STABILITY OF ARCHES. WOODBURY. No. 7. PAPERS ON PRACTICAL ENGINEERING, PUBLISHED BY THE ENGINEER DEPARTMENT, FOR TIIE USE OF TIlE OFFICERS OF THE UNITED STATES CORPS OF ENGINEERS. PAPERS ON PRACTICAL ENGINEERING. Nto. 7. TREATISE ON THE VARIOUS ELEMENTS OF STABILITY IN TIIE WELL-PROPORTIONED ARCH. WVITHI uiroerua. gab laes o'f rte fIttimate anld ActuaI gihrcst. BY CAPTAIN D. P. WOODBIURY, U. S. CORPS OF ENGINEERS. NIEW YORK: D. VAN N OSTRAND. 1858. TtEE first part of this paper, ending with Section V., is devoted mainly to the theory of the arch first proposed by Coulomb, and subsequently developed by Audoy, Petit, Poncelet, afnd other French authors. New developments and illustrations are here given, and new and extensive tables have been added. The thrust may be due to the tendency of the upper voussoirs to slide down their beds, or to tile tendency of some upper segment of the arch to rotate on the interior edge of its lowest joint.'Both of these tendencies are investigated as questions of maxima; and the greatest of the two resulting forces is the true thrust. In almost all practical cases this true thrust is due to rotation, and is Lyveloped at the instant of equilibrium preceding rupture and fall. For convenience we have called this the ultimate thrust. It is obviously much less than the actual thrust of the well-established arch; nor has any relation between the two been hitherto pointed out. The defective character of this theory was indicated by Coulomb, and has been clearly seen by many authors. Moseley, Mery, and others, have written able treatises in search or support of a better principles but no one seems to have seized the final idea, or combination of ideas, which places. the theory of the actual thrust upon a perfectly definite mechanical basis. ~We have before us an "Examen historique et critique des principales th6ories concernant l'Equilibre des Vouftes," by Poncelet, drawn up with the characteristic ability and learning of that great man, in which no allusion is made to any attempt to furnish a definite and exact theory of the actual thrust of circular and elliptical arches. Approximations to this thrust, 188 INTRODUCTION regarding the curve of pressure or curve of equilibrium as a sort of catenary, and the'intrados as nearly parallel thereto, have indeed been attained by English mathematicians of the last century, and, of late, far more completely, it is said, by Yvon Villarceau; and Carvallo, in a beautiful and highly practical treatise, published in the Annales des Ponts et Chaussees, 1853, has given, approximately, the actual thrust and all other elements of semicircular and elliptical arches surcharged horizontally. VWe say approximately, for Carvallo, to facilitate calculation, regards the joints of the arch as vertical, instead of perpendicular to the intrados, and determines the angle of greatest thrust by supposed rotations of the upper segments upon the intrados, instead of the lowest of the two curves which divide the joints into three equal parts. The last part of this paper, beginning with Section VI., furnishes an original and probably a new theory of the arch, with numerous tables giving the actual thrust of most arches in common use without calculation. This actual thrust is found to differ largely, in most cases, from the ultimate thrust above mentioned, their ratio varying from 1 to nearly 2. This new theory is based on the principle that the curve of pressure shall not approach the intrados or extrados within certain prescribed limits, and that it shall touch, at the three or five joints of rupture, so-called, the two curves which pass through those limits. Every case, as in the ultimate thrust, is a question of maxima. Coulomb pointed out the several modes of rupture to which arches are liable; in one of which the crown rises. Most writers upon the arch repeat his remarks; but no one seems to have given the subject any particular consideration —much less to have determined the limits within which such rupture is possible or probable. Indeed, without a knowledge of the actual thrust no practical or useful solution was possible. The reader will be surprised to learn that most of the light beautiful stone bridges of Great Britain are inclined to this mode of rupture. Our remarks on this subject, in relation to the limits of possible and practicable arches, and in the discussion of Table I, will be found, we trust, both interesting and useful. Every proposed bridge of large span should be investigated in view of this third mode of rupture, in which the arch may fall without disturbing the piers. If not unnecessarily heavy, the bridge will, in general, be in the neighborhood of the practicability connected with this mode of rupture. We have taken great pains to. prepare simple and comprehensive INTRODUCTION. 189 forlnulse for thickness of piers. These formulie are the same in form, whether we muse the actual or ultimate thrust; the only difference being in the magnitude of the thrust, and in its point of application or lever arm. Both the ultimate and actual thrusts of many arches consist of two parts: first, the thrust of the arch proper and a certain part of its load; second, the effect of a surcharge of constant vertical depth. These effects have been investigated separately, and the two maxima, taken directly firom the tables, give, when added together, the entire thrust in very slight excess. The accuracy of this method may be proved by comparing the results of Table F with those of a table given by Moseley at the end of his " Mechanics of Engineering." Although the tables and the reduced formula provided for the actual thrust cover most of the cases likely to occur in practice, still the subject admits of further development and illustration, as the field is supposed to be entirely new. Additional tables are desirable to make the theory more completely available, and somne of the given tables require enlargement. In connection with this theory we have said nothing of the sliding thrust; because that had already been disposed of in connection with the ultimate thrust. The sliding thrust, when it exceeds the actual rotation thiust, is, however, given in the tables at the end of Section VI. Cases of this kind are very rare. Very few arches conform precisely to the conditions which we are coinpelledl to assumne in the preparation of tables and formula. The geometrical method, given at the end of the paper, is independent of all conditions, and will furnish in a short time all the elements of the most complex case. It affords facilities for discussing the plan of an arch, which can hardly be found in calculation alone. It can be modified so as to give, by successive approximations, an intrados and an extrados at equal distances fiom the curve of pressure, with joints proportional to the pressure upon them; the pressure per unit of surface on these joints being constant throughout the arch, and, if we desire it, throughout the pier, which may always be treated as a part of the arch. The segmental arch, bounded by arcs of different circles, is itself, in most cases, a good solution of this problem of least material. The curve of pressure, in the same arch, at different stages of its load, is liable to great changes. Provided the extreme variations beallowable, 190 - INTRODUCTION. comparing the unloaded with the fully loaded arch, all intermediate variations may be kept still nearer the mean by putting on the load, in clue proportion, at the several parts of the arch, simultaneously. WVe are indebted for this idea to MI. Carvallo who makes perhaps extreme use of it. To prevent any change at all in the curve of pressure, in the arch wit h out any load and the same arch fully loaded, lie goes so far as to make the depth of the arch in constant proportion to the depth of the load on the same vertical lines. This may be very proper in aqueduclvt and other heavy bridges; but in light arches without much load at the key-the case of common bridges-it would give an extrados nearly horizontal. Certainly, there is no objection to a change in the situation of the curve of pressure, provided it remain within the limits prescribed by Navier; since, in that case, the change cannot be for the worse, and by allowing it we greatly diminish the quantity of material, without in any degree impairing the stability of the work. No attempt has been made to investigate the change. of form and Circumstances which, in some measure, must always result from the compression of the material. If we suppose the piers immovable, the extent of joint everywhere proportional to the pressure, and the curve of pressure everywhere at the middle of the joint, compression will convert the circular into an elliptical arch-a change in most cases favorable to the stability of the light arch, although it increases the thrust. These conditions can only be fully realized under a very peculiar load; they will, however, often be nearly realized, and will cause, approximately, the change above mentioned. In connection with the actual thrust, Section VI., nothing is said of adhesion of mortar, because this force can never act in the well proportioned arch, the curve of pressure passing within such limits that every part of every joint is compressed. The reader who may consult this paper for practical purposes only, is advised to begin with Section VI., and to refer to the first part of the work only when necessary to understand the second. Lie is also advised, as the arch must be regarded from many points of view, to acquire a general knowledge of the whole of that Section, before studying critically the particular patts. It may be asked, Why not, then, invert the work, and place the more important part first? The reason will be given. The first part was INTRODUCTION. 191 nearly completed before the author had discovered an exact mechanical foundation for a theory of the actual thrust-a foundation on which tables of that lthrust could be calculated, and definite results obtained, without any sacrifice of principle to facility of calculation. The second part, contrary to the original intention, began to expand, and finally assumed, well nigh, a character of completeness by itself. It was too late to re-write the whole. The author had neither time nor health to do so. Nor could the first part be dispensed with: it is necessary as a complement to the secondc. It is to be regretted, however, that the fullness of illustration andc demonstracion given to the first part of the work has not been given rather to the second. The great variety of dimensions adopted by different engineers for arches of nearly equal span and rise, is shown by Table I. No better proof could be given of the utility of some little attention to the theory of the arch. The engineer may thereby avoid, on the one hand, a large wasteful expenditure for useless excess of strength, or, at the other extreme, the mortification of seeing his work fall down in consequence of impossible proportions. The preparation of this paper has involved long and arduous labor, in the reduction of formule to numerical forms, and in the calculation of tables. The author gratefully acknowledges the assistance of DR. D. W. WVITrEHURST and MR. GEORGE B. PHILLIPS, in making the numerical calculations. Without their efficient aid, he could not leave undertaken so large a task. CONTENTS. [Suci parts as are supposed to be entirely new, are given in italics. Thle numbers refer to the paragraphs.] SECTION I. GENERAL INVESTIGATION WITHOUT REGARD TO PARTICULAR CURVES. Definitions and general remarks, 1; influence of common and hydraulic mortar, 2; general discussion of the thrust in the first mode of rupture, by rotation, 3 —11; stability of the pier, 11, 12; effect of surcharge upon the ultimate thrust, 13, 14; the effect upon the thrust of any supposed adhesion of mortar at the joint of rupture and at the vertical joint, 15; general formulae for the ultimate thrust in the first and common mode of rupture, by rotation, and for thickness of pier, 17; the coefficient of stability, as used by Audoy, Poncelet, and others, 18, 19; general discussion of the thrust due to sliding, 20-22; point of application of the thrust, 23; the manner of expressing the effect of adhesion, 24; general discussion of the third mode of rupture, by rotation, the key rising, 25; fourth mode of rupture, sliding, 26. Page 199-215. 19 4 CONTENTS. SECTION II. THE SEMICIRCULAR ARCH OF EQUAL THICKNESS THROUGHOUT, Application of the general formula, 27-47; arches without rotation thrust, 30; effect of mortar, 31, 32; compcarative elfect of mzortar upon7 large and small arches, 33; effect of surcharge of a constant depth, 34; the entire ultimate thrust obtained by addin~g two maxima together, 34; the effect of a column or weight upon the' key of the arch, 35; the thrust due to sliding, 36-39; the entire sliding thrust obtained by adding two maxima together, 39; the true thrust, 41; thickness of pier, formulse, 42; limit thickness of piers, 43; discussion of table A, 46; limit thickness of the circular ring, 47. Page 215-234. SECTION III. THE MAGAZINE ARCH, INCLUDING THE EXTREME CASE IN WHICH THE TWO PLANES WHICH FORM THE ROOF, ARE ONE AND HORIZONTAL, OR THE CIRCULAR ARCH SURCHARGED HORIZONTALLY. Application of the general formule to the muagazine arch, 48-67; the ultimate thrust, due to rotation, 48-51; effect of surcharge of a constant depth, 50; the entire rotation thrust obtained by adding two naxima together, 50; effect of a single column resting upon the ridge of the arch, 51; the sliding thrust, 52 —54; the entire sliding thrust obtained by addcling two mzaxinca together, 54; remarks on tables C, D, F, 14, 55; the roof inclined 45~, 56; the roof horizontal, 57; discussion of table C, the roof inclined 450~, 58; table D, the roof horizontal, 59; discussion and use of table F, which gives directly, or by proportional parts, the ultimate rotation thrust of all circular magazine arches in common use, with and without surcharge, also the sliding thrust when this exceeds the other, 60-63; thickness of piers, universal method, 64; particular cases, 65, 66; examples, 67; limit thickness of piers, 64. Page 235-264. CONTENTS. 195 SECTION IV. SEGMENTAL ARCHES-SCARP WALLS. General remarks, 68; the segmental ring, 69-73; thickness of pier, 72, 73; example, the Monocacy bridge, 73: segmental arches surcharged horizontally, 74-79; thickness of pier, 79; segmental arches with inclined roofs, 80-82; thickness of pier, 82; stability of a scarp wall, 83, 84; segmental arches, approximate formulce, 85-89. Page 264-290. SECTION V. ELLIPTICAL ARCHES. General remarks, 90; the unloaded elliptical ring, 91 —95; the elliptical arch with elevated ridge, 96-98; elliptical arches surcharged horizontally, 99-107; at the end of section V. tables A, B, C, D, E, from volume 12 Memorial de l'officier du Genie; and tables E', F, G, H, prepared for this paper; these tables give the ultimate rotation thrusts or the sliding thrusts when the latter exceed the former. Page 290 —323. SECTION VI. THE CURVE OF PRESSURE, NEW THEORY OF THE ACTUAL THRUST, ETC. ETC., BASED ON KNOWN MECEIANICAL PRINCIPLES. [The whole of this, as a definite system, is supposed to be new: the particulars are not given in italics.] Objections to the theory of the ultimate thrust, 108; general.principles, 108-111; pressure, per unit of surface, on the joints, 112; fundamental principles of the theory of the actual Thrust, 111-113; actual thrust of the unloaded circular ring, 114; thrust of semicircular arches surcharged horizontally, 115, 116, 118; the magazine arch, 117; segmental arches surcharged horizontally, 119, 120; elliptical arches surcharged horizontally, 121; the coefficient of stability, 122 —126; thickness of pier, universal method, 128; particular cases, 129-134; examples, 135; thickness of arch, how determined, example, the Monocacy bridge, 136; proper increase of thickness at and below the joint of greatest thrust, 137-139; relative pressure, per unit of surface, at the key and at the weakest joint below the key, 140; pressure, per unit of surface on the joints of the pier, 141; third mode of rupture, by rotation, the key rising, 142; limit thickness of possible circular arches surcharged horizontally, 142; limit 19 6 CONTENTS. thickness of possible segmental arches surcharged horizontally, 143; limit thickness of practicable circular arches surcharged horizontally, 144; ditto, segmental arches surcharged horizontally, 145; effect of surcharge upon the practicability of arches, 146; equation of the curve of pressure in the arch, 147; point of application of the actual thrust at the key, 148; point of application of the ultimate thrust at the key, 149; remarks on table I. 150; general remarks upon the determination of the thrust and upon the thickness of the arch at the key and at the weakest joints, 151, 152; geoinetrical methods of universal application, 153 —159; rupture of masonry by compression, 160; curve of pressure, in the pier, 161; in the arch, 162. Page 324-423. TABLES OF THE ULTIMATE THRUST, ETC. A. Semicircular arches, unloaded: ultimate rotation and sliding thrusts limit thickness of piers, etc. etc., page 310-11; explanation, paragraph 46. B. Formule for thickness of pier, unloaded semicircular arches, page 312-13; explanation, paragraph 42. C. Magazine arches, the roof inclined 45: ultimate rotation and sliding thrusts, etc., etc., page 314-15; explanation, par. 58. D. Semicircular arches surcharged horizontally: ultimate rotation and sliding thrusts, page 316-17; explanation, par. 59. E. The unloaded segmental ring; seven systems: the ultimate rotation, or the sliding thrust; in all cases the greatest of the two, page 318; explanation, par. 69. E'. Segmental arches surcharged horizontally; seven systems: the ultimate rotation, or, the sliding thrust; in all cases the larger of the two, page 319; explanation, par. 74. F. Magazine arches, ten systems, with columns for surcharge: the ultimate rotation, or, the sliding thrust; in all cases the larger of the two, page 320-1; explanation, par. 60 —63. G. Elliptical arches surcharged horizontally: the ultimate rotation thrust in eight systems; the sliding thrust, being less than the former, is not in any case given,-page 322; explanation, par. 99. CONTENTS. 19 IH. Elliptical arches without load: the ultimate rotation thrust in two systenis; the sliding thrust, being less than the former, is not given,page 323; explanation, par. 93. A. Corrections, to be used in computing the ultimate thrust of the unloaded elliptical arch from table A, page 294, par. 92. Table of the ultimate thrusts of elliptical and segmental arches, compared, page 304;. par. 101. Table of additions to the ultimate rotation thrust of elliptical arches of all kinds, caused by a surcharge of constant depth, page 306; par. 104. TABLES OF THE ACTUAL THRUST, ETC. ETC. AA. Semicircular arches, unloaded: ultimate and actual thrusts, ratio of the two; the sliding, being less than the actual rotation thrust, is not given,-page 424; explanation, par. 114. DD. Semicircular arches surcharged horizontally, the curve of pressure starting at the middle of thile key and passing through the middle of the joint of greatest thrust: the ultimate and actual rotation thrusts and the ratio of the two; the ultimate and actual effects of suircharge also compared; the sliding, being less than the actual rotation thrust, is not given,-page 425; explanation, par. 116. FF. Magazine arches, four systems: actual rotation thrusts, with a column for surcharge; coefficient of stability or ratio of the actual to the ultimate thrlust; the sliding, being less than the actual rotation thrust, is not given,-page 426-7; explanation, par. 117. DDD. Semicircular arches surcharged horizontally, the curve of pressure starting at one third the length of the joint from the extrados at the key, and passing the weakest joint at the same distance from the intrados: actual rotation thrusts with a column for surcharge; angles of greatest thrust, on three different hypotheses, compared; coefficients of stability, compared; the sliding, being less than the actual rotation thrust, is not given, —page 428; explanation, par. 118. 198 CONTENTS. EE. Segmenttal arches surcharged horizontally: actual thrlust in seven systems, with columns giving the effect of surcharge, and the ratio of the actual to the ultimate thrust, or the coefficient of stability; the sliding, there exceeding the actual rotation thrust, is given in the upper part of the last column, —page 430-1; explanation, par. 120. I. Elements of celebrated bricdges, page 432-; explanation, par. 150. Table.; Limit thickness of possible segmental arches, page 385; par. 143. Table. Limit thickness of practicable segmental arches, page 389; par. 145. ERR t A T ATA Page 287, line 14 from top, for K==- read C=1.-+ 9r z 4 5 Page 319, over 2d column, for r=5~/ read q'=-J./ TIIE THEORY OF THE ARCH. SECTION I. DEFINITIONS AND GENERAL RE5M1ARKS. 1. The arch proper consists of several parts, commonly called vqoussoirs, which press upon each other in surfaces meeting at one or more central lines. It is distinguished from the beam by exerting, upon its piers or abutments, an outward thrust. It may be upright or reversed, horizontal or inclined. Cylindrical arches are fully represented by a section taken at right angles to their general direction. The surfaces of this section give the relative masses of all the parts of the arch and its piers; and to obtain the actual weight of a unit in width, it is only necessary to multiply the surface, expressed in feet or any other unit, by the weight of a cubic unit of masonry. The arch, when submitted to calculation for the purpose of determining the requisite dimensions of its piers and other parts, is supposed to be a unif in width. The extrado6. is the exterior outline of the arch proper, whether made up of curves or straight lines. 2 200 THEORY OF THE ARCH. The intradacos is the interior line, and the corresponding surface of the arch is the soffit. The wedge-like pieces of which the arch is composed, are called voussoirs. In this paper it will be taken for granted, unless otherwise specially mentioned, that all. arches have a joint at the summit, midway over the opening between the piers, called the key-stone or vertical joint. The sides of the arch are called the reins. The line or bed at which the arch begins, or springs from its piers, is called the springing line. The blocks of masonry, or other material, which support two successive arches, are called piers; the extreme blocks, which generally support, on one side, embankments of earth, are called azutmrents. A pier strong enough to withstand the thrust of either adjacent arch, should the other fall down, is sometimes called an ab6trnen t pier. Besides their own weight, arches usually support a permanent load or surcharge of earth or masonry, or both. 2. Common mortar in heavy masses of masonry, hardens very slowly-remaining comparatively soft long after the centering has been removed and the arch left to its own support. We must not, therefore, in our calculations for large arches, look for any element of strength or stability in the adhesion of mortar made of common lime. We must suppose the voussoirs, whether large or small, to press upon each other without adhesion to resist either rotation or sliding. The joints are always rough, and any tendency to sliding is resisted by ordinary friction. On the other hand, arches of all sizes laid ill good hydraulic mortar,sand thin arches laid in common mortar, may derive some fncrease of stability from the adhesion of the mortar which unites the joints. Fortunately we can, by calculation or geometrical con ROTATION. 201 struction, easily determine the whole effect of this adhesion, when we assign to it a particular value, that is, so many pounds per square inch. Perfectly flat arches, with vertical joints, are entirely supported by this adhesion. They are not properly arches,.but beams. So far as any arch derives strength from the adhesion of its parts to each other, it partakes of the nature of a beam. The thrust of the arch, due to rotation in the case of actual rupture and fall, is, for convenience, sometimes called the ultimnate t7hruist. THEORY OF COULOMB. 3. We shall now indicate, in its most general terms, the theory of the arch first proposed by Coulomb, and subsequently developed by Audoy, Petit, Poncelet, and other distinguished French engineers; a theory hitherto eminently useful, notwithstanding its great defects. We shall hereafter, under the head of curve of pressure, notice these defects, and try to give a more exact and practical theory of the circular, segmental, and elliptical arch. FIRST AND ORDINARY IMODE OF RUPTURE-ROTATION. 4. Let figure 1, plate 10, represent a section of any symmetrical arch in a condition of stability. We may suppose this to be composed of two equal half-arches, meeting, and mutually supporting each other, at the vertical or key-stone joint. From the perfect equality of the two halves, it is evident that the pressure of the one is counterbalanced by the resistance of the other, and that the two forces are equal, horizontal, and directly opposed. This mutual pressure is exerted, generally, all along the vertical joint, from the intrados to the extrados. Its re 202 THEORY OF THE ARCH. sultant is applied at some unknown point ordinarily above the middle of that joint. The position of the resultant, however, becomes known, as we shall soon see, when the arch, no longer able to maintain itself, begins to fall, from insufficient resistance in itself or its supports. Let fig. 2 represent a section of the arch in a condition of instability-that is, all but able to stand upon its springing lines, and just beginning to fall. Fig. 3 represents the piers as adhering to the lower parts of the arch, and the whole as just beginning to fall. This is by far the most common mode of rupture,-almost the only mode, according to the French authors cited above, to which the circular arch of common use is at all exposed. 5. In this' first mode of rupture, the arch divides into four pieces; the crown settles down; the reins spread out; the vertical joint opens at the intrados, the adjacent segments touching only at the extrados; the reins open at the extrados, the adjacent segments touching only at the intrados; the weakest joint below the reins opens on the inside. The two lower segments revolve, outwardly, on the exterior edge of what we have called the weakest joint below the reins-thereby leaving rooan for the two upper segments to revolve, towards each other, on the interior edges of the joints at the reins. If the first rotation be prevented, by the weight of the piers or lower parts of the arch, the second rotation will also be prevented; for the crown can not settle down unless the reins spread out. What we have called the weakest joint below the reins, that is, the joint most disposed to open under the horizontal force or pressure acting at the crown, is, in almost all practical cases, the joint at the base of the pier, or at the springing line if there be no pier. 6. Let us suppose one half of the arch removed, fig. 4, and apply at a, the highest point of the arch proper, a ROTATION. 203 horizontal force, F, just sufficient to keep the remaining semi-arch in its place. This force is the thq-yst of the arch, and is equivalent to the mutual pressure which before existed. It is evident that the arch, in its tendency to fall, canlnot generate a pressure or force greater than that which will prevent its falling. 7. If the semi-arch were in one solid, indivisible piece, standing on its springing line, A B, the force. F would be known at once; for, calling /the moment of the semiarch, in reference to A, the interior edge of the springing line, we should have PX aCi=-l;/ or F — 8. But the arch is not solid: it may separate at any joint. If we regard any segment of the arch cta 6 n, a, fig.. 4, we see that a certain force, F', is necessary to prevent its fall, or rotation forwards, on the inner edge of the joint gn n. From g, the center of gravity of the segment of tlhe arch proper and its surcharge, drop the vertical line g' upon the horizontal, n' inM'. Let the surface cab n n r a= —S. 4" lever al1rm a qn'-y. We then have the equation of moments F',y=Sp; or F'_ 9. Now, if we suppose the position of the joint n n to vary, in succession, fronm the key to the springing line, the force TF', required to prevent the fall of the corresponding segments, will also vary. It will be very small near the summit, gradually increase towards the reins, become a maximum about 50 or 60 degrees from the key,. and grad 204 THEORY OF THE ARCH. ially diminish to the springing line. This maximum value of F,, whichl we shall call 1F is the thrust of the arch. As it is sufficient to sustain that segment which required the greatest support, it is more than sufficient to sustain all other segments, including the semi-arch itself. 10. The magnitude of this thrust, and the position of the joint corresponding to the angle of maximum thrust, are, it is easy to see, entirely unaffected by the piers and the lower parts of the arch,-that is, by anything below the joint itself. This remark is important, for it announces one of the few simple principles on which the whole theory of the arch depends. 11. The tendency of the horizontal thrust is to push over the semi-arch, causing it to turn round the exterior edge of its lowest joint. But if the moment of the thrust taken in reference to that center of rotation, be less than the moment of the semi-arch and pier, in reference to the same point, no Notion can ensue: the arch is stable. On the other hand, if the moment of the thrust prevail, the arch will begin to fall as represented in figures 2, 3. The equation of moments will be MiL representing the sum of the moments of the semi-arch and its pier, and I the lever arm of the thrust. 12. If we take into consideration the adhesion of mortar upon the joint of the springing line, or base of the pier, the equation of moments will become, art. 15 and following, Ex l=if-H+-ce, c representing the force of adhesion upon a unit of surface, and e the length of the lower joint. In this expression ROTATION. 205 there is nothing unknown except the thickness of the pier, which enters illto M. 13. If the arch have any surcharge whatever, not already included in X, we can always represent its weight by a saitface of masonry. Let S'- aa's', fig. 4-, represent such weight in magnitude and position; and let p' repre-.sent the horizontal distance of its center of gravity from the vertical passing through rn, the center of rotation. The horizontal force necessary to prevent the rotation of any particular segment and its load, will now be F' S9p+S'P Y Y The maximum value of F' as before, art. 9, will be the true thrust. We might, in arches of large span, wish to ascertain the effect, upon the thrust, of several weights in given positions. Let S', S" &c., represent their several magnitudes; p', p" &c., the respective distances of their centers of gravity from the vertical line through qn. We have Fw.=SP~fS'X+s'?" +... y 14. In general, however, the surcharge is continuous, and bounded by one or more straight lines. It is hardly necessary to say that, in the equation above given, the weights S', S", &c., are supposed to be located over the segment a b m nz r a, to which F' relates; and that, if the center of gravity of any one of them, as S", is on the left of the vertical through In, the'lever arm,'p", will be negative, and the product S'jp", therefore, negative. The surcharge adds, generally, but little to the difficulty of the investigation, as will hereafter be seen. 15. To complete the general formulae for the first and most common mode of rupture, it is necessary to add expressions for the adhesion of mortar upon the joints. 206 THEORY OF THE ARCH. The ultimate resistance of mortar to compression is much greater than its resistance to extension. Still, we can, without error, regard these forces as equal, provided we determine their value by experiments upon rectangular prisms of the same material as the arch,-that is, by observing the weight necessary to break a beam supported at one or both ends. The product of this weight by its lever arm makes known the effective resistance opposed by the mortar. The expression -cc12, which, as we shall see, measures the entire effect of the mortar when the ultimate resistance to extension is supposed to be equal to the ultimate resistance to compression, is identical in value with the expression lc'd2, which measures the effect of mortar when the resistance to compression is infinite, and the neutral axis at the edge of the joint. But c as determined on the first supposition, will be twice as great as c' determined on the last. Let us suppose, then, the ultimate resistance of mortar to extension and compression to be the same. Let c represent that resistance. When the joint rn n, fig. 4, has so far opened that the mortar at n is about to separate, the adhesion or effective force then applied at that point will be c -the ultimate strength of the cement. At the same instant the mortar will be compressed at in to the full extent of its capacity to resist; and there will be a neutral axis at the middle of the joint, where the mortar is neither extended nor compressed. The resistance, X, to extension, is proportional to the extension, and increases uniformly from the middle of the joint, where it is nothing, to the extrados, where it is c. It is given, therefore, at any point at the distance x from the neutral axis, by the proportion id': c"'::: X —l-; (d'mn) The elementary force is therefore x dx, and the entire resistance to extension ROTATION. 207 rd''cx cd' 2 o ldd= 4 R In like manner we find the resistance to compression to be, cd' 4 When we refer these two forces, which act in opposite directions, to any center of moments not between their respective resultants, we nmust regard them as having contrary signs. IReferred to m, the common center to which all the forces of the system are referred, the force R, tending to prevent rotation, is negative; the force R', tending to cause rotation, is positive; the lever arm of R is,rmn k6d'; the lever arm of _' is 6 mn= c'. Their combined moment, tending to prevent the rotation of the voussoir a 6 m n r a, around the point m, is ted' 4. (x f 6' In like manner, we find, at the vertical joint, above the neutral axis, a resistance to compression, C, acting with the lever arlnm (y —.d), tending to prevent rotation around the center, m; and below the neutral axis, the force C 4, acting with the lever arm (y -- d), tending to cause rotation around the point mn. The combined moment opposed to rotation is cd c(y - I - uy + I )= d 2 The two joints, therefore, oppose to rotation the moments -c(d2+d'2). We might have shortened the demonstration by assuming at once this obvious principle, that the algebraic sum 208 THEORY OF THE ARCH. of the moments of any two equal, parallel, and opposed forces, is constant for all centers of rotation in the same plane, and always equal to either force multiplied by the distance between their respective directions; a principle which might have a very extensive application in calculating the total effect of certain combinations of timber in wooden bridges and other structures. The French engineers, regarding the neutral axis as at the edge of the joint, give, as the moment of resistance due to mortar, -ed 2. We have been the more particular in the above reasoning, because we differ from Poncelet in estimating the effect of mortar upon the vertical joint.* X Poncelet says, page 201, vol. 12, Mlemorial de l'Officier du G6nie, after giving as the effect of mortar, c(d2+ d'2) cd; in which, fig. 4, d'-mn; d=-ab;;3Y 3(H+ h) y=am'; h=-EE'; H-aC; c=the force of adhesion upon a unit of surface," We are led inevitably to this result, by the principle of virtual velocities, in considering, at the same time, the cohesion on the joint of rupture of the reins, and on that of the key, and in supposing, after Mariotte and Leibnitz, that this force is proportional to its distance from the point of rotation of each joint. The justification of this result presents no difficulty except in that part which relates to the second of these joints, which is not usually considered, Now, we must observe that the resultant of the elementary forces x dx, which ant along ab, in opposition to a movement of rotation at a, and of which the moment, in reference to that point, is ( tcx2 evidently fd -dx=-cd2, can, as to this movement, be replaced by a horizontal force cd2 -, supposed to be applied at the edge of the intrados, m, of the lower joint; and 3y as the latter is opposed, at the same time, to the action of the horizontal thrust, pF, in its tendency to overturn the entire system of the semi-arch and its pier around the exterior edge of the base of the latter, with a lever arm, H+ h-y, whilst that of Fis H+ h, we see that, in relation to this movement, it must, in its turn, be replaced by a horizontal force cdcl(H+ — y) acting at a, to resist the movement in question, 3y(H+I ti) conjointly with the force which is capable of overcoming the cohesion, on the lower joint m n, with the lever arm y, and of which the known expression is cd' We 3y have, therefore, as the total resistance, the quantity, cd2(Hf+ h —y) cd'2 c(d2 + d'2) cd(2 3y(H+ h) 3+ y 3y (H+ h)' UNIT OF WEIGHT. 209 16. We llust caution the reader that the unit of weight in the foregoing equations, is the weight of a cubic unit of masonry. To make the terms involving c homogeneous with the other terms, we must regard c as the ratio of the strength of adhesion, in pounds, upon the unit of sunLface, to the weight of the same unit cubed. ~For instance, suppose the force required to separate a square foot of mortar to be 3,000 pounds, the cubic foot of masonry weighing 150 pounds, we must have c- 3~00 ~-20. 17. Collecting the above formulae, we have, as the genwhich makes the position of the joint of rupture at the reins depend upon the height of the arch and that of the pier." In the above translation we have changed the notation, to suit our diagram and text. This very distinguished philosopher has, we think, here fallen into an error. The force cd which the mortar opposes to rupture at the crown of the arch, ~~~~2~~~~~~~~~~Cd2 and of which the moment, in relation to the upper edge of the vertical joint, is cd is not opposed to rotation around in, the inner edge of the joint of the reins. Its whole tendency is to cause such rotation. The equal and opposite force, or resistance to compression, acting at a, the sunmmit of the crown, is alone opposed to rotation. The point of application of the force cd, or resistance to extension, is situated upon the vertical joint at two-thirds its length from thle extrados. Its lever arm, in reference to in, is (y - 2d); and the lever arm of the equal and opposite force, or resistance to compression, acting at the extrados of the crown, is y. The combined moment, in relation to in, is therefore d-( y-(y —2d) )- d. Wherever we suppose the neutral axis to be, the total resistance to compression must necessarily be equal to the whole resistance to extension; and this does not cease to be true even if we suppose one of these resistances, per unit of surface, to be infinite,-that is, if the neutral axis be at the edge of the joint. MI. Poncelet seems to have fallen, inadvertently, into one or two other errors, which make his final result nearly correct. The force cd, which we can substitute for the "resultant of the elementary forces 8y cx ed dx," or_-, in relation to the opening of the joint of the crown, for that very read 2 2 son we can not so substitute in relation to any other center of rotation. Finally, we are not at liberty to refer the moments of a system in equilibrium to different centers. 210 THEORY OF THE ARCH. eral value of the horizontal thrust, in the common mode of rupture, Without surch. ) the maxi- ) or adhesion of Imum val- F' — Sp (1) mortar, ue of ) Y With surcharge, do F= Sp +S' I + S9"+ (2) With adhesion do F',_Sp c(d2+d'2) of mortar, -y With both, do.F S__+' S'p" + c(d'2+d2) (4) y2 Y And, for the thickness of pier, F representing the thrust, however determined, Without adhesion of mortar at x Fx 1-=M (5) the base of the pier, With adhesion of molrtar at _=_ +~ce2 (6) the base of the pier, e is the thickness of pier, 6 the coefficient.of stability. 18. No arch could stand in bare equilibrium. Experience has shown that the senmi-arch and pier must have a certain excess of stability over tihe thrust; and the French engineers have provided fori this excess, in the equations which determine the thickness of pier, by assigning to the thrust an increased value-multiplying it by the coefficient of stcaility, generally assumed at 1.90 or 2 in the heavy arches of fortifications, but which may be safely assumed at less in establishing the piers of ordinary bridges. We shall give a discussion of this important subject hereafter. 19. The thrust at the crown, clue to the mutual action of the semi-arches, though not sufficient to turn over the semni-arch and pier, may under some circtmstances cause the whole mass to slide outwardly on the base of the pier. The equation of equilibrium, calling TV the whole surface of the semi-arch and pier, and f the friction or ratio of SECOND MODE OF RUPTURE-SLIDING. 211 resistance to pressure, will obviously be, F= WXf; and the practical formula, calling 6' the coefficient of stability, will be'_F'= Jxf. If we wish, however, to take account of the adhesion of mortar upon the base of the pier, we shall have'lF -W/xf+c x e. (7) As to this particular danger, the weakest joint is generally at the springing line, or very near it. Applied to that case, Wwould, of course, represent the surface of the semi-arch, and e its thickness at the springing line. If the masonry be well constructed, even of poor mortar,-that is, well bonded togetlher, —no sliding can ever take place at the springing line or base of the pier. The thickness of pier should always be determined in view of rotation alone. In those very rare cases in which any danger of sliding can still remain, care must be taken in the preparation of the foundations to render such motion impossible. The most economical and effective expeclient is, probably, that of giving to the base of the pier a slight inclination-say one foot in ten. SECOND MODE OF REUPTURE-SLIDING. 20. When the thickness of an arclh compared with its span is very great, the horizontal thrust no longer arises from any tendency to rotation, the lever arm p, equation (1), being very small, or even negative. The thrust arises from the tendency of the upper voussoirs to slide down their beds or joints. It is precisely like the thrust of an embankment of earth, and is determined in the same manner, viz., by the "prism of maxinmm thrust." Let f represent the friction along the joint r n2,, fig. 5. a c " the angle of friction, measured from a horizontal. 212 THEORY OF TtIE ARCH. Let v represent the angle between the joint qn n, and a vertical line.' S " the surface of the segment a b qn n a. " P' " the horizontal force acting on the joint a 6, necessary to keep the segment from sliding clown the joint m nz. We have, perpendicular to the joint m n, the force P' x cos. v+Ssin. v; and, parallel to the joint m n, the force S cos. v -Pt sin. v; hence the equation of.equilibrium, (P' cos. v +S ill. V)f cos. v - P sin. V. Taking account of the adhesion of mortar along the joint mn=d', we have, (P' cos. + sin. ) f+c x d' cos. v- P' sin. V. We have, therefore, as the general expression of the horizontal thrust, adVithout thle maximum},= S(cos. v —f sin. v) -x cotang.(c+ v); (8) aesion, value of sin. v f cos. v adhesion, do P'-S cotang. (a+v) -cd' os. (a) adhesion, sin. (a+v)' ( As in the case of rotation, we must suppose the angle v to vary from the summit towards the springing line, and ascertain the maximum value of P' in (8) or (9). This value, which is the sliding t?,uqst of the arch, we shall designate by P. 21. Equations (8), (9), include the case of a surcharge, if we suppose S to take in all the load which rests vertically upon the segment a b6 n n._ 22. In arches of common use, this mode of rupture can never take place. The resistance necessary to prevent rotation, which one half of the stable arch necessarily opposes to the other half, is, in almost all practical cases, THIRD MODE OF RUPTURE-ROTATION. 2 13 greater than the force necessary to prevent sliding. If it were otherwise, if P were greater than F, the former would be the proper thrust of the arch, and would take the place of Fin equations (5), (6), (7), arts. 17, 19, when we wish to determine the thickness of pier. 23. We commit an error in favor of stability, ill supposing the force P to be applied at a, the extrados at the crown. Its true point of application is always intermediate between ca and b, and generally near the middle of that joint. 24. The effect of adhesion on any joint, in resisting a force acting parallel to that joint, we have, in the usual manner, measured by the product of the force of adhesion -upon a unit of surface into the length of the joint expressed in the same unit. It does not follow that the force c, which will separate a square foot of cement when acting at right angles to the joint, will tear asunder neither more nor less when acting parallel to the joint. The rule, however, is the only one that has been offered, and it is, without doubt, sufficiently. correct. It would perhaps be better, proceeding experimentally, to regard the adhesion solely as giving a greater value to the friction. THIRD AMODE1 OF RUPTUR:E-ROTATION. 25. This is still more uncommon than the second. Gothic arches, fig. 6, and arches very light, and lightly loaded at the crown, and overloaded at the reins fig. 7, are liable to this mode of rupture. As compared with the usual mode of rupture, arts. 4, 5, figs. 2, 3, every thing is reversed. The crown rises; the reins fall in; the vertical joint opens at the extrados, the 214 THEORY OF THE ARCH. adjacent segments touching only at the intrados; the reins open at the intrados, the adjacent segments touching only at the extrados; the arch still divides into four pieces; the upper segments turn outwardly on the exterior edges of the joints at the reins; the lower segments turn inwardly on the interior edges of the lower joints. The active force which pushes over the upper segments, acts in this case at the intrados of the crown. It is generated entirely by the effort or tendency of the semiarch, or some segment of the arch above the springing line, to revolve under its own weight, turning on the inner edge of its lowest joint. To obtain the exact value of this thrust, we must determine the maximum value of the expression -, fig. 4, differing from the horizontal thrust in the first mode of rupture only in the lever arm of the thrust, which is now b6n'-my', instead of amn'-=y. Now, the horizontal force at b, fig. 4, necessary to cause any other segment to commence rising at the crown, turning outwardly upon n, is -,, y" representing the lever arm bn', and A' the distance of n, the center of rotation, from the vertical, ~g', dropped from the center of gravity of the segment. If any segment be caused to rise, it will obviously be that which offers the least resistance. We must therefore by trial find the ni7zimun value of q. If the horizontal thrust, as defined above, or rather if the maximum value of Sp,, which is a little greater than that thrust, exceed this minimum value, the arch will fall. This may take place in very light circular or segmental arches, surcharged horizontally; as will be proved hereafter. FOURTH MODE OF RUPTURE —SLIDING. 215 FOURTI-I MODE OF RUPTURE-SLIDING. 26. The thrust at the key, generated by the mutual action of the semi-arches, might, it would seem, cause the arch to slide outwardly upon some of its lower joints. The horizontal force at the key, necessary to cause such movement is Scotang. (v —a): for notation, see art. 20. Its least value, in practice, is always at the springing line, where, if anywhere, sliding will take place. The equation of equilibrium of the semi-arch upon its springing line, supposed to be horizontal, has already been given, art. 19. SECTION II. 27. We have given in the first section, the theory of the arch in its most general termns. We shall hereafter give geometrical methods of determining the thrust at the crown, and all the other elements of any case that is likely to arise, in terms equally general; that is, independently of the particular nature of the curves of the extrados and intrados. As circular arches are by far the most common, and admit of the most precise calculations, we shall first apply the theory to them. The tables calculated by M. Petit, Capitaine du genie, and the more. extensive tables prepared for this work and now published for the first time, will enable us to dispense with calculations in many cases, and greatly abridge them in nearly all. The brevity and simplicity which characterize the general expressions of the horizontal thrust, unfortunately vanish when we submit the most simple cases to calculation. 3 216 THEORY OF THE ARCH. CIRCULAR ARCIHES, INTRADOS AND EXTRADOS PARALLEL. 28. Referring to fig. 8, let I be the radius of the extrados; ir that of the intrados; v the arc which, in the circle whose radius is unity, measures the angle of any joint gn n with a vertical. Resume equation (1), F'=. We have,' -= I - (RI1); f = p sin. v —_ (R2 -i)1 s); y=? —7r cos.; These values substituted give,Sp 3 $r(_R2?9-VSi:.qg —2(,s —3)( cos. ). (10) ~~~S~ r(R-2)v sin -2(kV ( cos. v)..g _1~ny 6 (1 -r cos. v) an expression of the horizontal force F' which, applied to the extrados of the key, can prevent the voussoir a b m n a, corresponding to the aingle v, from turning round the point qn. This we can simplify by introducing the ratio of the two radii, -i R —, which gives K j 2 _t)V sin.y - I 1 ) (1.- v) () K — cos. v The value of v, or the inclination of the joint of rup" Distance from the center of the circle, fig. 8, on the bisecting line Cg, of the center of gravity of the sector Can R 2 x R 2 sin. vcv yR6 0 b-n - 2 sin. av 6bm= -...... sin.v ~ V surf. c a n x d-surf. C rn x d' "ring a b m n a=surf. of ring. _,(R3 —r3)2 sin. }av (R2-_r.) p=mg'=Lmm'- g'm'=r sin. v- -g x sin. ~u=.. as aLove Surface Ca n=IvR2; 2 sin. -,v=I —cos. v. CIRCULAR ARCHES. 217 ture which corresponds to the maximum value of F', may be obtained in the usual way by differentiation. The numerator placed equal to zero, gives, by reduction, -;2W" I cos. v+(1 — Jcos. v) siV - (12) sin. lf1 and the corresponding value of F' is F=-, (z'-1(T71(. cos.')1(T3-i)) (13) These expressions establish the highly important generalizations that, in all similar arches having the ratio I1 of the two radii constant,1st. The thrust is proportional to the square of the radius of the intrados, or to the square of any other linear dimension; 2d. The angle of rupture is constant. Let nt represent the numerator of the second member of equation (11). Let z represent the denominator of the second member of equation (11). The condition that F',=, shall be a maximumm gives ZC6 —tcz=-O; or -=-. In this way (13) was obtained. The value of v obtained by trial from (12) and substituted in (13), gives the true thrust. In this way, M. Petit has calculated table A, giving, either directly or by proportional parts, the horizontal thrust and the angle of rupture for all values of K, that is, for all semi-circular arches which have a constant thickness, or the intrados and extrados parallel. 29. To illustrate the march of F' through the quadrant, as we assign to K a particular value, and to v in equation (11) a succession of values, we have calculated 218 THEORY OF THE ARCH. the following table, of which the last column, to make the results more tangible, gives the values of _F' in pounds; r' being taken at 10 feet, and each unit multiplied by 150, the assumned weight of a cubic foot of masonry. K=1.20; r==10 feet; =-12 feet. v=100 F'= —2 x.013855=10 x 10 x 150 x.013855= 208.00 pounds. " 200 " r2 x.044678= 15000 x.044678= 670.00 " " 300 " r'"x.075111 — 15000 x.075111 =1127.00 " " 40 " 2 X.096668- 15000 x.096668=1450.00 " " 50~ " r2 X.108370=- 15000 x.108370 —1625.55 " " 550 " 2 x.110980- 15000 x.110980=1664.70 " " a570" 2 rx.111470=- 15000 X.111470= —1672.05 " 4 600 " r2 x.111700=- 15000 x.111700=1675.50 " " 63~ " r x.111390= 15000 x.111390 —1670.85 "," 65~ " r.2x.110746 = 15000 x.110746 —1661.19 " " t70O " 2 X.108160= 15000 x.108160-=1662.00 " " 800 " r x.099360 — 15000 x.099360 —1490.00 "' 90~ " r2 x.085660= — 15000 x.085660= —1285.00 " The angle of maximum thrust is about 60), and near that angle the variations of F' are very small,-only 53jpounds from 50~ to 70~. The exact value of the angle of rupture or angle of maximum thrust, is a matter of no importance. 30. If we suppose v-0 in equation (10), the thrust becomes -nothing. The same supposition, v=-0, in (12) gives. 3f2 — 8 — 2 from which we deduce two positive roots, Ji-[l; and, I_7-2.732 For these two values of ]iV' there is no thrust, and no angle of rupture, except that of the key. The first value, -=_ =r 1, corresponds to an arch infinitely thin, without weight, and therefore without thrust. EFFECT OF MORTAR. 219 The second value, K-2.732, corresponds to arches of great thickness, in which, however small we make the angle v, there can be no thrust, because the center of gravity of the segment resting on any joint, will fall within the intrados of that joint. For all values of K greater than 2.732, the thrust would be negative; that is, it would require a positive force to turn over any voussoir if one half the arch were to stand by itself. We are here speaking of rotation. Such arches have a very decided thrust from the tendency of the upper voussoirs to slide upon their beds. EFFECT OF M1ORTAR. 31. Let us now resume equation (3). FrISp l (6t +d4 ) 1 Y an expression of which the maximum value will give the thrust as modified or diminished by the adhesion of mortar. We have d=d'- -r r(K-A —- 1); and y-,_(x-cos. v); hence C(d2+(cl2) (K- )2 6y 1K —cos. v' To illustrate the effect of the mortar, let us agrain take up the arch corresponding to K —1.90, art. 29. Let us suppose the adhesion of mortar to be 3000 pounds per square foot, and the weight of a cubic foot of masonry 150 pounds; 3000 giving, art. 16, ce 150 =90; and reducing (r (K T)2 0.8 1 3g-cos. 3 1. 0- cos.v 220 THEORY OF THE ARCH. Recapitulating the table given in art. 29, and subtracting from each value of Pf' the effect of mortar corresponding to the same value of v, we havePounds. Pounds. Pounds. v —100~ F'=r2 X.013855 — r x 1.2392= 208.00 -1651.00= — 1443.00 " 20 " r2x.044678 -rxl.0244- 670.00-1537.00= — 867.00' 300~ " rx.075111 —rx.8000-1125.00 —1200.00= - 75.00 ".400 " r2 x.096668-rx.6145-1450.00- 922.00 —+ 528.00 " 500 r2 x.108370 —rx.4800=1625.55- 720.00 —+ 905.55 "' 60~ " r x.111700 —rx.3810=1675.50 — 571.50= 1104.00 " 70~ " r2 X.108160 —r X.3110-1622.00- 467.00= 1155.00 " 80~ " r'2x.099360 -rx.2600 —1490.00- 390.00= 1100.00 90~ " r2 x.085660 —r x.2200 —1285.00- 330.00- 955.00 The true thrust or greatest value of iF' corresponds now, we see, to about T0~, and is reduced from 1675 pounds to 1155 pounds. 32. We simplify the calculation, without any sensible error, by supposing v, in the term involving the adhesion of mortar, equal to 600, which, in general, differs but little from the angle of maximum thrust. This gives cos. v-. We shall therefore be able, with little labor, to correct the thrust, as given by the 4th column of table A, or obtained by a direct calculation, when we assign to the adhesion of mortar a particular value. Let C represent the decimal of that column, or the decimal obtained by calculation,-such that'2 X C is equal to the maximum thrust without regard to mortar. We have, as the thrust diminished by mortar, =? 2 X C_ 27, (K_ 1)2 (14) The numerical factor, 2 -9I_ = l t', is very easily calculated when we know the value of ]K. For instance, K- 1.20, gives C'=.01905; and (14) becomes, substitut EFFECT OF MORTAR. 2 1 ing for.' this value, and for C the value given by table A,F=r2 X.1114 —r x c x.01905, which reduces, when c-20, to F=r2 x.1114 —r x.381, tC" " "r=10 feet to F=11.14-3.81_=7.33=1100 pounds, differing slightly from the thrust above given for the same case; our calculation not giving precisely the tabular value of the thrust. 33. Equation (14), which can be put under the form = 9'2x C-rX c C', leads to this generalization. While the thrust of similar arches ( / — being constant), independently of the mortar, increases as the square of the radius of the intrados, the effect due to the mortar increases only as the first power of that radius. Consequently, in arches of large span, the effect of mortar becomes insensible; and in arches of small span, this effect may reduce the thrust to nothing. Placing the second member of (14) equal to zero, we have at once the radius of the intrados corresponding to an arch without thrust, (~-1)2 C 2 (K C_ r2C=-re K -2 giving r=0; and C- 2K —1 ) CI 3 2K-1i C 2K- - If K=1.20, we have, from table A, C-.1114, and by calculation C'=.01905. If we furthermore suppose c=20, we have -- 3'.42. This corresponds to an arch of 6'.84 span and a little more than 8 inches thick. If K-1.50; c=_20, we have as the radius of an arch without thrust, r-=9'.66. If K-2; c=20, we have r=34'.14. The general principles announced above relative to the effect of mortar upon similar arches of different spans, are, it is evident, of universal application, whatever be the curves of the extrados and intrados. 22 2 THEORY OF THE ARCHI. EFFECT OF SURCHARGE. 34. Equation ('2) Sp l+Sipjl+ C Sp +sp2,"+ _Fp —S'p'+.., gives the force Y' Y Y F' necessary to sustain any supposed segment of the arch when the extrados is loaded with the weights S', S", &c., with their centers of gravity in vertical lines at the distances p', p", &c., from the center of rotation, or point mn of the intrados. If the surcharge be continuous, and nearly constant in vertical depth, find in table A the value of P>:2s C, and to this add SP'+S" p +&c. 7y R-r cos. v calculated according to the circumstances of the case. The sum will be, with sufficient exactness, the thrust increased by the surcharge. Let the surcharge, reduced in depth, if necessary, to give it the density of the masonry, be of the constant vertical depth t. We shall havep'=Sil. v(r- -R)=r sin. v(I —-JK); S-=rtKsin v; and S) tI(2 -K) Sill.2v an X -cos.; which becomes zero, IY 2 X cos. I for all values of v, when ]=-2, or R= —'. The center of gravity of the surcharge then falls upon qn. Sill. V The variable factor, fw —cos.' we find by differentiation to be a maxitnunl when cos.v -- /f'-2 1; and KS v (K-r — ll ). K( Cos. ~, We have, therefore, with a very slight error in favor of stability, for the thrust increased by a surcharge of uniform depth, — EFFECT OF SURCHIARGE. 22 3 FI- 20C+rt XK(2 K)(fr- VK21 ) (15) in which C is obtained froln table A, for the given value of IV and the numerical factor K(2 - K) (K- VK2i) =NV is easily calculated when we know the value of ME. Example; K=1.20, which gives 1XV.5152. Table A gives C —.1114; hence F=- -2X.1114+rtX.5152; suppose tGar; we have FP)-2.1114-+2.1030 —r2.2144; that is, when K- 1.20 the thrust is nearly doubled by a surcharge of uniform thickness t=l-. The value of v, which renders sIn. v a maxinmm, K-cos. v never differs more than four degrees from the value of v corresponding to the same value of K]in table A; and it is the property of a maximum to exceed but little the adjacent values of the same function. We may therefore regard equation (15) as exact. Table F gives the values of N for all values of r between 1.0.2 and 1.42; these values being the same in all circular arches of equal thickness throughout, whatever load they may sustain in addition to this surcharge of constant depth. Tables A and F, therefore, give the thrust required without calculation. See discussion and use of those tables. 35. Let us take up another variety of the surcharge. Suppose a single column, represented in weight by a surface one unit in width and fH in height, to rest upon the crown of the arch. We have, F' — + sin. y TL-cos. v The maxilum value of =1'2, is given, without regard to surcharge, by table A. The maximum value of Hsin. v II — cos.' obtained by the calculus, is H/___1 The K- Cos. V /AIP 224 THEORY OF THE ARCH. true thrust, therefore, with some error in excess, is H 9,lsz2 ( -;I+ (16b It is remarkable that this addition to the thrust, caused by a weight upon the crown of the arch, is independent of r, the radius of the intrados, and therefore constant while K remains the same-that is, in all similar arches. This is also evident from general considerations. Example. On the crown of an arch, 20 feet in span and 2 feet thick, we wish to throw a column one foot square and 10 feet high, one half being supported by each semi-arch. We have = 1. 2 0; C, from table A, =. 114; =7.54; X- r.ll1 14+-7. 54 18.68. The angle v, in table A, corresponding to JIT 1.20, is 590.4. The angle v which renders sn-. V a maximum, is K- cos. V 1 r given by the relation cos. v —. —-- 83333; or v=33~. 35,. The true thrust would correspond to v=about 50~. In the above example we commit an estimated error in excess, or in favor of stability, of about five per cent. The error will be less as KT increases or I[ diminishes. The effect upon the thrust, of a weight placed upon the crown of an arch, is evidently the same in all circular arches, whether otherwise loaded or not. INTRADOS AND EXTRADOS PARALLEL. 295 TIIRUST OF SEMI-CIRCULAR ARCHES, SLIDING.-INTRAIDOS AND EXTRADOS PARALLEL. 36. Resume equation (8), art. 20. P'=Scot.(a+v), an expression for the horizontal force requirecl to prevent the segment whose surface is S from sliding down its bed or lower joint n m fig. 5. We have S —v(R2_ —.); and, substituting XK for R, pt 1.2(tK2 1) v cot. (a+v). The angle of friction according to Boistard is 37~. 14'; according to Rondclelet, 30~. Admitting the latter value, as more favorable to stability, we have. P'-=11,2(K'- 1) X x cot. (v+30~) (17) The angle v, corresponding to the maximum value of P', deduced by trial from the condition } sin. 2(v+30~) =v, or directly from equation (17), is a little over 26~, and it is obviously the same for all circular arches which have the extrados and intrados parallel, whatever be the relative parts. Giving to v this value, 26~, in equation.(17), we have, P -.2X (_K2 21) X.15304;. (18) from which M. Petit has calculated the sliding thrusts in table A. It will be seen in that table, of which a full discussion will be given hereafter, that the thrust due to rotation is greater than that due to sliding, for all values of XC from 0 to 1.44; and that the thrust due to slidcling exceeds the other for all values of KI greater than 1.44. This value of K corresponds to an arch of 20 feet span and 4'.40 thick throughout. 22 G THEORY OF THE ARCH. EFFECT OF MORTAR. 37. Resume equation (9):P Scot. (J + v)-_ed Cos. Ga sin. (a +v)' which now becomes 2(K oc7t. cr(KT —1) CoS. 300 19,a -1)v (a+v)- sin. (v +30) let us suppose K_=1.50, r==10', c=2, and find the maximum value of P'. POUNDS. v —20 P'r2 x 306 - x 1.1 305 —18.306 - 11.305-7.001 = 1050 4 26~" Ir2 x.19130 - r x 1.0444 —19.130 - 10.444 —8.680 —1302 "L 30 44 r2 X.1S890 -r x 1.0000 —18.890- 10.000=-8.890 —1333 4 350 x r2 X.17803-r x.9555=17.800- 9.550 —8.250=1237 The angle v, corresponding to the maximum value of P', is now about 30~. The effect of the mortar for that angle is rc(K —1); and it is nearly the same for v-26~. We have, therefore, this simple formula for the sliding thrust diminished by the adhesion of mortar:P =2 rcr(lK-1)K (19) in which C is taken directly from the fifth column of table A, and c is the adhesion of mortar upon a unit of surface — art. 16. The remarks made in art. 33, relative to the effect of mortar upon similar arches of different spans, are equally applicable here. The second member of (19) placed equal to zero, gives, as the radius of an arch without sliding thrust, r-0, and i- c(yK- 1). For c=2; K=1.50, which gives, table A, C —=.19130; we find r=:5.227; For c=2; K=1.20, " " C" —.06733; " r=5'.940. If we suppose c-20, as in art. 33, we have, as the EFFECT OF SURCHARGE —SLIDING. 2 27 radius of an arch without sliding thrust, for ]X —1.50, r —52'.27; for ]I —1.20, r-59'.4. The effect of mortar to prevent sliding is, we see, far greater than its power to resist rotation. In the latter case it has full effect only at the outer edges of the opening joints, and its influence is in most cases still further reduced by the small leverage with which it acts. 38. It is only in heavy arches, in which the thickness is nearly half the radius of the intrados, that the effective thrust is determined by any tendency to sliding. And in such arches we can not rely upon any adhesion of joints, unless the mortar is strongly hydraulic,'and considerable time has been allowed for it to set before removing the centering. EFFECT OF SURCHARGE UPON THE SLIDING THRUST. 39. The general equation applicable to this case is P' = cot. (v+ —30~) +fS' cot. (,v+ o30). We can always suppose the vertical depth of the surcharge, as far, say, as 25~ or 30~ from the crown, to be constant. Let t be that depth. We have S'-R sin. v x t - r=tKin. v, and S' cot. (v+30~)=-rtKgin. cot (n 4-.00) The maximum value of this expression corresponds very nearly to v=25~; while the maximum value of S' cot.(v +30~) corresponds to v —26~. Adding the two nmaxinma together, we have, with a very slight error in favor of stability, the sliding thrust increased by a surcharge of uniform vertical depth,P=2-C+~ —tKx..29592. (20) C is to be taken from the fifth column of table A, opposite the given value of K. 22S THEORY OF THE ARCH. The numerical coefficient fKX.29592 is given in the last column of table F, for values of lI ranging from 1.35 to 1.50, corresponding to arches of large span in which the sliding exceeds the rotation thrust. The thrust, increased by surcharge and diminished by mortar, is, P -,2 C+rtKX.29592 -rc(I- 1), (21) C being taken, as before, from the fifth column of table A. Example. Y -2; c —2; t- 5'; Ir-1 0'; weight of a cubic foot of masonry = 150 pounds. Table A gives C4(.45912..P= /r2 X.4591 —,rt X 2 X.29592 —2ir -45.91-2+29.592'- 20. 8325 pounds. If c-o, P-11395 pounds. 40. Table A begins with l-' 2.732, above which the rotcttion thrust is less than nothing. The sliding thrust, however goes on increasing as lif increases. Equation (21), expressed more generally, becomes PJ-2X (x2-1) X.15304+r.tKx.2s959s2-c( -1) (21)' in which r', ITd t, c, may have any values whatever: t is the mean depth of the surcharge, whether consisting of one or more masses; but in estimating the value of t, we should confine our attention to that part of the surcharge which is over the extrados within 30~ of the summit. 41. In the investigation of the true thrust, as modified by surcharge and mortar, three distinct cases may arise. I. It may be evidently due to rotation. See art. 31, and following. II. It may evidently be due to sliding. See art. 36, and following. III. It may be doubtful, and require an investigation of both cases. THICKNESS OF PIER. 229 The greatest of the forces, F, P, required respectively to prevent rotation and sliding, will, ill all cases, be the true thrust. The third or doubtful class will be very small in the hands of those who have made themselves soniewhat familiar with the subject, and especially with the tables contained in this paper. THICKNESS OF PIER. 42. The horizontal thrust at the key enters as an element into two questions: 1st, the thickness of the pier; 2d, the ability of the material of the arch to stand the pressure at the summit and at the reins. The latter question we shall take up hereafter. We have shown how to obtain this thrust, both for rotation and sliding. The greatest of these two, which are found all calculated in table A, is the true thrust. We defer for the present the supposition of any adhesion of mortar, or of a surcharge. Let F represent the thrust; C the greatest of the decimals in colum-ns 4, 5, opposite the given value of K. We have F —2 X C. Let h represent the height of the pier from the base to the springing line, fig. 8; 1 the lever arm of the thrust; e the thickness of the pier, 3 the coefficient of stability. We have I-k+ C a-h+l-. We must give to the pier such dimensions that its moment, increased by the moment of the semi-arch, shall be equal to the moment of the thrust multiplied by the coefficient of stability. Let nl= _1jt(Ki2-1); Tm -- (1-1); then n.2 - surface of semi-arch;?.S -- moment of semi-arch in reference to C, (n - 9h) r' —moment of semi-arch in reference to A4 7zu.2 X e+-(z —Mn)) - -3-moment of semi-arch in reference to Ei or F'. 230 TIIEORY OF THE ARCH. WTe have, therefore, expanding equation (5),- 7h2+.r2e+ (inb- m)- ra(2(K ( +h) (22) of whiclh the solution gives e + / 917 (6CK+ mn-n) +26 (23) Table B, calculated by M. Petit, gives the reduced form of this equation for all values of K between 1.10 and 2, on the supposition of strict equilibrium, = 1; the value of C being taken in each case from table A. To modify any one of these equations by introducing the coefficient S, we have only to add, under the radical, to the coefficient of A, 2 CK( — 1), and to substitute for the term independent of 2, the same term multiplied by the coefficient of stability. Example. K=1.50. C, being always the greatest of the decimals in columns 4, 5, is.19130 r t The coefficient of T under the radical, is.19370 To this we must add 2 CK( —1) (-1) X.57390 The term independent of, is.38260 For this we must substitute X.38,260 The equation will then read, e r /..9817 +h.9638- + (.1937 + (8 - 1).5739) +9 x.3826. LIMIT THICKNESS OF PIERS. 43. Mere inspection of equation (23) shows that when the height of the pier becomes infinite, we have e=-rx 2C THICKNESS OF PIER. 231 that is, the thickness of the pier whose height is infinite must be equal to the square root of double the horizontal thrust multiplied by the coefficient of stability. This interesting principle was discovered experimentally by Rondelet. It is universal-applicable to arches of every form, and under every variety of circumstances. The moments of the thrust and of the pier increase in nearly the same proportion with the height of the pier. This limit thickness is not very much greater than the thickness required for moderate heights. Slightly changing his radius, we take from M. Petit the following illustration; r=10'; _/- 12'.50; 1] 1.25. Strict equilibrium. Coefficient of 1.90. h - 7.60 feet. e —2.5000 feet. e- 5.6111 feet. " 10.00 " L' 2.8190 " " 5.7995 " 15.00 3.3180 " 6.0 87 I 20.00 " " 3.6407 "' 6.2640 " " 25.00 " " 3.8657 " " 6.3825 " 50.00 " " 4.4012 " " 6.6550 " " 250.00 " 4.9235 " 6.9147 " " infinite. " 5.0687 " " 6.9865 " The whole increase of the practical thickness, =_1.90, from Jh-10 feet to Ainfinity, is but little lnore than 14 inches. Table A gives the limit thickness for all values of K from 1.10 to 2.00, both for strict equilibrium and for a eoefficieent of 1.90 - s1, and 61.90. WVe thus have the means of criticising many existing cases, and nmay often be spared much labor and research. 44. If we wish to take into account the adhesion of mortar, or a load upon the back of the arch, the thrust, r2Cq which enters equations (22), (23), will no longer be furnished by table A alone, but must be determined according to the circumstances of the case, as already fully explained in this section. 4 23 2 THEORY OF THE ARCH. 45. If we take into consideration the effect of mortar only, the thrust is given by the greatest of the two forcesF=-2C2 — ~rc(2 1) ); eq. (14), art. 32; C from column 4, table A. P=rq2CU —rc(K-1); eq. (19), art. 37; C from column 5, table A. /Frepresenting the greatest of these forces, the thickness of pier is given by ('23), when we have substituted F for r2C( or for'. (23) becomes e -n2~ n 2 ___ 2F (23)' The effect of surcharge upon the thickness of pier, will come up in the discussion of other arches. DISCUSSION OF TABLE A. 46. This table gives, either directly or by proportional parts, for all values of K=- between 1 and 2.732. 1st. The angle of rupture. —Rotation. 2d. The ratio, C, of the thrust to the square of the radius of the intrados, in the case of rotation and the case of sliding. 3d. The ratio, /2/C, of the limit thickness of pier to the radius of the intrados, for the case of strict equilibrium, and for the coefficient of stability of 1.90, —_1, and 3=1.90. It will be seen that the angle of rupture, beginning with zero for K=2.7832, becomes 54~ 27' for K=2.10, DISCUSSION OF TABLE A. 233 attains its greatest value, 64~ 9' for ]i_1.50, varies between 54~ 27' and 64~ 9! for all the fortification arches ia common use, that is, for all the values of K between 2.10 and 1.12, and ends with zero again for K- 1. It will also be seen that the thrust due to sliding is greater than the thrust due to rotation, for all values of X greater than 1.44; and that the former is less than the latter, for all smaller values of KI. We must in all cases, to obtain the true thrust, select the greatest of these two values. Calling the greatest of these values (or the greatest of the decimals in columns 4, 5) C the limit thickness of pier is e=rl/2Ct for.strict equilibrium. e=rI/3.80C for the coefficient of-stability 1.90. e=-r/OGC for any coefficient of stability 6. As we have rV/2C=r ~ x 4/C, it is obvious that, while the thrust increases in a geometrical ratio, the limit thickness increases only in an arithmetical ratio; and that a small error in the thrust becomes much smaller in the pier. We are at liberty to suppose ri, the radius of the intrados, to remain the same throughout the table. Assuming this, we see that the greatest possible thrust that can be caused by rotation in any arch of the radius ir, is )r2 X.17535, and corresponds to E= 1.58. For instance, if r- 10 feet, we shall have I= — 1.58, R=15'.80, and P —r, or the thickness of the arch - 5'.80. We must not fail, however, to notice that the greater thrust, when IK= 1.58, is given in the column of sliding. 47. By means of this table, M. Petit has settled this question: What is the thinnest arch that can stand upon its springing lines? 234 THEORY OF THE ARCH. It is evidently necessary that the moment of the thrust should not exceed the moment of the semi-arch, both taken in reference to the exterior edge of the joint of the springing lines. Now, the moment of the semi-arch is jSi31t IE(2C; 1) -ne(ll 1)) The thrust is. 6C; its moment Kolk Valne of V Ratio of the diameter Moment of stability of the semi- Moment of the K=-. to the thickness. arch upon its springing lines. horizontal thrust. 2.000 2.000 r3 x 2,379075 r3 X 0.918240 1.500 4.000 r3 x 0.680956 r3 x 0.286250 1.300 6.666 r 3x 0.305502 ra x 0.186290 1.200 10.000 r3 x 0.172024 r3 X0.133608 1.150 13.333 r3 x 0.117659 r3 x 0.105528 1.120 16.666. r3 x 0.088806 r3 x 0.087763 1.114 17.544 r3 X 0.083320 r1 3 X 0.083434 1.100 20.000 r.3 x 0.071093 r3 X 0.074292 1.050 40.000 r'3x 0.031954 r3 X 0.040034 1.010 200.000 r3 x 0.005844 r3 X 0.008980 The inspection of this table shows that the arch has abundant stability for 17 — 1.30 and upwards; that its moment is in equilibrium with the moment of the thrust, for K-=1.114 nearly; and that the thrust prevails mpre and more for smaller values of K. This value, J+-=1.114, corresponds to an arch of which the span is a little more than 1 7- times the thickness; and that is the thinnest or lightest arch that can possibly stand upon its springing line. A thinner arch would be impossible. This fact has been confirmed by experiment. THE POWDER-MAGAZINE ARCH. 235 SECTION III. SEMI-CIRCULAR ARCHES OF 1800 WITH A ROOF-SHAPED SURCHARGE UPON TH.E CROWN. 48. The powder-magazine arch, fig. 9, belongs to this class. To make the inclosure bomb-proof, a surcharge of concrete, or other hard material, and sometimes of earth, also, is added to the covering of the arch proper. We suppose, for the present, the plane of the roof on each side of the central ridge, to be tangent to the proper extrados of the arch, as it frequently is in practice, and the the joint of rupture to rise vertically above the extrados through the surcharge. We also suppose the thrust to act always at a, the extrados at the crown, the mass of masonry or other material above acting only as a weight. Let I be the angle between the slope of the roof and a vertical; v still the angle between the joint of rupture and a vertical; r the radius of the intrados; _R the radius of the extrados, supposed to be equally distant from the intrados throughout; — R. We have, as the general expression of the horizontal force required to keep any voussoir A' cat b n p A' from falfing by rotation round the point mn, sin22v ( 2(K-co ) siln. I(6-31i-(3-2 ) sin. (I +v))- (24) sin. v' COSi. l \ o R - -R isin. (I+-v) Fig 9. CA -=B; np- =B; Rsin. v=h = perpendicsin. I sin. I B~-2B' ular distance between CA' and np; h3(B+ 2Bthe distance of the center of gravity 3(B+B') of CA'pn from a(A'; h x B —2(,. sin.- -h(+ / —)moment of trapezoid ( CA'p~ on =n- =( I(+)2R — R sin. x3r(.2vin 2s (3R-2eRsin.(I+v))) -sin2 236 THEORY OF THE ARCH. The maximum value of this expression must be found for any particular arch by trial. M. Petit makes an application of this formula, to the powder magazine of Vauban, of the following dimensions, viz.: I-=49~.'. 171"; R —5.035 metres-16.5148 feet; r=4.0605 metres=13.3184 feet; K=1.24. These values of I and K substituted in the preceding equation, give for v=53~.. F'=0.229290 X'2; t v —=54~.. F' —0.229381 Xr2=F;" " -- 55~.. _F'0.229295 X'.2 Near the angle of maximum thrust, and as far as six or eight degrees on both sides of that angle, the variations of 1F are very small. The exact determination of that angle is not important. It may be regarded in the present case as 54~. The thrust is _F0.229381 X2=40.69 feet. If one cubic foot of the masonry weigh 150 pounds, the thrust is equivalent to the effort necessary to sustain 40.69 X 150 pounds=6103 pounds. EFFECT OF 3IORTAR -UPON THE ROTATION THRUST. 49. To obtain the thrust diminished by the adhesion of mortar upon the joints, we have only to subtract from the value of the second member of (24), corresponding to each assigned value of v, the expression (see equation (3) and art. 15, 31),{, K' (6-3K-(3-2K)sin. It+v }). Moment of sector, Cb6, on m=-vr2(r sin. v r22 r3Sin.2v 1 )4 4sin.2 vv i.." xa_\'= F xr 7V sin-v cos.26V C cos.2 iv' (K-cos. v)=moment of trapezoid CA' p n —moment of sector CGb n; hence F'Bas above, (24). EFFECT OF MORTAR. 237 c(d2~+'2)_ _C(K-1)2 6y 3(K-cos. v) The resulting maximum will be the true thrust. But we can simplify this expression, as in art. 32, by supposing v, in the term involving the adhesion of mortar, to be always 60~, which differs but little from the angle of maximum thrust in heavy semi-circular arches, however loaded. See the various tables contained in this paper. Suppose we have obtained by direct calculation, or from tables A, C, D, or F, the thrust r2 C without regard to mortar. The thrust diminished by mortar is given by equation (14), Er.2 C{rC(K 1) (14) The effect of adhesion depends solely upon the dimensions of the arch proper, and is not affected by its load. It is, therefore, the same in all equal arches however different may be their loads; and the general conclusions of art. 33 are applicable to all arches, as already stated. If the joints are unequal, as often happens, we may still suppose v -60~, and d' representing the length of the joint of rupture, d the length of the vertical joint, the thrust will be (dZ2~d'2) C being the number which, multiplied by r2, gives the thrust independently of mortar; c the force, in pounds, required to separate a square unit of the joint, divided by the weight of a cubic unit of masonry. Example: the magazine of Vauban, art. 48. F=-r2 C- c (K- 1)21 X 0229381 -r X 0.519= 40.69 feet - 6.91 feet = 6103 pounds- 1036 pounds= 5067 pounds. 238 THIEORY OF THIE ARCH. THE EFFECT OF SURCHARGE UPON THE ROTATION THRUST. 50. To obtain the thrust as modified by a load or surcharge of the density of the masonry, and of -the constant vertical depth t, we must add to the second member of (24), see art. 34, S' rtK(2- K) sin.2 v =Y - 2 X — cos. v of which the maximum value is, art. 34, rt K(2 -1))(K- K2 1 tN. The effect of this surcharge depends solely upon the dimensions of the arch proper, and is not affected by any other load. Suppose we have obtained by calculation, or from tables A, C, D, or F, the thrust,20Q without regard to surcharge, we have P r2G+rt1 (15)' We have given, in the 13th column of table F, the values of N corresponding to all values of K between 1.09 and 1.42, applicable to all semi-circular arches, whatever load they may carry in addition to the surcharge of constant depth. This mode of treating the surcharge, leads;to a small error in excess, or in favor of stability; for while r2 C is the thrust independently of surcharge, calculated at a particular angle, rtN is the maximum effect of the surcharge, corresponding, generally, to some other angle. The true thrust would correspond to some intermediate angle, and would be somewhat less than the sum of the two maxima. The difference, however, in tlhe two angles, is very small; in the heavy arches of fortifications never exceeding six or eight degrees.-See table F. The error, therefore, always in excess, is very small, it being the THE MAGAZINE ARCH. 2 39 property of a maximum to exceed but little the neighboring values of the same function. This principle has been illustrated in art. 29. 51. The weight of a single column resting upon the crown of the arch, we can always represent by a surface one unit in width and Hin height. The thrust increased by such a weight, see art. 35, is, -F r2U~ H+ ra2 IC being the thrust independently of surcharge, obtained from table F or by direct calculation. For a more extended discussion of this case, see art. 35. All the. remarks there made are equally applicable here. THE SLIDING TIIRUST. 52. The general expression of the horizontal force necessary to prevent the segment whose surface is S, from sliding down its lower joint nt m, is, equation (8), P'=S cot. (a+v); which, in the present case, becomes, fig. 9, sP'r2 sKy *v_~ sin. isI_..v(~25)' sin. I tang. (a+x) v sin v Assuming a= 30, I and fiT being known by the conditions of the problem, we must find, by assigning, in succession, different values to v, the maximum value of P' in (25). The greatest of the two maxima, iF P, (24), (25), will be the true thrust. 240 THEORY OF THE ARCH. THE EFFECT OF MORTAR UPON THE SLIDING THRUST. 53. The sliding thrust diminished by the adhesion of mortar, is the maximum value of P' in equation (9)d' cos. a P'=XS cot. (a+iv) n. ( + ) The angle v which renders X cot. (a+v) a maximum, or which corresponds to the true sliding thrust independently of mortar, varies in the several arches, from 29~ when the roof is horizontal, to 2~2 when the roof is inclined 45~ on each side of the central ridge. As in art. 37, we can, without sensible error, suppose v in the term involving the adhesion of mortar, always equal to 30~. We have then, this formula for the sliding thrust diminished by the adhesion of mortar, P=r2- rc(K- 1), in which C is taken from table C, D, or F, or obtained by direct calculation, and the last term is the effect of mortar. For illustration of this subject, see art..37. The effect of adhesion depends solely upon the length of the joint, and is therefore the same in all equal arches however loaded. The above formula will seldom be needed; for it is only in very heavy arches that the sliding can exceed the rotation thrust, and in such arches it would hardly be safe to rely upon any adhesion of mortar. EFFECT OF SURCHARGE UPON THE SLIDING THRUST. 54. The general equation applicable to this case is P'.S cot. (v+30')+S' cot. (v+30'). We can always suppose the surcharge 8', which is entirely above the roof of the arch, to have a constant vertical depth as far as 25~ or 30~ from the crown. Let t be that depth. We have TEE MAGAZINE ARCH. 241 S'=tX RXsin. v-=rtKsin. v; and S'cot. (v+300~)-ltK sin. v X cot. (v+300), of which the maximum value, corresponding very nearly to v-25~, is, rtKX 0.29592. Adding the two maxima together, we have, with a slight error in favor of stability, P= r2C+rtKX 0.29592 (20)' q2C" being the sliding thrust independently of surcharge, obtained from tables C, D, or F, or by direct calculation. This formula has already been given, art. 39, equation (20), and is applicable to all circular arches however loadedalways giving a result slightly in excess. We have given in table F the value of Kx 0.29592, corresponding to values of 1K which render the slidinq greater than the rotation thru.st, applicable to all semicircular arches, tables A, C, D, F, and to segmental arches which have the angle at the center greater than 25~. This mode of treating the surcharge is precisely like that adopted in art. 50. Here too, the error, always in excess, is very small; for the angle v, which gives the sliding thrust without regard to surcharge, varies, in the several arches, from 22~ to 29~, never differing more than 4~ from the angle 25~, which renders the effect of the surcharge a maximum. The true thrust will be the greatest of the two maxima F. P, (15)', (20)'. GENERAL REMARKS ON TABLES C, D, F. 55. M;: Petit has applied the general equations (24), (25), to the two extreme cases in which the roofs are, respectively, horizontal and inclined 45~; limits between which probably all powder-magazine arches are comprised. Tables C and D contain his results. We have filled up the interval, between — 45~ and I-90~, with eight 242 THEORY OF THE ARCH. columns, I varying at intervals of 5~. The results are embodied in table F, which gives directly, or by proportional parts, the thrust of all circular arches of common use, whose loads are bounded on top by two symmetrical planes extending as far, at least, as firom the crown to the reins, or angle of rupture-applicable to all semi-circular roof-covered magazine arches, and to almost all full-circle stone and brick bridges. See discussion of that table. For the purpose of general illustration, and as introductory to tables C and D, we give the application of (24), (25), to the two cases mentioned above, I-45~, and -- 90~. ROOF INCLINED F'ORTY-FIVE ]DEGREES..56. Make i-45~ in equation (24). It becomes.. sin. v K2 6(k-co. ) 1 K2V2(6-3K-(3.-2K) sin.(45 +v))3sinv cos21(26) Suppose, furthermore a —30~ in equation (25), and for v substitute 22~, which corresponds, every nearly, to the maximum value of P', or the horizontal sliding thrust, we have P= —2(0.2234 X Ki2-0.14999) (27) By means of these two formule, and of another, e-rlV/2C, art. 43, M. Petit has calculated table C, analogous to table A, giving, for all values of K between 1.05 and 2, the angle of rupture, the thrust —rotation and sliding, and the limit thickness of pier required for strict equilibrium and for the co-efficient of stability 2. See discussion of that table. M. Petit has proved conclusively, by a course of reasoning like that given in art. 47, that this kind of arch, SURCHARGE HORIZONTAL. 243 1=45~, is always stable upon its springing line; that is, that however thin the arch may be, the moment of the thrust is always less than the moment of the semi-aich in relation to the exterior edge of the springing line. ROOF HORIZONTAL. Figiture 4. 57. Make 1-90~ in equation (24). It becomes F'-=r6( —os.in v) ] K((6-3EK-(3-2K)cos. v)-( ) }(28) Make I_90~, a=30~, in (25), and for v substitute 29~, which corresponds very nearly to the maximum value of P', we have P=r'2(0.16391 XK 2- 0.15206) (29) By means of these two formules, and of anothere-=1'/2C, M. Petit has calculated table D analogous to tables A and C. See discussion of that table. By a table similar to that given in 47, M. Petit has demonstrated that the moment of the semi-arch exceeds the moment of the thrust, both taken in reference to the exterior edge of the springing line, for all values of _K greater than 1.0435; and that the moment of the thrust is the greater for all smaller values of K. The thinnest arch therefore, of this kind, having a surcharge limited to the horizontal line tangent to the extrados at the crown, figs. 4, 7, that can stand upon its springing lines, is one whose span is about 46 times its thickness. 244 THEORY OF THE ARCH. DISCUSSION OF TABLE C-THE ROOF INCLINED FORTY-FIVE DEGREES. 58. After explaining the use of tables C, D, and F, we shall show how to apply the results to the determination of the thickness of pier. Table C gives, either directly, or by proportional parts, for all values of K between 1.05 and 2:1st. The angle of rupture, or of maximum thrust.Rotation. 2d. The decimal C, column 4, which, multiplied by the square of the radius of the intrados, gives the horizontal thrust on the supposition of rupture by rotation. 3d. The decimal C, column 5, which, multiplied by r2, gives the thrust on the supposition of rupture by sliding. 4th. Columns 6, 7; the value of the radical, 1/ 26C( which, multiplied by the radius of the intrados, gives the limit thickness of pier in the case of strict equilibrium, 6=1, and with the " stability of Vauban," -=2. It will be seen that the thrust due to sliding is greater than the thrust due to rotation for all values of K greater than 1.42, and that the rotation thrust is the greater for all smaller values of Ki Calling the greatest of these values (or the greatest of the decimals in columns 4, 5) C the limit thickness of pier, or the thickness required for an infinite height, is e=-r/2 U; for strict equilibrium; u= 1. e=rV/4(Y; for the stability of Vauban; - 2. The greatest of the decimals in columns 4, 5, must always be selected as giving the true thrust. The thrust given in table C for K 1.29, is double the thrust in table A for the same value of K. For larger values of K the thrusts of table C are more than double, TABLES C. AND D. 245 and for smaller values of X less than double, the thrusts of table A. Of all arches having the same span and radii, the magazine arch has the greatest thrust. Its span, however, is small, seldom exceeding 30 feet. DISCUSSION OF TABLE D-THE SURCHARGE HORIZONTAL. 59. Table D, analogous to A and C, gives directly or by proportional parts, for all values of K between 1 and 2. 1st. The angle of rupture, or of maximum thrust.Rotation. 2d. The decimal C, columns 4 and 5, which, multiplied by the square of the radius of the intrados, gives the horizontal thrust on the supposition of rupture by rotation, and by sliding. 3d. The value of the radical 42SC-, columns 6, 7, which, multiplied by the radius of the intrados, gives the limit thickness of pier in the case of strict equilibrium, -= 1 and, with the " stability of Lahire," — 1.90. It will be seen that the thrust due to sliding is greater than the thrust due to rotation for all values of K above 1.34, and that the former is less than the latter for all smaller values of K. We must in all cases, to obtain the true thrust, select the larger of these two values. The angle of rupture for all the usual values of ]K, in which the thrust is due to rotation, that is, from i= 1.10 to K= 1.34, differs from 60~ by only 5~; while, for the most common values of K, the difference is less. The thrust given by table D is equal to the thrust of table A when K= 1.30. The thrusts of table D exceed those of table A when K is less than 1.30. 246 THEORY OF THE ARCH. The thrusts of table A exceed those of table D when K is greater than 1.30. The greatest possible thrust that can be caused by rotation in any arch of the radius r, is I,2X X0.14506, and corresponds to K'=1.36. The thrust of table C is about double the thrust of table D when K-1.34. DISCUSSION AND USE OF TABLE F. 60. Columns 3 and 12 have been extracted from tables C and D, calculated by M. Petit. The remaining columns have been calculated for this paper. This table gives directly, or by proportional parts, the thrust, for all values of K between 1.02 and 1.50, of all semi-circular arches which carry loads of masonry, or of equal weight with masonry, rising to two planes meeting along a central ridge, and tangent to the extrados of the arch, or to surfaces parallel to such tangent planes. Column 3 gives the thrust for that case in which the two planes become one and horizontal, tangent to the extrados at the crown. Column 4 gives the thrust for I-85~, the two planes making each an angle of 5~ with a horizontal, and being each tangent to the extrados 5~ from the vertical or central joint. Column 5 gives the thrust for I=80~, the two planes making each an angle of 10~ with a horizontal and touching the extrados 10~ from the summit. Columns 6, 7, 8, 9, 10, 11, and 129 give the thrust for I=successively, 750, 70~, 65~, 60~, 55~, 50~, and 45~. Column 13 gives the addition to the thrust (rotation) caused by a surcharge of uniform vertical depth t above the roof. EXPLANATION OF TABLE F. 2-47 Column 14 gives in like manner the addition to the sliding thrust caused by a surcharge of uniform depth, t, above the roof. Near the top of each column of thrusts, will be seen a horizontal line. Below that line, the true thrust is dclue to rotation; above, to sliding. When the ratio of the two radii, A-K, places the thrust below that line, the addition A, caused by surcharge, must be taken from column 13. If the thrust is found above the horizontal line, A nmust be taken from column 14. Columns 3, 9, and 12 give the angles of maxiraun thrust: for sliding, dclown to the horizontal line; for rotation, below. These angles, for want of room, are not given in the other columns. In columns 3, 9, and 12, the thrusts are calculated within I degree of the anlle of maximum thrust; in the other columns within one degree. Columns 13 and 14 give results generally a little in excess, never too small. Column 13 gives the angles corresponding to the maximum effect of the surcharge in the case of rotation. Comparing these angles with the angles of maximum thrust, in columns 9 and 12, we see that, from K-l1.10 to K_ 1.42, they differ in no case more than 9~, while the mean difference is much less. We therefore know that the error in excess, arising from adding these two maxima together, is exceedingly small. The difference in these angles is very small in all the mgaraztine arches in common use, never, it is believed, exceeding 7~. It is only in very light arches, K=1.02, 1.03, &c., loaded nearly or quite horizontally, that the error becomes of any consequence. The error in excess in using column 14, is never of any consequence. 5 2 48 THEORY OF THE ARCH. RULES FOR USING TABLE F. 61. From the dimensions of the given arch, determine the ratio -/=-],6 the angle 1; and the elevation E of the ridge above the springing line. If JI, and I or E are found in the table, the thrust is given at once at the intersection of the two colunms. But one or both of these quantities, -K, and I or E, will generally be intermediate between the tabular numbers. The thrust must then be determined by proportional parts. We suppose the thrust to vary uniformly between any two successive horizontal columns, say from ]Ai- 1.20 to XK=1.21; and also uniformly between any two successive vertical columns, say from 1=50~ to I —45~, or rather fronm E=RX1.30541 to _ERX1.41421. This sort of interpolation can offer no difficulty to those who understand common arithmetic. Rule I. Suppose 1, of the given arch, to have one of its values in table F, K/being between two tabular values of that ratio. Under the given value of 1; take the difference of the decimals opposite the adjacent values of K. Call that difference dd (difference of decimals); let d K represent the difference between the adjacent values of 1KE, and d' 1the excess of the given value of K over the adjacent smaller value of L. The proportion d K: d d:: d'K: x gives the correction; which must be cadded to the decimal under I and opposite the smaller adjacent value of K, when the decimals increa&se ascending; subtracted when they deckrease ascending. Rule II. Suppose K'T, of the given arch, to have one of its values in table F, I or E coming between two tabular values of those quantities. USE OF TABLE F. 249 Opposite the given value of ],, take the difference of the decimals under the greater and less adjacent values of I or E. Call that difference d c. Let d 7 represent the difference between the adjacent values of X, and cl' Af the excess of the given value of E over the adjacent smaller value. The proportion dc E: d:: cd' F": gives the required correction; which must be added to the decimal opposite Kf and under the smaller adjacent value of EE, when the decimals increase from left to right; subtracted, when they decrease from left to right. We may use I or E according to convenience. Using the former, the proportion will be 5~: ci d:: d' I: x. The results will be nearly the same. As I diminishes from left to right, cij, in this last proportion, wmill be the excess of the preceding adjacent value of I over the given value of I, expressed in degrees and decimals of a degree. Rule III. Suppose ]K and I both to be intermediate between tabular values of those quantities. By rule I. determine the decimal corresponding to the given value of K and the precedinlg adjacent value of I or F.; then by the same rule determine the decimal corresponding to the given value of Ji, and the following adjacent value of I or E. Call the difference between these two decimals d d. Let dciF represent the difference between the adjacent values of E, and ci' - the excess of the given value of E over the preceding adjacent smaller value. The proportion. d E: cd c:: c' E: x, gives the required correction, to be added when the decimals increase from left to right, subtracted when they decrease from left to right. 25 0 THEORY OF THE ARCH. If we use I instead of E, the proportion will be 5~: dd:: d' I: x. Calling C the final value of the decimnal, we have the thrust F 9.2 X 6. In practice, the roof is hardly ever more inclined than 45. Still, we can, if necessary, obtain from the table the thrust corresponding to smaller values of 1, on the principles above explained. The rate of variation from 1=50~ to 1-45~, we may, without sensible error, suppose continued a few degrees below 45~. iRule IV. If there be a surcharge in masonry rising to the lines APt' D', I' ifi, fig. 10, draw the parallel lines AR D, Ai Ty, tangent to the extrados of the arch. Then determine the thrust of the arch R b A E D A, limited by these tangent lines, by the rules already given; and add, for surcharge, from columns 13, 14, as directed in those columns. The addition to be made is A.l= X decimal, in which t=AR'A=D'JD; r=the radius of the intrados. The decimals in columns 13, 14, are the same for all values of _i but are supposed to increase or diminish uniformly between any two consecutive values of Mi. They must be corrected by rule I. when the ratio, _i, of the given arch is not found in the table. PRule V. If there be a surcharge of earth above the masonry, reduce its thickness over the key and over the reins, in the proportion of its density divided by the density of the masonry. WVe can then regard all that comes below the reduced line as composed of masonry alone, and the case falls under rule IV. 62. If the two sides of the arch be unequally loaded, look for the thrust on that side which carries the greatest load,-for the thickness of pier on that side which carries the least. USE OF TABLE F. 251 63. Esxample 1. The magazine of Vauban. r=:13'.3184.; -R=16'.5148; hence IV=-=1.24 very nearly; I- 49~ 7' 7"; E 21'.85R- X 1.32258. Referring to rule II. and to columns If —1.24; 1=45~, I —500, we have clE d d d' E 1.41421.26850 1.32258 1.30541.22219 1.30541.10880.04631.0171717: —.00731.22219 F —1r2 X 0.22950 M. Petit gives, from an independent calculation,.. F — X 0.22938 The difference i 2 X 0.00012 is in effect nothing. Had we interpolated in reference to 1; the result would not have been quite so accurate, as the thrust varies rather with the elevation of the ridge thanl with the angle I. Example 2. The magazine at Fort Jefferson. =:14'; _=] —17'.50; T —1.25; tan1g. -=-7-5; J=56'(.3. 17.5 23"; E- -i AX 1.20542 = 21'.0948. sin. I Referring again to rule II. and to columnms I —1.25, I- 60~ 55, 55~, e have dE dc d cd'ET 1.2"2078.19027 1.20542 1.15470.16492 1.15470.06608:.02535:.05072: — 0.01946 0.16492 The thrust is.. 2 X 0.18438-36.13848 252 THEORY OF THE ARCH. Brot. forwardl, 36.13848 To this we must add for a surcharge 5'.90 deep (see column 13, opposite I- 1.25), A-14. X 5.90 X 0.46875 38./1875 Giving as the total thrust, say,. F-4.86000 Example 3. An arch constructed at Fort Porter, fig. 11. This arch has a surcharge, both of masonry and earth. Reducing the height of the latter in the proportion of 2 to 3, suplposed to be the relative weights of a cubic foot of earth arnd masonry, we find the reduced ridge to be 14'.28 above the springing line, and the reduced roof, on the side most loaded, to make an. angle of 86~ 46' with a vertical line. The data are r=6'; 2=7-'.668; T- 1. 278; i=-86~46'; E -- --'_ sill.IRX 1.00137 7'.68; t-14'.28 —E-6'.60. This case comes under rules III. IV. V. By rule I. dI —lI:c1 —.00085: d'J[i=.8: x-0.00068 Add.. 0.1.4101 Giving, as the decimal for KT- 1.278; I=90~,..... 0.14169-0.14169 Fincl, in like mannier, the decimal for I- 1.2T8, I 85~... 0.13486 dcl=z 0.00683 dE=.00382: cdd.00683: dc-'E-.00137: ~=. 0.00245 Giving as the thrust, without surcharge, ri2X 0.13924 -5.01264 To this add for surcharge — 6 X 6.60 X 0.44495-17.62000 Total thrust,:=F':2.63000 THICKNESS OF PIER.-GENERAL FORMUL1E. 253 THICKNESS OF PIER, THE ROOF HAVING ANY INCLINATION. fFiifmre 9. 64. We suppose the surcharge, if partly of earth or any light material, to be reclducecl in height in the proportion of its density compared with the density of the arch proper. Let A-the mean height of the pier from its base to the surface of the reduced surcharge over its top —E' 9), fig. 4; O 0' fig. 10; to be estimated if not known. E the elevation of the reduced ridge above the springing line- C ca, fig. 4; C R', fig. 10, 11. E'-the depth of the arch and its reduced surcharge at the springing line= A ca' fig. 1 0; C a or -A', fig. 4. n=the surface of that part of the, arch and its reduced load which lies directly over the half-span..,.,the moment of that surface in relation to the interior edge of the joint of the springing line. l-the lever-arn of the thrust=a I, figs. 4, 10. J? —the horizontal thrust however deternmined.:r=the half span=the radius of the intrados in semicircular arches. =-the distance between the exterior edge of the base of the pier, and the intersection of this base with the curve of pressnre, that is, the point of application of the resultant of all the forces which act upon the base. S=the coefficient of stability. e-the unknown thickness of pier. We have in all arches n-=-~r(E+-E') -the curvilinear surface A 6 C7 figs. 4, 10. in-=r r2( + Il E;L') -the moment on A of the s-urface A b C, figs. 4, 10. Let the resultant pass through the exterior edge of the base, x=0; we have gho7-e+neq+-=F1; giving (30) n nA 28n 21_Fl e= + $- (31)~ 254 THEORY OF THE ARCH. All the other quantities being known, the value of x is he2+ne+nF (39) $=X dn+, (32) Let the resultant pass through] the base at ~ its length from the exterior edge, x-A.e; and let - 1; we have c2h +,ne+ +-: F1 (303) e -— + 4 -6 +6f - A 2 h h (3l) 3 Let x —e, and S=l; we have Ye2%+-{Lne+ r —F1 (30 2) e=-3 +7 9$+/ -1~0-+lO (310) Let the resultant pass at any proportional distance, pe, from the exterior edge, x=pe, and let — 1, we have (I-pO)62hr+(1 (-p)ne =FZ (309p) -- 1'22 2 2 t) 2 (31p) / — p x.V//1-2pv, 1_ X + 1 s X h i Let the resultant pass through the middle of the base, x -e, and let =:1, we have I ne+n_2GF (30k) The values of e drawn from (31), (31), (312), (311), 7' * *n' A1 differ only in the numerical coeffcients of W' ~, T, I, so that, having solved one of these equations, we can readily solve others. It has been customary in large arches, to assign to ~, the coefficient of stability, the value of 1.90 or 2, aidcl determine the thickness of pier from an equation equival THICKNESS OF PIER.-GENERAL FORBMULIE. 255 ent to (31). But this is not always enough. We may still want to know where the resultant of the thrust and of the weight of the semi-arch and pier cuts the base. This is given by (32) for any assigned values of 7 and e. When we have determined e from (30), (31), for any particular coefficient of stability S, we can substitute, in the numerator of the second member of (32), for thes-lnze6+ rn, the equivalent (30), &FI, which gives to (32) a form more convenient for computation, viZ.: _(8 —1)8 (32S) Ell When 8 -2, its usual value, we have xGiving to fh and I in (32) the particular values which correspond to the springing line, we learn where the curve of pressure cuts that line. If we suppose i, the lever-arm of the thrust, to be variable in (31E), e also being variable, that formula becomes the equation of a right line, very easy to construct, whose intersection with the base gives the middle of the pier. The thickness of pier should never be less than that which is given by (31k); for if the resultant cuts the joint of the base within one third of its length from the exterior edge, that joint will open, or tend to open, at its inner edge. According to the rule deduced by Audcloy, from an examination of the magazine of Vauban, we suppose the thrust to be doubled, assume -- 2, and determine e from equation (31). This rule is perfectly safe in almost all practical cases; but it gives, to piers of great height, a thickness too small, by causing the curve of pressure to pass within the proscribed limit just mentioned, and by leaving the surface of the base too small to bear the superincumbent weight. 256 TIIEORY OF THE ARCH. To piers of small height, the rule gives a thickness unnecessarily large. Poncelet recommends as a rule for piers of small height, that the resultant should pass through the middle of the base. The required thickness is then given by the very simple formula (31k), which we here repeat, This last rule would seem to be a good one, so far as the pier is concerned, provided the thickness thus determined is sufficient for the superincumbent weight; but Poncelet did not intend that this rule should be followed blindly. It will often give results much too small. At the springing line of most segmental arches, it becomes illusory, for there we have Fl-nm, or, by the rule, e=O. If we knew the real acting pressure at the crown and its point of application, the rule would be perfect. To construct (31B): on the profile of the magazine of Vauban, fig. 13, from a', the intersection of the inner face m of the pier with the horizontal a a', lay off a' nA= —; on the. horizontal ca a' prolonged lay off np'= F, and, on the vertical through n, lay off nlp=. The diagonal n q is the required line. It is easy to give to (31) such a form as to establish, in its utmost generality, the principle already announced for a particular case, art. 43, relative to the limit thickness of pier. Calling i the difference between h and 1, we have 2aFl 2AF(1b+i) 9-, 9_i_ h 1b - Suppose h=infinity; (31) becomes ej/2SF. In like manner, we find the limit thickness to be, when x= —e, (3l3), e=VGF; THICKNESS OF PIER.-SEMICIRCULAR ARCHES. 2 5 when xZle, (31p), e- 2 F "c X e, (31~), e —infinity. The formulae given in this article are all of universal application, whatever be the load, and whatever the curves of the arch, circnlar, elliptical, segmental, &c. But the values of nb and m, given above, are based on the supposition that the surface of the reduced surcharge over the half-span is a single plane. Should that surface be irregular, it is only necessary to say that n, is the sum of all the surfaces over the half-span, and mz the sum of the moments of all those surfaces in reference to A, figures 4, 10, the interior edge of the joint of the springing line. Should the value of e resulting from (31), show that we have not estimated A very correctly, we shall be able to correct the estimate and calculate e anew. Strictly speaking, A is not precisely the samle inl (31), (32), (30), &c. In (32), h is properly the mean height; in (31), fig. 10, the height measured along a vertical cutting the base of the pier at 1 its thickness from the interior face. It would be easy in any particular case to make the proper correction; but such correction will hardly ever be necessary, for e changes very little with small variations of h. THICKNESS OF PIER. —THE INTRADOS A SEMICIRCLE. 65. The formulse of the preceding article all remain unchanged: but we have (figs. 4, 10), the curvilinear surface A b C, -7Tr.2=2 X 0.7854 moment on A of " " _8X 10=i 3X0.4520G5 Hence q47(E+ E') - x2 X 0. 8 4 mr =.2(Et_+E').'3 X 0.452065 258 THEORY OF THE ARCH. *E and E' are always given by the conditions of the problemn. They stand in this relation, E'-E-r cot. I: If the roof of the arch be inclined 45~ we have E'If the roof is horizontal, we have E'-E; and, introducing the ratio K-, n=IIJ2(I-0o.7s54) Mrn9r3(!ffK-0.452065) en (R K-0.452065 IH- 0.~854 THIE ROOF GREATLY INCLINED, AND WITH LITTLE OR, NO SURCHARGE. 66. When the roof of the arch is so steep and the arch so thin that we can regard the triangle 2D EP, fig. 13, as forming a part of the semi-arch or pier, we can give to mq and zn, in the equations of art. 64, a meaning which, without changing the form or purport of any of those formulse, shall render their application somewhat easier. Let 9z'=-the surface of the whole serni-arclh and its reduced load, and of that part of the pier which lies above the springing line. en'-the moment of that surface in relation to the interior edge of the joint of the springing line. h'=the height of the pier from its base to tlke springyiy line. Let E, 1,, r, x, S, e be the same as in. art. 64. We have'' — tang. Ix E — r2 x 0.7854;' — tan (rtang. IX tang. IX E) -' X 0.452065 If the roof be inclined 45~, we have'- - (,2(K2 _ 0.7854); i?'-7'(-(K2 a X 0.4714-0.452065). THICKNESS OF PIER. -THE ROOF STEEP. - 259 To determine the thickness of pier, we have h7'6~2+n'e+m'-=&F1 (30)' /2 2n'"'?'W 76 12 7(s7)' ill which 6 is ordinarily taken at 1.90 or 2, and the resultant of the thrust thus increased, and of the weight of the semi-arch and pier, passes through the exterior edge of the base. The point where the resultant of the true thrust and of the weights just alluded to cuts thle base, is given as follows: IT{' —n inP (32)' in which we substitute for e the value determined by (31)7 or other values according to circumstances. If we wish to determine the thickness of pier on the condition that the resultant of the true thrust and of the weight of the semi-arcll and pier shall intersect the base of the pier at a distance equal to one-third its thickness from the exterior edge (X w~e), we have /27' +l2 o,' 7 e -2- /+ 4/V -6 +6-t (31 )j If x-.e we have, -9+2h' T +6o'e+m' _FE. (30-)' e=-3a+ 9 - 10 -o+1 (312)' A' /&12 h A If x= pe, 1 being any fraction whatever, we have ( —e )eVa' +(1- r2)'e)+ n' — F(; (30p2)' / 1-pn\' // -1-\29 n'~ 2 m 2 F1i e — -,-2p;.'x2 -2Vp h'x +- - 2p hx (,p). ~~~~~~~~~~~-1 -- 9pI 260 THEORY OF THE ARCH. Finally, if the resultant pass through the middle of the base, x-le, we have n'~+. E' —Fl. (30,)' e=- 2 (-,. (31~)' When we have determined e, (30)', (31)', for any particular value of A, we can give to (32)' the following more simple form:x=((-15)e, (32+)' eh which, when — =2, becomes X67. Example 1. The magazine-arch, with the roof inclined 45~, without surcharge. r=10'; -=-12'; K= —-_1.20; 1=45~; n', art. 66, r 65.46; in', art. 66, - 173.356; 7h'10'; -0 h'+R_ 22'.; F table C, -- r2X 0.25806='25.806; from these data we have, by calculation, 7, —6.546;; 1 — 42.85; -7.3356; -=56.7732; hence, by the formulae of art. 66, for strict equilibrium,:-1, (31)', e= —6.546~+/42.85 —2 x 17.3356+2 x 56.7732-4'.487. for 8=2, (31)', e= — 6.546+V/42.85 - 2 x 17.3356+4 x 56.7732=8'.792. X z-le, (31~)', e- -2 x 6.546+4- 4 x 42.85 6 x 17.3356+6 x 56.7732 =7'.107. x —e, (31)',e-3x6.545+ 9x42.85-10x17.3356+10x56.7732 =8'.29. " x=e, (312)', e=12'.05. We may infer without further calculation that the rule of Audoy, which consists in doubling the horizontal thrust, or assuming S=2 in (31)', is in this case perfectly safe; for THICKNESS OF PIER. -EXAMPLES. 261 it gives a greater thickness than (312)', which requires the resultant to pass 2-e from the exterior edge. For e=8'.792 we find, by (32)', x-3'.7=-eX0.421. Making h' —o in (30)', we have, for strict equilibrium, or AS-, e —2'.08; for -2, e-G6'.81. By comparing (30)' with (31-)', we see that the thickness required for strict equilibrium at the springing line is precisely half the thickness required for the resultant to pass through the middle of the base. Example 2. The magazine at Fort Jefferson, of which we gave the thrust in art. 63, fig. 12: r —14'; R_-1'.50; -:= 1.25; _= 56 3'3"; t - depth of surcharge above the tangent planes = 5'.90; h- 16'.5 below the springing line+13'.5 above, =30; 1 —16'.5+17'.5-34'; F-74.86; -E=.i -+ t= 27'; ='-_E — cot. I-_17'.58 (arts. 64, 65); =-158.1:2; e-=-109.82; — =6.943; 5.27; -, -7. 8; =-36.594; - -84.841. With these data, the formlula of art. 64 give us, for strict equilibrium, or -1l, (31), e —-5.27+- /27.78-2 x 36.594+2 x 84.841- 5'.88. For _20, (31), e —5.27+V27.78 —2 x 36.594+4 x 84.841 —11'.88. x =e, (311), e=-2 x 5.27+/4 x 27.78-6 x 36.594+6 x S4.84L _9'.48. " x-\-e (31-2), e —-3 x 5.27+9 x 27.78-10 x 36.594+10 x 84.841 =11'.26. We see that the thickness required by the rule of Audoy, 11'.88, is amply sufficient, as it is greater than 11'.26, corresponding to x= —e. But the pier of this magazine has a running gallery 3' wide, 18' high, with a 4' solid wall outside, and with its floor nearly on the level of the base. 262 THEORY OF TIIE ARCH. As this case may occur again, we subjoin the necessary modification of (30), (31). It is evident that the pier will not go over in a solid mass, but that the divisions each side of the gallery will revolve separately. Let e' represent the thickness of the solid wall outside the gallery; e" this same thickness increased by the width of the gallery; e, as usual, the unknown whole thickness of the pier; h' the height of the gallery, supposed to have its floor on the-level of the base. The equation of moments corresponding to (30) is 1he2+ne+q12 - i7b'(62 -e'2) + Ih'(e - 1c)2 AFl givln g e= (fl-$) +-(h)2(r+ ~2h (e+e"2)) +2Fl (33) In the present case we have e'-4'; e" e'+3' —7'; h'-18S'. Substituting these values, we have, for strict equilibrium, or 6 1, e-6'.59; for 6-2, e-14'.04. Let us still further take into consideration, the effect of mortar upon the thrust. The arch is covered with concrete so as to mnake the vertical joint, extended through this concrete, 9' long, and the joint at the reins over 5' long. We suppose c, art 16, to be 25, and ttake only 4 of that effective force. We must subtract from the thrust, art. 49, (d:=9, d'-5', y=R — -=10'.50), xI d+ 14.02, which reduces F to 60.84. Substituting this value for F in the last term under the radical, we have, for — 2, e-11'.77. The actual thickness adopted for this magazine is 12'. We take no account of the effect of adhesion upon the base of the pier; for, in consequence of the division of this pier into two parts, and of the great length of the leVer ar-m of the thrust, this effect is almost nothing. THICKNESS OF PIER. —EXAMPLES. 2 (G 3 Example 3.-An arch at Fort Porter, fig. 11. The thrust of this arch has been given in art. 63, example 3. r —6'; -12-'.668; K —1.278; _F-22.63; l:18' below the springing line+-'.668 above-25'.668; h=18'+12'= 30'; E CR'-i14'.28; JJ', on the side least loaded,_ A a'- 12'.59; n, arts. 64, 65, -52.3356; qvz, arts. 64, 65,149.254; hence, by calculation, 3322 2/ nFl — 2.852; — 1.7445; — 3.0435; i74.975; — 19.3625. With these data, the formunla of art. 64 give us, For strict equilibrium, -=1, (31), e:3'.90. For =-2 (rule of Aucldoy), " e 6'.65. For — =e, (31~), e 6'.44. For.x-e, (312 ), e-'.86. Thle thlickiess given by the rule of Audoy, 6'.65, is barely sufficient, as it exceeds but little the value of e corresponding to x-A-. But the pier of this arch is not solid. It has counterforts on the inside, 3' wide and 6'.50 apart, united by arches at top starting from the level of the springing line of the main arch. The pier outside, 4' in thickness, is continuous and solid. Let e'_the known thickness of the solid wall on the outside; pj)the proportion of the vacant space between two counterforts to the same distance increased by the width of one of the counterforts; h' -the mean height of the vacant spaces between the counterforts. The equation of moments corresponding to (30) is ~ he2 + ne+m - p7- b/'(e 2_-2) - 1, giving /A -(:) 2-p h'(2-P- (34) 6.5 13 In the present case we have 2-9.519; ]'=19' nearly; e'-4'. We have, therefore, for strict equilibrium, 6 264 THEORY OF THE ARCHU. or S-1, e-3'.85; and by the rule of Audoy, -2, e= 7'.71. In (34) e is supposed to exceed e'. Example 4.-The magazine of Vauban, fig. 13. 9-12'.50; R — 15'.50; K= 1.24; E-=altitude of ridge above the springing line_20'.50; I-49C 7' 1T7"; I 8'; l'-7h'+-R1 23'.50; the thrust, already given for this arch, arts. 48, 63, is F- ~2X0.929381=35.84. From these data, referring to art. 66, we have by calculation, n' —120.039;'-=235.06; Fl' —842.26. 77i15.005;, —225.1463; r=29.3825;, -105.2823. With these values, the formulms of art. 66 give us, for strict equilibrium, 3=1, (31)', e=4'.410. For (S-2, rule of Aundoy, (31)', e-9'.234. For x-'e, =-1, (31~)', e- 6'.814. For x-2 e, — 1, (31%)', e:7'.761. For x-:le, (-1, (31-)', e-10'.117. Whene=9'.234, ruleof Audoy, (329)', x=4'.34 eX 0.47. We shall give a discussion of this arch hereafter. SECTION IV. ARChIES IN SEGMENTS OR PARTS OF A CIRCLE, USUALLY CALLED SEGMENTAL ARCHE-IS. 68. These arches are very common in fortifications, and still more common in bridges of large span. Indeed, the semi-circular arch of large span, and of the usual thickness at the key, which is about f-v- of the span, has a great tendency, after the removal of the centering, to settle down at the key and spread out at the reins about 60~ from the key, so that such arches can only be safely used when their thickness is greatly increased below the reins, or when their piers are continued above the SEGMIENTAL ArECHES. 265 springing line, in solid and almost incompressible masonry, as higlh as the reins. Such arrangements in effect reduce semi-circular to segnnental arches. Segmental arches are fully given when we know the span —, the rise of the intrados above the springing line =f, and the thickness at the key=l. Let r, as usual, represent the radius of the intrados, v' the half-amgle at the center. We have r_- sin. " COS.''V f 2 8f icos. 1 —. As the thickness of the arch at the key is given, wve know the value of the ratio of the two radii,. R d When not otherwise mentioned, we shall suppose the thickness of the arch to be the samze throughout. Shoulcl the thickness increase towards the reins, the formulse and the tables to be explained hereafter will give a slight excess of thrust. SEGMIENTAL ARCHES WITHOUT SURCHARGE, —INTRADOS AND EXTRADOS PARALLEL FIG. 14. 69. Look in table A for the angle of rupture cor'responding to the given value of If. If that angle be less than V', the thrust is evidently given at once by the table. But if the angle' be less than the angle of rupture in table A, it is easy to see that the prism of maximum thrust extends to tllhe springing line. The thrust (rotation) will in this case be given at once by (11) art. 28, wrhen we have substituted for v in that formuLla the known value of v'. In like manner the sliding thrust will be given by table A vwhen V' exceeds 26~. If v' be less than 26~, this thrust will be given by (17), art. 36, when we have substituted for v in that formula the known value of v'. 260 THEOPRY OF THE ARCH. Table E, calculated by M. Petit, gives, for all the values of K between 1.01 and 1.40 inclusive, the actual thrust in seven systems of segmental arch, being the varieties in most common use. These varieties are as follows: s 4, 5, 67,, 8, 10, and 16 times f. Above the horizontal line in each column, the sliding exceed the rotation thrusts, and the former only are given. Below the horizontal line the rotation thrusts only are given. If the angle of rupture in table A, corresponding to systems not given in table E, that is, to'segmental arches of which the half span is less than four times the rise, exceed v' by only six or eight degrees, the thrust may still be taken from table A without sensible error. Ilhtl.stcration.-Second column of table E,.=:4f, v'530 7' 30". For KJC-1.18 table E gives F=1,2 X 0.10313 " A "4 v —58~40', F-r' 2XO0.1041 Difference in the angles, — v' - 5~ 32' 30"; error in the thrust, always in favor of stability, P2 X 0.00104 70. The rotcation threu&st, cli)nnizsied bIy ~nortarc, is F=,~,C2' ( 6,+ (35) in which d'-the thickness of the arch at the springing line, r2Cd —the thrust, without adhesion, obtained from table E or by direct calculation. But if the thrust has been taken from table A, that is, if v' be nearly equal to v or exceed v, the effect of mortar and of surcharge has already been given in the discussion of the semicircular arch, art. 31 and following. 71. The sliding thrust diminished by the adhesion of mortar is, art. 37, F —2' X C_ ed' cos. 300 (36) sin. (V'+300) SEGMENTAL ARCHES WITHOUT SURCHARGE. 0 G ~2 C being tlhe thrust obtained, directly or by proportional parts, from table E, or by an independent calculation. This last formula is, of course, to be used only when tle dimensions of the given arch point to a decimnal above the horizontal line in one of-the columns. If v' exceed 26~, the slidiing thrust as affected by mortar and surcharge has already been given: art. 37 and following. THICKNESS OF PIER. T2. Let -tnsurface of semi-arch ca 6 n n a, c fig. 14; r" ~nzmoment of that surface in relation to mqn; "' -the lever arm of the thrust, or elevation of ac above the base of the pier; Let h7}the mean known or estimated height of the pier from its base to the upper surface of its surcharge; " F-the horizontal thrust however determined; " x=the distance between the exterior edge of the base of the pier and the point where that base is crossed by the curve of pressure; " 6-the coefficient of stability; " e —the unknown thickness of pier. We suppose the small triangle in n a' to belong both to the semi-arch and pier; thereby greatly simuplifying the formulse, while the very slight resulting error is always in favor of stability. We have, Fg2_ (J~-21)*; -— 3.( _ )(1 o; (3) When the angle of rupture extends to the springing line, qn and F stand in this relation, mn=F(f+dc); * arc of 1~ —0.017451; log. of ditto —-'2.2418'7; r —1. It will be most convenient, in calculating the value of nt, to express v' in degrees and decimals of a degree. 26 8 TTHEORY OF THE ARCHI. so that, knowing one of those quantities, we can obtain the other without a separate calculation. The subjoinecd formulbe are identical in form with those of art. 64, and only differ in the values of nz and qn, which we have given above. - A0 +II + n2.2 F (30)s - /tI 72 7 (31) 162 +vr e+n-F1 When e has been obtained from (31)S, for any particular value of S wre have t_( _-1 __- (32)S) /When - 2 e --- Let the resultant pass througlgh the base so as to malke x=-e, ewe have r 27+2cm F1; (30 - +< e= i? 1b2 -G-+6; (31 3)S eF-$ lb,,4a+%2, +,,lb= 3 r 1o 21i + -6o 9+)?2F (30 2 )S I"1,/ / 1 "0r'+1 (31) 5 I e —a-I 9 e (FI_ -~i) (312)S The discussion of these equations given in articles 64, 65, 66, need not be here repeated. It is necessary to determine, once for all, in every arch, the values of F. mn, and n. That done, the above equations are solved with great ease. THICKNESS OF PIER.-EXAMPLES. 269 73. Example. W~e will take the case reported by Mr. Haupt, in his veiy excellent work on Bridge Construction, page 130. "The Monocacy, a very violent stream, is crossed by a beautiful stone bridge (aqueduct), of nine arches, each 54 feet span, and 9 feet rise; arclhes 23 feet thick, abutments 10 feet thick and 10 feet high, on a foundation 3 feet high and 13 feet wide. "Some arches and piers had been built up and backed in; but, before the whole could be completed, a great flood swept the last center from under the arch just turned and not backed in, except partially on one side. The rise of this arclh being only one sixth part of the span, must have pressed with tremendous effect upon its last pier, especially as the supports were very suddenly knocked from beneath it, and it was brought to bear very suddenly upon the pier. This had been well built with hydraulic cement of tolerably good quality, only eight or ten rmonths before. The arch stood triumphantly, contrary to the expectation of all that witnessed it, who looked for nothing but the destruction of every arch then built, one after another." The pier had "lost much of its specific gravity by imnmersion." We have in this case s-54'; f:9'; c1-2'.50; j — 45'; R=47'.50; v'=36~ 52' 10"; K=-1+-=1.055555; h-10' This arch belong2 to o'.0;e of the systems of table E (see This arch belongs to one of the systems of table E (see column 4). The thrust given in that column, is, for K — 1.06, Ji_ — 2 X 0.04280 "' _i 1.057 F=2- 2 X 0.03709 2> X 0.03709'2 X 0.00571X: X 0.0031 Hence for 7- 1.05+ _F —.2 X 0.04026 8=1.5265 — 74.41; n-=F(9-+2.5)=937.5547 2 70 MTHIORY OF THE ARCH. The exact values of m and F are a very little larger than those obtained above by interpolation, but the differences would produce no sensible effect upon the results. We have, by calculatioli from the above data, - 7.441; — 55.37; -=93.15555; -1 5.2820 Substituting these values in the formulte of 72, we have, for strict equilibrium, or — =1, (31)S, e= —7.44+V/55.37 —2 x 93.7555+2 x 175.282=7'.34. 52For ___________,=-2, " e=-7.44+~4/55.37-2 x 93.7555+4 x 175.282-2=16'.41. =1].25, " e= —7.44+$/'55.37-2 X 93.7555 +2 x 175.282-10'.05. x-=e (31)S, e —2 x 7.44+/ 4 x 55.37-6 x 93.75,55+6 x 175.28211'.78. X-=e(31)S, e - 3 x 7.44+V/9 x 55.37-10 x 93.7555+10 x 175.282 =13'.92. x=-e (31 )S, e=-( - 1'.91. Had the pier lost one half its weight by immersion, we find, substituting 4ke2 for lze2 il (30)s, the thickness necessary for strict equilibrium to be only 8'52. We learn from (32)S, that the resultant of the thrust and of the weight of the semi-arch and pier crossed the base at the distance x-2'.46-e X 0.246 from the exterior edge. Consequently, the foundation-joiAnt of the pier was open on the inside as far as (10'-3 X 2.'46)-2'.62 from the inner edge. We have, in fact, overrated the stability of this pier; for the thrust given in our tables is the horizontal pressure acting at the crown of the arch at the moment of rupture, and is not so great as the existing pressure where the thickness of pier is such as to prevent rupture. It is interesting to remlark that, had the opposite half of this arch been loaded in masonry up to the horizontal tangent to the extrados at the crown, the thrust, table E', would have SEGMENTAL ARCHES SURCHARGED HORIZONTALLY. 2[71 been increased fifty per cent., while the elements of resistance would have remained the same. Consequently, the pier would have been overturned, for we have found that an increase of twenty-five per cent._, -:1.25, requirecl a thickness e- 10'.05. Had the surcharge been only one half as heavy as masonry, the pier would have been, ~-1.25, almost exactly in equilibrium. SEGMENTAL ARCHES SURCHARGED HORIZONTALLY. Fig1re 15. 74. This is the most commlon form of the river arch. The surcharge of masonry and earth usually rises to a horizontal plane passing a little above the extrados of, the key. For the present we shall suppose this horizontal upper surface to be tangent to the extrados at the key; and we shall continue to suppose that the load between this plane and the extrados is of equal density with the masonry of the arch. For notation, see art. 68. Look in table D for the angle of rupture, v, corresponding to the given value of KC If that angle be less than v', the thrust is given at once by the table; and thle effect of mortar and of surcharge will be the same as in semi-circular arches, arts. 14, 49, 50. But if v' be less than v, the prism of maximLan thrust extends evidently to the springing line; and, F'as usual denoting the rotation thrust, we shall have, after substituting the known value of v' for v in (28) -1 — - i()E-eos in.v' {cos2.v ( Sike manner the sliding thrust Swill be given by In like manner tle sliding thrust will be visen bny 272 THEORY OF THE ARCHi table D, if v' exceed 29~; and the effect of mortar and of surcharge will be the same as in semi-circular arches. But if v' be less than 29~, the sliding thrust will be given by (25) when we have substituted for i; in that formula, 90~, and for v the known value of v'. Table E', calculated for this paper, gives, either directly or by proportional parts, for all values of K between 1.01 and 1.40, and for all relations of the rise to the span between s —4f and s=16f, the horizontal thrust at the extrados of the key. Table E' is altogether analogous to table ]E. Above the horizontal line in each column, the sliding exceed the rotation thrusts, and the former only are given. Below the horizontal lines, the rotation thrusts only are given. Should the angle of rupture in table D exceed v' by only five or six degrees, the thrust may still be taken from that table without any sensible error; and the effect of mortar and of surcharge will be the same as in semicircular arches. 75. The rotation thrust diminished by mortar, is FC-1 ((d + d'2)'2 C being the thrust obtained from table E' or by direct calculation, d and cl' respectively the length of the upper and lower joint, both extended, if we please, beyond the true extrados of the arch, through a cover of masonry or concrete. If there be no such cover at the vertical joint, we have =f/-cl. In all cases,y=f+the thickness of the arch proper at the key. T/e 9-otatio" n tlruGwt incr-eased by a sqlhc7zarye of the constant vertical depth t, is o accordin to onenien — or, according to convenience, SEGMENTAL ARCHES SURCHARGED -IORIZONITALLY.'273 F= —'2 C+ t sin.2v' X (38)' in which q2 C is the thrust given by table E', or obtained by direct calculation, t the constant depth of the surcharge above the horizontal drawn tangent to the extrados at the key, d the lepgth of the vertical joint, d' the length of the joint at the springing line. 76. fThe siiding thrust increasecl iy sqvrc7acrge ancd climinihsed by mortar, is cd' cos. 30~ P =r2c+ t(r+d') sin. v' cot. (300 +v')-.); (39) sin.(v'~+300)..2 C being the thrust obtained firom table E' or by direct calculation. This formula is, of course, to be used only when the dimensions of the given arch point to a decimal above the horizontal line in table E'. If v' be nearly equal to 25~, or exceed 25', the sliding thrust and the effect of surcharge are given in table F; the effect of mortar becoming at the same time, art. 37, Qc(K —l 1), or rather, cd'. 77. It generally happens that segmental. arches of large span increase inz thickiess from the summit to the springing line. In such cases our formulm, and the tables founded upon them, give thrusts a little in excess, for we neglect the small trapezoid n, n' r"' r, fig. 15, whose weight is in favor of stability. Table E' will give the thrust of such arches, very slightly in excess; but, in estimating the value of lIT for the horizontal column, we no longer make _-_ R. for these radii may be drawn from different centers; but we have The effect of surcharge and mortar may be obtained 2 7 4 THEORY OF THE ARCH. from the formule of arts. 75, T6, which apply accurately to the cases under consideration. Those who wish to attain entire accuracy have only to subtract, from the thrust r, C, as given by table E', the following expression: Sill, 2 v'(d2-d) — 3 sin.2 V' cos,' 3) -78. Example. Casemate arch of Fort Jefferson, supporting the second tier of guns. Figure 16. The data are, s15'; f=2'; d-1'.50; c'-1'.50; from which we deduce, r= 15.06; v'- 30~ nearly; IE= 1.10, nearly; and clS=7f. Table E' gives for Ki1.10 and -s=f, F —r72X0.06784'" K 1.10 and _s8f, F-r2X0.05967 hence " K=1.10 and 8s=7f, F-r2 X 0.063315514.4635 A direct calculation gave F_]- -2 X 0.06 390 the difference, r2X 0.00015, is, in effect, nothing. But the arch has a surcharge, 6 inches deep throughout, which adds to the thrust, art. 7 5, if we suppose v'-300, 4.00987. By way of illustration, let us attribute to the mortar of the arch and of the concrete which covers it, an adhesive force of 3000 pounds per square foot. The mnasonry weighs, say 120 pounds per cubic foot; hence, art. 16, C —W305~ -25. We have cl, the depth of the arch at the key, 1'.50; and d', the depth of the arch and concrete at the springing line, a little over 4', say 4'. Substituting these values, we have as the effect of mortar (c cl 21].72; andtle finalthrust F_14.46335+4.00987-21.92 — -3.25. That is, the arch has no thrust. If we disregard the effect of mortar upon the vertical joint, we have F= —.57; still no thrust. SEGMENTAL ARCHES SURCUARGED IIOPRIZCNTALLY. 2 5 TIICKNESS OF PIER. 79. Let s=the span; f- the rise; d-the thickness of the arch at the key; t the depth of the surcharge above the key, the upper surface being horizontal. _l=the surface of that part of the arch and its load which lies directly over the half-span; q-nthe moment of that surface in relation to the vertical passing through the interior edge of the joint of the springing line; =-the lever-arrm of the thrust, or elevation of the point a above the base of the pier. h-the entire height of the pier from its base to the top of the surcharge over it-E'D' fig. 15. F7 x, 7, e, the same as in art. 92. We have WP- 1 9'TO,n= s(f+d+t) l'(2v' -sin. 2v') (40) 3-'' (~LIi'rCos.z qx''~Vi sill. V + — 3 (41) The formulse which give the thickness of pier under various circumstances, are precisely the same as in art. 72, and need not be here repeated. Example. The lower casemate arch of Fort Jefferson, regarding the floor, eight and one half feet below the springing line, as the base of the pier. The data aree /7:12'.50; s=-15'; f-2'; dl-'.50; t-0'.50; =: 15'.06; v'-300; I —.10; — 12'. This is the arch of which we obtained the thrust, F 14.46 in art. 78. The above data give us z - 30. -2' X 0.04529- 19.273;?z —112.50 — 3 X 0.0141 64.34. n, 1 ~n2 1n F7J — 1.58; 2-h2.49; =5.15; 18 Hence, for strict equilibrium, S-1, in (31)S, e=S'.89 for S=1.40, in (31)S... e=.99 w-Ae, (31A)8, e e * e./ 2076 TTHEORY OF THE ARCH. WVe see that, disregarding the effect of mortar, the pier should be at least 4'.74 thick. SEGMENTAL ARCtIES "WITH A SURCHARGE ON EACH SIDE OF THE CENTRAL RIDGE, RISING TO A PLANE OR ROOF AS IN THE MAGAZINE ARCH, FIGURE 19. 80. Let a-the span; f/-the rise; t-the depth of the surcharge, if any, above the two planes parallel to the roof and tangent to the extrados; v'-the semi-angle at the center; 9 =the radius of the intrados; cl-the thickness at the crown; cl'=the thickness at the springing line; I= the angle between the roof and a vertical. We have d / -1 -; sin. v'=; cos. -'. Look on table F for the angle of rupture, v, corresponding to the given values'of IC and l. This angle is given in three columns only, viz., under 1-90~, 60~, and 45~. Its value for other values of I may be estimated with sufficient accuracy by inspection, as it will be sufficient for our present purpose if we know that angle within six or eight degrees. If that angle be less than v', or exceed v' by only six or eight degrees, the thrust and the effect of surcharge are given at once by that table, precisely as if the intrados were a semicircle. The effect of mortar will also be the same as in semicircular arches. But if v' be less than v, the angle of greatest thrust extends evidently to the springing line, and the thrust itself will be given by (24), when we have substituted for v and 1, in that formula, the known value of v' and fin the given arch. In like manner, if v' exceed say 25~, the sliding thrust, if greater than the rotation thrust, will be given by table F; and the effect of surcharge will also be given by that table. SEGMENTAL ARCHES WITH INCLINED ROOFS. 2 But if v' be less than 250, the sliding thrust will be given by (25), when we have substituted for v and I in that equation, the known values of v' and I in the given arch. The rotation thrust, diminished by the effect of mortar, is F-, r2 C c-F+ C1 9P2C being the thrust obtained from table F, or by direct calculation, and the last term being the effect of mortar; d and cl' may be the whole length of the vertical and lower joints extended through any cover of masonry or concrete. When there is no such cover at the vertical joint, we have y-f+d. At all times we have y=r(K( — cos. v'). The rotation thrust, increased by a surcharge of uniform depth, t, above the roof of the arch, which last we suppose to be tangent to the extrados, is q P2 _ C1 r2 F=-r -2 C+-sit sin?.2 X 2' f3d 2 C being the thrust independently of surcharge. The sliding thrust, increased by surcharge and dilinished by the effect of mortar, is C=2' cot. 30~ PrUStC~ +iJ +cd') sin.' cot. (30~+v') l' o 30) 192C being the sliding thrust without regard to surcharge or mortar. In this formula we suppose v' to be less than 25~; if greater than 25', the sliding thrust is the same as in semicircular arches, and is given at once by table F, whenever the sliding is greater than the rotation thrust. 81. Example. The upper casemate arch of Fort Jefferson, figure 17, upper part; surcharged with concrete up to the roof, A 0, and above that roof with earth up to a horizontal line 85 feet above the springing line. The data are, s=15';f/3'; -- 43~ 36'; r- 10'.8T5; d:lR — = v=2'.28; 278 THTEORY OF TIlE ARCII. ]i= 1.21; relative weights of equal volumes of earth and masonry as 3 to 4. Reducing the elevation of the surcharge of earth in the proportion of 3 to 4, we may regard all below the reduced surface D' A' O' as having the density of masonry. Drawing the line P D parallel to 1' D' and tangent to the extrados of the arch, we divide the figure of the semi-arch into two parts; the one including all below this tangent; the other a surcharge of uniform depth above that line. The angle, 1, between R D or A' D' and a vertical, we find to be 82"~ 24' 20". As the angle of rupture in table F correspondiag to Ki- 1.21 and 1-90, is 63~, and the angle corresponding to 1[ 1.21 and 1=60 is 54~, ve know that the prism of maximum thrust extends to the springing line. Substituting for ]' V', and i in (24), the values above indicated, we obtain FP'r2 X.10727-12.687. Table F gives, for the same values of IKand 1, FX-r2 X 0.12141. The addition to the thrust caused by a surcharge of uniform depth, t-2'.725, is, art. 80, - 13.886; giving as the entire thrust X= 26.573. The effect of the adhesion of mortar upon the thrust is (d=4'; d'=4'; y:5.28; c=25.), 12_l+ d'2 -c. —-=25.25; leaving, as the final thrust, F-6.573 - 25.25=- 1.32. In assu ming c=2_5, we have not over-estimated the effect of good mortar, and may regard the arch in question as without thrust, provided there be no cracks in any of the joints. Unfortunately such cracks are very apt to occur, even during the construction of the arch. We have supposed the vertical and lower joints to extend into the concrete covering, making d and d' each 4 feet. SEGMENTAL ARCHES. —THICKNESS OF PIER. 929 TIICT(NESS OF PIER. 82. Let s=the span; f/the rise; cld-the thickness of the arch at the crown. E-thle elevation of the reduced. ridge above the springing line-v- n'R', fig. 19. E'=the elevation of the reduced roof above the springing line, meashred on the inner face of the pier, prolonged, — n a', fig. 19. n —the surface of that part of the semi-arcl and its load which lies directly over the half-span..m —the mnoment of that surface in relation to the iiner face of the pier. I=the lever armni of the thrust=a q, fig 19. ht-the mean height of the pier from its base to the top of the reduced surcharge upon it, to be estimated if not known. ~=the coefficient of stability. F tl-the thrust. e —the unknown thickness of pier. PE and E' are always known. We h av e,, =s(E+E') -' r2(2v' sin. 2v'). (42) /44 __3',2S"(Ft- a 1COS. Vq i I_2( 3 l r8(-v'sin.v+ -. The formlul which give the thickness of pier under various suppositions, are precisely thle same as in articles 64, 72, and need not be repeated. The formnule of 64, we have already said, are universal. The reader is referred to that article for a discussion of the formulm, and for the equation of the curve of pressure or resistance ill the pier. Exalmple. The upper casemate arch of Fort Jefferson, of which we obtained the thrust in art. 81. Let the floor 10 feet below the springing line, be the base of the pier, fig. 17. We have _E r'/'-H- 8'.125; _E' — ca- 7'. 125; 2 r 10'.875; v'-430 36' 10"; s= —15'; -17'; lz-15'.28; f- 3'; 7 280 THEORY OF THE ARCH. n_ 57.19 —r2 X 0.1308=41.73; m=n219.14 —' X 0.05566-= 147.55; F7-26.573. -=2.455; =6.026; T-=8.68; — =23.882. it A2 h2 h With these data, the formulae of art. 64 give us, For strict equilibrium, A=1, (31),.. e 3'.58 For AS2, rule of Audoy for large arches, (31), e-6'.72 For A-1.50, (31),..... e-5'.31 For xe, (31),...... e5'.83 For x=2-e, (31-),.... e= 7'.00 For x-e, (31),...... e12'.40 The thickness given by the rule of Audoy would seem, in this case, to be about right; as it is nearly equal to that which corresponds to x=-e. Were we to take into consideration the adhesion of mortar, and give to that force one half the value assigned in art. 81, we should find the actual thickness, 4', to be, amply sufficient. THRUST OF THE COMMUNICATION ARCHES OF A FORT UPON THE SCARP WALL, AND THE CURVE OF PRESSURE IN THE LATTER. 83. This is one of the most important applications of the theory of the arch. The scarp should be able to resist the thrust of the communication arches without any lateral motion whatever; and to this end the curve of pressure in the scarp should pass through the middle of the foundations, or very near that point. Each communication arch, besides its own proper load, supports through its entire span the weight of one half of each of the adjacent casemate arches, with all the surcharge of earth and masonry which may belong to the latter. The thrust, therefore, of the communication arches, and particularly of the upper one, is very great; and the effect of the latter is still further increased by the great leverage STABILITY OF SCARPS.-EXAMPLE. 281 with which it acts, that is, by its great elevation above the base of the scarp. These arches, on the outside, rest upon small piers carried up in contact with the scarp wall, but nowhere bonded in with it. 84. Example. Communication arches and scarp of Fort Jefferson, figs. 17, 18. Lower arch. Span-s-12'.25; rise=f=1'.75; thickness at the key=cl 1'.88; depth of surcharge above the keyt= 2'.87; radius of the intrados —q=11'.59; K-= 1+ -=1.16-; s= 7f; elevation of the extrados of the crown above the base of the scarp=Zl11'.63; surface rn 6 rn of the segment of the adjacent casemate arch-a= -22(2' — sin. 2v')- 20.5T4. But we advise the reader, in all problems of this kind, to regard the segment of a circle as the segment of a parabola standing on the same span, and tangent to the circle at the summit. This greatly diminishes the labor of the calculation, without leading to any sensible error in the results. According to this supposition we have a=-2 span Xrise=(in the case presented above) - X 15 X 2 - 0. The center of gravity of the semiparabolic segment standing on a horizontal base, is at the distance of 3 of this base from the altitude or axis of the whole segment. Let F as usual represent.the thrust, we have F=4r2x 0.0832..table E', the thrust without surcharge,.... =43.16 t6,~ + 4 X- +.. effect of surcharge firom ref. (17'.13) to ref. (20'),.... =59.34 4X15-a s.2 - + -d.X..effect of the adjacent casemate arch from (16') to (20'),.. =206.70 F= 309.20 28 THEORY OF ThE ARCH. We hlave computed the thrust -upon the whole length of one pier, which is 4'. In computing the surcharge we suppose the load to be limited to the inner face of the pier, and not as usual, to extend over the skewback. Uj/pel arch. s-12'.25-; f 3'; d- 1'.88; t-12'; r= 7'.7527; I(C-1.241; - 4.083f; l'=25'.63; a=215X3 =30. F'=4r02 X 0.13233.. table E', thrust without surcharge,.... =31.81 +4 X - Xd_. effect of surcharge from (31'. 13) to, say, (43'.13),.... =184.50 15x13-ac s. + X..effect of adjacent casemate ar~ch, (30') to (43'),.... =634.23 F'-850.54 Let NY=the volume of all the solids between the interior face of the scarp and the parallel vertical plane passing through the crown of the communication arches and above the floor of the lower casemates, which is at tlhe reference (7'.50). Mi=the sum of the moments of these solids in reference to the interior face of the scarp.'N=4 X 2 X 37.50.,the pier proper froml (5.50) to (43),... 300.00 12.25 12.25 X1.75 the +4 6.5X — -- X 2 X1.75 tlhe -~ 2 / lower arch from (13'.50) to (20'),. — 130.67 +4 (16.75 1225 -- X 2 x3) the upper arch and load, (26's5) to (43'), =361.36 Carried to page 283,. 792.03 STABILITY OF SCARPS.-EXAMPLE. 283 Brot. forward.. 79. 9.03 +(15 X 4 -15 x 2) (+1222 lower casemate arch and load, (16') to (20'), =325.00 (IV) +(13X15 — 15X3) (2+ 2 tpper casemate arIch and load, (30') to (43'), -1340.63 (T) A- 2457.66 2X —2 X 4 X 2 X 3g7.50, the pier from (5'.5) to (43'), -300.00 ~ 1.2.25/ 1. 2.2\ 4(6.50X- 2 + 4 ) +$-~~~~~~ lower -4 X —-2 X.5 X 2 +- J communication arch and load froml (13'.50) to (20'),...... -639.64 12+) /( upper -4X2 X X32+1X- - - corm. arch from (26'.25) to (43'),.. =191.97 +(IV) 325 X (2 +192.25) lower casemate arch from (16') to (20'),.. - 130.31 + (V) 1340.63 X (2+ -.25) upper casemate arch friom (30') to (43'),... -5446.31 tf=- 9498.2 3 We have calculated the values of F F', Ni, and Jf for 19' in length of the scarp. Dividing each by 19 we have their mean values corresponding to one foot in length of scarp. Let N_ 211 F F' 19 —129.35 —n; — 19499.91=m; — =16.274=F; 19=44.765-F'; 19 19 19 19 the known height of scarp from (5'.50) to (43') = 37'.5 —; the known thickness of scarp 8'= e. 284 THEORY OF TIE ARCH. — the distance between the exterior face of the scarp and the point where the curve of pressure cuts the base. Let us first suppose the pier 2' by 4', which supports one half the weight of the communication arches and their respective loads, to form an integral part of the scarp. We have, art. 64, 7ike2+qn+ -FlF- F' - = 13+' o =3'.256. (44) This is the equation of the curve of pressure in the pier (scarp), in which e, qt,., n/, and F' are constant, and h, I, and 1' vary by equal differences. Giving to 7A, 1, and 1', the values which correspond to the bottom of the foundations, viz., h-48; 1-22'.13; 1'= 36'.13, we find x —2'.13. As the foundations extend 4' in front of the scarp, we see that thte curve of pressure passes very nearly through the middle of the lowest course of masonry, its best possible situation. Consequently, the scarp is in no danger of rotary motion. Let us now suppose the piers 2' X 4' to be entirely separate from the scarp, as in fact they are. We have e2~?rn-F -F'l' (45) Giving, when hA 37'.50; 1 —11'.63; l'-25'.63, x-1'.21 " — 48'.00; 1=22'13;'-36'.13, x=0'.15 "' h —32'.00; 1=6'.13;' —20'.13, x-2'.04 The lower portions of both of these curves are sketched on fig. 18; the outer curve, c'', corresponding with these last results, the inner curve, c c, corresponding to the first supposition. The distance of this curve from the surface, at the point where it approaches the surface llost nearly, is the best measure of the stability of the sustaining wall. At t' the distance c' t' is 1'.21 e(8') X 0.151. There is no danger of the pier or scarp overturning; but there are two other points to which we must direct our attention. STABILITY OF SCARPS.-EXAMPLE. 285 (I). The horizontal joint t t', reference (5'.50), may open on the inside and allow the scarp to move laterally through a certain angle around c', near t', as center. (II). The bricks at t', the part most compressed, may be crushed by the superincumbent weight. As to the first, we can make no estimate of the extent of angular motion, not knowing the rate of compression of brick and concrete masonry under a given pressnre. As to the second, the entire weight supported by the joint t t' is, he 300 cubic feet of masonry. The pressure is greatest at t'; it is 0 at the distance 3x-3'.63 from 1'; the mean pressure is the divided by three times x; the pressure, per unit of surface, at t', is double the mean pressure. Calling this pressure per unit of surface at t', p, we have p = 2X h-=165.29=18,182. 3w pounds, supposing one cubic foot of the mixed masonry to weigh 110 pounds. This pressure, about 126 pounds per square inch, is rather too great, but probably does not exceed the allowed limit, one tenth the crushing force. There are, however, some elements of stability which we have not taken into consideration. We are warned by the cracks often seen in old works, not to rely, in any degree, upon adhesion of mortar in the communication and casemate arches. But there is another force which can hardly fail; viz., adhesion in the joints of the scarp. This force is C X 62 X5 X (8)2- 66.67, and the value of x becomes, at the joint t t', where,- 37'.50, lV6e2+rn+&-Fl-F / =-'.10 (46) which reduces the pressure, per square inch, at t', to 72.75 pounds. To this last pressure we ought to add the reaction of adhesion. 8 6 THEORY OF THE ARCH. PRESSURE UPON THE OUTER PIERS OF THE COMMUNICATION ARCHES. These piers have an area in the horizontal section of 2' X 4'-8, and each one of them has to sustain the whole value of N computed above. They are subject, therefore, to a pressure of 30.'21 per square foot; that is, each square foot bears a weight equal to that of a column of the same material one foot square and 307'.21 high; a pressure of S33,793 pounds per square foot, or 234.67 pounds per square inch. SEGAIENTAL ARCHES-APPROXIM3IATE FORINUL/E. 85. Tables E and E' give, in imost cases with sufficient accuracy, either directly or by proportional parts, the thrust of the segmental ring of equal thickness throughout, table E, and of the same ring loaded in masonry up to the level of the extradclos at the crown, table E'. B3ut those tables have not been extended to very flat arches, the last column in both corresponding to &-16f, and v'-14~ 15'; nor do they apply very well to cases in which s exceeds 10f, or v' is less than 22~ 37' 10"/ When it becomes necessary to make an independent calculation, and to ascertain the thrust without the aid of these tables, the exact methods already given are, it must be confessed, rather complex and tedious. XWe shall now give a much shorter method, sufficiently exact for all those cases in which v', the semi-angle at the center, does not exceed 30~; and applicable, with little error, to much larger values of v', particularly when the span is small, as it usually is in fortifications. The circular arc qn, fig. 15, departs but little from the parabola having its vertex at b, and passing through the point m. The equation of moments which determines the thrust, is SEGMIENTAL ARCHES. —APPRO XIMATE FORMULIE. 28 7 Fx a mn'-mnolment a n' in nx r aCt -lmnolm ent mn b qn' in, inz being the center of moments. Now the moment of the parabolic surface In 6 n' In, in relation to Iz, is,. in nw' X 6 in' X X n in'= -5 —sf ( -2 mnZ'); and the parabolic surface sin 6 in' Tr, is 32 -m In' X in' b. As the parabola is wholly below the circular are at all points between b and qn, we in effect suppose the arch to be a little heavier than it really is; and shall neutralize the error in part, or more than neutralize it, by adding the small triangle in, t, to that part of the arch which is on the left of the center of rotation. Let Cba n':y; 9 — radius of the intrados; PR=radius of the extrados; ]i= —; if the arch increase in thickness towards the pier or springing line, IE —, d being the thickness of the arch at the crown; v':the semi-angle at the center. We have, in relation to'in as center, monment a m' in t n zr ca-y x AR sin. V'(r sin. v'- -R sin.')_y x Ix.s2(1 -'I) -y822(K-_ 2). We have, therefore, as the thrust of the segmental arch loaded up to the level of the extrados at the crown, 4 2 12yb S5f)p (47) Illustrations. Suppose s910f; giving r —13f; V'v 220 37' 10"; and let KA L.10. The above fo1rmula gives, F 6,2 X 0.07 845 — 2 X 0.04642 Table E' gives for the same case,.. F'2 X 0.0465.5 Error, r,2 X 0.00013 Suppose s-6f; i-5f; v'=-36~ 52' 10"; I- 1.10. The above equation gives,. F. F- 2 X 0.07820 Table E' gives for the salme case,. F=-i 2 X 0.07724 Error, -r2 x 0.00096 * The point r, omitted on the left side of fig. 15, is vertically over n on the horizontal through a. 288 THEORY OF THE ARCH. This approximate formula, (47), gives with all desirable accuracy the thrust of the casemate arches of Fort Jefferson, art. 78. 86. The thrust of the segmental arch loaded horizontally up to a plane passing at the distance t above the crown of the arch, is F= L4Cs\(_I-22 12 + J ( - ) (48) 4 1 y In all cases y=f+dl. 87. The thrust of the segmental arch loaded up to any plane 1D' _?', fig. 19. Let in' R'-E; in a'=-A'; n r9=Ey"; the known distance n t-m n x sin. v'=D. We have 62.2 + I (= El +- - (E+ ) (9) The last term. is usually small, and in very light arches mray be omitted altogether. D and EF" can always be taken with sufficient accuracy from a drawing of the arch. Elquation (49) of course includes (48), but is more general in its character. It does not contain the ratio K, and is not founded upon the supposition that the arch is of equal thickness throughout. Moreover, it is strictly accurate as to the moment of that part of the arch and its load which overlies the skewback; the little triangle in t 7n no longer forming any part of this moment. Applied to the case in which the surcharge is horizontal, we have E'-=-]E; E"':E-m nX cos. v'; in all cases y f+d=a in'. Illustration. Let us apply (49) to the upper casemate arch of Fort Jefferson (see arts. 81, 82). The thrust given by (49) is. F 26.81 The exact thrust, art. 81, is.. F-26.57 The difference, = 0 24, always SEGMENTAL ARCHES.-APPROXIMlATE FORMULAE. 289 in favor of stability, is, we see, very small, notwithstanding the great extent of the semi-angle at the center,'v' —43~ 36' 10". We are therefore disposed to recommend formula (49) for exclusive use in calculating the thrust of the segmental arches of fortifications, when the thrust can not be obtained from the tables of circular or other arches contained in this paper. 88. The sliding thrust of segmental arches, when the angle of rupture extends to the springing line, is, using the notation of the preceding article, P- ('s(E+E'- /)+ (E'+E")) x cotang.(v'+ 30) (50) We can generally tell in advance, whether the true thrust is due to rotation or sliding; if not, it will be necessary to calculate both. T'IICKNESS OF PIER-APPROXIMATE FOREMULE. 89. For notation, see art. 82. Still regarding the intrados ~m b, fig. 19, as a parabola, we have - m,-142'2(E -KE'-&f/) (52), is in all cases the span, f the rise. Applying formulhe (49), (51), (52) to the upper casemate arch of Fort Jefferson, We find, when -=1, equation (31),. e 3'.59 The exact formulse, art. 82, gave us,.. e 3'.58 When 6-2, equation (31)... e=6'.74 The exact formulse, art. 82, gave.. e-6('.72 It thus appears that while the approximate methods require far less labor than the exact, they lead to almost identical results. The error committed in obtaining the 290 THEORY OF THE ARCI. thrust, is balanced in part, or more than balanced, when we apply (51), (52) to the determination of the thickness of pier. Equations (51), (52) may be used when the thrust has been obtained by exact methods; but the error in the thickness of pier will be somewhat increased. The formulse given for the thickness of pier in art. 64 are, as we have repeatedly stated, universal. The principle of these approximate methods has already been applied in calculating the stability of the scarp wall of Fort Jefferson; and we have gone over the same ground by the more laborious, exact modes of computation. The results were almost identical; but we could not always look for such close approximation. SECTION V. ELLIPTICAL ARCHES. 90. Elliptical arches are but little used on fortifications, where economy and stability are more regarded than architectural effect. They are, however, sometimes used in stone and brick bridges on great thoroughfares, and particularly in the neighborhood of large cities. The rise of the arch of almost all long bridges, is less than the half-span. There are three principal varieties of the intrados: 1st. The ellipse; 2d. The segment of a circle; 3d. The 3, 5, 7, &c., centered arch. The 2d, or segmental arch, is the strongest, the most economical, and, in general, the best. It is more easily built, less liable to change its form after the removal of the center, on receiving its final load, or any variable and occasional load, and has less horizontal thrust at the key ELLIPTICAL ARCHES. 291 than any other arch of the same span and rise; and its appearance, when the rise is small in relation to the span, is more agreeable to the eye than that of the flat ellipse. This variety has already been disposed of. We will here add the remark, that the segmental arch or ring should increase in thickness, from the key to the springing line, at such a rate as to become, if continued to 60~ from the key, about one half greater than it is at the key. Such increase will, in general, not only insure the requisite stiffness or stability of form in the arch itself, but will nearly equalize the pressure, per unit of surface, upon the joints of the key and springing line. Should the intrados of the segmental arch extend more than 60~ from the key, the augmentation of thickness must continue. Arches of the 3d class are all approximations, more or less close, to the ellipse having the same rise and span. With five or more centers, the approximation becomes almost an identity; and we may regard the ellipse as representing all these arches, that is, as having the same thrust and requiring the same thickness at the key and the same thickness of pier. THRUST OF THE ELLIPTICAL ARCH WTITHIOUT LOAD. Figztre 20. 91. Let A C, the half-span and semi-transverse axis, =1r; C b, the rise and semi-conjugate axis,=f; ca b, the thickness at the key, =- Let us compare this arch with a circular arch of the same span, and of a thickness, c'b' —d', at the key, as much greater than the corresponding thickness of the elliptical ring, as the half-span is greater than the rise. And let us further suppose the vertical depth of the elliptical ring to bear a constant ratio to the depth of the 292 THEORY OF THE ARCH. auxiliary circular arch, both measured on the same vertical line; so that, on any vertical line m r.', we shall have, mr: m' r':: a:' b'::f: r. This requires the extrados of the elliptical arch to be another ellipse, having, for its axes, Ca=o f+d, and CBCa' —r+- X d. This arrangement gives a continual augf mentation, not too great, to the thickness of the elliptical arch. Comparing together the segments m r a b, z'r' a' b', included between the vertical of the key and any other vertical vn' r', we see that their horizontal dimensions are the same, while their vertical dimensions bear the constant ratio of f to r. Consequently, the surfaces of those segments sustain the same constant ratio, and their centers of gravity are in the same vertical line. Projecting m and mn' horizontally on C a' at t and t', and disregarding, for the present, the negative influence upon the thrust of the surfaces, nearly triangular, m n r and m' n' r'; designating by S, the surface in r a b; by p, the distance of the center of gravity of this surface from the vertical through mn; by y, the lever arm a t; by S', p', y' the corresponding surface, distance, and lever-arm, a' t', of the circular arch,-we have for the thrust, F, of the elliptical arch, corresponding to any position of the vertical line in', ~F=-1- and, for the circular arch, P' S= But we have already found p=p', S —/', and we have f r:: Cb~: Cb'/:: Ca: Ca':: Ct: Ct': Ca-Ct (y) Ca'- Ct' (=y'). Consequently, y= —Xy'; and F=Fi'. This relation exists for all positions of the joint of rupture; hence, the maximum or true thrust of the two arches will be the same, and we are able to announce this principlhe: ELLIPTICAL ARCHES. 293 Tlhe thlrut of a senicircular arch, of equcza thickness throughout, and without load, is nearly equal to the thruct of an elliptical crchl of the same span, and of a vertical dep2t7 at the key and at every other point acs 7much less thanc the dqeth of the cirucular arch, on the same vertical lines, as the rie of the elliptical arlch i less than the hcaf-cspan. XTe shall therefore be able, with little error, to obtain the thrust of unloaded elliptical arches, fronm table A. 92. The result, however, thus obtained, requires a slight addition. We have neglected the difference in effect of the small surfaces In n Ir, in' n' r', which tend to diminish the thrust of the arches to which they respectively belong. If the moments of these two surfaces in relation to the vertical through m, like the moments of the segments t rct a, m' ri' a' b', stood in the proportion of f to. r, no correction would be necessary. But such is not the case. Drawt the tangents in o, m' o, to the intrados of the elliptical and of the auxiliary circular arch, meeting on the transverse axis or horizontal of the springing line at o; draw the normal inp' to the ellipse intersecting A C at p'; prolong the linep' in to n, the extrados of the elliptical arch: and the line p' m' to n" the extrados of the circular arch. n and n" are evidently on the same vertical line. Let v represent the angle between any joint in n-supposed to be normal to the intrados-and a vertical; v' the corresponding angle n' in' r', or m' C a', of the circle; and V the angle n" m' r'. We have v-angle p o rn; v'=angle p om'; __r angle p int p'; tang. v=ftang. v'; tang. V=Ltang.v V'':-. "tang. — /2 tang. v'. From these relations v and V are easily calculated when v' is given. The triangles in n r, in' n" r-', have the same altitudes, and bases m r, in' r', in the proportion of f to ar. Consequently, they have the same effect upon the thrust; and 2094 TIIEORY OF THE ARCH. the required correction consists in adding to the thrust, as given by table A, the effect of the triangle n' nn' q". Let A represent the required addition. We have m% ~III'(inI 7')2 sin.2 V'_(MI'')2 Sin.2 at'\ v'@'") (53) Now, in all the cases likely to occur in practice, the angle of rupture corresponding to the maximulu thrust, is in the neighborhood of 60~; and we shall calculate the value of A on that supposition. if representing the ratio of the two radii of the circular arch, we have, when v'60~,0 rn r(VKJ2P-_0.75 - 0.50); at''= r(-I- 0.50); n"'n'(K 1); qn f"?(1 2 ssin(60- YV)~-cos. (600,a2 - V)); tang. V. We subjoin the values of A corresponding to f=l r, and to all values of if between 1 and 1.60. TABLE A'. A VALUES OF - TO BE ADDED TO THE COEFFICIENTS OF T~ GIVEN BY THE 4TIH COLUMN OF TABLE A, CALCULATED ON THE SUPPOSITION THAT f _ Value of A F A _ A d A d Value of AKx = 1 + I + A c 1.01 0.00000 1.16 0.00092 1.31 0.00510 1.46 0.01343 1.02 0.00000 1.17 0.00108 1.32 0.00552 1.47 0.01414 1.03 0.00000 1.18 0.00126 1.33 0.005961 1.48 0.01487 1.04 0.00002 1.19 0.00144 1.34 0.00642 1.49 0.01562 1.05 0.00004 1.20 0.00165 1.35 0.00690 1.50 0.016839 1.06 0.00006 1.21 0.00188 1.36 0.00740' 1.51 0.01718 1.07 0.00009 1.22 0.00212 1.37 0.00792 1.52 0.01799 1.08 0.00014 1.23 0.00238 1.38 0.00845 1.53 0.01882 1.09 0.00019 1.24 0.00266 1.39 0.00900 1.54 0.01967 1.10 0.00025 1.25 0.00295 1.40 0.00957 1.55 0.02054 1.11 0.00033 1.26 0.00326 1.41 0.01016 1.56 0.02143 1.12 0.00042 1.27 0.00358 1.42 0.01077 1.57 0.02234 1.13 0.00052 1.28 0.00393 1.43 0.01140 1.58 0.02328 1.14 0.00064 1.29 0.00430 1.44 0.01206 1.59 0.02423 1.15 0.00078 1.30 0.00469 1.45 0.01274 1.60 0.02519 ELLIPTICAL ARCHES. 295 The value, A', of A corresponding to ally other relation of f to r, will be given, with sufficient accuracy, by the following formula: This last formula will give thrusts slightly in excess for values off between 1 and -, and thrusts a little too small for values off less than. 93. Recapitulation. To find the thrust of the unloaded elliptical arch, the extrados being an ellipse similar to the intrados; r=the half span; f=the rise; cd-the thickness at the key: Look in table A, 4th column, for the coefficient, C, of r2, opposite Ki=1+); to this, add the product C'=2(14-) x 4-, being taken from the above table A' opposite the same value of Ki. Then F-=r2(C-+ C'). Example. r —10'; f=6'8"; d-1'.50; hence i- 1+ d_ f-1.225. Value of C, 4th column of table A, mean between K=1.22 and K=1.23 — 0.12044 Value of C'-2(1- 2-) X0.00225 A being the 3 /14 v~vvu~ rr ) g2 mean between fC=1.22 and 1.23, table A' -0.00150 Total thrust Fr2X 0.12194 M. Audoy gives as the thrust of a threecenter arch of the salme span and rise and thickness, the intrados being described with three arcs of 60~ each, F= —2 X 0.12569 Difference, = 2X 0.00375 =about three per cent. of the true thrust. 8 996 THEORY OF THE ARCH. Example 2. 1r=10'; f-=5'; d-2', hence K=1 —+2 1.40. Value of C, 4th column of table A, opposite K-= 1.40 — 0.16167 Value of UC- 2(1 —) x 0.00957 table A' opposite Kr_ —1.40 -0.00957 pF=9 2 X 0.17124 M. Audoy gives as the thrust of a five-center arch of the same span, rise, and thickness, FZ r2 X 0.17914 Difference F=r2 X 0.00790 -about 41 per cent. of the true thrust. The rule stated above gives immediately, and with all desirable accuracy, the thrust of elliptical arches, unloaded. Table H contains, all calculated, the rotation thrusts in two systems of elliptical arches, corresponding respectively toJ=L and =-. The first column gives the quotient of the span divided by the thickness at the key, this quotient being the proper measure of the lightness of the arch. That table seems to require no explanation. SLIDING THRUST OF UNLOADED ELLIPTICAL ARCHES. Figure 20. 94. It appears from table A, that the rotation thrust of the unloaded circular arch exceeds the sliding thrust for all values of K less than 1.45. It appears from table A', art. 92, that the rotation thrust of the elliptical ring, bounded by similar ellipses, is greater than the rotation thrust of the circular arch resting upon the salme span and having a thickness at the key as much greater than the thickness of the elliptical arch as the half-span is greater than the rise; and that the difference increases as If increases. ELLIPTICAL ARCHES. 297 Without stopping to demonstrate it, we here state the fact that the sliding thrust of the elliptical arch is always less than that, of the auxiliary circular arch above described. Putting these facts together, we can give the following rule as perfectly safe, though liable to give a thrust too great: Find the rotation thrust, art. 93. Should this thrust be less than the sliding thrust found in table A, opposite d Iir= 1 ~-, adopt the latter as the true thrust. THICKNESS OF PIER-ELLIPTICAL ARCHES-UNLOADED. Figure 20. 95. Let F=the thrust; h=the height of pier from the base to the springing line; I=the lever-arm of the thrust or elevation of a above the base of the pier; n=the surface of the semi-arch A B a 6; m=the moment of that surface in relation to A; S -the coefficient of stability; e-the unknown thickness of pier. 7t d: c ~d d2 n =-dc/(2 +); Fe= =1, d(.sT08-.2146 X- 1X ); (54) 1ze2 +ne+ —sn = Fl e= -+ -2- +2 7THRUST OF ELLIPTICAL ARCHES SUSTAINING A LOAD OF MASONRY, OR OF EQUAL WEIGHT WITH MASONRY, RISING ON EACH SIDE OF THE CENTRAL RIDGE, TO A ROOF TANGENT TO THE EXTRADOS. Figure 21. 96. Let us compare the given arch with a circular arch of the same span, and of a thickness at the key as much greater than the thickness of the elliptical arch as the half-span is greater than the rise; and let us suppose the 298 THEORY OF THE ARCH. loads of the two arches to susta in this same relation in their vertical depths. We suppose the thickness of the two arches at the springing line to be the same, which requires the extrados of the elliptical arch to be an ellipse similar to the intrados. XJ P, the roof of the given arch intersecting the horizontal of the springing line at P; draw P R' tangent to the extrados of the circular arch. The two arches thus constructed, sustain to each other the relation described above. We have, by supposition, a: a' 6':: rf:'. Draw the vertical d D' passing through the points of tangency D and D)'. Draw any other vertical line, _p u' cutting at mn the intrados, at r the extrados, and at u the roof of the elliptical arch; at n,', r', u' the corresponding parts of the circular arch. We have _pu:p u':: d D:'::p r:p':: Ca: Ca'" C&: Clb'::f::: n:pm'. Hence mubu: m'u'::f: r. Moreover projecting in and rn' horizontally on the vertical through Y, at t and t', we have, as in art. 91, a t: ac t':f: r. Pursuing precisely the samle course of reasoning as in art. 91, we see that the thrusts due to the segments in u Rb?, m' t' R' b' are the same in the two arches, wherever the vertical m in' be drawn. Consequently, the maximum thrust of the two arches is the same, and we can announce this principle: 7The thrust of an elliptical arch loaded in maasonry rp to a pclane tangent to the extrcados, scpjposed to be similar to the intrados, is nearly equal to the thruSt of a circular arch7 of the caine span, and of a thicknzess of ring and of load at the key as, much greater than the correspondiny depths of the ellip tical arch, as the half-spjan is yreater than the rise; the load of the circular'arcsh also rising to a platne tangent to its extrados. We can, therefore, in most cases, obtain the required ELLIPTICAL ARCHES.-LOADED. 299 thrust from table F. The result, however, thus obtained, will require a slight addition, viz: moment on m' of the surface n' n' i' if' n" m' A. (55) lever arm a' t' which may be calculated on the supposition that the angle of rupture of the circular at'ch is 60~. It will be best to construct the diagram and obtain the elements of this calculation by protraction. For obvious reasons the addition required will generally exceed but little the values given in table A', art. 92. R-ule.-Suppose the angle CR P I' to be given by its tangent. Let angle R'P= —I; half-span-gr; rise=f; thickness at the keyab=cl. We have tang. I — -tang. I', which gives:. Obtain from table F the coefficient C of i-' corresponding to this value of I and to K=l +-; to this add the product 2 1 — X- -=C', -- being taken from table A', art 92, opposite the same value of IK. Then the thrust F=-2(C-+ C'). To this we ought to add, when n' i', in" i", 60~ from the key, have any considerable magnitude, the effect upon the thrust due to the trapezoid n' i' " n o 97. Example: 1r=10 feet; f=62 feet; I'=60~; dl 1'.30; Y=1-+ —=1.195; _=49~ 6' 24"=49~.102; 50~3 = 0~0.8 9. Coefficient of r2, table F, forl-= 1.195, I-45~:= C- 0.25676 "; 466'" ". 150~ " 0.20942 Difference, 0.04734 300 THEORY OF THE ARCH. 5: 00.89:: 0.04734: x =0.00846 Add a, as above, for R=1.195; I:50~, 0.20942 Add from table A', art 92, 2(1 —n) X 0.00155, 0.00103 Total thrust, F= 92 X 0.21891 M. Audoy gives as the thrust 6f a three-center arch of the same rise, span, and load, r2 X 0.22075 Difference-I of 1 per cent.,.. r2 X 0.00184 The rule given above can only be used when the angle I is greater than 45~, or but little less than that limit. In other cases it will be best to investigate the thrust geometrically. The effect of a surcharge of uniform vertical depth may be obtained from the table in art. 104; but the addition thus found will be a little too large. Let t represent the depth of the surcharge. Look in that table opposite K 1=l+- for the value of C; the required addition will be A:=2x -C. THICKNESS OF PIER. Figuire 21. 98. The general formulse of art. 64'are all applicable to the elliptical arch. E', I', 1, &c., must be taken from the given elliptical arch. The values of n and mn will be as follows: n=-r(E+l')-frX0.7854 o (.56 T112(-E10 E+-~E') -f 2x 0.452065 ELLIPTICAL ARCHES.-SURCHARGED HORIZONTALLY. 301 ELLIPTICAL ARCHES LOADED HORIZONTALLY UP TO THE LEVEL OF THE EXTRADOS AT THE KEY. _Figure 22. 99. This is the most common form of the elliptical arch, almost the only form in practical use. Let r-A C-the half-span and semi-transverse axis of the ellipse; f C6bthe rise, and semi-conjugate axis; d- cb=the thickness at the key. This thickness we suppose to be constant. The calculated thrust will be a little larger than it would have been had the extrados been another ellipse similar to the intrados. Let us compare the given arch with a circular arch, surcharged in like manner horizontally, having the same span, and a thickness at the key as much greater than the thickness of the elliptical arch as the half-span is greater than the rise. All the vertical dimensions of the auxiliary circular arch bear a constant proportion to the corresponding dimensions of the elliptical arch; so that, drawing any vertical, p m m' it', we have f ccl) Cib Ca p ppn ent 7- c'b'- CYb'- Ca6' pl- pm' a en n Consequently the surfaces m ng a b4, m' it' a' b', are also in the proportion off to gr; their centers of gravity are on the same vertical line; their lever arms, a t, a' t', or mn, mn' I', are in the proportion of f.to r. These surfaces, therefore, have the same thrust wherever the vertical line be drawn. Their maximum1 thrusts will be the same; and we come to this conclusion: The thrust of can elt tical arch suqstacining a load of mast~onrmy, or of equctG weigyht with masfnonry, rising to the hzorizontcal linue tange.nt to the extrados at the key, is nearly equal to the thrtst of the semnicircular arch, loacded in like mnanner, hayving the same span and ca tlzic7kness at the key as much grpeater than the thickness of the elliptical arch as the half-spcn is greater than the rise. 302 THEORY OF THE ARCH. We shall therefore be able to obtain from table D, with little labor, the thrust of elliptical arches. The thrust thus obtained would be perfectly correct if the moments, in relation to the vertical through nm, of the two surfaces m n i u, Z' n' i' I', stood like the segments on the right of that vertical, in the relation off to is. But this is not the case; and it is necessary to correct the result. In calculating table D, we have, in effect, subtracted, from the thrust due to the segmlent rn' u' a' I', the negative influence of mn' n' i' a', both taken at the angle of maximnurn thrust. For our present purpose we should have subtracted only the effect of mn' I" i" It', which stands very nearly in the required relation to the surface m n i X. The joint n - -a 6 — d is supposed to be normal to the intrados;?Mz' bq" parallel and equal to rm n. It is evidently necessary to add to the results of table D the difference of the effects of the two surfaces, 9q' n' i' i,'v,' in" i"';1,; or, increasing this difference slightly, the effect of the surface s 8'i' i", of which the altitude, m' tI', is also the lever arm of the thrust. Naming v the angle between n in and a vertical, v' the angle between mn' vn' and a vertical, we have, tang. =- r tang.,'; and for the thrust, 1 (; sin.' - 2, sin.2) (5 C+ f)~;~. (57) in which C' is taken from table D, and v' is supposed to be the angle of maximum thrust. This angle never differs much from 60~. We are at liberty, therefore, in all cases, to suppose v'-60~; which gives f2 Sin.2v' —a' Sin.2? _ 1+3 S ELLIPTICAL ARCHES.-SURCHARGED HORIZONTALLY. 303 Table G, calculated from the above formula, gives, either directly or by proportional parts, the thrusts of all elliptical arches in practicable use. The first column is the quotient of the span divided by the thickness of the arch at the key, this quotient being the proper measure of the massiveness or lightness of the arch. 100. The table on the following page gives the horizontal thrusts of elliptical and segmental arches of the same span, rise, and thickness, in four systems, all the arches being surcharged horizontally. The reader will perhaps be surprised to see that there is but little'difference in the thrusts of the two kinds of arches, and that, in very light arches, the difference is in favor of the elliptical intrados. NWhen the rise is one fourth or one fifth of the span and the thickness about one twentyfifth of the span, the thrust is nearly the same in elliptical and& segmental arches. To explain briefly the manner of comparing these thrusts: In tables E' and D, Ad is the radius of the intrados; in the following table, SUB is the half-span. Let 7=-the half-span; f-the rise; dcl=the thickness at key; r' —the radius of the intrados of the segmental arch of the same span, rise, and thickness; ]K-the ratio of the two radii of the segmental arch; K'- We have 2 /); (58) _IF~rz.,?'2 X =+e2 X C F=r'+ x Cd. x From this last formllula the coefficients of r'2 in the following table have been computed from the coefficients of.,'2 (9.2), in tables E' and D. 304 THEORY OF THE ARCI. 101. Talbe of horizontal c tha usts of elliptical and segmental archesq of the samne spcan, rise, and thiekness at the key, in four systems; surccharged horizontcally to the horizontaglplacne tangent to the extrados at the key; each arch7 of the same thickness th#rotughout.; r- the half-span; d=t7he thickness at the key/; F'-thruqst of elliptical arches,; F= thrust of segmental arches. Rise -= the span. Rise = ~ the span. d Thrust in Thrust in Cd Thrust in Thrust in =12. 1ld _K the elliptic- the segmen- P 1+ —= the elliptic- the segmen- F' 1 + K. al arch = F'/tal arch =F. r l arch - F1 tal arch=F _ =r2 X. =2 X C. K. =r2 X C =r2 X C. K. =r. 60 1,02300 0.09874 0.10747 0.920 1.02- 0.09075 0.09840 0.920 50 1.02760 0.10628 0.11334 0.940 1.0320 0.09703 0.10322 0.940 40 1.03448 0.11687 0.12169 0.960 1.0400 0.10600G 0.11021 0.960 30 1.04600 0.13308 0.13479 0.990 1.0541 0.11977 0.12101 0.990 25 1.05520 0.14469 0.14450 1.0001 1.0640 0.13005 0.12911 1.010 20 1.06900 0.16054 0.1]5798 1.020i 1.0800 0.14359 0.14040 1.020 15 1.09000 0.18355 0.17610 1.040 1.102 0.16349 0.15692 1.040 10 1.13800 0.22374 0.20880 1.070 1.1600 0.19564 0.18300 1.070 8 1.2000 0.21684 0.19737 1.100 Rise = ~ the span. Rise = 8 the span. 60 1.03077 0.08238 0.08478 0.970 1.0320, 0.07943 0.07905 1.000 50 1.03690 0.08727 0.08837 0.990 1.0384 0.08386 0.08250 1.016 40 1.04615 0.09448 0.09373 1.010 1.0480 0.09047 0.08764 1.030 30 1.06150 0.110521 0.10222 1.030 1.0640 0.10011 0.09580 1.045 25 1.07380 0.11336 0.10869 1.043 1.0768 0.10747 0.10174 1.056 20 1.0923(0 0.12453 0.11719 1.060 1.0960 0.11758 0.10988 1.070 15 1.12300 0.14065 0.12999 1.080 1.1280 0.13225 0.12194 1.085 10 1.18460 0.16564 0.14965 1.110 1.1920 0.15474 0.14008 1.100 8 1.23077 0.17977 0.15966 1.130 1.2400 0.16680 0.14906 1.120 6 1.30770 0.19848 0.16872 1.170 1.3200 0.18082 0.15690 1.150 102. In the preceding table we have supposed every arch to have a constant thickness throunghout. In practice, all light arches should increase in thickness from the key to the springing line, as already explained. Such increase, applied to the arches of the preceding table will diminish the thrusts of elliptical and segmental arches, without sensibly changing their relative magnitudes. ELLIPTICAL ARCHES.-SURCHARGED HORIZONTALLY. 305 ]EFFECT OF SURCHARGE UPON THIE ROTATION THRUST. Figure 22. 103. Suppose a surcharge of the density of the arch and of the uniform vertical depth t-atl. On the auxiliary circular arch lay off a' d': x t. Let us take into account in reference to any joints m n, mn' i', only that part of the surcharge which overlies the segments upon the right of the vertical, mmin'. The effect of the surcharge will be precisely the same in the two arches. But the addition to the thrust caused by this surcharge in the circular arch will be A - -- 1r-2,; of which the maxf K- cos.f imum value is A=}2XyX(K-We (59) The following table gives the values of the factor (K4/XK-1) for values of JKranging from 1 to 1.40. 306 THEORY OF THE ARCH. 104. Tc7ble of additions to the rotation thrust caugsed by a Surchzarge of con.etant vertical depth; d-the thiclknee8 of the eYii/tical arch at the key; r=the hcaf-span; fthe rise; t —the depth of the eurclharge on the ellipticcal cc6vch; A —te acdition to the thrwtst of thle elliptical arch,; C- the decimal in any colzmnn; i — the ratio of the two radii of thAe auxiliary semicircular arch..d ~ 2 t We have 1 — - + AI rt2 X X. d x- + C' zc. A=r2x -C. A=r2xx A=r2xxC. 1' + —I 1.00 1.00000 1.11 0.62823 1.22 0.52114 1.33 0.45313 1.01 0.86823 1.12 0.61562 1.23 0.51383 1.34 0.44804 1.02 0.81900 1.13 0.60379 1.24 0.50679 1.35 0.44308 1.03 0.78322 1.14 0.59264 1.25 0.50000 1.36 0.43826 1.04 0.75434 1.15 0.58211 1.26 0.49345 1.37 0.43357 1.05 0.72984 1.16 0.57212 1.27 0.48712 1.38 0.42900 1.06 0.70843 1.17 0.56263 1.28 0.48100 1.39 0.42455 1.0o 0.68934 1.18 0.55358 1.29 Q.47508 1.40 0.42020 1.08 0.67208 1.19 -0.54494 1.30 0.46934 1.09 0.65630 1.20 0.53668 1.31 0.46378 1.10 0.64174 1.21 0.52875 1.32 0.45837 105. Example. The Waterloo Bridge: span=l 120 feet -2r; rise_3 2 feet=f; thickness at the key=4.50 feet c-d. These dimensions give, the number in the first column of 2r table G, K'=- -263; f-r X 0.5 3. Thrust, 3d column table G, for If' 26, it r X 0.50,....:.Fr2 X 0.12767 Ditto for'-9-28 f-i: X 0.50 7. ~ e-F22 X 0.12 3 4 ELLIPTICAL ARCHES SURCHARGED HIORIZONTALLY. 307 Subtracting one third of the difference from the former, we have, for K' -26, fr X.50,..... 2 X 0.12627 In like m anner we find for K'= 262, j= 7' =r= X 0.55,..., _F2 X 0.12 068 50 Adding one third of the difference to the latter we have the required thrust, corresponding to K.'-26}, J=rX 0.531, F'lr2X 0.12254 Suppose a surcharge 4 feet deep throughout, we find, art. 104, opposite K- 1 + d 4.50 t S-1+ 3' 1.14, A= r2-X C= —'2 4 X 32 X 0.59264...2 X 0.07408 Total thrust, _F —2 X 0.19662 =707.83; that is, the thrust upon one foot in width of the bridge at the key, is equivalent to the weight of a column, of the material of the bridge, one foot square and 707.83 feet high. Dividing this by the thickness of the arch, 4feet, we have, as the mean pressure upon each square foot of surface at the key, 157.30. This mean pressure, according to the best authorities, should never exceed one twentieth of the ultimate strength of the material. Example II. ~ —10'; f=62feet; d=l'; giving K' 20; f r; Thrust, 6th column table G, opposite K' =20,.... F 2 X 0.12453 Three-center arch of the same rise, span, and thickness, M. Audoy,.. 2 X 0.13089 Difference, alout 5 per cent. of the thrust, =-)2 X 0.00636 308 THEORY OF THE ARCH. SLIDING THRUST OF ELLIPTICAL ARCHES SURCHARGED HORIZONTALLY. 106. This thrust will always be less than the rotation thrust, unless the arch have an enormous thickness at the key. For reasons substantially the same with those given in art. 94, we are justified in offering the following rule as perfectly safe, but liable to give a thrust somewhat too large. Rule. Find the rotation thrust as above explained. Should this thrust be less than the sliding thrust in table D, opposite I-= 1 -f adopt the latter as the true thrust. f THICKNESS OF PIER. 107. The formulae of art. 64 are applicable to all cases. Appliedcl to elliptical arches surcharged horizontally, we have E=E'=f+cd+t; n=rx-E — fxx0.7854; n( 2 qE-r2f/x 0.452065; (60) in all cases l=the elevation of the extrados at the crown above the base of the pier. Example. Waterloo Bridge, dimensions and load given in art. 105. Depth of the pier below the springing line= 19'.50; A- 60'; E- 32'-+4'.50+4' —40'.50; = 60 X 40.50 - 60 X 32 X0.7854=- 922.03; n — (60)2 X 40.50 — n12 (60)2X32X0.452065=20822. 1=56'; -=15.36; / -= 236.15; A =347.03; 7-=660.64; F, art. 105, =707.83. 2615 i h 4r0 ELLIPTICAL ARCHES SURCHARGED HORIZONTALLY. 309 These values substituted in (31), art. 64, give, for strict equilibrium, - 1,. 6=.... e 14'.02 (31), art. 64, give, for -_2,... e=31'.3 (31l, " = e, e-22'.43 (31 E) " _21... e-s6'.43 e=26'.43 The thickness of the existing piers is, at the bottom 30', at the springing line 20', and the piers extend above and below the bridge about one-fourth the width of the bridge, each way. Every pier, therefore, of this celebrated bridge, is an abutment pier, with very nearly the excess of stability prescribed by the French engineers. Comparing the moment of the thrust with the sum of the moments of all the elements of resistance, we find A, the coefficient of stability, to be very nearly 1.79. 310 THEORY OF THE ARCH. CIRCULAR ARCHES OF 180~, WITH PARALLEL EXTRADOS. (A) ~Dbse jiuviy the aongle of rupturq e, tke tlruSt, Can1ZC t7he limit thickness of piers. Value Ratio'V2. C. of the limit thickValue Ratio of the Ratio C, of the thrust to the ness of Pier to the radius o,f the of the of the Allgle square. of the radius r. of the in- intrados. ratio diameter of trados. 1 _ R to the Rupture. -. thickness., - _ Strict Co-efficient of Rotation. Rotation. Sliding. Equilibrium. stability, 1.90 2.732 1.154 00 00' 0.00000 ().98923 2.70() 1.176 13 42 0.00211 0.96262 2.65 1.212'22 00 0.00319 0.02168 2.60 1.250 27 30 0.00808 0.88151 2.50 1.333 35 52 0.o0283 0.80346 2.40 1.428 42 06 0.04109 0.72847 2.30 1.538 46 47 0.06835 0.65651 2. 20 1.666 51 04 1 0.08648 0.58767 2.10 1.810 54 27 0.10926 0.52186 2.00 2.000 57 17 0.13017 0.45912 0.9582 1.3223 1.90 2.282 59 37 0.14813 0.39943 0.8938 1.2320 1.80 2.500 61 24 0.16373 0.34281 0.8280 1.1414 1.70 2.857 62 53 0.17180 0.28924 0 76(06 1.0484 1.60 3.333 63 49 0.17517 0.23874 0.6910 0.9525 1.59 3.389 63 52 0.17533 0.23386 0.6839 0.9427 1.58 3.448 63 55 0.17535 0. 229t'1 0.6768 0.9329 1.57 3.508 63 58 0.17524 0..22434 0.6698 0.9233 1.56 3.571 64 01 0.1749 0.21940 0.6624 0.9131 1.55 3.636 64 03 0. 17478 0.21464 0.6552 0.9031 1.54 3.703 64 05 0.17445 0.20991 0.6479 0.8931 1.53 3.773 64 07 0. 173397 0.20521 0.6406 0.8831 1.52 3.846 64 08 0.17352 0.2)0054 0.6333 0.873) 1.51 3.920 64 08 0.17310 0.19590 0.6259 0.8628 1.50 4.000 64 09 0.17254 0.19130 0.6185 0.8527 1.49 4.081 64 08 0.17180 0.18673 0.6111 0.8424 1.48 4.166 64 08 0.17095 0.18218 0.6036 0.8320 1.47 4.255 64 07 0.170,,8 0.17766 0.5961 0.8216 1.46 4.347 64 06 0.16915, 0.17318 0.5885 0.8112 1.45 4.444 64 05 0.16798 0.16872 0.5809 0.80()7 1.44 4.545 64 03 0.16683 0.16430 0.5776 0.7962 1.43 4.651 64 00 0.16568 0.15991 0 5756 0.7934 1.42 4.761 63 56 0.16448 0.15555 0.5735 0.7906 1.41 4.878 63 52 0.16317 0.15122 0.5713 0.7874 1.40 5.000 63 48 0.16167 0.14691 0.5686 0.7838 1.39 5.128 63 43 0.16014 0.14264 0.5659 0.7801 1.38 5.263 63 38 0.15845 0.13841 0.5629 0.7760 1.37 5.406 63 32 0.15672 0.13420 (1.5598 0.7717 TABLES. 311 CIRCULAR ARCI-HES OF 180, WITHI PARALLEL EXTRADOS. (A) J2ble qiviny tke canvgle of riptlure, t17e tf rul, and t7Ie limit tAiclknes. of pier8. Valou atoatooValue Rati o \/'2. (. of the limit thickof tle Ratio of the tRatio a of the thrust to the ness of pier to thle radius of the o ate of t he aeR oe squares of the radius, s', of the in- intrados. ratio diameter t f tradfs. K -- to the rupture. r thickness. _ I- -. Strict Co-efficient of Rotation. R.otation. Sliding Equilibrium. stability 1.90. 1.36 5.55,5 630 26' 0. 15482 0.13002 0.5564 0.7670 1.35 5.714 63 19 0.15287 0.12587 0.5529 0.7622 1.34 5.882 63 10 0.15(196 0.12176 0.5495 0.7574 1.33 6.060 63 0 ) 0.14896 0.11767 0.5458 0.7524 1.32 6.264 62 50 0.14678 0.11362 0.5418 0.7468 1.31 6.451 62 33 0.14510 0.10959 0.5387 0.7425 1.30 6'. 666 62 14 0.14330 0.10559 0.5353 0.7379 1.29 6.896 62 09 0.14013 0.10163 0.5294 0.7297 1.28 7.142 62 03 0.13691 0.0977 0 0.5233 0.7'213 1.27 7.407 61 47 0.13430 0.09379 0.5183 0.7144 1.26 7.692 61 30 0.13157 0.08992 0.5130 0.7071 1.25 8.000 61 15 (.12847 0.08608 0.51)69 0.6987 1.24 8.333 61 01 0.12516 0.08227 0.5003 0.6896 1.23 I 8.695 60 40 0.1220)1 9.07849 0.4940 0.6809 1.22 9.090 60 19 o.11887 0.07474 0.4876 0.6721 1.21 9.523 60 00) (1.11516 0.0710(2 0.4799 0.6615 1.20 10.000 59 41 0.11140.046733 0.4720 0.6504 1.19 10.526 59 10() 0.10791 0.06368 0.4646 0.6404 1.18 11.111 58 40 0.10417 0.06005 0.4564 0.6292 1.17 11.764 58 019 0.10021 0.05646 0.4472 0.6171 1.16 12.500 57 40 0.09593 0.05289 0.4350 0.61038 1.15 13.333 57 01 0.09176 0.04935 0.4284 0.590)5 1.14 14.285 56 23 0.08729 0.04585 0.4178 0.5759 1.13 15.384 /55 45 0.08251 0.04237 0.401;3 0.5601 1.12 16.666 54 48 0.07789 0.03984 0.3947 0.5144 1.11 18.181 54 10 0.07273 0.03552 0.3S14 0.5259 1.10 20.000 53 15 0.06754 0.03213 0.3675 0.5066 1.09 22.222 52 14 0.06177 0.02879 1.08 25.000I 51 07 0.05649 0.02546 1.07 28.571 49 48 0.05065 0. 22 17 1.06 33. 333 48 18 0.04455 0.01891 1.05 40.000 46 32 0.03813 0.01568 1.04 50.000 44 04 0.03139 0.01249 1.03 66.666 41 04 0.02459 0.00932 1.02 100.000 38 12 0.01691 0.00618 1.01 200.000 32 36 0.00889 0.00308 1.00 Infini. 0 00 0.00(100 0.00(000 CIRCULAR ARCHES OF 180-0 EXTRADOS AND INTRADOS PARALLEL. (B) Tc6le of Thickne.ss of Pie)s. Value tio of Ratio -of atio of the thickness of piers to the radius of the intrados, of the the ratio diameter in function of the ratio -. of this radius, to the height of the =,to tthe I r thickness piers. CASE OF STRICT EQUILIBRIUr3I. 2.00 2.000 -2.3562 + Vt(5.551!7 5 + 1.7907 + - 0.9182) 9r r 1.'90 2.222 -2.0449 —+ I /(4.2021 + 1.3240 - + 0.7988) /I /t2 it 1.80 2.500 -1.7593 -+ t (3.0951 0.9 68- + 0.68+6) 1.70 2.857 -1.4844 - + V(2.2034 - + 0.6933 - + 0.5785) 1.60 3.333 -1.2252, +V ( 1.5012 0.375o + 0.4775) 1.59 3.389 — 1.2001 + - /(1.4404 + 0.3o66 + 0.4677) r92 r 1.58 3.44-8 -1.1752) — +/(1.3812 +0.3361 - + 0,4580) 1.57 3.508 -1.l1513 + - /(1.32 2 + 0.3151 + 0.4187) h 1'2 9h 1.56 3.571 -1.1261 - + 4 (1. 2677h+ 0.2966- + 0.4388) 1.55 3.636 -1.1015 (1.213 + 0.2783 + 0.4293) Ir + a'2' 1.54 3.703 -1.0772 - + 1.16052 + 0.2603 + 0.4198) h +V(1.1605-+09603 - 9. 9'2?' 1 3.773 -1.1031 7 + -' (1. l 091 2. ] O. 42g - O 0.4104) 1.52 3.8 46 — 1.092 t- t/(1.0592 - + 0.2224 +.401) i, /92 r 0/4011) 1. 51 3.920 — 1.0079 - + /(1.0146 + o. 2056 - + 0. 3.918) 11' r2 + 9 1.50 4.000 -0.9817- - /+ (0.9638' + 0.193' - + 0.3826) 1.49 4.081 -0.95837 +- /(0.9184 - + 0..1684 + 0.3735) 9'2 9' 1.48 4.166 -0. 9349, + 4/(.0. 741 -+ 0.1659 + 0.3644) 1.47 4.255 -0.9125- + V/(0.8328? 2- 0.1482 - + 0.3553) 5 /t2 h 1.46 4.347 -0. 8887 + 4/(0.7899 + 0. 162 h- + 0.3464) 1.45 4.444 -0.8659 + f(0.7498 + 0.12352 + 0.3374) 1.4 4.444 /92 92 1.44 4.545 -0.8432 - + V/(0.7110 + 0.1181 -+- 0.3337) 1.43 4.651 -0.8206 + (0.6735 + 0.11 + 0. 3314) r —2.0.73 r (9' 1.42 4.761 -0.793 - +/ (9.637- + 0.1143 + 0.3290) 1.41 4.878 0.7610, — 0. 0 / + v/(0. 6023 + 0.1102 - + 0.3263) 1.40 5.000 -0.7540 -+ O (0.5685 2 + 0. 1074 - + 0.3233) 1.39 5.128 -0.7321 + ~1(0.5359 - + 0.1048 - + 0.3203) 92 r 1.938 5.26.3 -0.7103, +- V(0.5045 /2 + 0.1021 - + 0.3169) /I$ ith. CIRCULAR ARCHES OF I 800-ExTRADOS AND INTRADOS PARALLEL. continuation of Ta6le (B). Thickness of Pier8. Value Ratio of Ratio - of the thickness of piers to the radius of the intrados, of the the r ratio di~ameter il function of the ratio r of this radius, to the height of the -_ to the r thickness piers. CASE OF STRICT EQUILIBRIUM. 1.37 5.406 2 1.37 5.406 -0.6887 - + (0.4743 + 0.0995 + 0.3134) r /2 r 1.36 5.555 — 0.6673, + v(0.4452 + 0.0969 0.3096) 1.35 5.714 -0.6460/ + / (0.4173 + oi2 0944 +> 0.3057) 1.34 5.882 — 0.6249 h +- (0.3904/+ 0.0926 - + 0. 3019) 1.33 6.060 — 0.6050 + v (O. 3660 - + 0.0903 - I 0.2979) r r.2 r 1.32 6.264 — 0.5831 + V (0.3400 - +0.0880- +0.2936) 9' h2 it r 2a I.31 6.451 -0. 5624 - + i/(0.3163 + 0.0875 + 0. 2902) 1.30 6.666 -0.5419 - + I(0.2937 + 0.0867 - + 0.2866) r 9'2 r 1.29 6.896 -0.6216- + (0. 2720 + 0.0828- + 0.2803) r -' 1. 28 77.142 -0.5014 -- + 4/(0.2520 2 0.08013 q O.2738) 1.27'7.40'7 -0.4926 + 4/(0.2426 -2 + 0.0778 h-f- 0.2686) /5 /52 it 1. 26'.692 — 0.4615 + / (0o.2130 0 -o. 0755 + 0.2631.) 1.25 8.000 — 0.4418 + 4/(0.1952 + O.0730 + 0.2569) Ai /s Is 1.24 8.333 — 0.42222 + /(0. 1783 + -0.0713 0.2503) 1.23 8.695 — 0.4028 - + (0. 1623 + 0.0684 - 0.2440) /r ra'2 t:O 237) 1.22 9.090 -0.38361 + / (0.1471 1 —0.0674. 277) 7'h 9'2 /' 1.21 9.523 -0.3645 - + (O..1329- + 0.0641 +0.2303) 1.20 10.000 -0.3456 /(0.119. 2228 -0.34561 q-I/ 4/ 01/+-0.0614 — 0.2228) 7' 1.,2 9' 1.19 10.526 -0.3268 + / (0.1068 - + 0.0600h + 0.2158) 7' r'2 7' 1.18 11.111 — 0.3082- + 4/(0.09502 + 0.0581- + 0.2083),2, 1.1'7 11.764 -0.289'7 + /(0.0840 2 +0.0561 + 0.2004) r 9.2 r 1.16 12.500 -0.2714 - + /(0.0734 - - 0.0559 1- 0.1919) 9. 7'2 1.15 13.333 — 0.2533 + ~/(0.0642 2 + 0.0536 - + 0.1835) 1.14 14.285 -0. 2353 - + 4/(0.0554 - + 0.0513 + -0.1745) it h2 it 1.13 15.384 — 0.2175 + 4(0.0473 + 0.0490 + 0.1651) 7' 7'2 7' 1.12 16.666 -0.1998 - + 4/(0.0399 + 0.0467 + 0.1552) 9. 9'2' 1.11 18.181 -0.1823- + / (0.0332/ 2+ 0.0426 - + 0.1455) 1.10 20.000 -0.1649 +4/(0.02'72- 0.0394 - 0.1351) 0.149t +0 09 -+0h 81 314 THEORY OF THE ARCH. CIRCULAR ARCHES OF 180, WITtI A SURCHARGE IN MASONRY INCLINED 450 ON EACH SIDE OF THE CENTRAL PIDGE. (C) cTable giving the angle of r6upture, the thrwst, acl the limzit tdchickmess of pieors. I ~alue Ratio V2. u of the limit thickalue Ratio Valne Ratio, C, of the thrust to the ness of pier to the radius of the of the of the of the square of the radius, r, of the in- intlados. ratio diameter anCgle of trados. -2f=R to the rupture. _ I thickness. - _ - Strict Coefficient of Rotat'n. Rotation. Sliding. Equilibrium. stability 2. 2.00 2.000 60~ 0.26424 0.74361 1.2212 1.72-46 1.90 2.222 60 0.28416 0. 5648 1.1458 1.6204 1.80 2.500 60 0.29907 0.57383 1.0759 1.5147 1.70 2.857 60 0.30867 0.49564 0.9956 1.4081 1.60 3.333 60 0.31245 0.42191 0.9186 1.2990 1.59 3.389 60 0.31249 0.41478 0.9108 1.2880 1.58 3.448 60 0.31257 0.40841 0.9038 1.2781 1.57 S3.508 61 0.31264 0.40067 0.8952 1.2660 1.56 3.571 61 0.31246 0.39367 0.8864 1.2548 1.55 3.636 61 0.31222 0.38673 0.8795 1.2437 1.54 3.703 61 0.31191 0.37983 0.8716 1.2318 1.53 3.773 61 0.31153 0.37297 0.8637 1.2214 1.52 3.846 61 0.31108 0.36615 0.8557 1.2102 1.51 3.920 61 0.31056 0.35938 0.8478 1.1989 1.50 4.000 61 0.30996 0.35266 0.8398 1.1877 1.49 4.081: 61 0.30928 0.34598 0.8318 1.1764 1. 8 4.166 61 0.30855 0.33934 0.8238 1.1650 1.47 4.255 61 0.30772' 0.33275 0.8158 1.1537 1.46 4.347, 60 0.30685 0.32621 0.8077 1.1422 1.45 4.44 60 0.30587 0.31971 0.7996 1.1308 1.44 4.545 60 0.30485 0.31325 0.7915 1.1193 1.43 4.651 60 0.30408 0.30684 0.7834. 1.1078 1.42 4.761 60 0.30296 0.30047 0.7784 1.1008 1.41 4.878 60 0.30173 0.7768 1.0986 1.40 5.000) 59 0.30001 0.28787 0.7746. 1.0954 1.39 5.128 59 0.29712 0.7709 1.0914 1.38 5.263 59 0.29706 0.7690 1.0914 1.37 5.406 59 0.29550 0.7688 1.0872 TABLES. 315 CIRCULAR ARCHES OF 1800, WITH A SURCHARGE IN:M[ASONRY INCLINED 450 ON EACH SIDE OF THE CENTRAL RIDGE. (C) ib5le giving the angle of rpture, tle thrust, and the limit thickness ofpie)rs. -alse to C, of the thrustRtothel Ratio A/2, C. of the limit thick-. Ratio Value Ratio,, of the thrust to the.. o pie' to the radius of the of the of the of the square of the radinus, r, of the ln- intrados. ratio diaioeter angle of trados..= __R to the rupture. Cfn r. thlickness., Strict Coe-ffcient of Rotat'n. -- Rotation. Slidiin,. Eqnilibriumn. stability 2. 1.36 5.555 59~ 0.29386 0.7665 1.0841 1.35 5.714 58 0.29285 0.7 653 1.0823 1.34 5.882 58 0.29037 0.7621 1.0777 1.33 6.060 58 0.28850 0.7596 1.0742 1.32 6.264 58 0.28654 0.7570 1.0705 1.31 6.451 57 0.28456 0.7544 1.0668 1.30 6.666 57 0.28231 0.221756 0.7514 1.0626 i.29 6.896 57 0.28027 0.7487 1.0588 1.28 7.142 56 0.27810 0.7458 1.0547 1.27 7.407 56 0.27578 0.7427 - 1.0503 1.26 7.692 55 0.27343 0.7395 1.0458 1.25 8.000.54 0.27102 0.7362 1.0412 1.24 8.333 53 0.26850 0.7328 1.0363 1.23 8.695 53 0.26608 0.7274 1.0316 1.22 9.090 52 0.26377 0.7263 1.0272 1.21 9.523 51 0.26074 0.7221 1.0217 1.20 10.000 50 0.25806 0.17172 0.7184' 1.0160 1.19 10.526 50 0.25546 0.7148 1.01091.18 11.111, 49 0.25277 0.7111 1.0045 1.17 11.764 49 0.25010 0.7072 1.0002 1.16 12.500 48 0.24742 0.7034 0.9948 1.15 13 333 47 0.24477 0.6997 0.9894 1.14 14.285 46 0.24218 0.6960 0.9842 1.13 15. 384 44 0.23967 0.6923 0.9791 1.12 16666 43 0.23732 0.6889 0.9743 1.11 18' 181 43 0.23502 0.6856 0.9695 1.10 20.000) 42 0.23292 0.12032 0.6825 0.9652 1.05 40.000 36 0.22902 0.6768 0.9571 316 THEORY OF THE ARCH. CIRCULAR ARCHIES OF 18 0, LOADED UP TO TItE LEVEL OF THE Top OF THE KEY. (D) Table giving the angle of ruFpture, the thlrust, and the linit thickness, of piers. Value Ratio Value Ratio, C, of the thrust to the1 Ratio of the limit thickness of of the of the of the square of the radius, r', of the in- pier to the radius of the intrados. Ratio. diameter angle of trados. R1 to the upture. _ _____ K(=-.V thickness. Strict Coefficient of RItotat'n. PRotation. Sliding. Equilibrium. stability 1.90. 2.00 2.000 36~ 0.05486 0.50358 1.0036 1.3834 1.90 2.222 39 0.07101 0.43966 0.9377 1.2925 1.80 2.500 44 0.08850 0.37901 0.8706 1.2001 1.70 2.857 48 0.10631 0.32164 0.8020 1.1055 1.60 3.333 52 0.12300 0.26755 0.7315 1.0082 1.59 3.389 52 0.12453 0.26232 0.7243 0.9984 1.58 3.448 53 0.12602 0.25712 0.7171 0.9885 1.57 3.508 53 0.12747 0.25196 0.7099 0.9784 1 1.56 3.571 54 0.12837 0.24683 0.7026 0.9684 1.55 3.636 54 0.13027 0.24173 0.6953 0.9584 1.54 3.703 55 0.13153 0.23667 0.6880 0.9483 1. 53 3.773 55 0.13289 0.23163 0.6806 0.9 381 1.5a2 3.846 55 0.13414 0.22664 0.6732 0.9280 3.920 55 0.13531 0.22167 0.6658 0.9177 1.50 4.000 56 0.13648 0.21673 0.6583 0.9075 1.49 4.081 56 0.13756 0.21183 0.6509 0.8972 1.48 4.166 56 0.13856 0.20696 0.6433 0.8868 1.47 4.255 57 0.13952 0.20213 0.6358 0.8764 1.46 4.347 57 0.14041 0.19733 0.6282 0.8659 1.45 4.444 57 0.14122 0.19256 0.6206 0.8554 1.44 4.545 58 0.14195 0.18782 0.6129 0.8448 1.43 4.651 58 0.14268 0.18312 0.6052 0.8341 1.42 4.761 58 0.14311 0.17845 0.5974 0.8234 1 1.41 4.878 59 0.14376 0.17381 0.5896 0,8126 1.40 5.000 59 0.14421 0.16920 0.5817 0.8018 1.39 5.128 59 0.14456 0.16463 0.5738 0.7909 1.38 5.263 59 0.14481 0.16009 0.5658 0.7799 1.37 5.406 60 0.14498 0.15558 0.5578 0.7689 1.36 5.555 60 0.14506 0.15111 0.5497 0.7577 1.35 5.714 60 0.14504 0.14666 0.5416 0.7465 1.34 5.882 60 0.14491 0.14225 0.5383 0.7420 1.33 6.060 61 0.14467 0.5379 0.7414 1.32 6.264 61 0.14460 O.5377 0.7412 L __ __ __ _ _-_ __ _ _ _ _ TABLES. 3 17 CIRCULAR ARCHES OF 180, LOADED UP TO THE LEVEL OF THE ToP OF THE KE Y. (D) iTa6e givinq the annle of rupture, the thru.st, and the limit thicknes8 of piers. Value Ratio Value Ratio, C( of the thrust to the Ratio of the linit thickness of of the of the of the squalre of the radius, r, of the in- pier to the radius of the intrados Ratio diameter ]angle of trados. I? to the rupture. - -- r-, thickness. I - Strict Coefficient of Rotat'n. RPotation. Sliding. Equilibriumi stability 1.90. 1.31 6.451 61~ 0.143900 0.5358 0.7394 1.30 6.666 61 0.143320 0.12495 0.5354 0. o79 1.29 6.896 61 0.142640 0.5341 0.7362 1.28 7.142 62 0.141860 0.5326 0.7342 1.27 7.407 62 0.141010 0.5310 0.7320 1.26 7.692 62 0.139880 0.5289 O.72990 1.25 8.000 62 0.138720 0.10405 0.5267 0.7260 1.24 8.333 62 0.131370 0.5235 0.7225 1.23 8;695 63 0.135930 0.5214 0.7187 1.22 9.090 63 0.134370 0.5184 0.7145 1.21 9.523 63 0.132630 0.5150 0.7090 1.20 10.000 63 0.130730 0.08397 0.5113 0.7048 1.19 10.526 63.12800 oo 0.50378 0.6993 1.18 11.111 63 0.126500 0.5030 0.6933 1.17 11.764 64 0.124150 0.4983 0.6868 1.16 12.500 64 0.121820 0.4936 0.680)3 1.15 13.333 64 0.118950 0.06471 0.4877 0.6723 1.14 14.285 64 0.116080 0.4818 0.6641 1.13 15.384 64 0.113030 0.4755 0.6563 1.12 16.666 64 0.109790 0.4686 0.6459 1.11 18.181 65 0.106410 0.4613 0.6358 1.10 20.000 65 0.102790 0.04627 0.4535 0.6249 1.09 22.222 66 0.098992 0.4449 0.6133 1.08 25.000 66 0.094967 0.4358 0.6007 1.07 28.5721 67 0.091189 0.4270 0.5886 1.06 33.333 68 0.086376 0.4156 0.5729 1.05 40.000 69 0.081755 0.02865 0.4044 0.5573 1.04 50.000 20 0.076852 1.03 66.666 71 0.071853 1.02 100.000 73 0.066469 1.01 200.000 74 0.061324 1.00 75 0.0o55472 0.0118. 318 THEORY OF THE ARCI-I. SEGMIENTAL AROCES —EXTRADOS AND INTRADOS PARALLEL. (E) atble of thrusts in sevene s8ystems; s thlbe pan; f the -ise; C( the cdecim al in any column; F 1 t7e tbZrust P2C The. thrust = the decimal x the square of the radlius of the intrados. of the ratio s=4/f 8=5F s=6f. s=/T s=Vf: s1 8=16/ Ir 5 5 17 I'=. =..f. r=./.'=5.. t'=-.f. =18f,.=82.5f. =B5" Ot"I. =4.36'1O-" =3G6. 52'; 10" ev=81,53'26" v284' 20". v=22037V10" v=14' 15'. 1.40 0.15445 0.14691 0.14691 0.14691 0.14691 0.14478 1.35 0.14717 0.13030 0.12587 0.12587 0.12587 0.12405 1. 34 0.14543 0.12987 10.12171 0.12171 0.12171 0.11999 1.33 0.14364 0.12781 0.11767 0.11767 0.11767 0.11767 0.11767 96 1.32 0.14173 0.12634 0.11362 0.11362 0.11362 0.11196 1.31 0.13975 0.12486 0.10959 0.10959 0.10959 0.10800 1.30 0.13764 0.12331 0.10682 0.10559 0.10559 0.10406 1.29 0.13543 0.12164 0.10563 0.10163 0. 10163 0.10016. 1.28 0.13311 0.11988 0.10437 0.0977 0.09770 0.09628 1.27 0.13068 0.118(03 0.10304 0.09379 0.09379 0.09244 1.26 0.12815 0.11609 0.10160 0.08992 0.08992 0.08862 1.25 0.12547 0.11402 0.10009 0.08668 0.08608 0.08483 0.0710S 1.24 0.12270 0.11251 0.09850 0.08549 0.08227 0.08108 }0.06862 1.23 0.12031 0.10958 0.09679 0.08423 0.07849 0.07735 0.06547 1.22 0.11675 0.10725 0.09499 0.08291 0.07474 0.07366 0. 06234 1.21 0.11354 0.10460 0.09305 0.08148 0.07102 0.06999 0.05924 1.20 0.11023 0.10196 0.09102 0.07999 0,.06981 0.06636 0.05616 1.19 0.10676 0.09915 0.08885 0. 07834 0.06859 O 0.06275 0.05311 1.18 0.10313 0.09617 0.08653 0.07651 0.06727 0.05918 0.05008 1.17 0.09934 0.0303 0.08408 0.0468 0. 06583 0.05212 0.04709 1.16 0.09537 0.08975- 0.08144 0.07264 0.06420 0.050o04 0.04411 1.15 0.09123 0.08634 0.0 7866 0. 070 50 0.062599 0.04904.0.04116 1.14 0.08690 0.08257 O. 07568 0.06812 0.06077 / 0.04803 0.03824 1.13 0.08238 0.07869 0.07251 0.06558 0.05890 0.04671 0.03534 1.12 0.07 764 0.07459 0.06911 0. (6297 O 0.05659 O.04451 O.03 247 1.11 0.07269. 07042 0.06548 0.06(026 0.05421 0.04384 0.02962 1.10 0.06737.06563 0.06158 0.05666 0.05160 0.04214 0.02681 1.09 0.06211 0.06077 O. 005739 (0.05345 0.04871 0.04023 0.02401 1.08 0.05636 0.05652 0.05288 0.04934 0.04552 0.03806 0.02192 1.07 (0.05052 0.05011 0.04804 0.04426 O.04200 0.03560 0.02111 1.06 0.04411 0.04428 0.04280 0.04058 0.03861 0.03276 0.02002 1.05 0.03776 0.0,804 0.0370(9 0.03550 0.03357 0.02944 0.01882 1.04 0.03096 0.03144 0.03095 0.02992 0.02862 0.02561 0.01720 1.03 0.02 0.02437 0.02424 0.02369 0.02293 0.02131 0.01524 1 90S? 0.01625 0.01681 0.01690 0.01673 0.01640 0.01546 0.01199 1.01 0.00834 0.00871 0. 00)88G6 0.0()889 0.00885 0.00862 0.00747 TABL-S. 319 TABLE E'. SEGMENTAL ArHIES LOADED UP TO THIE LEVEL OF TIIE SUMMIT OF TIIE KEY. 7Ta6le of thruQts inq seveo 6ystern&. Multiply the decimal by the square of the radius of thle intlradclos; F=r2 x a. Value _ of tile ratio s=4f. s=5b. s=6f. s=T7. s=: s=1fq s=l6J I, 29 50 1T K=-. r=I f., =-8J'=f r=e'f. r= —/f'=: 3f r=2.5f v=53T3 730". v=43i 3610" vl=3G52' 10" v=31 53 26" ev=2S8 4 20". v=22 3T'10" vs=14' 15'. 1.40 0.16920 0.16920 0.16920 0.16920 0.16920 0.15932 0.12760 1.39 0.16463 0.16462 0.16463 0.16463 0.16463 - 0.15490 0.12391 1.38 0.16009 0. 16009 ()0.16009 0.16009 0.16009 0.15052 0.12035 1.37 0.15558 0.15558 0.15558 0.15558 0.15558 0.14617 0.11611 1.36 0.15111 0.15111 0.15111 0.15111 0.15111 0.14185 0.11322 1.3S5L 0.14666 0.14666 0.14666 0.14666 0.14666 0.13756 0.10969 1.34 0.14225 0.14225 0.14225 0.14225 0.14225 0.13330 0.10619 1.33- 0.14138 0.13787 0.13787 0.131817 0.13787 0.12908 0.10271 1.32 0.14090 0.13353 0.13353 0.13353 0.13353 0.12488 0.09926 1.31 0.14032 0.12922 0.12922 0.12922 0.12922 0.120713 0.09583 1.30 0.13964 0.12499 0.12495 0.12495 0.12495 0.11659 0.09243 1.29 0.13885 0.12425 0.12071 0.12071 0.12011 0.11250 0.08906 1.28 0.13794 0.12342 0.11650 0.11650 0.11650 0.10843 0.08572 1.27 0.13693 0.12250 0.11232 0.11232 0.11232 O.10439 0.08241) 1.26 0.13519 0.12148 0.10817 0.10817 0.10817 0.10039 0.07910 1.25 0.13454 0.12036 0.10456 0.10405 0.10405 0.09643 0.07583 1.24 0.13316 0.11914 0.10359 0.09997 0.09997 0.09249 0.07259 1.23 0.13166 0.11780 0.10254 0.09592 0.09592 0.08858 0.06937 1.22 0.13002 0.11635 0.10138 0.09190 0.09190 0.08469 0.06618 1.21 0.12824 0.11478 0.10012 0.087192 0.08792 0.08085 0.06302 1.20 0.12632 0.11309 0.09816 0.08527 0.08397 0.07704 0.05988 1.19 0.12426 0.11121 0.09128 0.08412 0.08I05 0.07325 0.056711 1.18 0.12204 0.10930 0.09569 0.08287 0.07611 0.06950 0.05368 1.11 0.11966 0. 10719 0.09396 0.08150 0.07232 0.06519 0.05062 1.16 0.11712 0.10492> 0.09209 0.08002 0.069417 0.06210 0.04758 1.15 0.11440 0.10248 0.090017 0.01840 0.06819 0.0545 0. 04451 1.14 0.11151 0.09987 0.08788 0.07664 0.06680 0.05483 0.04159 1.13 0.10842 0.09710 0.08553 O.07473 0.06521 0.05124 0.03864 1.12 0.10514 0.09408 0.08298 0.01263 0.06359 0.04911 0.03511 1.11 0.10166 0.0908 O.08022.07034. 061'73 O. 0491 O. 03281 1.10 O.09796 0.08744 O.07724 0.06784 0.05967 0.04655 0.02993 1.09 0.09403 0.08316 0.01401 0.06510 0.05138 0.04503 0.027s)8 1.08 0.08986 0.0982 0.01051 0.06209 0. 0548 0.04329 0.02425 1.01 0.08544 0.01559 0.06671 0.05878 0.05202 0.04129 0.02251 1.06 0.08016 07.0106 0.06251 0.05511 0.04884 O 0.03897 0.02168 1.05 0.01579 0.06620 0.05806 0.05106 0.04526 0.03629 0.02064 1.04 0.1)053 0.06098 0.05314 0.04654 0.041210 0.03313 0.01929 1.03 0.06495 0.05536 0.04775 0. 04149 0.03656 0.02935 0.01156 1.02 0.05904 0.04931 0.04182 0.03583 0.03123 0.02479 0.01499 1.01..05277 0.0429 0.03530 0.02942 0.02505 0.01915 0.01125 320 THEORY OF TIIE ARCH. TABLE F.-CIRCULAR ARCHES OF 1800, WITH A LOAD OF ON EACH SIDE OF THE CENTRAL RIDGE, TO A ROOF [I/the angle between the roof and a vertical; r=the extrados; C=the decimal in any column; F=-the springing line. The last two columns give the addition with masonry, and'of the unifornm depth t above the'[~ r 1-90 /= 60o E=: = f5o I=590~ = TI=S I=70~ I=65~ E=R x 1.15470 _ _ _ E=R x E=R x E R x E-Rx E= R x'3'.. 1.00382 1.01543 1.03528 1.06418 1.10338 F=ir2C. F=. ==r2C F2 = r2C0. F <() (4) (5) fi z (9)F=r 1.50 4.000 29 ~0.21673 0.20535 0.19883 0.19787 0.20289 0.2147(:) 23~ 0.23408 1.48 4.166 29 0.206960.19588,,).189520.188580.193460.2049823 0.22388 1.46 4.347 29 0.19733 0.18654 ).18033 0.179410.18416).1953923 0.21381 1.44 4.545 29 0.18782 0.17733 0).17127 0.17036 0.17498 0.18594 23 0.20387 1.42 4.761 29 0. 17845 0.16824 0.16234 0.16144 0.16594 0.17661 58 0.19510 1.40 5.000 29 0.16920 0.15928 0.15353 0.15265 0.15877 0.17307 58 0.19291 1.39 5.128 29 0.16463 0.15485 0.14917 0.14840 0.15783 0.17201 58 0.19170 1.38 5.263 29 0.16009 0.15045 0.14484 0.14768!0.15688 0.17086 58 0.19041 1.37 5.406 29 0.15558 0.14608 0.14221 0.14685 0.15583 0.16962 58 0.18903 1.36 5.555 29 0.15111 0.14'1740.14157 0.14593 0.15468 0.16829 58 0.18755 1.35 5.714 29 0.14666 0.14094 0.14083 0.14492 0.15344 0.16686 58 0.18599 1.34 5.882 60 0.14491 0.14044 0.140020.14380.152100.1653458 0.18433 1.33 6.060 61 0.1446710.139840.139080.142580.14258 50660.1637258 0.18257 1.32 6.264 61 0.14460 0.13913 0.13805 0.14126 0.149140.1620058 0.18070 1.31 6.451 61 0.14390 0.13830 0.13691 0.13984 0.14750 0.475016017 57 0.17877 1.30 6.666 61 0.14332 0.13736 0.13567 0.138320.145710.1582457 0.17674 1.29 6.896 61 0.14264 0.13631 0.13431 0.136660.14385 0.1562057 0.17458 1.28 7.142 62 0.14186 0.13512 0.13282 0.13490 0.141880.1540557 0.17232 1.27 7.407 62 0.14101 0.13381 0.13121 0.13303 0.1397.9 )0.15178 56 0.16996 1.26 7.692 62 0.13988 0.13242 0.12948 0.13103 0.13761 0.14940 56 0.16750 1.25 8.000 62 0.13872 0.13089 0.12762 0.12891 0.13525 0.14693 56 0.16492 1.24 8.333 62 0.13737 0.12923 0.12563 0.126660.132770.1443455 0.16224 1.23 8.695 63 0.13593 0.12743 0.12350 0.12427 0.13018 0.14163 55 0.15946 1.22 9.090 63 0.13437 0.12550 0.12124 0.12175 0.12745 0.13878 55 0.15654 1.21 9.523 63 0.13263 0.12339 )0.11883 0.11909 0.12458 0.1357854 0.15353 1.20 10.000 63 0.13073 0.12116 0.11628 0.11628 0.12155 0.13264 54 0.15038 1..19 10.526 63 0.12870 0.11877 0.11357 0.11332 0.11840 0.12938 53 0.14713 1.18 11.111 63 0.12650 0.11623 0.11071 0.11020 0.11515 0.12602 52 0.14375 1.17 11.764 64 0.12415 0.11352 0.10768 0.106930.111660.1224951 0.14027 1.16 12.500 64 0.12182 0.11063 0.10450 0.10349 0.10804 0.11876 51 0.13669 1.1513.333 64 0.11895 0.10759 0.10119 0.09989 0.10426 0.11498 50 0.13294 1.14 14.285 64 0.11608 0.10443 0.09768 0.09609 0.10028 0.11106 49 0.12913 1.13115.384 64 0.11303 0.10105 0.09398 0.09212 0.096120.1068548 0.12521 1.12 16.666 64 0.10979 0.09749 0.09009 0.08796 0.09178 0.1024946 0.12119 1.11 18.181 65 0.10641 0.09373 0.086010.0860 108360.087250.0979245 0.1110 1.10 20.000 65 0.10279 0.08978 0.08175 0.07903 0.08250 0.09312 44 0.11290 1.08 25.000 66 0.09497 0.08140 0.07264 0.06925 1.06 33.333 68 0.08638 0.072130.06281 0.05865 1.04 50.000 70 0.07686 0.06186 0.05207 0.04707 1.02 100.000 73 0.06647 0..05112 0.04034 0.03422 TABLES. 321 MASONRY OR OF EQUAL WEIGHT WVITH MASONRY, RISING TANG:ENT TO THE EXTRADOS. radius of the intradclos; R=-iK-r:the radius of the thrust; BE-the elevation of the ridge above the to the thrust caused by a surcharge of equal weight roof, in the case of rotation and the case of sliding.] I=45~ Add for a Add for a 1=55~ I=50~ E=R x 1.41421 a 3 [surch'ge ofsunrcharge E=R x E —R x uniform of uni- [: 1.22078 1 830541 -0. depth t, the forml II =j2C. E F=r2QC. = additionl= depth t, -pt F- r 2 0. A=:6tC. A=rtC. [; [ I =r1 b. 1 Rotation. Sliding. (10) (11) el _ (12) K18 (18) (14) )0.26225 0.30085 220 0.35266 The addition to be 0.44388 1.50 taken froe this col0.25133 0.28891 22.33934 m when the thrs........43796 1.48 0.240560.27712 22 0.32621 comes below the hori 0.43204 1.46 0.22993 0.26551 22 0.31325 top ofenh chlumn. 0.42612 1.44 0.22178 0.25665 60 0).30296 650 0.33918 0.42020 1.42 0.21941 0.25418 59 0.30001 65 0.35297 0.41429 1.40 0.2181010.25284159 0.299712 64 0.35998 0.41133,1.3'a9 0.21671 0.25138 59 0.29706 64 0.36705 0.40837 1.38 0.21522 0.24985 59 0.29550 64 0.37421 0.40541 1.37 0.21365 0.24826 59 0.29386 64 0.38146 0.40245 1.36 0.21199 0.24653 58 0.29285 63 0.38880 0.39949 1.35 0.21023 0.24473 58 0.29037 63 0.39625 1.34 0.2083810.24284 58 0.28850 63 0.40379 1.33 0.20643 o.24086 58 0.28654 62 0.41143 1.32 0.204387 0.23876 57 0.28456 62 0.41920 The adar 1 31 tion to be 0.20227 0.23670 57 10.28231 62 0.42711 mnde from 1.30 0.20009 0.23451 57 0.28027 61 0.43513 wthen theoln 1.29 0.19780 0.23220 56 0.27810 61 0.44329 thrust comes 1.28 above tie 0.1954010.22983 56 0.27578 60 0.45161 lorizonttnl 1.27 /0.19289 0.22732 55 0.27343 60 0.46009 line near the 1.26 top of each 0O.19027 )0.22478 54 0.27102 60 0.46875 column. 1.25 0.18757 0.22219 53 0.26850 59 0.47760 1.24 0.18481 0.21948 53 0.26608 59 0.48665 1.23 0.18192 0.21665152 0.26377 58 0.49592 1.22 0.1789010.21385 51 0.26074 58 0.50543 1.21 0.17588 0.21095 50 0.25806 57 0.51520 1.20 0.172730o.207950 o o0.25546 56 0.52527 1.19 0.16944 0.20493149 0.252'77 56 0.53564 1.18 o.16617o.20182 49.25010 55.54637 Angle.1 0.1627380.19857 48 0.24742 55. Angl5748 rptre 25of, 1.16 0.15913-0.19515 47 0.2447 54 0.56901 vey r.15 46 0).24218 53 0.58102 the maxi'm 1.14 effect to the 44 0.23967 52 0.S9359asurcharge. 1.13 43 (.23732 52 0.60676 1.12 43 0.23502 51 0.62063 1.11 42 0.23292 50 0.63532 1.10 47 0.66778 1.08 44 0.70588 1.06 41 0.75313 1".04 35 0.81867 1.02 322 THEORY OF THE ARCH. T-A3LE G. ELLIPTICAL ARCHES OF 1800, WITH A LOAD OF MASONRY, OR OF EQUAL WE[GHT WITH MASONRY, RISING TO THE LEVEL OF THFE TOP OF THE KEY. [r7-the half span; f=the rise; C-the decimal in auy column; F-Sthe thrust -.2 C. The true thrust clue, in every case, to rotation.] l26le of T]iuqsts in Eig47t Sy.tems. 2 1 271 3 2 3 4 9,,f=r r.,= i =,/=- fr,f= r f= -r.f= r 5 2 50 5 3 4 1 o _1 1 27 3 1 =3 2 9.~-t] 5 4 - 00 10 3 8 5 20 p w the span. the span. the span. the span. the span. the span. the span. the span. (1) (2) (8) (4) (5) (6) (7) (8) (9) 6.00 0.235330.21612 0.19840.180820.119001506 6.50 0.22332 0.20903 0.19312 0.17714 016904 0.15523 7.00 0.23222 0.21598 0.20277 0.18830 0.17358 0.16605 0.15330 7.50 0.22387 0.20932 0.19250.18390 0.17014 0.16304 0.15081 8.00 0.21684 0.20347 0.19221 0.17977 0.16680 0.16015 0.14846 8.50 0.21064 0.19824 0.18779 0.1759410.16369 0.15716 0.14606 9.00 0.20513 0.19351 0.18359 0.17231 0.16049 0.15436 0.14367 9.50 0.20005 0.18915 0.17971 0.16904 0.15755 0. 1164 0.14132 10.00 0. 22374 0.19564 0.18511 0.17604 0.16564 0. 15474 0.14901 0.13907 10.50 0.21809 0.19143 0.18134 0.17283 0.16255 0.15199 0.14643 0.13686 11. 00 0.21298 0.18756 0.17780 0.16939 0.15969 0.14936 0.14401 0.13451 12.00 0.20393 0.18050 0.17133 0.16340 0.15422 0.14452 0.13946 0.13066 13.00 0.19633 0.17420 0.16555 0.15798 0.14926 0.14006 0.13526 0.12695 14.00 0.18956 0.16855 0.16029 0.153()6 0.14477 0.1 3599 0.13141 0.12368 15.00 0.18355 0.16349 0.15552 0.14863 0.14065 0.13225 0.12795 0.12035 16.00 0.17817 0.15881 0. 15116 0.14455 0.13687 0.12889 0.12476 0.11745 17.00 0.17312 0.15449 0.14718 0.14076 0.13341 0.12581 0.12167 0.11472 18.00 0.16861 0.15060 0.14348 0.137300'.13027 0.12278 0.11892 0.11226 19.00 0.16442 0.14696 0.14008 0.13410 0.1274() 0.12009 0.11636 0.10997 20.00 0.16054 0.14359 0.13692 0.13120 0.12453 0.11758 0.11399 0.10788 21.00 0.15690 0.140461 8 033980.128540.12200 0.1152 0.11180 0.10589 22.00 0.15355 0.13753 0.13134 0.12584 0.11959 0.11310 0.10977 0.10405 23.00 0.15042 0.13480 0.12885 0.12341 0.11737 0. 11109 0.10789 0.10233 24.00 0.14746 0.13231 0.12635 0.12118 0.11529 0.1092410.10609 0.10073 25.00 0.14469 0.13005 0.12407 0.11906 0.11336 0.10747 0.10444 0.09924 26.00 0.14208 0.12767 0.12197 0.117080 0.11154 0.10581 0.10286 0.09781 28.00 0.13730 0.12347 0.11810 0.11346 0.10823 0.10278 0.10001 0.09528 30.00 0.13308 0.11977 0.11463 0.11026 0.10527 0.10011 0.09745 0.09321 33.00 0.12740 0.11491 0.11016 0.10604 0.10139 0.0-9659 0.09431 0.09031 36.00 0.12250 0.11076 0.10625 0.10240 0.09804 0.09382 0.09172 0.08755 40.00 0.1168)7 0.10600 0.10185 0.09827 0.09448 0.0904i 0.08823 0.08460 45.00 0.11110 0.10108 0.09728 0.09428 0.09069 0.08673 0.08484 0.08169 50.00 0.10628 0.09703 0.09383 0.09089 0.082)7 0.08386 0.08217 0.07922 55.00 0.10222 0.09392 0.09065 0.08770 0.08455 0.08148 0.07987 0.07720 60.00 0.09874 0.09075 0.0879 0.0885200.08238 0.07)943 0.0 796 07.0550 TABLES. 323 TABLE IT. THRUST OF TH:E UNLOADED ELLIPTICAL RINTG BOIUNDED BY SIMILAR ELLIPSES. [r -the span; f=the rise; dl-the thickness at thle key; C-the decinmal in any column; 1F —the thrust-=2GC; semi-axes of the intrados, f and r; semi-axes of the extradosf+d, and r +y/l.] Thrutst in lTwo Systems. Rlise=4 the Rise= the rise the ise — the Value of span. span. Value of span. spal. 2r 1 2 2r 1 2 d fr. f=3. d f= -r. f=-r. F=7'2 X F=r2 X F=r2 X X, X2X 6.00 0.18273 19.00 0.11725 0.09559 6.50 0o.1777 2 20. 00 o 0.11305 0.09222 7.00 0.19773 o.179255' 21.00 0.10953 0.08904 7.50 0.19412 0.16761 22.00 0.10614 0.08592 8.00 0.18893 0.16265 23.00 0.10291 0. 08304 8.50 0.18431 0.157782 24.00 0.09981 0.08050 9.00 0.17970 0.15341 25.00 0.09685 0.07815 9.50 0.17544 0.14938 26.00 0.09419 0.07575 10.00 0.17124 0.1.4621 28.00 0.08924. 0.0'7144 10.50 0.16776 0.14132 ( 30.00 0.08468 0.06770 11.00 0.16310 0.13730 33.0( 0.07889 0.0(6241 12.00 0.15574.13030 36.00 0.07364 0.05835 13.00 0.14968 0.12376 40.00 0.0(6779 0.05364. 14.00 0.142'33 0.11799 45.00 0.06137 0.04867 15.00 0.13686 0.11242 50.00 0.05663 0.04459 16.00 0.13142 0.10783 55.00 0.0&235,0.04172 17.00 0.12621 0.103a5 60.00 0.04870 0.03815 18.00 0.12174 0.09941 324 THEORY OF THE ARCH. SECTION VI. CURVE OF PRESSURE. 108. In the preceding part of this worlk we have followed the theory of Coulomb, Aucloy, and Poncelet, and calculated the thrust of arches at the moment of rupture. We have called this the maximum thrust of the arch. At that imaginary moment, the horizontal thrust at the key, in the ordinary mode of rupture, acts upon a single point or line of the extrados; the resultant of this thrust, and of the weight of that part of the arch and its load which lies above the joint of rupture, comes upon the intraclos of that joint; and the resultant of the same thrust, and of the weight of the whole semi-arch and pier, falls entirely upon the exterior edge of the base of the pier (see figures 2, 3). No masonry could withstand this pressure exerted upon mere points or edges. Not only must rupture be avoided, but we must take care not to approach that condition; that is, we must, if possible, so proportion and build the arch and its pier, as to keep the curve of pressure near the middle of the joints. CURVE OF PRESSURE. 325 Let us suppose any arch to be on the point of falling in consequence of the insufficient thickness of its pier. The curve of pressure, touching the extrados at the key, and the intraclos at the reins about 60' from the key,. passes finally through the exterior edge of the base. Before reaching this condition of rupture, the vertical joint has gradually opened on FIG. 23. the lower side, the joint at the reins on the exterior side, the pier has slightly yielded to the pressure of the arch, the key has settled down, the reins have spread out. These movements, from which the best of arches are not entirely free, are often developed, in badly proportioned works, so as to exhibit wide cracks at the key and reins, without any immediate danger to the structure. They are due to two principal causes: in single arches, to deficiency of mass in the pier; in continuous arches, to the absence of suitable arrangements for preventing lateral motion at the reins. 109. In single or actutne'n tv acfhes, te magnituce of the horizontal thrust, acl the place of the curve of pressue in the arch, depend lctrgey 2upon the climensions of tle pier. Let us for a moment admit that r b the pier or abutment is absolutely immovable, and that the material of the arch is susceptible of but very little compression. And let us fur- ther admit, what has been abundantly proved in this paper, and con- firmed by numerous observations, that the joint of rupture, about 600 from Fi,24. 326 THEORY OF THE ARCH. the key, is the weakest joint, or joint first to open at the extrados. As the arch below that joint, mn z, being firmly attached to the pier, is imnnlovable, and the masonry above that joint, by supposition, nearly incompressible, it is evident that the pressure at the key, ca b, and at the joint of v'a/tmre, m n, will act all along those joints; in other words, those joints will be everywhere in contact. In the most perfect condition of stability, the resultant of the horizontal forces acting along the key, a 6, will pass through the middle of that joint; in like manner the resultant of all the forces acting along rn an, normal to that joint, will pass through its middle point. Let us now suppose the thickness of the pier to be gradually diminished luntil its top begins to move away from the arch: the crown will begin to settle, the reins will spread out, the curve of pressure will approach the extrados at the key and the intrados at the reins; finally, when the pier has been sufficiently reduced, the curve of pressure will pass through the extremities of those joints. Thus, by mere external changes in the pier, we have caused the curve of pressure in the arch to move by degrees, from the place of most perfect stability, to that of final rupture and fall. In this final condition, the thrust of the archl is the horizontal force which, applied at the extrados of the key, is just sufficient to prevent the rotation of the segment above the joint of rupture around the intrados of that joint; this force acting with a lever arm equal to the vertical distance between these points. On the other hand, in the condition of most perfect stability, the acting thrust is the horizontal force which, applied to the middle of the key, is just sufficient to prevent the rotation of the same upper segment around the middle of the joint of rupture; this force acting with a lever-arm equal to the vertical distance between these middle points. This real thrust, acting when the arch is firmly estab CURVE OF PRESSURE. 327 lished upon its piers, is much greater than the final thrust exerted at the moment of rupture,-sometimes more than double. We thus perceive that what has been called the maximurm thrust in the former part of this work, is really the least thrust that can ever act at the crown of the arch, and that this minimum is attained at the moment of rupture. The effective thrust is increased in the same arch as its lever-arm is diminished, or as the curve of pressure falls at the key and rises at the reins. In ordinary circular, segmental, and elliptical arches, surcharged horizontally, or surcharged more at the key than at the reins, the curve of pressure can never fall below the middle of the key, or rise above the middle of the reins. 110. The effective horizontal thrtust, the place of the cutsve of preeSre, anld the stability of continuous a rches, restin9 on inztermediate piers depend large/ly upon the material of the top of the pier let weem the arche.s. If this filling be of earth, or of indifferent masonry, the reins of the arch will spread out, the curve of pressure will rise at the crown and fall at the reins, the key will settle down, cracks make their appearance, and the whole assume an appearance of instability, or even worse. The remedy of this is simple and certain. FIG. 25. The arch, if light, should increase in thickness from the key towards the springing line, so as to add, 60~ degrees from the key, at least fifty per cent. to the thickness at the key; and the spaces between the arches, over the 10 3 8 THEORY OF THE ARCH. piers, must be filled with closely jointed, solid masonry, in horizontal courses, abutting, in vertical joints, upon the adjacent voussoirs, and extending as high as, say, within 45~ of the crown. These precautions are best illustrated in the London Bridge, the most remarkable structure of its kind, perhaps, in the world. A reversed arch, of equal thickness with the arch proper, is laid upon the top of each pier, abutting, in vertical joints, upon the voussoirs of the reins and lower parts of the adjacent arches. The joints are as thin as possible; and no other motion can occur in the arch than the little which arises from the compressibility of the granite. These precautions secure to the semicircular or elliptical arch, all the stiffness and stability of the segmental arch. ill. The curve of pressure at anzy joint shouzld not pass witlhin one thir d of its /,ength7 fot either eclde. Suppose the pressuLre to be nothing \ at the intrados, a, and to increase unifor mly from that point to the extra- FIG. 26. dos, I. It is plain that the pressure at any point along a6, will be represented by the ordinate -of a certain triangle. The whole pressure will be represented by the surface of that triangle; and the point of application of the resultant of all the pressures will be at c, opposite the center of gravity of that triangle. We then have c baW a b. Vice versa, if the point of application be at c, c b= a 6, we know that the pressure is nothing at a. If the point of application be at c, c6b being less than 1a b, c being still opposite the center of gravity of the triangle whose ordinates represent the pressure, we know that the vertex of c that triangle, and point of no pressure, are at e, 1 e 3 X ~ c. In this casie, the joint ca 6 will open at a, as far as e; the adjacent joints FIG. 27. PRESSURE PER UNIT OF SURFACE. 329 will also open until we come to one where the curve of pressure passes within the prescribed limit. This reasoning is, of course, applicable to all the joints; and we readily conclude that the curve of pressure should lie entirely between two other curves which divide the joints into three equal parts. The foregoing reasoning is based on the principle, first applied, we believe, by Navier, that the material of the arch is perfectly elastic, and that the pressure upon any joint varies uniformly from the extremity most pressed to the point of no pressure. In fact, the last condition alone is sufficient, as it allows us to represent the pressure, upon the several points of any joint, by the ordinates of a triangle or trapezoid. PRESSURE, PER UNIT OF SURFACE, UPON THE JOINTS. 112. Let F represent the total perpendicular pressure upon any joint; d, the length of the joint; 1, the distance between the resultant or curve of pressure, and the nearest edge of that joint; P, the pressure per unit of surface at the edge most exposed. If the curve of pressure pass through the'middclle of the joint, we have P=. If the curve of pressure pass at one third the length of the joint from _=. either edge, as 6, we have the pressure at b equal to twice the mean pressure 2~ " along a b, or P=- d FIG. 26. If 1 be less than: cd, or c b less than a 6, the whole pressure comes upon e b, and we have P= e2-. We have no good means of esti- 6L mating the distance eb; but, what is FIG. 330 THEORY OF THE ARCH. more to the purpose, we can generally confine the curve of pressure within the two curves above mentioned, or even keep it still nearer the middle points of the joints. The pressure being nowhere b p nothing upon the joint itself, will be represented by the ordinates of some trapezoid cb P P'. P, P, representing the pressures, pelr unit - of surface, at b and ca, respectively, we shall have, P: P': 6 P: Ca; P+P1 moreover, F- x-d. The resultant of all the pressures will pass through the center / of gravity of the trapezoid. FIG. 28. Let p,p', c, be the projections, upon a b, of the centers of gravity of the two triangles a b P, a P P', and the whole trapezoid a 6 P P', respectively. The point c is found by dividing p' p in the inverse ratio of a P' to b P. We have _' c: c:: bP: ac P':: P: P; pl c _p.:: P: P+-P'; giving p='c(P+P'); butp' c 2d-l;P- -P':::; P pp=Ac. Hence P =- X 3 ) (59) is the mean pressure per unit of surface.. From (59) the values of P already given, are readily deduced. The formula, however, is not applicable to the case in which the point of no pressure comes within the extremities of the joint. ACTUAL THRUST. 331 THE TRUE THRUST OF TEE ARCHI. 113. We have reminded the reader that almost all arches, on the removal of the center, show a tendency to settle down at the key and spread out at the reins. This FIG. 29. tendency often results in the production of cracks, at tihe intrados of the key, anld at the extrados of the reins onl each side of the key. Suppose the crack at the key to extend from a to a certain point e: the whole pressure comes upon e b, the remaining part of the joint, and all that part of the arch near the key which lies below the joint e, is worse than useless; in like 1manner, supposing the joint at the reins to open from n to e', that part of the arch which is external to e' is useless. If mere weight be wanted at any point, it will be better to load the arch with some cheap material. We have also relninded the reader of the self-evident truth, that the movements and cracks in question will always be developed unless the pier opposes a sufficient resistance. If the pier, though large enough to prevent actual rupture and fall, is still weak, the crown will continlue to settle, and the 7horizonztal t7rust to diinish7b, until the pier is able to withstand the diminished thrust. The arch, in dilimiishing its thrust, tries, as it were, to acco-mmnodate itself to the weakness of the pier. Between the condition of most perfect.stability, the curve of pressure passing near the middle of the joints, and the condition of final rupture and fall, the existing thrust becomes less and less, varying 332 THEORY OF THE ARCH. sometimes, as we shall hereafter see, in a ratio as great as 2 to 1, or larger still. This condition of most perfect stability is highly favorable to the joints of the arch, the pressure being nearly equally distributed; but it is the condition which gives rise to the greatest thrust, and requires the greatest magnitude of pier. We can not say that the pier ought, in all cases to be large enough to withstand so great a thrust; but it is very certain that the pier ought to withstand that diminished thrust which is developed when the curve of pressure at the crown and at the reins, passes through the limits already fixed; viz., at thle key, - the length of the joint from the extrados, and at the reins, 1 the length of the joint from the intrados. If the pier cannot withstand this thrust, the joints of the arch will certainly open. We have here a perfectly distinct point of departure for a new calculation of the thrust of arches. Draw the curves ca' n', b' r ns', dividing all the joints into b three equal parts. Suppose the horizontal thrust to be n /, applied at I' on the key, 6' 6 1a. Draw any joint mn' n' n, the vertical n ~r through the cv surcharge, and the line ir t re- _/ presenting the top of the sur- FIG. 80. charge; project in' horizontally at x on the vertical, b6, which divides the arch into two equal parts; and project y, the center of gravity of the segment irn n r c a on m' at g'. Let y',-6' x, represent the lever arm of the thrust, and p', =-n' I', the lever arm of the segment. As the resultant of the thrust, applied at 6', and the weight of the segment resting on in an, applied at g, its center of gravity, passes, by supposition, through en', in in' - ACTUAL THRUST. 333 -}n- n, there must be, in case of equilibrium, an equality of moments in relation to that point. Hence, F' representing the force, and S' the surface m n r? t 6 a mn, we nmust have F'x y'-S'xp' or F' —' This force F' will be small when the joint 1m n is near the key; it will increase as the joint departs fromn the key, and become a maximum at the reins, about 60~ from the key. Suppose n on, to be the joint of rupture corresponding to the maximlum value of F'. The curve of pressure between the key and the joint of rupture will be situated entirely between the limit curves, a' n?', b' n', which divide the joints into three equal parts. From b', where it is neares-t to the extrados, it will gradiually depart front the extrados, pass through the middle of some joint about midway between a 6 and n nz, continue to approach the intrados, and come nearest, relatively, to that curve at in', n 3MI'=M- in; it will there be tangent to the inferior curve ac' e', and begin to recede fronm the intrados. In light arches, the curve of pressure after leaving b', may at first pass a little above the superior limit, b','; but it never can pass within the inferior limit, a' in', either above or below the joint of maximum thrust. The reader, by a little reflection, will see the truth of this last remark, and will also see its importance. 334 THEORY OF THE ARCH. THRUST OF THE UNLOADED CIRCULAR RING OF EQUAL THICKNESS THROUGHOUT, ON THE SUPPOSITION THAT NO JOINT SHALL OPEN, THE CURVE OF PRESSURE AT THE KEY BEING ONE THIRD THE LENGTH OF THE JOINT FROM THE EXTRADOS,-AT THE REINS, ONE THIRD THE LENGTH OF THE JOINT F'ROM THE INTRADOS. 114. The point of ap- -S plication of the thrust is at b',b 6=-Iab. The arcs, c''f, 6'11, dividing the joints /, into three equal parts, the segment rn n 6 b, corre- sponding to any joint m n, / is exactly equal to three times the central segment, i,' n'' b'; and the center of gravity of the whole is I near the center of gravity FIG. 81. of this central part. In the light arches of ordinary use these centers may be regarded as coinciding. Suppose them to coincide. The thrust of the arch,m n 6 a&, on the condition expressed at the head of this article, is precisely equal to three times the thrust of the arch yn' n' 6' a', calculated on the condition of actual rupture. This last thrust for all proportions of the two radii is given by table A, flrom which the following 1has been deduced. We briefly explain the mlode of computation. Let K represent the ratio of the two radii of the given arch; KZ' the ratio of the radii of the central arch; F' the thrust of this last arch taken from table A; F the required thrust of the given arclh. We have K —1 if =1 +r - F=3 x F' x (3 +I3 KT)'-_rX 3 x (2 +1K)2 X C. K3-2' 1.-3 \+ T\ 3 3s ~f4R\ TABLE AA. 335 Table AA, at the end of this paper, gives the values of F, or the actual thrust of the circular ring, for all the values of Ifbetween 1.01 and 1.40 inclusive. For explanation see the head of that table. This table proves that the effective thrust acting when the arch is firmly established upon its piers, is much greater than the thrust at the momnent of actual rupture. The ratio, ~, of these two thrusts, beginning at 1.065 for K —1.01, becomes 1.204 for K-1.08, 1.35 for K=l.17, 1.50 for K-1.25, 1.83 for K=1.40. It attains its greatest value, 1.94, when KI —1.45. The values of F corresponding to I — 1.01, 1.02, have been calculated by the law of differences, table A not giving those values. THRUST OF SEMICIRCULAR ARCHES SURCHARGED HORIZONTALLY. I 15. This is by far the most common form of the arch. All arches carry loads, and these loads most frequently rise to a surface nearly horizontal. It is plain that the pier should oppose to. the thrust of the arch a resistance sufficient to prevent the formlation of cracks or openings at the weakest joints, or joints which actually open in case of rupture and fall. Figs. 2, 3. The actual thrust of the arch, when the joint at the key is about to open at the intrados, and the joint at the reins is about to open at the extrados, is evidently the very minimum which the pier is required to oppose. If the pier is unable, in the slightest degree, to meet this thrust, it is evident that the joints in question will open, and continue to open, until the pier is able to withstand the diminished thrust, or until the structure falls. Let us assume that the pier is able to withstand that greater thrust which is developed in the condition of most 336 THEORY OF THE ARCH. perfect stability, when the curve of pressure, or resultant of all the pressures, passes through the middle of the key and the middle of the reins. By the latter we mean the weakest joint, generally about 60~ from the key. In the investigation of this thrust we shall suppose the thickness of the arch to be the same throughout; while in practice, as we have repeatedly stated, this thickness should gradually increase firom the key, so as to become about fifty per cent. larger at the reins. This increase will slightly diminish the thrust, and slightly elevate the curve of pressure at the reins. The curve of pressure passing through the middle of the joint at the reins, supposed to be equal in length with the joint at the key, will pass through the inferior limit of the former joint when increased by fifty per cent.; viz., within one third of its length from the intrados. It will be a little below the superior limit at the key; that is, by the difference between 2 and I, or by -6 the depth of that joint. Almost all large bridges are exceedingly light in their proportions; they are made generally of the most incompressible stone; and it is not too much to say that their piers should be able to withstand the thrust in question developed in the condition of most perfect stability. Applied to arches very heavy in their proportions, whether large or small in their actual dimensions, this would perhaps be an exaggerated thrust. Such arches have an excess of thickness throughout, and require no increase at the reins. Assume the curve of pressure to pass through c, the middle of the key, and to touch the line drawn through the middle point of all the joints, supposed to be equal in length, ait c,, on somze joint, in rn, such as to give the greatest possible thrust on the condition imposed. To illustrate this method in advance of more particular calculations, suppose the thickness of the arch to be ~o ACTUAL THRUST. 337 the span, or Kil.10. Let us 52 -t find the value of the hori- t zontal force, F', which, ap- t plied at c, the middle of a b, shall hold in equilibrium any segment, a vz n - b ca, on c, the middle of the lower joint..,' Let v the angle, rn Ca, between the joint m n and a vertical; r-the radius of the c intrados; m nc-a b; in all casesmc_ - n-say m n,, / the joint of the practical arch. 32. lFia. 32. Values of s. Values of F'. Values of's. Values of F'. 00 r2 x 0.10492 50~ r2 x 0.13563 5 " 0.10544 55 " 0.13746 10 " 0.10698 58 " 0.13795 15 " 0.10945 59 6 0.13801-=F, the max'm. 20 " 0.11270 60 " 0.13800 25 "' 0.11654 65 " 0.13708 30 " 0.12073 70 " 0.13457 35 " 0.12504 75' 0.13038 40 " 0.12914 80 45 " 0.13277 90 " 0.10825 The angle of maximum thrust is in this case 59 or 60 degrees, and the curve of pressure corresponding to that maximum, passing through the middle of the key and the middle of the reins, 59~ firom the key, is traced, necessarily, outside or above the central point of every other joint. If it were inside of the middle of any joint, as at v -20~, it would immmediately follow that the value of F' corresponding to the middle of that joint, must be greater than P1 the actual maximum thrust. We may learn from the above table that the curve of pressure corresponding to the maximlum F passing through the middle of the key and of the reins, is, everywhere be 338 THEORY OF THE ARCH. tween those joints, very near the central points, since the maximum, F so little exceeds the other values of F'. In the upper parts of the arch, a very small change in the lever arm of F', or vertical distance between c and c,) would make a large change in the value of F'. These remarks are applicable, though in different degrees, to the curve of pressure corresponding to all values of K. CALCULATION OF THE MAXIMUM THRUST OF SEMICIRCULAR ARCHES SURCHARGED HORIZONTALLY; THE CURVE OF PRESSURE PASSING THROUGH THE MIDDLE OF THE KEY AND OF THE JOINT OF GREATEST THRUST; THE JOINTS OF EQUAL LENGTH THROUGHOUT. 116. R-the radius of the extrados; r —the radius of the intrados; d=the thickness of the arch at the key; / —-=1~-; v-the angle between any joint andl a vertical. By a course of investigation similar to that explained in the note appended to equation (24), art. 48, we find, as the general expression of the horizontal force, F', which, applied at c, the middle of the key,-shall hold in equilibrium Rany segment a m n r 6 ac on c, the middle of the lower joint, /COS.'IEv(2 - cos.v)1+C2 +os.-vc eos.vK'2 +F -r x K(' cotalpg. -iv (60) In like manner we find, under the same supposition as to the curve of pressure, as a general expression of A', the addition to the thrust caused by a surcharge of constant depth, t, above the extrados of the key, (+cos. ) (61) The maximum value of A' corresponds alwrays to v-O; it then becomes, KE1 K+P1 ACTUAL THRUST. 339 The total thrust obtained in any given case by adding this value of A to the maxi- r. _ mum value of i', in (60), would involve an error in excess, very proper in light arches, but not required per-, haps in heavy ones. The words light and heavy, here' / used, refer to the proportions j of the arch, not to its absolute dimensions..We give below the numerical forms of (60) and (61) FIG. 32. corresponding to values of v beginning with zero and in. creasing by 5~ to 750, and to v-90~.- By three or four substitutions the reader can obtain the maximum 9sum of F' and A' when r, ][ and t are known. In practice, the thickness of the arch generally increases from the key to the reins or to the springing line. In this case, -I _ 1 —-. The results will be a little in excess. The sum of F' and A' thus obtained will correspond to the angle of maximum thrust within 21 degrees; and this is quite near enough. v:=;' —r~(K'+~K + 1. ) v-~5~; F':r.[0019x 2~.33143 x3+ Coefficient of K2, log. 0.000824.,, it K3, "6 1.520395. A'=rt( 1.9962 x K) 340 THEORY OF THE ARCI. v_=10; ~,_r (r1.00748 x ]K+0.32578 x K 3- _ 0.99746; v 0; F (1 K+1 Coefficient of K2, log. 0.003325. " " K3, " 1.512918. A'2t(1.98481 x K). v=l15;,( 1.01646 x K'+0.31649 x - 099428); Coefficient of K2, log. 0.007088. " K3,s " 1.500361. /1.96593 x K ) 1.02834 x K2+0.303785 x K3+2 v=20~; F'=?g 3 K 1 -0.98982) Coefficient of K2, log. 0.012135. " " K3, " 1.482567. 1.93969 x K A'=rt K 25~; Fr2(104246 x K2+o.28795 x i+?0.8408); Coefficient of K 2, log. 0.018058. 4" " K3, " 1.459319. 1.90631 x X I 30; 1.05801 x K+0.26934 x Kr3+ 0.9 Coefficieit of Ji, log. 0.024490. Coefficient of ~2, log. 0.031035. "4 1" K 3, " 1.395084. / 1.8191O x K A':rtJ'K+-1 / K+1I ~ -0.9 Coefficient of K], log. 0.031035. ", K3,, 1.395084. I K+1 Cofiin f].8,1g 00490.x NUMERICAL FORMS. 341 vz400; F' —r?( 10896 x K2-+0.22548 x 3 +3_ 095905); Coeficient of K, log. 0.037273. cc "c K3, " 1.353105. A'-t(t 1.76604 x K) v=450; F'= 2(1.10355 x K'+0'20118 x K+- 3 _0.94806)' Coefficient of K', log. 0.042792. ic " KK3, " 1.303594. A' =t(.70711x K). 1.1148 x K"2+0.17599 x K'2V —~500; K+r 79xK+ 3-0.935715 -1 I Coefficient of K2, log. 0.047199. " " K3, 3 1.245498. A' rt( 1.64279 x K) - K+1' v=55; F 1X 1.1223 x K-1t+0.15043 x 23 0.92200); Coefficient of K2, log. 0.050106.' " i K3 " 1.177328. A't 1.57358 x Kr) vt=600; F0 r c 2(8 Kj ~+ 3 S +Y_0.90690); Coefficient of K2, log. 0.049995. - " " K3, " 1.000885. 1.42262 x I - 342 THEORY OF THE ARCH. v —70~; F'=r2(1.1 253 x K0.0765 < -0.872404\; Coefficient of K2, log. 0.046310. "( ".'K3, " 2.883661.,l~ t( 1.34202 x K ) A'-rt( Kq-1jK v —75~; F=r2( 1.09592 x K2+0.0543 x K3 85296) Coefficient of K2, log. 0.039778. 4" " 4K3, " 2.734794. 1.25882 xKi Vz90~; OFIr( K+3-0.78540); From (60) we have calcularted table DD, giving From (60) we have calcullated table ADD, giving what we may regard as the actual thrust of the semicircular arch,' surcharged horizontally, under the conditions expressed at the head of this article. The table also gives the maximum effect of the surcharge of any constant depth, t, above the summit of the arch. In drawing up the table we have reduced (60) to its numerical forml for every value of v, in whole numbers, from 40~ to 75~, inclusive. But knowing the resulting maximum thrusts to be somewhat greater than they need be in heavy arches, we have supposed v —60~ for all values of i Texceeding 1.'2-2. Being once on the track we have generally found the maximum value of F', corresponding to any particular value of AL, by three or four substitutions. NUMERICAL FOR3iS. 343 Column 1 gives the value of K= 1+. 2 ~ ~ 2r'$ 2 4' 66 2d r or ratio of the span to the thickness. 3 " angle of maximum thrust down to IC-l1.22, v-45~; below that, v is assumed at 60~. " 4 " maximum value of F' down to If' 1.22; below that the value of F' corresponding to — 60~. 5 " for the purpose of comparison, from table D, the maximnum and actual thrust in the case of rupture and fall. ", (; " 8F ~ or ratio of these two thrusts, properly the coefficient of stability.'7 " the value of A, or maximum effect of the surcharge, v-0. 8 " for the purpose of comparison, from table _F the values of A2, or maximum effect of the surcharge in case of rupture and fall. "I 9 " A= A6 - or ratio of these two effects.!11 344 THEORY OF TIIE ARCH. CALCULATION OF THE MAXIMUM THRUST OF THE ROOFSHAPED SEMICIRCULAR ARCH; THE CURVE OF PRESSURE PASSING AT ~ THE LENGTH OF THE JOINT FROM THE EXTRADOS AT THE KEY, AND FROM THE INTRADOS AT THE JOINT OF GREATEST THRUST. FIG. 88. 117. -R=the radius of the extrados; r-the radius of the intrados; d=the thickness of the arch at the key; R d _KyT - =1+-; I=the angle between the roof and a vertical; v —the angle between any joint and a vertical. By a course of investigation similar to that referred to, art. 116, we find, as the general expression of the horizontal force, FI' which, applied at c, on the central joint, a b, c b= 3 ba, shall hold in equilibrium any segment, a mn np R a, on c1, a point of the lower joint m,, in c,- I n, F'- = r2sin.2v x K2 x 2(2 -sin.(I+v))-K'3(1 -sin.(I+v)) v(K+2)+ 1 sin.I sin.v COS. V (62) K(2 - cos. v) + 1-2 cos. v In like manner we find, under the same supposition as to the curve of pressure, as a general expression of A', the addition to the thrust caused by a surcharge of the constant depth t, above the tangent roof, ZD R, THE MAGAZINE ARCH. 345 _i' ( —I (4-) (63) A' tKsin2 -VK(2 -cos. v) + 1- 2 cos. v (3) of which the maximum value is 4KKX-2 2K1 + I 3K64-3 A +rt - X J'2, (G4) corresponding to an angle whose cosine is 2K+1 3- /3K2 - 3 COS. — + (65) From (62), (64), and (65), wre have calculated table EF, giving, directly or by proportional parts, under the conditions expressed at the head of this article, the actual thrust in all the isolated magazine arches int common use. In drawing up the table we have reduced (62) to its numerical form for values of I respectively equal to 60~, 550 50~, and 45~, and for values of v increasing by 2S~, from 30~ to 60~, that is, as far as necessary, both ways, to ascertain the maximum thrust. The results under each value of i, are not exactly the maximum thrust, but, in general, a little less; the difference, however, is practically nothing. EXPLANATION OF TABLE FF. Colunn 1 gives the value of ][I 1+-. l 1 under each value of I, gives the angle of greatest thrust. 2, under each value of i; gives the decimal C; F: the thrust=r2. 3, under each value of i, gives the coefficient of stability, S, or ratio of the actual thrust to the diminished thrust at the moment of rupture and fall, the latter being obtained from table F. " 1, under " surcharge," gives the angle of maximum thrust of the surcharge. " 2, under "surcharge," gives the decimal C; A=rtC=the maximum effect of the surcharge. 34~6 THEORY OF THE ARCH. REMtARKS. It will be seen th'at the angle which renders the effect of surcharge a maximum, differs but a few degrees from the angle of maximum. thrust in the arch proper; consequently the error, in excess, which we commit, by adding the two maxima together, and taking their sum as the actual thrust of the arch and its load, is exceedingly small, and the table may be regarded as practically exact. It will also be seen that the coefficient of stability, ~, is nearly the same, for the same values of IE, in all the arches; consequently, we can obtain, from table F, the actual thrust, on the conditions announced at the head of this article, for values of I some degrees above 60~, by multiplvying the thrust, computed from that table, by the value of 6 found opposite the given value of /[ in table FF. The maximum effect of surcharge, given by the last column of FF, is independent of i; and the same in all arches. For rules for using table FF, see rules for using table F, arts. 61, 62. CALCULATION OF TIlE THRUST OF THE SEMI-CIRCULAR ARCH(I SUCI?.IIHARGED HORIZONTALLY; THE CURVE OF PRESSURE PASSING AT A THE LENGTH OF THE JOINT FROM: THE EXTRADOS AT THE KEY, AND FROMI THE INTRADOS AT THE JOINT OF GREATEST THRUST. 118. This is a particular case of the roof-shaped arch discussed in art. 117. We need not repeat that discussion. Formnloe (62), (63), (64), and (65), remain the sa me. In the first it is only necessary to mlake I-90~ which reduces (62) to 2v(K~2 2) 1 2 X 2(2-cos. v)-K3(1 - cos.v )- sinv s.v (62)' F'=~ I sin. v 2 os — ~tt2 Ki(2 - cos. v) + 1- 2 cos. v TABIE DDD. 347 From this formula and froml other sources, we have calculated table DDD, giving the muaximum or actual thrust of the arch in question under the coclditions stated above. EXPLANATION OF TABLE DDD. The 1st column gives the value of K= l1+ —; cd=the thickness at the key. The second column gives the decimal C; FC, E-2C-the thrust on the condition stated at the head of the table. The 3d column gives the decimal C;, A- rt CUthe addition to the thrust caused by a surcharge of the constant depth t above the key. F+A - the entire thrust, with an excess arising from adding two maxima together. The 4th column gives the joint of rupture or angle of greatest thrust on the supposition of actual rupture and fall, from table D. The 5th column gives the angle of greatest thrust on the supposition that the curve of pressure is at ~ the length of the joint from the extrados at the key, and from the intrados at the joint of greatest thrust., calculated at intervals of 21 degrees. The 6th column gives the angle of greatest thrust on the supposition'that the curve of pressure is at the center of the joints of the key and of greatest thrust, taken -from table DD. Below _K=1.22 the angle is not given. The 7th column gives the coefficient of stability, ~, or the ratio of the actual thrust, on the conditions stated at the head of the table, column 2, to the calculated thrust at the instant of rupture and fall, table D. The 8th column gives the coeffcient of stability on the supposition that the curve of pressure passes through the middle of the joints, from table DD. 348 THEORY OF THE ARCI. REMARKS ON TABLE DDD. Columns 6 and S have been taken from table DD. They are added here that the reader may see at a glance, how the angles of greatest thrust and the thrusts themselves vary with the suppositions which we make upon the curve of pressure. The greatest value of A in column 7, is 1.92; in column 8, it is 2.59. CALCULATION OF THE THRUST OF SEGMENTAL ARCHES, SURCHARGED HIORIZONTALLY, THE CURVE OF PRESSURE PASSING THROUGH THE MIDDLE OF THE KEY, AND THE MiIDDLE OF THE JOINT OF GREATEST THRUST, ~WHICH IS GENERALLY AT THE SPRINGING LINE. 119. If the semi-angle at the center be as great, or nearly as great, as the angle of maximum thrust in table DD, art. 116, we shall find in that table the actual thrust on the condition announced above. This table may, indeed, be used in all cases without any great error; for we have shown that the curve of pressure, passing through the middle of the joint at the key and the middle of the joint of greatest thrust, will continue near the centers of all intermediate joints. The error will, of course, be always in excess. The exact thrust, when it is less than that given in DD, will be obtained from equation (60) art. 116, when we have substituted for the constants which enter into that formula, their known values. The effect of a surcharge of constant depth may also be obtained from table DD, with an error, always in excess. SEGMENTAL ARCHES.-TABLE EE. 349 CALCULATION OF THE THRUST OF SEGMENTAL ARCHES, SURCHARGED HORIZONTALLY, THE CURVE OF PRESSURE PASSING AT A TIlE LENGTH OF THE JOINT, FROM THE EXTRADOS AT THE KEY, AND FROM THE INTRADOS AT THE JOINT OF GREATEST TIIHRUST, WVHICH IS GENERALLY AT THE SPRINGING LINE. 120. Notation: s-the span; f=the rise; r=the radius of the intrados; c-the thickness of the arch at d the key; ]i=11+-; C-=the decimal in the first column under each value of v; iF-the thrust-r2C; v=the semi-angle of the whole arch. If one half the angle subtended by the given arch be as great, or nearly as great, as the angle of maximum thrust in column 5, table DDD, we can obtain the required thrust directly from that table. But if this half-angle, v, be less than the angle of maximum thrust, substitute its known value, and the value of KWin (62)': the resulting value of F' will be the actual thrust on the condition stated above. To obtain the effect of a surcharge of the constant depth t above the key: if v be as great, or nearly as great, as the angle which renders the effect of surcharge a maximum, see last column but one of table FF, the required addition will be obtained from the last column of that table, or from the 4th column of table EE. But if v be less than the angle in question, the required effect of surcharge must be computed from formula (63). The sum, F+A, as usual, will be the entire thrust. Table EE gives, either directly or by proportional parts, the actual thrust of segmental arches in common use. 350 THEORY OF THE ARCTH. EXPLANATION OF TABLE EE. Column 1 gives the value of K=1 +-. 1, under each value of v, gives the decinal C; Fi=q20C=the thrust of the arch loaded up to the level of the extrados at the key.'" 2, under each value of v, gives the ratio, B, of this actual thrust to the thrust of the same arch at the moment of rupture and fall, taken from table E'. 3, under each value of v, gives the decimal C; A =-rtC-=the addition to the thrust caused by a surcharge of the uniform depth t. ELLIPTICAL ARCHES. 121. In the first part of this paper, section V., art. 90, and following, we have shown how to obtain the ultimate thrust of the elliptical arch, loaded and -unloaded. The actual thrust may be found in a similar manner. ELLIPTICAL ARCItES SURCHARGED HORIZONTALLY. Let us compare the given arch with a circular arch, surcharged. in like m-nanner horizontally, having the same span, and a thick- / " - ness at the key as much -1 greater than the thick- / ness of the elliptical / arch as the half-span is / greater than the rise. /, x Through c, the middle B c A - of ac b, and c8, the mid- Fri 84 ELLIPTICAL AIRCiIES. 351 die of A i, draw the ellipse c c2 c8 similar to the intrados. This curve will cut the several joints of the elliptical arch near their central points, and will never pass within the inferior limit, one-third the length of a joint from the intrados, unless we give an unnecessary extension of these joints at the reins. Draw the curve c' c'2 c3, dividing into equal parts the joints of the auxiliary circular arch, supposed here to be of equal thickness throughout. Drawing any vertical t' tp, let S- the surface i t b6 a, S' —the surface n' t' 6' a', y=c e, or vertical distance between c and C2,' —C'', p-the horizontal distance bletween the vertical Itp and the center of gravity of,S p' the corresponding distance in the circular arch. The horizontal force F which, applied at c, shall hold in equilibrium the surface sS in relation to c2, as the center of rotation, is F= SXP; in like manner, we have VF'= 5' xp' But wherever the vertical be drawn, the depths in t, in' t', stand in the constant relation of the rise to the halfspahi, or f to r; the surfaces 8, 8', and the lever-arms y and y' stand in the same relation; the centers of gravity of S and 8' are on the same vertical line, so that p=lp'. We have, therefore,': q"'::: y', and =, or F=F'. These surfaces having the same thrusts wherever the vertical be drawn, their maxilmum thrusts will also be the same; and we arrive at this result: Th/e actualc t7hs8t of an el}ptical acrch s8zstaininq a load of masonry or of eqcal weight with inacsonry, rising to the ho-rizontal line tangent to the extr-aclos at the key, iq qvearly equact to the thru,6st of the seinicirczular arceh, loa(ded in~ like nanner, having th/e So ame Spat, andz c thickness at 352 THEORY OF THE ARCH. the key as munch greater than tJhe thickneqss of the elliptical arch, as the hacf-,scn is greater th7an the rise. We shall, therefore, be able to obtain immediately from table DD, the actual thrust of the elliptical arch. Strictly speaking, a slight correction-addition-would be necessary, as we have disregarded the influence of the small surfaces c2 A' t, ec'2n r' t'; but on the other hand we have provided for some exaggeration of the thrust, by supposing the curve of pressure to pass through the middle of the key, and near the middle of the joint of greatest thrust. Example. Central arch of the London Bridge. rhalf-span —6'; f-the rise-38'; d —the thickness at the key- 5'; D)-the thickness of the auxiliary circular archi Xd —10O'; KC1+ —=1.1316; hence from the 4th column of table DD, by proportional parts, F.r X.16723966.45 cubic feet. Suppose a surcharge one foot deep above the key, t-l'; the corresponding surcharge of the auxiliary circular arch is -2', and its effect is, 7th column of table DD A 2' X 76' X 1.061t-161.38. THE COEFFICIENT OF STABILITY. 122. In the first part of this paper, we have shown how to find, by tables or calculation, the actual thrust of most arches at the supposed moment of rupture and fall, when all the forces in the system act upon three points or edges of masonry; viz., the extrados at the key, the intrados at the reins, and the exterior lower edge of the semiarch or pier. We have shown, art. 109, that this ultimate thrust is also the minimum or least possible thrust that can ever exist in the arch. COEFFICIENT OF STABILITY. 353 This minimum, Audoy, Poncelet, and' others, multiply by what they call the coefficient of stability: 2, or some smaller number, according to the proportions of the arch; -and they determine the thickness of pier on the condition that the resultant of the thrust thus increased, and of the weight of the semi-arch and pier, shall pass through the exterior edge of the base of the pier. The value of this coefficient was determined by an examination of a powder-magazine of Vauban, which, having stood the test of ages, was presumed to have all the necessary elements of stability. The value of the co-efficient thus determined, was found to be about 2; and this is evidently applicable to all similar structures, that is, to structures identical, or nearly so, with the magazine of Vauban, in all their proportions, and only different in their absolute dimensions. The rule has been of great service, for it so happens that most magazines have been modeled after that of Vauban. But the idea was entirely empirical, and was so understood by its authors. The co-efficient of stability thus defined andl used, is by no means a correct index or measure of the stability of any pier. In piers of great height, it will generally give results too small,-in- piers of small height, results too large; nor is it sure to give correct results in any work not strictly similar to that of Vauban from which the rule was deduced. The proper value of this coefficient will change, not only with every variation of the proportions of the arch, but, in the same arch, with every increase or reduction of the height of the pier. Nor is it possible to know this proper value without first learning the actual thrust of the well-established arch; and when we have attained this knowledge, we have only to mlake proper use of it: thet coefficient has become useless. We know the thrust itself. As it is evident that the curve of pressure in the arch should be everywhere traced between two other curves 3 54 THEORY OF THE ARCH. which divide the joints into three equal parts,-so, in the pier, this curve must lie between corresponding lines; and at the base, where it approaches nearest to the exterior face, it must not come nearer than - the length of the lower joint, or A the thickness of pier. This conclusion is inevitable, if we admit the principle of Navier as to the distribution of the pressures upon the joints; and this principle is generally admitted, we believe, by engineers. 123. Let us examine the magazine of Vauban anew, fig. 13. Its dimensions are: Radius of the intrados= = 12'.50; radius of the extrados=R-= 15'.50; i=_-X=1.24; inclination of the roof to a vertical=- 490~' 17"; heiglht, of the pier from the base to tlle springing line- =1 -'; thickness of pier given by Vauban 8'; on the exterior face, counterforts 6' long and 4' deep, with intervening spaces of 1.2'; whole thickness of pier and counterfort- 12'. The ultimate thrust of this arch, arts. 48, 63, is F_ r2X0.2294. But we learn from table FF that its least actual thrust, consistent with the condition that no joint shall open at either the extrados or intrados, is q.2 X 0.3496. Substituting this value in formulla (311-)' art. 66, we obtain, as the least thickness of a solid pier whose lowest joint will not open on the inside, 10'.56. The mean thickness of Vauban's wall is 9'.33, and the thickness given by the rule of Audoy, S-2, formula (31)', art. 66, is 9'.23. We come at once to this unexpected resnlt,-that the rule of Audoy does not in this case give a pier of sufficient thickness to withstand the thrust of thearch without any openings of the joints. We are authorized and compelled to conclude, that the magazine of Vauban derived some additional stability from adhesion of mortar in the alchl or pier, or both. During the construction of an arch it is almost impossible to prevent cracks or openings at tlhe reins, and MAGAZINE OF VAUBAN. 355 although they may close on the removal of the center, still the adhesion of mortar must be forever impaired or destroyed. It would be unwise to rely upon any adhesion of mortar in tIe terclh. On the other hand, the pier is subject to no disturbance during its construction; its mortar has time to set; and we may in some cases rely with confidence upon this element of stability. The calculation of the effective resistance of the piers in Vauban's magazine is complicated by the existence of abutments. Let us still assume that the lowest joint, under the action of the horizontal thrust, is on the point of opening at its inner edge, where the pressure is consequently zero, and that the pressure increases uniformly from that point to the exterior edge of the abutments. The mean pressure will be represented by the ordinates of the oppo —- site Figure, and the B point of application of FIG. 35. the resultant of all the pressures will be at C, B C' —5'.255. The mean center of gravity of the pier is 7'.143 firom B, the exterior edge. Taking the moment of the pier and of the semi-arch in relation to C, we find the thrust, acting horizontally one foot below the extrados at the key, which stands in equilibrium with these elements of resistance, to be 59.70 —r2X0.3373, less than the actual thrust before given by 1.2X0.0123. The supposition of a very slight adhesion of mortar upon the base of the pier, viz., 11 pounds per square inch, not more than a fifteenth part of the adhesion of good mortar, will equalize the two thrusts. It thus appears that the magazine of Vauban illustrates and confirms in a remarkable manner the new theory of the arch. Its piers present almost precisely the resistance which this theory requires. 356 THEORY OF THE ARCH. COEFFICIENT OF STABILITY-NEW DEFINITION. 124. The coefficient of stability is the ratio of the actual thrust of the well-established arch to the ultimate thrust existing at the moment of rupture and fall, or the quotient of the former divided by the latter. This ultimate thrust for most arches likely to occur in practice, is given by the tables which belong to the first part of this paper, viz., A, C, D, E, E', F, and G. If we had tables equally extensive of the actual thrust, we should no longer need this coefficient. But at present such is not the case. M. Cavallo has given an extensive table of the actual thrusts of semicircular arches, based on the supposition of vertical joints, and of a surcharge bounded horizontally at top with a depth on all vertical lines bearing a constant ratio to the depth of the arch proper on the same lines. And we have given in this paper tables of the actual thrusts of the unloaded circular ring, AA, art. 114; the magazine or roof-shaped arch, FF, art. 117; the semicircular arch surcharged horizontally, DDD, art. 118, the segmental arch surcharged horizontally, EE, art. 120; all based on the supposition of joints of equal length perpendicular to the intrados, and of a curve of pressure at the key - the length of the joint from the extrados, and at the joint of greatest thrust I the length of the joint from the intrados; also, a table, DD, art. 116, of the actual thrusts of semicircular arches surcharged horizontally, with joints as above, and with a curve of pressure passing through the middle of the key-stone and weakest joint. We have given in all of these tables the value, A, of the coefficient, as defined above, to show at a glance how the actual compares with the ultimate thrust when both are known, and to exhibit a guide which may lead to the former when the latter only is known. For instance, we learn, from table FF, that the value COEFFICIENT OF STABILITY. 357 of A, corresponding to KA'-1.20, is 1.47 when I-45~; 1.46 when I-50~; 1.46 when — 55~; 1.45 when I-60~; and we learn from table DDD that the value of ( is 1.47 when K= 1.20 and I-90~; we may thence conclude, by analogy, that ( does not exceed 1.47 for roofs of any inclination between a horizontal and 45~. In like manner we learn the value of 8 for other values of X, and may deduce the actual thrust from table F, when the given value of I does not place the case in table FF. In all this we use the ultimate thrust as a convenient and almost indispensable standard of comparison; not only because that thrust, as already remarked, has been extensively calculated and published, but also because it can be determined with great accuracy by experiments upon models. The actual thrust cannot, with equal accuracy be thus determined. Regarding this ultimate thrust as the standard, we may with propriety call the ratio by which it must be multiplied to give the actual thrust, the coefficient of tclbility. DISCUSSION OF THE COEFFICIENT OF STABILITY, OR RATIO OF'THE ACTUAL TO THE ULTIMATE THRUST, WVHEN THE CURVE OF PRESSURE LIES, AT THE KEY, - TIIE LENGTIH OF THE JOINT FROM TILE EXTRADOS, AND AT TIIE REINS' THE SAME DISTANCE FROM THE INTRADOS. 125. Semicircular arcces surckarged ho'rizon tally. —We learn from column 7, table DDD, that this ratio, beginning with 1 for 1= 1, gradually increases by nearly arithmetical differences, to its maximum, 1.92, corresponding to i-=1.35. It then begins to diminish, and would finally become 1 again. Ti7e mnagazine arch.-We learn from table FF, that the value of 3 constantly increases from ]_i 1.15 to K= 358 THEORY OF THE ARCH. 1.40. It attains, in fact, its greatest value when I, c'2... c'; c, c1, c2... c6, dividing the joints of the arch into three equal parts. The curve of pressure must not pass outside of these limits between the key and the joint of greatest thrust. III. Division of tAhe archl into fictitious vonUssoieqs. Divide the semi-arch into 4, 6, 8, or 10 segments, according to its absolute size, and to the degree of accuracy 406 THEORY OF THE ARCH. which we wish to attain. Let these segments have a common altitude, so chosen as to effect the desired division. Make the division as follows: from e, with the common altitude as a constant radius, escribe a small arc and draw the joint il el tangent thereto. The joints are generally perpendicular to the intrados. Next, from e1 as a center and with the same radis, dclescribe another are, and draw the joint il c'2 C2 62 tangent thereto... and so on to the springing line. WTe need not waste time in trying to make this division come out exact. The last segment, fig. 50, may have an altitude, eap, a little more or less than the constacnt, ap. IV. Pcartiel and total areas of t7te fctitious. votssoivs. Draw the parallels i c1 to e i1, meeting e1 i, prolonged at d; il c42 to el i, meeting e2 i2 prolonged at 12... and so on to the springing line. If the last altitude, ep, fig. 50, comes out unequal, lay off CC a on p2 e, prolonged if necessary, p a being the constant altitude, which we will designate by a. Draw i5 a' parallel to e5 i6; then e5 d6 parallel to a a', and eb d'6 parallel to a e6. The surface i, i6 e6 e, is measured by a X ~ c1 d'6. The area of any other segment, as i2 i3 e e2, is measured by.a X d, e,. It will be seen that we regard the right lines supposed to connect i il, il i2... e el e e, &c., as representing the outlines of the arch. This supposition is generally, in practice, favorable to stability. On the vertical C;E, from the assumed point of application of the thrust, which, in fig. 48, is c, the upper limit, lay off c s- I-l el, s1 s2-9 d2 e2, and so on to the springing line. Then the partial segments will be measured as follows, viz.: the surface i i1 ei e by axe CS, surface i ie2 e by ca X 1 82, &c.; and the total segments, i i2 e2 e by a Xe s2, i is e e by aXcs, &c. Area of the entire semi-arch — ca X c s, or a X c s,... as the segments may be 6, 8, &c., in number. If the line c 8s or c s... is found to extend too GEOMETRICAL METHOD. 407 far on the drawing for convenience, all the distances c s,,,'l2 s.... 5 s6 may be reduced by one half in which case the areas of the several segments will be measuredby c Si, 91 s2... 82, &c., multiplied by 2ca; that is, by a into the reciprocal of the fraction of reduction. Draw the indefinite horizontals s, u1, s2 2. *. s6 %16. V. Centei-s8 of gr'avity o f t7he partial 8egnents or vows0soirs. Of any quadrilateral, fig. 49, draw the diagonals intersecting at gn; lay off e4 a=?fl i, i4p=- e,, and mark c, the middle of e4 i,, and n, the middle of i4 e8; join a n, cp,; their intersection, 04, is the center of gravity of the figure. This point is sometimes given more conveniently by taking C 04- Cop, or 1? 04 —~ n a. In this manner, or in any other more expeditious manner, determine the centers of gravity o0, 02, 0o, &c., of all the segments. VI. Cenzters of grcavity of tAe totalg 8eg9n, te. Draw a horizontal through some point, iT', about as far below the point of application of the thrust as the extreme center, o6, is distant from the vertical C c. Project vertically on that horizontal the centers o0, 02, 03... respectively at o'l, o, 0'8, &c. Prolong o' c until it intersects the horizontal through a1 at in1; thence draw n71 i12, parallel to o'2 c, meeting s2 %.2 at In2; thence sn2 n3, parallel to o'3 c, meeting s8 vt at in3, and so on to the springing line. Project o0 vertically on the horizontal c 27, at n1. Lay off c ( —c ]K'; draw K~n2 parallel to rn2 c meeting c n7 at t2,; KvT parallel to zon c, meeting c 12, at,; and so on to n6, corresponding to in6, and to the springing line. The points, n,, n%7 n, 123, &c., on the direction of the horizontal thrust, are directly over the centers of gravity of the total segments, n12 corresponding to i i2 e2 e, or segmzents 1 and 2 comlbined; 93a to segments 1, 2, and 3 com15 408 THEORY OF THE ARCH. bined, etc.;'n6 is over the center of gravity of the whole semi-arch. (a) VII. T/ie acitcal ttrulst. Let us determine the horizontal force, which, acting at c, shall hold the first segment in equilibrium on c'1, i c' _A i el. ni c'l is obviously the direction of the resultant of this force and of the weight of the segmlent, represented by cs. Draw c tl parallel to c'1 z, meeting e ml at t1; slt is the force required. In like manner draw c ts parallel to c'2 n2, meeting 8s2 n2 at t2; c t3 parallel to c'3 i3 meeting S3, n3 at 1;' and so on, until the ordinates sl t4, s2 t2, &c., having attained their maximum, begin to diminish. Draw a vertical, u1'u3, tangent to the curve tl t3 t t4... If t3 be the point of tangency, and t=the greatest horizontal ordinate of the curve (in this case t- s t3), the actual thrust expressed in cubic feet, or cubic units, if other than feet be used, will be F — aX t. We must not forget to double a if the liles c s1, 8l,2, &c., have been reduced by.. See IV. VIII. TAhe curve of prsess8ure. Mark ul, u22,'ts,...,, tlhe intersections of the vertical just mentioned with 8s ml1, 82 i2, 83 rn3,..86 in6. Draw n, xt parallel to %a c, meeting the joint i el at x1;?2, x2 parallel to'2 c, meeting i2 e2 at 4x.. ~.~6 4 parallel to t(6 c, meeting the springing line at x,. (a) Let sl, S2, S3, &c.,=the areas of the combined segments, 1; 1 & 2; 1,2, & 3, &c. "m ml,rm2, m3, &c.=mom'ts on C " " " " " Then S2-sl=area of the 2d segment; sa3-s2=area of 3d segment, &c. M2- 1m=M01ot m' " a —)m2=mom't " " We have, by construction, x -Ol CS1 s'1, eK':KV, c~s, cK' e lf' 1' 02 X 882 S lfl -fl?1 cK': K'o'2:: 8182: s2?n2-Sl/ l= -812 S2 2 - cK' 5c.K" n 2 Hence, s2n-2=. In like manner sm3-=- &c. We also have CS2: s2m2:: cK: cn2= cKx S2Mf2 m2 *173 374 cKx-s - In like manner, cn 3-, ca,4 -, &c. CS2 82 83 84 GEOMETRICAL METHOD. 409 The curve of pressure passes through the points c, x,, X2, 3...x6. (b) It cannot come within the inferior limit, c', c'1, c'2, &c.; but it will generally approach the outer limit above the springing line, if the arch be full circle. If the curve of pressure runs above the upper limit immediately after leaving the key, we must regard the assumed point of application of the thrust as impracticable. A lower point must be taken. If the curve corresponding to this last point still pass beyond the outer limit. before totucl7ing th7e inner, a still lower point must be taklen. If the curve of pressure, starting from the upper limit at the key, fall immediately below the upper limit curve, we need not investigate the thrust or the point of application any further. IX. OCaC/inzg thee poi?nt of ca1)picCtio?, of the tvruet. Suppose we have obtained, as above, the actual thrust and the curve of pressure corresponding to any point of application, say c, ic — ie, and wish to determine the variations in both, consequent upon a change to any other point of application, say c', ic' I ie. Project the points nP, 2i n, %n4, etc., already determined, vertically on the horizontal passing through the new point of application, c'. Marlk these projections as n'1, nt'2, n'8, etc. Determine a new curve of thrusts (VII.) by drawing ct', parallel to c', n',, meeting sl in1, at t'1; ct'2 parallel to C2 n 2, meeting s2 n22 at t'2, etc. Draw a vertical tangent to the new curve t'l, t'2, t'3, etc., and malk its intersections, u'1 with sl in1, i't with s %,27 etc. The greatest horizontal ordinate of this curve-s. u'1= (b) We have css=the vertical force due to the total segment i is e3 e; s. 2t3=the actual thrust or constant horizontal force. The diagonal uz c is the direction of the resultant; n3 is one point in it. The same considerations apply to the other points. 410 THEORY OF THE ARCH. &, n'2, etc., is the actual thrust under the conditions ilposed. Determine the new curve of pressure by drawing',l'1, parallel to u', c, meeting i, el at m',; n'2 X'2 parallel to 1U1'2 C, meeting i2 e2 at x'2 and so on, to the springing line. These changes are made in a few minutes by a repetition of steps already explained. The points sY, 82, s,... 1a,''112, etc., are laid off once for all. To avoid confusing the diagram, the new curves of thrust and of pressure, are not indicated in fig. 48. In like mannei, any number of points of application may be assumed and their corresponding thrusts and curves determined. But it will rarely be necessary to assume more than two or three points. X. Lmhits of 0po0ssbie and placticable acrc7he,. If we suppose the curve c', c'l, 62,... which marks the nearest approach of the curve of pressure to the intrados, to be the intrados itself, and, assuming i, the lowest point of the key,. as the point of application, find the resulting curve of pressure to pass outside of the extrados between the key and the joint of greatest thrust, we know the arch to be i;nz208o.iSe, whatever the resisting power of the materials. If the curve of pressure starting firol c', the lower limit, and touching that limit again at the joint of greatest thrust, is found, at some intermediate joint, to pass beyond the uplper or outer limit, we regard the arch as impracticable. The curve of pressure can not, in such a case, be confinedl within two other curves which divide the joints into three equal parts, and the joints of the arch must open at the intrados, or extrados, or both. XI. Th e case of. two unequacIl ca'ches, or achC7tes beCatrinZ unezquac loadc7s2, witA ac conmtoJ key. Figure 48 representing the semi-arch of least thrust, let sl, u"1the thrust of the other part. First, to test the GEOMIETRICAL MIETHOD. 411 possibility of the smaller semi-arch, assumne i as the point of application; on a horizontal passing through i project 9n2, %2,?4, etc., already determined in relation to any other point of application. Call these projections Wn",'"2, n"8, etc. Contilue the vertical through a"l meeting 92 9?.2 at n "2, 83 m23 at W'"3, etc. Construct the curve of pressure in the usual malner, by drawing n", x", parallel to it", c meeting i el at x"1; n"2 X"2 parallel to. Ut"2 c meeting 1i2 e,2 at X" 2, etc. If this curve pass outside of the extrados2 the arch is impossible. Second, to test the practicability of the semi-arch, assume c', i c' — i e, as the point of application. Oin the horizontal through c' project vertically the points.1 at 9n'1, 122 at 7n'2, 913 at Wn', etc., and determine points in the curve of pressure by drawing, from these last points, parallels n1', x', to u,", c, meeting i?; e, at x'1, fl'2'2 to i,"t2 C, etc. If the curve thus determined, pass beyond the outer limit, C, C1, c2, CO,... etc., the arch is impracticable. At the point where it crosses that limit, it will be necessary to begin to increase the thickness of the arch. XII. Direction and macgnitzde of the press.,re 2poqn the several joinst. As the actual thrust, 83 t3, fig. 48, is the constant horizontal force acting upon all the joints, while c ls, 0 s2,c S3.. etc., represent the surfaces or weights of the total segments, 1; 1 and 2; 1, 2, and 3, etc., it is obvious that the diagonals,'c10, Z2c, (0c, U 4C, etc., represent, both in magnitude and direction, the resultants which press upon the several joints; viz., 110c on i1el, 112C on i2e2, 3c0 on ie,3 etc. If the angle between any one of these resultants and the corresponding joint should be less than the complement of the angle of fiiction, say less than 60~, rupture may take place by sliding. Lay off these diagonals froml the saLme point on any line, as c' In', fig. 54; make t v=i e; continue the line c' v. The intercepted ordinates, 2(s2 V2,?'3 s3, etc., will give the required lengths of joint on the condition that these shall be proportional to the resultants. 412 THEORY OF THE ARCH. XIII. Tbce thrm/st by slidincg. The horizontal force P' necessary to hold any surface S in equilibrium on a joint mlaking the angle v with a vertical is, a being the angle of friction, P'-Sxcotang. (aC+v). Assume a:30~. Then P =Sx tang.(60~ - ). Draw the line Cv, fig. 48, 600 from a vertical; draw c t"-1-angle 81 c t"1=-ang. v Ci=-60~-0v, meeting sl mn at t" t; 2-ang. t ang. a2 ct Ci meeting s2 2 at t2, and so on. The greatest horizontal ordinate of the curve t"2 t",..., is the thrust by slidling The corresponding joint of greatest thrust is generally inclined about 30~ to the vertical. It wouldc be well to interpolate one or two joints near 300, with altitudes - a. Should the thrust due to sliding exceed that cldue to rotation, the former will be the actual thrust. The point of application may be assumed, generally, as near the center of the key, and the curve of pressure may be determined precisely as in XI. XIV. 5TVickzeesm of P'ier'. Continue v6 x,, corresponding to the springing line, fig. 48, to the base, x,. A B- A. x7 will, if the height of the pier be small, be very nearly the thickness required on the condition that the curve of pressure shall meet the base at one third its length from the exterior edge. 2 X A x, —A B, is the thickness required in order that the curve of pressure shall pass through the middle of the base. The exact distance may be obtained most easily by a curve of errors. Let x=the distance between the exterior edge of the base of the pier and the intersection of this base with the curve of pressure; e-the required thickness of pier; p=the ratio of x to e; so that x=pXe. p is usually assumed at one third, sometimes as large as one half. Having already obtained the provisional thickness A B,, mark'1, such that i1", x, —) X'X1 A. If p —I,': xT xA. GEOMETRICAL METHOD. 413 ABl being too large, lay off the error B1 r1 above B,; this givesf,. Assume AB2 as another thickness. Determine the distance B2 c/7,, precisely as if the surface 4AB E2 q6 were a segment of the arch, such that a X B2 d7 A B2 x B2 Lx. Lay off I B2 cd7 from s6 reaching to s,, draw the horizontal S7'7.. Project the center of gravity of A B2 E2 i6 on -'o'6- at o'7; draw mn6 n7 parallel to o',c, meeting es uw at mn1; draw fnvT parallel to.m7c, thence 7~ xs parallel to qu7c. This gives xs. Mark r2 such that ~2 W cs-p X 92A, and, AB2 being too small a thickness, lay off the error, B2 9r2, below B2.. giving f2. Joinf, f2 meeting the line of the base at B. AB is the thickness required, very nearly. Verify this by assuming AB as the thickness; determine eI, T 12,,and x anew; this will at least give a new point in the curve of errors very near the base. Connect this point with the more distantf, or f2 and mark the new intersection with the base. Further trial will rarely be necessary. If we wish to cover the pier above the springing line, by a mass of masonry sloping gradually to the extrados of the arch, we must take into account this new surface and its moment as a part of the pier, adding its surface to cs7, projecting its center of gravity on il'o',, anid so on, in the usual order to the intersection of the curve of pressure with the base. XV. 5Tle pre.ssure pev nit of su?/Cface. The pressure at the key being F/ -t X a, the mean pressure at that joint, cl representing its length, is, t X a multiplying this by w, the weight of a cubic unit,, we have the pressure in poundsW1 X ca X WV The mean pressure at any other joint, as: i, e, fig. 48, is q14 X a X wt The mean pressure should not the ultimate esisting exceed -:: the ultimate resisting power of the material. 414 THEORY OF THE ARCH. The greatest danger will be at the joints of rupture, so called, where the curve of pressure approaches nearest to the outlines of the arch. We here use the pressure or resultant itself, not its normal component. We -might, with more accuracy, use the latter, if we took notice also of the component parallel to the joint which generally tends to cause the rupture which we wish to prevent. XVI. Increase of the crch, below the joint of gpreatest thriust. The proper tllickness of the full circle or semielliptical arch at the springing line, we can best estimate after laying off the curve. of pressure down to the joint above. The last exterior segment must be at least equal to one third the length of the joint. THE LOADED ARCH. 155. Let fig. 51 represent the proposed arch, with any intrados and extrados, and sustaining any load rising to p2, P, P2, etc. As to the point of application and the situation or range of the curve of pressure (see I. and II. of the preceding article. III. Division of the cah into fictitious vousssois or sqegments. Divide the semi-arch into 4, 6, 8, or 10 segments, with a common altitude drawn from the upper angle of each perpendicular to the opposite side or joint, precisely as in the preceding article. From the points of the extrados thus determined, raise the verticals ep, e1p1 e2 P2, etc., through the mass of the surcharge, supposed to have the density of the arch proper. Bear in mind the last part of III., 154. IV. Pcartial acnd total areas. Draw the parallels, i c[L to e il, meeting el i1 at d1; i, cl2 to el i2, meeting e2 i, at c2, GEOMIETRICAL MIETIIOD. 415 and clso on to the springing line. Draw the parallels p k, to e Fp, meeting el p, at 7,, thence Jk cl', to el e, meeting i, el at c'l; P kc2 to e1 192, meeting e6212 at i2,, thence k2 cd'2 to e1 e2, meeting i2 e2 at d'2, and so on, to the springing line. On the vertical Cc passing through the middle of the key, beginning at c, the assumed point'of application of the thrust, lay off c = -1d', 8I — ~c1 d22, 2, and so on to the springing line. Then, area of the 1st segment including its loadcl a X c s; area of 2d segment=ca X 8 a2, etc.; 1st and 2d combined-c a X c s2; lst, 2d, and 3d combinedCC X C s8 etc. In fig. 51, for want of room, we have reduced these distances,, 8c sl, 2 S2, etc, by one half, so that, in. the formnula just given, ca must be replaced by 2a. Draw the indefinite horizontals 8s 21, 8s2 %2, s3 e,3 etc. V. Centers of gmvctvity of the pacrtiacl seygqnets. Determine o', fig. 52, the center of gravity of any segment of the arch proper (see V. 154); and 0, the center of gravity of the sperincumrnbent load. Drawing perpendiculars o'a', o a, to the line o o', mnake o' a'-e3 Ci'3, o aC-(C e3; join Ca a' meeting o o' at o,03. This gives the center of gravity, o,, of the 3d segment and its load. (a) In like manner the other centers, o,, 02, 4,, etc., may be found. All the remaining parts of the construction are precisely the same as in the preceding article, exceptXIV. Thicle88ss of Piev. A B being the line of the base, add to the semi-arch that part of the pier which underlies the assumed thickness at the springing line. Lay off 8, 8s representing its surface; project its center of gravity at o',, and determine in thle usual manner the corresponding points, 9/2, vn and finally T6. (a) The parallels o' a', o a, are not necessarily perpendicular to o o' but may be drawn in any other more convenient direction. 416 THEORY OF THE ARCH. Mark r%, x6- -2x6A. On the line e5 B,, above B1 if A B1 be too large, below if too small, lay off the error B1, f- B1A. Assume another thickness, say A r1, and, corresponding to the addition thus made to the pier, B,', p6 p ),, determine the suite of points o',, s,, n7i, X7 Lay off x7 — 2 - 1A x, and 91 f2 — r1 q2% the corresponding error. f2 would have been laid off above the base if the supposed thickness had been too great. If A -B had been too great, the line 9r p0 would have been on the right of B1, s6 s7 laid off fiom.6 downwards, m6 m1% drawn backwards, parallel to c o0. Continue ff2, meeting the line of the base at B. A B is the required thickness, nearly, corresponding to x=-e, or 19 — (see XIV., art. 154). In like manner the thickness required for any other value of p, may be determined. In general, X6- I1= x 6 Al, x722 Pp1 XA. SEGMENTAL ARCHES. 156. These may be treated precisely as semicircular or semi-elliptical arches. Light arches, however, loaded or unloaded, admit of a more simple method, of which the only error consists of a slight exaggeration of the thrust, and of the dangers to which the structure is exposed. This method is recommended for universal use in light arches of large span. We have taken for illustration, an arch which Capt. Meigs is now building on the Washington aqueduct; fig. 53 representing the arch as unloaded; fig. 54, the same with its final load. The span is 8s220 feet; the rise f-57'.266; the thickness, at the key d=4 feet, at the spring d'=6 feet; the semi-angle of the opening v,550; the load, supposed to rise to a horizontal, and to have the density of the arch, is 12 feet deep over the key, t 12'. GEOMETRICAL METHOD. 417 157. T7he uznloac7ed segnentlz ctarch, fig. 53. Divide the arch by vertical lines i, e1, i2 e27,3 e38 etc., at equal horizontal distances apart, into four, six, or more segments, according to the extent of the angle and to the absolute size of the structure. Midway between these joints lay off, through the arch, the verticals l( cd'l, d2 c'2,, (73 ('3, etc. Points in the direction of all these verticals may be obtained at once by dividing the half-span, or any equal and parallel line, into twice as many equal spaces as we make segments in the semi-arch. Calling a the common horizontal distance between the joints ix e1 and i2 e2, etc., any one of the segments, as i 4 e4 e3, will be measured by a X c14 c'4. At the same time, these midway mean depths of the segments pass very nearly through their centers of gravity; we assume them to pass through these centers, the error here, too, being in favor of stability. Project vertically on the horizontal through Ar', df at o't, c72 at O'21 C(3 at o13, etc. Trace the curves c' c' c', *3...; c c c2 c3... dividing the actual joints (perpendicular to the intrados) into three equal parts. Make c 8s —cl d'a, 81 s,2 72 dc', etc. All the remaining steps are precisely as in art. 154. In this particular case wo see that the curve of pressure starting (as it generally may in the unloaded arch) from the upper limit, and touching the lower curve at the joint of greatest thrust between i4 and &i, passes over to the upper linlit again at the springing line. The arch is practicable, and that is all. All the error wme commit in supposing the joints to be vertical, is that of neglecting the little surface c' cd' e,, which tends to diminish the thrust, and adding the effect of the still smaller surface d c', i6. At other joints the error will be still less. Th/e thickness. of pier may be obtained by construction, as already explained, or by the formule of art. 128; in using the latter, remember that t c = e X aC, qn-n X Pu6. 418 THEORY OF TI-IE ARCH. 158. Tlhe Jloacl segymentcal arceh, fig. 54. Divide the semi-arch precisely as in 157, the vertical joints i e, i1 e, ie2, etc., being at eclual horizontal distances apart, and the broken lines c, cdl', c12 d'2, 13 cl',, etc., midway between the former, extending from the chords i i1,, 42i3, etc., to the upper surface of the surcharge. On the horizontal IK' o's project vertically cld at o'l, 1c2 at o'2, 13 at ~0', etc.; lay off c' s=-c d''; - s c,-2- c d'22, etc.; extend o' c' to m,, thence draw P 1 mT2 parallel to o'2 c', etc., all precisely as in art. 154. For want of room, we have reducedl' 8s, 1,2 etc., by one half. K(nowing, in advance, that arches of this kind are inclined to the third mode of rupture, we assume c', the lower limit, as the point of application. The resulting curve of pressure justifies the assumption. It nearly touclhes the outer lilit midway between the key and the spring, while at these points it coincides with. the lower limit. Comparing the unloaded and the fully loaded arch, figs. 53 and 54, we see that the'curve of pressure has made the greatest possible change in its place, consistent with the condition of remaining within the prescribed limits. In this case the intrados and extrados are parts of different circles, and it is obvious that no catenarian curves could furnish a more economical structure. Whatever curves be adopted, allowance must be made for variations in the curve of pressure corresponding to variations in the load, bearing in mind all partial removals for repairs, which may be made at a future day. The thrust - 2a X t — 2 X the half span X st-= 2 706, taken firom the drawing, is about four per cent. more than the thrust computed from art. 116, v-55~, K- 1.0298_ 1~-6. This, if we suppose each cubic foot to weigh 1i70 pounds, gives a mean pressure at the key, of 115,000 pounds per square foot-1,600 pounds per square inch, GEOMIETRICAL METtIOD. 419 and a pressure of 3,200 pounds per square inch at the edge of the key most exposed. This is more than twice as great as any pressure given in table I, but is little over one tenth the strength of the material as determinedl by Capt. Meigs. The pressure at the spring, F 2 SaX c' t5, gives a somewhat larger pressure per unit of subrface. Were this arch, after the removal of the center, loaded progressively in horizontal layers, it would, at one jstage, be surcharged horizontally up to the level of the top of the key, or nearly to that level. At that stage it would be far firom practicable and barely possible (arts. 143, 145). It would undoubtedly fall. Hence the necessity of putting on the load in due proportion at the reins and at the key simultaneously. Ordinary prudence would require a larger arch, and in fact the plan of Capt. Meigs provides for a larger arch of rubble masonry resting on the cut stone of the inner arch. 159. We have supposed the curves, which mark the nearest approach of the curve of pressure to the outlines of the arch, to divide the joints into three ecqual parts. The geometrical method does not, however, in any measure depend upon these proportions. We can adopt any other proportions; for example, lay off the limit curves each at one tenth the thickness of the joint fr'om the central point, leaving the exterior parts each two fifths the joint. PTPTU'RE OF 3MASONRY BY COMPRESSION. 160. Let the vertical a b rising from a solid foundation, and the indefinite horizontal bf, be the outline of a piece of mlasonry supporting a weight or pressure distributed 4 0 THEORY OF THE ARCH. uniformly and indefinitely r t along the upper surface bf. The width measured at right angles to the plane of the section —one unit. p23the pressure per unit of surface on b f..... 7 =the height a 6. S - the surface a b i corresponding to any line of rupture a n.'- -the weight of each unit square in S. FIG. 55. v =the angle b a n. f=friction=cot. a. a —the angle 6 af between the vertical and the natural slope-90~ — the angle of friction; f-cot.cathe ratio of fiiction to pressure. g-the coherence of the material, per unit of surface, or the force which, acting along any line acn, shall tear asunder one unit in length. TV-p X7 X tang. v-the whole weight or pressure on b n. P=the horizontal force which, assisted by friction and coherence, shall hold in equilibrium the weights Sxjj and WV corresponding to any prism a b n. We have, parallel to a n, the components Pxsin. v, WX cos. v, S9Xp' X cos. v' X /v; and, perpendicular to cos. 71 a n, P x cos. v, Wx sin. v, SXp' X sin. v. For equilibrium we have P sin. v+gqX f-(P cos. v+- F sinl.v+-SX cOS. v p' Xsin. v) f= W cos. v+SXp' cos. v, which, substituting for W, ph X tang. v, and for S, hi'2 tang. v gives P=ph tang. vtang.(a-v)'h tang. v X (76) tang. (aC-v) -. h sin. a( cos. vx cos. (a-v)' GEOMETRICAL METHOD. 42 1 We readily find, by the calculus, or trigonometry, or plane geometry, that tang. v X tang. (ct-v) and cos. v X cos. (a —v) are both maxima when v)-ca. Hence, the greatest horizontal force which the weights TW and Sxl)' can cause, is P=,ph tang.2 Ia+'p'h2 tang.2 Ia -2sc7b tang. -c. (77) If p2 or W be nothing, we have the thrust of an embankment of earth; and we see that the angle of greatest thrust is not affected by any supposed cohesion in the mass. If W be so large that S in comparison may be neglected, we have the thrust of a column. P= —ph tang.2 a -27g, tang. Ca. (78) Let b-the width of the columnlllr tang. c. WTe have P= TVx tang. 1a-S 2Sg (79) Let us suppose the weight or pressure ITjust sufficient to overcome the tenacity of the material; we have -P- O, and W X tang. ICa- 2gb; (80) giving the weight that may be supported when the tenacity and the angle of friiction are known, and either of the latter when the other and the weight are known. The angle of friction in stone, is from 30~ to 36~, giving ct(600 to 540, and the angle of rupture or angle of least resistance from 30~ to 27~. We are at liberty to suppose the line ca to be the surface of an arch —the intrados or the extrados, b in any joint of the arch, W the pressure at that joint; and we learn, from the above, that the masonry, if too weak to withstand the thrust, will give way, by sliding, in a direction inclined about 250 or 300 to the line of the intrados or extrados. We are indebted to Mlosely for the application of this principle to columns, and to experinents to determine 422 THEORY OF THE ARCH. the resistance of materials to crushing, as it has been improperly called. He has, in substance, given (p79); and his remarks thereon have suggestedl the whole of this article. THE CURVE OF PRESSURE IN THE PIER. 161. The equation of that curve, alt. 64, is = 27 ~ne+rn Fl (32) Suppose 7h as well as x variable, e constant, d the constant difference between I and h7i, so that l=?hld. Develop (32), for x substitute x'+-e — F, and for 7A, e?7'1 ~ There results X'7= -?t +-n+ TC/ -aC constant. This is the equation of an equilateral hyperbola referred to a horizontal axis n-:Td above the point of applhcation of the thrust, and to a vertical axis outside the e exterior face of the pier, inside, should that quantity be negative. If F'-62 or e —$/' the vertical asymtote will coincide with the exterior face which the curve of pressure will meet at an infinite distance. If F be less than 4e2 the astymtote will be within the pier. Mosely, in a different manner, has obtained the same result, a fact of which we were not aware in writing the above. 162. The curve of pressure in an arch very heavily loaded, is very nearly the common parabola. THE CIRCULAR RING. 423 The most economical curves of the intrados and extrados would, in this case, also be parabolas. At the other extreme, an unloaded arlch infinitely thin, the curve of equilibrium is the catenary. 16 424 THEORY OF THE ARCH. ACTUAL THRUSTS. TABLE, AA. Tab~le of ThrwtstS of the Unloaded CirculCar Riny. The first column gives the ratio of _R, the radius of the extrados, to r, the radius of the intrados. The second column gives the ratio of the diameter, 2r, to the thickness at the key. The third column gives the new thrust based on the condition that no joint shall opew; the curve of pressure approaching the extrados at the key, within one third thle length of the joint, and the intrados at the reins within one third the length of the joint; this thrustPF. The fourth column gives the thrusts of table A, for the same values of W], calculated on the supposition of actual rupture, the curve of pressure passing through the extrados at the key and the intrados at the reins; this thrust F'. The fifth column gives the ratio --, of these two thrusts. The values of fF and b are a little in excess, the excess increasing with lI. __ P ~ ~ Values of F' Values of E _ II V~alues of F. from 1 K Yalues of F. friom Table A. CA Table A. (1) (3) (4) _ 5 () (l) (2) (3) (4) (t) 1.01 200. 00 r2 x 0.00946 T2 x 0.00S89 1.065 1.21 1 9.52?-2 x 0.16431 r2 x O. 11516 1.42i 1.02 100.00 0.01818 0.01691 1.075 1. 22 9.09 0.11151 0.11887 1.443 1.03 66.66 0.02693 0.02459 1.095 1.23 8.69 0.17 860 11.464 1.04 50.00 0.08504 0.03139 1.116 1.24 8.33 0.18555 &., as in 1 1.482 1.05 40.00 0.04348 0.03813 1.140 11.25 8.00 0.19255 table A. 1.499 1.06 33.33 0.05178 0.04455 1.162 1.26 7.69 0.20012 1.521 1.07 28.517 0.05988.05065 1.182 1.2 7.40 0.20624 Anle of rp- 1.536 1. 08 25.00 0.06804 0.056-191.204 1. 28o 7;14 0.21281 ture the 1.554 1.09 22.22 0.07607 0.06177 1. 232 1.29 6.89 0.21925 same as in 1.565 1.10 20.00) 0.08369 0.067 54 1.239 11.30 6.66 0.22632 table. 1. 51 1.11 18.18 0.09105 0.o 2a 1.1.252 1.31 6.45 0.23338 1.608 1.12 16.66 0.09845 0.01789 1.264 1.3 2 6.26 0.24050 1.638 1.13 15.38 0.10581 0.08254 1.283 11.33 6.06 0.24965 1.616 1.14 14.28 0.11335 0.08729 1.298 1.345.88 0.25501 1. 68a. 1.15 13.33 0.12076 0.09 1.7 6.16 1.35 5.71 0.26236 1.716 1.16 12.50 0.12826 0.09593 1.331 1.36 5.55 0.26784 1.13(0 1.171 11.76 o0.13553 1. 352 1.375.40 0. 27 5 33 1.175 1.18s 11.11 0.14281 &c., as in 1.371 1.38 5.26 0.28286 1.785 1.19 10.53 0.15024 table A. 1.392 1.39 5.13 0.28849 1.801 1.20( 10.0o( 0.15721 1.412 11.40 5.00 0.29616 1.812 TABLE DD. 425 TABLE DD. Table of the actaal thrests of yemicirc-ular crchAes sulqcharged horizontally, t7,e curve of presseue passing thvrougA the middle of the key and the niddle of the wzc-dest joint (see art. 116, an explanation of the columns). Ratio of Valnes of the span, o Values of F2 F Values of A from table to the Values of from table D= - A= the maxi- F,=the Values Of thick- the maximum the maximum or mum effect maximum of /- ness at c thrust down thrust in the coeffi- of asurcharge effect of the A 1+ the key o I: to v=60. case of rupture cent of of const harge i-A 1 2r I e 2 and fall. stay. depth t. the case of -d stbl rupture snd fall. 1.00 1730 t 2 X 0.05563 12 x 0.05547 1.00 -rt x 1.0000 1.00 1.01 200.00 71 0.06315 0.06132 1.03 1.0050 1.16 1.02 100.00 70 0.07083 0.06647 1.06 1.0099 st x 0.8187 1.23 1.03 66.67 69 0.07865 0.07185 1.09 1.0148 1.30 1.04 50.000 67 0.08668 0.07686 1.13 1.0196 0.7531 1.35 1.05 40.00 66 0.09484 0.08175 1.16 1.0244 1.40 1.06 33.83 65 0.10317 0.08638 1.19 1.0291 0.7059 1.46 1.07 28.57 63 0.11165 1.22 1.0338 1.50 1.08 25.00 62 0.12029 &c., from 4th 1.27 1.0385 0.6678 1.55 1.09 22.22 61 0.12908 clof 1.0 1.0431 1.60 1.10 20.00 59 0.13801 tabl. 1.34 1.0476 0.6353 1.6 1.11 18.18 58 0.14709 1.38 1.0521 0.6206 1.70 1.12 16.67 57 0.15633 1.43 1.0566 0.6068 1.74 1.13 15.38 56 0.16571 1.47 1.0610 0.5936 1.79 1.14 14.28 55 0.17522 1.51 1.0654 0.5810 1.83 1.15 13.33 53 0.18490 1.56 1.0698 0.5690 1.88 1.16 12.50 52 0.19471 1.60 1.0741 0.5575 1.93 1.17 11.76 51 0.20468 1.65 1.0783 1.97 1.18 11.11 50 0.21479 1.70 1.0826 2.02 1.19 10.53 49 0.22504 1.75 1.0868 &c., froml the 2.07 1.20 10.00 47 0.23544 1.80 1.0909 13th column 2.12 1.21 9.52 46 0.24598 1.86 1.0950 oftable. 2.17 1.22 9.09 45 0.25667 1.91 1.0991 2.22 1.23 8.69 60 0.25960 1.91 1.1031 2.297 1.24 8.33 60 0.26935 1.96 1.1071 2.32 1.25 8.00 60 0.27916 2.01 1.1111 2.37 1.26 7.69 60 0.28900 2.07 1.1150 1.27 1.40 60 0.29893 2.12 1.1189 1.28 7.14 60 0.30889 2.18 1.1228 1.29 6.89 60 0.31892 2.24 1.1266 1.30 6.67 60 0.32899 2.30 1.1304 1.31 6.45 60 0.33911 2.36 1.1342 1.32 6.26 60 0.34929 2.42 1.1379 1.33 6.06 60 0.35952 2.49 1.1416 1.34 5.88 60 0.36980 2.55 1.1453 1.35 5.71 60 0.38013 r2x 0.14666 2.59 1.36 5.55 60 -0.39051 0.15111 2.58 1.37 5.40 60 0.40095 2.58 1.38 5.26 60 0.41143 &co fomt o 2.57 1.39 5.13 60 0.42196 table D. 2.56 1.40 5.00 60 0.43254 2.56 426 THEORY OF THE ARCIT. TABLE FF. (See art. 1 1t.) THE MAGAZINE OR ROOF-COVERED CIRCULAR ARCH OF 1800, WITH A LOAD OF MASONRY, OR OF EQUAL WEIGHT WVITHI MASONRY, RISING, ON EACH SIDE OF THE CENTRAL RIDGE, TO A ROOF TANGENT TO THE EXTRADOS. Actual thrust in four.systemns, the curve of p/2reure pcas&sing at one third the length of the joint fromz the extrado8 at the key, and fromn the intrado at the joint of gireatest th&rust. [I-the angle between the roof and a vertical; r-the radius of the intrados; R-the radius of the extrados; d=the thickness of the arch proper at the key; K —]= r +; CU-the decimal; F-the thrust-r2X C; A-the addition to the thrust caused by a surclharge of the constant depth t above the tangent roof; C-the decimal in the last column; A — rt C; the coefficient of stability= the quotient of the thrust in this table, divided by the thrust of the same arch in table F the ratio of the actual thrust, on the conditions expressed above, to the theoretic thrust at the instant of rupture and fall. The angle of maximum thrust is measured from a vertical.] 1= 60~ I=55 I=50~ I=45 Surcharge. Value of 4 4i 4 A= Ii=__= 0t;, F=; -FF = 1-_R _ g F = v effectof ] a Kthrust a thrust= d thrust= a thrust= a surch'ge cd r2,. Cr %. r2C r2C. - %' 2 =rCC. r c E C E C a S G 1.40 500 0.3549 1.84 50~ 0.3961 1.81 47 10.4523 1.78 47~ 0.5273 1.76 520 0.6622 1.39 " 0.34821.82 " 0.38921.78 " 0.44521.76 " 0.51961.75 " 0.6654 1.38 " 0.3415 1I. 79 " 0.3823 1.76 " 0.4381 1.74 " 0.51191.72 51 0.6 687 1.37 " 0.3347 1.77 " 0.37541.74 " 0.4308 1.72 45 0.5042 1.71 " 0.6720 1.36 47 ~00.3281 1.75 " 0.3685 1.7345 0.42341.71 " 0.4968 1.69 " 0.6753 1.35 " 0.3214 1.73 47~ 0.3615 1.71 " 0.4159 1.69 " 0.4893 1.67 " 0.6788 1.34 "!0.3147 1.71 " 0.3546 1.69 " 0.40891.67 " 0.4818 1.66 50 0.6822 1.33 " 10.3079 1.69 " 0.3476 1.67 " 0.4018 1.65 " 0.4742 1.65' 0.6857 1.32 " 0.3011 1. 67 " 0.3406 1.65 " 0.3946 1.64 " 0.4666 1.63 " 0.6893 1.31 " 0.2943 1.65 " 0.3336 1.63 " 0.3873 1.62421 0.45901.61 49 0.6929 TABLE FF. —CONTINiUED. 42 7 I= 60~ I=55~ I=50~ 1=45~ Surcharge. PI,- F= F = s= M th= o- = o2 eSffectof thlrust= o thrust thrust= ~ thrust= surch' 1 d S-,, $ 9,2 r2 d r~ C. 2 C.' i 8 E C 8 a 8 a 8 | O C Ci C 1.30 47~ ~0.2874 1.63 45~ 0.3266 1.61 45~ 0.3798 1.60 42~0 ).4517 1.60 49~ 0.6966 1.29 45 0.2805 1.61 " 0.3196 1.60 42~ 0.3729 1.59 " 0.4443 1.59 " 0.7004 1.28 " 0.2737 1.59 " 0.3126 1.58 " 0.3658 1.58 " 0.4369 1.57 48 0.7042 1.27 " 0.2669 1.57 " 0.3056 1.56 " 0.35861.56 " 0.4294 1.56 " 0.7082 1.26 " 0.2599 1.55 0.2986 1.55 " 0.3514 1.55 40 0.4219 1.54 " 0.7122 1.25 42~ 0.2529 1.53 42J 0.2916 1.53 " 0.3440 1.53 " 0.4148 1.53 47 0.7163 1.24 " 0.2461 1.52 0.2846 1.52 40 0.3372 1.52 " 0.4076 1.52 " 0.7205 1.23 " 0.2392 1.50' 0.2776 1.50 " 0.3303 1.50 0.4003 1.50 " 0.7249 1.22 " 0.2323 1.48 " 0.27061.49 " 0.3233 1.49 " 0.3930 1.49 46 0.7293 1.21 " 0.2253 1.47 " 0.2636 1.4737 0.3161 1.48 37J 0.3860 1.48 " 0.7339 1.20 40 0.2184 1.4540 0.2566 1.46 " 0.3089 1.46 " 0.3790 1.47 45 0.7387 1.19 " 0.2115 1.44 " 0.2498 1.45 " 0.30231.45 " 0.3720 1.46 " 0.7435 1.18 " 0.2045 1.42 " 0.2430 1.43 35 0.2956 1.44 35 0.36501.44 44 0.7486 1.17 371 0.1976 1.41 " 0.2362 1.42 " 0.2889 1.43 " 0.3584 1.43 " 0.7539 1.16 " 0.1907 1.40 " 0.2294 1.41 " 0.2820 1.42 " 0.3518 1.42 43 0.7594 1.15 " 0.1837 1.38 35 0.2226 1.40 " 0.2750 1.41 32% 0.3451 1.41 " 0.7651 1.14 35 0.1770 1.37 " 0.2164 321 0.2690 " 0.3389 1.40 42 0.7712 1.13 32~ 0.1701 1.36 " 0.2100 " 0.2629 " 0.3327 1.39 41 0.7775 1.12 " 0.1635 1.35 0.2036 30 0.2567 30 0.3266 1.38 " 0.7841 1.11 30 0.1568 1.34 " 0.1971 " 0.2505 " 0.3211 1.3740 0.7912 1.10 " 0.1504 1.33 30 0.1904 " 0.2441 " 0.3155 1.35 39 0.7988 4 8 THEORY OF THE ARCH. TABLE DDD. Tac7le of the actual thruststt of sdnicircarm bqarches surchar'ged horizontally, the curve of pressure pass8ing at 4 the length of the joint from the eeetactos at the key, and from the intrados at the joint of yreatest t7h1rust, &c., &c. See the explanation in art. 118. Angle of maximum thrust. 1, or coef't of stability. Values of F=r2C. A=rtC'. The Curve of Curve of Curve of 1 In the case curve of pressure pressure at pressure /=R-= of actual pressure at at the one third at the d rupture oue third middle of the length middle of +. d.. and fall. the length, the joints, of the the joints, r Table D. &c., &c., joints, &c. &c., as above. Table DD. as above. Table DD. (1) (2)_ (8) (4) (5) (6) (7) (S 1.00 0.0556 1.0000'75 12~ 30' 1730 1.000 1.00 1.01 0.0625 0.9248 74 C" 71 1.020 1.03 1.02 0.0694 0.89'71 73 70~'70 1.040 1.06 1.03 0.0763 0.8772'71 67~ 30' 69 1.060 1.09 1.04 0.0832 0.8612'70 " 67 1.080 1.13 1.05 0.0903 0.8476 69 c" 66 1. 100 1.16 1.06 0.09'73 0.8358 68 65~. 65 1.126 1.19 1.07 0.1042 0.8252 67 " 63 1.140 1.22 1.08 0.1112 0.8156 66 " 62 1.170 1.27 1.09 0.1181 0.8069 66 62~ 30' 61 1.190 1.30 1.10 0.1250 0.7988 65 " 59 1.220 1.34 1.11 0.1319 0.'7912 65 " 58 1.240 1.38 1.12 0.1387 0.7841 64 60~ 57 1.260 1.43 1.13 0.1455 0.7775 / 64, 56 1.290 1.47 1.14 0.1523 0.7712 64 c" 55 1.310 1.51 1.15 0.1590 0.7651 64 " 53 1.340 1.56 1.16 0.1657 0.7594 64 " 52 1.360 1.60 1.17 0.1723 0.'7539 64 " 51 1.390 1.65 1.18 0.1788 0.7486 63 57~ 30' 50 1.410 1.70 1.19 0.1853 0.7435 63 4" 49 1.440 1.75 1.20 0.1918 0.7387 63 " 4'7 1.470 1.80 1.21 0.1982 0.7339 63 " 46 1.490 1.86 1.22 0.2046 0.7293 63 " 45 1.520 1.91 1.23 0.2109 0.7249 63 " 1.550 1.91 1.24 0.21'71 0.'7205 62 t" 1.580 1.96 1.25 0.2233 0.7163 62 i" 1.610. 2.01 1.26 0.2294 0.7122 62 t" 1.640 2.07 1.27 0'.2355 0.7082 62 " 1.670 2.12 1.28 0.2415 0.7042 62 " 1.700 2.18 1.29 0.2475 0."7004 61 t" 1.730 2.24 1.30 0.2534 0.6966 61 " 1.''70 2.30 1.31 0.2593 0.6929 61 " i 1.800 2.36 1.32 0.2651 0.6893 61 1.830 2.42 1.33 0.2709 0.6857 61 " 1.870 2.49 1.34 0.2766 0.6822 60 i" 1.910 2.55 1.35 0.2823 0.6788 60 t" 1.920 2.59 1.36 O0.2879 0.6753 60 " 1.900 2.58 1.37 0.2935 0.6720 60 i 1.880 2.58 1.38 0.2990 0.6687 59 " 1.870 2.57 1.39 0.3045 0.6654 59 " 1.850 2.56 1.40 0.3099 0.662'2 59 4" 1.830 2.56 430 THEORY OF THE ARCH. TABLE EE. SEGMENTAL ARCHES LOADED UP TO THE LEVEL OF THE TOP CURVE OF PRESSURE PASSING WITHIN ONE THIRD TIlE AND FROM THE INTRADOS AT THE SPRINGING LINE OR [C-the decimal inl any column; r-the radius of the caused by a surcharge of the constant depth I above the key; 1+-; dcl=the thickness at the key. See art. 120. s=4f; r=2-f; s=5f; "=83-gf; s=Qf; r=5f; s= r; =68f; v=538~ 7' 80". =48~ 86' 10". v=86~ 52' 10". v=81~ 58' 27". 1+. F=r2 X C. A=rtC. F=r2 xC. A=rtC. F=r2 x C. A=rtC. -=r2 X C. A=rtC. C. C. C.. 3 C. C. C C. 1.40 0.3091 1.827 0.6624 0.2938 1.736 0.647(1 0.2700 1.5960. 6067 0.2441 1.443 0.5563 1.39 0.3036 1.845 0.6654 0.2887 1.754 0.6511 0.2655 1.613 0.6115 0.2403 1.460 0.5615 1.38 0.2982 1. 862 0.6687i 0.2836 1.771 0.6552 0. 2610 1. 630 0. 6163 0. 23641.477 0.5668 1.37 0. 2926 1. 880 0.6720 0.2784 1.789 0.6593 0. 2564 1.648 0.6212 0.2325 1.494 0.5723 1.36 0.2871 1.900 0.6753 0.2732 1.808 0.663.5 0.2518 1. 666 0.6262 0.2285 1.512 0.5778 1.350.28141.918 0.6788 0.2679 1.8260.6677 0.2471 1.6840.63130.22441.5300.5835 1.34 0.2758 1.938 0.6822 0.2626 1.8460. 6720O. 242411.705 0.6365 0.2203 1.548 0.5893 1. 33O. 2700 1.9090 0.6857 0. 2572 1.865. 6764 0. 2376 1.723 0.6418. 2162 1. 568 0.5952 1.32 0.2643 1.876 0. 6893 0.2518 1.886 0.6807 0.2327 1.743 0.6471 0.2120 1.588 0.6012 1.31 0.2584 1.842 0.6929 0.2463 1.906 0.6852 0. 228 1.763 0.6526 0.2078 1.608 0.6074 1.30 0.2526 1.809 0.6966 0.2407 1.926 0.6897 0.2229 1.783 0.6581 0. 2035 1. 628 0.6138 1.29 0.2466 1.775 0.7004 0.2351 1.893 0.6942 0.2178 1.804 0.6638 0.1991 1.650 0.6202 1.28 0.2407 1.745 0.7042 0.2294 1.859 0. 6988 0.2127 1.826 0.6695 0.1946 1.670 0.6269 1.27 0.2346 1.714 0.70820.2237 1. 826 0.7035.2076 1.849 0. 6754 0.1901 1.693 0.6337 1.26 0. 22851 1.683 0.7122 0.2179 1.793 0.7082 0. 2023 1.870.6814 0. 1851 1.715 0.6406 1.25o 0.2224 1.654 0.7163 0.2120 1.761.7130 0. 1970 1. 883 0.6875 0.1808 1.73'7 0.6478 1.24 0.2162 1.623 0.7205 0.2061 1.730 O.71'79 0.1916 1.850( 0.6937 0.1761 1.761 0.6551 1.23 0.20993 1.5940.724910.2001 1.700 0.7228 0.1862 1.817 0.7001O. 1712 1.785 0.6627 1.22 0.2036 1.566 0.7293 0.1940 1.668 0.7278 0.1806 1.781 0.7066 0.1663 1.810 0.6704 1.21.1972 1. 538 0'.7339.1878 1. 636 0.7329O. 1750 1.7480. 7132 0.1613 1. 835 0.6784 1. 200.1907 1.510 0.7387 0. 1816 1. 606 0.7380 0. 1692 1.7130.7200.1561 1. 830.6865 1.19 0.1842 1.482 0.7435 0.1753 1.575 0.7432 0.1634 1.680 0. 7269 0. 1509 1. 794 0.6950 1.18 0.1776 1.456 0.7486 0.1689 1.546 0.7485 0.1575 1.646 0.73400,.14561 1.756 0.7036 1.17 0.1710 1.429 0.7539 0.1624 1.515 0.7539 0.1514 1.611 O.7413 o.1401 1.719 0.7125 1.16 0.1643 1.403 0. 7594 0.1558 1.4850. 7594 0.1453 1.578 0.7487 0.1346 1.682 0.7218 1.15 0.1575 1.377 0.7651 0.1492 1.456 0.7651 0.1391 1.544 0.7564 0.1289 1.644 0.7313 1.14 0.15061. 351 0.77120.14251.4250.77120. 1327 1.5100.76420.12311.607O.7411 1.13 0.1437 1.326 0.7775 0.1356 1.397 0.7775 10.1263 1.477 0.7722 0.1171 1.568 0.7512 1.12 0.1367 1.301 0.7841 0..1287 1.368 0(.7841 0.119'7 1.442 0.7804 0.1110 1.529 0.7617 I 11 0.1296 1.276 0.7912 0.1216 1.338).7912 0.1129 1.408 0.7888 0.1047 1.489 O.7726 1.10 0.1225 1.250 0.7988 0.1145 1. 310 0.7988 0.1061 1. 3740.7975 0.0983 1.450 0.7839 1.09 0.1152 1.226 0.8069 0.1072 1.2790.8069 0.0991 1. 339 0. 8064 0.09171.4090 0.7956 1.08 0.1079 1. 199 0.8156 0.0998 1.251 0.8156 0.0919 1.304 0.8156 0.0849 1. 36t 0. 8077 1.07 0.1005 1. 177 0.8252 0.0923 1.221 0.8252 0.0846 1.268 0.8252 0.0779 1.325 0.8203 1.06 0.0930 1. 151 0.8358 0.0847 1.191 0.8358 (1. 0771 1. 232 O. 8358 0.0707 1. 283 0.8334 1.05 0.0855 1.128 0.8476 0.07701 1.163 0.8476 0.0695 1.196 0.8476 0.0633 1.239 0.8470 1.040.07781 1.03 0.8612 0.0691 1.133 0.8612 0.06161. 160 0.8612 0.05561. 196 0.8612 1.03 0. 0700 1.077 0.8772 0.0611 1.103 0.8772 0.0536 1.124 0.8772 0.0477 1.1500.8772 1.020. 0622 1.054 0.8971 0.05291.073 0.89710.0453 1.084 0.8971 0.039,51.103 0.8971 1.01 0.0542 1.027 0.9248 0.0446 1.042 0.9248 0.0368 1.042 0. 9248 0.0310 1.05 0.9248 1.000.0462 1.0000 0.0361 1.00000.0281 1.00000.0222 1.0000... TABLE EE. 431 TABLE EE. OF THE KEY. ACTUAL THRUST IN SEVEN SYSTEMS: THE LENGTH OF THE JOINT FROM THE EXTRADOS AT THE KEY, JOINT OF GREATEST THRUST. intrados; Fithe thrust; A-the addition to the thrust F- C2X; A-X C1xC; s=the span; f=the rise; I= s=8f; r=8-f; s=1lf; r-=13; 8s=lQf; r=822-f; 2' 2 v=28~ 4' 20". v=22~ 83' 10". v=14~ 15'. K=8 4 20. 1~-. d =r2x G. A= rtC. F=.2x C/. A= rtC F-=r2X. A=rtC.|. j Ca. d. C. _ _ 1. 40 0.2188 1.293 0.5038 0.1745 1.095 0.4070 0.127'6 1.000 0.3538 1. 39 0.2156 1.310 0.5092 0.1722 1.112 0.4123 0.1240 1.000 0.3512 1. 38 0.2128 1.326 0.51438 0.1699 1.129 0.4178 0.1204 1.000 0.3487 1.37 0.2090 1.343 0.5205 0.16,76 1.146 0.4235 0.1168 1.000 0.3462 1.36 0.2056 1.361 0.5264 0.1652 1.165 0.4294 0.1132 1.000 0.3437 1.35 0.2022 1.378 0.5324 0.1628 1.183 0.4354 0.1097 1.000 0.3411 1.34 0.1988 1.398 0.5385 0.1603 1.203 0.4417 0.1062 1.000 0.3386 1.33 0.1953 1.416 0.5448 0.1578 1.222 0.4481 0.1027 1.000 0.3361 1.32 0.1917 1.436 0.5513 0.1553 1.243 0.4548 0.0993 1.000 0.3335 1.31 0.1881 1.456 0.5579 0.1527 1.265 0.4617 0.0958 1.000 0. 3310 1.30 0.1844 1.475 0.564710.1501 1.287 0.4688 0.0924 1.000 0.3285 1.29 0.1807 1.497 {0.5717 0).1474 1.310 0.4762 0.0891 1.000 0.8 260 1.28 0.1769 1.519 0.5789[ 0.1447 1.335 0.4838 0.0857 1.000 0.3234 1.27 0.1739 1.541 0.5863 0.1418 1.358 0.4918 0.0824 1.000, 0.3209 1.26 O.1690 1.562 O.594010, 0.1389 1.383 0.5000 0.0791 1.0000.3184 1.25 0.1649 1.585 0.6019 n 0.1360 1.411 0.5085 0.077 1.020 0.315 9 1.24 0.1608 1.608 0.6101 0. 1330 1.438 0.5173 0.0762 1.050 0.3133 1.23 0.1566 1.633 0.6185 0.1298 1.465 0.5266 0.0749 1.080 0.3134 1.22 0.1523 1.657 0.621 10.1267 1.496 0.536410.0736 1.112 0.3220 1.21 0.1479 1.1 683 0. 6361 0.1234 1.527 0.5465 0.0723 1.147 0.3312 1.20 0.1434 1.707 0.6454 0/.1200 1.558 0.5571 0.0709 1.184 0.3410 1.19 0.1388 1.735 0.6550 0.1166 1.593 0.5681 0.0694 1.222 0.3516 1. 18 0. 1341 1.760 0.6650 0.1131 1.627 0.5797 0.0679 1.264 0.3629 1.17 0.1293 1.788 0.6754 )0.1094 1.663 0.5918 0.0663 1.310 0.3750 1.16 0.1243 1.788 0.6861 0.1055 1.699 0.6045 0.0646 1.35[7 0.3880 1.15 0.1192 1.74810.6971 0.101511.738 0.6180 0.0629 1.410 oo 0.4022 1.14 0.1140 1.706 0.7086 0.0974 1.777 0.6321 0.0611 1.469 0.4175 1. 13 0. 1086 1.663 0.7207 0.0931 l1.818 0.647010.0592 1.534 0.4342 1.12 0.1030 1.619 0.7333 0.0886 1.805 O.6627 0.0570 1.597 0.4524 1.11 0.0972 1.575 0.7464 0.0839 1.751 (0.6794 0.0547 1.668 0.4725 1.10 0.09131.529 0.7601 0.0791 1.701 0.6971 0.0523 1.749 0.4947 1.09 0.0852 1.484 0.7744 0.074011.64410.7160 0.0496 1.830 0.5192 1.08 0.0789 1.440 /0.789410.068611.58410.7360 0.0467 1. 93010. 5467 1.07 0.0723 1.390 0.8051 0.0630 1.525 0.757410.0436 1.938 0.5775 1.06 0.065411.340 0.8216 0.0570 1.46210.780410.0400 1.84310.6125 1.05 0.0583 1.287 0.8389 0.0507 1. 3,97 0o.8050 0.0361 1.752 0.6524 1.04 0.0509 1.235 0.8572 0.0440 1.329 0.8315 0.0316 1. 637 0.6984 1.03 0.0432 1.180 0.8765 0.0368 1.256 0.8601 0.0265 1.506 0.7521 1.02 0.0351 1.125 O.89 1 0.0291 1.173 0. 8911 0.0206 1. 73 o0.8155 1.01 0.0267 1.068 0.9248 0.0209 1.094 0.9248 0.0135 1.205 0.8916 1.000.oo 01781 1.0000ooooo.0121 1.00000.0051 1.ooo0 432 THEORY OF THE ARCH. ELE3IENTS OF PARTICULAR BRIDGES- THE DI)IENSIO] [D —the half-spaln in elliptical arches Thickness Rlatio of the d diam. curs 0 NAME OR SITUATION.:Dato. Architect. Intrados. Rise. Span. atthe the in. key. r trados J 4 t 5 d at tile " the ke to the Feet. Feet. Feet. Feet thick's[ 1 Bishop Auckland, over the Wear.................1388..........Segmental.. 22.00 1 100 ~1 5 4 4.7 2 Llanwast, in Denbighshire. 1636 Inigo Jones.... do... 17.00 58.00 1.50. 38.66 44.3' 3 Westminster Bridge, central arch.Oa. 150 Labelye Semcrcle..... 38. 00. 60 14. 0010. 00 10.0 4 Taaf, Soutis Wales 1755 Edwards...... Segental... 5.00 140.00 2.50 2.50 56.00 10.0. 5 Wellington Bridge, over the Aire, at Leeds.............. Rennie.'. do......15.001000 3.00 7.00 33k 60.5 I6 Waterloo Bridge, nine equal 1 arches............... 1811 do >........ ff Semi-ellipse.. 35.00 120.004.15. 8.00 25.26..... I London Bridge, central arc 1831 George Ronnie. do 38.00 152.00 5.00 10.00 30.40..... I8 Staines Bridge, five equal arches............832Ren........ Segmental 9.25 7400300 6. 24 52. 9 Clester Bridge.............. 825 Harrison..... do 42.00200.004.00 6.0.50.00 0. C 10 Edinburgh.................. Mylne........Semicircle... 36.00 12.00 2.75..... 26.18 26.1 1I1 Hu teeson Bridge, Glasgow..... Robt. Stevenson Segmental... 13 1 9.00 9 3. 50 4.50 22.51 37.2 12 Wlhi;tadder Bridge, Allanton.842Do&Sons... do........50 500 2.50 3.00 30.00 53.5 13 Railway Bridge, at Maidenhead....... Brunel........ Semi-ellipse.. 24.25 128.00 5.25 7.16 2. 38 14 Bridge at Neuilly, over the Seine, five equal arclhes.. 1774 Perronet.... do 32.00 128.00 5.25 24.38. isl 5 Bridge of Pesmes, on. t.he Oulgnon............... 1712 Bertrand. Segmental... 3 44.753.....10 11.61.35.C' 16 Chateau Thierry......... 1786 Perronet. Semi-ellipse.. 17.00 51.00 4.00. 12.175 17 Louis IX. e............ 17191 do Segmental... 9.15 94. 00 3..... 25.64 64.4 18 Neours. o............ 1805 do do. d 3.15 53.253..... 16.81 60. l19 Turin................... Mos.. 00....... do3 1.0.7 617.. ELEMENTS OF PARTICULAR BRIDGES. 4,3 COMPILED MAINLY FRO0M CRESY'S ENCYCLOPEDIA. 3 —the radius of the intraclos in segmental arches.] Thrust at the O Nature of' keyf or /j /fthe }. _ each foot of material R EMRKS. bridge in when I width. ] known. p u ccl r2x 0.06356 25869....... 4000 11.00 Probably with little surcharge. 0.08960 10560....... 4000 27.00 "The road-way approached a horizontal line" in consequence of the substitution of vehicles for pack horses. 0.19180 5908.... 4000 49. 00 Thickness taken from the drawing in Cresy's Encyclopedia. Surcharged horizontally. 0. 0157 35072.. 4000 8.00 Fell on the removal of the center, the cown rising; but stood after being rebuilt with hollow spaces in the surcharge over the reins. 0.05764 25362....... 4000 111 Surcharged horizontally. 0. 17113 21905 Granite. 6000 20.00 Counter arch over the piers to receive the horizontal thrust. Settled but a few inches on removing the center. 0.16723 30926 do 6000 14.00 Settled at the key only 2" on removing the center. Counter arches 6 feet thick over the piers. 0.057 21 18862...... 5000 20.00 0.06406 50223 o.. 6000 86o The crown settled only 21 inches on removing the center. 0.11683 8809....... 400 33.00 0.07810 15177 o...... 4000 19.00.06195 1140....... 4000 16.00 o0.25292 24666 Brick. 1200 31 Six longitudinal walls support the railway. 0.. 19 870 24804... 4000 11i.61 Settled 2 feet at the crown on removing the center. 0.06726 12644.... 4000 23.00'Settled considerably at the crown on removing the center, the abutments moving laterally. 0.26476 68861....... 400042.06 0.04513 27498....... 4000 10- In calculating the last column, the pressure, per unit of surface, at the extrados of the key, has been assuimed at twice the mean pressure. 0.03871 18200.... 4000 16.00 0.04887 42294.....6000 10'...... The weight of a cubic foot of stone is assullled to be 160 pounds. Brick masonry 125 pounds per cubic foot. I.oo~,.h~ o..u o o. ~o uo INDEX. Paragraph and Table. Page. Arch-how distinguished from the beam 1, 2. 199, 201.." " represented.. 1. 199. Circular arches of 180~ without loadthrust, ultimate, by rotation,. 28-35, 38, A. 216-224, 227...".. " including effect of mortar,.... 31, 32. 219-221. thrust,ultimate, by rotation, including effect of surcharge,.. 34, 35. 222-224. thrust, actual, by rotation,.. 114, AA. 334, 335... 64 sliding,. 36 —38. 225-227.. ".. including effect of surcharge, 39. 227-228. thrust, actual, by sliding, including effect of mortar,.. 37, 38. 226, 227. thrust-when due to rotation, when to sliding,.41. 228. thrust by rotation, when zero,. 0, 33, 40. 218, 221, 228. thickness of pier,.. 42-45, B. 229-232... " limit,.. 43, 46. 230, 233. " arch "... 47. 233. geometrical method,... 153, 154. 404. Circular arches of 180~, surcharged horizontallythrust, ultimate, by rotation,. 57, D. 243. " actual, ". 115, 116, 118. 335-343, 346........ DD, DDD...." by sliding,. 57, D. 243. thickness of arch-general rules,. 136, 137, 151, 152. 372-376, 401-404.... I limit,.. 57, 142, 144. 243, 380, 386. pier,.... 64, 65, 128-135. 253-258, 359-371. geometrical method,. 155. 414. (For further particulars, see magazine arch.) Coefficient of stability,. 17-19, 122-126. 210, 211, 352-359. Coulomb-theory of,..... 201. Curve of pressure,.. 108-162. 324-423....." limit of approach to the extrados and intrados,. 111. 328. Curve of pressure-equation of, in the arch,... 391. Curve of pressure-equation of, in the pier,.. 161. 422. 436 INDEX. Paragraph, and Table. Page. Curve of pressure, inecessary situation of, 152. 402. ((" " in the arch —limits, 162. 422. Definitions and general remarks, 1. 199. Elliptical arches.without loadultimate rotation and sliding thrusts, 90-94, A', H. 291-297, 294, 323. actual thrusts,. 121. 350. thickness of pier,.. 95, 128. 297, 359. "6 $arch,.. 136, 138. 372, 376. Elliptical arches with elevated ridge — ultimate thrust-rotation, 96, 97. 297. thickness of pier,... 98, 128. 300, 359, geometrical method,. 155. 414. Elliptical arches surcharged horizontally — ultimate rotation thrust, etc.,. 99-106, G. 301-308, 322. actual thrust,.. 121. 350. thickness of pier,... 107, 128, 133, 135. 308, 359, 363, 370. "L arch,. 136, 138, 151. 372, 376, 402, compared with segmental arches,. 100-102. 303, 304. geometrical method,.. 155. 414. Geometrical method, of universal application,... 153 —159. 404-419. Joint of rupture, rotation,. 5, 8, 9, 10. 202, 203, 204... " sliding,.. 20. 212...... rotation, circular ring, 28, 114, A, AA. 216, 334. 6-.. "':sliding, " 36, A. 225. r" otat'n, magazine arch, 48, 117, F, FF. 236, 344. I. " 1 sliding;... 52, F, FF. 239. rotation, circular arch surcharged horizontally,.. 57, D, DD, DDD. 243. Joint of rupture, sliding, circular arch surcharged horizontally,.. 57, D. 243. Joint of rupture, rotation, segnmental ring, 69, A, E. 265. Joint of rupture, sliding, segmental ring, 69, A, E, 265. Joint of rupture, rotation, segmental arch surcharged horizontally,. 74, 119, 120. 271, 348, 349. Joint of rupture, sliding, segmental arch surcharged horizontally,. 74. 271 Joint of rupture, rotation, segmental arch, with elevated ridge,.. 80, 117. 276, 344. Joint of rupture, sliding, segmental arch, with elevated ridge,... 80. 276. Limit thickness of pier, circular ring, 43, 46. 230, 232..c..... in all cases,. 64. 256.... " of arch, circular ring, 47. 233. INDEX. 437 Paragraph and Table. Page. Limit thickness of possible circu. arches, surcharged horizontally,.. 57, 142. 243, 381. Limit thickness of possible segmental arches surcharged horizontally,. 143. 383. Limit thickness of practicable circular arches surcharged horizontally,. 144. 386. Limit thickness of practicable segmental arches surcharged horizontally,. 145. 388. Limit thickness of arches as affected by surcharge,....146. 390. Magazine arch, with elevated ridgethrust, ultimate, by rotation, 48-51, F. 235 —239, 320-1. " actual, ". 11, FF. 344. cc it by sliding, 52-54. F, FF. 239-341. " roof inclined 45~, 55, 56, C, F, FF. 241, 242. horizontal,.. 57, 115, 116, 118, D, 243, 335 —343, 346. DD, DDD. thickness of pier,.. 64-67, 128-135. 253-264, 359-371. "' " examples,. 67, 135. 260, 365. iMonocacy Bridge.. 73, 136. 269, 373, Mortar-commonn-no element of stability,. 2. 200. tIorltar-hvdraulic-llay add to stability, 2. 200. Mortar-effect of, 15, 16, 17. 205-210. (See thrust of circular, magazine, etc., arches.) Pressure per unit of surface,.. 112. 329..... 44 limit, 136. 37 2. " relative at the key and at the reins. 140..379. Pressure on the joints of the pier,. 141. 319. " determined geometrically, on the several joints,.. 153 —159. 404. Point of application of tle thrust,. 4,6,23, 111, 148, 149. 202, 202, 213, 328, 392 —396. Pier-thickness of-general conditions, 11, 12, 17, 19. 204, 210, 211..... " fornulT, 64, 128. 253, 359. Pier-thickness of-linmit,. (See circular, elliptical, etc., arches.) 64. 256. Rupture of masonry by compression, 160. 419. " joint ot (see joint). " third mode, the key rising,. 25, 142-146. 213, 380-390. Segmental ring without load,.. 68-73, 156, 157, E. 265-271, 416. thickness of pier,. 72, 73. 267-271. geometrical method,... 156, 157. 416. Segmental arches surcharged horizontallyultimate rotation thrust, etc.,. 4-179, E'. 271. 438 INDEX. Paragraph and Table. Page. actual rotation thrust, etc., 119, 120, 158, EE. 348, 349, 417. thickness of pier.. 7 9 275. 4" arch, limit, 143, 145. 8 -3S 9. Segmental arches with elevated ridge, 80-82. 276 —80..... approximate formulae. 85-89. 286-290. Scarp walls, stability of,. 83, 84. 280. Surcharge, its effect on the ultimate rotation thrust,. 13, 14, 17. 205, 210. Surcharge, its effect on the practicability of arches,.146. 390. Surcharge, its effect on the actual sliding thrust,.. 21 212. Thrust, ultimate, by rotation,. 1-107. 199, 33. 4' " obj ections to the theory,.. 108. 324. Thrust, ultimate, by rotation, general formul.. 17. 210. Thrust, actual, by rotation, 108-162. 324-423.... " dependent on the pier,. 1.. 109, 110. 325-328. Thrust, actual, by rotation, of the wellestablished arch, 113. 331. Thrust, actual, by sliding, general formulre,.... 20, 21. 212. Thrust, actual, by sliding, 1 107. " true, generally due to rotatio'n, 22. 212. Tables (see table of contents). Thickness of pier (see pier). arch, 136, 151, 152. 272-389, 401-404. (" "tr? lincrease of, at and below the reins,... 137, 138, 139. 375-378. Ultimate thrust (see thrust).