CHAPTER XIII.

=131. Cantilever and Stiffened Suspension Bridges.=—There are two other types of bridges of later development which have, in recent years, become prominent by remarkable examples of both completed structure and design; they are known as the cantilever and stiffened suspension bridges. Both are adapted to long spans, although the latter may be applied to much longer spans than the former. A cantilever structure, with a main span of 1800 feet between centres of piers, is now in process of construction across the St. Lawrence River at Quebec, while the well-known Forth Bridge across the Firth of Forth in Scotland has a main span of 1710 feet. The longest stiffened suspension bridge yet constructed is the New York and Brooklyn Bridge, with a river span of about 1595.5 feet between centres of towers, but the stiffened suspension system has been shown by actual design to be applicable to spans of more than 3200 feet, with material now commercially produced.

[Illustration: FIG. 41.]

[Illustration: FIG. 42.—Monongahela Bridge—Pittsburgh, Carnegie & Western Railroad (Wabash), at Pittsburgh.]

=132. Cantilever Bridges.=—Figs. 41 and 42 exhibit in skeleton outline two prominent cantilever designs for structures in this country. That shown in Fig. 41 was intended for a bridge across the Hudson River between Sixtieth and Seventieth streets, New York City. The main central opening has a span of 1800 feet, and a length of 2000 feet between centres of towers. Fig. 42 shows the Monongahela River cantilever bridge,[3] now being built at Pittsburgh, Penn. Both figures exhibit the prominent features of the cantilever system. The main parts are the towers, at each end of the centre span, which are 534.5 feet high in the North River Bridge and 135 feet high in the Monongahela River structure, and the central main or river span with its simple non-continuous truss hung from the ends of the cantilever brackets or arms which flank it on both sides. These cantilever arms are simply projecting trusses continuous with the shore- or anchor-arms. They rest on the piers at either end of the main span, as a lever rests on its fulcrum. This arrangement requires the shore extremities or the anchor-arms to be anchored down by a heavy weight formed by the masonry piers at those points. Recapitulating and starting from the two shore ends of the structure, there are the anchor-spans, continuous at the towers, with the cantilever arms projecting outward toward the centre of the main opening and supporting at their ends the suspended truss, which is a simple, non-continuous one. It is thus evident that the cantilever bridge is a structure composed of continuous trusses with points of contraflexure permanently fixed at the ends of the suspended span. The greatest bending moments are at the towers, and the great depth at that point is given for the purpose of affording adequate resistance to those moments by the members of the structure. The following statement shows some elements of the more prominent cantilever bridges of this country and of the Forth Bridge:

[3] This bridge was designed by and is being constructed under the direction of Messrs. Boller and Hodge, Consulting Engineers, New York City.

Length of Cantilever Name. Opening, Centre to Total Length. Centre of Towers.

Pittsburgh 812 feet. 1504 feet. Red Rock (Colo.) 660 ” 990 ” Memphis (Tenn.) 790.48 ” 2378.2 ” Forth 1710 ” 5330 ”

The arrangement of web members of cantilever structures is designed to be such as will transfer the loads from the points of application to the points of support in the shortest and most direct paths. Both Figs. 41 and 42 show these general results accomplished by an advantageous arrangement of web members.

It is interesting to note that the first cantilever bridge designed and built in this country was constructed in 1871. This structure was designed and erected by the late C. Shaler Smith, a prominent civil engineer of his day.

=133. Stiffened Suspension Bridges.=—The stiffened suspension bridge is a structure radically different in its main features and its mode of transferring load to points of support from any heretofore considered, except arched ribs. When a load is supported by a beam or truss, the stresses, either in the web members of the truss or in the solid web of the beams, travel up and down those members in zigzag directions with a relatively large amount of metal required for that kind of transference. That metal is represented by the weight of the web members of the truss and of the solid web of the beam. Again, there are two sets of truss members—the chords or flanges, one of which sustains tension and the other an equal amount of compression. The greater part of this metal must be so placed and used that the working intensities of stress are comparatively small. This is particularly the case in compression members of both chords and webs which constitute the greater portion of the weight of the truss. All compression members are known as long columns which sustain not only direct compression but bending, and the amount of stress or load which they carry per square inch is relatively small, decreasing as the length increases. For all these reasons the amount of metal required for both beams and trusses is comparatively large. In suspension bridges, however, the conditions requiring the employment of a relatively large amount of metal with relatively small unit stresses are absent. The main members of a suspension bridge are the cables and the stiffening trusses, the latter being light in reference to the length of span. The cables are subjected to tension only, which is the most economical of all methods of using metal. A member in tension tends to straighten itself, so that it is never subjected to bending by the load which it carries. The opposite condition exists with compression members. Again, grades of steel possessing the highest ultimate resistance may be used in the manufacture of cables. It is well known that wire is the strongest form in which either wrought-iron or steel can be manufactured. While the ultimate tensile resistance of ordinary structural steel will seldom rise above 70,000 pounds per square inch, steel wire, suitable to be used in suspension-bridge cables, may be depended upon, at the present time, to give an ultimate resistance of at least 180,000 pounds per square inch. The elastic limit of ordinary structural steel is but little above half its ultimate resistance, while the elastic limit of the steel used in suspension-bridge cables is probably not less than three fourths of its ultimate resistance. It is seen, therefore, that the high resistance of steel wire makes the steel cable of the suspension bridge a remarkably economical application of metal to structural purposes.

The latest example of stiffened suspension-bridge is the new East River Bridge reaching across the East River from Broadway in Brooklyn to Delancey Street, New York City, now being built, with a main span of 1600 feet between centres of towers. The entire length of the metal structure is 7200 feet, and the elevation of the centres of cable at the tops of the towers is 333 feet above mean high water.

Fig. 43 shows a view of this bridge. Its three principal divisions are the cables, the stiffening trusses, and the towers. The latter afford suitable points of support for the cables, which not only extend over the river span, but are carried back to points on the land where they are securely attached to a heavy mass of anchorage masonry. These anchorages must be sufficiently heavy to prevent any load which may come upon the bridge from moving them by the pull of the cables. It is usual to make these masses so great that they are capable of resisting from two to two and a half times the pull of the cables.

[Illustration: FIG. 43.—New East River Bridge.]

=134. The Stiffening Truss.=—The function of the stiffening trusses is peculiar and imperatively essential to the proper action of the whole system. If they are absent and a weight should be placed upon the cable at any point, a deep sag at that point would result. If a moving load should attempt to pass along a roadway supported by a cable only, the latter would be greatly distorted, and it would be impossible to use such a structure for ordinary traffic. Some means must then be employed by which the cable shall maintain essentially the same shape and position, whatever may be the amount of loading. It can be readily shown that if any perfectly flexible suspension-bridge cable carries a load of uniform intensity over the span from one tower to the other, the curve of the cable will be a parabola, with its vertex at the lowest point. Furthermore, it can also be shown that if any portion of the span be subjected to a uniform load, the corresponding portion of the cable will also assume a parabolic curve. It is assumed in all ordinary suspension-bridge design that the total weight of the structure, including the cables and the suspension-rods which connect the stiffening trusses to the cable, is uniformly distributed over the span, and that assumption is essentially correct. So far as the weight of the structure is concerned, therefore, the curve of the cable will always be parabolic. It only remains, therefore, to devise such stiffening trusses as will cause any moving load passing on or over the bridge to be carried uniformly to the cables throughout the entire span. This condition means that if any moving load whatever covers any portion of the span, the corresponding pull of the suspension-rods on the cables must be uniform from one tower to the other, and that result can be practically accomplished by the proper design of stiffening trusses; it is the complete function of those trusses to perform just that duty.

=135. Location and Arrangement of Stiffening Trusses.=—It has been, and is at the present time to a considerable extent, an open question as to the best location and arrangement of the stiffening trusses. The more common method in structures built is that illustrated by the New York and Brooklyn and the new East River bridges. Those stiffening trusses are uniform in depth, extending from one tower to the other, or into the land spans, and connected with the cables by suspension-rods running from the latter down to the lower chords of the trusses. It is obvious that the floor along which the moving load is carried must have considerable transverse stiffness, and hence it may appear advisable to place the stiffening trusses so that the floor may be carried by them. On the other hand, some civil engineers maintain that it is a better distribution of stiffening metal to place it where the cables themselves may form members of the stiffening trusses, with a view to greater economy of material.

Figs. 44, 45, and 46 illustrate some of the principal proposed methods of constructing stiffening trusses in direct connection with the cables. The structure shown in Fig. 44 illustrates the skeleton design of the Point Bridge at Pittsburgh. The curved member is a parabolic cable composed of eye-bars. This parabolic cable carries the entire weight of the structure and moving load when uniformly distributed. If a single weight rests at the centre, the two straight members of the upper chord may be assumed to carry it. If a single weight rests at any other point of the span, it will be distributed by the bracing between the straight and curved members of the stiffening truss. Obviously the most unbalanced loading will occur when one half of the span is covered with moving load. In that case the bowstring stiffening truss in either half of Fig. 44 will make the required distribution and prevent the parabolic tension member from changing its form.

[Illustration: FIG. 44.]

[Illustration: FIG. 45.]

[Illustration: FIG. 46.]

The type of bracing shown in Fig. 45 possesses some advantages of a peculiar nature. Each curved lower chord of the stiffening truss corresponds to the position of the perfectly flexible cable with the moving load covering that half of the span which belongs to the greatest sag of the cable. The two parabolic cables thus cross each other in a symmetrical manner at the centre of the span. If the moving load covers the entire span, the line of resistance or centre line of imaginary cable will be the parabola, shown by the broken line midway along each crescent stiffening truss. The diagonal bracing placed between the cables is so distributed and applied as to maintain the positions of cables under all conditions of loading.

The mode of constructing the stiffening truss between two cables, shown in Fig. 46, is that adopted by Mr. G. Lindenthal in his design for a proposed stiffened suspension bridge across the Hudson River with a span of about 3000 feet. The two cables are parabolic in curvature and may be either concentric or parallel. This system of stiffening bracing possesses some advantages of uniformity and is well placed to secure efficient results. The same system has been used in suspension bridges of short span by Mr. Lindenthal at both St. Louis and Pittsburgh. The stiffening bracing produces practically a continuous stiffening truss from one tower to the other, whereas the systems shown in Figs. 44 and 45 involve practically a joint at the centre of the span.

In all these three types of vertical stiffness the floor is designed to meet only the exigencies of local loading, being connected with the stiffening truss above by suspension bars or rods, preferably of stiff section.

When stiffening trusses are placed along the line of the floor, as in the case of the two East River bridges, to which reference has already been made, those trusses need not necessarily be of uniform depth, and they may be continuous from tower to tower or jointed at the centre, like those of the New York and Brooklyn suspension bridge. This centre joint detracts a little from the stiffness of the structure, but in a proper design this is not serious.

=136. Division of Load between Cables and Stiffening Truss.=—In a case where continuous stiffening trusses are employed it is obvious that they may carry some portion of the moving load as ordinary trusses. The portion so carried will be that which is required to make the deflection of the stiffening truss equal to that of the cable added to the stretch of the suspension-rods. In the old theory of the stiffening truss constructed along the floor of the bridge this effect was ignored, and the computations for the stresses in those trusses were made by the aid of equations of statical equilibrium only. That assumption, that the cable carried the entire load, was necessary to remove the ambiguity which would otherwise exist. In modern suspension-bridge design those trusses may be assumed continuous from tower to tower with their ends anchored at the towers, or they may be designed to be carried continuously through portions of the land spans and held at their extremities by struts reaching down to anchorages, so that those ends may never rise nor fall, but move horizontally if required. If there are no pin-joints in the trusses at the centre and ends of the main span, equations of statical equilibrium are not sufficient to enable the reactions under the trusses and the horizontal component of cable tension to be found.

One of the best methods of procedure for such cases is that of least work, in which the horizontal component of cable tension is so found that the total work performed in the elastic deflection of the stiffening trusses, suspension-rods, cables, and towers is a minimum. After having found this horizontal component of the cable tension and the reactions under the stiffening trusses, the stresses in all the members of the entire structure can be at once determined. It is obvious that the stiffening truss and the cables must deflect together. It is equally evident that the deeper the stiffening trusses are the more load will be required to deflect them to any given amount, and hence that the deeper they are the more load they will carry independently of the cable. It is desirable to throw as much of the duty of carrying loads upon the cables as possible. It therefore follows that the stiffening trusses should be made as shallow as the proper discharge of their stiffening duties will permit.

=137. Stresses in Cables and Moments and Shears in Trusses.=—The necessary limits of this discussion will not permit even the simplest analyses to be given. It is evident, however, that the greatest cable stresses will exist at the tops of the towers, and that if the horizontal component of cable tension be found by any proper method, the stress at any other point will be equal to that horizontal component multiplied by the secant of cable inclination to a horizontal line, it being supposed that the suspenders are found in a vertical plane.

If the stiffening trusses are jointed at the centre of the main span, as well as at the ends, the simple equations of statical equilibrium are sufficient in number to make all computations, for the reason that the centre pin-joint gives the additional condition that, whatever may be the amount or distribution of loading, the centre moment must be zero. If _l_ is the length of main or centre span and _p_ the moving load per linear foot of span, and if the stiffening trusses run from tower to tower, the following equations will give their greatest moments and shears both by the old and new theory of the stiffening truss.

_p_ = load per lin. ft., _l_ = length of span in ft., Old theory. New theory.

Max. moment _M_ = 0.01856_pl_² _M_ = 0.01652_pl_²}no centre Max. shear _S_ = ⅛_pl_ _S_ = ⅛_pl_ } hinge. With centre hinge _M_ = 0.01883_pl_² and _S_ = ⅛_pl_

The details of the theory of stiffening trusses for suspension bridges have been well developed during the past few years and are fully exhibited in modern engineering literature. The long spans requiring stiffened suspension bridges are usually found over navigable streams, and hence those bridges must be placed at comparatively high elevations. This is illustrated by the clear height of 135 feet required under the East River suspension-bridge structures already completed and in progress. Furthermore, the heights of towers above the lowest points of the cables usually run from one eighth to one twelfth of the span. These features expose the entire structure to comparatively high wind pressures, which must be carefully provided against. This is done by the requisite lateral bracing between the stiffening trusses and by what is called the cradling of the cables. The latter expression simply means that the cables as they are built are swung out of a vertical plane and toward the axis of the structure, being held in that position by suitable details. The cables on opposite sides of the bridge are thus moved in toward each other so as to produce increased stability against lateral movement. Occasionally horizontal cables are stretched between the towers in parabolic curves in order to resist horizontal pressures, just as the main cables carry vertical loads. This matter of stability against lateral wind pressures requires and receives the same degree of careful consideration in design as that accorded to the effects of vertical loading. The same general observation applies also to the design of the towers.

=138. Thermal Stresses and Moments in Stiffened Suspension Bridges.=—All material used in engineering structures expands and contracts with rising and falling temperatures to such an extent that the resulting motions must be provided for in structures of considerable magnitude. In ordinary truss-bridges one end is supported upon rollers, so that as the span changes its length the truss ends move the required amount upon the rollers. In the case of stiffened suspension bridges, however, the ends of the cables at the anchorages are rigidly fixed, so that any adjustment required by change of temperature must be consistent with the change of length of cable between the anchorages. The backstays, which are those portions of the cables extending from the anchorages to the tops of the towers, expand and contract precisely as do the portions of the cable between the tops of the towers. As the cables lengthen, therefore, the sag or rise at the centre of the main span will be due to the change in the entire length of cable from anchorage to anchorage. In order to meet this condition it is usual to support the cables at the tops of the towers on seats called saddles which rest upon rollers, so as to afford any motion that may be required. Designs have been made in which the cables are fixed to the tops of steel towers. In such cases changes of temperature would subject the towers to considerable bending which would be provided for in the design.

The rise and fall at the centres of long spans of stiffened suspension bridges is considerable; indeed, for a variation of 120° Fahr. the centre of the New York and Brooklyn Bridge changes its elevation by 4.6 feet if the saddles are free to move, as intended. In the case of a stiffened suspension bridge designed to cross the North River at New York City with a main span of 3200 feet a variation of 120° Fahr. in temperature would produce a change of elevation of the centre of the span of 6.36 feet. Such thermal motions in the structure obviously will produce stresses of considerable magnitude in various parts of the stiffening trusses, all of which are invariably recognized and provided for in good design.

=139. Formation of the Cables.=—At the present time suspension-bridge cables are made by grouping together in one cylindrical mass a large number of so-called strands or individual small cables, each composed of a large number of parallel wires about one sixth of an inch in diameter. The four cables of the New York and Brooklyn Bridge are each composed of 19 strands, each of the latter containing 332 parallel wires, making a total of 6308 wires, the cables themselves being 15½ inches in diameter. The wire is No. 7 gauge, i.e., 0.18 inch in diameter. In the new East River Bridge each of the four cables is 18¼ inches in diameter and contains 37 strands, each strand being composed of 208 wires all laid parallel to each other, or a total of 7696 wires. The size of the wire is No. 6 (Roebling) gauge, i.e., 0.192 inch in diameter. These strands are formed by laying wire by wire, each in its proper place. The strands are then bound together into a single cable, around which is tightly wound a sheathing or casing of smaller wire, 0.134 inch in diameter for the New York and Brooklyn Bridge. The tightness of this binding wire insures the unity of the whole cable, each wire having been placed in its original position so as to take a tension equal to that of each of the other wires. The suspension-rods are usually of wire cables and are attached by suitable details to the lower chords of the stiffening truss, also by specially designed clamps to the cable. The stiffening trusses are usually built with all riveted joints, so as to secure the greatest possible stiffness from end to end. The stiffened suspension bridge has been shown by experience, as well as by theory, to be well adapted to carry railroad traffic over long spans.

=140. Economical Limits of Spans.=—In the past, suspension bridges have, in a number of cases, been built for comparatively short spans, but it is well recognized among engineers that their economical use must be found for spans of comparatively great length. While definite lower limits cannot now be assigned to such spans, it is probable that with present materials of construction and with available shop and mill capacities the ordinary truss-bridge may be economically used up to spans approximately 700 to 800 feet, and that above that limit the cantilever system is economically applicable to lengths of span not yet determined but probably between 1600 and 2000 feet. The special field of economical employment of the long-span stiffened suspension bridge will be found at the upper limit of the cantilever system. So far as present investigations indicate, the stiffened suspension type of structure may be employed to advantage from about 1800 feet up to the maximum practicable length of span not yet assignable, but perhaps in the vicinity of 4000 feet. Obviously such limits are approximate only and may be pushed upward by further improvements in the production of material and in the enlargement of both shop and mill capacity.

PART III.

_WATER-WORKS FOR CITIES AND TOWNS._