UC-NRLF 531 PRACTICAL TREATISE ON THE PROPERTIES OF CONTINUOUS BRIDGES. BY CHARLES BENDER, C. E. MEMBER OF THE AMERICAN SOC. CIVIL ENGINEERS. NEW YORK : D. VAN NOSTRAND, PUBLISHER, 23 MURRAY AND 27 WARREN STREETS. 1876 COPYRIGHT, 1876, BY D. VAN NOSTRAND. CONTENTS. INTRODUCTION AND HISTORICAL NOTES ON THE THE- ORY OF THE ELASTIC LINE, AND ESPECIALLY OF CONTINUOUS GIRDERS. I. The general principle of continuity of girders and trusses. The theory reduced to a combination of single spans, the theorem of three moments being nothing but its algebraic expression. The principle of continuity only modifies the law that the total load of and upon a span must be taken up by its |two piers. The moments over the middle piers expressed by these elastic reac- tions. Correct mode of calculation of deflections of trusses indicated. II. The modulus of elasticity depending upon special ex- periments. Tensile experiments by Bornet, Ardant, Hodgkinson, Ed. Clark, Malberg, Morin, Styffe, Staud- inger, and by the author. Experiments on compression by Hodgkinson, Dulean, the engineers of the Cincin- nati Southern Railroad and Bauschinger. Experiments on flexure by Morin, Woehler, Staudinger. Experi- ments on shearing or torsional Moduli by Duleau, Woehler, etc. III. Deficiencies of continuous girders and trusses. The assumption of a constant moment of inertia. Con- tinuous trusses should have but one web system. Dis- turbances of strains from manufacture, settling of ma- sonry and foundations. Building continuous girders on false works in point of quality preferable to pushing them over the piers. Disturbances in strains of continuous bridges by un- equal expansion. Continuous bridges require the best class of substruc- ture. The change in the strains of continuous bridges from tension into pressure must be provided for by proportion- ing the members to the sum of both strains. Different systems of bridges must be compared by giving to each its proper and most advantageous proportions, but not by supposing all to be of the same depth. Continuous plate girders not only are really more economical than single span plate gilders, but the the- ory also is more in conformity with the latter mode of construction. Many French engineers, in accordance with the supposition of theory, applied uniform cross- sections of flanges. Economy of such girders, with constant flanges, by lowering of middle supports, or in case of three and more spans by suitable proportions of spans. Economy not enhanced by these methods if the chords are properly varied. IV. Development of the theory, without higher calculus. Deflection of a conti lever beam. " truss resting freely on two supports. Henry Bertot's equation. Correction for unequal heights of support. " expansion, of chords. Deflections of trusses by considering the webmembers as well as the chords. V. Example of two continuous spans and single spans. Influence of each separate panel load. Consideration of excess load of .locomotive. Maxima chord and web- strains. A simple mode of calculation of single spans. Discussion of weights. Theoretical quantities of single and of continuous spans of two and more spans. Influence of dead load. No theoretical saving of continuous trusses over single spans of most economical depths. The saving in chords neutralized by loss in the webs of continuous trusses. Correction of strains on account of deflections by webs. Error to amount to 29 per cent. VI. Possible irregularities of the strains of the two calcu- lated continuous bridges. Influence of heights of support, and of unequal expan- sion of chords. Modes of adjustment. Experience as to influence of heat of sun observed on the Tarascon bridge in France, the Tfceis bridge in Hun- gary, South wark stone bridge, Britannia and Victoria bridges in England and Canada. Drawbridges in this country as observed by Mr. 0. Shaler Smith and by Mr. John Griffen. Diminution of deflection of continuous trusses of no practical moment. Length of continuous bridges limited by longitudinal expansion and contraction. Deflections of continuous bridges form a test of their strains being properly calculated or agreeing with real- ity. This quality recognized by learned French engineers. Difference of calculated and actual deflections pro- portional to difference of calculated and actual reac- tions. VII. RECAPITULATION AND CONCLUSIONS. Continuous draws, first investigated by Professor Stenberg in 1857, who also in his lectures gave the idea of weighing the reactions. C. Shaler Smith, the first engineer who investigated and built continuous draws without end reactions un- der dead load. Continuous draws to be calculated as two continuous spans. Drawbridges consisting of two single spans to be uni- ted for each movement and severed if in place. Large spans with limited depths can be built as con- tinuous trusses with hinges in alternate span. These trusses first patented by De Bergue in England in 1865. PEEFAOE. A paper by the author, being a critical ex- amination into the merits of continuous gird- ers, was prepared six years ago, but its pres- entation to the American Society of Civil Engineers wa^s delayed till the last spring. In this paper the subject for the first time is found to be based without the use of higher calculus on one simple geometrical relation, forming the connecting link between single spans and continuous bridges. The same paper, increased with further data resulting from its discussion, is compiled into the short treatise now presented. It is hoped that it may contribute to clear opinions as to the real merits of the systems of single and continuous spans, and would lead to a more thorough understanding of the nature of each. The subject having never before been treat- ed in this light, it is believed that railroad engineers will not unfavorably receive the results of special studies which have occupied vra a period of many years, and which in the main are: That in addition to the sensitiveness of continuous bridges, the economy claimed for them does not exist either theoretically or prac- tically in all instances in which the construc- tion of properly designed compound single span trusses is not limited as to their depths. As a result of these conclusions, there would seem to be one more good reason that the most valuable time of polytechnic students should not be unnecessarily wasted by enter- ing deeply into a theory which more essen- tially is of mathematical and historical in- terest. C. BENDER, C. E. Member American Society of Civil Engineers. NEW YORK, October 12, 1876. . APPLICATION OF The Theory of Oob^feadus -Girders TO ECONOMY tt Lately the introduction in this country of continuous girders has been suggested on the plea of greater economy than, it is asserted, can be obtained under application of the highly perfected and simple Anferican truss- es. The majority of advocates of that sys- tem who are now devoting their time to the- ory, though well acquainted with the mathe- matical part of the subject, have omitted some practical bearings which would be nec- essary to enforce their assertions. In reality, mathematical investigations of the subject of continuous girders require not a very high degree of training in analysis. It needs but the execution of the integration of one single equation, which execution may become leng- thy and tedious, and may require much pa- tience. But once the one mathematical idea 10 of that equation be understood, the rest of the work is common and mechanical alge- braic labor. If it wei*e tPiie hat continuous girders give more economy thari the system in use in the C'mt^ct States, it ; would rcertainly be a heavy charge against the engineers who have made the art of bridge building their specialty, and who study their profession with all earnestness. Many will deny, at the outset, that this charge is just ; for the sake of others, the author pro- poses to show that the American practice of bridge building hitherto, has been in the prop- er direction*towards other improvements, and that the theorists who wish bridge builders to follow their advice have studied the sub- ject in but one of its bearings, and have omitted to examine closely their premises as well as their conclusions. The author believes it is not only desirable, but necessary that this question should be fully discussed, from various reasons. Practical en- gineers generally do not place much confi- dence in long formulas, and if they once have studied mathematics thoroughly they lose the taste for these studies after some time of practice, since they have convinced them- 11 selves as to the futility of ultra refined theo- retical speculations. These engineers will not be very likely to adopt structures whose calculation of strains would waste so much valuable time. But these engineers could not prevent a new method of construction in time becoming fashionable, whether correct or not, as long as it were founded on some elegant theory and seemingly led to econ- omy. For, under our large factors of safety, we can commit many sins in construction be- fore they are found out. Again, there is always a number of men who, because they do not understand abstruse calculations and formulae, rather than admit this fact, publicly endorse them warmly. And finally, when in polytechnic schools for a number of years, a certain theory has been thoroughly studied with zealous assiduity, a little army of its admirers will fill positions in railroad and in public engineering offices, anxiously waiting for the first opportunity towards introducing into practice what they consider the finest jewel of their technical knowledge. The au- thor frankly admits once to have been of this number. But after studying the subject of continuity of trusses for several years, and a 12 careful examination of its suppositions he found himself compelled to admit that the theory is not correct scientifically, and does not agree with the physical laws of elasticity of iroif.* We are now prepared to prove, that for me- dium spans, say of 200 feet, the construction on the principle of continuity leads to greater truss weights in addition to greater cost of workmanship than are required by the use of single spans with improved details. This last result is very important indeed, * Six years ago, in a paper written for the German Soci- ety of Engineers, Verein JDeutscher ingenieure in Berlin, which was translated into English and published two years ago in the Railroad Gazette of New York, the author stated : 44 The writer of these lines himself had for some time thought that it might be possible, by application of pin joints, by reducing the number of parts, by the use of proper scales and adjustments for the regulation of the pressures on the three or more piers of a continuous bridge, and by the use of scientifically correct and complete formulae, to pro- duce reliable continuous trusses, by means of which the large rivers of this country could bespanned without the use of false works." "With a great deal of labor he had constructed an analyti- cal expression, which embraced the relation of the moments of flexure over three consecutive piers of a continuous gird- er. In this formula, due attention was given not only to the deflections caused by the chords, but also to those due to the tensile and compressive members of the web system ; also the actual section of each separate member was intro- duced. It therefore did away with two errors of the formu- lae generally quoted in books, which are only applicable when the girders are very shallow and when the web is a 13 for if it were possible, under application of the principle of continuity to arrive at an economy in weight and cost, there would be a large market for this article however objec- tionable the mode of construction; for, rail- road officers in the majority of instances will be led by the consideration of first cost; es- pecially since, in bridge building (thanks to our factor of safety) many errors remain un- punished for a long time, continuous girders with their delusive theory and deceptive stiff- ness under application of lattice and rivets would gain a wide market. plate, and which even under those suppositions do not coin- cide very satisfactorily with experiments." u Notwithstanding the theoretical improvements men- tioned, it was finally found that the labor spent in finding said formula had been in vain, from a reason which in Eu- rope, as far as known, has not received any consideration. It is the great variability of the modulus of elasticity, which in the formulae of the books is supposed to be a constant value of about 25,000,000 pounds per square inch." " But the writer has tested , during his presence at the Phoenix Iron Works, many thousands of eye-bars, made for actual use in bridges, and found that the modulus of these members is very changeable, namely from 18,000,000 to over 40,000,000 pounds per square inch, so that small sections give the lowest and large sections the greatest figures. The same result was obtained by the Canadian engineer who inspect- ed the iron for the International bridge near Buffalo, as well as by Mr. B. Nicholson, who was sent to Phoenixville by the government officers of the United States to inspect the iron for the Mississippi bridge at Rock Island." 14 The theory of continuous girders, as given in text-books, does not always permit the philoso- phy of the principle involved to be clearly seen : its representation generally is rather obscure. In order to explain this principle as clearly as possible, the author worked out a new method of treating the subject. The results under this treatment naturally must agree with those derived from the application of the general theory of the elastic line which, in the last century (1744), was first given by Leonard Euler of Basel, then member of the Academy of Science in Berlin, which, by Navier, early in this century, was propagated among engineers, and lately was somewhat simplified by Henry Bertot in France.* * Jacob Bernoulli! having in the year 1695 given ike no- tion of the " neutral line," tried in 1705, shortly before his death, to find the equations and properties of the " elastic line." In this he did not succeed, but Leonard Euler in his book "de curvis elasticis" (Lausanne and Geneva), 1744, solved this problem, showing that for flat elastic curves the second differential co-efficient is proportional to the moment of flexure of exterior forces. P. S. Girard, in his work " Traite analytiqiie de la resistance des solides, 1798," page 50, &c., translated Euler's treatise from the Latin into the French language, and he adds as an application of Euler's theory the investigation of a beam fixed at both ends. Ey- telwein and Navier extended this labor to the beam continu- ous over three and more supports. Henry Bertot in France, in the year L855 (Comptes Rendus de la societe des logo- 15 I. THE GENERAL PRINCIPLE INVOLVED IN THE THEORY OF CONTINUOUS GIRDERS. We first consider a number of single spans of the lengths 1? 4> 4> 4? touching each other re- spectively over the piers B> C, D, E. We suppose each span to be loaded in any con- ceivable or desired manner ; in consequence, each span would deflect so as to form certain curves as indicated by dotted lines. The lower chord would not remain straight, the end-posts would not remain vertical. Differ- ing with the nature of the material with the sectional areas of the members of the bridge, with the loads imposed upon them, the trusses would show certain angles y iy o\, j/ 2 , # 2 , ;/, nienrs Civils cle Pans, page 278, &c.), tor the iirst time gave what is called ' the theorem of the three moments," which later (1857), and independently of Bertot, was found by Cla- peyron and Bresse in France, and by the English engineer, Heppel, in the year 1858, in India. In the United States Colonel Long, the well known in- ventor, in the boos on his patent (1838) truss, edited in 1841 in Philadelphia, probably for the first time has applied the principle of continuity to wooden skeleton trusses and num- bers of such wooden bridges since then were built in this country. Mr. Pole in 1852 applied the theory to a bridge over the Trent, consisting of two continuous spans of 130 feet each, and this engineer also did the analytical part of the calcula- tion of the strains of the Britannia bridge, such as contained in Mr. Edwin Clark's famous work. (Vide Transactions I. C. Engineers, Vol. xxix.) to dg, &c., , E, and press apart the bottom joints of these posts, so that not only the top but also the bottom chords of the adjacent spans would touch each other; or, in other words, insert certain tensile forces in- to the upper chord, and equally large compressive forces into the lower chord of each truss. Thus each truss by a certain unknown moment of flexure would artificially be bent up- ward in such a manner that certain angles a l9 /?,, ^ 2 , /? 2 , a v fis? would be produced, were the dead and the . live loads of the trusses removed. When at each central pier, the desired continuity, consist- ing of connection of the top and bottom chord ends, separ- ated under the dead and live loads alone, but overlapping each other un- ^--- der the action of the mo- ments M 19 MV M 3 , , hold- ing the truss-end A to the pier. Though Fi, 3. / the force />, holds down the end A, yet the moment M^fli will cause a convex elastic curve, so that the end posts which originally were vertical, are made to form 20 angles a and /?, to their vertical positions. Since a moment of flexure can only be neu- tralized by another opposite moment, there must exist another force, p l9 acting on the pier .Z?, which in "combination with -\-p^ on the lever Z, equals precisely M l =.fh 9 in other M words, />! must be equal to - ; - and M^=pJ, r *i For any section (7, of the beam A J3, the moment of flexure is equal to the force p l9 multiplied by the distance x, and the greater x is, the greater the moment of flexure in C. When x becomes equal to 1 19 the maximum of the moment is reached, namely J/j=^) 1 ^, whilst at A, the moment of flexure acting on the beam is zero, because x is zero. And as the curvature of a beam increases directly as its moment, the beam is not bent at all at A, but is gradually bent more and more the nearer we come to J3. From what has been said, this law can be deduced: that the application of a moment M v , to the central pier of an end truss span of a continuous girder does call forth two forces +/?!> p^ which, however, do not alter the sum of the reactions A and IB of this span in whatever manner it may be loaded. 21 The moment M, reduces the pressure on the pier A, but only by increasing with the same amount p l9 the pressure on the pier B. From this observation it further follows that by the principle of continuity, no load rest- ing on an end span can be carried over to the next span, but that the sum of these loads always is neutralized by the two nearest piers between which it acts. The distribution only of the reactions, which for single spans is governed by the law of the lever, in the end spans of continuous girders is modified. What has been said of a single span acted upon by one moment M iy is equally true, if on its other end another moment Jf 2 , would act. All we need do is to add together the effects due to each separate moment. There- fore, also, any load acting on a middle span of a continuous girder is taken up by the two nearest piers. Also, in this instance, the sum of the two partial reactions belonging to this span on these piers, equals the total load between them. Only the proportion between these reactions, by the principle of continuity is modified. This result could have been anticipated from the following consideration : The strut 22 C D, Fig. 4, carries down to the pier D the loads due to the span G. As long as the bearing D is inelastic, the diagonal D E can not be drawn down and the vertical pressures carried by the members C D and E D must be directly annihilated in D. The case would be very much different if the pier D were elastic,* for then there would arise a deflec- tion of the truss G Fin D, and a portion of the shearing force from one truss could travel o to the next one. Having learned that the moments M^ J/ 2 , J/3, cause the existence of pairs of forces + Pi Pv +P P + PB Pv + P* Pv /, so that the (n-l) mo- ments are sufficient to calculate the n forces ^/^ . . . p n . After these prepar- ations we proceed to the values of the angles a\ _ft l y l 6 l ; 0/ 2 P2 Y2 V 2 5 ^ C< 5 & C> s^ ^ ^ Can these values f >> rft C > --i 1 moduli I s a s S a "p .2 6 S ' fl 8 J| T3 |.S rS C I . s a a *S CO CO J ^ 8 ^ 1 O 1 ^CO H O ^ , sa O 5 c^ O ^ * s s 00 >i-H S rs o s 6 d QJ ^H ^ 3 8-t3 a S g 3 ! fl ** rrt " S 00 O rj 00 ^!I? ^ *"< JlH 3 -a EH I 'o 'g a H cr 4! 5 > 6 contains 0. modulus th At the Vienna Exposition, a set of test- pieces could be seen,* which showed as fol- lows : Carbon per cent 1. 0.75 0.5 0.28 012 Specific gravity 7.837.84 7.85 7.86 7.88 Tensile modulus (mil- lions pounds. .. 25.1 24.6 27.7 24.9. 26.1 Ultimate tensile strength 90 000 80 000 70 000 67 000 65 000 pounds The modulus of this class of steel and iron was, in the average, noticably lower than those for Ternitz iron and steel, the difference being about 20 per cent. B. Baker made some, experiments on steel bars previous to his experiments on crippling strength and found the modulus from 29,100, 000 to 37,330,000 pounds, which result shows a difference of 28 per cent. He saysf " Every practical man who has noted the behaviour of iron girders under bending stresses, knows whilst one girder may deflect a certain amount under the test, another one precisely similar and placed apparently under precisely the same condition, may deflect some 30 per cent, more or less." Hodgkinson made two direct experiments * Bessemer metal from the Reschitza Works in Hungary. t In his book on Beams and Columns. 36 on the compression moduli of iron, and found 19,200,000 and 21,000,000 pounds. These two experiments strengthened Hodgkinson in his belief of the correctness of his theory as to a weakness of wrought iron under crushing stresses, whilst they only prove how easily an experimenter may be misled. Du- leau, in France, directly measured the com- pressions of fibres in comparison with their extensions, which he, differing from Hodg- kinson, found to be exactly equal. Very valuable hints as to the qualities of iron can be derived from experiments on flexure, which can be conducted easily with sufficient accuracy. Morin, by such, deter- mined the following moduli : For iron from works near Rouen 31,800,000 pounds. " " " Jackson Petin &Gaudet... 28,400,000 u k . u Ale , Lik (Algeria) 28.960,000 " English crown bars 23,440,000 " French I beams with equal flanges. . 29,330,000 " * " " u unequal " .. 24,400,000 " beams also from Dupont & Drey- fussin Ars sur Moselle 26,00,0000 " " beams also from Dupont & Drey- fussin, equal flanges 23,600,000 k ' ** beams also from Dupont & Drey- fussin, unequal flanges 23,000,000 ** " beams also from Dupont & Drey- fussin, same beam reversed 23,000,000 " Here again we have differences of moduli 37 amounting to 39 per cent., and for the same class of iron (Lorraine beams) of 27 per cent. Thomas D. Lovett* has lately furnished an elaborate series of experiments on compres- sion members, such as actually used in the bridges of the Cincinnati Southern Ry. Up to the time of his report t 30 compression members had been tested and broken; their moduli varied from 19,300,000 to 34,600,000 pounds. J Experiments on hollow wrought iron tubes made by Hosking gave these results : Moduli of a rectangular tube 20.405,000 pounds " " round tube 24,500,000 " " elliptic 24,300,000 " Moduli of rails, experiments made by Mo- rin: Tredegar iron, double headed, maximum modulus 27,730,000 pounds Vignole's French rails, average modulus. . 26,400,000 *' Dowlais rails, double headed minimum modulus 21,100,000 " * Consulting Engineer of the Cincinnati Southern Ry. t November 1st, 1875. + These experiments, which conclusively prove the supe- riority of the American system of bridge details, are very complete, and will, doubtless, attract much attention. There were below 20 millions pounds 1 modulus from 20 to 25 " " 7 moduli " 25 to 30 " " 15 " 30 to 34.6 " " 5 the greatest difference being 31 per cent. Morin believed that the great variations of moduli (even of rails of same section and make) should be explained by the quality of the iron, and he judges that the better metal should show the higher modulus. But the great variations also of moduli of bars of un- doubtedly excellent make and of great uni- formity seem to disprove his judgment. He states that he has met with moduli, as low as 17,000,000 pounds, while the author has ob- served 18,000,000 as a minimum.* * General Morin acknowledges that it is pretty difficult to determine with exactness the average value of moduli to suit the results of old and new experiments, and he says: But, moreover, it must not be lost sight of that it happens pretty often that iron bars of the same manufacture, fur- nished by the same works, present notable differences in their resistance to flexure. The distinguished French officer proposes a classification of iron of high grade (average modulus of 30,000,000 pounds), ordinary iron (modulus 25,000,000 pounds), and soft ductile iron (modulus from 21 to even 17,000,000 pounds). But this classification can hardly be upheld, since the very best irons (for instance, Swedes, Russian, &c., brands) also are the softest and most ductile (and as regards ultimate strength somewhat weak) irons which, according to the classification, would have to belong to the highest and also to the lowest class. The experiments on eyebars, now parts of existing bridges, as made by the author, the iron being double refined (best-best) have given moduli from the low- est to the highest class. There seems to exist this law that the moduli of bars of same section made from double refined iron bars (rolled three times, packeted and welded twice), such as called best-best, are more uniform than bars made from best iron, such as were used by Herr Malberg in the Muhlheim suspension bridge. At least, many thousands of bars tested at Phrenixville, Pa., proved to be remarkably uniform in their moduli as long as they were of the same section, whilst the moduli were very variable when bars of different sections were compared. On the other hand, Styffe's experiments, which were made on excellent steel and iron, gave a maximum tensile modulus of 34,584,000, and a minimum of 27,585,000 pounds, which is similar to that of an iron rail from Avon, in Wales. Morin's experiments on flexure of unhard- ened steel gave the following results : Maximum. Minimum. From Petin & Gaudet, refined. . . 28,800,000 28,100,000 " " " puddled.. 31,800,000 29,200,000 " " u crucible... 32,300,000 29,200,000 Kvupp's 32,200,000 28,700,000 " mean of 17 experiments.. 30,300,000 English 28,900,000 It should be noticed, that even for the finest metal that we know, such as crucible 40" steel, the variation in its modulus amounted to 11 per cent., and for the renowned Krupp steel, 12 per cent. The most accurate experiments are still to be mentioned (those of Herr Woehler), made by suspending the test-piece A A, so that it Fig. 7. I * B (7 Si could expand freely ; the lever L L carried the load P and the arms A B, A B were ex- actly equal; the piece B J?, acted upon by a constant moment of flexure, carried the meas- uring apparatus C v, v C, by which the de- flection of C C could be very exactly found. In this apparatus the deflections of the test- pieces become comparatively very large, and piece C C was free from the influence of knife edges. These are Herr Woehler's results : Modulus of iron from Laura huette 28,930,000 pounds. " " Phoenix huette.... 29,360,000 " " " Minerva huette. .. 31,680,000 " Low Moor iron 31,230,000 " 41 Modulus of "homogeneous iron from Pear- son, Coleman & Co 32,340,000 pounds. Bochum steel 32,000,000 ** Kmpp steel ...31,600,000 * 4 All these materials were of unusually ex- cellent quality, and the maximum difference still was 12 per cent.* The same variability which we have found to exist between iron and steel, not so much as to the quantity of carbon contained, as from imperfections of manufacture and other causes unknown to us, was noticed with the shearing or torsional moduli. Thus Duleau found for iron from Perigord, moduli of 14,450,000, and again of only 7,980,000 pounds. Iron from Arrieges gave 8,450,000, English Iron 9,860,000, and also 12,800,000 * Redlenbacher quotes the moduli of iron from 21,300,000 to 35,500,000 pounds, of steel from 28,500,000 to 34,100,000 Ibs Reuleaux (Der constructeur) for wire bars and ordinary steel gives 28,500,000, for cast steel (crucible steel) gives 42,700,000. Kupffer, in St. Petersburg, by experiments on sound and flexure of small specimens, gets from 25,000,000 to 30,000,000 pounds. Coulomb, Tredgold, Lagerhjielm and Woshler found the modulus of hardened steel exactly equal to that of unhar- dened steel. Kupffer in some instances finds the modulus of hardened steel 6) p. c. higher. Styffe finds that the modulus of cold worked iron is low, but can be raised by exposure to a glowing heat. He also says that phosphorous lowers the modulus. 42 pounds. Wiebe in Berlin quotes the shear- ing moduli thus : Soft' wrought iron 9,000,000 pounds. Bariron 10,250,000 " Steel 9,000,000 " Finest cast steel 14,000,000 " These figures prove that we cannot know the shearing modulus of any class of steel or iron without direct special experiment. Our conclusions with reference to the sup- position of a constant modulus of elasticity for the calculation of deflections and of con- tinuous girders are from known and un- known causes : First, .plain iron and steel bars vary in their moduli very considerably. The smallest modulus of iron was found to be 17,000,000, the maximum above 40,000, 000 pounds. Single refined bars of same stock, manufacture and section vary in their moduli by 35 per cent. Double refined (best- best) bars vary little, as long as bars of same section are tested, but considerably with the sections, the minimum being 18,000,000, and the maximum above 40,000,000 pounds. The moduli of rails vary by 30 per cent., and similar results must be expected from com- mon angles, beams, channels, &c. Second, consequently, riveted bridge members com- 43 posed of angles and plates of various thick- ness and manufacture, interrupted in their homogeneousness by punched holes, covering, reinforcing splice-plates, &c., must neces- sarily show still greater variations in their moduli than was found for plain integer bars. Third, the hypothesis of a constant modulus of elasticity of the material of a bridge being unfounded, the theory built on such hypo- thesis should be abandoned.* Having arrived at such conclusion, we nevertheless must expect to hear an objection against its logical consequences, namely this that numbers of continuous bridges do good service in practice. So they hitherto have done, not because the principle of conti- nuity is admissible, but because the factor of safety used in their construction has hidden the error made in their design. For the same reason, the Victoria bridge in Canada stands, which is made continuous, but simply by * Mr. Baker most pertinently remarks with reference to continuous girders : " The most expert mathematician would have to devote a month or more to the preliminary calculations of a very ordinary bridge, and the result de- duced would not after all be more reliable in practice than those arrived at by comparatively simple modes of investi- gation, chiefly on account of the varying elasticity of different portions of even the same plate of iron." 44 combining two single spans whose greatest chord-sections are in their centres, whilst the greatest chord-strains, according to theory, would fall where the cross-sections are made the smallest. For the same reason, conti- nuous draw bridges stand, which we find composed of two halves, each designed as a single span. The author knows of one instance, that a Hodgkinson cast iron beam was put in place upside down, so that the heavy tensional flange was under compression while the com- pressional flange of only one-fifth the area of the tensional one was strained under tension, and yet, on account of 'the factor of safety, the beam stood. III. OTHER DEFICIENCIES OF CONTINUOUS GIRDERS, as regarding the imperfections of the theory, the danger from defective manu- facture, from settling of piers, and the in- crease of strains by the action of the heat of the sun and the omission of the influence on the strains caused by deflections due to the web systems. In order to find the exact extension or com- pression of a member of a bridge, we must know not only the modulus and the total strain of the member, but also its cross-sec- 45 tion. The problem of continuous girders, however, is to find this very section. The theory assumes that all sections are equal, or at least that the moment of inertia of a girder or a bridge is a constant throughout. Under this supposition we get smaller chord strains over the middle piers than exist in reality. In the case of two equal continuous spans under full load, with uniform moment of inertia, we find the moment of flexure over a middle pier to be equal 0.125 I 2 p, where p represents the total load per lineal foot, and I denotes the length of each span in feet. But if we suppose that the same bridge, under full load, shall be strained equally per square inch, the co-efficient 0.125 becomes 0.1464, which indicates strains over the mid- dle piers 15 per cent, higher. In reality, the continuous bridge being not perfectly varied in chord sections, the difference will be less ; but it may be remarked that the chords of a continuous bridge, properly designed accord- ing to specification, would only be about 10 per cent, lighter than those of equal single spans. With an enormous amount of labor, this deficiency of the ordinary theory can be corrected, and it has been done in a . few 46 bridges. But considering the irregularity of the moduli, such labor seems superfluous. A serious cause of errors in the construc- tion of continuous girders refers to the distri- bution of strains over the posts and ties in case that two or more web systems have been adopted. In a single span bridge, a load brought on a panel joint of one separated web system, being split into two shearing forces in accordance with the law of the lever, there cannot be any mistake about the strain in a web member, as long as the end posts are vertical, and if they are inclined, the error can amount to only one increment of one panel load. The problem of web strains with conti- nuous girders depends not only on the law of the lever, but also on the angles of deflection a, ft, y, 6 not only of one, but of all spans together. We remember that by the moments M^ MV MV &c., forces +p^ p <&c., were originated, which disturb the law of the lever. If, therefore, in a continuous bridge there are two or more web systems, we are utterly ignorant as to the distributipn of the reactions over the two or more systems which, at every end pier and at every middle pier are connected. How much of p iy p^ p^ do not change their lengths. Fourth. The web systems of the girders are so arranged that there is no doubt of the office of each separate system of struts and ties, which con- dition can only be fulfilled in case of but one system of diagonals and posts. Fifth. The temperature of all members is alike, and can- not change in any separate member. We use this notation : E is the modulus of elasticity in pounds per square inch, which, as known, is the measure of stiffness of material, the greater the modulus or the less the value of -j-j the less proportionally are the elastic de- formations. I is the moment of inertia of the girder, equal for a skeleton truss, to the cross-section of one chord multiplied by one half the square of depth of the truss, all dimensions taken in inches. Like E, the value I stands in inverse geometrical propor- 61 tion to the deflection of a beam or girder. Pv Pz Pn+i denote the elastic reactions in pounds caused by the unknown moments N <8i? / y^- rfW^ )//j///cyFM(nwrz \ \ ' k 11 'ft* 1 II I / 1 11 / <^^ / -4f --O - - "/1{ over the middle piers of continuous girders. ^ / 2 / 4 . . . i n are the lengths of the spans in inches, conse- quently M l MZ - -^f n -i must be measured in pound inches. The above figure represents a truss A B, which is supposed to be acted upon by no other forces but the pair + ^ S 19 which create a moment M= Sh counteracted by a force p in A. This force p in combination with p in B on the lever / (= A B) has the tend- ency to turn the truss A B in opposite direc- tion to My and to produce equilibrium; con- sequently pi must equal M. The sum of the horizontal as well as of the vertical forces being zero, no movement of the truss A 13 will be possible; nevertheless its elasticity will cause a flexure which increases in curva- ture from A to B. This is due to the mo- ments of flexure increasing in geometrical progression from A to B, which moments in the triangle A B C are represented by the straight line A C. The maximum moment occurs at B and is =. M^=. Sh"=.pl. For any distance x, from A the moment will be MX =. px. The above figure also represents that the 63 tx) tal strains in the chords increase in geo- metrical proportion from A to B. At B the total strains will be S and 8, in A the strains will be zero. The chords being sup- posed to be equally strong in section, the strains per square inch likewise increase in a geometrical progression from A to B. The web strains, however, remain constant through the whole girder, because, according to the nature of this problem, the shearing force has a constant value ==/>.* We know that the expressions for the an- gles a and y6> must contain E and I as divi- sors, and I and the maximum moment as multipliers, so that we only need find the co- efficient to this expression. Actually the development gives: __ ~ 6 E T~6 E l) and } (III.) so that ft is twice as great as , the angles ft and y# 1? are known as well as the angles a and or r The angles q> -f- ?/? together must equal ft + ft^. (Consider that (p + if> + angle D = 180, and that + /? t + D also = 180). The angles of deflection being very small, can be considered as equal to their tangents, namely : d d (p -- ; tp ---T / and cp : ip - b : a. But on Ma Mb the other hand ft nr-and ft l = - so 67 that /3 l : fi=b : a 9 and sine?

> apparently /? t ^? and ft ?/?, so that sim- p!y : __ ff M a . M 6 M 6 Jlf a Pab (2 In case the girder A 13 should have carried any number of loads, P l _P 2 &c., with distan- ces ! ^> 1? 8 2 , 3 5 3 , &c., there would have been 1 v f I ^ e sum ^ a ^ i ~~ 6 1 E~J II expressions ) , 2 Pa b (a + 2 b)} )= 6 / A 7 1 f ( the sum of all | ~~ 6lTT~i ^11 expressions ) Now we are prepared to write the final formula of continuous girders. The equil- ibrum of the moments M 19 M a and M 8 with the forces P 19 P 2 , &c., and Q 19 Q Z9 3 , is found from (Eq. /.) # 2 + y 9 =n // 2 -|- a 3 where 68 /, X -1 2 ~- - ?3 $2 -f- y s are the angles of deflection due to angles of elevation due to M 1? M, and M 3 of the spans considered as single ones. Their values are : tf, = -==-= 2 [Pa b (2 a -f- b)~\ for span / 2 , 7s = -Q-JzTj' 2 [ 6 ^ ( 2 )] for span / 8 . __ P * ~~6 E I~ consequently 6 J7 J (tf, + . _. E I> 3 ~6 (I, ( which actually is the equation of Henry Ber- tot ; also : ~ 2 [Pa b(a + 2b)+~- 2 [Qab (2 a + ft)] = M, I, + 2 Jf f (/, + 3 / 3 ) + M, 1 3 ( VI.) This equation is of the first degree and con- tains three unknown quantities, viz.: M 19 M a , 69 M 8 , whilst the expression on the left side is fully known since the loads JP 19 P^ P^ *~ d <*-i ) . \ I'm ' the positive sign to be taken if the leg of the angle is below and the negative if the leg is above the horizontal line through the middle pier under consideration. In all other re- spects the problem is the same as the one we have just described; namely, this is the gen- eral equation: 6 E I (y m + ;/vh) = M^ l m + 2 M m For n spans, there are (n 1) equations of this kind, and the moments M and M n are equal to zero. The values y are to be sub- stituted with their proper signs. In the special instance of two equal spans e?, being an elevation of the middle pier above the line A, C, we have Fig. 12. I* / ^ 7 _. * * * .- li' it ' 2 = Oj o 2 =.- --. Both angles are > the line A C, and consequently posi- tive; we therefore have 6 E I ( d^\ 3 E T d so that this elastic end reaction becomes p= 3 E^Id (FZZZ) If d had been negative, M would have been a moment, causing (above the middle pier) pressure in top and tension in bottom chord. We now proceed to our last theoretical problem, namely, to calculate the influence of heat on one chord. Suppose, therefore, a properly manufactured girder resting on sup- ports A B C D, etc., with the bottom chord covered by floor-planks: the top chord ex- pands by the heat of the sun, the difference 75 of temperature between both chords being t degrees Fahr. In this instance, the uniform- ly heated top chord will expand 150,000 of its length for each degree Fahr. If the girder is first considered to be without weight it must assume a flat arc, whose radius is easily found. Two posts which originally were Fig. 13. V ^ 150,000 -, parallel have spread apart of the pan- t el length, and consequently 1 : p : : : h or p = 150,000 --. But the radius p be- t ing found, it is easy to also calculate the ele- vation of this flat circle above each middle pier, and this known the problem is at once reduced to the previous one. Especially for two equal continuous spans there if an elevation of the girder equal to I 2 , which is the natural position of the girder 2p considered without gravity, the bed plate be- Z 2 ing placed - - below the bottom chord ; we 2/J 2 have theref ore d= - ^^ and Eq's 7? T t nmches),Jf= - P ~ ~ 1,200,000 I h ' where the minus sign indicates regarding M, that the moment causes compression in the top, and tension in the bottom chord and regarding p, that the end piers really are pressed by this elastic reaction; in other words, that p increases the pressure on the end piers as caused by dead and live loads on the girder. Dead and live loads, however, actually press down the girder to the middle pier either partly or wholly. The moments of correction, M 19 J/" 2 , M s , fi> are not influenced by the deformations due to the web systems, which assumption was about justified in the calculation of homogeneous plate girders such as we know to have been first used in Eng- land and France. Is such simplification of theory also jus- tified in case of continuous trusses of great depth ? It is impossible to investigate by direct analysis this cause of error. For we do not know the sections of the web members, nor can we consider them of equal value, nor could we estimate this unknown quantity even if we would assume it as a constant value, as was done with the unknown chords. 78 All that we could do would be : first to pro- portion a continuous bridge under considera- tion of the chords only, thereupon to calcu- late the correction due to the web and then make another calculation founded upon the corrected sections. In this manner with a great deal of labor we could finally succeed to proportion a continuous bridge properly. This labor indeed would be immense. We now shall develop the necessary for- mulas towards consideration of the deflections due to the web system of continous and other trusses. Fig. 17. Let A B C be a triangle whose sides a b c have been altered by very small quantities Z/a, A b 9 Ac; the problem is to find the alter- ation A oC of an angle. We have a* = 2 + c* 2 b c . cos. OC, which, by inserting the differences, leads to (a + A a) 2 = (b + A b)* + (c + A c}* 2 (b + A b) (c + A c) cos. (oC Moc). By developing this equation and consider- ing that the squares of differences are very small quantities in comparison with their first powers, we get: 79 a A a=b A b -f c A c (b A c + c A b) cos. OC -f b c sin. oC ^ ex Hence we derive the value of A oG aAa bAb cAc-^tyAc+cAb) cos. QC b c . sin. oc By applying this formula to a rectangular triangle the formula is simplified into : Fig. 18. A a Ad a A a = - ---- .- . -=- b d b 6 a A R _ * Ab A d jl_ b~ ~jjT ; ? a b } The sum of A oc -f A /3 -f A JR = 0, as ex- pected. These few formulae (l) are sufficient to cal- culate the angles of deflection at a joint of a properly built skeleton bridge. . 80 Fig. 19. Fig. 19 represents part of a quadrangular truss, whose panels have the length, c, whose height is h, and whose diagonals are of the length d. The truss being un- der transverse strain, receives alterations of the lengths of its members and consequently of its angles. The angle A originally was 180 degrees, we now have to calculate the small alteration y\ of this angle. The alter- ation is the sum of the alterations of the three angles around A, namely: A c n _, A c l n A h nl A h n ~] h - - A he In this expression A c n h (2.) repre- senting the influence of the chords c and c 1 , is a sum, because, if 415 X 20 = ~- 2,966,000 pound feet. Any moment M m , is found by considering the m panel loads acting at their joints. There is namely, in accordance with the law of the lever: M m 40,170 X me? (1+2 +3 + otcr^. -^-i < CClOOO^CSC^COGOO SOffl-^lOt^ P ocicjxoioa O H fc I Q I Q -tj H q s s Slilllll ilii &8^iS: |g glilil ;n5ooS3Scoocoao s^S. 92 from the value A = 40,170. This operation is represented on the diagram of shearing forces. Fi. 14. Loaded Yimrmet^ I*- 4 (b.) Maxima live shearing forces acting in the direction of A. For this purpose, the second span is supposed to be unloaded, and a train to extend from the panel m to the middle pier. Fig. 15. Oh**** ^,-r | 1 _= <.. /__ ^ M (c.) Maxima live shearing forces acting up- ward in the center. For the calculation of these forces the second span is supposed to be loaded, and the first span also loaded from the end pier to panel m. The bridge of which we are calculating the 93 forces is shown with two web systems. It has been explained* why it is impossible to calculate exactly, the strains occurring in each one of these systems. It must now be added that this uncertainty also necessarily attends the chord strains, whose determination is especially difficult in those cord pieces which, at each passage of a train, have to bear com- pression as well as tension. There is no method to overcome this imperfection of the theory. In the following calculation we have separated the systems, and have supposed that each system would act independently. For the calculation of the forces V c there arises another difficulty. The value p name- ly, is inflenced considerably by the total load on the second span. Two methods are possi- ble, either to combine the system, 1, 3, 5, &c., 11 of the first span, with the system 1, 3, 5, COC31CD * OS -a en os cc -q CD CO CO GO GO OS 4^ h-OOOOSCOI- tOCDOSCOl-'' OH-ccoo-a- GO OS ~3 CO CO e H K IT 1 O tocoosoc^ i ^ H doco^osi-'; ^ oM OOCOGOGOOO'l iC i* H-o-a^oi. j Og ~~ i J? w ; c oa 95 The shearing maxima A and V are now to be combined, so as to obtain the total max- ima in each panel, such as represented on the diagram of forces. For the calculation of the diagonal and post strains, the two sys- tems again must be treated separately. The diagrams for the chord and web strains in combination with the two tables referring to the web strains of each separate system, can now be used to calculate in the usual manner the members of the proposed bridge. For this purpose we choose a height of 25 feet (one-eighth of the span) and after con- sideration of the web strains in the end pan- els we arrive at the diagram of strains repre- presented. An examination of these strains will give proof that the chord strains cannot be properly determined without consideration of the diagonals, and that consequently the mere theoretical comparison of curves of mo- ments and shearing forces may lead to con- siderable errors. Top and bottom chords of continuous gird- ers after all, cannot be calculated with per- fect certainty, even under the objectionable suppositions made. It is, therefore, advisa- ble to construct them to resist tension as well 96 as pressure. A proper section for these chords would be two built channels connect- ed with lattice bars at top and bottom. The pins can be put with mechanical correctness through the centre lines of these channels, and the re-enforcements can be placed so that the pins bear against the metal added to the web plates. We construct the diagonals of weldless eye bars, and the counter rods with swivels. This arrangement has a scien- tific advantage. Each web member carries but one kind of strain; whereas, in bridges with diagonal web members only, diagonals, at least near the centre of the span, have to resist tension as well as pressure, and there- fore must be designed to sustain the sum of both. Moreover, vertical posts are more con- venient, with reference to the construction of the joints. The built chord channels are calculated to be 16 inches deep, and the angle bars 3x3 inches, the latticing to be double top and bottom. The posts are also designed on this basis, with 2 rolled channels and latticing; their bearings are made flat against the bot- tom and top chords; the radii of gyration have been duly calculated. 97 The strains upon compression members are in exact agreement with the formula, the ( H* 1 factor 1 X rTTT^ increasing from 1.12 v 5,000 ) of chords to 1.82 of posts; the section of the lightest post is taken at 8 square inches. On this basis the sections of chords, diago- nals, posts, counters, &c., have been determin- ed in agreement with the specification. From the strain sheet thus obtained (see Plate) this bill of materials is calculated : Chords, latticing, joint and reinforcing plates 85 912 pounds. Posts with latticing, top and bottom bear- ings, rivets 35 560 " Diagonals and swivels 43292 Pins and rollers with cages 3 850 Cross-beams, hangers and washer-plates 16 000 Stringers, 25 600 Struts and portals 6 000 Lateral rods 4 000 Castings (end post feet and heads bed- plates, &c,) 5 000 Floor bolts and washers 3000 Total weight of one iron 200 feet span 228 214 " Iron, per lineal foot 1.141 pounds. Timber and rails 300 " Total dead load, per foot 1.441 " Assumed weight per foot.. 1.200 " A too light dead load, therefore, was as- sumed ; but the error amounts to less than 98 5 per cent, on the truss weights proper, say about 5,000 pounds in the span. The cor- rected weight of the iron-work of this con- tinuous bridge would then amount to 1,166 pounds per lineal foot. For the sake of comparison under precisely the same specification, for the same form of truss, for the same details and the same num- ber of panels, a 200 foot single span has been calculated. This is the strain sheet with data and mode of computation ; * Span 200 feet ; 12 panels, 16 feet 8 inches long ; diagonals for 27 feet depth, 31 x and 43 feet ; secants, 1.17 and 1.59 ; tangents, 0.61 and 1.23 ; dead load, l,200X- 4 = 10,000; live load, 2,240 X^r= 18,666 24 pounds per panel ; excess of locomotive load on a joint, 12, 444 Chords. 3X0.61X28,666=53,000; 2X1- 23, X 28,666 =70,600 ; IX 1-23 X28,666 = 35,300; 2KX 1 - 23 X 28 666 = 87,000 ; * This example will show the wide difference as to the time required for the calculation of the strains of a single span bridge compared with continuous bridges* 99 g QQ r From dead " live " locor 88S fff o'^ ' Maxima Diogonal From dead ' live * Maxima Diagona 100 I 1 /! X !- 23 X 28,666 X 53,000; /, X !- 23 X 28, 600 = 17,600. Addition. 53,000, 87,000, 70,600, 53,000 35,300, 17,600 ; whence the chord strains, 53,000, 140,000, 210,600, 263,600, 299,000, 317,000 pounds. The compression members are designed first, as done with the continuous span ; and second, to consist of hollow segment columns. The same sections are adopted for both cases, but with the hollow posts we gain all latticing and still have a greater factor of safety in regard to ultimate strength. BILL OF MATERIALS. Top chords 49,500 pounds. Bottom chords 28,000 Pins and rollers 4,500 Posts 34,000 Diagonals 37,000 Cross bearers, &c 16,000 Stringers 25,600 Lateral struts and portals 6,000 Lateral rods 4,000 Castings 5,000 Floor bolts 3,000 Totol weight of iron 212,600 Weight per foot 1,063 Timber and rails ... 300 Total weight per foot 1,363 Weight assumed 1,200 101 Hollow colum chords 39,000 pounds. Bottom chords 28,000 Pins and rollers 5,000 Hollow posts 27,300 Diagonals 37,000 Cross bearers 16,000 Stringers 25,600 Lateral struts and portals 6,000 Lateral rods 4,000 Castings 12,400 Floor bolts 3,000 Total weight of iron 203,300 Weight per foot 1,017 Timber and rails . 300 Total weight per foot 1,317 Weightassumed 1,200 The weight assumed, 1,200 pounds, con- sequently was too light also for a single span, and the truss weight shoiddjlje iricfreased by ,4 per cent., so that the actual* weights woulcf lie respectively 1383 andfl33^,ppun4^|)^^o<)t; These still are respectively 5 S'ahcTlO^ pounds less than we obtained for the continuous girders. Having now seen that in the construction of two continuous spans there is no economy, if compared with properly designed single spans, it will be well to examine the weights in detail. These following, are the percentages of 102 weights as calculated for a supposed dead load of 1,200 pounds per foot. CONTINUOUS GIRDERS. 25 FEET DEEP. SINGLE SPANS 27 FEET DEEP. Latticed Posts. Hollow Columns. Chords .... 37.7 34.6 36.4 33.3 33. 31.6 Webs Both Balance 72.3 27.8 69.7 30.3 64.6 35.4 This comparison shows that the more per- fect the .detail design, the smaller the per- centage of weight taken up by the chords and '.&ej?/ Tbe sir-gle span, with hollow wrought iron segment columns, gives the best result. The single span, with latticed posts, is super- ior to the continuous girders with latticed posts and chords, for the chords and webs still contain 2.6 per cent, less of the total weight in the first design than in the second. This is not alone due to the height, 27 feet, of the single span, for an increase in height would hardly reduce the chords of the continuous 103 girder, since 5 /i 2 f these chords cannot be re- duced in section without lowering the heights of chord members, and therewith reducing the admissible chord pressure. The panel length being taken at 16 feet 9 inches, the truss height could be increased to 33 feet without losing weight in diagonals, but the posts would become considerably heavier. The maximum height of a truss is reached if an increase in height causes an increase of total weight ; that is, if by an increase of height the web and lateral bracing in- creases more than the chords decrease. The most perfect compressional members permit the use of the greatest depth, since the weight of the posts is a large part of the total weights. The single span, with hollow wrought iron segment posts (Phoenix col- umns) therefore, has the smallest dead weight. From the variability of strains in their chords and webs, continuous girders require contin- uous riveted chords. Under this construction, loss of material seems unavoidable, because these members cannot be made without it, in practically too small sections at the points where the moments became zero. On the 104 other hand, it will be found advisable in con- tinuous girders to avoid too great a variety of riveted members intermixed with eyebars. This construction has been tried several times in this country with drawbridges, but it is doubtful whether any gain actually is obtain- ed by such design. The continuous girders, such as here de- signed, have one advantage over the fixed span, because the poste have been arranged with two flat ends, whereas the single span was designed with posts of but one flat bear- SINGLE SPAN; CONTINUOUS GIRDER. Latticed Hollow Members. Columns. Posts, pounds 35,560 34000 27 300 Diagonals, * 4 43,292 37 000 37 000 78,852 71,000 64,300 Ratio 1.23 1.11 1.00 Chords Pounds 86,000 78,000 74,000* Batio 1.16 1.05 1.00 * With castings. 105 The foregoing comparison shows that the web of the best designed single span is 23 per cent, lighter than the web of the continuous girder. Theoretically (compare strain sheets) this advantage of the single span amounted to only 12 per cent. Theoretically the chords compare thus : continuous girder 4,403, to single span 5,026, or as 1.00 to 1.14. In other words, for the same height of trusses, 27 feet, though the continuous girder appears theoretically to save 14 per cent, in the chords, in reality it causes a loss. While a oontinous girder of three spans, proportioned according to theory, would show a gain in the chords of 33 per cent., in fact (see Laisle and Schubler) execut- ed examples of acknowledged excellence of design gave only 15 to 20 per cent, and this gain is only comparative since obtained under sacrifice of height, the depth being one twelfth instead of one eighth the average length of the three continuous spans. Having given the practical figures and weights for 200 feet spans, it yet remains to show whether there actually was any theoret- ical advantage in favor of continuous skeleton structures. To this end, from the foregoing 106 tables, the theoretical quantities were calcul- ated, consisting of the products of the length of each member into its maximum strain, re- spectively into the sum of positive, plus neg- ative maximum strains (in case a member has to carry tension as well as pressure). In order also to show that three spans, of what in books is usually claimed as a more economical ar- rangement of length of spans, do not give any greater advantage than two continuous spans, a bridge of 600 feet total length has been calculated. Its outer spans have 1 1 panels of 16 8 ' each, its middle span has 14 panels of 16' 8", so that the same panel length is con- sidered which we assumed in the previous examples. This example will give evidence that Laisle end Schubler are correct in stating that two continuous spans are about as econ- omical as three, and consequently also as economical as an arrangement of more than three span ; in other words, that it is sufficient to confine our calculations to two continuous spans. We get the following theoretical quantities in pound-feet, and hence by multiplying with -r-, and dividing by the unit strain of 10,000 3 107 pounds per square incfe, also the theoretical weight of the trusses. THEORETICAL QUANTITIES. 718,670 I Single Spans 27 7 deep (not deep enough). Webs Two continuous spans Chords 790,500 \ 25'- deep Webs Three continuous spans Chords 25? deep (a little too deep).. Webs Pound, Feet. Chords 837,670 \ Total. Ibs 1,555,340.519 100 100.5 77^830 | 1,563,330 521 : 88^000! 1.708,000570109.7 Per It might be rejoined that the advantages of continuity better present themselves in case of heavier dead loads. Therefore, under pre- cisely the same conditions, but for a dead load of 2,400 Ibs., we have calculated two single spans, and three continuous spans. These are the (N. B. In the three continuous spans the eflect of the heavy locomotive is not yet considered.) 108 THEORETICAL QUANTITIES. I- P* CQ Pound, < Total. .^f <*H Feet. ^_ lb Per cent Single Spans, 200 7 , 27 7 ( 1,130,000 \ 915,000 ) 2,045,000 682 101.4 Locomotive considered ( Webs Three continuous spans \ 25 7 deep j Chords 958,400 \ 1,058,200 j 2,016,609 672 100.0 Locomotive not consid- 1 ered . i^Webs In this instance the continuous girders are designed too deep, and the single spans too shallow, for their proper heights the quantities of the chords should have become nearly- equal to those in the webs ; and by also con- sidering the locomotive load the theoretical advantage of 1.4 per cent, would have been turned the other way. In all these examples, the chords of the continuous girders can be noticed to be light- er than those of single spans, whilst, revers- edly, the webs of continuous girder are heavier than those of single spans, in such proportions 109 that the gain in the chords is just about neu- tralized by the loss in the webs. It must be expressly stated that, if it were possible with continuous girders to save so much in the chords, that this saving, less the extra weight in the webs, would leave a final saving ; this would only indicate a saving in the theoretical value of the trusses. The con- necting parts, as latticing, rivets, reinforcing plates, then the lateral strutting, lateral dia- gonals, and the whole floors remain constant quantities unaltered by the principle of con- tinuity. In our example, about one-third of the iron of the whole bridge is a constant quantity, and a theoretical saving in the trusses of three per cent, could only realize two per cent, on the iron work of the whole bridge. Most writers on continuity, however, only mention the theoretical saving in the chords, without substracting the loss in webs, or without considering the quantity of con- stant weight of iron. The reason why the webs of continuous bridges must become heavier than those of single spans, can easily be demonstrated by the following examination of a uniformly and fully loaded continuous bridge : 110 Be A a span of a continuous brige, uni- formly and fully loaded by p pounds per lineal foot, C D may represent the line of shearing forces, A C being the reaction on Fiff. 21. AC ~p. . (l-X) Ill A, equal to px, and B D being the other re- action, =. (I x) p. It is well known that of a continuous bridge, even if uniformly loaded (for instance by dead load), save in the middle span in case of an odd number of spans, the reactions A and B arising from the load on A B are not equal (on account of the couple of forces + Y Y* see above, part I). Consequently the line of shearing forces CD does not pass through the centre point of A B. The quantity required in the web of span A B is equal to the sum of the triangles A GE + EBDy multiplied with a certain constant co-efficient dependant on the nature of the design of the web system. Since then A E=x and BE^=i I x y there will be the web co-efficient X [ f +(^ a*) f ] This is no constant function ; it has a minimum which occurs for x = l x-=. in other words : The theoretical value of the web of a single span, even for uniform load, is lighter than it would be for a continuous bridge. For single spans there is the web ~ co- l* efficient X ~^> an( i f r two equal continuous 2 112 spans each one is = co-efficient X 0.531, 2 , this being 6.2 per cent, more than for single spans. Practically, in case of many spans the webbing of continuous bridges would only be slightly greater than for single spans ; where it not that the movable load influences con- tinuous girders very materially more than it does single spans. This is due to the fact that the point E in the previous figure, on ac- count of the variable and moving live load, moves over a considerably greater part and further from the center of the span AB, than happens for single spans (compare strain sheet, pages 94 and 99). Now, taken as granted that the theoretical web material of continuous bridges is greater than that for single spans, it follows directly that single spans can always be designed, which require not more total material than that of a continuous bridge. For it is known that, theoretically speaking, in trusses with parallel chords the least web material is in- dependent of the height of truss. Consequent- ly, even if in both systems, the theoretical web material would be alike, all that would be necessary would be simply to make the single span trusses correspondingly deeper. 113 Since, however, the webs of continuous gird- ers require more material than single spans ; there is only a want for a slight increase of height of truss of single span in order to bring it on the same footing with the continuous one. Practically the total minimum is about reached if the web material and the chord material are equally heavy, and this limit happens therefore sooner for continuous than for single span trusses. Hitherto we have developed the strains, sections and weights of two continuous rail- road girders of 200 feet length. We have made the incorrect supposition that the mo- ment of inertia of these girders is a constant value. But in reality the effective sections of the parallel chords vary from 16 to 38 square inches. Therefore, the curves of mo- ments and the values of shearing strains given by our formulae, and on which we based the estimate of weight, are not correct. We have stated that the maximum moment over the middle pier would be 15 per cent, greater, were the moment of the inertia of the girders so varied as to produce an equal maximum strain of 10,000 pounds per square inch of the bridge totally loaded, and this difference in 114 continuous girders with three openings sinks to seven per cent. In the given example, however, under the suppositions made, the difference between actual and calculated max- imum moments is less than 15 per cent. What it actually is, can be determined by the use of a similar corrected theory, but would involve great labor, without bringing us much nearer to the actual strains. Indeed the chords being not theoretically varied, the discrepancy is small as compared with other shortcomings of the usual theory. This tedious correction was made in some instances, as for the Vistula bridge, near Dirschau, and the error was taken into ac-. count in the design of the Krementschug bridge over the Dnieper in Russia. This bridge was designed in 1866, with all possible economy.* It consists of two parallel and separate structures, one for roadway and one for double railway tracks. There are four * By H. Sternberg, Civil Engineer and Professor in Karls- ruhe. Through his kindness the author received copies of the design, strain sheet and estimate, so that the figures quoted deserve the more confidence, as referring to the completed worJt of not a mere theorist, but one of much practical ex- perience in matters of design and manufacture. The bridge was, however, built upon another design. 115 through spans, 118 metres or 3 87 feet between centres, bridged by two pairs of continuous girders. The calculation was based on a dead load of 3,710 and on a live load of 3,600 kilos per meter for each track (respectively 2,480 and 2,400 pounds per foot). The weight of the iron work proper, amounts to 2,340 pounds per lineal foot of each track. There are two trusses, 10.5 meters or 34.5 feet deep, which is a depth of the span. The webs are de- signed as stiffened lattice work arranged in ten systems, the diagonals running at angles of 45. The diagonals of the meshes are 2 meters or 6.56 feet long, equal to the distance of cross bearers of 4 feet 2 inches depth. The rail-stringers are of wood. The following weights have been calcu- lated : Chords of one span of two tracks 946,000 pounds. Webs " " " " " 508,000 " Bracing and floor beams, &c., 360,000 " 1,814,000 Or 2,343 pounds per foot of each track. These weights are obtained under maximum strains of 8,500 pounds to the square inch (600 kilos per square centimeter) both for com- pression and for tention of compressional 116 diagonals and chords as well as of parts under tension, for which latter ones the net areas are considered. The lightness of webs was much furthered by the diagonal system and by adopting the same strain throughout as far as practicable. This lightness of course is secured, but only under sacrifice of certainty as to the diagonal strains and of the postulate of scientific de- termination of these strains. Nevertheless, the Krementschug design is certainly one of the best proportioned and most economical lattice bridges, and gives evidence of its being designed by an engineer who knows how to appreciate expense of manufacture in the mill as well as in the shops and field. The bridge when compared with other European struc- tures is light, and comparatively much ligther than the Mainz bridge on Pauli's plan, so much the more since the latter is designed for 20 per cent, lighter live load and for 33 per cent, greater strains. Yet compared with single quadrangular trusses, designed with the most improved American details, the Krementschug bridge does not show economy in material, and in regard to manufacture and erection it cannot compete with single spans. 117 A quadrangular double track truss bridge of proper proportions and details, 400 feet span, can be built with the same weight of iron per lineal foot for the same live load, and the average maximum strain. In the calculation of the Krementschug bridge, the moments and shearing forces were first determined according to the usual theory, whereupon the correction was introduced due to the varied moment of inertia of the struc- ture. The maximum moment over the central pier under full load of the bridge amounted to 12,300,000 kilogrammeters, but was correct- ed to 14, 420,000 kilogrammeters. The maxi- mum moment, however, towards the middle of each span was only reduced by 2 1 /, per cent. The chords, therefore, by the correc- rection were increased, as also were the web strains, slightly* We shall now apply the formulae gained * The design of Herr Sternberg's bridge has this advantage over many newly built lattice bridges, that the lengths of compressional members are so chosen as to permit their be- ing proportioned for crushing and not for crippling. The chords, 40 inches wide and 32 inches deep, are made only 6 feet long between panel joints, are braced laterally every 6 feet in the bottom chords (cross-bearers), and every 12 feet on top, so that the top and bottom lines of each chord are held in position. The compressional diagonals form dia- 118 for the calculation of the influence of the webs on the reactions p of our two continu- ous spans of 200 feet. We shall only con- sider one part of this labor by supposing the bridge to be fully loaded while an exact cal- culation would require to suppose all those modes of loading which we have examined under consideration of the chords, accord- ing to the common theory. The chords of our design are almost of equal section. For the sake of simplicity we there- fore apply the general formula for the angles s and tf under supposition of an average chord section. The dead load is 1200 pounds and the live load is 2240 pounds per foot. There results for deflection : I '__ _3440X200X200X200X2 ' O\/O/< "2X24X30,000,000X25X25X25 = 5.87 2400 phragms between the vertical chord plates, 32 inches deep, for panels of 13 and 15 feet, the chords must be proportioned against crippling, their radii of gyration must be correctly calculated and inserted in the formula. Thin vertical plates of channel shaped chords should be secured in position by diaphragms, and the lateral bracing should be properly cal- culated and proportioned. The radius of gyration for chan- nel shaped chords of lattice bridges is very small, and the chord sections will increase considerably, if the specifica- tion is duly enforced. 119 where the divisor 2400 is the length of the span in inches. This angle of deflection e r\H '- must be increased by the influence of 2400 the posts and diagonals. The result of the 3 79 calculation is ' ~ showing a correction amounting to about 61 per cent, of the angle due to the chords alone. The angle of eleva- tion caused by the chords under action of the total force p = 42,630 pounds, such as found with the ordinary theory, equals 42.630 200* 5.82 3 " 30,000,000. 25 s ~ 2400 2 This value of elevation is so nearly equal to 5 87 , found to be the angle of deflection due 2400 to the chords alone as caused by the full load on the two spans, that were it not for the web system we should feel very much satis- fied with the exactness of the theory. But the force p =. 42,630, also causes ex- tensions and compressions of the web mem- bers which result in the angle of elevation 120 1 7 ' amounting to 30 per cent, of the angle caused by the chords. We have now these angles : Caused by chord. Caused by web. Total. 5.87 3.79 9.66 of deflection 241)0 2400 2400 5.82 1.70 7.52 of elevation The total angles should be equal, but they differ by 29 per cent. The angle of elevation is too small, in other words the force 42,630 is by 29 per cent, too small, or hence the max- imum moment over the middle pier which we found to be 8,130,000 pound feet is by 29 per cent, too small, and the chord sections at this point should be 49 inches instead of 38 inches. The webs are too strong at the end piers and too weak at the central pier. The chords in the middle of the spans such as designed are too large. The point of contrary flexure is nearer to the center of each span than antici- pated. The force p being increased from 42,630 to 55,000 pounds., the end reaction A under full load from 115,036 pounds de- creases to 102,666 pouuds. While according to the common theory 121 0.375 of the total load should be carried by the end piers, the corrected theory only gives 0.34 ; in other words, instead of 8 / 8 th only 8 / 9 th of that load are carried by the end piers and the maximum central moment for two equal continuous spans under the ordinary theory expected to be - p I* becomes only p. Z*. We should now also calculate the influence of the web under other suppositions as to the position of live load. We then should have to calculate anew the strain sheet, we should have to correct the sections, and finally make the whole calculation over again, when again the claims for economy of continuous bridges were to be examined. We will not enter into this labor, the inevi- table conclusion being that the common the- ory is not sufficient for the calculation of continuous skeleton structures. Its use is con- fined to the proportions of homogeneous shal- low plate girders. In some instances contin- uous rolled beams and plate girders of uniform section may be used with advantage in build- ings and for floors of bridges if vertical stiffness must be secured and if the head-room is very 122 limited. But true economy always points to single spans. The most economical structures require but very little calculation, so that estimates can be made within a few hours without formu- lae or drawings. A practical engineer will get along without any formulae. All that is necessary towards making a good esti- mate is a piece of paper and a pencil in the hand of a bridge engineer, who in the school of practice has learned to sift rub- bish, both analytical and graphical, from the few principles of natural philosophy which are really needed, which are commerciably applicable and from which, by plain reason- ing, special rules readily can be derived when- ever desirable. It must not be understood as if analysis were considered to be worthless; on the con- trary analysis, if not superficially applied^ is a most powerful thought aiding machinery and always logically gives the correct answer to a question, but it does not criticise the hy- potheses, unless several contradictory hypoth- eses had been underlaid to the calculation. The proper appreciation and limitation of the power of analysis from an engineering point 123 of view most lucidly has been given by the late Professor Rankine in the preface to his ap- plied mechanics, which we recommend to the readers. VI. EsiMATE OP POSSIBLE IRREGULARITIES IN THE STRAINS OF THE TWO CONTINUOUS 200 FEET RAILROAD SPANS investigated in the previous paragraphs. We will now consider the irregularities caused in the strains of continuous girders if, from any reason, they do not fit to their bed plates. For this purpose, we refer to Fig. 11 and to Eq's (VIII), and assume first^ that the masonry of the middle pier has set- tled one inch. We have for the moment of correction and for the correction of the pier ... ZEId , M __ reactions M = and^? = ; E being l> I the modulus, assumed at 30,000,000 pounds, I the average moment of inertia, equal to say 7,200 inches pounds, arid I the length of the span in feet, consequently, M = 3.30 000 000.7 200.1 12.200.200 =1,350,000 pounds feet. This moment of correction will produce pres- sure in the top chords and tension in the bot- tom chords over the middle pier. The re-ac- 124 tion of each end pier will be increased by l ~^<&~- - 6 - 75 Pounds. The bridge be- 2UU ing fully loaded, by reason of the settled pier the moment over this pier will decrease from 8,430,000 to 7,080,000, that is, by 16 per cent. If the bridge were fully loaded only on one span, the maximum moment within this span would increase by/) -1= 6,750 . . 200 = 562,000 pounds feet. The maximum moment of the fifth panel (see Plate) was 5,952,000 pounds feet, and increases to 6,514- 000 pounds feet. This is an increase of 9V S per cent., or about as much as it was expect- ed to save in the chords under application of the theory of continuity. If from any reason defective construction in the shops, or the middle pier being built too high, or the end piers having settled the bed plate on the middle pier should lie comparatively too high by one inch, the cen- tral maximum moment would be increased by 1,350,000 pounds feet, and the total strains over the middle pier would be in- creased by 44,000 pounds, or by 16 per cent. 125 of their calculated values, and the moments within the span would increase or decrease correspondingly. For every other inch of difference between bed plate and girder bear- ing the same correctional strains would arise, for Eq's ( VIII) teach that the corrections are proportional to the values of elevation or depression. It follows then, conclusively, that the in- troduction of continuous girders requires the best class of foundation and masonry for the piers. Alone from this reason, practical en- gineers would not like to use delicate super- structures like continuous girders, even if these would afford some economy of mat- erial, which, as we have seen, is not the case. It was proposed, long ago, to improve con- tinuous girders by weighing the reactions. But the question arises, whether this improve- ment, which involves some additional cost, is any longer necessary when we know that the theory is of so little practical value. It is also questionable whether a continuous girder, regulated by scales for one mode of loading, would still be properly adjusted under any other position of a moving train ; for we know 125 that with each other position of the load, other diagonals will come into action.* We will next examine the influence of the sun on continuous bridges whose upper or lower chords are covered by roadway planks or otherwise.** In this climate, the power of the sun is great, as any one may feel on a hot summer afternoon by laying his hand on iron exposed to the direct rays of the sun. The difference in heat of iron thus exposed or shaded may be 30 or 40 Fahr. Suppose, therefore, the bottom chords of our 200 feet spans to be covered by planks, what would be the correction needed for a difference of temperature equal to 30 Fahr. ? Under this supposition, the girders, consid- ered without weight, would rise so as to form 25 part of a circle whose radius is 150,000 . 80 = 125,000 feet. The rise in the centre of a * For an ingenious mode of regulating the reactions of the Boyne bridge, see Theory of Strains by B. B. Stooey. Vol. II., p. 460. ** Single span bridges without counter diagonals, whether with one or more web systems, are entirely free from this influence. Those with counters experience in their region extra strains of about 2,000 Ibs. per square inch, immaterial since the counters and diagonals in the 'center of single spao bridges are made stronger than indicated by calculation. 121 chord of 400 feet woud be, consequently, ' 2" oino ~ ' > or ' 9 from Eq's ( VII T), is known to cause a moment M 1,350,000 . 1.92 = 2,592,000 pounds feet. This has a tendency to reduce the moment over the middle pier, which, for the unloaded bridge, was found equal to 374,000 pounds feet, leaving still a pressure in the bottom chords over the mid- dle pier of 393 pounds per square inch of sec- tion. The moment M, 2,592,000 pounds feet, would cause additional pressure on the end piers of 12,960 pounds, and additional strains in end diagonals equal to about 10 per cent. of their maximum values. If one span were fully loaded the maximum moment of 5,952,000 would be increased by 13,000.. 200=il,083,000 pounds feet, which is 18 per cent. At the points where the moments change from positive to negative comparatively very great moments would be produced. Thus, at the third panel- joint from the middle pier, the greatest positive moment is 2,200,000 pounds feet, which would be increased by 128 / o 18,000. . 200 = 1,950,000 pounds feet, so that the greatest positive moment at that point would be 4,150,000 pounds feet. In case the top chord were covered by floor planks, the moment M^ 2,592,000 pounds feet, would cause tention in the top and compression in the bottom chords over the middle pier. The maximum moment, 8,430,000 pounds feet, would be increased more than 30 per cent. ; so that each degree Fahr. would cause one per cent, of additional strain. The negative moment at the second panel- joint from the middle pier would be increased from 3,2^0,- 000 to 5,400,000 pounds feet, that is, by 67 per cent. If the temperature in the top chords were raised to 40 Fahr. the central bearing could no longer act under the dead load only, for the truss would be i inch above the bed-plate. Arch bridges without hinges must be de- signed under the theory of continuity, with its defects and unfounded suppositions. Some of the objections against this theory as here applied have peculiar force as the dif- ficulty of proper manufacture, of close fit to the piers, and especially the influence of tern- 129 perature. A few exclusively theoretical writers on the authority of Oudry's experi- ments, and of thermometric measurements at the Tarascon bridge in France deny the influence of heat on iron arch bridges with flat bearings. But they ignore the experience gained with the Theis bridge in Hungary, which might set at rest experiments with the thermometer.* This bridge changes its bearings on the abutments daily, and it is ob- served that the pressure moves from the lower chord bearing to the upper and back again. The bridge being by design very stiff, leaves its thrust bearings in winter, and unloaded, acts as a beam. It was anchored *AIr Stoney thus remarks about the effect of temperature : The rise in the crown of one of the cast iron arches of South- wark bridge for a change of temperature of 50 Fahr. was observed by Mr. Rennie to be about 1.25 inches; the length of the chord of the estrados is 24G feet, and its versed sine, 23 feet 1 inch, and accordingly the length of the arch, which is segmental, is 3,020.8 inches. The range of temperature to which open work bridges, through which the air has free access are subject: in this country, seldom exceeds 81 W Fahr. The range of temperature of cellular flanges, may, however, exceed that mentioned above, as Mr. Clark mentions that the temperature of the Britannia tubular bridge, before it was roofed over, differed " widely from that of the atmos- phere in the interior, for the top during hot sunshine has been observed to reach 120 Fahr., and even considerably more ; and on the other hand, a thermometer on the surface of the snow on the tube lias registered as low as 16 Fahr." 130 to the abutments the next summer after this observation was made, but during the follow- ing winter the piers commenced to move, wherepon the connections were removed. This example illustrates one of the practical difficulties inherent to continuous girders. Most all European continuous bridges, as well as single span bridges, have either their bottom or their top chords protected from the sun's rays. The high iron viaducts in Switz- erland and France (Freiburg, Busseau, Cere, 3 72 \ P , 4 ^l j!toi.a t ^x l --i xj ^ajj ^ Now, E, I, ojj, ^ and/) Jein^ constant quanti- ties, Aj cannot be equal to A, because d l is not equal to d. They only could become equal if E were different from what it has been sup- posed in the calculation. This supposition here falls away, because the actual deflections of the Augst Bridge do not correspond propor- tionally with the theoretical ones. Hence, by subtracting the equations, we get : E I (dr-dj= ^-^(xSPxJ in words : The difference of actual and theoretical de- flection would be proportional to the difference 139 of actual and theoretical reaction, provided we were right in using the theory in the calcula- tion of continuous trusses. And we find : .6. E.I. The actual reaction is greater than the theoretical reaction by ~ . 6 . E I, or consequently. x^ I x The actual reaction not being equal to the theoretical one, the actual strains are not the theoretical ones, or the calculation of theory does not correspond with reality. Whether this proves incorrectness of the theory, or improper execution, is not open to a conclusion from the mere experimental results without the values E and I. But it is suffi- cient to know that remarkable differences of theory and practice do exist, and may be ex- pected again. VII. RECAPITULATION AND CONCLUSIONS. 1. The mere theoretical calculation of the curves of moments and shearing forces of girders or arches without proper consider- ation of proportions, details and cost of man- facture, is exceedingly fallacious, and this fallacy will be the greater, if the theory by 140 which the moments and shearing forces are calculated, stands upon false premises. 2 There is theoretically no saving in con- tinuous bridges ovej most economically arranged single spans. Whenever single spans can not be built economically there is the place for continuous bridges of which the points of reversion of curvatures are fixed by hinges. 3. The theory of continuity is based on the hypothesis of a constant modulus of elas- ticity, which, as proved, does not agree with the nature of the material. It has been shown that the modulus of wrought iron varies from 17,000,000 to over -40,000,000 pounds per square inch.* 4 Even if it were assumed that the mo- dulus had a constant value, still a correct theory would require that there be but one system of diagonals in the web of a continuous girder. Under an arbitrary supposition, the strains in the diagonals and posts of contin- uous girders with two or more systems can- not be calculated, but only guessed. 5. The theory neglects the influence on the moments and shearing forces caused by *Page 161. 141 the deflections due to the extensions of the web ties, and to the compressions of the web struts.! The theory also needs a correction if the chords are varied. 6. The correct application of the prin- ciple of continuity involves an exceedingly tedious labor, and, if generally introduced, would greatly impede the business of bridge construction in this country. 7. In the determination of the section of chords and webs, it must be considered that a member exposed to tension as well as to pres- sure, must be proportioned to resist the max- imum tension plus the maximum pressure. 8. Continuous girders require very ac- curate workmanship, both in the shops and in the field, which, if exacted by the inspect- ing engineer, will cause a greater expense than that for single spans. The connections at the points where the strains change from the positive to the negative must be made with more care than if tension or only com- pression had to be resisted. Especially, in case of riveting, the holes must in the field be t We have proved that this influence is considerable, and upsets the common theory. 142 rimmed to match perfectly, the rivets placed more closely and driven thoroughly. 9. The foundations and masonry of piers on which continuous girders shall be placed, must be of excellent quality. Single span trusses may, without injury, be placed on piers which have settled several inches; but this is not the case with continuous girders. Engineers contemplating the use of continu- ous girdere should realize the necessity of this provision, and previously estimate the additional cost of substructure. 10. If it is intended to roll continuous girders over the piers, ordering and inspect- ing engineers should examine carefully whether the contractor has calculated the extra strains arising from the weight of the projecting cantilever, has properly reinforced the posts and introduced additional diagonals and chord material at the points of change of flexure. 11. Continuous girders improperly built or placed on their bed plates, have to resist greater strains than contemplated, which, for one inch difference in height of location of bed plate, on the middle pier of a 200 feet span, is increased by 16 per cent. 143 12. If the upper or the lower chords of a continuous bridge are protected from the di- rect heat of the sun, the strains are much dis- turbed and (for 30 difference of tempera- ture) may, over the middle pier be increased 30 per cent, and at the points of change of flexure more than 50 per cent., and the struc- ture may even rise from the middle piers, notwithstanding its dead load, 13. The proportions of depth of span to height depend essentially on the system and on the details used. The lighter theoretical- ly and practically the web oan be made, the greater the height can be chosen, which is only limited by the practicable length of web members and by the calculation of the strains, sections and weights due to the effect of wind. Practically, the best depth is obtained if an additional foot increases the weight of the total structure. Continuous girders, requir- ing more material in their webs than single spans, cannot be built as high. European bridges having been built too shallow for single spans, as far as economy is concerned, were better proportioned when built continu- ously. This is one of the reasons why, in 144 Europe, continuous bridges proved to be the lighter. Properly proportioned single spans on the same system, at least should be no heavier. 14. We have found by investigating the example of two 200 feet spans that properly designed single spans with American details, are actually lighter than continuous girders. The bridge of Buda-Pest and the Krement- schug bridge are examples of large continu- ous structures of economical European con- struction, but they do not compare either in cheapness or quality with single spans, well proportioned, having the most scientific American details. 15. Continuous bridges deflect as much as single spans of correspondingly greater depths. It has now been proved that the theory of continuity, most interesting as it is in a scientific point of view, nevertheless forms only a part of pseudo-science; being based on false suppositions, it is too delicate in execution and under use, and finally, be- cause it is not economical. Practical con- structions are designed with a certain factor of safety. The truer the theory, the easier its suppositions can be fulfilled, the less its re- 145 suits are modified by disturbing influences; the less it is influenced by the unreliable in- cidents of the application of human labor : the more reliable a construction will be, and the smaller the factor of safety may be taken. It has been shown, that by theory we can- not gain greater perfection in practice, unless we constantly are comparing the results of our deductive investigations with experimen- tal facts.* Results of such experiments on executed continuous girders are not known. All that we have are some notes on deflec- tions of finished bridges under test loads. We usually learn that the deflections were much less than expected or calculated, which is communicated as a ' proof of the excellency of workmanship, as if workmanship could re- duce the extensions or compressions of iron in other words, could raise the modulus. In this country, continuous girders have only been used for draw-bridges. The calculations of the strains of these continuous girders are still more complicated, more delusive, and more untrustworthy than those made for fixed bridges, principally on account of the * Compare what Mr. B. Baker says, pages 221, 228 and 313, " Strength of Beams, Columns and Arches. London. 1870. 146 compressibility of the turning apparatus and masonry, as well as on account of irregulari- ities of end supports. It is not intended to enter into these math- ematics. It only is mentioned that, proba- bly, Prof. Sternierg was the first engineer who applied the theory of continuity to the various suppositions as regards dead load of fixed anS swinging draw, of partial and full live load, the dra^jbeing screwed up at ends or loose, &c. The pivot bridge, by him was considered as a continuous girder over three openings, two large outer spans and a short middle part. Herr Sternberg applied the for- mula thus obtained to the draw of Kustrin in Prussia.* The central part of the draw really constituting a separate part of a lattice beam composed of chords and lattice diagonals, this calculation was justified. With large draws, * His whole investigation was published in the report of the Kreutz Kustrin R. R. of 1857. In his lectures the profes- sor gave it in all its essential features, when he also men- tioned the idea of weighing the reactions of continuous gird- trs. In this country Messrs. Channte and Morrison in their work on the Kansas bridge applied the theory to continuous draw bridges. Mr. Shaler Smith in a very lucid article in the transactions of the Am. Society Co. E., 1874, has explained his mode of building continuous draws without end reactions when unloaded, as first introduced by him. 147 as built in this country, the calculation based on three openings can be reduced to that for two spans so long as the two centre diagonals, only necessary to give stiffness during the movement of the swing bridge, are provided with elastic sling loops, or any other suitable elastic medium. The half loaded draw will depress somewhat the drum, the wheels and even the masonry, and the light diagonals be- ing incapable of taking up any great amount of strain without stretching, the other bearing on the round pier will remain in action. The two chords above the round pier will sensi- bly experience the same amount of stress. It is even admissible to slacken these diag- onals when the draw is fixed, so that the bridge may act by a scientifically correct gen- eral arrangement of modified continuity, and before the bridge is to be turned, by some arrangement, the diagonals might be brought into action. By such construction the greater part of the weight would not be thrown on only a few wheels. The great and unneces- sary complicity of the calculation of a bridge resting on four supports can be dispensed with, so much the more since the theory 148 of continuous trusses deserves but little confidence. In the discussion on this subject (Transac- tiens American Society of Civil Engineers, 1876, Vol. V. pages 227 and 228), the author has proved, mathematically, the correctness of this construction. It had already been put under test in Mr. A. P. Boilers draw span of 258 feet over the Hudson at Troy, New York; which, having no center diagonals at all, even swings around without central bracing, simply relying upon the stiffness of the riveted chord. However, it sways a little more than the designer wished. Continuous draws can be entirely avoided by building draws consisting of two single spans to be united when the water way has to be opened. This construction was studied and worked out in details at the same time by the Keystone Bridge Company, and by the author. The detail constructions differ in that point, that the Keystone Company applies hydrau- lic presses at the ends of the single spans to be worked from the centre, so that the ends being raised the tensile central top chord bars, with oblong pinholes, are released of their tension, whereas, the author raises the ends 149 from the centre pier by shortening the dis- tance between the central top pins, or by lengthening the central bottom chord by means of inserted hydraulic presses. (For sketches and details see the discussion refer- red to above). The investigation which is now finished will surely not impair confidence in the con- struction of bridges whose design is exclu- sively based on the plain unmistakable law of the lever, which can be calculated in a short time, be easily manufactured and erected. If engineers wish to build continuous gir- ders, they will do better to use continuous bridges with hinges, (one kind of such struc- tures was first proposed by Professor Ritter, hingei in alternate spans were first patented by de Bergne in England in 1865, then rein- vented by Gerber in Munich 1866, and by the author in 1867 in this country, each of these inventions having been made indepen- dently from the other,) when they will escape all uncertainties caused by defects of theory, by inequality of moduli, by several systems of diagonals, by inequality of heights of sup- ports, of additional strains caused by heat of 150 sun, &c* Practical men will welcome such simplicity, notwithstanding it may not satisfy a few mathematicians, because the problems connected therewith, to them, may not seem sufficiently interesting. * Mr. C. Shaler Smith is just about finishing a bridge of this kind over the Kentucky River, consisting of 3 spans oi 375 each. 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Revised edition, 8vo, cloth... 5 oo MOORE. Portrait Gallery of the War. Civil, Military and Naval. A Biographical record, edited by Frank Moore. 60 fine portraits on steel. Royal 8vo, cloth ............................. .............. 6 oo 16 D VAN NOSTRAND S PUBLICATIONS. PRESCOTT. Outlines of Proximate Organic Analysis, for the Identification, Reparation, and Quantitative Determination of the more commonly occurring Or. ganic Compounds. By Albert B. Prescott, Professor of Chemistry, University of Michigan, izmo, cloth... x 75 PRESCOTT. Chemical Examination of Alcoholic Li- quors A Manual of the Constituents of the Distilled Spirits and Fermented Liquors of Commerce, and their Qualitative and Quantitative Determinations. By Albert B. Prescott; 12 mo, cloth 150 NAQUET. Legal Chemistry. A Guide to the De- tection of Poisons, Falsification of Writings, Adul- teration of Alimentary and Pharmaceutical Substan- ces ; Analysis of Ashes, and examination of Hair, Coins, Arms and Stains, as applied to Chemical Ju- risprudence, for the Use of Chemists, Physicians, Lawyers, Pnarmacists and Experts Translated with additions, including a list of books and Memoirs on Texicology, etc. from the French of A. Naquet. By J. P. Battershall, Ph. D. with a preface by C. F. Chandler, Ph. D., M. D,. L. L. D. i2mo, cloth. ... 2 oo McCULLOCH. Elementary Treatise on the Mechan- ical Theory of Heat, and its application to Air and Steam Engines. By R. S. McCulloch, 8vo, cloth.... 3 50 AXON. The Mechanics Friend; a Collection of Re- ceipts and Practical Suggestions Relating to Aqua- ria Bronzing Cements Drawing Dyes Klectri- city Gilding Glass Working Glues Horology Varnishes Water-Proofing and "Miscellaneous Tools, Instruments, Machines and Processes con- nected with the Chemical and Mechanics Arts; with . umerous diagrams and wood cuts. Edited by Wil- ,atn E. A. Axon. Fancy cloth i 50 17 D. VAN NOSTRAND S PUBLICATIONS. ERNST. Manual of Practical Military Engineering, Prt pared for the use of the Cadets of the U. S. Military Academy, and for Engineer Troops. By Capt. O. H. Ernst, Corps of Engineers, Instructor in Practical Military Engineering, U. S. Military Academy. 192 wood cuts and 3 lithographed plates. i2mo, cloth.. 500 BUTLER. Projectiles and Rifled Cannon. A Critical - Discussion of the Principal Systems of Rifling and Projectiles, with Practical Suggestions for their Im- S-ovement, as embraced in a Keport to the Chief of rdnance, U. S. A. By Capt. John S. Butler, Ord- nance Corps, U. ti. A. 36 plates, 4to, cloth 7 50 BLAKE. Report upon the Precious Metals: Being- Sta- tistical Notices of the principal Gold and Silver pro- ducing regions of the World, Represented at the Paiis Universal Exposition. By \Villiam 1 J . Blake, Commissioner from the State of California. 8vo, cloth 2 oc TONER. Dictionary of Elevations and Climatic Regis- ter of the United States. Containing in addition to Elevations, the Latitude, Mean, Annual Temperature, and the total Annual Kainfall of many locaiiiie?; with a brief introduction on the Orographic and i hysical Peculiarities of North America. By J. M. Toner, M. D. 8vo, cloth 3 75 MOWBRAY. Tri-Nitro Glycerine, as applied jn the do observation and pn five hundred thousand pounds of this explo ive Mica, Blasting Powder, Dynamites; with an acc< unt of the various Systems of Blasting by Electricity, Priming Compounds, Explosives, etc., etc. By George M. Mowbray, Operative Chemist, with thirtee n illustra- tions, tables and appendix. Third Edition. Re- written. 8vo. cloth 3 01 18 YA 01395 T 6-3 55 UNIVERSITY OF CALIFORNIA LIBRARY