UC-NRLF SB < CQ o fo i ( a o o CH > L w U) O I O X UJ m o UJ > CJ THE) VAN NOSTRAND SCIENCE SERIES. o. 13. GASES MET \VITH IN COAL MINES. By J. J. Atkinson. Third edition, revised aid enlarged, to which is added The Action of Coal Dusts by Edward H. Williams, Jr. o. 14. FRICTION OP AIR IN MINES. By J. J. Atkinson. Second American edition. o. ir. SKEW ARCHES. By Prof. E. W. Hyde, C.E. Illustrated. ' Seco.nd edition. 1C. GRAPHIC METHOD 'FOR SOLVING Certain Questions in Arithmetic or Algebra. By Prof. G. L. Vose. Second edition. ] o. 17. WATER AND WATER-SUPPLY. By Prof. W. H. Corfield, of the University Col- lege, London. Second American edition. 13. SEWERAGE AND SEWAGE PUR1FI- cation. By M. N. Baker, Associate Editor "Engineering 1 News." Second edition, re- vised and enlarged. < O. 19. STRENGTH OF BEAMS UNDER Transverse Loads. By Prof. W. Allan, author 1 of "Theory of Arches." Second edi- tion, revised. ,o. 20. BRIDGE AND TUNNEL CENTRES. By John B. McMaster, C.E. Second edition. ' /o. 21. SAFETY VALVES. By Richard H. Bael, C.E. Third edition. 22. HIGH MASONRY DAMS. By E. Sherman Gould, M. Am. Soc. C. 'E. Second Edition. 10. 23. THE FATIGUE OF METALS UNDER Repeated Strains. With various Tables of Results and Experiments. From the Ger- man of Prof. Ludwig Spangenburg, with a Preface by S. H. Shreve, A.M. o. 24. A PRACTICAL TREATISE ON THE Teeth of Wheels. By Prof. S. W. Robinson. 3d edition, revised, with additions. r o. 25. THEORY AND CALCULATION OF Cantilever Bridges. By R. M. Wilcox. o. 2G. PRACTICAL TREATISE ON THE PROP- erties of Continuous Bridges. By Charles Bender, C.E. o. 27. BOILER INCRUSTATION AND CORRO- sion. By F. J. Rowan. Now edition. Re- vised and partly rewritten by F. E. Idell. o. 28. TRANSMISSION OF POW T ER BY WIRE Ropes. By Albert W. Stahl, U.S.N. Fourth edition, revised. THE VAN NOSTRAND SCIENCE SERIES*. . ^ Jk No. 20. STEAM INJECTORS, THEIR THEORY and Use. Translated from the French by M. Leon Pochet. No. 30. MAGNETISM 'OF IRON VESSELS^ AN?) Terrestrial Magnetism. By Prof. Falrman Rogers. No. 31. THE SANITARY CONDITION OF CITY and Country Dx^lling--houses. By G^brge E. Waring-, Jr. Second edition, revised. No. 32. CABLE-MAKING FOR SUSPENSION Bridges. B. W. Hildenbrand, C.E. No. 33. MECHANICS OF VENTILATION. By George W. Rafter/ C.E. . Second edition; re- vised. No. 34. FOUNDATIONS. By Prof. Jules Gaudar^ C.E. Translated from the French. Second edition. No. 35. THE ANEROID BAROMETER: ITS Construction and Use. Compiled by Cffebrgre W. Plympton. Tenth edition, revised and enlarged. No. 36. MATTER AND MOTION. By J. Clerk Maxwell, M.A. Second American edition. No. 37. GEOGRAPHIC AL SURVEYING: ITS Uses, Methods, and Results. By Frank De Yeaux Carpenter^C.E. No. 38. MAXIMUM STRESSES IN FRAMED Bridges. By Prof. William Cain, A.M., C.E. New and revised edition. No. 39. A HANDBOOK OF THE ELECTRO- Magnetic Telegraph. By A. E. Loring. Fourth edition, revised. No. 40. TRANSMISSION OF POWER BY COM- pressed Air. By Robert Zahner, M.E. New edition, in press. No. 41. STRENGTH OF MATERIALS. By Wil- liam Kent, C.E., Asspc. Editor "Engineering News." Second edition. No. 42. THEORY OF STEEL-CONCRETE Arches, and of Vaulted Structures. By Prof. Wm. Cain. Fourth edition, thoroughly re- vised. No. 43. \VAVE AND VORTEX MOTION. By Dr. Thomas Craig-, of Johns Hopkins University. No. 44. TURBINE WHEELS. By Prof. W. P. Trowbridge, Columbia College. Second edi- tion. Revised. SUSPENSION BRIDGES AND CANTILEVERS THEIR ECONOMIC PROPORTIONS AND LIMITING SPANS Submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy, in the Faculty of Pure Science, Columbia University. D. B. STEINMAN, C.E., PH.D. Assistant Professor of Civil Engineering at the University of Idaho. NEW YORK D. VAN NOSTRAND COMPANY, 23 MURRAY AND 27 WARREN STREETS. 1911. COPYRIGHT, 1911 BY D. VAN NOSTRAND COMPANY PBEFACE. In recent engineering literature there is frequent reference to the question of the relative adaptability of the cantilever and suspension bridges to long span con- struction and to the dearth of adequate data from which the limiting and eco- nomic spans for the two bridge-types might be deduced. In order to supply this deficiency and to determine as defi- nitely as practicable the length of span at which the suspension bridge becomes economically superior to the cantilever, the author has undertaken the investiga- tions which are summarized in the fol- lowing pages. In connection with these investiga- tions there have arisen several subsidiary problems of design. It was found nec- essary to determine the economic rise- ratio for suspension bridges, the mini- mum depth of stiffening trusses for iii 228654 IV adequate rigidity, the economic depth of stiffening truss, the best span-ratios and the minimum width for cantilevers and allied questions of design or construction. The solutions of these problems, together with outlines of the methods of designing the different parts of the bridge struc- tures are included in this book. The author desires to express his in- debtedness to Professor W. H. Burr of Columbia University for his valuable sug- gestions and helpful guidance, to Profes- sor C. N". Little of the University of Idaho for help in correcting the manuscript and other kind assistance, and to the Depart- ment of Bridges of New York City for considerable information and data freely placed at the author's disposal. COLD SPRING ON THE HUDSON, August 1, 1911. CONTENTS CHAPTER I. INTRODUCTION ART. PAGE 1. Statement of Problem and Proposed Method of Investigation 7 CHAPTER II. STUDY OF SUSPENSION BRIDGES 2. Suspension Bridges Notable Spans ... 20 3. Wire Cable vs. Eye-bars 22 4. Economic Ratio of Rise to Span 35 5. Minimum Depth for the Stiffening Truss 50 6. Economic Depth of Stiffening Truss. . . 53 Bibliography on Suspension Bridges ... 59 CHAPTER III. DESIGN OF SUSPENSION BRIDGES 7. Principal Data 65 8. Design of the Stiffening Truss 66 9. Design of Suspenders 80 10. Design of Cables 81 11. Design of Towers 85 12 Design of Masonry Piers 95 v VI ART. PAGE 13. Design of Anchorages 97 14. Estimates of Cost 99 CHAPTER IV. CONCLUSIONS FOR SUSPENSION BRIDGES 15. Empiric Formulae for Weights 106 16. Maximum Span for Cable 109 17. Maximum Span for Suspension Bridges . 113 18. Empiric Formula for Cost of Suspen- sion Bridges 119 19. Economic Span for Suspension Bridges. 120 CHAPTER V. STUDY OF CANTILEVERS 20. Cantilever Bridges Historical Sketch Notable Spans 129 21. Economic Span-ratios for Canti- levers 136 22. Minimum Width for Cantilevers 142 Bibliography on Cantilevers 145 CHAPTER VI. DESIGN OF CANTILEVERS 23. Principal Data 149 24. Estimates of Cost 150 CHAPTER VII. CONCLUSIONS FOR CANTILEVERS 25. Empiric Formulae for Weights 157 26. Theoretical Limiting Spans for Canti- levers 161 27. Theoretical Limiting Span for a Simple Truss 164 vn ART. PAGE 28. Maximum Practicable Span for Canti- levers 165 29. Empiric Formula for Cost of Canti- levers 171 30. Economic Span for Cantilevers 172 CHAPTER VIII. FINAL COMPARISONS AND CONCLUSIONS 31. Costs of Suspension Bridges and Cantilevers 174 32. Span of Equal Cost 177 33. Summary 181 34. Conclusions 183 SUSPENSION BRIDGES AND CANTILEVERS CHAPTER I INTRODUCTION ART. 1 STATEMENT OF PROBLEM AND PRO- POSED METHOD OF INVESTIGATION EACH type of bridge construction has some limiting span-length which it cannot, physically, exceed. This maximum span may be defined as the length at which the ratio of the intrinsic weight to the applied weight becomes infinite. In other words, it is the length at which the structure cannot carry any load in excess of its own 7 weight ; any attempt to increase the load resulting in members of infinite cross-section. Under the stress of neces- sity, the span of any type of bridge may be pushed as close as may be desired to the maximum span-length; but this can be done only at a very great sacrifice of economy, as the cost of the structure increases very rapidly when the span approaches the limiting value. The length of the maximum span for any form of bridge may be determined with sufficient definiteness from theoretical considerations applied to the data of actual designs; but it must be borne in mind that the results of such determination are subject to expansive revision when the methods of design or the materials of construc- tion undergo improvement. In addition to a maximum span- length, each bridge type has an economic range of spans, within which it will be less costly than any other form of construction. If, for any two com- parable types of structure, a span of equal cost be determined, that span will be the inferior economic limit for one of the bridge-types and the superior economic limit for the other. The term economic span will be used to designate the span at which the cost of a bridge would exceed the cap- italized value of its usefulness to the builders. Although this is a problem of great practical significance, it does not lend itself to accurate treatment in any general manner owing to the large possible variations in local conditions, such as the magnitude and financial importance of the traffic expected, the cost of real estate for land spans and approaches, and the difficulty encoun- tered in locating suitable foundations. The results of a general determination of the economic span, as here defined, will therefore be of doubtful practical value except for purposes of illustra- tion or comparison. The maximum and economic span- lengths for bridges of ordinary span have been fixed pretty definitely by 10 the results of numerous designs and comparative estimates. We may take our values of these limiting lengths from the consensus of opinion among engineers as evidenced by their uni- formity of practice. Thus the ordi- nary truss bridge finds its range of usefulness between the limiting spans of about 120 and 550 ft., there being but three truss spans exceeding the latter value. Below this range, the steel girder or concrete arch is more economical; above these limits the steel arch enters into competition. The latter construction, in turn, ceases to be economical at about 800 ft., although the proposed Hell Gate Arch is to have a span of 978 ft. For longer spans, the selection of a bridge structure narrows down to a choice between the suspension bridge and the cantilever. The relative econ- omy of these two forms of construction has long been a mooted question. It is true that the longest span in existence, viz., the Forth Bridge, is of the canti- 11 lever type, but it is a question whether the selection of this type was not a mistake. "It is not a design which would ever be imitated. Its propor- tions are very injudiciously taken, and there is a failure to reach the degree of economy which ought to exist even in the cantilever/ 71 In the preliminary investigations for the Quebec Bridge, the Phoenix Bridge Company made comparative estimates of a cantilever and a suspension bridge for the ISOO-ft. span. " Although the cantilever type exhibited the more economic results of the two as the members were then computed, at the present time the economy of the adopted type is not so clear as it was originally thought to be." 2 The col- lapse of that ill-fated structure, and the investigations following the completion of the Queensboro Bridge have shaken 1 W. H. Burr, Proceedings of the Engineers' Club of Philadelphia, December, 1899. 2 Editorial in Engineering Record, Sept. 5, 1908. 12 the general confidence in the cantilever type of construction and have directed the attention of engineers to the more adequate design of compression mem- bers in such structures. When these members are designed in accordance with the recent disclosures, it is a question whether the previously ac- cepted economic limit for cantilevers, viz., about 2000 ft., will not have to be considerably reduced. The limiting economic span for canti- levers is placed by Prof /Burr 1 at 2000 ft. for railway bridges and at 1400-1600 ft. for highway bridges; by Prof. Merriman 2 at 1500 ft.; and by Prof. Melan 3 at 500 meters. Gustav Lin- denthal 4 (M. Am. Soc. C.E.) would 1 Proceedings of Engineers' Club of Phil- adelphia, December, 1899. Ancient and Modern Engineering (New York, 1903), p. 177. 2 Roofs and Bridges (New York, 1905), Part IV, pp. 110, 155. 3 Handbuch der Ingenieur-Wissenschaften (Leipzig, 1906). II. Band, V. Abteilung, s. 206. 4 Engineering (London), December 19, 1890 also Proceedings Am. Soc. C. E., Sept. 21, 1904. 13 use the cantilever for spans between 500 and 2000 ft., the steel arch between 2000 and 4000 ft., and the suspension bridge for all greater spans. Jos. Mayer l (M. Am. Soc. C.E.) considers the minimum economic span for the sus- pension type to be 800 ft. for highway bridges and 1300 ft. for railway bridges. No mention is made of data upon which these estimates are based. The question of maximum span has also been under discussion. In 1894, a prominent bridge engineer declared before a congressional committee that "a bridge of 2800 ft, would be just capable of holding its own weight with- out carrying any live or moving load." 2 A few months later, a board of engineers approved a design for a suspension bridge of 3100-ft. span and showed its " feasibility of manufacture and cost. 7 ' This Board, appointed by act of Congress to determine the prac- ticability of a long-span bridge across 1 Proceedings Am. Soc. C. E., October, 1904. 2 Engineering Record, Nov. 18, 1899. 14 the Hudson River at New York City, and consisting of Major C. W. Raymond, W. H. Burr, G. Bouscaren, Theodore Cooper and Geo. S. Morison, conducted a series of investigations and reported that, at the proposed site, a 2000-ft. clear-span cantilever could be built for $27,000,000, a 3100-ft. clear-span cantilever for $51,000,000, and a 3100- ft. clear-span suspension bridge for $31,000,000.! These results indicate that the maximum practicable limit for suspension bridges is above 3000 ft., and that the economic limit for canti- lever bridges is far below that value. In the same year the Secretary of War directed the formation of " a Board of Officers of the Engineer Corps who shall investigate and report their conclusions as to the maximum length of span practicable for suspension bridges and consistent with an amount 1 Senate Executive Documents, 53d Con- gress, 3d Session, No. 12, Report of Board of Engineers on the N. Y. and N. J. Bridge, Aug. 23d 1894. 15 of traffic probably sufficient to warrant the expense of construction/' Assum- ing that the bridge of maximum span is supported by sixteen 2 IJ-inch cables, and has to carry a uniform live-load of 27,540,000 Ibs.^L. the Board ob- tained a value of L = 4335 ft. for the practical maximum span. 1 The above assumptions, however, were rather ar- bitrary; any other combination of assumed values for cable-section and loading would have resulted in a different value for the maximum span- length. Although comparative designs of the two types of long-span bridges have been prepared in individual instances for the particular local conditions obtaining, the writer has been unable to find any general comparison of the two bridge-types for the purpose of establishing their relative economic or 1 Senate Executive Documents, 53d Con- gress, 3d Session, No. 12, Report of Board of Engineer Officers to Make Investigations of Certain Bridges, Sept. 29, 1894. 16 limiting spans. He has therefore under- taken the determination of these values, namely : 1. The maximum practicable span for the suspension-bridge form of con- struction. 2. The maximum practicable span for cantilevers. 3. The maximum economic span for suspension bridges. 4. The maximum economic span for cantilevers. 5. The span of equal cost for the two types; in other words, the span at which the cantilever ceases to be economically superior to the suspen- sion bridge. The method proposed for the solu- tion of these problems is to prepare designs and estimates of a wide range of cantilevers and suspension bridges and to deduce therefrom the laws of variation of weight and cost with length of span. The relations between span and weight, thus established, will fix the maximum feasible span for 17 each form of construction. The cost- curves of the two types will indicate their comparative economy at different spans as well as the critical span of equal cost. Finally, an estimate of the maximum probable traffic returns r compared with the costs of different spans, will determine the economic limiting length for each type. The cost of a structure for a given span will, of course, be affected by local conditions such as prices of material, specified unit stresses, depth of founda- tions, etc. In order that the com- parison between the cantilever and the suspension bridge may be an absolutely fair one, it is essential that all such arbitrary or varying factors be chosen with extreme care and be kept exactly the same for both types of construc- tion. Furthermore, in order that the results for the limiting spans may be the true maximum values, it is neces- sary to carefully determine and use the most favorable proportions, material and form of construction for each design. 18 TABLE I. NOTABLE Date. Name. Location. Engineer. 1903 Williamsburg. . EastR., N. Y... L. L. Buck. . 1883 1909 1869 Brooklyn Manhattan. . . . i Niagara EastR., N. Y.... EastR., N. Y.... Niagara Falls Roebling Dept. of Br. . Keefer 1867 Ohio R Cincinnati O Roebling 1851 2 Niagara Lewiston N Y Serrell 1900 Miampimi. . Mexico 1848 3 Wheeling . Ohio R W. Va . Ellet 1903 1834 1855 1899 Elizabeth 4 Freiburg Niagara Ry . . . Niagara Budapest Switzerland Niagara Falls .... Lewiston, N. Y . Dept. of Br. . Chaley Roebling R S Buck . 1900 Rochester Ohio R , Pa 1877 s Point Pittsburg Hemberle 1902 Vernaison France 1896 E Liverpool . Ohio 1864 Clifton Bristol Eng 1845 Lancz ' Budapest .... Clark 1855 Morgantown W Virginia 1825 Menai Wales Telford 1905 1904 Villefranche. . . Caperton France. W Virginia .... (Gisclard) . . . Cooney 1868 6 Moldau Prague 1852 7 Charleston .... W. Virginia Ellet 1818 Tweed Grand Ave. . . . Berwick, Eng. . . . St Louis Brown 1862 Lambeth . . London 1826 8 Conway Chester Eng i Strengthened 1888. Wrecked and rebuilt 1889. Re- placed by arch 1896. 2 Removed 1864. 3 Rebuilt 1854 and 1862. 4 Two cables added 1881. 5 Reconstructed 1905. ^Rebuilt with cables 1900. ? Failed 1904. 8 Rebuilt 1904. 19 SUSPENSION BRIDGES. Span. Rise Cables. Truss. Loading. =/ D T, I h No. Size. D'p'h W't' L.L. D.L. HIT 1600 596 177 4 181" 40 118 6300 16620 2.6 1595 *930 t!28 4 15| 9, 17 85 2600 8200 3.2 1470 *725 160 4 21i 24 122 8000 18000 2.2 1268 1057 *281 t90 4 12* 28 52 3600 1042 87 8 21 1030 -j- 9 5 100 300 3 1010 4 8 951 145 95 4 20"E.B. 23 66 5500 14500 2.6 870 174 63 6 2,8 5 21 821 t54 4 10i 16 25 1800 1800 1.0 800 34 53 4 10 14 28 2400 2850 1.3 800 416 72 2 7 18 800 145 88 2 8"E.B.J 8 34 1700 764 *172 f 2 4 17 680 600 O.ft 705 9 7| 702 Chains 663 *285 48 4 Chains 608 150 42 6 31 5 20 2000 578 *260 43 16 Chains 2500 512 128 t 2 tWire 20 2000 2000 1 510 # 37 2 1ft 6 120 100 482 8 Chains 478 25 4 17 450 30 12 Chains 400 *150 40 4 JChains 60 363 .4. JChains 327 22 Chains * Provided with suspenders in side spans. t Stiffened with diagonal stays. j Braced cable construction. Railway bridge. Replaced by arch in 1897. 20 CHAPTER II STUDY OF SUSPENSION BRIDGES ART. 2 SUSPENSION BRIDGES THE economic utilization of the mate- rials of construction demands that, as far as possible, the predominating stresses in any structure should be those for which the material is best adapted. The superior economy of steel in tension and the uncertainties involved in the design of large-sized compression members point emphat- ically to the conclusion that the mate- rial of long-span bridges, for economic designs, must be found to the greatest possible extent in tensile stress. This requirement is best fulfilled by the suspension-bridge type. The superior economy of the sus- 21 pension type for long-span bridges is due fundamentally to the following causes: 1. The very direct stress-paths from the points of loading to the points of support. 2. The predominance of tensile stress. 3. The highly increased ultimate resistance of steel in the form of cable- wire. With the exception of the Forth Bridge, the Queensboro Bridge and the Quebec Bridge (under construction), all structures exceeding 1000 ft. in span have been suspension bridges. Table I gives a list of the most notable structures of this type, with their principal dimensions. It is seen from this table that all suspension bridges erected after what might be called the experimental period (1796-1876) have a minimum span of about 800 ft. and a maximum span of 1600 ft. In order to secure information needed in determining the maximum and economic spans for suspension bridges, 22 in addition to the data supplied by existing structures, the writer has undertaken the design of three suspen- sion bridges, having span -lengths of 1500, 2250 and 3000 ft., respectively. To justify the large expenditure in- volved in structures of this magnitude, they will be assumed to be railroad bridges, with additional provision for electric cars, driveway and footwalks. Before we can proceed with the designing of the structures, it is nec- essary to find the most favorable solutions of the following problems: 1. The choice between wire-cable and eye-bar construction. 2. The economic ratio of cable-rise to length of span. 3. The best ratio of depth of stiffen- ing truss to length of span. ART. 3 WIRE CABLE vs. EYE-BARS One of the first questions to be decided in the design of a suspension 23 bridge is the choice between a steel- wire cable and a chain of eye-bars for the principal carrying member. The latter enables the bracing for the pre- vention of deformation under moving load to be incorporated in the cable system; the other requires a separate stiffening truss for the reduction of these deflections. A third method of bracing the suspension bridge against deformation is the introduction of diagonal stays between the towers and the roadway, as was done in the Ohio River, Brooklyn and Vernaison Bridges. This method, however, has been aban- doned in recent designs as the stays have been shown to be of doubtful utility and, furthermore, fail to act in unison with the cable and suspen- ders under changes of temperature. The earliest suspension bridges were built with chains. James Finley, the pioneer builder of suspension bridges in America, used common wrought- iron chains for all of his bridges from the 70-ft, span at Uniontown, Pa. 24 (1796), up to his greatest achievement, the 309-ft. span of the Schuylkill Bridge (1808). In 1818, Brown sub- stituted a chain of eye-bars, bolted together, in building the Tweed Bridge of 450-ft. span. This construction was followed in the Menai Bridge, built by Telford in 1825, and the Hammersmith Bridge (London) built by Clark in 1827. The following year an advance was marked by the use of open-hearth steel for the chains of the Karl Bridge at Vienna (312 ft. span). Steel eye-bar chains re- mained in use for bridges of constantly increasing span, including Brunei's Hungerford Bridge (London, 1845), Clark's Bridge at Budapest of 663 ft. span (1845), and the Clifton Bridge of 702 ft. span (Bristol, 1864). In the meantime, John A. Roebling was at work at Saxonburg, Pa., invent- ing and developing the manufacture of wire rope. He soon conceived the possibility of its application in the construction of suspension bridges, in 25 place of the eye-bar chains, on account of its superior strength and ease of erection. Beginning with the use of wire cable for the suspension of canal aqueducts, he proceeded to apply it to more ambitious structures. In 1848, Chas. Ellet built the wire-cable bridge over the Ohio River at Wheeling with a span of 1010 ft. which was blown down in 1854. In 1851, Roebling commenced his 821-ft. suspension bridge over Niagara Falls, the first and only sus- pension structure to be built for heavy railroad traffic. In 1867, he completed the Covington and Cincinnati Bridge over the Ohio River, with a span of 1057 ft. These achievements of Roeb- ling, however, were but a preliminary training for " the monumental work that was to cost him his life while crowning it with glory." When he presented his plans for spinning his steel wires over the vast span of the Brooklyn Bridge, he had to defend his ideas against the scoffing of the whole world and had to fight his 26 opponents inch by inch before the right to try was given him. The bridge was finished by his son in 1883, and held the record for length of span (1595.5 ft.) ujitil the Williamsburg Bridge (1600 ft.) was completed by L. L. Buck in 1903. With the third span across the East River, the Manhattan Bridge of 1470 ft. span, opened in 1909, we are brought to the present day in the his- tory of suspension bridges. Since the time of Roebling, wire cables have been used in all suspension bridges with but one exception: the Elizabeth Bridge over the Danube River at Budapest (1903). Its span of 951 ft. is the largest of any suspension bridge outside of America. In nearly all of the above bridges, a stiffening truss is the means employed to prevent the deformations due to live-load, wind and changes of tem- perature. Another practicable method is to build the cable as a trussed struc- ture like an inverted two-hinged or three-hinged arch. This method was 27 used in the Point St, Bridge at Pitts- burg (span = 800 ft.) which was built by Hemberle in 1877. In this bridge the cable is composed of 8-inch eye- bars and is trussed on its upper side by bracing connecting it to two straight chord-members running from the ends to the middle of the cable. In 1894, Gustav Lindenthal offered a design for a suspension bridge over the Hudson River, with a clear-span of 3100 ft., in which he used two pairs of parallel cables connected by a system of bracing in a vertical plane. 1 He proposed building the cables of pin- connected wire links, claiming for such construction the advantages of accurate work and close inspection in the shop, rapid erection, and possibility of vary- ing the cable-section as required. In preparing the design of the Man- hattan Bridge over the East River, the New York Department of Bridges followed LindenthaPs scheme, using 1 Report of Board of Engineer Officers on the N. Y. and N. J. Bridge, 1894. Appendix D. 28 braced cables built up of eye-bars. The following year, with a change of city administration, that plan was abandoned and the structure rede- signed with wire cables. In September, 1904, Lindenthal explained and de- fended his design in a paper read before the American Society of Civil Engineers, 1 which gave rise to a long series of very fruitful discussions. 2 The principal advantages claimed for the two types may be summarized as follows: ADVANTAGES OF THE BRACED-CABLE TYPE OF SUSPENSION BRIDGE 1. By having the greatest depth of the bracing at the one-quarter points, where the maximum moments occur, the stiffness of the bridge with a given expenditure of material is greatly increased. 2. The small depth along the middle third of the span reduces the tem- perature stresses. 1 Transactions Am. Soc. C. E., Sept. 21, 1904. 2 Ibid., Oct., 1904 to Mar., 1905. 29 3. The stiffened-cable construction saves one chord of the truss as the cable itself forms the upper chord. 4. The section of an eye-bar chain may be varied with the stress, whereas the entire wire cable must have the maximum section. 5. Greater weight, if it does not increase the cost, is an advantage in a bridge, as it serves to increase the rigidity of the structure. 6. The pin-connections facilitate the speedy erection of the cable. DISADVANTAGES OF THE BRACED-CABLE TYPE 1. An unpleasant appearance is pro- duced by the lattice-work up in the air. 2. Since the bottom chords of the bracing run to the top of the towers, special wind-chords at the floor-level become necessary for lateral stiffness, and these may be as heavy as the bot- tom chords themselves. 3. It is impossible to calculate the 30 stresses accurately on account of the difficulty in adjusting the members. The safe working stresses should there- fore be reduced by at least 5 per cent. 4. The braced-cable type exposes a large area to the wind at the highest elevation, thereby greatly increasing the wind-stresses in the bracing and in the towers. 5. If the floor is not stiffened verti- cally, every suspender receives a heavier concentrated load and every shock from moving loads is transmitted directly to the cables instead of being absorbed by the floor system. 6. The braced system introduces many elements of uncertainty and complexity in the structure, and the history of bridge design shows that " the lines of progress have been in the direction of eliminating uncertain ele- ments and holding fast to those features which secure certainty in the deter- mination of stresses. " 7. The practical difficulty, hazard and expense of making satisfactory 31 connections between a cable and the web-members of an overhead bracing system preclude the use of wire cables, and to abandon the wire cables is to abandon all the essential advantages of the suspension bridge. The superior economy of the wire cable over eye-bar construction rests on the following considerations: (a) Steel wire with an elastic limit of 180,000 Ibs. per sq.in., is obtainable at a cost of but twice as much per pound as nickel-steel eye-bars with one-fourth the elastic limit. 1 (6) The eye-bar heads and pins add about 25 per cent to the weight of the cable. (c) An eye-bar chain will therefore weigh about four times as much, and cost about twice as much as a wire cable to carry an equal load, if the same factor of safety is to be main- tained. (d) The increased dead-weight of 1 Cf. R. S. Buck and Jos. Mayer, Trans. Am. Soc. C. E., Oct., 1904. 32 the cable , when eye-bars are used, results in increased stresses in cable, towers and anchorages. (e) The wire cable is self-supporting during erection and all the problems involved have been worked out and successfully demonstrated. The eye- bars, on the other hand, would require a temporary supporting cable; and the manufacture and erection of eye- bars of suitable size for very long spans present many unsolved difficulties. Many of the advantages enumerated above, particularly those bearing on the economy of the respective types, practically balance each other, so that, in consequence, there is really no mate- rial difference between the costs of the two forms of construction. In a comparative design of the two types for a proposed bridge at Cologne, of 722 ft. span, O. Erlinghagen found that the eye-bar chain would require about twice as much material as the wire cable but, on account of the difference in unit prices, the total cost of the two 33 structures was almost exactly the same. 1 Jos. Mayer (M. Am. Soc. G.E.) compared the two types for a span of 3000 ft., and found the braced-cable type to weigh 5000 Ibs. more per linear foot than the other. 2 Prof. J. Melan found the difference of cost between chain and cable for the Elizabeth Bridge at Budapest to be negligible, the eye-bar chain being adopted for other considerations. 3 The two designs for the Manhattan Bridge showed a difference in cost of 7 per cent in favor of the eye-bar type, although the chain in the latter weighed 2^ times as much as the wire cable. 4 Omitting from consideration the pos- sibility of using the overhead sys- tem of bracing in conjunction with a wire cable, a construction which would combine the essential advantages of 1 R. R. Gazette, Nov. 20, 1903. 2 Transactions Am. Soc. C. E., Vol. 48, p..' 371. - 3 Transactions Am. Soc. C. E., Feb., 1905. 4 Engineer (London), Aug. 28, 1903. 34 both systems but of which the feasibil- ity has not yet been demonstrated, and without presuming to decide be- tween the relative advantages of the two types of cable construction, the writer will adopt the wire-cable type in the present investigations for the following reasons: 1. As both types are practically equal in cost, this choice will not affect the results for the economic lengths of span. 2. As the wire cable affords a great saving in weight, it will yield a larger value for the maximum practicable span. 3. As this choice conforms with the accepted past and present practice in suspension bridge design, and as no radical changes need be expected in the proximate future, the results obtained will be better adapted for comparison with the data of existing structures and of greater value in estimating future designs. 35 ART. 4 THE ECONOMIC RATIO OF RISE TO SPAN (a) Cost of Cable. Let ft = the ratio of the rise (/) to the span (I) of the cable. The total load per linear foot carried by the cable consists of two parts: #0 = the weight of the cable itself, and <7i = the weight of the total suspended (dead and live) load. As is well known, if a is the inclination of the cable at the towers, the maximum tension in the cable will be (1) The weight of the cable per horizontal linear foot is given by (2) where SQ is the unit working stress and fo is the weight per linear foot per square inch of cross-section. Sub- 36 stituting the value of T from Eq. (1), Eq. (2) may be written, fl f o == ^(s f o + 9 f i) *Z vl-f-16n 2 . (3) H where ! )^7 W The factor (1 + fn 2 ), for all practical values of n, varies only from the value 1.02 to 1.05; hence the quantity k may be considered as practically in- dependent of the rise-ratio n. The solution of Eq. (3) gives (5) If Z/o = the total horizontal length of cable and c = the cost per pound of the cable material, the total cost of the cable will be, 37 Substituting the value of g Q from Eq. (5), we obtain /c-vr+^ . (6) -kl-> n where a=^-c . ... (7) Numerical Values. Allowing for the weight of cable wrapping, a mean value for ro is 3.5. Hence, by Eq. (4), for a range of we may take, 0.455 f4t\ ^ = . . . (ft ) SQ If c =15 cents per Ib. and y has its minimum value of 1.5 (for no suspenders in side spans) , a = 22.5. . . . (70 38 If the side-spans are suspended, =2 y 6 hence a = 30.0 .... (7") (b) Cost of Suspension Rods. If L 2 is the total length of bridge provided with suspension rods, the total load carried by them will be giL 2 . Since the cable is a parabola, the average length of the rods is //3. If $2 is the unit stress, ^2 the unit weight, and 02 the unit cost, the total cost of the rods will be ^Z *"JS / A q This may be written where Numerical Values . If s 2 = 30 ,000 Ibs . per sq.in., 7-2 = 3. 4, c 2 = 12 cents per 39 lb., and 1/2/2=1 (i.e. no suspenders in the side-spans), then b =.00045. . . . (90 L 2 If -y =2 (side-spans suspended), then b =.00090. . . . (9") (c) Cost of Towers. With the entire bridge fully loaded, the total compres- sion in each tower will be twice the end-shear in the cable or P s = (g +gi)-i. . . (10) If 3 is the working intensity of stress, the required section for each tower is A,-g--> + gl) *. (11) S3 3 or, substituting the value of g Q from Eq. (5), A.-21? --- J_ -. (12) Let 7-3 = the weight per linear foot of tower per square inch of cross-section, c 3 = the unit cost of the material, T-/= 40 total height of tower; then the total cost of the two towers will be or, substituting the value of A 3 from Eq. (12), , (13) 1 1 kl< n where Numerical Values. A comparative study of existing structures yields the following mean values for the constants: 7-5 = 6.5 (including diagonal members, details, etc.), c 3 = 5.6 cents per Ib. (price of structural steel and erection), s 3 =8000 Ibs. per sq.in. (for the direct stress exclusive of bending stresses), r=1.6 (a mean value for the longer spans). With these values, Eq. (14) gives c=.0145. ... (140 41 Thus far, in the analysis, there is noth- ing essentially new. We now pro- ceed to apply the above results to our immediate purpose. Total Cost. The total cost of the structure, exclusive of truss, anchorages, substructure, etc. (which are inde- pendent of n), may be obtained by adding together the right-hand members of Eqs. (6), (8), (13), giving the result, or, -expanding the radical, The condition for a minimum total cost then becomes 128afcn 3 I . (1 lo/cm -r LAOKM . .; , ]- 42 Clearing of fractions, and arranging the terms according to the ascending powers of n, we obtain, (bkW-ak)-(2ckl+2bkl)n+(8ak+b+c+lGbkV)n* -(16&M)n 3 -(96afc)n 4 -f . . .=0. (16) It may be readily shown that the terms containing n 3 or higher powers of n are absolutely negligible in comparison with the remaining terms, and they will therefore be dropped from the equation. Solving the resulting quadratic equa- tion for n, we obtain (b + c)2kl V 4 (6 + c)dk 4- 32a 2 fc 2 (17) On account of the relatively small values of b and &, the terms containing bk 2 and b 2 k 4 may be neglected without any appreciable error for any feasible length of span. The equation is thus simplified to (18) 43 This equation shows: 1. That the economic rise-ratio (ri) increases with the length of span. 2. That the economic rise-ratio in- creases with any increase in the cable- factors (a, k) and decreases with any increase in the tower-factor (c) or the suspension-rod factor (6). See Eqs. (4), (7), (9), (14). Substitution of Numeral Values. Case I. No suspenders in the side-spans. By Eqs. (4') (7'), (9') and (14') k = b =.00045. so a = 22.5 c=.0145. Substituting these values in Eq. (18), we obtain .0299Z ^2.96180 + 16210 + .0008676* .0658s +360 (18a) If s = 60,000 Ibs. per sq.in., this becomes 44 where I is the span-length in thousands of feet. Hence, if 7 i nnn' ^ 110 i UUU , II -L AV/ x-v i 2000', n- .118= JL [ Economic ratios 8.5 of rise to span. 3000', n=. 126=^ 4000', n =.135 = =-^ Case //. Suspended side-spans. By Eqs. (4'), (7'0, (9") and (14'), so a = 30 c=.0145. Substituting these values in Eq. (18), we obtain .0308? \/4.06so + 28,800 + .000894Z2 .0677s (186) 45 If s = 60,000 Ibs. per sq.in., this becomes n= .0068Z V.0133 +.000044P, (18'6) where Z is the span-length in thousands of feet. Hence, if 1 Z=1000', n=. 123 = ^2 orvrw * _ ion_ Economic ratios ^v/UU , /& . 1OU ._ _ r /. . 7.7 of rise to span. 3000', n = .138 = ^ 4000', n =.146 = 7^ The above results justify the value of n = \ generally recommended for spans of the usual length. WORKING STRESS IN THE CABLE In the preceding treatment, a value of 60,000 Ibs. per sq.in., was assumed for the unit stress in the cable. This con- forms to generally approved practice, 46 and affords a safety factor of three (on the elastic limit). The following considerations indicate that the safety factor may be judiciously reduced as the span is increased: 1. On a longer span it takes more time for a maximum load to come upon the bridge, so that the application of stress is more gradual. 2. On a longer span with the same traffic, the combination of loads pro- ducing maximum stress will be much rarer in occurrence. 3. As the span increases, the dead- load becomes a greater percentage of the total load, so that the range of stress variation in the cable is diminished. 4. The resistance of tension members to suddenly applied stress increases with their length, on account of the increase in the resilience of the members. 5. It is reasonable to expect some improvement in the cable material before the larger spans are built. Furthermore, the best material can be afforded in the largest spans. 47 For the above reasons ; the unit cable stress should be specified as an increas- ing function of the span. For this pur- pose, the following formula will be convenient : this gives s = 60 ,000 at 1= 1500 } s = 75,000 at 1 = 3000 . (19') s = 86,500 aU = 4500 j and causes s to approach the limiting value of s = 180.000(=E.L.) as I ap- proaches oo . For substitution in the general for- mulae of the preceding investigation, it will be more convenient to replace (19) by a linear equation which will give essentially the same values for s for all practical values of L Such an equation is s = 45,000 +10Z . . . (20) This gives, as before, s = 60 ,000 at Z=1500 SQ = 75,000 at 1 = 3000 * (20/ 48 Economic Rise-ratio, Corrected for Variable s - Instead of assuming the constant value of s = 60,000, let the value of s be specified by the linear formula so = 45,000 +10L . . (20) Substituting this value in Eqs. (18a) and (186) we obtain the two formulas _ .0093? + V.01359 + .00271+ .0000787*2 1+0.198Z (21a) and _ .00874? -f V.Q17Q1 + .Q033Z+ .000072ft 1+0.192Z (216) Case I. If there are no suspenders in the side-spans, Eq. (2 la) gives for Z = 0, (s = 45,000), n= . 116 Z=1000, (s = 55,000), n=. 115 Z = 2000, (s = 65,000), n = . 113 Z=3000, (s = 75,000),n=.lll Z = 4000, (s = 85,000), n=. 110 49 Hence, for suspension bridges of this type, the economic cable-rise is about one-ninth of the span. Case II. If the side-spans are also suspended from the cable, Eq. (216) gives for Z=1000', (SQ = 55,000), n=. 127 Z = 2000', (s Q = 65,000), n = . 124 Z = 3000', (s = 75,000), n = . 122 Z = 4000', (SQ = 85,000), n = . 120 Hence, for suspension bridges of this type, the economic cable-rise is about one-eighth of the span. The above rise-ratios will be used in the following designs. The versed-sine (fi) in the side-span is fixed by the relation ^--frV (22) necessary for equal cable inclinations at the towers. In the absence of any governing conditions, the side-spans will 50 be assumed one-half the length of the main span, so that we must have /i={ .... (220 ART. 5 MINIMUM DEPTH FOR THE STIFFENING TRUSS If a simple truss of span I is covered with a uniform load q, the deflection at the mid-point will be QQ/I 7? T > * * * ^ ' the inclination (or slope) at the ends of the span will be dN r~2M 16 oP the bending moment at the mid-point is M=f; (3) 51 and the corresponding chord-stress is Md where d is the depth of truss and / is its moment of inertia (assumed constant) . Eliminating M and q from Eqs. (1), (3) and (4), we obtain (5) Eliminating q from Eqs. (1) and (2) we obtain dN N Eliminating N from Eqs. (5) and (6), we find dN_2 si l_ f7 . dx 3 Ed' In the stiffening truss hinged at the towers, the maximum stresses occur with a load extending over one-half of the span. Each half of the truss then acts very nearly as a simple beam carry- 52 ing a uniform load ( = actual load suspender forces). We may therefore apply the preceding formulae, (1) to (7), upon replacing I by the half -span of the suspension bridge (= ]. Eq. (7) then \ ^ / becomes dN_li I , 7 ,x dx~3 E'd' Hence, if the maximum allowable grade (-T-) is specified, the minimum \dx / depth of truss will be defined by If 5i = 20,000 (i.e., 30,000 minus wind- stresses, secondary stresses, column-flex- ure stresses, etc.), # = 30,000,000, and -r- = 1 per cent (a limiting value for rail- ctx road bridges), we find Min. ~ = . (9) 1 45 53 ART. 6 ECONOMIC DEPTH OF STIFFENING TRUSS It can readily be shown that the cost of the cable is unaffected by any change in the depth of the stiffening truss, within the limits of practice. The effect upon the weight of the web-members is also negligible. We may therefore define the economic depth of stiffening truss as the depth which will render the chord-areas of the truss a minimum. By Eq. (5) of the preceding article the intensity of stress in the chords of the truss produced by a cable deflection = Af is ip j 8-4.8^. J/. . . . (1) If AL is the total elongation of the cable,, then 1 1 Cf . Melan, Theorie der eisernen Bogen- und Hangebrucken (Leipzig, 1906), p. 14, Eq. (39.) 54 For n=~=, tM s equation reduces to I o J/=1.62.JL . . . (20 The cable-stretch, JL, is composed of the elongation due to live-load plus the temperature expansion, or where s is the total cable stress pro- ducible by the combined effect of the live-load (p) and the dead-load (g). The total length of the cable is given by 1 cai (4) For n = J and n x =^ ; this yields L = 2.Ul. . . . (40 1 Cf . Melan, Theorie der eisernen Bogen- und Hdngebriicken (Leipzig, 1906), p. 14, Eq. (43). 00 Substituting the values of (2'), (3) and (4') successively in (1), we obtain If M is the bending moment produced by the live-load, and k is the correspond- ing unit stress, the chord-section will be - ..... The value of the moment may be written M = m.p-P, . . . (7) where m is a factor nearly constant for all sp&ns ( = ^Q approximately l ). If 1 is the total allowable chord-stress, deducting the stress (s) caused by the deflections, there remains, k = si-s .... (8) 1 Cf. Burr, Ancient and Modern Engineering' (1903), p. 175. 56 Substituting the values of (7), (8) and (5) successively in (6), there results (9) For a minimum chord-area, the de- nominator of the above expression must be made a maximum. The necessary condition is Si-32.88/s \ 9 + P or Ec0n mic = Substituting Qnf ^-U 26000, and ^^=12000, Eq. (10) becomes . d 791 Economic y = . (100 + 12000 57 Assuming a cable-stress increasing with the span-length (cf. Art. 4) and taking - from the results of the writer's designs P we find, with the aid of Eq. (10'), for Z= 1500, =1.8, s =60 ; 000, p =2250, =2.3, =67,500, =3000, =3.1 =75,000 -JL oo Mean=-L, This value is somewhat higher than the average of past practice, probably be- cause most designs have been a com- promise between the demands of economy and those of aesthetics. The Williams- burg Bridge is the only long-span struc- ture conforming to the above economic ratio, but its appearance is undoubtedly marred by the excessive depth of the stiffening truss. 58 Eq. (9) yields T =M>, or, d . *! 1 16.44 (s 07 hUutf) = 2.(Econ.y^ \ ^/ That is, the cost of the stiffening truss approaches infinity as the depth departs from the economic value either toward a value twice as great or toward a zero value. This shows that the truss depth is a very important factor in the economic design of a suspension bridge. For the purpose of this investigation, economy of weight and cost is of greater significance than any aesthetic con- siderations. We shall therefore adhere to the depth-ratio derived above, even at a sacrifice of appearance. It will be noted that the above value of I } is but little in excess of the 40. 59 minimum permissible depth ( = /) as \ 45 / found in the preceding section. BIBLIOGRAPHY ON SUSPENSION BRIDGES Williamsburg, (N. Y.) Engineering Record, 1895 (II, pp. 127, 142); 1898 (II, p. 228); 1902 (I, p. 148); 1903 (I, pp. 80, 482; II, pp. 277, 756, 809); 1908 (I, p. 658). Engineering News, 1896 (II, p. 126); 1897 (II, p. 173); 1898 (I, p. 114); 1899 (I, p. 330); 1901 (I, p. 289; II, p. 2); 1902 (I, p. 182; II, pp. 124, 198); 1903 (I, p. 81; II, p. 535). Railroad Gazette, 1896 (July 31); 1898 (Feb. 11). Iron Age, 1896 (Sept. 24). Zentralblatt d. Bauverwaltung, 1896 (p. 442). Zeitschrift d. Ver. deutscher Ing., 1904 (p. 1213). Engineering, 1905 (Oct. 27). Brooklyn, (N. Y.). Deutsche Bauzeitung, 1870, 1873, 1876, 1878, 1880. Journal of the Franklin Institute, 1873. 60 Scientific American, 1873, 1876, 1877, 1878, 1882. Engineering, 1873, 1876, 1877, 1878, 1879, 1890. Schweiz. Bauzeitung, 1883. Engineer, 1874, 1877, 1881. Annales des ponts et chausse'es, 1874. Trans, of the Amer. Soc. of Civ. Engrs., 1877. Nouv. ann. de la constr., 1879, 1880 Harper's Magazine, May, 1883. Johnson's Cyclopaedia (1896); " Bridges." Railroad Gazette, Nov. 23, 1894; Dec. 9, 1898; Jan. 10, 1902. Van Nost rand's Science Series, No. 32. Engineering News, 1901 (Oct. 10); 1902. Manhattan, (N. Y.). Engineering Record, 1900 (May 12); 1898 (Jan. 22); 1904 (July 2, Dec. 17); 1905 (July 29, Nov. 25, Dec. 2); 1906 (Aug. 25); 1908 (Nov. 21, Oct. 31); 1908 (Apr. 4, Aug. 8, Dec. 5); 1909 (Apr. 3, Oct. 2); 1909 (July 31). Engineering News, 1905 (Aug. 3); 1909 (Oct. 14). Scientific American, 1908. Niagara. Ann. des ponts et chausse'es, 1877. Engineering (London), 1881. Nouv. ann. de la constr., 1880. 61 Zeitschr. f. Bauk., 1883. Railway Age, 1898 (Dec. 30). Engineering Record, 1897 (Apr. 24), 1899 (Aug. 26). Engineering News, 1899 (Jan. 12). Ohio R. (Cincinnati). Engineering Record, 1898 (Sept. 10, Nov. 26) Miampimi (Mexico). Engineering Record, 1900 (Oct. 20). Zeitschr. f. Transportw. u. Strassenb., 1902 (p. 348). Elizabeth (Budapest). "Die Eisenkonstruktion der Elisabeth- Kettenbriicke im Budapest" (Published by the Government), 1904. Deutsche Bauzeitung, 1895. Schweiz. Bauzeitung, 1895 (I, p. 48); 1897 (I, p. 148; II, p. 168). Engineer, 1895 (II, p. 106); 1904 (I, pp. 379, 429, 438, 503, 514, 579, 628). Engineering Record, 1895 (Oct. 5); 1895 (Aug. 24); 1900 (Nov. 10). Oesterr. Monatschrift f . d. Oeffent. Bandienst, Apr., 1899. Zeitschr. d. osterr. Arch. u. Ing. Ver., 1904 (pp. 261, 277). Zeitschr. f. Transportw. u. Strassenb., 1904 (p. 366). Engineering News, 1905 (Aug. 24). Engineering Mag., 1906 (March). 62 Freiburg, (Saane R., Switzerland). Chaley, "Pont suspendu sur la Saane & Freiburg," Paris, 1835. Allgemeine Bauzeitung, 1836. Zeitschr. f. Bauw., 1863. Nouvelles annales de la construction, 1881. Riese, "Die Ingenieurbawerke der Schweiz." Rochester, (Ohio R.). Engineering News, 1897 (I, p. 194). Point (Pittsburg). Engineering News, 1877. Railroad Gazette, 1878. Ann. des ponts et chausse"es, 1879, II. Deutsche Bauzeitung, 1879. Engineering Record, 1903 (I, pp. 2-10); 1901 (I, pp. 424, 455); 1905 (I, p. 517). Vernaison (France). G&rie civil, 1903 (I, p. 145). Engineer (London), 1903 (p. 62). Nouv. ann. de la constr., 1903 (p. 162). Sci. Am. Supp., 1903 (Oct. 10). Engineering Record, 1904 (Sept. 10). E. Liverpool (Ohio). Engineering News, 1897 (I, p. 198). Clifton (Bristol, Eng.). Zeitschr. d. osterr Ing. u. Arch. Ver., 1863. Zeitschr. d. Arch. u. Ing. Ver. zu Hannover, 1865. 63 Lancz (Budapest). Clark, "An Account of the Suspension Bridge across the River Danube," Lon- don, 1853. Morgantown (Monongahela R., W. Va.). Engineering News, 1905 (Mar. 9); 1907 (Apr. 18). Menai (Wales). Smiles, "Lives of the Engineers" (Telford). Villefranche de Conflent (France). Engineering Record, 1905 (I, p. 171). Caperton (W. Va.). Engineering Record, 1904 (Aug. 6). Moldau (Prague). Kostlin, "Ueber die neue Moldau-Briicke in Prag," in Zeitschr. d. osterr Arch. u. Ing. Ver., 1868. Schmitt, "Der neue Kettensteg iiber die Moldau in Prag," Prague, 1870. Charleston, (W. Va.). Engineering News, 1905 (Feb. 2). Engineering Record, 1904 (Dec. 24), (Dec. 31). Tweed (Berwick, Eng.). American Encyclopedia, " Suspension Bridges." Lambeth (London). Heidmann, "Die Lainbeth-Brucke in Lon- 64 don," in Zeitschr. d. osterr. Ing. u. Arch. Ver., 1863. Conway (Chester, Eng.). "The Conway Suspension Bridge," Engineer, 1881 (p. 27). Engineer, (London), 1904 (May 20). Hudson R. (Projects). Report of Board of Engineers, Washington, 1894. Ann. des travaux publ., 1890 (p. 175). Engineering News, 1894 (Sept. 13); 1901 (May 16). Engineering Record, 1894 (II, pp. 375, 390); 1895 (I, p. 275, II, p. 20). Railroad Gazette, 1894 (Sept. 14). Ge*nie civil, 1894 (I, p. 193), 1895 (I, p. 57). Stahl u. Eisen, 1896 (p. 174). Scientific American, 1896 (May 2). Proc. of the Am. Soc. of Civ. Eng., 1896 (p. 469). Trans. Assn. of Civ. Engs. of Cornell Univ., (G. Lindenthal) 1896. Engineering Magazine, 1898 (Dec.). Proc. N. Y. R. R. Club, 1901 (Apr. 18). 65 CHAPTER III DESIGN OF SUSPENSION BRIDGES ART. 7 PRINCIPAL DATA FOR DESIGN OF SUSPENSION BRIDGES No. 1. No. 2. No.3. l=Span =1500, 2250, 3000ft. j 1= :Side-span=J/2 = 750, 1125, 1500ft. /=versed-sine=0.12Z = 180, 270, 360ft. /i= versed-sine in side- span^/ 4 = 45, 67.5, 90 ft. d=depthof truss=.024Z = 36, 54, 72 it. a=panel length (be- tween suspenders) =18.5, 22 . 3, 24 . 8 ft. Type: Stiffening truss hinged at the towers. Suspension rods in side-spans. Loading: 4 railroad tracks at 3000 Ibs. = 12000 Ibs. p.l.f. 2 lines of cars at 1000 = 2000 40 ft. of roadway at 75 = 3000 20 ft. of roadway at 50 = 1000 Total congested load =18000 Ibs. p.l.f. Total width of structure = 115 ft. +2 cantilevers at 10 ft. 4 Cables, spaced 37'.5+40' + 37'.5 c. to c. 4 Trusses, spaced 37'.5+40' + 37'.5 c. to c. 66 ART. 8 DESIGN OF THE STIFFENING TRUSS By proper adjustment of the suspen- sion rods after the erection is completed, all strain may be taken out of the stiffen- ing truss before live-load is applied. The dead-load is thus carried wholly by the cable and may be entirely omitted from consideration in designing the stiffen- ing truss. The latter will therefore be designed merely for the uniform live-load of 4500 Ibs. per linear foot, a temperature variation of 60F.,and a lateral wind pressure of 30 Ibs. per square foot. The formulae used will be those based on the Theorem of Least Work, as de- veloped by J. Melan in his "Theorie der eisernen Bogenbriicken und Hange- briicken." l This theory has been found 1 Handbuch der Ingenieur - Wissenschaften (Leipzig, 1906), II. Band. V. Abteilung. XII. Kapitel, pp. 17-50. Cf. Miiller-Breslau, "The- orie der durch einen Balken versteiften Kette." Zeitschr. d. Arch, und Ing. Ver. zu Hannover, 1881, p. 57. Am Ende, "Suspension Bridges with Stiffening Girders," Proc. of the Inst. of 67 by the writer to give results essentially the same as those yielded by formulae established in a different manner by M. Maurice Levy in his " Calcul des ponts suspendus rigides/' l and by those of C. Schwend in his treatise on suspension bridges. 2 The above theories involve the follow- ing common assumptions: 3 1. The cable is supposed perfectly flexible, freely assuming the form of the equilibrium polygon of the suspender forces. 2. The truss is considered a beam, initially straight and horizontal, of con- stant moment of inertia and tied to the cable throughout its length. C.E., 1899, p. 306; and A. J. Dubois, "The Stresses in Framed Structures," New York, 1896. 1 Annales des Ponts et Chaussees, 1886, II., p. 179, et seq. 2 Schwend, Ueber Berechnung und Konstruktion von Hangebrucken unter Anwendung von Stahl- draht-Kabeln und Versteifungsbalken (Leipzig, 1887). 3 Cf. M. T. Godard, Annales des Ponts et Chaussees, 1894, II., and Melan, p. 17. 68 3. The dead-load of truss and cable is assumed uniform per linear unit, so that the initial curve of the cable is a parabola. 4. The form and ordinates of the cable curve are assumed to remain unaltered upon application of loading. All of these assumptions, with the ex- ception of the last, are ver,y near the actual conditions even in the case of flexible trusses. The last assumption is admissi- ble only with trusses of sufficient rigidity, and its exclusion results in the " Genauere Theorie," developed by Melan. 1 For the purposes of this investigation, the Least Work Theory will be deemed sufficiently accurate, its variations from the Exact Theory being negligible and on the side of safety. Live-load Stresses. The bending mo- ment at any section (x) of the main span (7), produced by a load (p) covering the entire length of the bridge , is 1 Melan, p. 50, et seq. 69 where The results of preliminary estimates of truss and cable gave for I = 1500 2250 3000 _/_ A _i JL A f 2 62 75 94* With these values, assuming /=/i, we obtain N - 1.82, 1.80, 1.78. and M=0.0322, 0.0278, 0.0230ps(Z-x). The same relation obtains for the side spans, upon substituting l\ for I. The above figures may be interpreted as indi- cating that the truss in each case serves to carry 6.44% 5.56% 4.60% of the total load, the remainder being carried by the cable. 70 The position ( = n-l) of a concentra- tion producing zero bending moment at any section (x) of the truss is called the critical point for that section and is given by * y = .218, .216, .214-. (3) y The roots of this equation for different values of x are most easily obtained from a graph of the function which, once plotted, can be used for all suspension bridge designs. For the greatest negative moment at any section (x) of the main span, the load should cover both side spans and the portion (I?) of the main span be- tween the farther end and the critical point. The resulting bending moment is 2px(l -x) K-F-fl < 4 > 71 A graph of the function f(ri) = (2 ft 4n 2 -f 3n 3 ) was used by the writer to simplify the labor involved in the repeated application of the above formula. For the greatest positive moment at any section of the main span, that span is loaded from x=0 to x=, giving M max = M to t - M min . . (5) There are no critical points in the side spans. For the greatest negative mo- ment at any section (x) in one of the side spans, load the other two spans, giving -x) '"mm r TVT which reduces to M min = -.455, -.459, -ASZpx^ -x). Loading the span itself produces the greatest positive moments: -Mrnax = M ioi M min = .487, .487, ASSpx^-x). 72 With all spans completely loaded, the shear at any section (x) of the main span will be (7) In the designs at hand, this reduces to S to t=.0644, .0556, . the coefficient in each case representing the fraction of the total load carried by the truss. The same relations apply to the side spans upon replacing I by Zi . The critical point ( = n-Z) for zero shear at any section (x) of the truss is given by This formula is solved by the same graph as that used for Eq. (3). For maximum shear at any section of the truss, the load should extend from # = to the given section, and from the critical 73 point to the farther end of the span. The shear will then be There will be no critical point for any section where x> l - j = .272^, .275Z, .278Z. For all such sections, n= 1, and the last term in the equation for /S max vanishes. There are no critical points in the side spans, the maximum shear at any section being simply -IK) '41! 74 A graph of the function serves to simplify the application of Eqs. (9) and (10). The stresses in the diagonal web mem- bers are obtained by dividing the shear at any section by the number of web members cut by that section, and multi- plying the quotient by the secant of their inclination to the vertical. Temperature Stresses. The cable-ten- sion produced by a rise in temperature (t) is given by ZEIwtL Ht== ~ For an extreme variation of t= 60 F., Ewt=l 1,720. Taking the value of / from preliminary estimates, Eq. (1) yielded #, = 260, 420, 560 kilo-pounds. The resulting bending moment at any section of the truss is given by M t = H t Y . . (2) 75 and the transverse shear by S, = # r tanr, . . (3) where T is the inclination of the cable at the given section. Wind Stresses. Let p the total hori- zontal wind-load per linear foot. The resulting lateral deflection (h) causes a displacement of the plane of the cable and suspenders from the vertical , thereby giving rise to a force of restitution (r) equal to the horizontal component of the suspender tensions. If v is the mean vertical distance of the truss and live load below the cable chord, then the reaction component of the displaced weight (W) will be r=\-W (1) Considering the truss as a beam acted on by a uniform horizontal load (p r), we have 76 From (1) and (2) , there results E.JLA r= ~w = T 'P- ' ( 3 ) 1+ T*384*^Z In the designs at hand, Eq. (3) yields r/P = 19%; 45%; 57% in the main spans, and r/p = 3.5%; 9.3%; 20.3% in the side-spans. The specified wind-load (p) consists of a pressure of 30 Ibs. per square foot acting on the exposed surface of one truss and on half that of the remaining trusses, also on a train of cars 14 ft. high. On the round surface of the cables, only one-half of the above pressure is con- sidered effective. The total wind pres- sure in our designs is thus found to be p= 1220; 1400; 1545 Ibs. p.l.f. and the effective horizontal pressure will be p-r=988; 772; 671 Ibs. p.l.f. 77 in the main spans, and p -r= 1177; 1270; 1231 Ibs. p.l.f. in the side-spans. The resulting lateral deflections, by Eq. (2), will then be &=18i"; 481"; 61" at the center of the main span, and at the center of the side -span. Each of the above pressures (pr) must be resolved into two parts acting in the planes of the upper and lower lateral systems respectively. On account of the wind pressure on the floor and train, the lower chords get the major share of the wind-load, amounting, in the present designs, to about 63% of the total pressure. The system of bracing adopted is simi- lar to that used on the Manhattan Bridge. The upper lateral bracing consists of diag- onals and struts connecting the inside to the outside trusses. There is no bracing over the roadway or sidewalks. The 78 upper lateral system is therefore com- posed of two independent horizontal trussses, 37.5 ft. deep, so that the chord- stresses due to the wind moments will equal Af^-r-75. The lower lateral system consists of the four lower chords tied together by the floor beams and special diagonal braces extending across the entire floor. Assuming the wind-stresses in the chords to vary as their respective distances (20 ft. and 57.5 ft.) from the neutral axis, the stress in the outside lower chords will equal M^-f-129. It is thus found that the wind-stresses in the upper and lower chords are almost equal, so that the same sections may be used for both. In designing the members of the lateral systems, the wind pressure is considered as a moving load. The max- imum shear at any section (x) is given by and the resulting stresses are obtained 79 by dividing this shear by the number of laterals cut by the section and multiply- ing the quotient by the secant of their inclination. Working Stresses. The chords of the stiffening truss are made of nickel steel and are designed for working stresses of 30,000 Ibs. per square inch in tension and / P \ . for 30 ,000 -*- (!+ Jin compression. The web members, bracing and con- nection details are of structural steel and are designed for working stresses of 20,000 Ibs. per square inch in tension and / I 2 \ for 20 ,000 -f- (1+ Jin compression. Computations. The limitation of space renders it impracticable to reproduce here the detailed computations of the stresses and weights of members. For the same reason, the design of the floor system will not be given here. The final weights, however, are tabulated in the following article: 80 ART. 9 DESIGN OF SUSPENDERS The load carried by the suspenders consists of the following items: Truss and bracing Floor Suspenders Total Dead-load Live-load Total load p.Lf . (-ft) Total load per suspender S. B. S. B. S. B. No. 1. No. 2. No. 3. 2,624 4,015 5,786 Ibs. 3,712 3,800 3,812 126 207 320 6,462 8,022 9,918 4,500 4,500 4,500 10,962 12,522 14,418 Ibs. 202,800 279,800 358,000 Ibs. With a specified working stress of 30,000 Ibs. per square inch, the required sec- tions for the suspenders will be: 6.8 9.3 11.9 sq. in. Their average length (=*/+*= 96 144 192ft. Hence, their mean weight= 126 207 320 Ibs. p.l.f . of truss, as assumed above. 81 ART. 10 DESIGN OF CABLES Preliminary Estimate. For an accu- rate design of the cable its weight must be known in advance. For this purpose, use is made of Eq. (5) of Art. 4, viz.: go =01 '- n With /c= - , and n=-=0.12, the 136000 I above reduces to .000068Z flfo "" 9fl l-.00006S f or, for Z=1500, 2250, 3000, 0o = . 1140!, .1810!, .25701. Taking the values of the total suspended load (0i) from the preceding article, we find the weight of the cable to be go=1250, 2270, 3710 Ibs. p.l.f. 82 After a first design these values were slightly altered to those given below for the final computation. Final Computation. The total load carried by the cables consists of the following items : s. B. s. B. s. B. No. 1. No. 2. No. 3. Dead-load on sus- penders (as above) 6,462 8,022 9,918 Weight of cable 1,286 2,160 3,720 Total dead-load (=0) 7,748 10,182 13,638 Live-load (=p) 4,500 4,500 4,500 Total load p.Lt.(=g+p) 12,248 14,682 18,138 For a live-load (p) covering the main span, the horizontal component of the cable tension will be In the designs at hand, this gives #=.9170; .9250; .9365 pi. For a live-load (p) covering one of the side-spans, the cable tension is given by l fl I h2 l 83 In the present designs this gives #=.0572; .0578; .0585 pZi. Combining the above values, we find the cable tension for a uniform live-load (p) covering all the spans to be H p =.9742; .9840; .9950 pZ. As the cable sustains the entire dead- load (g) without any relief from the stiffening truss, the corresponding hor- izontal tension will be _gl 2 _gh 2 g ~8f~8fr' ' ' * or, in the designs at hand, #0=1.04170*. Substituting the values of (g) and (p) in the above expressions for H, we obtain #0 = 12,100,000, 23,880,000, 42,600,000 Ibs. H P = 6,135,000, 9,960,000, 13,440,000 Ibs. As previously given (Art. 8), the cable tension producible by a fall of tem- perature is H t = 260,000, 420,000, 560,000 Ibs. 84 Hence, Total H= 18,495,000, 34,260,000, 56,600,000 Ibs. Multiplying this value by the secant of the inclination of the cable at the towers, (sec. = 1.109), we find the maximum cable tension to be Tmax = 20,510,000 Ibs., 38.000,000 Ibs., 62,770,000 Ibs. With the specified unit stress of 60,000 Ibs. per square inch, the required cable section (per truss) will be A = 342, 633, 1046 sq.in. This section will be provided as follows: S. B. No. 1. One cable of 37 strands, each containing 319 wires of 0.192 in. diam. Diam. = 23f in. Total area = 342 sq.in. S. B. No. 2. Two cables of 37 strands, each containing 296 wires of 0.192 in. diam. Diam. = 22-| in. Total area = 634 sq.in. S. B. No. 3. Three cables of 37 strands, each containing 326 wires of 0.192 in. diam. Diam. = 24 in. Total area = 1047 sq.in. 85 Each cable will be wrapped with a single layer of No. 10 (B. W. G.) iron wire. Allowing for catenary, cable wrapping, etc., the mean weight of the cables will be 2160, 3720 Ibs. pj.f. as assumed above. ART. 11 DESIGN OF TOWERS Each tower will consist of four box- columns, one for each cable system, rigidly tied together by transverse brac- ing in a vertical plane. Comparative designs by the writer indicated a small economy of material in favor of pin- bearing columns, but this saving is more than outweighed by the more expensive construction, greater dif- ficulty of erection, uncertainty of action and aesthetic inferiority due to the impression of instability. The more 86 usual design, namely with the columns rigidly fixed at the base, will therefore be followed. Assuming the required clearance of the truss above M. H. W. to be 135, 160 and 185 ft. for the respective bridges, and assuming the top of the masonry piers to be 30 ft. above M. H. W., the total height of the steel towers will be: for S. B. No. 1, 180+36 + 135-30 = 321 ft. S. B. No. 2, 270 + 54 + 160-30 = 454 ft. S. B. No. 3, 360 + 72 + 185-30 = 587 ft. Deducting the height of the pedestal castings, the effective height of the towers will be A = 316; 448; 580ft.; and the height up to the stiffening truss will be Ai = 100; 124; 148ft. The maximum fiber stress in the tower columns will occur when the live-load covers the main span and the farther 87 side-span at maximum temperature. Under this condition of loading, the top of the tower will be deflected toward the main span as a result of the following deformations : 1. The upward deflection (J/\) at the center of the unloaded side-span. 2. The elongation of the cable between the anchorage and the tower due to the elastic strain produced by the applied loads. 3. The elongation of the cable due to thermal expansion. These deformations are computed as follows : (1) The upward deflection (J/i) is found by considering the side-span as a simple beam subjected to a downward loading equal to the suspended dead-load (p) and an upward loading equal to / P\ the suspender tensions It = H -*- J . If Pi(f= P) is the resultant of these loadings, the central deflection will be jA-JLEilli m 4/1 "384 El ' ' ' W 88 For the spans under investigation, this gives J/i = 2.630; 3.760; 4.885 ft. (2) The elastic elongation of the cable in the side- span is given by H Cds* l ET -Jdtf which reduces to JL 1= = 1.915; 2.875; 3.860ft. (3) The temperature expansion of the cable in the side -span is given by -T- ax (3) t I -T- J ax which reduces to JLi = 0.390; 0.585; 0.780ft. The length of the cable in the side-span is given very closely by ) (4) 89 from which we find dLi 77 ^ A ~r ^ y~2 n~ ^ van. ULI j 1 . 125, (5) and The deflection of the top of the tower is then given by ,_ Substituting the values from (1), (2), (3), (5) and (6) in (7), we obtain the max- imum tower deflection, ?/o = 2.798; 4.150; 5.515ft. Considering this deflection as pro- duced by an unbalanced horizontal force P applied at the top of the tower, this force may be calculated if the sectional dimensions of the tower are known. As these are not known in advance, the following procedure is adopted : 90 Assume the moment of inertia of the section of the tower to have a maximum value (/o) at the base and to diminish regularly toward the top according to some appropriate law. A study of actual designs indicates the applicability of the law of variation represented by the empirical formula 7 = /o(l-vV/0, . . (8) where x is the distance of any section above the base. Substituting this rela- tion in the differential equation of the elastic curve of the cantilever and integrating, there results P 30 ^ 7 o /m P = 23^' ' ' (9) In the designs at hand, this reduces to P = 3.4707 ; 1.736/ ; 1.094/ . The other loads acting on the tower are the vertical reaction (F = 2#-tan a) 91 at the saddles, and the end-shears (Vi) at the point of suspension of the stiffen- ing truss. In the designs at hand, for the condition of loading under considera- tion, we find 7= 17,150,000; 30,700,000; 52,100,000 Ibs. F!=- 1,037,000; -1,852,000; -2,640,000 Ibs. At any section (x) of the tower, the horizontal deflection (y) from the initial vertical position of the axis is given with sufficient accuracy by the equation for the elastic curve of the cantilever: This gives, for x = hi, ?/i = 0.376, 0.436, 0.492ft. The maximum fiber stress at any section of the tower will be = 7 M-d where 92 The moment of inertia for the form of section used here is given approx- imately by / = 0.50A.d 2 . . . . (13) Substituting (12) and (13) in (11), and applying the resulting equation to the base of the tower, we find /P\ 2Vyo2V iyi s== IT l*o+"ji-jH T~T \/o/ Aodo ^Mo Using an allowable fiber stress of s = 24,000 Ibs. per square inch, and substituting the numerical values of P//o, V, Fi, h, 7/0, and yi, as given above, there results the following relation : S. B. No. 1: 24,000= 1096d 0+ 960070+16113000 S. B. No. 2: 24,000= S. B. No. 3: o^ AAA aAvj , 800000000 , 49460000 24,000= 6474+ - -r-j - + - - A - . AQ 93 This relation is satisfied by the follow- ing values: S. B. No. 1: ^o = 3240 sq.in., do = 15 ft.; /. 7 =365,000 in 2 ft. 2 S. B. No. 2: ^o-4260 sq.in., d = 17 ft.; /. /o =625,000 in. 2 ft. 2 S. B. No. 3: A = 7200 sq.in., do = 20 ft.; .-. 7 = 1,440 ,000 in. 2 ft 2 The horizontal force P is now deter- mined from Eq. (9) as P=694,000; 1,085,000; 1,575,000 Ibs. All the remaining sections of the tower may now be proportioned, using Eq. (8) as a guide and checking the maximum fiber stress by Eqs. (11) and (12). The mean section of the tower is thus found to be Mean A = 2210; 3240; 5550 sq.in., and the total weight, including connec- tion details and bracing (about 100 per cent) , is found to be Weight = 4,465,000; 9,720,000 ; 22,406,000 Ibs. per column. (Structural steel.) 94 Pedestals. The pressure at the foot of each tower column is a maximum when the live-load extends over all the spans, &nd is then made up of the following items : Max. Cable Reac- S. B. No. 1. S. B. No. 2. S. B. No. 3. tion (V) Max. Truss Reac- tion (Vi) Wt. of Tower Wt. of Pedestal 17,500,000 326,000 4,465,000 420,000 32,500,000 422,000 9,720,000 810,000 53,830,000 466,000 22,406,000 1,200,000 Total load (Ibs.) 22,711,000 42,452,000 77,902,000 Bearing area re- quired at 15 tons per sq. ft. 757 1448 2597 In order to evenly distribute the above loads over the requisite area of masonry, the tower legs will rest on pedestals of cast steel (annealed) having the follow- ing dimensions: S.B.No.l. Top of casting 10 X 30 Bottom of casting 20X40 Height 5 ft. Weight of casting (Ibs.) 420,000 S.B.No.2. 18X34 30X48 6ft. S.B.No.3. 24X40 36X72 7ft. 810,000 1,200,000 95 ART. 12 DESIGN OF MASONRY PIERS The steel castings at the bases of the tower columns are anchored on pedestal blocks of selected granite. The top of the pier is made just large enough to hold these pedestals, except where a larger section is required to reduce the pressure in the masonry to 12 tons per square foot. The pier is built of 1 : 2^ : 5 concrete with limestone (ashlar) facing. The upper pier, above the starling, is given the section of a rectangle with semicircular ends and has a batter of 1 : 20 on sides and ends. In the lower pier, the sides (batter 1:20) are con- tinued at each end in two circular arcs to form a cutwater (batter 3 : 20) . Below the masonry is a rectangular cribwork, used as a coffer-dam during construction. It is built of 12X12 in. timber, and is filled with concrete as 96 the sinking progresses. It rests directly on the pneumatic caisson which has a steel working chamber 7 ft. high, and a reinforced-concrete roof 3 ft. thick. After reaching rock, the caisson and shafts are carefully filled with 1:2:4 Portland cement concrete. Since the economic depth of founda- tions increases with the length of span, it will be assumed that the pier-depths for the bridges under design are somewhat greater for the longer spans. The fol- lowing are the principal elevations as- sumed : S. B. S. B. S. B. No. 1. No. 2. No. 3. Top of pier 30 30 30 Starling 5 5 5 Mean high water ( = datum) 000 Base of pier=topof cribwork 15 15 15 River bottom -20 -25 -30 Base of cribwork=top of caisson -80 -90 -100 Cutting-edge of caisson = surface of rock -90 -100 -110 The principal sectional dimensions of the piers, determined as outlined above, are as follows: 97 S. B. S. B. S. B. No. 1. No. 2. No. 3. Top of pier 177X42 195X50 225X74 Base of upper pier. 179X44 197X52 227X76 Top of lower pier . 200 X 44 222 X 52 259 X 76 Base of pier 206 X 46 228 X 54 265 X 78 Cribwork 212X52 234X60 271X82 Caisson 212X52 234X60 271X82 A complete design and estimate of the piers yielded the following quantities of material required: S.B.No.l. S.B.No.2. S. B. No. 3. 12,890 15,490 37,400 Pier masonry Concrete filling in cribwork .. . . 25,300 Concrete filling in caisson 4,080 Timber in cribwork 620 Steel in caisson and shafts 1,150,000 2,320,000 2,650,000 Ibs. Earth excavation. . 28,600 39,000 65,000 cu.yds 5,200 790 26,300 cu.yds. 67,800 " 8,230 " i nnJ M -ft- lt08() (B.M. ART. 13 DESIGN OF ANCHORAGES The anchorage, as a mass, is required to offer sufficient frictional resistance to sliding to resist the tension of the cable. Dividing the maximum horizontal tension (H) by the coefficient of friction (/*=0.6), 98 the quotient will be the necessary weight of anchorage masonry. Introducing a factor of safety of 2, the masonry re- quired for the anchorage is thus found to be 63,500; 109,000; 183,200 cu.yds. This will be built of concrete with lime- stone facing as in the case of the piers. Enclosed in the base of the anchorage are heavy box girders to which the cables are anchored by means of chains of eye-bars. -The total amount of steel work in each anchorage is found to be 1,037,500; 2,472,500; 4,090,000 Ibs. The anchorage will be supported on a concrete foundation resting on bearing piles if necessary. The volume of mate- rial in this foundation is found to be 5,290; 10,400; 16,650 cu.yds. 99 ART. 14 ESTIMATE OF COST In order to establish an equitable schedule of unit prices for the two types of bridges, a critical study was made of the different bids received by New York City for the Williamsburg, Manhattan and Queensboro Bridges. These struc- tures were selected as combining the qualifications of recentness of date with similarity of construction and magnitude to the designs under investigation. The prices of the successful bidders were adopted except where they differed so widely from the following bids as to indicate their having been unbalanced for some special reason. The unit prices determined by this comparison are in- corporated in the following estimates of cost. 100 s 10. GO CM- o s O> * ^ Oi ' 1 rH H fe 00 CO CO O5O CTi ' 00 rt< GO 00 O5 Oi I s *. Tf OO 8 OlOiOOt^rHT^ COi-H COOOTf, O^^OT-l T^t^ w O "'-' ti 03 "^ ^ 2 t*_Q ^ r" H3 d "Q B-.S^lJBto ^ g ^CliO o 'r!W3C(3 -si; 42 ^ tHtf}^ 3" 1 - 1 CC ^ O CD A >> F 102 SIT! ^52 ^S-s O fH O> gs EH 103 CO LCf S8 3 s g C* oo o o o o o O O 00 ^O T^ *O rf O O O ss 00 O (M 'O rf CO" t^ 00 COCO O 10 (M Tt< CO(M r-? s 00 (M iO O 00 (N $ O O> CO ^ ^-^ g oC,fl .s-cilg-^ C^ 'C w g rt fl C -| -* - rj w o K^ (V) (T).S X >^^ ^ ^ d liilfe i Gra 104 c^ * *0" . CO O CD 01 o co" oo~ oo~ oo~ of o" t^. O Ol TF rH CD Tf O rH Ol s 8^ (M rH co~ CO^ QO^ (M^ O^ O^ O^ CD" i>^ oo" r-T cT iT (M CD O CD CD 8S (M CD S3S I I ai ^^ \4 ^ s ^^ o pj ill >H O O Jg So 1 106 CHAPTER IV CONCLUSIONS FOR SUSPENSION BRIDGES ART. 15 EMPIRIC FORMULAE FOR WEIGHTS OF SUSPENSION BRIDGES FROM the results of the preceding designs, we may construct semi-empirical formulae for the weights of the different parts of a suspension bridge. Such formulae will be useful in preparing estimates for other spans and loadings; also in drawing general conclusions as to maximum and limiting spans. Let the weights (per linear foot) of live-load, cable, suspenders, truss (in- cluding bracing) and floor be represented by L, C, S, T and F, respectively. The weight of the cable is evidently pro- portional to the total load it sustains and to the length of span, provided the 107 rise-ratio remains unchanged. The weight of the suspenders is likewise proportional to the load they carry and to the span, as a longer span requires a proportionally greater average length of suspenders. The weight of the truss may be considered in two parts: the larger portion provides for the live-load and is proportional to that load and to the span length, if the depth ratio remains constant; the other part of the truss-weight represents the contribu- tion of the wind-stresses and will vary with the square of the span if the width of bridge remains constant. The weight of the floor system is evidently propor- tional to the assumed live-load, and is practically independent of the length of span. Introducing the necessary co- efficients of proportionality, the above relations may be formulated as follows: 108 Substituting the values of weights and spans from the preceding designs, the spans being measured in units of 1000 ft., the above relations yield the following values for the undetermined coefficients. S. B. No. 1. S. B. No. 2. S. B. No. 3. Mean. a= .070 .066 .068 .068 b= .0077 .0073 .0074 .0075 c= .35 .34 .36 .35 d= 105 105 105 105 e= .83 .84 .85 .84 The uniformity in the above coefficients for the different spans confirms the rationality of the relations assumed above. Introducing the empirical constants in those relations, we obtain the following formulae for suspension bridges: . (2) 109 ART. 16 MAXIMUM SPAN FOR CABLE The theoretical limiting span for a suspension bridge is the span at which the cable-section becomes infinitely large in proportion to the load it can carry. The defining equation for this condition is, therefore, = oo (1) 9i where g\ is the intensity of the total load suspended from the cable. If <7 = the weight of the cable per linear horizontal unit, and n = the rise- / ratio =, the maximum tension in the L cable will be 6w*. (2) The weight g$ is given by the equation 110 where s is the unit working stress and 7-0 is the weight per linear foot per sq.in. of cross-section of the cable. From Eqs. (2) and (3) we obtain T (4) Introducing the condition for maximum span, as defined by Eq. (1), we obtain the relation Hence, Qo == - (5) This equation shows that the maximum limiting span is independent of the live- load or weight of the stiffening truss. Equating the first derivative of the second member of (5) to zero, we find Ill the rise-ratio giving the absolute max- imum span to be ... (6) With this value, Eq. (5) becomes Zmax= 1.085^. . . (50 To Numerical Values. For very heavy cables, the weight of the cable-wrapping may be neglected; hence ^ is simply the density-factor of steel =3. 4. The extreme limit which no cable-span can ever exceed is that for which s equals the elastic limit of the best cable material or s = 180,000. Eq. (5') then gives Extreme Z max = 65,520 ft. . (7) For practical purposes it is necessary to introduce a safety factor in the working stress; reducing the latter to = 60,000, we obtain, by Eq. (50, Z max =21,S40 ft . . . (70 This span with a rise-ratio of n = .306, would require towers about 7000 ft., 112 high; the above result, therefore, is of no practical significance. Replacing the above value of n by the economic ratio and retaining s = 60,000, Eq. (5) yields Practical J max =15,160 ft . (1") This value gives the limit of span which may be approached, but not exceeded, by successively reducing the amount of extraneous load for any given cable-section or by increasing the cable- section for any given load so that db oo . 01 At the maximum span given by Eqs. (7), (7') or (7"), the cable-section may have any finite value so long as there is no load suspended from it. The addition of the smallest load will neces- sitate reducing the span or else increas- ing the section A to infinity. It may be noted that the limiting span increases directly with the working 113 stress SQ and inversely with the cable- weight factor fo- In comparison with the above limit- ing values, it is of interest to consider the largest existing cable-span. This is a cableway at Caperton, W. Va., built in 1898, and having a span of = 2100 ft., or only about one-seventh of the span- length which may yet be attained. ART. 17 MAXIMUM SPAN FOR SUSPENSION BRIDGES From Eq. (2) of Art. 15, we obtain the following empirical expression for the weight of the cable (C) in terms of the suspended loading and the length of span (I) : Substituting the empirical expressions for 8, T and F in terms of the live-load L, Eq. (1) becomes, practically, .068i(1.84L+ .35ZJ + 1Q5Z 2 ) . -.0075Z) 114 The theoretical maximum span for the suspension bridge is the span at which the cable-section becomes infinite. This condition is realized when the denominator of Eq. (2) reduces to zero, or Jmax = - = 14,700 ft. This is the upper limit of feasible spans for suspension bridges and repre- sents the span at which the suspension bridge ceases to be self-supporting. It is somewhat smaller than the maximum span-limit (15,160 ft.) for a simple cable, as determined in the preceding article, simply because the weight of cable wrapping and fastenings was not considered in that investigation. Another method of deducing the limiting span, possessing the advantage of greater generality inasmuch as it is applicable to other forms of bridge- structures, is as follows: Let Ci= the weight of the cable and W\ =the total suspended load, per linear foot, for any span (li). For any other span 115 (Z), the weight of the cable is given by the proportion C (C+W)l ' Solving this equation, we obtain C WC * (4) - For C = GO t the denominator of the above expression must reduce to zero, or This formula enables the limiting span to be calculated from the results of any single design. 1 In the designs at hand, 1= 1,500 2,250 3,000 C= 1,286 2,160 3,720 TF = 11,162 12,522 14,418 .'. Zmax = 14,300 15,300 14,600ft. 1 The same formula, with C denoting the weight of the arch-rib, was used by the writer in determining the maximum spans for steel and concrete arches. See Thesis submitted for the Degree of C.E. at Columbia University. 116 The practical agreement between these values indicates the reliability of the above method of determining the max- imum span from a single design. Taking the mean of the above values, there results a value identical with that yielded by the first method of this article. Although the above is the limiting feasible span, defined by C= oo, it is evident that the sections will become too large for practical construction long before that length of span is reached. In order to determine the practical maximum span, a value for the maximum cable-section must first be fixed. The cables of the three largest sus- pension spans, viz., the Brooklyn, Wil- liamsburg and Manhattan Bridges, are 15f, 18| and 21^ inches in diameter, respectively. Cables with a diameter of 24 inches were used in one of the designs in this investigation, but it is doubtful whether any larger diameter 117 can be put together without excessive difficulty of manipulation and uncer- tainty of proper distribution of stress among the different strands. With the exception of some old chain bridges, there is no bridge on record with more than six cables. Twice that number, or four groups of three, were used in one of the writer 's designs, but it is extremely improbable that a greater number than 16 could be practically combined in one structure. Assuming, then, a section of 16 cables of 24 inches diameter for the bridge of maximum span, we have max = 20,000 Ibs. Solving Eq. (2), Art. 15, for the length of span in terms of the cable- weight and live-load, there results ^ Substituting in this equation the max- imum value of C, as just established, and .assuming different values of the live- 118 load, we obtain the following values for the maximum span : For L = 0, Zmax= 9,500 ft. L= 10,000, Z max =4 , 900 ft - L= 15,000, Z max =4,000ft. L = 20,000, Cax^ 3 ; 500 ft - The first of the above results simply signifies that 9500 ft. is the span at which the cable will just be able to support the wind-bracing. As this condition is one of zero loading, it may be omitted from practical consideration. Furthermore, it is hardly probable that a structure of the magnitude under investigation would ever be planned for a lighter loading than about 10,000 Ibs. per linear foot, particularly when it is remembered that this would require about 10 Ibs. of steel for every pound of useful load. It will therefore suffice to restrict the conditions of the problem to the practical limits of L = 10,000 to L = 20,000 Ibs. of live-load per linear foot. We thus find that the maximum practical 119 span for suspension bridges ranges from 3500 to 4900 ft., depending upon the assumed live-load. ART. 18 EMPIRIC FORMULA FOR COST OF SUSPENSION BRIDGES The expression for the cost of any span (/) will be assumed of the general form ... (1) Since we have but three values deter- mined for C by actual design, the for- mula is limited to an equal number of terms. From the results of the preceding estimates, we have C = $ll,645,000 for Z= 1500 ft. C = $23,760,000 for I = 2250 ft. C = $46,785,000 for Z = 3000 ft. Substituting these values in (1), and solving the resulting three equations 120 for the unknown coefficients, we obtain C = 8900Z-3.77Z 2 + .0020Z 3 . . (2) as the general cost-formula for suspen- sion bridges. This gives the combined cost of steel work and substructure for any span for an assumed live-load of 18,000 Ibs. per linear foot. For any other loading, the above coefficients should be changed in proportion. ART. 19 ECONOMIC SPAN FOR SUSPENSION BRIDGES As established in the preceding article, the cost of a suspension bridge is given by the expression C = 8900Z-3.77Z 2 + .0020Z 3 . . (1) Of this cost, about 65 per cent repre- sents the steel work and the remainder provides for the masonry and anchorages. A study of the contracts for recent long-span bridges shows an additional cost of about 20 per cent for pavements, 121 tracks, railings, ornamental work, elec- tric lighting, etc. To this must be added the cost of terminal structures and real estate for the approaches. In a city structure the last item, as in the case of the Manhattan Bridge, may amount to more than the cost of the bridge itself. A fair average value for this item is about 100 per cent. Adding the above items, we find Total first cost of bridge = 220% C. (2) The rate of interest will be assumed at 5 per cent. The cost of maintenance, including repairs, painting, lighting, etc., averages about 4^ per cent of the cost of the superstructure. In addition, a certain annuity must be set aside for the periodic renewal of the superstructure. The foundations may be permanent, but the steel work has a limited life. The life-periods of various suspension bridges, or the periods before recon- struction or removal, whether terminated by failure or increased traffic demands, have been as follows: 122 1827 1 Hammersmith (London) .55 years. 1829 2 Regnitz (Bamberg) ... .59 " f 1 O it 1834 2 Freiburg (Switzerland) fir , , 1839 * Weser (Hamlin) 51 li 1845 l Neckar (Mannheim) 46 " 1850 Fairmount (W. Va.) . . .40 " 1851 3 Niagara (Old S. B. at Lewiston) 13 " 1852 4 Charleston (W. Va.) .... 52 " 1855 3 Niagara Falls (Railway Bridge) 42 ' ' 1867 5 Cincinnati (Ohio R.) . . .31 " 1868 * Moldau (Prague) 32 " ( 1 Q 1869 3 Niagara Falls (Hwy. Bridge) 1877 6 Point Bridge (Pittsburg) .28 Mean Life Period 31 years. x Melan, " Konstruktion der Hangebriicken >k (Leipzig, 1906), p. 204. 2 Nouv. aim. de la constr., 1881. Also Riese, "Die Ingenieurbauwerke der Schweiz." 3 Eng. Record, 1897 (Apr. 24); 1899 (Aug. 26). 4 Eng. News, 1905 (Feb. 2). Eng. Record, 1904 (Dec. 24). 5 Eng. Record, 1898 (Sept. 10, Nov. 26). 6 Eng. Record, 1905 (May 6-13). 123 This value , or 30 years in round numbers, will be adopted as the probable life of the steel work of a suspension bridge. It is true that the newer bridges possess the advantages of improved material and construction, but the more severe traffic to which they are subjected and the smaller margin of safety provided in their design prevent them from attain- ing as long a life as some of the old structures. For a railroad bridge strained to its full capacity, the above value of the life-period is certainly not too small. At 5 per cent compound interest, a sinking fund to meet the cost of re- newal in 30 years will require an annuity of 1.505 per cent. (See Annuity Tables.) The annual charge against the bridge will therefore consist of the following items : Interest charge = 5% X 220% C = ll%C Repairs and mainte- nance =4.5% X 65% C= 3% C Depreciation = Annuity for renewal in 30 years = 1.505% X 65% C = 1% C Total annual charge =15% C 124 The limiting economic span is that at which the revenue from traffic (T) just balances the annual cost of the structure. We may therefore write 15%C=T. ... (3) as the defining condition for the economic span. In determining the maximum traffic returns (T) to be expected from a long- span bridge, we are guided by the following considerations : The Brooklyn Bridge now carries about 118,000,000 paid passengers per annum. 1 The max- imum daily is 30 per cent greater than this rate and the maximum hourly is 500 per cent greater still. 2 The Williams burg Bridge, opened 20 years later, is a close competitor, with a pas- senger traffic of 75,000,000 per annum 3 ; at the present rate of increase, it will 1 Report of Public Service Comm., 1st District, N. Y., 1909. 2 Engineering Record, 1910 (June 11). 3 Engineering News, 1910 (Jan. 27). 125 soon equal the older structure in volume of traffic. The usefulness of both these structures is steadily growing, despite the opening of several competing routes of communication across the same river. Each bridge helps to build up the dis- tricts which it connects, thereby creat- ing increased traffic for itself. Thus there were 9,000,000 passengers crossing the Brooklyn Bridge in 1884, 1 42,000,000 passengers in 1893, 1 and 118,000,000 passengers in 1909. 2 The total number of passengers annually crossing the East River increased 500 per cent in the 17 years after the opening of the Brooklyn Bridge, 3 and 240 per cent more since the Williamsburg Bridge was opened. 4 These facts indicate that 1 Statement of Chas. Macdonald to Bd. of Eng. Officers, 1894; also Engineering News r 1893 (Feb. 23). 2 Report of Public Service Comm., 1st Distaict, N. Y., 1909. 3 Lindenthal in Discussions at the N. Y.. R. R. Club, Apr. 18, 1901. 4 Engineering News, 1910 (Jan. 27). 126 any large bridge, if judiciously located, will ultimately get all the traffic it can .accommodate. In the absence of any better guide, let us take the amount of travel on the Brooklyn Bridge as the maximum traffic to be expected on any other long-span structure. This value may justly be augmented by 50 per cent for the greater capacity of a six-track bridge, but this additional profit will be disregarded to compensate for the early years of un- developed traffic. We will therefore assume 118,000,000 passengers using the bridge in a year. For a span of the length under consideration, a toll of 5 cents per passenger would be an equitable rate. The same charge is now made for ferry or tunnel transporta- tion across the Hudson River, and bridge travel would surely be preferred for its greater speed and comfort. At night, when the passenger traffic is a minimum, the tracks can be used for the transportation of freight. There are about 6000 cars of freight, inbound 127 and outbound, at Jersey City daily, and about half of this belongs to New York. We may therefore safely count on at least 1500 cars of freight daily crossing a six-track bridge over the Hudson River or any similar location. The total annual traffic over a long- span bridge may therefore attain the following value: 118,000,000 passengers at 5 cents = $5,900,000 547,500 cars of freight at $4= 2,190,000 /. T= total annual revenue =$8,090,000 Substituting this value in eq. (3), we obtain C= $54,000 ,000 as the maximum economic cost for a long-span bridge. With this value of C, the solution of eq. (1) yields Economic 1 = 3170 ft. In the case of the proposed Hudson River Bridge, of somewhat shorter span, calculations indicated a profit of less 128 than 1 per cent on the investment, 1 thus confirming the above result. Conclusion. The limiting economic span for suspension bridges is about 3170 ft., and will be less wherever the probable traffic returns are smaller than assumed in this investigation. 1 H. G. Prout in Discussions at the N. Y. Railroad Club, Apr. 18, 1901. 129 CHAPTER V STUDY OF CANTILEVERS ART. 20 CANTILEVER BRIDGES HISTORICAL SKETCH IN adaptability to long spans and possibility of erection without false- work, the cantilever is the sole compet- itor of the suspension bridge. Both of these types have attained prominence by remarkable examples of design and construction. With but one exception, however, the suspension type has never been employed for fast railway traffic. Its use has been confined to highway bridges or wherever aesthetic require- ments prevailed. The banner bridges of the suspension type, those over the East River, have been attributed to " an ingrained fad of the New York 130 Department 01 Bridges." For long rail- way spans, the cantilever has almost invariably been given the preference on account of its superior rigidity at a given cost. The longest span in the world (Forth Bridge 1710 ft.) is of the cantilever form, and the Quebec Bridge, now under construction, will raise the record for length of span to 1800 ft. Although cantilever design is a com- paratively recent development in en- gineering, the idea is by no means a new one. Bridges of logs, put together on the cantilever principle, have been used in tropical countries since prehistoric times. 1 In 1783, a wooden cantilever bridge of 112-ft. span was reported by travelers in Thibet. 2 An 1800-ft. " fly- ing pendant lever bridge " to cross the East River and a 3000-ft. span across the North River were proposed by Pope in 1810. 3 Fairbairn's proposal 1 Van Nostrand's Magazine, Jan., 1886. 2 Pope, Treatise on Bridge Architecture, New York, 1811. 3 Ibid. 131 for the Britannia Bridge, in 1845, was a cantilever design. 1 Stephenson, in 1846, and Edwin Clark, in 1850, sug- gested the cantilever idea. 2 In the latter year Sir John Fowler built a wooden model to illustrate the form of construction. 3 In 1859 Prof. Ritter of Hanover proposed cutting the chord of a continuous truss at the points of con- traflexure and worked out the stresses in the resulting cantilever structure for a span of 526 ft. The first canti- lever design actually constructed, how- ever, was a bridge of 124 ft. span over the Main River near Hassfurt, designed and built by Gerber in 1867. 4 On this account, cantilevers are known as " Ger- ber Bridges " on the continent. In 1871, Fowler and Baker built two cantilever spans of 800 ft. over the Severn, and in 1873 Baker designed a 1 Engineering (London), Mar. 5, 1886. 2 Engineering (London), Feb. 28, 1890. 3 Proc. of the Inst. of C. E., IX, p. 256. 4 Mehrtens, "Der deutsche Briickenbau im 19. Jahrundert," Berlin, 1900. 132 cantilever ferry bridge of 650 ft. span over the Tees. 1 The first cantilever bridge for railway traffic, a span of 148 ft., was built in 1876 over the Warthe at Posen. 2 In the same year, the first American cantilever, the Kentucky Via- duct, was built by C. Shaler Smith. 3 In this structure and in the Niagara Cantilever built in 1883 by C. C. Schneider for the Michigan Central Railroad, the possibility of erection without false- work was first demonstrated. 4 In 1881, four years after the completion of the Kentucky Viaduct, the final designs for the Forth Bridge were approved. The successful completion of that re- markable structure in 1889 5 marked the end of the experimental period for the cantilever and served to fix that type in its present dominant position in long-span construction. 1 Engineering News, Nov. 24, 1904. 2 Engineering (London), Feb. 28, 1890. 3 Trans. Am. Soc. of C. E., Nov., 1878. 4 Trans. Am. Soc. of C. E., Nov. 1885. 5 Engineering (London), Dec. 6, 1889. 133 A table of the most noted cantilever bridges with their principal dimensions is appended. It will be observed that only three cantilevers exceed 1000 ft. in span, whereas eight suspension bridges have exceeded this value. It is mainly between the limits of 500 and 1000 ft. that cantilevers predominate. The high values of the ratio of dead-load to live- load are significant, indicating a greater expenditure of metal than is required in suspension bridges. It will also be noticed that there are comparatively few cantilevers below 500 ft. span, that being the domain of the ordinary truss. With increasing span- length, however, the cantilever bridge becomes superior to the simple truss be- cause of the increasing significance of the following advantages. 1. No obstruction of the channel during erection and the saving of the cost of falsework. 2. Lower economic depth of truss. 3. Smaller required width resulting in a saving in the piers and in the floor system. 134 TABLE II NOTABLE Date. Name. Location. Engineer. 1908 1 Quebec St Lawrence R. Cooper 1889 Forth Scotland .... Fowler & Baker 1908 Queensboro. . Lansdowne EastR., N. Y... India Dept. of Br. 1903 j[ Monongah ela Pittsburg Boiler & Hodge 1892 Memphis .... Mississippi R. . . Morison 1910 Beaver Ohio R., Pa 1903 Ohio R Boiler & Hodge 1905 Thebes Miss. R., 111. ... Modjeski & Noble 1904 Ruhrort . . . Rhine R , Ger- 1891 Red Rock Colorado R 1902 Marietta Ohio R Ohio 1902 1887 Czernavod . . . ^ Poughkeepsie DanubeR Hudson R 1906 Long Lake New York 1903 1897 Connel Ferry. Francis Joseph .... Scotland Budapest Barry 1883 1876 1883 Niagara 3 Kentucky R. Fort Snelling Niagara Falls. . . Ohio Mississippi R., Minn Schneider C. S. Smith Schneider 1 Collapsed 1908, before completion. 2 Reconstructed 1906. 4. More favorable distribution of the dead-load, the material being massed toward the piers. 135 CANTILEVER BRIDGES Loading. Sp'n = L S.S. =1 C.A. = 771 A.A. = n Width Depth. L.L. D.L. D.L. L.lT 1800 675 562 500 67 98-315 13000 26000 2.0 1710 350 680 680 32-120 50-350 4480 21000 4.7 1182 591 630 60 45-185 8440 27000 3.2 820 200 310 248 812 360 226 346 32 60-126 9000 9000 1.0 790 452 169 621 30 78 4000 7000 1.75 769 285 242 320 11000 700 310 195 298 9000 8000 0.9 671 366 152i 521 32 50-75 8000 10000 1.25 667 443 112 390 36 46-82 660 25 650 270 300 600 27 623 548 208 170 525' 30 38-75 6000 525 175 175 0~ 24 20-60 920 840 0.9 524 232 146 106 21 30-118 4000 7620 1.9 514 102 ,206 257 6000 495 120 187i 20 7i 28 3000 375 300 75 375 18 38 2530 0.85 315 105 105 105 3 Replaced by truss bridge, 1910. 5. More favorable distribution of the wind-load, for the same reason. 6. Decreased wind-load stresses. 136 In order to establish data for deter- mining the economic and maximum spans, complete designs and estimates will be made for cantilevers of three different spans: 1000, 1500 and 2000 ft., respectively. The condition of load- ing and specifications for allowable working stresses will be assumed the same as for the suspension bridges pre- viously designed. Before we can proceed with the design of the cantilevers, it is necessary to find the most favorable solutions of the following problems. 1. Economic span-ratios for canti- levers. 2. Economic width for cantilevers. ART. 21 ECONOMIC SPAN-RATIOS FOR CANTILEVERS Let Z = length of suspended span; m = length of cantilever arm; n = length of anchor arm. 137 Then the total channel span will be L-Z+2m .... (1) and the total length of structure will be ... (2) Assuming that the weight per linear foot of truss in any span or arm is equal to some constant factor times the length of that span or arm, let a = suspended span factor; 6 = cantilever arm factor; c = anchor arm factor. Then the weight of the whole structure will be W=aP + 2bm 2 +2cn 2 . . (3) This will be a minimum for al = bm = cn ... (4) Hence the economic lengths of the sus- pended span, cantilever arm and anchor arm are inversely proportional to the 138 respective dead -load factors; in other words, the economic lengths are such as to make the weight per linear foot uniform over all parts of the bridge. From (1), (2) and (4) we obtain l 1- b T I/ 7 ^ * JLJ . ffli 7 ^ LJ . li ^ - b+2a b+2a Numerical Values. The weight fac- tors a, 6 and c should be obtained by actual computation of weights of struc- tures similar to those under considera- tion. The resulting values will, of course, be found to vary more or less with dif- ferent loadings and span-ratios. By omitting from consideration all spans of unusual proportions, and reducing the results to a common assumed load- ing (=18,000 Ibs. per linear foot as in the following designs), the writer obtained 1 Cf. Burr's solution of the problem in his " Stresses in Bridges and Roof Trusses, etc." (Wiley, 1908), App. V., p. 472. 139 the following average values of the truss- weight factors: a =14, 6 = 42, c = 21.! Substituting these values in Eqs. (5), we obtain Z=0.60 L, ra = 0.20 L, n = 0.40 L, for the economic span-ratios of a canti- lever bridge. Prof. Burr 2 recommends values of Z=0.5 to 0.55 L; ra=0.20 L; n = 0.42 to 0.5 L. An exact theoretical solution of the problem, 3 eliminating the use of em- pirical constants, shows that the total moment areas will be a minimum for Z=0.68L; w=0.16L, n = 0.37 L. 1 Compare these values with those yielded by the writer's designs, viz.: a=13.6, 6 = 42.3, c=21.7. 2 Burr's " Stresses in Bridges and Roof Trusses, etc." (Wiley, 1908), App. V, p. 472. 3 Marburg in Proc. of Eng. Club of Phila., July, 1896. 140 This solution is practically defective, however, in disregarding the web-mem- bers and lateral bracing and in assuming uniform dead-load over the whole struc- ture. A study of the spans of the cantilevers tabulated in a preceding article, omitting . those of extraordinary form, yields the following results : Extreme values: l/L m/L n/L .20 -.66 .17 -.40 .28 -.78 Mean values: .38 .31 .45 These values compared with those established above indicate that, in past practice, the suspended span as a rule has been made too long and the cantilever arms too short for the best economy. For the designs in this investigation, the writer has adopted a compromise between the dictates of theory and those of conformity with past practice. The 141 lengths of the suspended spans will be made for L=1000 1500 2000, 1= 500 650 800, so that l/L= 0.5 0.43 0.4. The suspended span is thus made a diminishing fractional part of the total span as the latter increases in length; otherwise the length of the suspended truss would become prohibitive before the limiting cantilever span is reached. (See Art. 26.) The lengths of the anchor arms will be made thereby conforming with the ratio indi- cated by both theory and practice. This determination of the best length for the anchor arm, however, is purely academic as, in any actual design of a cantilever, the location of the piers is 142 determined by natural conditions and the requirements of navigation rather than from any theoretical investigations. ART. 22 MINIMUM WIDTH FOR CANTILEVERS 1 On account of the saving in piers and floor members, it is desirable that the width of any bridge structure should be a minimum consistent with the demands of lateral rigidity. In simple truss bridges it is generally considered that adequate stiffness is provided by a distance c. to c. of trusses of ^ of the .span. The minimum width for canti- levers may then be defined as that width which will insure the same degree of rigidity as the above. Using the same notation as in the preceding article, the total length of span for a cantilever bridge is denoted by L = l + 2m . . . .(1) 1 Adapted from an analysis given to his classes by Prof. Burr. 143 If a lateral load, p per linear foot, acts on the structures, the central deflection will be ~El384, 2 3 8 This reduces to and for Z = 0.5L for l = OAL . (3) In a simple truss of span V the central deflection is given by h--~r* (& ~384 El With equal rigidity, the simple truss and the cantilever must suffer the same deflections. Equating the right-hand members of Eqs. (3) and (4), there .result Z' = .668L for Z = 0.5L ] and [ . (5) r = .701L for 1 = AL 144 The distance c. to c. of trusses should not be less than r "=18' or, substituting the relations of Eq. (5) Minimum w== to L. . (6) i Zo It is interesting to note here that the mean width-ratios for the structures listed in the Table of Cantilevers is a value on the safe side of the required ratio. In the designs at hand a ratio of w = L will be adopted except in the o shortest span where the required space for the roadway will necessitate a slightly greater width-ratio, namely, 145 BIBLIOGRAPHY ON CANTILEVERS Quebec. Engineering (London), 1900 (II, 189, 241); 1902 (II, 419); 1903 (I, 92); 1905 (II, Sept. 22). Canadian Engineer, 1902 (May). Engineering News, 1897 (Oct. 14); 1902 (Nov. 20); 1905 (Sept. 14); 1906 (July 6); 1907 (Oct. 31). Engineering Record, 1905 (Mar. 4, Apr. 1, Apr. 8, Sept. 16); 1906 (June 23, Dec. 29); 1908 (Mar. 14, Apr. 25); 1910 (Sept. 24, Oct. 1, Oct. 15). Railroad Gazette, Vol. XXXIX, No. 11. Forth. Engineering (London), 1882 (Sept. 1, 8); 1883, 1884, 1885 (Feb. 6); 1886, 1887, 1888, 1889 (Dec. 6); 1890 (Feb. 28, pp. 213-283). Queensboro. Engineering Record, 1901 (Mar. 16); 1902 (Dec. 13); 1903 (Feb. 28, Aug. 22); 1904 (Mar. 3); 1905 (Mar. 4, 18, May 20, 27, June 10); 1906 (Jan. 6, 27, Feb. 10, 17, 24, Mar. 3, 17, Sept. 15); 1907 (Dec. 21); 1908 (Apr. 11, May 28, Aug. 8, Nov. 14). Engineering News, 1907 (Nov. 23); 1908 (Nov. 12, 19). Iron Age, 1907 (Mar. 7). 146 Railroad Gazette, 1901 (Aug. 16); 1903 (Oct. 2). Scientific American, 1903 (Sept. 19). Marietta (Ohio). Engineering News, 1902 (Nov. 20). Engineering (London), 1902 (II, p. 418), Engineering Record, 1903 (Sept. 26). Railroad Gazette, 1902 (July 11). Monongahela (Pittsburg). Engineering News, 1902 (Nov. 20). Engineering (London), 1902 (II, p. 417). Railroad Gazette, 1902 (Mar. 14); 1903 (Aug. 14). Engineering Record, 1902, 1903 (Jan. 3); 1904 (Mar. 5, Apr. 2, Apr. 9). Memphis (Tenn.). Engineering News, 1892 (Feb. 27, Mar. 12, May 12, June 16, Aug. 11, Sept. 15). Mingo Junction. Engineering News, 1902 (Nov. 20). Engineering (London), 1902 (II, p. 417). Engineering Record, 1904 (June 25, July 2); 1903 (Oct. 3). Thebes (111.). Railway Age, 1902 (June 20). Engineering News, 1902 (Nov. 20); 1905 (May 11). Engineering (London), 1902 (II, p. 416); 1903 (I, p. 326). 147 Engineering Record, 1904 (Nov. 12, Dec. 24); 1905 (Mar. 4); 1905 (July 22, Sept. 16). Trans. Assn. of Civ. Eng. of Cornell Univ., 1905. Engineer (London), 1905 (July 21). Railroad Gazette, 1903 (Jan. 9); Vol. XXXVIII, No. 21. Ruhrort (Germany). Schweizerische Bauzeitung, 1904 (June 18). Engineering (London), 1904 (II, 7, 35). Engineering News, 1904 (July 14). Engineer (London), 1907 (Nov. 1). Zeitschrift d. Ver. Deutscher Ing., 1904 (July 2). Connel Ferry (Scotland). Engineer (London), 1903 (Sept. 11). Engineering (London), 1903 (July 31). Engineering Record, 1904 (July 2). Poughkeepsie (N. Y.). Engineering (London), 1887 (May 13). Engineering Record, 1905 (Oct. 28); 1906 (Aug. 18). Long Lake (N. Y.). Engineering Record, 1906 (Sept. 29). Francis Joseph (Budapest). Zeitscrhrift d. Oesterr. Ing. u. Arch. Ver., 1897 (Feb. 26). G 1- O II N 250 500 750 1000 1250 1500 $ 2,023,000 3,758,000 5,395,000 7,130,000 9,140,000 11,645,000* $ 4,040 3,760 3,595 3,565 3,655 3,880 32.3 15.0 9.6 7.1 5.8 5.2 4.5 4.7 4.7 4.8 4.9 5.2 5.5 5.8 6.1 6.4 $ 1,343,000 2,288,000 3,410,000 4,-905,000* 7,035,000 - 10,570,000* $ 2,980 2,540 2,525 2,725 3,130 3,910 21.6 9.2 6.1 4.9 4.5 4.7 5.0 5.5 6.1 6.8 7.5 8.3 9.1 10.0 10.7 11.5 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 14,130,000 18,720,000 23,760,000* 29,950,000 37,390,000 46,785,000* 57,710,000 71,050,000 85,780,000 103,200,000 4,040 4,680 5,280 5,990 6,790 7,780 8,870 10,160 11,450 12,900 15,370,000 22,125,000* 30,990,000 42,570,000 56,910,000 74,650,000 95,740,000 121,120,000 150,710,000 184,900,000 4,880 6,150 7,640 9,450 11,500 13,850 16,360 19,240 22,300 25,680 C=8900Z-3.77Z 2 +0.002Z 3 C * Values obtained by aptual design. Other values interpolated or extrapolated by aid of empiric formulae given below. 3. The cost per foot of length for cantilevers is a minimum for a span of about 600 ft., indicating a rapid decrease of economy in the use of the type for shorter spans. 176 4. Between the limits of 1500 and 3000 ft. for suspension bridges, and of 1000 and 1750 ft. for cantilevers, the value of K( = C + l 2 ) remains fairly con- stant. Hence, for all normal spans, the cost of either type may be estimated as varying approximately with the square of the span. Within the above ranges, as an average value, C = 4.7l 2 for suspension bridges and C = 4.8 L 2 for cantilevers. 5. The value of K for the suspension bridges is a minimum for a span of about 1800 ft., indicating that that is the span for which the suspension type is economi- cally best adapted. 6. The value of K for the cantilevers Is a minimum for a span of about 1250 ft., indicating that that is the span for which the cantilever type is economically best adapted. 7. The limiting economic span, i.e. the greatest span for profitable erection, defined by C = $54,000,000, is shown 177 by the table to be about 3200 ft. for the suspension type and about 2700 ft. for the cantilever type. The tabulated costs, where they lie outside of the spans actually designed, are subject to the inherent errors of any method of extrapolation; but the above results are sufficiently accurate for the purposes of this investigation. ART. 32 SPAN OF EQUAL COST PLOTTING the costs for the three sus- pension spans designed above, a smooth curve passing through the three points and the origin constitutes a cost-graph for suspension bridges. In the same manner, the graph representing the costs for different cantilever spans is con- structed. The point where the two curves intersect marks the span of equal cost for the two types of construction > and is found at 1=1670 ft. 178 Below this span, the cantilever bridge is cheaper; above this span, the sus- pension bridge exhibits the greater econ- omy. As this value lies within the range of spans actually designed for each bridge, i.e., between the known points on each graph, the error of the above method, involved in extending the results of individual designs to other spans, is negligible. The analytical equivalent of the above process consists in comparing the cost- formula for the two types of bridges. The cost of a suspension bridge, designed for a live-load of 18,000 Ibs. per linear foot, as established in Art. 19, is given by C=890(M-3.77P +.0020P, (1) and that of a cantilever bridge for the same loading, as established in Art. 39, (2) where I denotes the total channel span. Equating expressions (1) and (2), and 179 COST GRAPH ^ FOR SUSPENS,ION|BRI|DGES AND iLEl VERS_ 1-jJI / oi u / of: 1 / m.r 1 ^ Z^ 250 500 750 1000 12501500 1750 2000 2250 25002750 3000 32503500 3750 SPAN (IN FT.) 180 solving, we find, for the span of equal cost, 1 = 1670 ft. exactly as before. The above comparison is somewhat unfair to the suspension type, as the side- spans in the above designs were 0.5Z in the suspension bridges and only OAl in the cantilevers. Assuming, therefore, that steel viaduct approaches are added to the cantilever bridges to make up the difference in total length, and esti- mating the cost of such viaducts at $1000 per linear foot/ the expression (2) becomes modified to C = 6550/-5.25Z 2 -f.0038Z 3 . . (3) Equating this to expression (1) and solving for the span of equal cost, we find 1= 1626 ft,, somewhat less than the value established above. If the comparison of costs had 1 This is the price assumed in a similar case by the U. S. Board of Engineers in 1894. 181 been made between bridges of equal total length instead of between bridges of equal channel span, the result would have been still more favorable to the suspension bridge, reducing the critical span to about J 1500 ft, Neglecting these differences in favor of the suspension type, we conclude that ^ = 1670 ft. is the extreme upper limit of spans at which the cantilever can com- pete with the suspension bridge, when economy is the sole criterion. ART. 33 SUMMARY The results of the preceding investiga- tions may be summarized as follows: 1. Maximum Span for a Cable. The .greatest span theoretically possible for a steel cable of any cross -sect ion is 65,520 ft. based on the ultimate resistance; 21,840 ft. based on a safe working- stress of 60,000 Ibs. per sq.in.; 15,100 ft* 182 if the rise is restricted to the economic ratio of one-eighth the span; and 14,700 ft. if the weight of cable-wrapping and fastenings is taken into consideration. 2. Maximum Span for Suspension Bridges. The last value, 14,700 ft., is also the maximum span theoretically possible for a stiffened suspension bridge. The greatest practicable span, defined by a maximum section of 16 cables of 24 inches diameter with a minimum live- load of 10,000 Ibs. per linear foot, is 4300 ft. 3. Economic Span for Suspension Bridges. The greatest suspension span, for which the necessary outlay would be warranted by the probable traffic returns is 3170 ft. 4. Maximum Span for Cantilever Bridges. The greatest span theoretically possible for a cantilever bridge is 5600 ft. In this maximum span, the sus- pended span will be 1850 ft., the canti- lever arms 1875 ft., and the anchor arms 2760 ft. The practical limit for canti- levers, denned by a maximum chord- 183 section of 3000 sq.in., with a minimum live-load of 10,000 Ibs. per linear foot, is 3060 ft. 5. Maximum Span for Truss. The greatest span theoretically possible for a simple truss is 1960 ft. 6. Economic Span for Cantilever Bridges. The maximum economic span for cantilevers, denned by the condition of zero net profit on the investment, is 2700 ft. 7. Span of Equal Cost. The critical span at which the suspension bridge becomes economically superior to the cantilever bridge is 1670 ft. Summary: s B c % Theoretical max. span. . 14,700 5,600 Practical max. span 4,900 3,060 Max. economic span. . . . 3,170 2,700 Span of equal cost 1,670 ART, 34 CONCLUSIONS In the foregoing designs, special care was taken to proportion the depths of truss in both types for equal and ample 184 rigidity , so that the single inherent advantage claimed for cantilevers, viz. greater stiffness for railway traffic, is eliminated from consideration. Con- sequently no advantage remains to the cantilever type above the limiting span of 1670 ft. The suspension bridge, on the other hand, is universally ad- mitted to possess greater aesthetic qual- ifications; and, for a long-span city structure, this is a factor of decisive importance. For this reason alone the above value for the span of equal cost should be considerably reduced in favor of the suspension type, to obtain the "span of equal merit." The preceding investigations have been restricted to designs for heavy railway traffic. For highway bridges, a rela- tively lighter stiffening truss may be used in the suspension type, thereby causing a considerable reduction in the span of equal cost. Confining our attention to economic considerations, our final conclusions may be stated as follows: 185 1. The range of economic usefulness for cantilevers extends from the upper limit for the truss or arch to a span of 1670 ft. Beyond this value , the cantilever would be more costly than the suspension type, although yielding a probable profit on the investment up to a span of 2700 ft. 2. The range of economic usefulness for suspension bridges begins at 1670 ft. (or less in the case of highway bridges) and extends to the upper economic limit of 3170 ft. Above this limit, the construc- tion of suspension bridges would be practically feasible, but not as a profitable investment, up to an extreme limit of 4900ft. Vita Born June 11, 1886. B. S. (Summa cum laude) , College of the City of New York, 1906. A. M. 9 C. E., Columbia Uni- versity, 1909. Ph. D., 1911. Fellow in Applied Mathe- matics, College of the City of New York, 1906-1909. Tutor in Surveying 1909-1910. James Scholar in Applied Science, Columbia University, 1907-1909. University Scholar in Engineer- ing, 1909-1910. Instructor in Applied Physics, Stuyvesant Eve- ning High School, 1909-1910. 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By Willifl Wallace Christie. Illustrated. Second e