000fs, and that formula can give no result greater than that limit, even when 1=0. Now, if 80OOOb3s. was less than the actual crushing load for the kind of iron used in Hodgkinson's experiments (from which the Gordon formula is understood to have been de.rived), it must follow that Gordon's formula gives results smaller than the true ones, for short pieces. This is probably the case, and, although Mr. Gordon's formula is very simple and ingenious, sliding smoothly and plausibly from one extreme in length to the other, it unquestionably gives closer approximations to correct results for the ordinary range of lengths, than when applied to the very short pieces. The numbers in the table are deduced upon the supposition that the thrust members in a bridge, will not act with less advantage than when bearing upon a pivot at each end of the axis of the pieces respectively; and it is not deemed proper to assume that, in consequence of having flat end bearings, the piece in any case can sustain a greater stress than is indicated by the numbers in the table. It will be observed that, in order to obtain the absolute strength of a piece, we should multiply its corresponding number in the table, by the denominator of the fraction (i or i) at the head of the column. LATERAL, OR TRANSVERSE STRENGTH. 161 LATERAL, OR TRANSVERSE STRENGTH. XCIV. The transverse strength of bars or beams, would seem to be deducible from' the positive strength of the material, in the following manner: Let ab, Fig. 26, represent a portion of a rectangular beam or bar, projecting from a wall in which it is 1FIG. 26 firmly fixed. If a weight be il FBI 26'.1 ~ ~ applied at w, the upper part 1/' iK (11tl of the beam will be extended, i,illll ___ \T_ and the lower, compressed; and, where these portions meet, is what is called the!I jlll III neutral plane. Experiment shows that this plane, in rectangular beams, is central between the upper and lower surfaces; or at least, very nearly so, for all elastic substances, until they approach rupture. The tendency of the weight at w, thel, is to produce rotation about the point c (or, the line of intersection of neutral plane and face of wall) and the cohesion of the upper portion cd, and the repulsion of the lower part, cb, tend to resist rotation. Now, to determine the amount of this resistance, which is the measure of transverse strength, we will first consider the upper portion; and it is obvious that, at every part of the cross-section, the resistance to rotation is as the resistance to extension, multipled by the distance of the part above the neutral plane. But the resistance to extension, by the law of elasticity, is as the degree, or armount of extension, which is determined by the distance from the neutral plane; parts at 2 inches from this plane, or the centre of motion, being extended 21 162 BRIDGE BUILDINGO twice as much as those at one inch, and resisting twice as much. Then, denoting the distance from this plane by the variable quantity x, the resistance to extension by any part, equals x multiplied by a certain constant (s), and may be denoted by sx, while the resistance to rotation about c, equals sx2. Again1 representing the horizontal breadth or thickness of the beam by 1, we have t.lx to represent the differential of the section (in its state of increase from c toward d), and s.t.x2dx, the differential of resistance. Then, integrating, aud making x = cd = h, we have the whole resistance to rotation, of the part above the neutral plane, equal to - s.t.h3 = toI.hxhxs.h. But s.h becomes equal to the positive strength of the material when x=cd = h, and t.h = the area of section above the neutral plane. Therefore the power of this part to resist rotation, is equal to ~ of the area, multiplied by half the depth of the beam, and by the positive strength of the material; in case the negative strength exceed the positive. Now, it is obvious that the part below the neutral plane exerts exactly the same amount of resistance to rotation, as thepart above. Therefore the whole power of resistance to rotation about e, in other words, the resistance to rupture, is equal to ~ of the whole crosssection, multiplied by I the depth of beam, and by the positive, or cohesive strength of the material; that is equal to I C.t.Dx> enough to throw the whole weight upon the extension platesee, and t" to 1" in width, except on the side next the beam proper, where it is to be clipped or drawn down to 4". This, however, is not an essential point. In case of bridges already erected, the ring will have to be left open as at R', and when used, heated and closed around the upright. 188 BRIDGE BUILDING. CXI. The Link Chord, composed of a set of links to each panel, connected by pins or connecting blocks (the latter affording also points of attachment for verticals, diagonals, &c.), both for Arch and Trapezoidal trusses, was originally adopted by the author, as the readiest means of putting the requisite amount of chord material in a manageable form, both as it regards manufacturing the parts, and erecting the structure. This form renders the whole section available for sustaining tension, avoiding any loss in rivet or bolt holes for forming connections. The experience of more than a quarter of a century, during which time many hundreds of bridges with link chords have been constructed, and used in almost all conceivable conditions, (in many cases, undoubtedly, the links having been but imperfectly manufactured and fitted to the connecting blocks), with a degree of success and satisfaction seldom exceeded, may reasonably be regarded as fairly establishing the efficiency and safety of this mode of construction, when proper care is used in the performance of the work. Continued and successful usage in a multitude of instances, is regarded as a better criterion as to the reliability of a plan of construction, than a small number of isolated tests, however severe; and such usage the link chord has been subjected to. CXII. The theoretical questions to be considered in this case, would seem to be, as to the possible deterioration of the cohesive strength of the iron, produced in forming the bends at the ends of links - the indirect, or lateral strain in those parts, resulting from imperfection of the fitting to the connecting block or pin, and, the imperfection of the weldings, both as it re ARCH TRUSS BRIDGES. 189 gards complete cohesion, and the tendency to crystallization under the welding heat, not being fully destroyed by subsequent hammering and working. The whole process of the manufacture and refinement of iron, is based upon the principle that disconnected pieces of iron brought in contact under intense heat, but without complete fusion, and subjected to violent compression, as by hammering or rolling, will unite, and become a single piece or mass. Every bar of refined iron found in the iron market, is composed of half a dozen or more parts, which were once separate and disconnected. Those having been " fagoted," or placed in juxtaposition, and submitted to a welding heat, and passed repeatedly between ponderous rollers, or subjected to the blows of heavy hammers, are united and drawn into bars of required sizes and forms for use. These masses, taken from the furnace and suffered to cool without hammering or rolling, would be found more or less crystaline and brittle. But the latter operations prevent such a result, and the iron becomes more or less soft and flexible, even in a cold state. Iron which has undergone the uniform process of rolling, is generally of uniform quality and strength throughout the whole piece; and, as far as it can be used in that state, without re-heating and re-working, it may be regarded as somewhat more reliable than when it has been forged and welded into different and more complex forms. The high temperature required in welding, demands experience and judgment in determining the proper time to " strike," that is, when the metal is hot enough to adhere firmly, but not overheated to burning. Moreover, though the hammering required to bring 190 BRIDGE BUILDING. the parts together and reduce them to proper form and size, may prevent crystalization immediately at the welded point, still on either side are portions which may have been heated so as to change the arrangement of particles, and not subjected to sufficient hamimering to counteract the deteriorating tendency. Hence, a break is more liable to take place a little on one side, than immediately through the welded part. To obviate this liability, the parts to be welded should be enlarged by upsetting several inches from the end, so as to admit of re-drawing under the hammer a little beyond where the intense heat has reached. But theory aside for the moment, although the avoidance of welding in work to be exposed to great stress is desirable, it is nevertheless a fact established by large experience, that welded parts will bear as great a strain as takes place in well proportioned bridge work, with as much certainty as ever has been realized in any department of the means of locomotion. Danger lurks everywhere at all times. In railroad travel, boilers burst, rails break, wheels and axles break, etc,, etc., but the failure of a weld in bridge work is rare indeed, and very few authenticated cases can be referred to. I would, however, prefer a weld in the straight part rather than in the end of a link, unless made with an excess of section around the bend. WThether a bend around a pin of 1~ or 2 times the diameter of the link iron is more liable to break than the straight sides of the link, I can refer to no reliable authority to determine. The longitudinal strain is no greater in the bended, than in the straight parts, if well fitted to the pin. But of course, it can not be expected to have a fit so close as to ensure a firm pressure quite round the ARCH TRUss BRIDGES. 191 semi-circle. Hence the bearing is mainly on the back side of the pin, until by a yielding to compression, and by a slight bending of the link end, a pressure is produced all around. This slight bending, good iron will undergo without having its strength impaired, when in its normal condition. But this condition is disturbed in the process of bending, the outside portion being extended, and the inside compressed, whereby the stiffness of the part is increased. In the outside portion the power of resisting extension is increased, while that of the inside portion is possibly diminished; and, whether the aggregate resistance to extension is increased or diminished, experiment alone can determine; and, undoubtedly, the more soft and flexible the iron, the better can it adapt itself to a bearing upon the pin. Hence, it should be allowed to cool gradually from a full red heat, after the shaping is finished. Hence, also, the necessity of extra section in welded ends, which, being less flexible, must obtain bearing surface by compression and yielding of contiguous parts, rather than by bending, and consequently, must undergo greater transverse strain in the end of the link. CXIII. A link formed of wire " in diameter, formed to a pin in" in diameter at one end, and brazed with a long lap at the other, suffered a permanent stretch in the straight part, of one per C. of its length, with no:apparent injury at the ends. Other analogous experimnents have shown similar results, namely, that the straight portions will yield before the bended portions. Now the same degree of disturbance in the metal takes place in a small, as in a large rod, bent to a curve whose radius has the same ratio to the diameter 192 BRIDGE BUILDING. of the rod. Hence, it is difficult to avoid the conclusion that rods of soft and flexible iron, such as ought to be used for tension members in bridge work, bent to a proper fit upon connecting pins of diameter about twice that of the rods, and formed into links by welding in the straight parts, are quite safe under any stress within the limits adopted in bridge work. But it seems to be more convenient to form the weld at one end of the link, if not both, and such has been the usual practice; and, as before remarked, if a, surplus of metal section quite around the bend be secured, and the work well performed, this plan can scarcely be regarded as faulty, especially, in view of the long, varied, and successful usage of such vast numbers of links made in this manner. Now, although the link chord is very simple, efficient, and convenient to make and manage, there are available alternative devices, some of which will be here described. THE EYE-BAR CHORD. CXIV. This is composed of two or more single rods, of oblong, square, or round section to each panel; connected by cylindrical pins passing through strong eyes at each end of the chord bars. This plan until recently, has involved quite as much welding as the link chord; the eyes having been formed in separate pieces, and welded to the body of the rod. But within a few years a process has been devised by the Phcenix Iron Co., of Pennsylvania, for upsetting and forming eyes upon rolled bars. A mold or die gives the desired form and size to the head, and aside from the fact that a violent disturbance of the normal condition of the iron is produced in the vicinity ARCH TRUSS BRIDGES. 193 of the head, there can be no question as to the excellence of the work produced; and it is undoubtedly, perfectly reliable, under any stress to which it is admissible to expose the material in bridge work. Figure 0SB represents the joint of an eye-plate chord at c, adapted to the arch truss. Upright and diagonals have each an eye to receive the connecting pin at the lower end. The upright has a washer above the eye to form a beam seat above the eyes of the chord plates. Perhaps the washer should be in the form of a saddle or stool, with downward projections bearing upon the pin outside of the diagonals; or, perhaps inside, in case the diagonals be in pairs, as before suggested. [cIII.] SIZE OF CONNECTING PIN. CXV. Considering the average bearing upon the pin, to be at the centre of thickness of the eye, or link end, as the case may be, the thickness of the eye indicates the leverage upon which opposite links act, when side by side upon either end of the pin. Estimating the strength of the pin, then, at 4,500fbs. to the square inch of section, with a leverage equal to the diameter of the pin [see xevIi,] we obtain the proper diameter of the pin as follows: Let a=area of section in link or chord bar. t=thickness of eye=leverage of action. x=diameter of pin, in inches. Then,.7854X= area of pin section; and this multiplied by 4,500 x 3= 433,z equal to the resisting power (/ t of the pin; while 15,000a=the power of the link; and putting these two expressions equal to one another, and deducing the value of x, we have the required diameter of the pin, V4.244.t inches,=x. 25 194 BRIDGE BUILDING. CXVI. If a=4 square inches, and t=1.5 inches, then a.t = 6, and x = /6x4.244 - 2.94 inches, This diameter of pin is required to withstand the action of the chord alone, which is the only stress upon the pin when the chord is at maximum tension. But when the diagonals running in the same direction horizontally, with the inside links, are brought into action, they act in conjunction with the links in producing stress on the pin. Now, the greatest stress upon bn, Fig. 11 [see xxxIv] occurs when the point b alone is loaded, and the links ab sustain 2 of their maximum stress from movable load, and bn sustains 5w", giving a horizontal pull of about 6.5w",theamountvarying with depth of truss. Again, besides the 6w" bearing at the point a, in virtue of the movable weight (w), at b, we have 3w' due to weight of structure, also bearing at a; and assuming 38w to be equal to 1w, or 7w" the whole pressure at a, equals 13w", when the horizontal pull of bn equals 6.5w". The tension of ab, in the usual proportion of arch trusses, equals about 24 times the bearing at a, whence the stress of ab with the point b alone under load, equals 13w"x2.25, = 29.25wv". Deducting from this, 6.5o" for horizontal pull of bn, it leaves 22.75w" - stress of be. Then, assuming the diagonal to act in the centre of the pin, and the length of pin between centres of bearing of outside links to be 27", we find the stress at the centre of the pin, by taking the moments with respect to the centre, of the action of the two links at either end of the pin. The difference of these moments, the forces being opposite, is the moment of the force producing stress at the centre of the pin; in other words, it is the force acting transversely ARCH TRUSS BRIDGES. 195 upon the pin, at a leverage of 1 inch, the inch being our unit of length. We found the pull of ab = 29.25w", or 14.625w" at each end of the pin, which multiplied by distance fiom centre (13.5t') gives a moment = 197.4375w", while for be, the moment is -x22.75x12" = 136.50w; and the difference = 60.9375w" = stress in centre of pin, upon a leverage of 1". Assigning such a value to w" as will give the assumed stress of 15,000bs. to the inch upon ab with the truss fully loaded, with a bearing at a, of 21w" for movable, and 7w" (= 3w'), of weight of structure, we find a stress of 28 x 2- (= 63)w" 8 x15,0000Tbs. = 120,000fbs; whence w" - 1,905lbs. which, being substituted in the above amount of 60.9375wtt gives the stress in pounds at the centre of the pin, on a leverage of 1"/ equal to 116,0861hs. We have seen [xcviII] that the resisting power of a projecting pin equals 4,500 -, which in this case, equals 4,500AD (L being = 1), equal to 4,500x.7854x3 Then, making this expression = 116,088Tbs. we have x = 3.2'; being 0.26" larger than is required to withstand the action of chord alone, at its maximum stress, as already shown [cxvi.]. By similar process we find very nearly the same results with respect to the shorter pins toward the centre of the truss. For, although the maximum action of diagonals takes place under greater stress upon chords, the difference is balanced by diminution in length of pins toward the centre of the truss. Should this mode of connection be adopted, the preceding illustrations and examples, it is hoped, will enable the proper proportions of connecting pins to be determined for trusses of whatever dimensions. 196 BRIDGE BUILDING. A RIVETED PLATE-CIORD. CXVII. May be formed of flat plates as long as may be conveniently managed, connected by splicing plates of a little more than half the thickness of the chord plates, one upon each side, riveted or bolted with such a distribution of rivets, &c., as may not weaken the plates by more than the width of one rivet hole. The area of rivet section should be at least A to as great as the net section of the chord plate, on each side of the joint; and, go, Figo 3841 denoting the splicing plate, the distance cd, from the joint to the centre of the first rivet hole, should be at least twice the diameter of the rivet (depending somewhat upon the size of rivet and thickness of plate, as well as the soundness of grain in the iron). The succeeding rivets, a, e, f, &c., should be placed alternately on opposite sides of the centre, so that the oblique distance ae (= O), may equal the transverse distance (= T), + the diameter of whole (= H ). Then, representing the longitudinal distance be, by L, we have T+H- = 0, and (T+A)" 02, = T2+L2 -2 T2STH+ 2; whence L V= /2T.H + 12. If the plates be 6" wide, and T = 31/" (which is regarded as in good proportion, the above formula gives L = 2-1' very nearly, for a 3'1 hole. Then, 5" being allowed for the space ce, and 2" each for ed and eg, the splice plates would have a length of 201", and 7 of the whole section of chord plates would be available for tension; since an oblique section through two holes, would quite equal a direct transverse section through one hole. The amount of rivet section above given is estimated upon the assumption that each rivet must be sheared ARCH TRUSS BRIDGES. 197 off in two places; and that it will resist, those shearings, each, with about & of the force required to pull the rivet asunder by direct longitudinal strain. It is obvious that the two rivets e andf, Fig. 341, sustaiinig a portion of the stress of the chord plate, relieve in the same degree the stress upon the portion between those rivets and the joint, or end of the plate; whence it is not necessary to preserve the same section in the portion thus relieved., as in other portions of the plate. Th refore the rivets a and c, nearer to the joint, may be larger than e and f, when the section of plates requires more rivet section; provided always, that the least net section of splice plates, have as great an area as the chord plate has through only one of the smallest rivets. For instance, four 1" rivets are sufficient for plates 6" X ". But plates 6" x5-/ require more rivet section - say 1-/ for e and f, and -" for a and c while, the same for the former and 1" rivets for the latter, give about the required section for plates 6" x i/" This leaves in each case, the same proportion of net available section of plates. Moreover, if rivets a and c be placed opposite to each other, and f be removed to a, the rivets being'" and 1" respectively. Then, the smaller rivets sustaining over ~ of the stress, while the others sustain less than 2, the latter may cut off - of the net section (which is, in this case a" less than the whole width of plate), and still leave enough to sustain more than their own legitimate share of the stress. This may be done by one rivet or two, placed opposite c; and thus the length of splice plates may be shortened to 151 inches, instead of 20-, as represented in the diagram. But, as in this case, the long plate has a net width of 54" and the splice plates, only 4" llL1) CUIdC~ YYLL~LIVIC4 198 BRIDGE BUILDING. the latter require 37~ per 0. more thickness than the former, so as to nearly or quite balance the saving in length. As to the proportions of parts, in this kind of work, I would suggest that the thickness of plates be from'th to -fith of their width, and the diameter of rivets, from 1 to 1- times the thickness of plates. If plates be very wide and thin, they may be liable to be strained unevenly, and if very narrow, an unnecessary proportion of section is lost in rivet holes. FIG. 34~.. The end connections of plate chords of this CX-VIII. The end connections of plate chords of this kind, may be effected by riveting on side plates at the ends, as seen at E, Fig. 341, so as to give a thickness that will allow about ~ of the width of plate to be cut away by a hole for the connecting pin P, either round or oblong with square ends for adjusting keys or wedges. Or, the side plates may be omitted, and two keyholes made in the middle of the plate, one for a key having a thickness equal to the diameter of the smaller rivets, and far enough from the end to admit of another hole nigher to the end, with about 2" between the holes, This may, if necessary, have twice the width of the other hole, and should leave at least twice the width of hole, between hole and end. The width of the wider hole,+twice that of the other, should equal about half the width of the plate; and the keys should be driven to an equal bearing before the work be subjected to use. ARCH TRUSS BRIDGES. 199 The connecting blocks used with this chord, sustaiinin only the horizontal action of diagonals, may be considerably lighter than those used with the links, especially in arch trusses. In order to transfer the horizontal action of diagonals to the chords, mortises may be made in the plates, as seen at?n Fig, 31, not wider than the smallest rivets used in splicing, to receive tenons of wrought iron cast in the block. As to the merits of the riveted plcte, as compared with the link chord, it may be assumed that two splices are sufficient for any truss not exceeding 100' long, and that the weight of splicing plates and rivets will equal 4 or 5 feet extra length of plates, say 6 per cent upon a chord 80' long. To this we have to add about 14 per cent for extra section to compensate for rivet holes, making 20 per cent of iron lost in forming connections. Links require about half as much extra material, to be taken up in bends, lappings, and enlargement of section at the ends; showing about 10 per cent less iron for the link, than for the plate chord. This would amount to about 400lbs. for two trusses of 80', with links of 1"' round iron. But this may be nearly or quite balanced by 500 or 600G bs. of castings, which may be saved in weight of connecting blocks. The economy of material being so nearly equal in the two chords, their relative merits must depend mostly upon the comparative cost of manufacture, and the relative efficiency of the chords in use. It is deemed far from improbable that the riveted plate chord might be found, on fair and thorough trial, to be worthy of extensive use in arch trusses, in place of the link chord. The fact that in the plate chord, the iron is used in its original condition, as it comes from the rollers, is certainly favorable. 200 BRIDGE BUILDING BRIDGES WITH PARALLEL CHORDSo CXIX. These may be constructed with or without vertical members, and in form, either rectangular, with vertical end posts, or trapezoidal, having inclined end members, or king braces, as exhibited in FigSo 12, 13, 18 and 19. TRAPEZOIDAL TRUSS BRIDGE, WITH TENSION DIAGONALS AND COMPRESSION VERTICALS. For short spans, less than 70 or 80 feet long, the simple cancel, as in Fig. 12, will generally be used, with trusses too low to admit of connection between upper chords, except in case of deck bridges. The same plan of lower chords composed of links and cast iron connecting blocks, may be used, as already described for the arch truss. The connecting blocks are shorter, and may be cast in connection with the upright, or the latter may be in a separate piece. In the latter case, the block should have a suitable seat to receive the upright, and keep it in place. As the upper chord depends upon the stiffness of the beam and upright for lateral support to keep it in line, the upright should be firmly attached to the beam, and at right angles therewith. There is no means of estimating exactly the transverse force which the chord may exert upon the upright. But if the ends of chord segments be properly squared and fitted, the lateral tendency will be quite small. It is recommended, that each upright have a transverse strength sufficient to withstand a force of 1,000os. acting at the upper chord; that it have a web and flange form of section, with a width of web at the BRIDGES WITH PARALLEL CHORDS. 201 connection with the beam, not less than sg of the dis. tance of upper chord from the beam. Fig. 35 will serve to illustrate the modes of conneco tion for most of the members of a bridge of the kind under consideration. That part of the upright between a and b, is contracted in length. Otherwise, the parts are represented [- l in nearly correct proportions. At c, is represented the connecL= 11 tion of the upright with the end At== ^^u^ of the beam, by means of a... double eye and bolt, as shown at Ell h._____ h This receives the web of the m beam, to which it is secured by the transverse bolt, which should be long enough to receive the eye of a sway rod under both head and nut. The stem of this fixture extends through the upright at its widest part (whence it may taper in both directions), and is secured by a nut upon a screw of about 1-" in diameter. The beam should rest with its lower flange upon a small projection cast upon the upright, and not hang upon the connecting fixture. If so preferred, the sway rods may be connected by a screw and nut cast in the end of the connecting block, as seen at d. This plan has been used, but the connection by the bolt at c is deemed preferable. The outer and inner flanges of the upright at the top, being increased to nearly an inch in thickness, according to size of bridge, and extending 3 or 4 inches above the web, terminate in semicircular concaves to receive the pin connecting the diagonals with the upper chord. A full view of the flange at the top of 26 202 BRIDGE BUILDINGo the upright, with the pin resting in the concave, is shown at e. A heavy cross-bar from flange to flange at a, and light cross-bars at intervals of 16 to 18 inches from a to b, serve to support the flanges, and stiffen the piece. The diagonals are formed with eyes to receive the connecting pin at the upper end, and screws and nuts to connect with the block at the lower chord, in the same manner as in the arch truss. The main diagonals, those inclining outward from the centre of the truss, should be in pairs, and in size, proportioned to the stress they are liable to, as determined by the process fully described in sections XXXIx, &c. The links acting in conjunction, horizontally, with the main diagonals, should go on next the end of the connecting block, as that arrangement obviously produces less stress upon the block. The upper chord, usually formed of hollow cylinders, has openings in the underside at the joints, for uprights and diagonals to enter, where they connect by means of the transverse pin already mentioned. The cylinders should have an extra thickness for 3 or 4 inches from the ends, and a strong collar around the opening, to restore the loss of strength occasioned by the opening; and the ends should be squared in a lathe, to secure a perfect joint and a straight chord. If it be required to give a cambre to the truss, the ends of cylinders should be slighly beveled at the ends, making the under side a trifle shorter. This is easily effected by throwing the end opposite the one being turned, out of centre more or less, according to the cambre required. An 8 panel truss requires an excentricity equal to J of the requiredr ise in the centre BRIDGES WITH PARALLEL CHORDS. 203 of the truss. For any even number of panels, make a series of odd numbers, 1, 3, 5, &c., to a number of terms equal to half the number of panels; add the terms of the series, and divide the required cambre by the sum, and the quotient equals the required excentricity to give the proper bevel. For an odd number of panels, take as many even numbers 29 4, 6, &c., as equal half the greatest even number of panels; add the terms and divide as before for the excentricity. For illustration, for 8 panels, the four odd numbers 1+3+5 7- = 16, whence the excentricity should be l' of the cambre, as above stated. For a 7 paneltrussthe three even numbers 24+6 = 12. Hence the excentricity should be, of the cambreo The reason for this rule will be obvious without more particular demonstration. At the obtuse angles of the truss, a hollow elbow is inserted (g, Fig. 35), reaching about 10 inches each way from the angular point, at the centre of the connecting pin, with an opening in the under side for upright and diagonals to enter, where they are fastened by a pin or bolt, as at the intermediate joints; the cylinders meeting the elbow, being shortened by as much as the elbow extends from the angle, either way. The vertical member connecting with the elbow, is exposed to tension only, sustaining a weight equal to the gross panel load of the truss. It may be composed of two wrought iron suspension rods, united in a single eve at the top, and diverging downward to a connection with the beam and connecting block; or, it may be of cast iron, like the intermediates, with wrought iron eye plates, in place of the cast iron flanges with concaves as seen at e. These should be fastened by efficient means to the cast iron part of the upright; which lat 204- BRIDGE BUILDING. ter should have a cross-section nowhere less than one square inch to each 2,000ibs. of the gross panel load. A complete wrought iron connection from beam to elbow, howevern is to be preferred. The thickness of web and flanges of the uprights, should be from 3 to ~ inch, and the cross-section of upper chord. cylinders should be about 20 per C. greater than that of the portion of bottom chord forming the opposite side of the oblique parallelogram included between consecutive main diagonals and included sections of chords; as d e k 1, Fig. 12. The upright should be so formed as to bring the centres of upper and lower chords in the same vertical plane. Sway rods in this class of bridges, should be about'" in diameter, with a turn buckle near one end for adjustment, and an eye at each end, for connection with the bolt at c. The screw working in the turn buckle is cut upon the short piece, which should be I" larger in diameter than the long piece which has no screw upon it. The lower chords, king braces, and sway rods of the endmost panels, connect with cast iron foot pieces FG. 36. upon the abutments, as represented SS~ ^in Fig. 36. The portion of lower /~NN~ ~chord in the end panels, usually =1^1 ~consists of single rods, instead of -links, with an oblong eye at one end to receive the connecting block, and a screw and nut for connection with the foot piece (Fig. 36), at the other end. This plan of construction will generally yield precedence to the Arch Truss plan, for short spans, except for deck bridges upon rail roads, in which case the BRIDGES WITH PARALLEL CHORDS. 205 structure will be secured laterally, by x ties, or sway rods between beams, and between king braces at the ends; no x bracing being required between lower chords. Low trusses constructed in the manner above described, have been used satisfactorily for supporting the outside of wide side walks; answering the purposes of a protection railing at the same time. For this purpose, the uprights are only 5 or 6 feet long, so as to bring the upper chord about 4 feet above the flooring. The first instance of this kind was in the case of the canal bridge on Genesee street in Utica, built 18 or 20 years ago, and repeatedly copied since. CXX. Bridges from 80 to 100 feet for common roads may be constructed with single canceled trusses, 13 to 14 feet high; in which case the panels will require to be wide (horizontally) in order to avoid an inclination of diagonals too steep for good economy. But for railroad purposes, the trusses require a depth of about 20 feet to afford sufficient head room under the top connections, unless the beams be suspended below the bottom chords. Hence, the -Double Cancelated Truss should be adopted for " through bridges " of spans exceeding 70 or 80 feet. Figures 18 and 20 exhibit in outline, the general character of the double cancelated trapezoidal truss bridge; and, it is only necessary in this place, to de_ scribe feasible modes of forming and connecting the various members; which may be done essentially as described in the preceding section, with such modifications as follow. 206 BRIDGE BUILDING. Cast iron Uprigyhs are composed of two or more pieces. When of two pieces, they may be connected by flanges and bolts at the centre, where they should have a diameter of about 0- of the length, and a cross-section determined by the maximum stress, and the power of resistance of the material, as indicated in the table [xcIII.] The upright may taper from the centre to either end to a diameter of 5 to 6 inches, internally. The lower end is to stand upon FIG. 37 a properly formed lj~l 7 l,,If seat (/ Fig. 37), ji U jl ^^ If,/,9 may have an openll l ___ ing at the bottom, ______il __ {C'\ upontheinnerside, ll'l ji ~where the beam may enter and rest upon a seat (e), inside of the upright, upon the connecting block. The strength destroyed by this cutting the post should be restored by additional metal in a band or collar (c, Fig. 37), around the opening, and, if necessary, by the wing flanges d d, extending 6 or 8 inches above the opening. To avoid too much cutting of the post, the flanges of the beams may be reduced to 3 or 38 inches in width. The post and beam seat upon the connecting block may be elevated 3 or 4 inches above the links, as may be required, so as to allow sway rods to pass through with simple screws and nuts for adjustment; thus dispensing with turnbuckleso BRIDGES WITH PARALLEL CHORDS. 207 Holes should be cast in the central part of the post, for diagonals to pass obliquely through. Or, what is perhaps better, the connecting bolts may be lengthened so as to permit the insertion of an open box, or frame, between the flanges, as seen at a, Fig. 37. This intermediate piece should be so constructed as to close the ends of the hollow pieces meeting it, and prevent the water from getting inside. The top end of the upright is forked, with concaves for the connecting pin to rest in, as described in the last section, and as seen at a, Fig. 38. The cap piece of the post may be cast separate, or in connection with the upper half of the column. Both plans have been satisfactorily used. All joints, when practicable, should be accurately fitted by turning or planing. This plan of a cast iron upright, composed of two principal parts, with or without the centre piece, is perhaps as good as any for general use; the principal disadvantage being the difficulty of giving a sufficient diameter in the middle for stiffness, without two much reducing the thickness of metal, or increasing the amount of cross-section beyond the proper theoretical proportions. To obviate this difficulty, the device adopted in the original model of the Trapezoidal bridge, was that of using truss-rods, or stiffening rods, to secure the post against lateral deflection, after the mannner shown in Fig. 38. In the case of using stiffening rods for the uprights, it may be recommended to form each half of the column in two pieces, somewhat in the manner above described for the whole one, without stiffeners; making the piece forming the end portion about Bth shorter 208 BRIDGE BUILDING. than the other, with a strong flange at the larger end, to afford attachments for the stiffening rods. FIG. 38. In Fig. 38, a c d exhibits the upper half of the upright; h, the stretcher at d;, the flange at c (enlarged), and i, j, enlarged sections of the two ends forming the joint at c. The piece running toward the centre has no flange at c, but has an increase of thickness for a short distance from the joint, as shown atj, and a diameter about 41' larger than the abutting piece, which latter has a small burr entering the former'1 or 1 to keep the ends in place. At d, each of the pieces meeting at that point, has a bi-furcation, so as to form an opening for diagonals to pass through, at the same time passing through the stretcher h. The lower half of the upright is the satne as the upper, except the end, which is squared to fit a fat bearing upon the connecting block. An enlarged vertical section of the lower end is shown at 1, Fig. 38. See also Fig. 37, where is shown t!he arrangement for the beam to enter the opening in the lower part of the upright, as described. f ew pages back. Floor beams of wood or iron may be suspended below the chords by bolts passing down through the connecting blocks, or, wooden beams nmay be in two BRIDGES WITH PARALLEL CHORDS. 209 parts, resting upon flanges cast upon the upright abo ut 3" above the lower end; the beam timbers being hollowed out upon the insides, so as to embrace the upright, in part, leaving a space of 2 or 3 inches between, and secured in place by bolts and separating blocks. The mode of inserting iron beams by means of openings in the uprights, has already been explained, Lateral > ties, or sway-rods may be inserted by bolting to the beams (Figs. 31 and 33), attaching to the inner end of connecting blocks, as at d, Fig. 35, or by passing through the block between the links and the post and beam seat, in the manner referred to two pages back. Diagonal ties of wrought iron, and transverse struts of wrought or cast iron, are also required between the upper chords, to keep them in line. Cast iron crossstruts may have the web and flange form of section, with shallow sockets at the ends, to admit the connecting bolts at the upper chord to enter, after passing through eyes upon the upper sway-rods and nuts to hold them in place. These sway-rods require turnbuckles for adjustment, when they extend across one panel only. But if the bridge be wide between trusses, the rod may extend only from the end of one crossstrut to the centre of the next, where it may pass through the strut, and receive a nut on the end. Thus, four rods meeting at the centre of the strut, each having its appropriate hole to pass through, all as near to one another as practicable, with sufficient space for nuts to turn (see a and e, Fig. 39), it forms a convenient arrangement for adjusting the rods to a proper tension, at the same time affording lateral steadiness to the cross-strut. The end-most struts, however, should have no rods connecting with theml in the centre, as they can have 210 BRIDGE BUILDING. no antagonist rods on the opposite sides to prevent the springing of the struts. The end panels should have two full diagonals with turn-buckles, and two half diagonals connecting with the centre of next strut. FiG. 39. In Fig. 39, a shows the middle of the cross-strut, with the upper flange removed; c, a joint of the upper chord, where the connecting bolt passes transversely, receiving eyes of sway-rods, and nut, and entering the end of the strut at d; the upper part of the strut being removed, down to the socket. The bolt bears upon a slight swell in the bottom of the socket, to ensure a central thrust: (see also, b, Fig. 38). At e is presented a side view of the centre of the strut, showing the arrangement of the holes. A similar device has been used with good effect for giving lateral support to posts or thrust uprights, of the web and flange form, so proportioned as to have greater stiffness transversely than lengthwise of the truss. It has been demonstrated that the weight sustained by these posts, increases toward the ends of the truss, while the tension of counter diagonals runs out to nothing, a little way from the centre of the truss. For instance, 4/6 Fig. 18, sustains 6o"-lo', which is a negative quantity whenever wi is less- than 4w', that is, BRIDGES WITH PARALLEL CHORDS. 211 when the greatest movable load is less than four times the weight of structure, as is usually the case. But instead of dispensing with that member, and other counters on the left, they may be made in two pieces each, of 5, or " iron, connecting with the upright at the crossing by screws and nuts, in the manner above described; thus preventing the uprights from deflecting lengthwise of the truss, where the greatest weights act upon them, and where otherwise, they would require to be heavier. GENERAL TRANSVERSE SUPPORT. CXXI. The system of cross-struts and diagonal ties serves to preserve the upper chords in line, but does not prevent the whole structure from swaying bodily to the right or left; a result which would be fatal to the structure. In the arch truss Fig. 27, the width of base at the bearings upon abutments, resulting from the peculiar form of the arch, affords the required stability in this respect. In case of the trapezoidal truss, when high, various devices have been resorted to for producing the same results. For deck bridges, cross tying between king braces at the ends, is an easy and efficient means of accomplishing the object. For through bridges, guys from the connecting bolt at the elbow of the obtuse angle, anchored in the abutment, may be employed. But this requires extra length of abutments and piers, and the effects of change of temperature, ar, t ttighten and slacken the guys, so as to ilmpair their efficiency. To obviate the latter objection, double acting guys (acting by thrust and tension), applied at one side only 212 BRIDGE BUILDING. of the bridge, have been employed; the effect of temperature being only to very slightly sway the bridge laterally, but not so as to be detrimental to stability. This also, requires 5 or 6 feet more length of pier, than what is necessary to bear the vertical pressure. Again, the king braces have been made with two branches diverging from the elbow to a base of 2 or 3 feet in width, according to height of truss. This plan has been used in a large number of bridges, with satisfactory results. But it contracts to a small degree, the available width of bridge; not, however, so as to produce material inconvenience. Another device is, the introduction of two or more long beams, extending 5 or 6 feet outside of the trusses, say at the first thrust uprights from the ends (as over Figs. 3, 381, Fig. 18), with guys extending from the connecting bolt at the upper chord, to the ends of said long beams (see g Fig. 38). Arches may also be introduced at the ends of the bridge, attached to the king braces, say a quarter 0o the way downl from thle t, ad witht conieetingbolt at the elbow. These may be made with a full, or an open-worl web, and fl anges of''2 or 2) inch angle iron upon both sides of th:e web, at the top, ancd around the arch, and either angle iron or plain flat bars, along the sid-es next the king braces. web of 3,, plates placed edg'e to edge, and battened upon both sides with plates of the s ame about 4" wide, riveted alternately on eachi side of the seam, with angle iron, etc., as above, riveted once in 6", forms a stiff and substantlal arch for the purpose under consideration, such as have been used effectively in a bridge of 160ft. span. BRIDGES WITH PARALLEL CHORDS. 213 Moreover, simple arch braces extending from the king brace to a stiff and substantial cross beam from elbow to elbow (see Fig. 40), will /1 // a AS ^ \ 0effect nearly the /same result as the _- _;p ^r M arch. In both \ _iL -~~~ ca ses, a considera-\ t a'.\\ / ble degree of lateral l 1- 0gtlllt- t stress is liable to \I gU~II!.lJ X1 I j^be thrown upon the!\ - } I II II ]'king braces, which l jll,llll,,,i accordingly should ported by truss rods. and struts opposite the feet of the arch or braces. Whether the truss rods be used or not, it is advisable that the connection with the king brace be made by means of a bolt running through the whole diameter of the king brace, with nut or shoulder bearing externally and internally upon both sides, to counteract any tendency to collapse. Fig. 40 presents an end view of a bridge, showing arch braces, with truss rods to sustain the thrust of arch braces against king braces. The internal figure gives an enlarged view of the connection at the elbow. A strap a (about 1"x5"), bent twice at right angles, is riveted or bolted to the flanges of an I beam (about 9" deep), leaving a space of about 4 inches from the end of the I beam, for eyes of two sway rods and a nut upon the large connecting bolt. This bolt in large bridges being from 3 to 4 inches in diameter through the elbow, is reduced to 2 or 2g inches in the part pro 214 BRIDGE BUILDINGO jeeting through the strap above mentioned, and the eyes of sway rods. The truss rods may not be necessary (with substantial king braces), for spans not exceeding 150 feet. But they will add to the security, in all cases of railroad bridges having cast iron king braces. These members being over twice the length of the cylinders in the upper chord, are usually cast in two pieces, and connected by bolts and flanges in the middle, where they have a diameter of about -1 of the length of brace, and taper to the size of the upper chord at the endso CXXII. WROUGHT IRON THRUST MEMBERS. The trapezoidal bridge, as described in detail in the preceding section, and as originally intended, is a wrought and cast iron bridge. But it will readily be seen that with slight modification of detail, it is easily adapted to the use of wrought iron upper chord, vertical posts, and main end braces; which latter, for convenience, have been designated in this work, as king braces. All of these members may be in the form of the patent wrought iron column of the Phcenix Iron Coo of Pennsylvania, formed of flanged segments, united by riveting; or of rectangular wrought iron trunks, as well as various other forms of sectiono For the Phcenix column, a cast iron connecting piece may be inserted at the joints of the upper chord, with ends formed to enter the squared ends of the chord cylinders, and receive them against a shoulder of the connecting piece. This piece may have an opening in the under side to receive the diagonals and uprights, where they are secured by a transverse WROUGHT IRON THRUST MEMBERS. 215 connecting bolt, in the same manner as at the joint of the cast iron chord cylinders, as before described. In this case the upright may have a cast iron top piece, formed as seen in Figs. 35 and 38 upon the top of cast iron uprights. A separate top piece has sometimes been used with cast iron verticals. FIG. 41. The connecting piece may also be formed as indicated in Fig. 41, with a downward branch like process to meet and receive the squared end of the vertical in the same'manner as the horizontal part connects with chord cylinders. In this case the connecting piece must have openings as at b b Fig. 41, for the eyes diagonals to enter. Fig. 41, shows an inside view of the joint piece, as it would appear it' cut vertically and longitudinally, and the near half removed. The horizontal part consists of a cylindrical shell a little thicker than the wrought iron chord cylinder, with ribs upon the outside corresponding with those of the wrought cylinders, and as shown in end view c. Upon the inside, the ring and flanges a a, project inward, leaving usually a space of about 5 inches (according to dimensions of 216 BRIDGE BUILDING. bridge), for eyes of diagonals. These are to ease the lateral strain of the connecting bolt or pin. The process meeting the vertical, may be rectangular in horizontal section, composed of two parallel fiat plates, in form as may suit the taste of the designer, united by two irregular plates formed to the profile of the parallel plates. The openings for diagonals, are, of course, through the irregular plates. These are drawn in at the bottom so as to form a square with the parallel sides, large enough to cover the flanges of the 4 segment column selected for the upright. See Fig. 41. The inside of the square d, is filled in to form a hollow round, about an inch less in diameter than the hollow of the column, that it may have a ring or collar (represented by the inner white ring around d), projecting about 2 inches beyond the shoulder into the wrought iron column. On the top of the joint piece may be an arrangement of oblique holes for the attachment of lateral x ties, and on the inside, facing the opposite truss, an abutting seat for the cross-strut, which may be in the form of a 6" I beam, or such other form as may be preferred. The foot of the post may stand upon a properly formed seat upon the connecting block of the lower chord, with an opening to receive the beam, in the same manner as described for the cast iron post. See Fig. 37. It will be necessary ror diagonals to pass through the centres of uprights, and for that purpose 10 or 12 inches in length, as may be necessary, may be left out of two opposite segments, and the strength thus lost, restored by additional metal, in such form as may be found convenient and efficient. Or, a cast-iron middle piece may be inserted in the upright. WROUGHT IRON THRUST MEMBERS. 217 In the case of an upper chord of rectangular trunks, and uprights of other than a cylindrical form, the joint piece will be correspondingly modified. The position of diagonals may be reversed, connecting by an eye with a wrought cylindrical connecting pin at the lower chord, and by screw and nut with the joint piece of the upper chord. This involves merely a question of practical economy and convenience. Sometimes, also, the connection is made by an eye at both ends of the diagonal, depending upon accuracy, as to length, in the manufacture, for the proper adjustment of parts. It is also practicable to provide means of adjustment in the length of vertical members. CXXtII. But, to enumerate all the changes, and peculiarities of detail admissible in the construction of the Trapezoidal Truss Bridge, even if practicable, could hardly be regarded as expedient in this place. The essential requisities are, to provide material enough of good quality in all parts, to withstand the forces to which they are respectively liable, with efficiet connections of parts, by the most direct and simple means, and with sulch an arrangement and adjustment as may produce the most uniform degree of strain upon all parts of each member. For instance, ecah section of the lower chord is usually composed of several bars, and it is important that each should sustain its proportionate share of the stress. In the link chord composed of two links to each panel, if the links be properly fitted, the two sides of each must act very nearly alike, while the connecting block acts as a sort of balance beam to equalize the tension of links acting upon its two ends; and, if the two links of a pair vary slightly in length, the connect28 218 BRIDGE BUILDING. ing block still secures equality of stress upon the two~ The same is the case with regard to a chord composed of two eye bars instead of links, to each panel. But the serious mistake is sometimes committed, of putting the two links or bars upon the same side of those in the succeeding panel, FIG. 42. as in Fig 42; where it is obdLJL2Lbl- vious that the inside links (a, ^":..: —. b, c), are exposed to more ac~ —~1- a e' -~@- ^ tion than d, e, f. For, if the inside links be 3", and the outside ones 4" from centre of pin, since a and b tend to turn the pin in one direction about its centre, and d and e in the opposite direction, the forces being in equilebirio - the moments (with respect to the centre), of forces tending in one direction, must be equal to those of forces tending in the opposite direction. Hence, representing the stresses of the several links by the letters designating them respectively on the diagram, we have 3 x (a + b) = 4 x (d + e), whence, a + b = - (d + e); showing - more stress upon the inside than upon the outside links. On the contrary, if the link e, be removed to e' upon the inside of a, then d and e' act in one direction, and a and b in the othier; and, assuming as before, the inside links to be 3", and the outside ones 4" from centre of pin, we have 4 + 3b = 4d + 3e'. But a + d = b + e', and if the force be communicated at the ends, equally upon the two sides of the chord, giving equal stress upon a and d, for instance, the tendency is to an even balanced action throughout the length of chordo Hence the two links of each panel should always act upon the connecting block or pin, at equal distances from centre of pin. MULTIPLEX CHORDSo 219 MULTIPLEX CHORDS. CXXIV. In very long or heavy bridges, the required amount of chord section in the middle portion of the truss, is so great, that it is deemed expedient to introduce more than two links or eye bars to the pallelo This is sometimes done by alternating them upon the connecting pin, increasing the number and sizes according to the.increase of stress from panel to panel toward the centre. This mode of construction, unless the bars be arranged and proportioned with almost impracticable care and nicety, is liable to be attended by an accumulation of lateral strain upon the connecting pin, beyond what it can bear without bending, or springing so much as to materially disturb the equality of stress upon the links, or chord bars. FIG. 43. 7 0 s 56, 401 3 C 2'....1-~lt,..~'~i...........'; 1!.J.^..^..1 ~__i'............................. k L_ s. i _ 2 i |< _8 9 144- =A- S- -1 2 To illustrate this subject, let Fig. 43, represent one quarter of the chord of a 16 panel bridge. The line CC may denote the central axis of the chord running through the centres of connecting pins; D, at a distance of, say 8" from C, the line in which the diagonals act upon pins, and the other parallel lines at intervals of 3" from D, and from one another (see Figs. 220 BRIDGE BUILDINGo on right hand of diagram), the centres of thickness of links, at which points the action of respective links is supposed to be concentrated upon the pins. Also, let a, b, c, etc., represent the panels of the chords. NTow, if 15N\T or 15, represent the stress upon the chord in the two first panels, a and b, that of the succeeding panels to the centre, will be as 22, 84, 44, 52, 58 and 62 (see lower figures in diagram), and the ciagonals (producing increments of action upon chord), will have a horizontal action represented by 7 in panel b, by 12 in panel e, and so on by 10, 8, 6, 4. These being added successively to 15, produce the numbers just stated for the chord in the several panels. The first three panels, a,b and c require only one link upon each side, as indicated by the oblique black lines. The 4th panel, d, may have 2 links on a side, and the most favorable position for them, as regards action upon connecting pins, will be as shown, diverging from the central axis, so as to bring the end toward the abutment, nearest to the main diagonal connecting with the same pin. The first pin, connecting a and b, having two equal forces acting in opposition, but at different distances from the centre line 7, we take the moments of these forces with respect to that line; which are, for a, 15 x14=210, and for b,15 x 11=165. The difference (45) between these moments, equals the moment of the resultant, or the lateral stress of the pin, exerted on a leverage of 1"o Assuming the value of W, our unit of stress (and always understood as annexed to the figures denoting stress), to represent 5,000Fis. we have for stress of pin in this case, 45 x 5,000-.L. The L, being 1" may be omitted in the expression. MULTIPLEX CHORDio 221 Then, making xdiameter of pin, its resisting-power= -Dx4,500 (see [xcviii])=.785 x x x 4,500 - 1"= 3,532.5x3; and putting this equal to 45x5,000 (the stress above found), we obtain x=4" (very nearly), =required diameter of pin. At the next pin we take the moments of one link 15x14"=210, and one diagonal, 7 x8"=5, making 266 in one direction, against that of one link, 22x11"= 242. Hence the resultant moment = 24, and 24 x 5,000=3,532.5x3, gives the required diameter of pin in thie centre, x=3}/, nearly. But this is the general stress in the portion of pin between diagonals, and may be greater or less than at certain points where forces are applied. For instance, if the aggregate moments of forces in opposite directions be equal, the resultant moment is nothing, and the middle portion of the pin, between diagonals has no stress, and might be cut out and removed, as far as strength of chord is concerned. In the case in hand, the moment of link b, with respect to link c, equals 15"x3= 45=stress of pin at centre of e. Hence the required diameter at this point is found by the equation 45x5,000=3,532.5 x 3, whence x=4", the same as pin No. 1. At the next pin, if we add another link, making 2 links sustaining 34W, at an average of 14" from centre9 giving a moment of 476, against one link, 22x14, + one diagonal 12x8 = 404, we obtain a resultant momeent of 72; whence, 72x5,000 = 3,532.5xz3 and x = 4.67 inches, = required central diameter of pin, and as will be readily seen on trial, the greatest required at any point. Again, assuming at the 4th pin 2 links and I diagonal against two links, we have for the former, 34x17" + 10x8 = 658, and for the latter, 44x14 = 222 BRIDGEi BUILDING. 616, whence the resultant moment is 42. Therefore the equation 42x5,000 = 3,532.5z3, gives x =.9 inches, = required diameter in centre, while for the outside link on this pin, the stress, 17, multiplied by 3 shows a momentof51 Heence, x =- (/(5x 5,000) = 4.16 inches 3. 8,532.5 = required diameter at that point. At the 5th pin, there are 3 links, against 2 links and one diagonal, giving moments for the latter, 44x17+ 8x8 = 812, and for the former, 52x17 = 884; whence the resultantmoment = 72 and x = /(. 25t )) 4.67 incheso The moments at pin No. 6, are, for 3 links, 52x209 + (for diagonal) 6x8 = 1088, in one direction, and for 3 links, 58x17 = 986, giving a resultant of 102; whence; x - (102x5',00) = 5.24. b,5t2.5 Lastly, adding another link at the 7th pin, the moments are 58x20 + (for diagonal) 4x8, = 1,192, against 62x20t' =1240, whence the resultant is 48, andx - 48(sx 5,000) - 4.08. 3,632.6 In this case the eyes, or link-ends are supposed to be bored in the direction of the pin, a little obliquely to the direction of the link, so as to bear through the whole thickness, as long as the pins remain perfectly straight. But the pins having a degree of elasticity, and considerable length, must yield to the action of links, springing' more or less in the direction of the greater sum of moments. It will be seen, moreover, that in each case, the consecutive ends entering the outside link, as 3 and 4, 5 and 6, &c., are always sprung toward one another; the inevitable result of which must be, a relief or relaxation of the outside link, whence it must sustain a less degree of strain than its fellows located farther from the ends of the pins. MULTIPLEX CHORDS. 223 Now, as a 12 foot link, under a stress of 10,000G s.o to the inch is extended less than 5- of a foot, a slight 1,000 springing of connecting pins would relax the outside links materially, especially when the pins tend to spring toward one another. Again, if the links run parallel with the centre of chord, and at right angles with the connecting pins, as indicated by the double black lines (Fig. 43), the moments of forces upon - pin No. 5, for instance, will be-for 3 links acting toward the right hand, 44 x 17 + (for diagonal)8 x 8 812, against 3 links acting toward the left, with moments equal to 52 x20 = 1,040, showing a difference of 228; whence x=^ (2s28 X 5,~) 3,5352.5 6.85 inches -- required diameter of pin at the centre. At pin No. 6, are 3 links with a combined moment of 52 x 20, - (for diagonal), 6 x 8, = 1076, against 3 links with a combined moment of 58 x 17 = 986, show" ing a difference of 90; consequently, x V( 90x 5s,00) 3',532.5 5.03 inches = required diameter of pin. Such would be the result as to stress and required diameter of pin, provided the pin remain perfectly straight. It is true that the spring of the pin in the direction of the greater moment, or sum of moments, will, in practice, produce an obliquity in its direction through the eyes, which will throw the centres of bearing upon the pin, nigher to the adjacent sides of the eyes, and thus reduce the difference of opposite moments, and consequently, the stress upon the pin. But such relief to the pin must be attended with a disturbance of the central and uniform strain of the chord bar; the strain being brought near one side of the bar. Moreover, as this can only result fromr actual springing of the pin, there must inevitably be a degree of relaxa 224 BRIDGE BUILDINoG tion of the outside link, whenever the pins at its two ends are deflected toward one another. On the contrary, an outside link or bar connecting with two pins springingfrom one another, is necessarily subjected to greater strain than those nigher the centres of pins, in the same panel. In this case, the forces tend to spring the pins toward one another at the ends, whence the outside link, must sutfer more or less relaxation. It seems unnecessary to carry these examples further. The above results show a decided advantage in the oblique position of links, diverging; toward the centre of the span, so as to have the inside link opposed to the diagonal. The arrangement of links, or eye bars, here assumed, and the amount of stress assigned to them, are no exaggeration upon what has been put in practice. But the preceding calculations must be sufficient to demonstrate the exceptionable character of such practice. Two links upon a side (4 to the panel), after two or three panels next the end, so thin as not to occupy an unnecessary length of pin-each taking hold of the pin outside of the succeeding one toward the centre of the truss, may be admissible. But a greater number, in the opinion of the author, for reasons already given, is not to be recommended. DOUBLE CHORD. CXXV. To obviate the difficulty attending the use of the multiplex chord, consisting of many links in a panel, we may make use of what may be distinguished as a Dotuble Chord. We have seen [LVI], that in double cancelated trusses with vertical members, there are two independent sets DIFFERENT MODES OF CONSTRUCTION. 225 of diagonals and verticals, which have no interchange of action between one another. Now, each of these sets may have its own lower chord, also acting independently, each of the other, but uniting at the same point at the foot of the king brace, which is common to both sets of web members. In such case, the two chords (which we may call subchords), may be one above the other, and composed of links or eye-bars, extending horizontally across two panels; the links or bars of one sub-chord connecting opposite the centre of those in the other, and the uprights in one set, being as much longer than those in the other, as the distance, vertically, between the upper and lower sub-chords. By this means, about one-half of the extra material in chord connections would be saved; and a more uniform stress upon the chord bars secured, than would be practicable, even with 4 links acting upon one connecting pin. DETACHED, AND CONCRETE PLANS OF CONSTRUCTION. CXXVI. In the plan of Trapezoidal truss had under consideration in the last few preceding sections, the several members are formed in separate pieces, to be erected in place, and connected by screws, bolts, connecting pins, &c., as the parts of wooden bridges and building frames are erected, after being framed and prepared, each for its particular place. There is another mode of construction, in which members and parts of members are permanently riveted together in place; or, in case of small bridges, the whole structure is permanently put together at the manufactory, and transported by water or rail to the place of erection and use. The former of these may 29 226 BRIDGE BUILDING. be called the detached, and the latter, the concrete mode of construction. The detached plan is probably the best adapted to wrought and cast iron bridges, and also, at least, equally adapted to bridges entirely, or essentially constructed of wrought iron, when vertical thrust uprights are employed. But it can hardly be regarded as advisable to construct iron bridges with independent members, without thrust verticals. For, although aswe have seen, [xLvI,] the latter plan shows a trifle less action upon the material than the plan with verticals, the oblique thrust members in the web, are 40 or 50 per cent longer (according to inclination), as well as being in greater number, and sustaining less average action to the piece. The 7 panel truss, Fig. 12, has 4 compression verticals, liable to an average action of 8w"; while truss Fig. 13, has not less than 6 diagonals, liable to an average compression of 4w" V/2 (when the inclination is 45~), equal to 5.65w". In the mean time, these members being over 40 per cent longer, and sustaining only about the same aggregate amount of action, can not be so economically proportioned to perform their required labor, when acting independently, as the fewer and shorter uprights. Still, the Trapezoid with individual members is practicable, probably with about the same economy of material without verticals as with them; and, if it be deemed expedient to adopt the former, the modes of forming and connecting the various parts may be so nearly like those already described for the latter, that particular specifications will not be given in this place. The essential conditions to be observed, are, besides proportioning the parts to the kind and degree of strain DIFFERENT PLANS OF CONSTRUCTION. 227 to which they may be exposed, to see that the forms of diagonals liable to compressive action, be made capable of withstanding such action, according to the table of negative resistances [xcIII]; and, that those liable to a change of action from tension to compression, and the contrary, be formed and connected in such manner as to enable them to act in both directions. CXXVIIo In the concrete, or rivet work plan of construction, the Trapezoid without verticals may, it is thought, be generally adopted with advantage. Upon this branch of the subject, however, but little of detail will be attempted at this time, the author having had very little direct practical experience in the premises. The first point to be attended to, of course, as in all cases of bridge construction, is, to arrange the general outline and proportions of the truss; that is, the number of panels, and depth of truss suitable for the particular case in hand. This being done, the amount and kind of force, whether thrust or tension, to which each part is liable, should be determined; for which purpose, the value of w, and of w' (the variable and constant panel load for the truss), must be assumed, or estimated according to the best data at command; when the stresses of the several parts are readily obtained by process already explained; [xLIv, &c.]. We are then prepared to assign the requisite crosssection to each part, and to adopt a suitable form of bar, or combination of bars and plates, for each member. Thrust members will usually (if long), be formed of several parts, mostly fiat plates, angle iron, T iron, and channel iron, united by riveting in such form of cross-section as may give the largest diameter practi 228 BRIDGE BUILDING. cable without too much attenuation of the thickness of material, a point upon which no certain rules can be given. Flat plates, when connected by riveting at the edges, may be of a width of 30 to 40 times the thickness perhaps, without liability to "' buckle" under reasonable compression. When riveted along the centre, a width of 12 to 20 times the thickness, will be in better proportion. UTPPER CHORD. CXXVIII. A good upper chord may be made in rectangular, or box form, of flat plates and angle iron; or, for small bridges, of channel iron, with flanges either inward or outward, upon the two vertical sides, with flat plates upon upper and under sides; the upper riveted, and the lower one either riveted, or put on with screws, tapped into the lower flanges of the channel bars. The upper plate, when flanges turn inward, may project half an inch, or an inch, and the lower one, come even with the sides. The channel bars should meet at the nodes, or connecting points, and a splice plate covering the joint may project below the chord far enough to form a connection with diagonals by riveting. (Fig. 44). Diagonals acting by tension only, may be plain flat bars of width from 8 to 10 times the thickness. Those acting by thrust principally, may be of T iron with short diagonal bars riveted to the mid rib, (e Fig. 44), giving a width corresponding with that of the upper chord, or with the space between tension diagonals, so that the latter may be riveted to the cross-plate of the T iron at the crossings, to give lateral support to RIVET-WORK BRIDGES. 229 the thrust members. Angle iron may also be used instead of T iron, in these members. FiG. 44.' " o ol oo 0 p 0 0 i 0!.................. _ot,,,,...ii............1 Diagonals acting by both thrust and tension, should be formed and connected with reference to the forces they are liable to. For small bridges, small plain I bars may be used for thrust diagonals with advantage. In all cases of tension, rivets should be so arranged when practicable, as to leave all the section available, except the diameter of a single rivet hole; that is, no section through two or more holes, including the one farthest from the end, should have less area than a square section through one hole. [cxvII, Fig. 31.] In Fig. 44, a, a, &c., represent tension diagonals, of plain flat bars, with cross-section proportioned to the stress in each case; b,b- thrust diagonals of T iron and short diagonal plates, as seen at e; c, c, the upper, 230 BRIDGE BUILDING. and d, the lower chord; the dotted line j, shows the meeting of lower chord plates, about 4 inches toward the abutment from the point of meeting of the several centres of chord and diagonals. The side plates of upper chord may meet at the centre of the node, or connecting point. The upper splice plates are of irregular form (or, they may be cut on a regular slant from upper to lower angle), but such as to cut without waste of iron. They may be clipped out upon the under side, as by the curved line, or not, as may be preferred. The lower splice plates may be rectangular, and of such length and width as to admit of a sufficient num. ber of rivets, properly arranged, to be equal in strength to the net section of chord plate and diagonals. It is scarcely necessary to repeat, that rivet section connecting two thicknesses of plate only, should exceed the net section of plate by as much as the direct tensile strength exceeds the shear-strength of iron. LOWER CHORD. CXXIX. The following plan of a flat plate bottom chord adapted to a connection of diagonals by connecting pins, is transcribed from the author's former work; and, by widening the splice plates, as in Fig. 44, is equally adapted to the concrete mode of construction; i. e., by rivet work. The plan contemplates each half-chord as composed of two courses of plates (except near the ends), spliced alternately, one at each node so as to "break joints." The two half chords are to be placed at such distance apart as to accommodate the connections with diagonals, and with uprights, when used in connection with uprights. iRIVET-WORK BRIDGES. 231 For a 16 panel truss, as arranged in Figures 18 and 19. Suppose w = 12m (m representing 1,000Ibs.); w' = 4m, and W = 16m, = w w; - diagonals (except the steep ones), inclining 45~. The end brace, then, sustaining?7W = 120m, [LVI], produces tension equal to 60m, upon the first and second section of chord, in Fig. 18, the proportions for which will be here considered. Allowing then, 10m to the square inch, each half chord requires a plate of about 8" by- ", up to the second node from the end. This plate may extend - say within 8" of the centre of the connecting pin at the 2d node, where it may be connected with a s" plate, by two splice-plates about 27" long (see A. Fig. 45), with a net section equal to the'I plate, or, say 41' thick. Fig. 45, exhibits a disposition of rivet and pin holes, at A, so arranged as to preserve the full section of plates, less the diameter of a single 1" rivet hole. Or, the splice-plates may be 7" shorter, and ~ thicker, and the two rivets next the joint (j), on either side, opposite one another, as at BB, Fig. 45; thus giving the same section (of splice-plates), through two opposite rivets in the thicker, as through one rivet in thinner and longer splice plates. In this case, the joint should be 41t, from centre of connecting pin (p), and a little more, when the rivets exceed 1" in diameter. At the third node, an increase of section is required, and a'" plate may be added on the inside, lapping 9 or 10 inches back of the pin, with a I" splice plate of the B pattern to balance the extra inch in width required for opposite rivet holes, and a 2" pin hole. The inside plate continuing past the next, or 4th node, the i" outside plate may be met by, and spliced to a t" plate, in either of the modes indicated by A and toa 232 BRIDGE BUILDING. B, Fig. 45. On plan B, the outside splice plate should be at least I'" thick, and the inside one, 5. In this as in other cases where a thinner plate meets a thicker one, the former is to be furred out to the thickness of the latter. At the 5th node, the outside plate may continue, while the inside one is succeeded by a A" plate, with a'" splice-plate inside, and one of 3' thickness upon the outside; splice-plates in all cases being intended to be upon the outside, and not between the two courses of plates forming the half chord. The same general process being continued, each course being spliced at alternate nodes, and breaking joints with one another, we introduce in the outside course, a 1" plate from the 6th node to the centre of the chord, and a'" plate from the 7th node, past the centre to the 9th node, and so on, with a reversed order of succession to the other end of the chord. The two 1" plates in the outside course, should meet at the centre connecting pin, and all other joints should be a few inches from the pin, on the side toward the end of the chord, as in diagram, Fig. 45. FIG. 45. B B A A 0 0 0 \ 0' ~ Each pair of splice plates should have a minimum net section, together with the net section of the continued plate, at least equal to the sections of the continued, and the thinner spliced plate, through one of the smaller rivets used in the splice; and the relative thickness of the two splice plates should, as nearly RlIVET-WORK BRIDGES. 233 as practicable, be inversely as the respective distances of their centres from the centre of the spliced plate. For illustration; at the 6th node, the continuous plate is A", and the thinner spliced plate ", making in the two, a thickness of 1k", by 7" for the net width; giving a section of 101 square inches. This splice requiring 1}" rivets next the joint, to give the necessary rivet section, the net width of splice plates and continuous plate through two opposite 1i" rivets, is only 51". Consequently, the aggregate thickness required to give 10~ square inches, is about 1.91 "; and, deducting 0.625" for the continuous plate, we have 1.285"t for thickness of the two splice-plates. Then, representing thickness of spliced plate by a (disregarding the furring plate, or including it in the quantity a), that of the continuous plate by b, that of the two splice-plates by c, and that of the thicker one by x; we form the following equation, as will be obvious on reference to Fig. 46, which is an edge view of splice at node 6. xx (a+x) = (c-x) x (b+- (a+c —x); whence, the formula x = c x (a+2b+c). 2 (a+b+e). This formula applied to the case represented in Fig. 46, gives x = 0.7804", and c-x = 0.5046/. FIG. 46. The letter a in the diagram shows the splicing of a 1" with a /" plate, the thickness being equalized by a furring plate. Figure 46 gives also, a general idea of the splices proposed for this kind of chord, in case of the adoption of 30 234 BRIDGE BUILDING. the short splice plates and opposite rivets, as seen at BB, Fig. 45. p indicates the connecting pin (which, in the concrete plan of construction should be replaced by two opposite rivets, as seen in Fig. 44), having a cross-section in the parts passing through the chord plates, about equal to that of one of the two main diagonals connecting with each pin respectively, at the several nodes. The body of the pin between chord plates, should have lateral stiffness enough to withstand the stress produced by diagonals horizontally, estimated upon the principles of the lever, which will be greater as the distance of diagonals from chord plates is greater, and the contrary. If the bearing of the upright upon the pin be between the diagonals and the chord plates, as by a bi-furcation like that at the upper chord (see a Fig. 38) the body of the pin will usually require a section about equal to that of the two main diagonals connected with it. But this is no certain rule. The ends of the connecting pin should extend through the chord plates so as to receive a thin nut upon each end, and also the eyes of sway rods upon the inside end, in case that mode of connection be adopted for those parts. In the case of trusses without verticals constructed in rivet work, the best balanced action will be secured by connecting diagonals between the splice plates, by means of rivets through both, thus bringing each diagonal bar directly over each half chord, and producing uniform stress, as nearly as is practicable. When diagonal bars do not fill the space between splice-plates, the deficiency may be made up by furring plates, or thimble rings. RIVET-WORK BRIDGESo 235 Tension diagonals will usually require from 25 to 33 per cent of extra section to make up the loss in rivet holes. In thrust diagonals, no allowance need generally be made for rivet holes, as rivets properly distributed, will not impair the efficiency of the member in withstanding compression. With regard to the relative merits of this kind of lower chord, it requires, in the proportions above assumed, namely, 8" width of plates and 1" diameter of the smaller rivets, about 14 per cent of extra section on account of rivet holes, through the whole length~ For splice plates and rivets, at least an equal amount should be allowed, making 28 per cent for waste material, over and above the net available length and cross-section. The corresponding waste in the link chord, and in the eye-plate chord [cxiv], can scarcely exceed 10 per cent, when the connections are made with wrought iron pins. Hence, the advantage as to economy of material, seems decidedly in favor of the latter plans; and the cost of manufacture can hardly be estimated in favor of the former. If the riveted chord, then, have any claim to favor and preference, it is mostly owing to the fact, that being manufactured cold, it escapes the deteriorating effects frequently resulting to iron in the process of forging and welding, and the risk of flaws, and imperfect cohesion of the welded surfaces. How far this consideration should be regarded as an offset, or an overbalance to 15 or 20 per cent, of material lost in rivet holes and splices, further experience and observation alone can probably determine. 236 BRIDGE BUILDING. SWAY BRACING. CXXX. The primary and essential purpose of a bridge is, to withstand vertical forces which are certain, and, to a large extent, determinate in amount. We can estimate nearly the weight of a train of rail road cars, a drove of cattle, or a crowd of people; and the amount of material required to sustain them. But the lateral, or transverse forces to which a bridge superstructure is liable, are of a casual nature, depending upon conditions of which we have only a vague and general knowledge; and, can not predetermine their effects with any considerable degree of certainty. We know full well from experience, that it is always expedient to provide every bridge superstructure with means of support against transverse horizontal forces; and we introduce certain parts and members for that express purpose. These have been frequently alluded to heretofore in this work, under the designation of sway-rods, lateral ties, or lateral braces. But no attempt has ever been made, to the author's knowledge, to point out the proper sizes and proportions of such members, upon any determinate principles or data. In this respect, reliance has mostly been placed upon "judgment," and general observation as to precedent and common practice; as was the case in fact, with regard to bridge construction generally, until within the last twenty-five or thirty years. Within this period, and since the extensive use of iron in bridge construction has been introduced, more attention has been given to scientific principles, in adjusting the proportions of the several parts and members designed to withstand the effects of vertical pressure. SWAY BRACING. 237 The modern bridge builder, if he has been properly educated for his business, having arranged the outline of his truss, makes his computations, and marks upon each line of his diagram, so many thousand pounds of tension upon this, so many tons of compression upon that and so much shear strain, or lateral strain upon each rivet, connecting pin, or beam, and assigns to each place a member containing such an amount, and such a kind of material, as experience has proved to be sufficient to sustain the given stress with safety. Thus far, his course is scientific and sensible. But in arranging his system for securing lateral stability and steadiness, science can lend him but little assistance. He knows the wind will blow against the side of his structure; but whether with a maximum force of one hundred pounds, or as many thousands, he has no means of knowing with any considerable degree of certainty, or probability. He knows, furthermore, that every deviation from a straight line by a body passing over and upon a bridge, even to changing the weight of a pedestrian from one foot to the other (unless his steps be directly in front of one another, and this could hardly form an exception), is attended by more or less tendency to lateral swaying of the structure. Every inequality in the line of a rail road track, laterally or vertically, unless both rails have precisely the same vertical deviation, produces a transverse motion in the centre of gravity of the load, and consequently a lateral sway in the structure. The passage of a carriage wheel over a stick or a pebble, raising one wheel above the opposite one, changes the centre of gravity of the load to the right or left, and impels the structure in the opposite direction. 238 BRIDGE BUILDING. These are some of the external causes generating transverse action, and motion of the structure. But in addition to these, the upper chord itself, acting by thrust, is, at best, in unstable equilibrio, and liable at all times to exert more or less transverse action, and, if not kept in line by an efficient system of transverse bracing or tying, will lose its equilibrium, and be deprived of the power of performing its appropriate functions in the structure. Now, these disturbing lateral forces are quite small, compared with the vertical action upon the trusses; and, the vertical strength of the truss does not necessarily imply any power of resistance transversely; the tendency of the lower chord to preserve a straight line, being essentially balanced by that of upper chord or arch to buckle laterally;* provided the chords be so dependent upon one another that both must sway to the right or left at the same time. Hence, it is always expedient to provide some especial means for counteracting these lateral forces, which is usually done by the introduction of a system of horizontal diagonal ties or braces (small iron rods in iron, and the same, or timber braces, in wooden bridges), below the track or platform, in the horizontal panels formed by consecutive beams, apd the chords of opposite trusses. Also, when trusses are sufficiently high, diagonals and cross-struts are introduced between upper chords, to prevent lateral buckling. No attempt will here be made to assign specific stresses as liable to occur in sway rods or braces, based upon calculations from the uncertain and indetermin ate elem ents upon which the lateral action upon * The only truss known to the author, not liable to this lateral buckling, is the Whipple Independent arch truss, shown in Fig. 27. SWAY BRACING. 239 bridges depends. But, judging from experience and. observation, it may be recommended that iron swayrods be made of iron not less than 8 inch in diameter, for bridges of five panels or under, i inch from six to ten panels, inclusive. For twelve and fourteen panels, 4 inch for ten middle panels, and 8 inch for the rest; and, for sixteen the same as last above, with the addition of a pair of 1 inch rods in the end panels. These are the least dimensions recommended (in all cases exclusive of screw thread), for ordinary bridges with panels not much exceeding 10 feet. For panels approaching or exceeding 12 feet, 8 inch may properly be added to the above specified diameters generally. If upper sway rods connect in the middle of crossstruts, with a longitudinal reach across two panels, [see cxx, and Figs. 38 and 39], they may safely be made smaller than when they cross one panel only. The action of wind is nearly a uniform pressure from end to end of the structure, and causes much the same progressive increase of stress upon sway-rods, as the weight of structure and uniform load produces upon diagonals in the trusses- a fact which was recognized in assigning larger sway-rods at and near the ends of long bridges. But the casual impulses resulting from unevenness in track or platform, giving slight lateral movement to passing loads, and acting at single points here and there, this way and that, do not produce an accumulation of effect toward the ends. Hence, as it regards withstanding the latter forces, no variation in sizes of sway-rods is required. CXXXI. Sway-rods acting by tension would obviously draw the opposite chords toward one another, but for the resistance of transverse beams or struts, 240 BRIDGE BUILDING. while they also exert a longitudinal action upon the chords, thereby increasing or diminishing the stress upon chords, due to the action of structure and load. Chords, however, are usually proportioned without provision for increase of stress liable to accrue from action of sway-rods; and, from the small sizes of the latter, as compared with the former, and the obliquity'of their action, seldom expending more than half their direct stress upon the chords longitudinally, this small action may be neglected, as forming one of the contingencies for which a large surplus of material is always provided in chords, over what is actually required to withstand the effects of any probable vertical action. Certain modes of inserting and connecting sway-rods have been previously alluded to, sometimes with the beams by means of eyes and bolts [CVIII, Figs. 31 and 33], and sometimes more directly with the chords [cxIx, Fig. 35, d, and Fig. 39, d.] The best connection is that which gives the nearest approximation to central and uniform action upon all parts of the chord, and also of the beam or strut. The plan described in section cxx, and seen in Fig. 37, when admissible, affords a good connection for bottom sway rods. Undoubtedly there may be better devices for the purpose under consideration, as well as for other details, than any that have occurred to the author. But such as are herein described have mostly been put in successful practice, and are thought not to be seriously faulty. COMPARISON OF PLANS. 241 COMPARISON OF DIFFERENT PLANS OF IRON TRUSS BRIDGES. CXXXII It is the purpose of this chapter to canvass the relative merits of most of the several systems of IRON BRIDGE TRUSSING, which have claimed and received more or less of public notice and approval during the last few years; and of which the distinctive principles have already been discussed in preceding pages; though not in the precise combinations here about to be presented. We may take the number, lengths and stresses (the latter governing principally the required cross-sections), of the several long pieces or members of the truss, in the manner employed in the fore part of this work, as affording a near criterion of the comparative cost and economy of the bridges respectively. Then, after reference to such peculiarities as may seem advantageous or otherwise, leave the reader to his own conclusions in regard to the relative merits. THE BOLLMAN TRUSS, FIG. 47, Is founded upon the general principle discussed in sections xxII and xxIII, with oblique tension rods, and a thrust upper chord, in place of the thrust braces and tension lower chord as represented in Fig. 9. Let Fig. 47, represent a truss 15' high, and 100' long; or, in the proportion of 1 to 6. Also, let w represent the maximum variable load for each of the points c, d, e, etc., and w' (say,= w), the permanent weight of one panel of superstructure, supposed to be constantly bearing at each of said points. Then making W= w x w, we have W = weight sustained by ac. 31 242 BRIDGE BUILDING. Now, we have seen [VII], that the stress upon an oblique in such case, equals the weight sustained, multiplied by the length, and divided by the vertical reach of the oblique; and, assuming that the member requires a cross-section proportional to the stress, it follows that (making ab = 1), the amount of material required in ac, will be as the weight it sustains, multiplied by the square of its length. Hence, the material required in ac, must be as - W x aC2. Then, diminishing be until ac coincides with ab, W x ab2 becomes W, which is still proportional to the material required in ae (which has now become = ab, = 1), and, being replaced by M, representing the actual material required to sustain the weight W, with a length equal to ab (our unit of length)9 in a vertical position, we have only to substitute M x ac2 for W x ac' to know the actual material necessary to sustain the weight W (at a given stress per square inch of cross-section), with any length and position, retaining the same vertical reach, equal to unity. It must be obvious, therefore, that M, with the coefficient used before W, to express the weights respectively sustained by the several oblique rods in truss 47T will, when multiplied by the squares of the respective lengths of those obliques, show the amountgof material required in their construction, under the conditions above expressed. Let m = -M, and h = be. Then, we manifestly have, for material in the 14 obliques of the truss in question 7 (h2+1)+ 6m (4h2+1) + 5m (942+1 ) ~ 4m (16h2+1) + 3m (25h'+1) -+ 2m (36Bh+1) + im (49ht+1) = (336h2~ 28)nm, for those meeting at a, and a like amount for those meeting at 1; making a total of (672h'+56)m. But h= 0.694, which substituted in the last expression, gives 522.368m, = 65.296M. COMPARISON OF PLANS. 243 FIG. 47. BOLLMAN TRUSS. a,..s r q p o n m L b c d e f g i j k The thrust of the chord al, equals the horizonta. action of the 7 obliques connected with tither end. Making then x = V~, and h == be, = bk, it is obvious that each oblique carries weight equal to x x the number of panels not crossed by it, while its horizontal reach equals h x the number of panels it does cross. Hence, the horizontal action of each oblique, equals hx x the pro. duct of the numbers of panels at the right and left respectively, of the lower end of the oblique. The compressive force acting from end to end, upon al, then, must be equal to hx (7, + 2x6, + 3x5, + 42+ 5x3, + 6x2, + 7), = 84hx, = 10-Wx0.833, - 8.75W. Multiplying stress by length, and substituting M, we have 8.75 x 6.66M = 58 M = materialrequiredinal, at a given stress per square inch ofcross-section; M being the amount required for a unit of length (ab), to sustain the unit of weight (W), at the same rate of stress. Add 7M for two end posts, with length equal to 1 and bearing weight equal to 7W, and we obtain 65KM as a total for thrust material in long pieces, not including 7 intermediate uprights, not properly to be classfled with other parts, as their action is merely incidental, except that of supporting the weight of upper chord. The parts above considered, mainly determine the character of the truss as to economy of material. 244 BRIDGE BUILDING. Other parts, such as short bolts, nuts, connecting pins, &c., although just as essential, are comparatively, of small amount and cost, except the intermediate uprights, which will be referred to hereafter. If the truss be used in a deck bridge, and the end posts be replaced with masonry, the intermediates will sustain the same weight as the ends sustain in a through bridge, thus giving the same representative of material as above found. THE FINCK TRUSS, FIGO 48, CXXXIII. Possesses several of the characteristics which distinguish the Bollman plan. Both dispense with the bottom chord, which is common to most, if not all other plans of truss, for both iron and wooden bridges. Both also employ a pair of tension obliques acting in horizontal antagonism to each other, at each of the supporting points c, d, e, &c. But while in the one, the members of each pair of obliques are of equal length and tension, in the other, the pairs consist of unequal members (except at the centre), as the diagrams will sufficiently illustrate. It will readily be seen that Fig. 48 exhibits three classes of obliques, consisting respectively of 2, 4, and 8 members tothe class. Supposing a trussof the same dimensions and proportions, and subjected to the same load, as in case of Fig. 47, and using the same notation, as far as applicable; it is manifest that each of the 8 short obliques, sustains 1W. The 4 next longer sustain upon each, a weight equal to W - one half directly, and the other, throiugh the short obliques and uprights. The two long obliques sustain 2W each, being the half of 1W, received directly atf, and 1 and 2 respectively COMPARISON OF PLANS. 245 through the upright, from members of the other classes, meeting at the point p. The material required for all the obliques, then, (ab being= 1, and be= h), is 8 x (h2 + 1)+ 4 x 1(4h +1) + 2 x 2 (16h2 + 1)M, being the number of pieces in each class multiplied by co-efficients of W in weights sustained, and by squares of length respectively, and the sum of products multiplied by M. Subsituting in the above expression the value of h2, (0.694), and, reducing and adding terms, we derive material in obliques = 70.296 M. FIG. 48. FINCK TRUSS. a s r q p o n m I o'-~le d e J'g # z k The compression upon the chord a 1, is equal to the horizontal action of one member of each class of obliques, communicated at each end; that is, equal to (I h + 2h + 8h)W, = 10 h W; and, multiplying by length (= 6.66), and substituting 0.833 for h, and M for W, we have (10.5 x.833 x 6.66)M - 58.k M, to represent the material required in al; - the same as in case of Fig. 47. The uprights of the Finck truss obviously sustain 12W, namely, 31 at each end, 3 in the middle, and 1 at each of the quarterings, r and n. But, in comparing this with the Bollman truss, it seems fair to offset 6 uprights, not including the end and centre ones, in the Finck, against 7 in the Bollman truss not estimated; hus leaving 10M for uprights in the former, making 246 BRIDGE BUILDING. a total of 681M9 for compression material, excepting the 6 intermediate uprights, excluded as above. Both of the above considered trusses exhibit a beau" tiful simplicity, and facility of comprehension in principle, and they will be left for the present, for a discussion of the POST TRUSS. CXXXIVo This, like the two preceding plans, is designated by the name of its distinguished designer and publisher, S.S. Post, Esqr., of Jersey City. Figo 49 gives a general view of the only specimen of this truss which the author has had an opportunity of examining. It is a sort of compromise between the trusses represented by Figs. 18 and 19, of which the object sought appears to have been, to obtain a nearer approximation to the most economical angle of inclination for both thrust and tension members (between chord and chord), by inclining the latter at an angle of 45~, and the former at a less angle with the vertical. These are both favorable conditions, considered alone and by themselves, as we have already seen [Lxv and LXVI]; and it is proposed to compare the economy of this particular arrangement, with that of a truss having vertical posts, with oblique tension diagonals; as well as with other plans, preceding and succeeding. Assuming the same length and depth of truss, and the same load, both constant and variable, as in the preceding cases, acting at the points x, v, u, &c., let w represent the greatest variable load for the length of one panel, and w' the weight of superstructure bearing upon one truss, for the same length, supposed to be concentrated at the nodes of the lower chord, and assumed to be equal to Jw. Also, let I equal the verti COMPARISON OF PLANS. 247 cal depth of truss (between centres of chords), and let tension diagonals incline 45~0 and posts lean 1 horizontally to 3 vertically; the space between posts being two-thirds of the depth of truss. FIG. 49. r. _Post's Truss z b G d e, g I x v u u r g j o m Then, omitting counter ties up to tf, from the left, as neutralized by weight of structure; we see that the weight at x, being only 4 as great as at the other nodes, on account of the short space xy, 3w. 80 (or 3w", substituting for the occasion, W" for w - 80), represents the proportionate part of that weight, tending to bear upon the abutment at m; and this, with 12w" for weight at v, and 20o" for weight at u, + 28tu" for weight at t, makes 63w" accumulated upon if, when x, v, u and t alone are loaded. Now, the action upon this truss is less certain and determinate than where the thrust pieces are vertical, or inclined equally with the tension pieces. But supposing that the weight of superstructure at s, or at s and r together, neutralizes, or reflects back a part equal to w', or ~80w", = 27w" nearly, of this 63w", we have abalance of 36&w', as the maximum weight for tf. Then, whether this 63w'* which must go to the * This full amount 63e" is used here; for, although it is assumed that only a part of it is transmitted through tf, the balance is restored from weight of structure which otherwise would pass to the abutment at Y. 248 BRIDGE BUILDING. abutment at m, in virtue of the loads at x, v, u and, is transferred throughfs to sg, or throughfr to rh; or whether it is divided equally or unequally between the two, is not quite obvious. But assuming, as what might seem probable, that it is transferred in equal portions to sg and rh, in that case, sg sustains as a maximum, 36w" for weight at s, + half of 63w", makingsay 67w"; supposing that sg and re sustain none of the weight of structure; which, though probably not strictly true, will not materially affect the result. Again, (we are now considering the nodes at the lower chord as being loaded successively from left to right), the weight at r gives 44to" to rh, in addition to, say 32w" tending to be transmitted from lf; and w, or 27w" for structure, making 103w". For maximum weight on iq, there is due to movable weight at q, 52w", + 67w" from sg, +27wv" on account of weight of structure, making 146w"; while pk sustains (60+103+27)w", = 190w", and ol sustains (68+ 146+27)w" = 241" The maximum weight upon nl, is made up of that ofpk + (w+w') at n, = 270w". Having thus determined the maximum weights which these diagonals are respectively required to sustain, disregarding some small matters of uncertainty, of little practical importance, we find the sum of these maxima, for the 6 pieces parallel with ue on the right, to be 783w", = 9.7875w. Then, multiplying by 2 (the square of the common length, ay being = 1), and substituting M' for w ( as M was substituted for W in the preceding cases), we derive 19.575M' = material required for the 6 pieces in question. Add to the last amount 3.7M' for the steep diagonal nl (being the square of length by weight sustained, and w changed to M'); COMPARISON OF PLANS. 249 and we have the whole material for tension obliques in the half truss; which doubled, exhibits for that class of members in the whole truss, 46.55x'; omitting 6 counter ties, not required to sustain structure or load, and the value of which will be considered (in general) hereafter under the head of counter bracing. The short section nn of the lower chord, has no determinate action. The section no has a tension equal to j of the weight acting on nl and kn, under a full load of the truss, equal to i the weights upon r, p and n, for nl, and I of those at r and p for /kn; the whole equal to i x2j (w+w')+~ x2 (w+w'), = 2.11w. To this, the diagonal ol adds at o, 2 (w+z'), and io adds ~ (w+w'), making 5.22w = tension of op; while a like addition at p, for the action of pk and hp, shows 8.33w for pq. Again, qi adds at q, w+w', equal to 1.33w, while rh contributes a like amount at r; making for qr and rs respectively, a tension of 9.66w, and 11w, restoring neglected fractions. It is probable that a small decussation of forces through re and sy, under a full load of the truss, would modify these stresses slightly, but not so as to produce a material difference in the final results of the present discussion. Summing up the stresses thus determined for different portion of the lower chord, counting like strains upon corresponding sections, and deducing the required material (as above done with regard to diagonals), remembering that the length of sections equals 2 of unity, we obtain 41.1M' = material required in lower chord. This added to 46.55M', the amount above determined for obliques, gives tension material for the whole truss, equals to 87.65M'. 32 250 BRIDGE BUILDING. Now, it is manifest that the quantity here represented by u', has the same ratio to that denoted by M in the estimates of material for trusses Fig. 47 and Fig. 48, as the weight w in the former case has to the weight W in the latter. But W was used to express I of the gross load of the truss, while w represents only - of the variable, assumed to be equal to I of the gross ]oado Therefore w: W:: x-': ~; whence, w = 0.66W; and M' =.6M. This equivalent substituted in the expression 87.65M', gives 52.59M = tension material for the post truss. The maximum weights sustained by the thrust braces, equal respectively those borne by the tension rods communicating such weights, and for the 5 pieces on either side of the centre, the amount is equal to w" x (36+67 + 103 + 146+ 190) = 6.77w, which doubled, gives 13.54w for the whole of that class of members. This aggregate weight, multiplied by the square of the common length of pieces (1.11), with w changed to Ml produces 15.02i = 9.01M. The end section (kl) of the upper chord, sustains compression equal to the weight upon ol and 4 of that upon nl, under a full load of the truss, = 2 (w+w), +4x24 (w+w'), = 3.88w. Add 2 (w+w') for weight on pk, and 4 of that amount for that on kn, and it makes a compression of 7.44w upon ki. Again, adding w+w' ( = 1.33w) for action of qi, and X of the same for that of io, makes 9.22w for compression of ih, while a like addition for action of rh and hp, makes 10.99w = compression of hgy and gf. We may call the last stress 11w, as some fractions have been neglectedo The above amounts of stress upon the several sections of the half chord, added together and doubled to COMPARISON OF PLANS. 251 represent the whole chord, and multiplied by the length of section (-), produce 56.72w,= 34.03 W; whence, material for top chord = 34M; very nearly. The two end posts obviously sustain the gross load of the truss (deducting what comes upon one half of the short spaces mn and xy), which equals 91 (w + w'), = 12.66w; and, the length being 1, the material equals 12.66M' = 7.6M. Summing up the amounts thus determined, of material for the several classes of thrust pieces, we have: For Braces, or inclined posts,......... 9.01Mo 1G Upper Chord,...................... 34.00M. 6 End Posts............................. 7.60M Total, for Thrust,.................... 50.61M 4 6 Tension,........... 52.58M. WHIPPLE'S TRAPEZOIDAL TRUSS. CXXXV. The distinctive characteristics of this plan are, an Upper Chord made shorter than the Lower, by the width of one panel at each end, giving to the truss a Trapezoidal form — dispensing with nonessential members, and proportioning the several parts in strict accordance with the maximum stresses to which they are respectively liable; principles and devices first promulgated in the original edition of this work, and applied by its author in the construction of trusses with parallel chords, with or without vertical members. Truss Fig. 50 has vertical posts and tension diagonals; and, using w and w' to denote the same quantities as in the last preceding case, and pursuing the method explained with reference to Fig. 18, [LVI], we have the maximum load for 3/5 equal to 4w" - w' 252 BRIDGE BUILDING. (making w" e w divided by the number of panels 0.1), =.4w- -w, since w' = w. For 4/6^ we have.6w, without increase or diminution on account of structure; while, for the 3 next diagonals on the right, we have successively, 9w + -w', 1.2w + w' and 1.6w + l1zwt, making altogether 3.7w + 3 w', = 4.7w; showing for the 5 pieces, 5.53w. This being doubled and multiplied by square of length (2.775), and w changed to m', gives material for 10 long diagonals = 30.69M'. FIG. 50. 1 2 3 4 5 6 7 8 9 1 2 4 6 9 12 16 20 25 2 1 1 2 The two steep diagonals togther, sustain 4 (w + w'), = 5w, which, multiplied by square of length (1.44), produces material = 7.68M'; while the two tension uprights manifestly require 22M'. We have consequently, material for the system of tension obliques and verticals = 41.03M/. The end brace obviously sustains 41 (w + w'), and exerts a horizontal stress = 4w (two-thirds of the weight borne), upon the two first sections of the lower chord. The steep tension oblique adds 2 of weight borne, making 5.76w for the next section, while the two succeeding diagonals toward the centre, adding 1~- times the weights borne successively (under a full load of the truss, of course), give 8.42w and 10.19w, for tension of second and first sections from centre, respectively. Then, adding, doubling, and multiplying by length of section, we obtain, material for-lower chord = 43.16m'. COMPARISON OF PLANS. 253 Add to this the amount for diagonal system as above found, and we have the whole amount of tension material for the truss = 84.18M' = 5.05M. The maximum weights sustained by obliques, and by them transferred to 7 thrust verticals, being in the aggregate = 6.62w, the length of members being unity, need only the substitution of M, to express the required material for said verticals; which, reduced to terms of M, equals 3.97M. The first and second sections of the upper chord, obviously sustain the same action respectively, as the fourth and fifth of the lower chord while the 4 middle sections of the former, receive the additional action of diagonals 3\5/7 (upper figures), under full load. Hence we cipher up, material for upper chord - 32.6M. The end braces, sustaining 9 (w+w') - 12w, with a length whose square is 1.44, obviously require material (12xl.44)M' - 10.37M. The truss, then, requires thrust material, for upper chord, 32.6M, for end braces, 10.37M, and for uprights, 3.97M; making a total for the truss, of 46.09M. Tension material as above, total 50.50. TRUSS WITHOUT VERTICALS. CXXXVI. Assuming a truss (Fig. 51), of same length, depth, and number of panels, and same load, variable and constant, as in the two cases last considered, with diagonals crossing one panel only, we have nearly the Isometric Truss,* adopted by Messrs. Steele and McDonald. Arranging the numbers over the diagram, as in Fig. 51, and using the process explained [XLVII, Fig. 19], it * In the Isometric, the diagonals incline at 30~, while in Fig. 51 they incline nearly 34~. 254 BRIDGE BUILDING. will be seen that either end brace, and the obliques parallel therewith, are liable to maximum weights as follows, preceding from end to end. FIG. 51. 1 2 3 4 5 6 7 8 9 1 2 4 6 8 12 16 20 25 25 20 16 12 9 6 4 2 1 Compression. Tension. End Brace 4.5(w+w') =6.000z.t Oblique No. l.................. 2.100 6 66, 2.................. 1.533 66 6 3...........o.... 1.066.233w 66 4......o............ 1.600 6.600'~ G6 " 5...................233 1.066 6 66 ~ 6.................. 1.533 1G G6 G6 7 *.................. 2.100 ~ SS 6G 8.................. 2.666 4 Totals 11.533w 8.200wo Then, doubling for the two sets, multiplying by square of length (1.44), and changing w to Mr, we have, to represent material....for compression 33.215M, tension 23.616M,/ The end brace, sustaining 4.5 (w + w') = 6w, exerts a tension of 4w upon the end section of the lower chord. The next brace sustains 1- (?w +') = 2w, making a tension of 5.333w for the second section. The tension and thrust diagonals meeting the chord { The small thrust action which the movable load tends to throw upon 6, 7 and 8, and the small tension upon 1 and 2, are neutralized by weight of structure. COMPARISON OF PLANS. 255 at the next node, sustain together (under a full load of the truss), 3 (w + w' = 4, adding 2 of which, gives 8wt= tension of the 3d section, while 2-w borne by the obliques meeting at the next node, makes a tension upon the 4th section equal to 9.777w; and 1Wl at the next node (the tension diagonal only, being in action, under a full load), gives for tension of the 5th section, 10,666w. Adding the stresses of the several sections of the halfchord, doubling, multiplying by the common length (2), and changing w to M' shows material for lower chord = 50.37M,. The end section of the upper chord sustains thrust equal to 2 x (weight on end brace, (= 6w), + weight on tension oblique meeting said brace), = 2 8.666t = 5,77t. The two obliques meeting at the first node from the end, sustain together 4w, adding 2.666w to the above, and making a compression of 8.444w upon the second section; while succeeding diagonals make the stresses of the 3d and 4th sections, 10.222w, and 11.1z respectively; whence, by process already employed and described, we derive: Material for upper chord =........ 47.392M' 28.435M Add for end braces,............... 17.28 " = 10.368"' "G other obliques,............. 15.935' = 9o561" Total for compression material, 80.607M' = 48.364M Tension, chord,............. 50.37M' Obliques,.................... 20.616 V erticals,.................... 5 -~- ~78.986M' = 47.391M. Grand Total, 95.755M. 256 BRIDGE BUILDINGO THE ARCH TRUSS. CXXXVII. A parabolic Arch Truss of the same length, depth and load as allowed in the five preceding cases, and having 9 panels, will compare, as to representative of amount of material, as follows: Let w, represent the variable, and wu,, = ~w,, the permanent panel-load. Then, taking the greatest depth of truss (15f.), as the unit of length, as before, the length of chord will be 6.666, and the verticals respectively 1, 0.9, 0.7, and 0.4. The length of panel (11.111. ), being divided by 15f, (the unit), gives 0.74074. Hence, tension of chord =.74074 4 (w, + w,,) x 4. == -1 x 7.4074w, which, multiplied by length of chord (= 6.666), and w,, changed to M,, gives representative of material = 9.8765 x 6M, = 65.843m,; in which M, is the unit of material, proportional to the unit of length (15',) x unit of stress, w,. The maximum tension of diagonals, as determined instrumentally by process explained [xxvii, &c.,] varies from 1.11w,, to l1w,; and, taking the highest, multiplying by the aggregate length (15.4), and changing w, to M, we obtain material = 20.52M,. The verticals sustain tension, each, = 1-w, with an aggregate length of 6, giving material = 8M,; making a total of tension material = 94.376M,. The horizontal thrust of the arch, must be in all parts the same as the tension of the chord (at the maximum under full load), and it is manifest that the material for each segment, must be to that of the middle segment, as the squares of respective lengths to unity; that is, equal to material in said middle segment, multiplied by squares of respective lengths. COMPARISON OF PLANS, 257 But the representative for the middle piece equals'th that of the lower chord, = 7.316M,. Hence, this amount multiplied by the sum of squares of all the others, +1 for the middle segment,found to be 9.058+1, =10.058? gives, to represent material for the whole arch, 73.584M,. Then, the vertical members are liable to be exposed to compressive action, represented by the small amount of 2.058M,, which added to the above, gives a total of compression material, equal to 75.642M^. Now, the factor Mi, here used, is to the factor M used in the preceding cases, manifestly, as Ix -, to., as l: 8 whence, 12M, = 8M; and we reduce the coefficients of M, by 3, and change MI to M, to bring the last results to the same standard measure as in the preceding. Effecting these changes, we have, for tension material, Chord 43.895M, + Diagonals 13.689M + Verticals 5.333M, equal to a total of62.917M. Forcompression, Arch, 49.056+Verticals, 1.372, = 50.428M. SYNOPSIS OF PRECEDING DEDUCTIONS. The following tabulated statement may promote the convenience of comparison: Trusses. Material required expressed in Ms. Compression. TenCmsioon. Comp. Grand Designated. total.. Total. Total. Chord. Ends. Posts, &c. Bollman,. 65.296 58.338 7.000 65.333 130.629 Finck..... 70.296 58.333 7. 3.000 68.333 138.629 Post,.... 52.590 34. 7.6 9.01 50.610 103.200 Whipple, 50.500 32.6 10.37 3.97 46.94 97.44 Isometric,. 47.391 28.435 10.368 t9.561 48.364 95.755 Arch...... 62.917: 49.056 1.372 50.428 113.345 * Actual, but not a determinate quantity. 1, Thrust Diagonals.: Arch. 33 258 BRIDGE BUILDING. CXXXVIII. The figures in this table are to be understood in all cases as prefixed to the quantity M, which, as far as relates to tension material, represents a determinate amount of worotght iron; while, as it relates to compression material, M represents an amount of cast or zwro ug/t iron, varying as the forms and proportions of parts vary. But, in the present discussion M may be assumed to have a uniform value in ex pressions relating to material under the heading of chords; and of ends, whether oblique or vertical. The quantities under the head posts, require in general, probable twice as high a value for M, as th-at required for the other classes of thrust members, as it regards all but the first named truss, while the first is not represented in that column at all, although the parts there referred to are as indispensible, practically, and require nearly as much material as corresponding parts in the other plans. With regard to plan NTo 2 (the Finck), 6 posts actually required (two of which, at the quarterings, sustain determinate weight equal to W each), are also omitted in the table, to place this plan upon an equal footing with the preceding one. There is also a consideration with regard to the effects of load upon these two trusses, especially the first, which render it pcarticlly necessary to use diagonal ties, or " panel rods" in the several panels; and such have usually been introduced wherever such bridges have been constructed. As any one pair of suspension rods in the Bellman truss may be under full load, while the others are without load, the loaded node would, in such case, be depressed, while that o'n either side would retain nearly its normal position. Thus would result an obliquity COMPARISON OF PLANS. 259 in panels adjacent to the loaded point, and consequently, a tendency to kink in the upper chord, by opening the joint above the loaded point upon the under side, and the next joint either way, upon the upper side. Hence the compression of certain chord segments would be thrown upon the extreme upper side at one end, and the lower side at the other end. This would be decidedly an unfavorable condition, which the panel rods are used to obviate by distributing the load of loaded points over adjacent, and more remote parts of the truss. Otherwise, the bridge would act under a passing load, somewhat in the manner of a pontoon bridge. By estimating a reasonable amount of material for posts and panel ties, the figures in the table, opposite the first two trusses would be materially increased. Hence, it must be obvious that the necessary material for the two above named trusses, is not so fully represented in the table, as in the case of the other four; with regard to which- assigning proper values to M in the different columns of the table, and assuming the members to adhere to one another as firmly as the different portions of each cohere among themselves, a complete truss would be formed in either case (of dimensions as above assumed), sufficient to be used in a bridge required to bear a gross load equal to 4 times the weight of superstructure; provided the proper ratio of safe variable load to weight of structure be as 3 to 1; as is nearly the case with regard to a 100 foot bridge.* M, in the preceding table, represents a piece of iron, 15t long susficient to sustain with safety, a weight W, equal to i of the gross maximum load for one truss of a 100ft. bridge. Allowing 1,0001bs. to the linea! foot for movable, and 3331bs. for permanent load, W, represents i X 133,3331bs.= 16,6661bs. Then, reckoning the safe stress of 260 BRIDGE BUILDING, In such case, the results already obtained, would show the releative cost of the several trusses (exceptingfl the first two)'with almost absolute exactness. But, as the parts of a truss can not be so connected and welded into a single pie-ce, without enlargements at the joinings, by any skill or process now in use,'we have to include as an item of cost, in all plans, a considerable amounlt of material above and beyond the net lengIths and cross-sections, as here before determined with regard to the trusses under discussion, requiredl for the lapping of parts, screws and nuts, eyes and pins, &c., to form the connections of the different memtbers with one another. Witth regard to the trusses under comparison, no obvious reason presents itself, why any one should require a percentage of allowance -for connections ma.terially greater than another. Leaving out the two filrst as perhaps already sufficient-ly tdiscussedl t-he others consist of about the same numnber of necessary memlbers, and with the exception of the arch truss, aedmit of nearly the same forms and connections of parts. The Isometric, or Tra pezoid without verticals, presen-ts the fewest lines in the dicagram- but some six of those lines represent both telsioln ad tirust memi. bers eithe sepatr, ate, or combined, which p'obah bly coimplicactes the iron (thrusit or tension), at — sy 10,001bs, to the inch of cross-sec;ion, it talkes lI- sqnuare incihes to sustain the wn eg!0ht s; bein n- abutoe ibs. to the foot, or 8A-Albs. for 15'. Thisi, increase d by - j sa, 0 er ci. for extra nmaterial in connections, oives the practical value 0of1 w* hich, nultipliled by the co-eOLicient of M'] in t1h table, produoce apl)roximately, the respelctive Awei'hts of tlruses. N ow, 1 X.37 0= lOIlbs. which multiplied by I138.38, the coefnicient for the Arch trusst,nives for the weiollt of that truss, 1,1.47 1ib. Ad i for 10 feet vwidth of platformi (with wooden bea ms), -- say,O>ft. b.,. of timber and plank, equal to about 30,000 lbs., and -ea 1)h 03.t71's. to represen' t the rpenoama bnent- load of lte truss. Bht we hv0 assumled a truess proportion0 to sustain with safety 13,3831b1, w, ic is a liittle lmore than 4 tims the weight of structure Lhere,aove estimlatedl as supported by the truss. COM:PAIISON OP PLANSo 261 details of connection, quite as much as the extra three members in truss Io. 4. The Post truss presents the larger number of acting members, even omitting six counter ties seen in the diagran, with apparently no advantage as to modes of connection. Both the Post and the isometric have 10 members represented in the 4th column of the table, whereas the WVhipple truss has only 7, and these the shortest of all; and, as the material in these parts manifestly acts at a disadvantage, they being comparatively long and slim, and sustainingr slight action, any excess in their number, would seem to be unfavorable to economy. It is believed, however, that the Post truss would be improved in economy by reducing it to a trapezoidal contour, as, for instance, by removing the parts outside of bx and kn (Fig. 49), and changing the tension pieces av and ol for others connecting b with v, and o with k; thus converting the figure to a trapezoid very similar to that of Fig. 50; and, by striking out one panel from the latter, and arranging parts as in Fig. 20, except as to inclination, the relative merits of inclined and vertical posts, as represented in these two plans may be fairly tested. Analysis of trusses modified as just indicated, show tension material slightly in preponderance with the vertical, and thrust material a little the greater with inclined posts; the average being about one per cent greater in the case of vertical posts. This balance though trifling in amount, is upon the side where it was to be looked for, in view of the result of investigations had with reference to Figures 12 and 13 [xxxIx-xLVI], as well as the case of the Isometric. Both the Post truss and the Isometric, as to principle of action, may be classed with Fig. 13, where 262 BRIDGE BUILDING. weight is transferred from oblique to oblique, and not from oblique to vertical, and the contrary. The same may be said of truss Fig. 15, sometimes called the Triangular, in which verticals are used merely to transfer the action of weight from the point of application to the connections of the obliques; after which, the weight has no action upon verticals. Now finally, we see by table of results, that if the Post truss be changed to the trapezoidal form, as above suggested, it will occupy a position, as to amount of material, or more strictly speaking, the amount of ae. lion upon material, between Fig. 50 and Fig. 51; which latter differ from one another less than 2 per cent; a, difference, which would undoubtedly be increased somewhat, under different general proportions of trusses. For instance, while Fig. 50, shows an inclination of diagonals used in connection with verticals, probably nearly approaching the optimum, Fig. 51, though superior to the true Isometric (with angles of 60~), in the greater inclination of its obliques, would give still better results with an inclination of about 40~. CXXXIX. On the whole, we must look to other quarters than the amount of action upon material, for plausible ground upon which to found a decided preference for either of the three plans in question. A difference of two or three per C., and even more, may easily result from greater or less facility of constructing and erecting the structure, while a regard for appearance may also be worthy of consideration. Hence, Engineers and builders will adopt one or another plan, according to individual taste and judgment, and the one who carries out the principles of either system with COUNTER BRACING. 263 the greatest skill, and the best materials and workmanship, will probably produce the best bridge. Judging from the preceding tabulated statement, the arch truss seems, prima facie, to labor under a somewhat formidable disadvantage in the fact that it shows an amount of action upon material 10 or 15 per cent. greater than the three preceding plans just es. pecially referred to. But for the light of experience, we might be led to discard the plan without a trial. But, having chanced to be the first plan of iron Truss successfully put in use, and having had its capabilities fully tried and demonstrated, before any formidable competitor appeared in the field, it could not be dislodged from its position, until a rival plan could not only theoretically, but also practically demonstrate its superior claim to public favor. The result has been such as to show that even a very considerable excess of action upon material, may be overbalanced by more advantageous action of thrust material, and greater simplicity and facility of constructtion; insomuch that the Whipple Patent Arch Truss, with trifling modifications from the original pattern, has competed successfully with all other plans, for the class of structures it was originally designed and recommended for (common bridges of 50 to 100 feet), during more than a quarter of a century, which has been fruitful in efforts at improvement in iron bridge construction. COUNTER BRACINGU The elasticity of solid materials, is manifested in bridge trusses, by their downward deflection under 264 BRIDGE BUILDING. load, and the recovery of their previous form and position on the removal of the load. This arises principally, from the temporary elongation of parts exposed to tension, and the contraction of those exposed to compression, according to laws and principles supposed to be understood. The deflection of trusses within the usual limits, whenl properly proportioned, is not essentially detrimental to their safety and durability; but rather enables them the better to resist sudden impulses,except in case of a regular succession of impulses, at intervals corresponding with those of the natural vibrations of the structure, or with some multiple or even division thereof; a result frequently noticeable, and sometimes, to a degree somewhat unpleasant to the eye, as well as suggestive of danger. Hence, great emphasis is often employed, in expressing the supposed advantages of "counter bracing," as a means of stiffening trusses, and preventing, or diminishing their vibration. What is technically called G" counter-bracing," as applied to bridge trusses, is the introduction of a set of diagonal, or oblique pieces or members, to act in antagonism to the main diagonals, whether acting by tension or thrust, which contribute toward sustaining the weight of structure and load; the object being, to retain in the truss when unloaded, more or less of the deflection produced by the load, when the truss is loaded. My object at the present time is, to exhibit the process and results of my investigations as to the theory and effects of this counter-bracing, as usually practiced in bridge building, and to state the conclusions arrived COUNTER BRACING. 265 at, as to the value of counter-braces, towards effecting the object proposed. FiG. 52. a b _e f q h i ~ r q p o n m I k I assume a truss (see Fig. 52) composed of horizontal chords (of equal lengths), at top and bottom, vertical posts, and diagonal tension rods, inclined at 45~, or at any other given inclination,-the truss being uniformly loaded from end to end, and so proportioned that all of the above named parts, in that condition of the load, shall undergo an amount of extension or compression, proportional to the respective lengths of parts, multiplied by a constant factor (E), equal to the elastic change effected in a length equal to that of the uprights between centres of chords, which is assumed as the unit of length for the occasion. Then, let L represent the length of truss, P, the number of panels, H, equal to L- P, the horizontal reach of diagonals, and D (equal to 2LE,), the difference in length, occasioned by extension of lower, and compression of upper chord, Now,9 assuming no change in lengths of diagonals and verticals, it is manifest that the chords assume, in these circumstances, the forms of two similar and concentric arcs of circles, of which the difference in length is to the mean length, as the difference of radii is to the mean radius, _R But the difference of radii manifestly equals the distance between chords, equal to 1. Using, then, the representative signs before adopted, we have D: L: 1:; whence...... R= L D. 34 266 BRIDGE BUILDING. Now, the depression at the centre of the truss, is evidently equal to the versed sine of half the arc made by the chords, and is found with sufficient nearness, by the equation. D. ep. -(L) 2B, = L2 B-. Then, substituting L - D for R, we have dep.= L2 -L (L D), = DL. Hence, if the length of truss equal 8 times the depth, or 8, the deflection due to this cause, will equal the difference in length of the two chords, produced by their extension and compression. Again, if length equal 6, then, dep.= D x 6, = 3D - 4 -= 9E. The depression resulting from extension of diagonals, may be illustrated as follows. If the points a and b of a rectangular panel abed (Fig. 53), be fixed, and ac be extended by an addition FiG. 53. equal to eh to its length, produced by a d the action of weight at c, either di\ g rectly, or through the upright dc; the points d and c will fall to g and A, and the very small triangle ceh (eh b \\ representing only the elastic stretch 1 of ac), will be essentially similar to abe " whence, ch: ac:: eh: ab, and eh. E:: ac: ab,: (1 + E2) 1... Therefore, eh = E/ (1 + E2). But ch: (1 + R2):: eh:,: EV (1 + 12): 1; consequently,..h c= E + EH,2. Now, if this represent one of the end panels of a truss, all parts of the truss between the end panels, must descend through a space equal to ch, in consequence of the extension of diagonals in the two end panels; and so for each succeeding pair of diagonals, to the centre of the truss. Therefore, the depression COUNTER BRACING. 267 in the centre, due to the stretching of diagonals, must be equal to ~P x (1 + H)E, =- (P +I- PH2)E. The depression in the centre of the truss, due to compression of uprights, is simply equal to the aggregate compression of all the uprights on either side of the centre one, and consequently, equal to P'E. This amount added to that produced by extension of diagonals, as above determined, makes, for uprights and diagonals together, a depression equal to (P + PBH 2)E. The value of this last expression, length and depth of truss being the same, varies slightly with variation in number and width of panels, but not so as to be a matter of practical importance. Assuming L = 8v, = 8, we find that, P =2, and P = 16, make (P + Pl2)E 18E. P=4 andP= 8 " =12E P =, makes " = 11E. 6 panels, therefore, seem to produce the least deflection. The deflection resulting from changes in lengths of chords, has been shown to be equal to ILD; and, substituting PH for L, & 2PHE for ), we have.. LD= IPP2g2E, =deflection from change in chords. The term E, then, with the following co-efficients, expresses the depression at the centre of the truss, resulting from all changes in length of parts, namely, for chords, iP2]2, for diagonals,...... P+APl2, and, for uprights, IP. Hence, the deflection of a truss, under the conditions here assumed, depends upon three simple elements, represented by the letters, P, H, & E; and is expressed in the following general formula; Deflection- (I P212, + iP + 2 P12, + ~ P)xE iV vuvv U 2 2 S2688 BRIDGE BUILDINGo The parts of this aggregate co-effcient of E, referring respectively to chords, diagonals, and uprights, are separated and distinguished by commas. The formula just given is equally applicable in case of thrust diagonals and tension verticals; as will be made obvious by a moment's examination of the principles involved. Now, if the truss could be anchored down, by ties and anchorage absolutely unyielding, to the point of its utmost deflection under load; the load might be removed, and replaced, without any rising or falling of the truss,-the load and the anchors alternately retaining the deflection, and preserving a constant and uniform strain upon the truss. The same effect is partially produced by counterbracing; and the object of the present investigation is, to determine, approximately, at least, to what extent this may be done, and what is the real advantage of counter-braces, in trusses with parallel chords; beyond where they are necessary, to counteract the effects of unequal variable load, upon the different parts of the truss. \We have seen that deflection results from three causes, all, of course, depending upon elasticity; namely; difference effected in lengths of —first chords, second, diagonals, and third, uprights. The theory of counter-bracing is, that by the introduction of antagonistic diagonals, the material is prevented from regaining its normal state on removal of the load; and consequently, that it yields to the reimposition of load, to much less extent than it would do, in the absence of counterSo As to the deflection due to the difference in lengths of chords, equal, as shown by the general formula one COUNTER BRACING. 269 page back, to one-half of the whole, for a truss in which L = 611, and to more than half, when L is greater than Gil; the counter-diagonals have no tendency to retain or diminish that difference, or the deflection produced by it. The diagonals and counters, simply contract or extend (according as they act by tension or thrust), the two chords equally, without affecting the difference between the two. On the contrary, the action of the counter diagonal tends to retain the tension (or thrust, in case of thrust diagonals), of the main in the same panel, and. also, the compression (or tension), of uprights; and, in as far as that is accomplished, the deflection due to the elasticity of those parts, is retained, on removal of load from the truss. Suppose, in a truss with tension diagonals, loaded and depressed as already explained, and all parts extended or contracted to the amount of E x respective lengths; a counter-diagonal to be inserted in each panel, crossing the mains, as shown in the diagram (Fig. 52), and of half the size of the latter, such being the usual proportion for counters. Now, the counters being adjusted so as not to act while the load is on, but ready to act immediately, as the main diagonals begin to contract, then, the load being removed, the main will contract by its elasticity, opposed by the counter, until they come to an equilibrium; each sustaining the same amount of tension. Still, the aggregate extension of the two beyond the natural state, must be essentially the same as that of the one, under the load; the one gaining, just as fast as the other loses. But the main, having a cross-section twice as great as the counter (chords and uprights retaining the same 270 BRIDGE BUILDING. lengths), must lose two-thirds of its tension, while the latter is acquiring strain enough to withstand the remaining third. Hence, 2 thirds of the deflection due to extension of diagonals, is recovered, on removal of the load, while the counter-diagonal retains the other 1 third. But the posts (the greater portion of them), do not remain stationary as to length, as above assumed; the main and counter diagonals together, exerting, obviously, only 2 as much action upon them in the new condition, as the former exert under load, they are relieved of 1 of their aggregate stress under load; but do not recover in the same degree, their original aggregate length; for the relief falls mostly upon the larger uprights, where the relative effects are less than the average. To illustrate the case as to uprights- if equal weights act at the nodes of the lower chord (Fig. 52), the compressive action upon the posts at p, q, r, and s, is obviously as 1, 3, 5, 7, respectively, or as 3n, 9n, 15n, 21n. [See analysis of Fig. 12]. Then,- Counterbracing, and removing load; sa is relieved of two-thirds of its stress, equal to 14n, while bs exerts a force of 7n upon br, making with 5n retained by bq, a total of 12n upon br, and showing a relief of 3n. Again; cq receives 5n through cr, and 3n through ep, = 8n in all, being a relief of In. But dp2, receiving 3n through dq, and In retained by do, sustains 4n, being an increase of In. Now, as these uprights are assumed to undergo the same contraction under load, 4 of the deflection on account uprights, is due to each. Therefore, sa being relieved of 2-3ds of its action, restores.. 2-3ds of {, ( 16.6 per C.), of that deflection. In like manner, br restores 1-5th of {, = 5 per C., and cq 1-9th of ~, or 2.8 COUNTER BRACING. 271 per C., making, for the pieces together, 24.4 per C., restored. This is diminished by 1-3d of {, or 8.3 per C. (on account of increased compression upon dp), leaving a balance of 16.1 per C. only, of deflection from contraction of uprights, which is restored in spite of counter-diagonals, in the case under discussion. Moreover, the main and counter diagonals, producing more or less effect of contraction upon the chords, according to the degree of inclination of the former, and the cross-sections of the latter, it may, perhaps, be reasonably assumed, that the contraction thus effected in the horizontal, is a full offset to the 16 per C. of expansion in the vertical sides of panels, as above shown; so that we may regard the whole deflection from uprights, as being retained by counter-diagonals. To state the full result of the foregoing investigation then, we find in case of Fig. 52, which is a fair representative of the average of trusses; that counter-bracing, obviates all the deflection due to compression of uprights, together with I of that resulting from extension of diagonals; and, making I = 1, in the formula for deflection (p. 267), we have - deflection saved by counter-diagonals, = (3 X 8 + 4) ~- 28, = a little less than 24 per C. of the whole deflection. If H= 0.75 (truss 52), the result would be about 311 per C. savedo But even these results are based upon conditions never occurring in practice. It has been assumed that all parts of the truss undergo equal degrees of change under a full load; which may be nearly true with respect to chords, but not to other parts. The maximum action upon od and dp (Fig. 52), requires those parts to be 2~ times as great, as they need be under full load; while p and cq require 4 more, and, qb and br, 1-20th more 272 BRIDGE BUILDING. cross-section at the maximlum than under a full load of the truss. Now the deflection resulting from elasticity in these parts, being less in proportion as the parts are greater, the saving by counter-bracingg must be less in the same degree, as far as it relates to such parts. This at once reduces the above computations for deflection retained, from 311 and 24, to 25 and 19 per C., for the two cases respectively; and, considering the increase of section required for uprights (in iron trusses), on account of great length and small diameter, as heretofore alluded to, it is deemed to have been fully demonstrated, that the efects of counter-diagonals, of half the size of the mains, are, to retain in the truss when unloaded, from one-sixth or less, to one-fourth of the deflection produced by a full movable load. But it has been seen in the progress of our investigations as to the action of load upon the different parts of the truss, that counter-diagonals are required in one or two panels on either side of the centre, and there, they can not be safely omitted. But, beyond the point where the weight of structure acting on the mains, begins to overbalance the effects of unequal and variable load upon the counters, I do not consider the advantages of counter-diagonals to be sufficient to warrant their use. In the case of rail road trains, gliding smoothly over bridges of ordinary spans, a quarter or a half of an inch more or less of deflection, is of slight importance, Iwhile, in bridges for ordinary carriage travel, the only objection to it is, that it slightly increases the degree of vibration produced by successive impulses, as of the trotting of animals, in time with the natural vibrations. Now, counter-bracing tends to shorten the intervals of COUNTER BRACING. 273 the natural vibrations by diminishing their extent; but can not destroy the liability to vibration; and the alteration of interval produced, may as often bring the vibrations nigher in tone with the gait of a trotting horse, as otherwise. In certain cases the effect would be one way, and in others, the opposite; and in general, the only result would be, to diminish the extent of motion; by one quarter, or less. Such is the result of the best reasoning and science that I have been able to bring to bear upon the subject of counter-bracing. To find the actual maximum deflection of a truss it is only necessary to know the value of P and 1, and to assign to E a value determined by the character of material, and the stress upon the several parts under full load. In Fig. 52, if H = 1 = 121ft., and the tension of wrought iron equal 15,000ibs. per square inch, the value of E for that material, will'be about 0.0075 ft.; and this will apply to the lower chord, and the obliques, ar and Ii. But the average value of E' for diagonals of wrought iron, would be about 0.006ft. For cast iron, 11,0001bs. to the square inch, requires about the same value for E, as 15,000 upon wrought I.; and, as that is a fair working rate of compression for cast iron in the upper chord,.0075ft. may be taken as the value of E for chords, in general. Uprights, for reasons heretofore explained, require a value for E", not greater than.005ft. The above values of E and fI, substituted in the formula (e P2_B, + P + PH29 + P,) x,it becomes P2R2_E + (~P + PH2)E' + - PE", equal to 4 x 64 x.0075, + (4 + 4).006, + 4 x.005, =, 0.188ft. about 24 inches. Hence, a well proportioned wrought and 35 274 BRIDGE BUILDING. cast iron truss, one hundred feet long, by 12~ feet deep, may be depressed 24" in the centre by a distributed load (including structure), with tension not exceeding 15, and thrust, not exceeding 11 thousand pounds to the square inch in cross-section of iron. WOODEN BRIDGES. STRENGTH OF TIMBER, &Co CXL. The qualities of wood as a building material, have been extensively treated of by authors whose works have long been before the public, with a degree of ability and research to which the present writer can make no pretensions. He will therefore at this time, simply state the conclusions arrived at from reading and observation (coupled with some experimental research) with respect to the average absolute strength, positive, negative, transverse, and to resist splitting, in certain cases; of the timbers principally in use for building purposes; as also, the forces they will bear with safety under various circumstances; leaving it, of course, for others to adopt his views for their own practice, or to modify and correct them, according as their greater experience or better judgment may dictate. At the same time, the author may be allowed to express his firm belief, that the views about to be presented, if fairly observed, will lead to the adoption or continuance of a safe and economical practice as to the proportioning of timber work in bridge construction. Pine timber in this country is perhaps to be ranked as among the most valuable timber in use for building purposes; especially in bridge building. White oak, WOODEN BRIDGES. 275 and some other varieties, are preferred for certain purposes, as being harder, stiffer, and especially better calculated to sustain a transverse action, whether tending to bend or crush it. But in what follows, reference will principally be had to the ordinary white pine of this country; and the deductions here made, may readily be modified so as to apply to other materials of known strength, when so required. The absolute positive, or tensile strength of pine, may be stated at about 10,0001ts. to the square inch of cross-section. It might therefore seem to be safely reliable in practice, at 15 or 16 hundred pounds to the inch, upon that part of the section of which the fibres are not separated in forming connections with other parts of the structure. And so it probably would be, when new, sound, and straight grained. But timber in bridges, is usually more or less exposed to wetting and drying, and deterioration in strength,- especially as it regards tension. Moreover, in forming connections of parts and pieces in a structure, it is difficult to secure a uniform strain upon all the uncut fibres;- one side of the piece being often exposed to much greater stress than the other. In view of such facts, it is deemed advisable to seldom allow less than one square inch section of unbroken fibre to each 1,000lbs. of tensile strain. NEGATIVE STRENGTH OF TIMBER, CXLI. The ability of pine to resist compression in the direction of the length of piece, is from 4 to 5 thousand pounds to the square inch of section, and this varies but little, whether the pieces be of length equal to once, or five or six times the diameter. It moreover 276 BRIDGE BUILDING. diminishes only about one-third with an increase of length up to 18 or 20 diameters. Now, if we take about I of the absolute strength, say 8001bs. to the inch for a length of 6 diameters, and 560 for 18 diameters, and substract 401bs. per inch for every increase of 2 diameters in length, between 6 and 18 diameters; and from 18 to 40 diameters, compute the quantities by the rule given [LXXXIX], in relation to negative resistance of cast iron, we shall form a table of negative resistances of timber, for a range of lengths which will cover the principal cases that will occur in bridge building, which the author feels confident in recommending for the adoption of engineers and practical bridge builders. If it be desired to extend the table to greater lengths than 40 diameters, the formula which makes the strength as the cube of the diameter divided by the square of the length, may properly be used. The following brief table of negative resistance of timber, has been constructed in the manner above inTable of Negative Resistance of Timber. Diameters. Pounds. Diameters. Pounds. Diameters. Pounds. 6 800 24 368 42 166 8 760 23 838 44 151 10 720 28 296 48 188 12 680 30 280 48 127 14 640 82 2481 50 117 16 600 34 227 52 1018 560 38 210 54 100 20 470 38 195 57 90 22 418 40 183 60 81 dicated, and exhibits at a single view, the number of pounds to the square inch of cross-section, which timbers of different lengths will bear with safety, at inter WOODEN BRIDGES. 277 vals of 2 diameters in length, for all lengths between 6 and 60 diameters. The first column gives lengths in diameters, and the second, the number of pounds to the square inch, borne, with safety. TRANSVERSE STRENGTH OF WOOD. CXLII. Pine timber will bear a transverse strain of 1500 or 1600tbs. to the square inch of cross-section; that is, the projecting end of a beam will bear 1500ibs. for each square inch of its cross-section, applied at a distance from the fulcrum equal to the depth of the beam; the force acting parallel with the sides. In other words, a beam 1 inch square upon supports 2 inches apart, will sustain 3,000 s. midway of supports, provided the timber be not split or crushed; as would certainly be the case with so short a leverage. It will therefore be proper in practice, never to expose this material to a greater transverse strain than 2501bs. (upon a leverage of 1 diameter), to the square inch; and, to calculate the strength of a projecting beam, this quantity should be multiplied by the crosssection and the depth. and the product divided by the distance of the load from the fulcrum. LxcIv.] For the safe load in the middle of a beam supported near the ends, take four times the above quantity (= 1,00ibs.), multiply by cross-section and depth, and divide by length between supports. A beam will bear twice as much load uniformly distributed over its length, as when it is concentrated in the centre, in case the beamn is supported at the ends, or at the end in the case of a projecting beam. But these are familiar principles and need not be dwelt upon in this place. 278 BRIDGE BUILDING. CLEAVAGE. CXLIII. In order that a piece of timber may act by tension, it is necessary that a portion of its fibres be separated, to form a heading for the stretching force to act against; and, that the strength of the piece may be made available for as great a part of its length as may be, without having the head split off, it becomes important to know the power of the material to resist such a result. Let ab Fig. 54 represent a heading by means of which the stick is made to act by tension. Now, as the timber is incapable of supporting upon the ends of its fibres with safety, for a great length of time, a force of more than 8 or 10 hundred pounds to the square inch, the area ab should contain at least one square inch-for each 1,000bs. to be applied to it. And, if the head ab be too nigh the end of the stick, the part abed will FIG. 54. split off, and be thrust over the end of the timber. It is found by experiment that to produce this effect upon timber of sound and straight grain, requires a force of nearly 600lbs. to the square inch of cleavage in the area efcb. It is therefore obviously necessary to safety, that the head ab, be at a distance from the end, equal to at least 10 times the depth (ae) of the head, that the area of cleavage may be sufficient to stand as great a force as the area of head can stand; i. e., there should be 10 inches of cleavage surface to one inch of head surface, WOODEN BRIDGES. 279 If the heading be formed in the central part of the stick, as by a mortice or pin hole, two cleavages must be made from the hole to the end in order that the part may be forced out. Hence, the hole need be only about five times the width of hole from the end; that is, an inch hole should be five inches, and a two inch hole, 10 inches from the end. TRANSVERSE CRUSHING. Timber is sometimes liable to be crushed by forces acting transversely to the direction of its fibres. If the pressure be applied to the whole side of the piece, it should not exceed 150, or at most 200lbs. to the square inch, in practice. If acting on one-half of the surface, it may perhaps, be 300lbs. to the inch, without yielding very injuriously; and, for a very small portion of surface, as under a bolt head or washer, a pressure of 5001tbs. to the inch may be admissible. These limits are taken with reference to pine timber. Hard timbers, will bear, probably, 25 to 50 per C. more with safety. CONNECTIONS OF TENSION PIECES, AND PROPORTIONATE AMOUNT OF AVAILABLE SECTION. CXLIV. From what has been already said, it follows that for a piece to act with the best advantage by tension, if the connection be made all at one point in the length, one-half of the fibres require to be cut off; so as to form an area of heading equal to the cross-section of the remaining part of the stick; since it has been assumed that the power to resist tensile strain with safety, is the same as the power to resist compression upon the ends of fibres. But if several headings, or shoulders be made at different points, or distances 280 BRIDGE BUILDING. from the end, a less portion of the fibres require to be separated. If, upon a piece 4 inches thick, instead of one shoulder 2 inches deep at 20 inches from the end, we make two of one inch deep, each, the one at 10, and the other at 20 inches from the end, we have the same area of shoulder, and 50 per C, more fibres to act by tension; which may be made available by another shoulder at 30 inches from the end. Thus a greater proportion of the fibres, but a less proportion of the length is available. In the same manner, if a piece be connected by pinning, requiring 2 pins of 2 inches in diameter, at 10 inches from the end, four 1 inch pins, two at 5, and two at 10 inches (if stiff enough), give the same shoulder surface, and require the cutting of only half as many fibres; and, two more pins at 15 inches from the end will give Jths of the whole area of section available for tension. In case the smaller pins be not stiff enough, they may be of an oblong section in the direction of the strain. A still further reduction of depth of shoulder or width of pin, will make a still larger proportion of the fibres available, but not so much length; and, experience and judgment, with a little calculation, must dictate as to the proper medium in this respect. The theoretical limit is, when the shoulders are infinitely small, in which case, the whole cross-section becomes available. But, as the resistance to cleavage must be equal to the force of tension, it follows that the loss in available length, is proportional to the amount of cross-section available for tension. In practice, it is usually not expedient to estimate more than one-half or two thirds of the whole section WOODEN BRIDGES - CONNECTING PINS. 281 as available for tension. This reduces the safe practical strain for timbers sustaining tension, to from 500 to 700lbs. to the square inch, for the whole cross-section; and the proper point between these limits should be determined by the mode of forming the connections in specific cases. PINS OF WOOD AND IRON, FOR CONNECTING TIMBERS IN BRIDGE WORK. CXLV. Perhaps no more suitable place will occur for making a few general remarks upon the merits and use of pins for connecting pieces of timber. While it is readily admitted that the plank lattice girder, put together exclusively with wooden pins, answered an excellent purpose in affording cheap and serviceable bridges in this country when timber was abundant, and the iron manufacture in its infancy, it is nevertheless believed that the use of wooden pins in bridge construction, is not destined to a long continuance. Where pins are required in wooden bridge work, it is thought that iron may be used with a decided advantage over wood - not in the lattice bridge of the usual form, composed of a great number of diagonals, and a legion of connecting pins; but in a modified form (as in Figures 13 and 19), with a greatly reduced number of pieces, and points of connection. Wooden pins for the purpose under consideration, do not possess sufficient strength in proportion to the surface, unless made so large as to require too much cutting of the timber. Moreover, the action upon the pin tends to crush it laterally, in which direction the hardest timbers available for pins, scarcely offer as much resistance as the ends of fibres to which they are opposed. 36 282 BRIDGE BUILDING. Where pieces are connected with their fibres parallel, wooden pins or keys with cross-sections elongated in the direction of the grain, to give them the necessary strength, may be employed without too much cutting of the timber. But, as just remarked, the key is liable to yield before the cut ends of the fibres are taxed to their full capacity. It is therefore poorly adapted to the purpose in any case where great strength is required. Moreover, when the pieces to be connected are placed across one another, the hole will not admit of elongation without too much cutting of at least one of the pieces. If it be required to connect a piece by a pin between two other pieces as seen in Fig, 55, upper diagram, the pin, as already seen, should be strong enough to bear as much strain as the opposed surface can sustain. Now, we have seen that this can scarcely be done by wooden pins. Still if sufficiently stiff, they may yield somewhat to compression, without material loss of strength. Taking the transverse strength of pin timber at 300lbs. to the inch, with leverage equal to diameter, the expression 4 x 300ad I1 (a representing the crosssection, d, the diameter, and 1, the length of pin, between centres of outside bearings), gives the amount which the pin will bear in the middle. Now, the two outside pieces, having each half the thickness of the centre one,* I must equal 1t times the thickness (t), of the middle piece; while the effect * The outside bearings may be regarded as concentrated at the centres.re; - of thickness of the pieces, while the stress....2/;-..."... of the pin in the centre, is the same as if::'""''-...' the force exerted by the middle timber,'"~..".... 300ad — 1t,= 1200ad t,= strength of the pin. But the opposed surface will bear 1,0001d; and putting this expression equal to the former, and deducing the value of d, in terms of t, it will show the smallest diameter of a wooden pin, strong enough to bear as much as the opposed surface. This equation gives d = 1.03; t whence, it appears that the wooden pin should be 3 per C. greater in diameter than the thickness of the middle timber. In the same manner, the strength of an iron pin in the same circumstances, is respresented by 4x1l x5,000ad -- 11t, = 20,000ad, t which made equal to 1,000td, gives d = 0.2521, hence, the most economical diameter for an iron pin in fastening one piece between two others, is about Ith the thickness of the middle piece; i. e., taking the stiffness of a round pin at 5,000lbs. But reducing it to 4,500Tbs. as proposed in another place [xcvIII], it gives d = 0.266t; whence, even upon this basis, it will be safe in practice to make d = 4t, and the whole length of pin = 2t, so that it may extend into the outside pieces to the extent of half the thickness of the middle piece. Since the outside pieces (Fig. 55), require half the thickness of the middle piece, and the pin requires a diameter equal to =t = 1 the thickness of outside pieces, it follows that in pinning or spiking a plank or timber to the outside of a thicker piece, the pin or spike should f Dividing the equation 1209ad —t -1,O000t, by 100d and multiply, ing by t, give 12a = 10t2. But 12a - 12X.7854d2, =9.4248d2, 10tawhence, d2 1.061t2, and dv/'1.061t' = 1.03t. 284 BRIDGE BUILDING. have half the thickness of the piece attached, that it may not bend with less force than the ends of the severed fibres can bear; and should extend into the thicker timber at least 6 times its diameter. For, as Fio. 55. i... ^ -.........-....... the inner portion of the pin or spike, must act upon the wood in the same direction as the part through the attached piece, it requires the same amount of surface to act upon, while the intermediate portion requires a surface equal to that acting upon the two end portions. And, even in this condition, the pressure is not uniform upon all parts of the length of the pin, since there is a neutral point, as represented by the upper dotted line (lower diagram, Fig. 55), where the pressure changes from one side to the other, and, near this point, must be very light in both directions. Hence, for the most perfect results, in such cases, the pin should probably enter the thicker timber to a distance of 7 or 8 times the diameter of the pin. When the end bearings of the pin act transversely to the grain, they require at least 50 per C. more extent of bearing, or even twice as much, when practicable. At 50 per C. 1 = l1t, and the effect of the pressure exerted by the middle piece, is 5ths that of the same force at the centre of the pin. The equation for the proper diameter of the pin, then, is 4xxx5,00ad — 3t = 1,000td; whence, d = 0.283t, and length of pin =2-t. WOODEN BRIDGES - SPLICING. 285 SPLICING. CXLVI. The term splicing, as applied to timber work, may be defined to be the uniting of two pieces of timber by their end portions, so as to form (in figure) a continuous timber upon a straight axis. The splicing of timber to withstand a thrust action, requires only the meeting of the squared ends of pieces; or, a half lap, formed by removing the half of each for a foot or two, more or less, from the end, and lapping the remaining halves, so as to have the extreme end of each, meet the shoulder of the other. But the splicing of pieces to withstand tension, obviously requires a more complicated process; and, from what has already been said, [cxLIv,] it is clear that only a part of the absolute section can be made available to withstand a tensile strain. FIG. 56. In Fig. 56, we have the profile of a lock splice, by which one-third of the section is available for tension; the depth of the locking being equal to one-third of the thickness of timber. Now, that the locking may not split off, we have seen that the lap should extend 10 times the depth of lock, each way, making a lap of 6- times the thickness of the timbers. By slanting the timber to a thickness at the end equal to that in the neck of the lock, we lose none of the cleavage required to split off the hook, while we gain in amount of section where it is required for bolt holes to secure the splicing. Otherwise, the bolt holes would 286 BRIDGE BUILDING. reduce the available section below one-third of the whole. It is proper to observe with regard to this splice, and also the succeeding one, that the power being applied upon the reversed shoulder, or hook, out of the line of the unbroken fibres which resist the power, the tendency is to throw the ends outward, and produce a degree of lateral action, which weakens the timber to a somewhat greater degree than in proportion to the amount of fibres severed. FIG. 57. With a double lock splice, as in Fig. 57, one-half of the section is available. This requires a lap of 10 times the thickness of the timber. By three lockings upon the same principle,. of the fibres may be utilized for tension, with a lap of 12 thicknesses (or 12t.), and, by a lap 131t, we make twothirds of the fibres available. Finally, by a lap of 201. and an infinite number of lockings the whole crosssection would be available. But this, of course, is a point not attainable in practice. From ~ to -- say an average of ~, is as much as can be reckoned on, and about as much as can usually be made available for tension, at the end connections of a single timber. Splicing may also be effected by a plain scarf, with bolting, pinning and spiking, as indicated in Fig. 58. With bolts, pins and spikes properly arranged and proportioned, a strong splice may be formed in this WOOODEN BRIDGES - SPLICING. 287 manner, with a less lap than what is required in the lock splice. In this case the fastenings should pass FIG. 58. through at right angles with the plane of the joint, that they may not be slackened by a slight yielding of the timber to pressure, in the holes. This, however, is a device which will probably, seldom be resorted to in bridge construction. Timbers may also be shackled together end to end by iron bolts and straps, as shewn in Fig. 59. The aggregate cross-section of straps should be about 1 square inch to each 10 to 15 thousand pounds of strain which the splice is intended to bear; and the diameter of bolts fastening the straps, about one-fifth of the thickness of timber, to secure the greatest effect for the amount of section destroyed in cutting the bolt hole. FIG. 59 To connect two timbers 10x12 inches, so as make half of the fibres available for tension, we may take 6 straps 2 feet long from hole to hole, and containing a cross-section of about 1 square inch, each. Also 6 bolts of 2" in diameter, and arrange the straps and bolts as shown in the figure, the straps being placed upon the 12" sides. This will cost, say for 170lbs. of Iron at 7cts., $11.90. 288 BRIDGE BUILDING. The expense of a double lock splice (Fig. 57), will be about 7 cubic ft. of waste timber,...... $3.60 40obs. of iron bolts, washers and plates,... 2.80 Labor in fitting the timbers, say,.............. 1. Total,...................................... $7.30. showing the shackle connection to be from 4 to 5 dollars the more expensive. CONSTRUCTION OF WOODEN TRUSSES. CLVII. With a thorough comprehension of the power of timber to resist the various kinds of strain to which it may be liable in bridges, and other timber structures, and of the general principles of forming connections in timber work, as attempted to be explained and set forth in the last few preceding pages; and a knowledge of the general forms of arrangement for the several members in bridge trusses, or girders, and of the manner of computing the stresses to which the several parts are liable to be subjected, as treated of in the first 100 pages or so, of this work, the details of practical construction of wooden truss bridges may be intelligently entered upon. Nothing more elaborate will be here undertaken, than a reference to general forms of trussing suitable for wooden bridges of different spans, and a description of what seem to be the most feasible methods of forming connections at peculiar and specific points, The method pursued will be, to proceed from the shorter spans, and more simple combinations, to structures of greater length, and requiring a greater number and a more complex arrangements of parts. WOODEN BRIDGES. 289 Two PANEL TRUSSESo CLVIIIo The form presented in Fig. 3, with rafter braces ad and de, and a tie or chord ac, together with an iron tension member db (in I or 2 pieces), is probably the best adapted to bridges from 20 to 25 feet in length. The braces should meet with a vertical joint at d (Fig. 3), and toe into the chord tie with two headings, and one or two small bolts, as in Fig. 60. FIG. 60. Assuming the brace to be capable of sustaining a thrust of 500lbs. to the inch of section, and the heading 1,0001bs, to the inch, the aggregate depth of heading, af, and de, should be one-half the depth cb, of the brace; and, the point f, should fall below the point d, by 1 V- ad, so as to give a length of cleavage fh, - laf or 10 dh. The shoulder de, then, should be, (1),... de =I cb - i ad, = cb -a I- ab + - -db, We here speak of a d b as a straight horizontal line, not shown. This is regarding af as equal to the vertical depth of cut at af; which will be sufficiently near the truth for our present purpose,'provided the brace be not very steep. But (2),...de = db. sin. dbe, -db. sin. cab. 37 290 BRIDGE BUILDINGO and, putting this value of de equal to the one above, and changing vulgar to decimal fractions, we have, (3),..db. siL. cab = 0.5 eb- O.lab + O.1db. Then, transposing, and uniting co-efficients of db. (4.)...(sin. cab -0.1) db =0.5cb -O.lab, whence, 0.5 cb-0.1 ab (5),...db -i. cab- 0.1 Now, from equation (2) we derive db= si-~ which value of db being substituted in equation (5), we have de 0.5 cb-0.1 ab (6)... sin. cb =sin. cab —. whence, multiplying by sin. cab. we derive, 0.5 cb 0.1 a -b (7),... de 0.1 Then, substituting for ab, its 1sin. cab cb equa l, s. cab the last equation becomes, 0.5 cb -0.1 b 0.1) o..c- 0.1 0.1 (8),..de = ~~ i sil. cab sin. cab Making the angle cab 26033g', which is regarded as a suitable inclination for the brace, being one, vertical, and two, horizontal reach, sin. cab = 0.447, which substituted in (8), gives de =.356 cb, and af —.144cb. This, it will be recollected, is deduced upon the supposition that the brace will sustain a compression of 500ibs. to the inch, and no more; which will depend upon the length as compared with the least diameter. If the brace be capable of bearing with safety, more or less than 5001bs. to the inch, the heading, or butting surface should be more or less than half the area of cross-section, in like proportion. For, if unnecessarily large, it requires too much cutting of the chord, and if too small, the pressure upon abutting surfaces becomes too great. With the inclination of brace above assumed, the compression upon the brace obviously equals the weight WOODEN BRIDGESo 291 sustained multiplied by V/5; and, for a rail road bridge, at 1~ tons to the lineal foot, the weight upon each brace, will be 6,2501bs. = v; or say, g (w + w') 7,5001bs. This by /5, gives 16,770lbs. = thrust of brace, while 15,000lbs. = tension of chord. Now, at 500F1s. to the square inch of gross section, the chord requires 30 square inches, and the brace 331 inches, being a little less than 6' square. But the length of brace being about 11ft. or 22 diameters of a 6" stick, we find by the table [cxLI], the brace is only capable of bearing 4161bs. to the inch. Hence, with a 6" least diameter, a section of 40.3 inches, or nearly 6" x 7", becomes necessary. Still the butting surface required is only 16.77 square inches - a little less than 2~k' depth (at right angles with the brace), by 7' in width. This 21 inches in depth may be divided between the two shoulders at a and d, in any manner that will leave a length of cleavage from a to the end of the chord equal to 10 af, or more strictly 10 afx cos. cab, which equals the vertical depth of cut atf. But the line df, should preserve a descent, equal to both of its length. The depth of shoulder being thus reduced from Icb (= 3" in this case), to 2k", de is diminished in the same degree, and from.356cb, becomes X. 356cb =.2966cb; and, substituting 6" for cb, we have de = 1.78 inches. In the meantime ctf becomes 5x.144cb, =.72". The vertical depth of cut for de, = 1.78", is 1.78xcos. cab, 1.78 X.894, = 1.59". Add to this the vertical cut atf, equal to.642" and it makes 2.233", = aggregate vertical depth of cut in the chord, whence the distance eg should be 22.33 inches, to afford the necessary resistance to cleavage. Now, we require in the chord 15 square inches of unsevered fibre, to withstand the horizontal thrust of 292 BRIDGE BUILDING. the brace while we require, as seen above, 1.59x 7 = 11.13 inches to be cut away to form foothold for brace, making aggregate section of chord = 15+11.13 - 26.13 sqr. inches, equal to about 7" x 39", by strict computation. Timbers so small, however, although capable of sustaining, without excessive stress, any action to which a bridge is legitimately exposed, is not to be reconmended in practice, as the structures might be destroyed by casualties which would but slightly affect the large timbers required in heavier and longer structtur es. The centre of bearing of the truss upon the abutment, should be directly under the point i, at the meeting of central axes of the brace, and the unsevered portion of the chord. Otherwise, an injurious lateral strain would result to the chord at its weakest point. The transverse beam at the centre of the truss, may be placed above the chord or below, as preferred, and sustained by 2 suspension bolts descending divergently from a saddle, or double 1 washer at the vertex of the braces, passi ng through the beam, and secured by nuts and washers upon the under side of the beam, as shown in Fig. 61. The divergence of bolts should be from 1th to Ith their length, and the section of bolts, a trifle more than what is required silmply to sustain the weight, as they may L I I 1 ^ act utequally, in consequence of a small L. lateral tendency of the braces..~ ~ 1g A slmall bolt should pass vertically through chord and beam, to preserve them in place. Also, a small bolster, or corbel block WOODEN BRIDGES. 293 (j. Fig. 60 and 61), under the chord at the end, affords some protection at the weak point in tlhe chord. A pair of horizontal x braces in each panel, between beam and abutments, or plate timbers upon abutments, are required to produce lateral steadiness in the structure. The idea of constructing the trusses of a rail road bridge, even of 20' span, of 6" timbers, to persons in the habit of seeing such bridges constructed with timbers 10 or 12 inches square, will undoubtedly suggest visions of catastrophe, courts and coroners; and, in view of liability to casualty, fretting at joints, alnd perhaps surface decay, it may be advisable to use in such structures, timbers somewhat larger than the above computations indicate as sufficient to withstand determinate forces. But, as an instance of what strength may be obtained with very small timbers, properly proportioned and put together, it may be here stated that a model of a 20 feet truss, upon a scale of 1 to 12, constructed as above explained, of "' x 5-" braces and chord, bore without material injury, 350fbs. at the centre, equivalent to 7Ofhs. distributed, and representing 700x144 - 100,8001bs. upon one truss, or over 100 net tons upon a 20 feet bridge; being some four times as much as a single track rail road bridge of that span is usually subjected to. With regard to the proper size of transverse beam, the formula (see rule [CXLII]), 1, = W, (a representing area of section, d, depth of beam, 1, length between supports, and W, the load in the centre), gives a 1wO Then, assuming = 15', W = 7,500bs. ( 15,0001bs. distributed), and d = 14"; we have a 294 BRIDGE BUILDING. 1,000x 70 96.4 square inches; which divided by depth (d), in inches, gives thickness (t) = 7 inches nearlyo Or the formula I = 1W gives the required thickness directly. But in this case, I and d must express length and depth in inches, since the co-efficient of d (1,000) refers to square inches of section. Otherwise, the coefficient must be multiplied by 144 to make it refer to the square foot of section; in which latter case the value of t will be obtained in feet. In the case of beams to sustain rail road track, we may let i' = length of beam exclusive of the portion between rails, and W = weight upon the 2 rails. If I' = 120" and W = 25,000lbs., and d = 14" the above formula becomes t - 0x,5000 _ 3,000,000, = 15.3 in. 1,000X 142 196,000 THREE PANEL TRUTSS CLIX. A three panel truss bridge of wood may be constructed upon the plan shown in outline by Fig. 7' The main braces b and ab' may connect with the chord in the same manner as in the two panel truss described in the last section, and illustrated by Fig. 60; while the upper end may be square, and the whole bevel to form the angle abb', given to the member bb'. Or, the bevel may be upon both members; in which case the saddle plates at b and b' should extend over the joint, so as to throw a part of the weight directly upon the brace. In case the bevel be all upon bb', the saddle need not bear upon the brace. The counter braces in the middle panel may box into the chord and the horizontal bb', in the manner shown in Fig. 62, either by the black or the dotted lines; the upper end of the counter toeing against the WOODEN BRIDGES. 295 end of the main brace, when the form of connection shown by the black line is used. As the counter braces cross, or meet in the centre of the panel, one may be in two pieces thrusting into the other as at c Fig. 62; or one member may be in two full length pieces, and the other a single brace between the former, of such width vertically, as to possess the required cross-section; say 21-' x 6" for the outside, and 4 x 8 for the middle one, and the whole connected by a small transverse bolt at the crossing. Fio. 62. \~^^i a / The stresses of the several parts of the truss may be determined in the manner explained in section XVIII, and the timbers proportioned accordingly, and in conformity to rules in relation to strength of timber [cXL and CXLI]. For a truss of 30 feet to carry a gross load of 15,000ibs. to the panel, with a horizontal reach of brace equal to twice the vertical - chord and " straining beam," (bb', Fig. 7), should be 7" deep x 9" wide; main braces 8" x 9". Counter-braces being subject to only one-third of the movable panel load, may properly be 4 x 8 or 5 x 6, if one be severed at the crossing, or as above specified, if one member be in 2 full length pieces. Two counter-braces might cross one another side by side, but this would not produce a well balanced action. 296 BRIDGE BUILDING. Bridges of this length of span are, moreover, often built with counter braces omitted, for common road purposes. But such practice is defective, unless extra depth of section be given to the lower chord, so that its stiffness may transfer a portion of weight over the quadrangular middle panel; and in no case is it advisable to dispense with counter braces in a rail road bridge of three panels. Beams may be suspended by divergent bolts as in Fig. 61, and bolted to the chord; while horizontal x ties or braces, as may be preferred, in each panel will prevent lateral swaying of the structure. The above is probably the simplest and best plan of wooden truss for bridges of 30 to 35 feet span. FOUR AND SIX PANEL TRUSSES. CLX. The same general arrangement, with the same kind of connections, in trusses of 4 or 6 panels, according to length of span, may be used with good effect for common road purposes, in any length up to 70 or 80 feet. In such cases, each panel should have one main brace, and counter braces may be entirely omitted; as the partial movable load is seldom so great as to neutralize the action of weight of structure upon the main braces. In the 6 panel truss, the movable must exceed the permanent panel load upon the two beams next either end, with no movable load upon the other beams, in order to neutralize the constant tendency to action upon the central pair of main braces. This is obvious from the fact that the greatest tendency to tension action upon the latter, is 3", = w, while the permonent load gives a constant opposite tendency, equal to 2w'. Should such cases occur, the transverse stiffness of :~[n WOODEN BRIDGESo 297 both upper and low chords must be overcome before a collapse could take place. In the case of iron trusses, the chords are supposed to have no lateral stiffness at the nodes; consequently, counterbraces, or ties, as the case may be, are always necessary in one or two panels each side of the centre. Fig. 63 represents a Six Panel truss, as arranged and recommended by the author 16 or 18 years ago, and adopted by the Canal department of the State of New York, for farm and country road crossings over the State canals, upon which several hundreds of them are in use. The arrangement of upper and lower chord timbers, and the divergent suspension rods, to maintain the erect position of trusses, as well as the assignment of correct proportions to all the parts throughout, are believed to have originated with the author of this work. The lower and longer portion of the bottom chord, is usually in two pieces, spliced with double locking and bolting (see Fig. 57), over the centre beam. Transfer blocks are also inserted between upper and lower timbers, to transfer a part of the stress of the longer to the shorter portion, and thus diminish the strain at the splicing. 38 298 BRIDGE BUILDING The long portion of the upper chord may also be in two pieces meeting with squared ends, or with a plain half lap, of a foot or so. Transfer blocks or packing pieces and bolts should likewise be inserted as indicated in the figure. The dimensions of the several members, of course, will depend upon the length and depth of truss, and the load it is required to bear. It is seen by processes explained heretofore [XL and LIII], that the portion of chord under the triangular end panels, and also the endmost sections of the upper chord, are liable to action equal to 21-W/ in which expression W = w w', h = the horizontal, and v = the vertical reach of braces. The next sections (top and bottom), are liable to 4W, and the lower chord under the two v middle panels, to 4W4-. -LlLl r V _ The end braces are liable to 25-W Vh2 + v2 v, the next braces, to (10w"+lw')/h2 + v2 + v, and the middle ones, to (6w"+1-') Vh + +v v; while the verticals are exposed to 2-W for the endmost, 1W for the middle, and 10 w' + lw' for the intermediates. Now, we have only to assign specific values to w and w', and to h and v, in order to obtain the actual maximum stresses the several parts are liable to, from the general expressions just found. Let h = 12', and v = 7'; which, though not an economical proportion, as we have seen [LXIV], may be admissible for bridges of light burthens, giving a better appearance, and the structure being less top heavy. The weight of a light superstructure of this description, is 18 or 20 tons - say, w' = 3,000bs. Then, assuming w = 6,000bs. which will be sufficient for the WOODEN BRIDGESO 299 lighter class of private and country bridges. Then, - = 2 = 1,714, and Vh2 + v v = 1.984. Substituting these values in the above expressions for stresses, we have 2~ x 9,000 x 1.714, 38,565, = tension of end section of bottom chord. For the next section, 4 x 9,000 x 1,714 = 61,704Tbs.; and, for the two middle sections, 69,417fbs.; while the compression of the two portions of the upper chord, is 38.565lbs., for the end, and 61,7041bs., for the middle sections. The maximum compression of the three sets of braces, is 44,653 for the ends, 14,880 for the middle ones, and 28,760 for the intermediates. The tension of suspension bolts, is, at the maximum, for the endmost 22,500, for the middle ones, 9,000 ( = W), and for the intermediates 14,500. The main portion of the lower chord, requires a lap at the splice, equal to 10 times its depth, [CXLVI.] Hence, the less depth, the less waste in splicing, and the more lateral stiffness of truss. But this also involves greater required section in the lighter braces, which become too thin, vertically, to act with advantage under compression. There is no ready means of determining the exact optimum in the ratio of depth to width of timbers in this case; and we shall not err greatly by assuming a ratio of width to depth as 3 to 2, or as 4 to 3; neither to be rigidly adhered to. The bottom chord mav suffer tension in the second panel, equal to nearly 62,000bs., requiring 62 inches of net section; while the second brace has a maximum horizontal thrust of nearly 25,000s., requiring the severing of 25 inches, whence this part of the chord should have a gross section of 62 + 25, = 87 inches. 300 BRIDGE BUILDINGa This amount may be furnished nearly, by a section of 8'" x 10', 8 x 11, or 7 x 12. Assuming the second, the end braces should be 81 X 11, the next 7 x 11, and the middle ones, 5~ x 11. We have seen above, that 62,0001bs. of tension, are communicated to the long timbers of the lower chord, while the splice at the middle is only good for 500lbs., to the inch of gross section, being 44,0001bs.; thus leaving a deficiency of 18,000lbs. to be sustained and made up by the upper timber. In the mean time, the middle braces exert about 8,000lbs. of horizontal action upon this piece, under a full load of the truss, and near 13,000ibs. at the maximum action of those braces. Hence that timber should have a mininum net section of 26 inches, + 18 inches to be severed for the insertion of transfer blocks. The timber should therefore be at least 4"' deep. The transfer blocks should be 1"f' thick, in this case, and 15 or 16 inches long, and be well fitted in position as indicated in Fig. 63. This mode is preferable to that of using blocks twice as thick, and letting one-half into each timber by a square boxing; because it leaves a larger section of timber opposite the middle of the block where bolt-holes are required. Otherwise it would be necessary to provide additional gross section on account of bolt holes. The same reason applies in the case of braces toeing into chords, &c.; where the boxing, instead of being as deep at the heel as at the toe of the brace, should taper out to nothing at the heel. See black line at foot of counter brace c, Fig. 62. This case has been given in pretty full detail, since the plan seems to merit, as it certainly enjoys, a high degree of popularity, for small bridges for ordinary use. WOODEN BRIDGESo 301 By increasing the depth to at least Ith of the length of truss, inserting counter braces in the two middle panels, and proportioning members to the respective strains to which they are liable; this plan is undoubtedly well adapted to rail road purposes in spans from 50 to 70 feet in length. For greater spans than 70 feet for rail roads and 80 for common roads, higher trusses, with top connections and lateral bracing or tying, should undoubtedly be adopted. CLXI. The bridge usually designated as Beardsley's Bridge, is identical with the one shown in Fig. 63, modified by the substitution of iron bottom chords, composed of two parallel rods (to each truss) in 5 pieces or parts corresponding in size with the stresses of chords under respective panels. The middle and largest part extending under the two middle panels, and the others, each under one panel only. These pieces or parts, being connected by turn-buckles, or screw couplings, pass through cast iron shoes, into and against which the several braces toe and thrust; the shoes being prevented from sliding outward upon the rods,. by the couplings. The shoe should in all cases be so formed and located that the axes of action of chord, brace and vertical, meet at the same point, as it regards the intermediates while as to those upon the abutments, the axes of chord and brace should meet over the centre of bearing upon abutments. This arrangement (understood to have been the suggestion and device of Mr. Geo. Heath), gives very satisfactory results, and the only practical question with regard to it, as compared with the one with wooden 302 BRIDGE BUILDING. chords, seems to be merely one of economy and convenience. If suitable timbers for chords can be readily and reasonably obtained, it is thought to be quite as advantageous to use wooden chords, THE HOWE BRIDGE. CLXII. A very popular plan of wooden bridges, which has, in fact, superseded most others in New York and New England for rail-road purposes from the time of the introduction of the rail road system, is known as the Howe Bridge. The trusses have upper and lower parallel chords, together with main and counter braces, of wood, tied vertically by wrought iron tension rods from chord to chord, the principle of action being the same as in the plan shown in Fig. 63. The braces act upon the chords and verticals through the medium of cast iron shoes or skewbacks, with ribs or flanges let into the chords to a sufficient depth to sustain the horizontal thrust of braces, and with tubes, or hollow processes, square externally, and having round holes to receive the vertical bolts. These tubes project downward through the lower, and upward through upper chord, between the courses of timber composing the chord, being boxed into the timber on each side of the tube, so as to leave about an inch between adjacent courses for ventilation; the tubes, extending through the chords, reach an iron plate upon the opposite side, which serves as a washer, or bearing for the nuts of the suspension bolts. By this means the vertical action of braces is brought directly upon the verticals, without a transverse crushing action upon the chord timbers. WOODEN BRIDGES, 303 The chords are formed of 3 or 4 courses of timber side by side, with a depth equal to two or three times the thickness; the joints in the several courses being so distributed that no two courses may have a joint in the same panel when avoidable. Fig. 64, represents a side view in the upper, and a top view in the lower diagram, of a portion of the botFIG. 64. tonm chord. At t is represented a view of the tube of the skewback as it would appear with the outside chord timber removed; at m m, the seats of the main braces, and c, the seat of the counter brace. Over a, is a clamp, or lock piece, and bb' are transfer blocks, or packing pieces, to secure the joint, and transfer the strain from one to another of the chord timbers. The transfer blocks may be placed obliquely as at b, or straight, as at b'. The latter is the more usual, but the former leaves the greater section of timber at the point where the bolt holes occur. The braces are usually placed with a horizontal about half as great as the vertical reach, and extending across one panel only. Counter braces used throughout, and the upper chord made of equal leugth with the lower, giving the truss a rectangular, instead of a Trapezoidal form. 304 BRIDGE BUILDING. Now, it is obvious that in a rectangular truss, as represented in Fig. 52, the end posts, and one panellength of the upper chord at each end, as well as one counter-brace, are entirely useless, as it regards sustaining weight of structure and load. It wiPn readily be seen, moreover, that no counter-braces except those of the two middle panels, in the 8 panel truss, Fig. 52, have any sustaining action, unless the variable exceed 4 times the permanent load of the truss. It is furthermore manifest that there is a large amount of surplus material in the portions of lower chord toward the ends; the tension of that chord being in the several panels, preceding from the end (in the case of Figo 52), as 38, 6, TS and 8. Hence, over onefifth of the material in a chord of uniform section, is in excess. But the greatest sacrifice of economy in the Howe Bridge as usually constructed, results from the steep pitch of the braces. For, while, as was seen [LXVI], braces act with about the same economy at an inclination giving a horizontal reach equal to the vertical, as when the former equals only one-half of the latter, that is, withl h = v and h = v, it was shown in the succeeding section, that the action upon verticals was nearly twice as great in the latter, as in the former case. For instance, suppose Fig. 18 to represent a 16 panel trnss, with thrust braces and tension verticals. Estimating successively the action upon verticals with diagonals crossing two panels, as in Fig. 18, and the same with diagonals crossing but one panel, we find the action over 85 per cent more in the latter than in the former case. Vith regard to chords, the horizontal effect is essentially the same in both cases, while the vertical thrust WOODEN BRIDGES. 305 of braces, being but little over half as great with the long, as with the short horizontal reach, may be sustained by the timber oi the chord, thus obviating the necessity of tubes extending through the chord from the cast iron skewback; and furthermore, may enable the iron shoe to be dispensed with altogether, in many cases. Hence would result a still further saving in expense, as well as in weight of structure. Take, for example a brace 10" square, capable of resisting a tnrust of 50,000lbs. in the direction of its length, and a vertical pressure of 35,0Os. when inclined at 45~o Whether the end be cut as at d, e, orf (Fig. 64), it covers a horizontal area of 141 square inches, giving a square inch for every 250lbs. of vertical pressure. This does not much, if any, exceed the capacity of timber for resisting transverse crushing, as estimated in section CXLIII, when acting upon a portion of surface so limited with respect to the whole. Perhaps, however, the propriety of dispensing with the iron shoe, should not be too strenuously urged. But there seems to be little excuse for incurring the sacrifice of iron required in suspension bolts in case of the steep braces, over what is required with the greater inclination. The interference of bolts with braces, when the latter reach across two panels, is perhaps the greatest obstacle in the way of adopting the latter arrangement; and this may be managed by either passing the bolts through the intervening braces (which does not materially impair their strength, when supported at intervals by counter-braces), or between main and counter braces, as may seem most favorable in respective cases, In view of the above considerations, the author can not avoid regarding the usual practice in the construction of Howe Bridges, as decidedly faulty. 39 306 BRIDGE BUILDING. TRAPEZOID WITHOUT VERTICALS.* CLXIII. This form of truss, Figs. 13, 15 and 19, has been shown [XLIV, &c.], to be liable to a less amount of action upon materials, in sustaining a given load under like general conditions, than any of the other forms analyzed in this work; and this advantage may be made practically available in wooden bridge construction, by a system of chords and diagonals connected by transverse iron bolts and pins at the nodes of upper and lower chords. The lower chords should be proportioned in their several parts, nearly in accordance with the stresses to which such parts are liable. This may be accomplished by a pair of parallel courses of timber of uniform section upon the outsides of the chord from end to end, placed at such distance asunder as to admit the ends of diagonals between them, and also, to admit of additional courses of chord timbers upon the inside of the former, to be introduced as required toward the centre, to give in each panel a section of chord, proportional to the computed strain for such part. The pieces composing the several courses, may be spliced with the double lock, Fig. 57, usually with the centre of the splice at the nodes, or connecting points of chords with diagonals; no two splices in the same half-chord to occur at the same node. The upper chord should be increased in section by enlargement of the section rather than the number of courses. Or, in some cases, timbers may taper in thickness toward the ends of chords, either upper or * The characteristic of this truss, is not that strictly speaking it has no vertical members, but that there is no general alternate transfer of weight from diagonals to verticals, and the contrary. WOODEN BRIDGES, 307 lower. For instance, if 5" in thickness be sufficient in the end panel, and 7" be required in the next a timber extending over the width of two panels, 6" at the smaller, and 8" at the larger end, will answer the requirement with perhaps less waste of timber and labor than would suffice under a different arrangement. But such matters must be left to the judgment of the designer. The upper chord acting by compression, the timbers may be connected by a half-lap of 1~ or 2 feet at the nodes, where the main connecting bolts will secure the ends. The diagonals which act principally by compression (represented as the narrower ones in Figs. 65 and 66), may be in pairs, while those mostly exposed to tension (the wider ones), may be single, and placed between the former. Thus usually three pieces are united at each node, FIG. 65. In some cases where the thickness of diagonals exceeds the space between half-chords, the thrust diagonals may be shouldered to fit a boxing upon the inside 9f the chord; as by either of the vertical dotted lines, Fig. 65. Sometimes also, the boxing may extend through the whole depth of the chord, so as to require no cutting of the diagonal; and again, the thickness of the diagonals may be reduced in the parts 308 BRIDGE BUILDING. between chords, and no cutting of chord timbers required. When cutting of timbers becomes necessary for purposes as above, it should be in the parts where the greater surplus over the necessary net section occurs, whether in chord or diagonals. Every part should have a square inch of net available section for each 1,000s. of tension, and a square inch of bearing upon bolt, pin, or shoulder, for each 1,0001s. of either tension or thrust to which the part is liable; and the bearing upon bolts and pins should be estimated as equal to the diameter multiplied by the length of hole through the piece; or, equal to the section of timber severed by the hole. CLXIV. Fig. 66 is a general representation of the half of an 8 panel truss, suitable for a 100 foot common road bridge. Let v = 14', = distance between centres of upper and lower chords, and h = 12~', =horizontal width of panel. Then, assuming w = 10,000Fs. (= movable panel load), and wu' = 4,000lbs. (= permanent panel load), we have. =.893 (nearly), and D =.m77' = length of diagonal; whence, v = 1.34; and, computing the stresses of the several parts and members by the process explained in sections [XLIV, &c.,], the maximum vertical pressure at a equals 49,000ibso giving a longitudinal compression upon ai, equal to 65,660Tbs., and a tension upon ab, equal to 43,750fbs. For the double member ai, 8" x 9" timbers are sufficient; while 4" x 12" (in each half), would answer for ab. But to give greater transverse stiffness for supporting floor timbers, it is preferred to have the outside course of lower chord timbers 5 x 12 inches. WOODEN BRIDGES. 309. ^^ ^ ^^.^...-~~-..................... L^A^ 310 BRIDGE BUILDING. The piece bj having no office but to fill the space at b, and to give support to a i, may be of any convenient dimensions. The maximum tension of other portions of the lower chord is, for be, 56,250; for cd, 81,250, and for de, 93,750ibs. For the upper chord, we have compression ofih, hg and gf, 62,509bs., 87,500bo., and 100,000lbs. respectively. The diagonals and verticals are liable to maximum tension and compression as shown in the following statement; and may properly be of dimensions as marked opposite each in the right hand column below; in case of double members, the figures indicate the width and thickness of each. Parts. Tension, lbs. Compression, lbs. Cross-Section, inches. bi 28j000 double 3 x 11 ci 28,140 single 5 x 12 dh 20,435 4 x 12 eg 12,730 670 " 3 x 11 fd 6,700 6,700 d. 3 x 6 yc 670 12,730 " 3x 6 hb 20,425 " 3 x 7 ja 65,660 8 x 9 The inside course of lower chord timbers may be - a 4" x 12" piece extending from d across the two middle panels of the truss, spliced at each end to a tapering piece 4 x 12 at d, and 2 x 12 at b; and consequently, 3 x 12 at c. Then, leaving a space of 8" between half chords at d and e, we have 10" at c, and 12" at b. Each half of the upper chord should be 8" x 12", in the two middle panels, and placed 9" apart; connecting with a tapering piece each way, from 8 x 12 at g, WOODEN BRIDGES. 311 to 6 x 12 at i; where the end should be beveled to a line bisecting the angle aih, and abut against a beveled shoulder upon the upper end of the king brace ai. The king brace is also cut away upon the inside, leaving only 1" in thickness, to make up, with bi and ic, a, thickness equal to the space (13") between the halfchords at i. The parts thus meeting at i are to be fastened by 2 transverse bolts of at least 2"' in diameter. These afford the requisite square inch of bearing surface for each 1,OOObs. of pressure, with an unimportant deficiency for the member ic, which may be eked out with a 1" pin through bi and ic only, if thought advisable, thus giving 55 square inches for vertical and diagonal together. These members should extend at least 14" beyond the centres of holes. At h and g, the three diagonal pieces just fill the space between chord timbers, and require at h, two bolts and one plain pin of 1k" in diameter, and at g, the same number &c., of 13" diameter. The diagram shows only two bolts at each connection. At the point f, where two pairs of braces meet, one pair may be cut off at the meeting, and a 4 by 6 inch piece introduced, lapping 2 feet between the cut pieces (reduced each ~ inch in thickness, inside, to the extent of the lap), and secured by 2 bolts and 1 pin of 1" diameter; the upper end passing between the opposite braces, the latter being boxed V" inside, to afford room for the 4" piece; and the whole secured by a single 1" or 13" bolt through chord and braces. The connections at the lower chord are somewhat more complicated, but involve little difficulty. The best connection at a, is made by cutting a vertical 312 BRIDGE BUILDING. shoulder or heading i" deep, upon both sides of the half chord, as shown by the vertical dotted line in the diagram A, Fig. 66; the brace being forked with counter shoulders upon the inside. This affords 36 square inches of shoulder surface, which, assisted by 2 bolts of 1" diameter, give 50 square inches, to withstand less than 44,0001s. The end of the brace is thus made to bear directly upon the abutment without any crushing action upon the chord. At b, the space in the chord is 12", while the verticals descending parallel, would occupy 11". But giving a divergence of 2~", and boxing'" upon the inside of chord timbers, leaves a space of 62" between verticals at b. Then, boxing bh }" upon the inside at the crossing with ic, there will be a 3" space between braces bh at b, and a thickness of 41" (of the pieces bh) between the verticals bi; also, a shoulder of 1_" upon the outside, which may be made to act vertically in a boxing upon the inside of bi, thus securing the requisite bearing surface for the thrust of bh. Thus arranged, the point should be fastened with 2 bolts and 1 pin of 1-"' diameter. The piece bj will have 3" in thickness at b, and will be furred out, if necessary, to fill the space atj. The space at c is 10 "; and, cy being shouldered' at the upper side of the chord at c, and boxed " at the crossing with hd, the point c may be secured by 2 bolts and 1 pin of 1i" or 2" diameter. A " boxing of df at d, upon the inside, leaves a thickness of 9"9 being 1" greater than the space in the chord, and the pieces df therefore require a further reduction in thickness upon the outside between chord timbers, of T" upon each. The point d, requires 13t" bolts and pin. WOODEN BRIDGESo 31O The two single diagonals meeting at e, may be halved into one another at the crossing, and a 3x11 inch piece lapped and locked on to each, as shown Fa. 67. by a a in Fig.67; thus serving to fill the space in the chord, and to restore strength to the diagonals. The lap pieces are to be reduced 3" to 2k" in thickness below the lock at 1. Two 14" bolts are sufficient at the point e. - Transverse joists, or floor beams may be placed upon, or suspended below either the lower or upper chords. Sway bracing a..f may be locked and bolted upon the upper chords, and iron x tie rods used at the lower chords; the beam timbers being shouldered against the inside of cords, so as to strut them apart against the action of the ties. Angle braces from the king brace ai, to a transverse beam from truss to truss at i, will aid in preserving the erect position of trusses. These braces should usually be lapped and bolted at the ends, so as to act by either tension or thrust The preceding specifications, it is hoped, will serve to make the peculiarities of detail in the kind of truss under consideration, properly understood. It may be deemed advisable to adopt the rectangular, instead of the Trapezoidal form of outline for the truss, by extending the upper, to the same length with the lower chord, inserting vertical posts at the ends, and exchanging the double vertical bi, to a single diagonal meeting the upper chord and end post at their point of junction; thus simplifying the connections at b and i. This modification, unlike the case of the trapezoid with verticals, involves no increase in amount of action upon materials, though it increases the number of members, and changes the manner of distribution of the action. 40 314 BRIDGE BUILDING. MODULUS OF STRENGTH, FOR BRIDGE TRUSSES. It is shown in preceding pages of this work, that, knowing by experiment the strength of the materials to be employed, we may calculate the necessary crosssection of each part of a bridge truss, in order that it may sustain a given load, with a given stress upon the materials. It is sometimes, however, a satisfaction to have a confirmation of the correctness of our calculations, by direct experiment upon the same combination complete, which we propose to employ for actual use. For this purpose, instead of applying the test to a full sized structure, which would involve a great deal of labor and expense, the test may be applied to a model, made in the true proportions, upon any scale. Now, it is obvious that with the same combination and arrangement of members, the stresses, whether positive, negative or transverse, produced upon the several parts by the acting forces, will be in proportion, throughout, to the weight sustained, whatever be the length of pieces; such stresses being determined by the positions and angles, and not by the lengths of pieces. It is further manifest that the ability of parts to withstand the effects of the acting forces, must be as the cross-sections of parts respectively; and in similar models, the parts, being similar solid figures, have their cross-sections as the squares of the magnitude of scale upon which they are respectively constructed, while the bulk and weight of each corresponding part, and of the combinations complete, are as the cubes of the magnitude of scale. MODULUS OF STRENGTH. 315 Then, assuming two similar models, the scale of one being mn times as great as that of the other, the weights which they will respectively bear, under the same stress of material, will be as W to Wmr2, while their respective weights will be as 1 to m3. Now, dividing the sustaining power of each by its own weight,the quotients are as W toW n2 or as W to ~. But the lengths being as L to Lm, if we multiply the quotients just found by respective lengths, we have WL for the one, and Lm W - m, = WL for the other; showing that the length of a model truss by the number of times its own weight which it can bear (with a given stress), is a constant quantity, whatever be the scale of such model. W Again, the quotients W, and -, multiplied by the lengths L and Lm, give the products WL, and -~ x Lm, equal to WL. Hence, the product of a truss medal into the number of times its own weight which it is able to sustain, is also constant, whatever be the relative values of the two factors. It follows, that making these two factors variable, and representing them by Q and L, the one increases at the same rate at which the other is diminished; and, when Q = 1, L must be equal to the greatest length at which a truss of the same plan and proportions, and under the same stress of materials, can sustain its own weight alone. This length, as we have seen, is determined for a model upon any plan, constructed upon whatever scale, by multiplying the length of model by the number of times its own weight it is capable of sustaining. 316 BRIDGE BUILDING. This product may be called the MODULUS OF STRENGTH and the plan of truss which gives the largest MODULUS, may fairly be regarded as the strongest plan. The Modulus may refer either to the actual breaking load, as found by experiment, or to the load producing given rates of strain upon materials, as determined by calculation. EXAMPLES. (1). A bar of cast iron 1 inch square and 12" between supports, will bear (at 6,000lbs. to the inch of section, upon a leverage equal to depth of beam), a distributed load of 4,000lbs. which divided by its weight, = say 3.12lbs. gives Q = 1250; and L being 1 foot, the Modulus = QL, = 1,250 feet. (2). A beam of pine timber 12' long and 6" square, at 1,5001bs. to the inch upon a leverage equal to depth, as above, bears a distributed load of 18,000lbs. LCXLII.] For the weight, say 3 cubic feet at 361bs. = 108bs.; whence, Q = 1,000 166.6, which multiplied by L (= 12') gives Modulus equal to 2,000 ft. By reducing the length of the beam just considered, to 6 feet in length, retaining the same section, it would give a Modulus of 4,000 feet, instead of 5,000, as given in the Appendix to my former work; the difference arising from the assumption of a smaller specific gravity for pine in the latter case. (3). The two panel model with chord and rafter braces, mentioned in the latter part of ~ [CLVIII], 20" long, and weighing 0.18Th. supported a load equivalent to 3,885 times its weight, while L = 1 feet; whence, 3,885x1- = 6,475 feet, = its Modulus. MODULUS OF STRENGTH. 317 (4). A model wooden truss 4 feet long, made many years ago by the author, on the plan of the truss Fig. 66, having 10 panels, and a depth equal to 1- of its length, weighed 0.9bs. and bore a distributed load of ~0o 600ibs. Hence, the modulus of the truss was ~0 x 4 0.9 = 2,664 feet, being more than half a mile. The model was somewhat strained but not broken; and recovered its normal shape and condition on removal of the load. It was subsequently sent to the U. S. Patent Office. These examples, however can not be taken as indices to the relative merits for general use, of the different forms of truss to which they refer. Each possesses qualities suited to special occasions. (5). A model of a 6 feet Trapezoidal Iron Truss (the first ever constructed), weight a little less than three pounds, sustained 700lbs. distributed, without any appearance of overstraining; thus showing a modulus of 720 x 6 = 1,400 feet, with an estimated stress upon the 3 chord, at the rate of about 16,000lbs. to the square inch. The model represents a truss of 144 feet, upon a scale of 1 inch to the foot. The sustaining power of a full sized truss in the same proportions, would be 700 x 242, = 403,200lbs, while the weight of truss would equal 3 x 243 = 41,472Tbs. Doubling this for two trusses, and adding, say 10,000s, for beams, &c., we have 92,944Fbs. for the weight of a 144 feet bridge, capable of sustaining, at a stress of 16,0001bs. to the square inch upon the chords, over 356 net tons beside reight of structure. CORRECTIONS. Page 18, line 18 from top, ( A+ )M, should be ( +~ )N. 2h 19, line 9 from bottom, 2 should be 3 3 20, line 3 from top, v/2 - should be v /2,.. 21, line 7 from top, h = al should be h = ab. i/h- / 1 V2 22, line 2 from top, 3 v -+, l shlouldbe 3 -/h2 + o V v 22, line 4 from top, contracted, should be counteracted. 29, line 2 from top, = stress, should be gives stress. 42, line 19 from top, X should be + 55, line 5 from top, reference letter y, should be g. 65, line 5 of Note, db'a, should be dba. 11.85 11.8E 66, line 14 from top, 1-8. should be 18E. 12' 12 73, line 1 below Diagram, four should be three. 88, line 13 from top, w' should be w". 91, line 8 from bottom, X should be -. 110, line 2 from note, s/16, should be mniu16. 116, line 4 from bottom of note, a, b, c, d, &c., should be a c", d', &c. 119, line 2 from bottom (.84- 2) v, should be (.84 —.20) v, 120, line 10 from top, (4-52, should be (4.5'), 121, line 11 from top, =.84, should be -.84. 144, line 16 from top, adjust should be adjustment. 785.78 5 171, line 7 from top,5, should be 6 6 196, line 13 from bottom, (T + A)', should be (T + H)). 198, line 1 from top, 37~, should be 311. 199, line 6 from top, Fig. 31, should be Fig. 34-. 252, line 5 from top, 9w, should be.9w. 253, line 3 from top, 5.05M, should be 50.5M. 288, line 2 from top, $3.60, should be $3.50. ScITENTIFITI BO.. PUBLISHED By D. VAN NOSTRAND, Q3 Murray aCd 7 Warrei St-.1, Jew York. RANCIS' (J. B.) Hydraulic Experiments. Lowell Hydraulic Experiments-being a Selection from Experiments on Hydraulic Motors, on the Flow of Water over Weirs, and in Open Canals of Uniform Rectangular Section, made at Lowell, Mass. By J. B. FRANCIS, Civil Engineer. Second edition, revised and enlarged, including many New Experiments on Gauging Water in Open Canals, and on the Flow through Submerged Orifices and Diverging Tubes. With 23 copperplates, beautifully engraved, and about Ioo new pages of text. I vol., 4to. Cloth. $I5. Most of the practical rules given in the books on hydraulics have been determined from ex periments made in other countries, with insufficient apparatus, and on such a minute scale. that in applying them to the large operations arising in practice in this country, the engineer cannot but doubt their reliable applicability. The parties controlling the great water-power furnished by the Merrimack River at Lowell, Massachusetts, felt this so keenly, that they have deemed it necessary, at great expense, to determine anew some of the most important rules for gauging the flow of large streams of water, and for this purpose have caused to be made, with great care, several series of experiments on a large scale, a selection from which are minutely detailed in this volume. The work is divided into two parts-PART I., on hydraulic motors, includes ninety-two exper. ments on an improved Fourneyron Turbine Water-Wheel, of about two hundred horse-power, with rules and tables for the construction of similar motors:-Thirteen experiments on a model of a centre-vent water-wheel of the most simple design, and thirty-nine experiments on a centre vent water-wheel of about two hundred and thirty horse-power. PART II. includes seventy-four experiments made for the purpose of determining the form of the formula for computing the flow of water over weirs; nine experiments on the effect of backwater on the flow over weirs; eighty-eight experiments made for the purpose of determining the formula for computing the flow over weirs of regular or standard forms, with several tables of comparisons of the new formula with the results obtained by former experimenters; five experiments on the flow over a dam in which the crest was of the same form as that built by the Essex Company across the Merrimack River at Lawrence, Massachusetts; twenty-one experiments on the effect of observing the depths of water on a weir at different distances from the weir; an extensive series of experiments made for the purpose of determining rules for gauging streams of water in open canals, with tf'les for facilitating the same; and one hundred and one experiments on the discharge of water Itrough submerged orifices and diverging tubes, the whole being iully illustrated by twenty-three double plates engraved on copper. 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