The static behavior of a cable-stayed girder can best be gaged from the simple, two-span example of Fig. 15.54. The girder is supported by one stay cable in each span, at E and F, and the pylon is fixed to the girder at the center support B. The static system has two internal cable redundants and one external support redundant.
If the cable and pylon were infinitely rigid, the structure would behave as a continuous four-span beam AC on five rigid supports A, E, B, F, and C. The cables are elastic, however, and correspond to springs. The pylon also is elastic, but much stiffer because of its large cross section. If cable stiffness is reduced to zero, the girder assumes the shape of a deflected two-span beam ABC.
Cable-stayed bridges of the nineteenth century differed from those of the 1960s in that their stays constituted relatively soft spring supports. Heavy and long, the stays could not be highly stressed. Usually, the cables were installed with significant slack, or sag. Consequently, large deflections occurred under live load as the sag decreased. In contrast, modern cables are made of high-strength steel, are relatively short and taut, and have low weight.
Their elastic action may therefore be considered linear, and an equivalent modulus of elasticity may be used [Eq. (15.1)]. The action of such cables then produces something more nearly like a four-span beam for a structure such as the one in Fig. 15.54.
If the pylon were hinged at its base connection with a stayed girder at B, rather than fixed, the pylon would act as a pendulum column. This would have an important effect on the stiffness of the system, for the spring support at E would become more flexible. The resulting girder deflection might exceed that due to the elastic stretch of the cables. In contrast, the elastic shortening of the pylon has no appreciable effect.
Relative girder stiffness plays a dominant role in the structural action. The stayed girder tends to approach a beam on rigid supports A, E, B, F, C as girder stiffness decreases toward zero. With increasing girder stiffness, however, the support of the cables diminishes, and the bridge approaches a girder supported on its piers and abutments A, B, C.
In a three-span bridge, a side-span cable connected to the abutment furnishes more rigid support to the main span than does a cable attached to some point in the side span. In Fig. 15.54, for example, the support of the load P in the position shown would be improved if the cable attachment at F were shifted to C. This explains why cables from the pylon top to the abutment are structurally more efficient, though not as esthetically pleasing as other arrangements.
The stiffness of the system also depends on whether the cables are fixed at the towers (at D, for example, in Fig. 15.54) or whether they run continuously over (or through) the pylons. Some early designs with more than one cable to a pylon from the main span required one of the cables to be fixed to the pylon and the others to be on movable saddle supports.
Most contemporary designs fix all the stays to the pylon.
The curves of maximum-minimum girder moments for all load variations usually show a large range of stress. Designs providing for the corresponding normal forces in the girder may require large variations in cross sections. By prestressing the cables or by raising or lowering the support points, it is possible to achieve a more uniform and economical moment capacity. The amount of prestressing to use for this purpose may be calculated by successively applying a unit force in each of the cables and drawing the respective moment diagrams.
Then, by trial, the proper multiples of each force are determined so that, when their moments are superimposed on the maximum-minimum moment diagrams, an optimum balance results.
(Guidelines for the Design of Cable-Stayed Bridges, Committee on Cable-Stayed Bridges, American Society of Civil Engineers.)
Static Analysis Elastic Theory
Cable-stayed bridges may be analyzed by the general method of indeterminate analysis with the equations of virtual work.
The degree of internal redundancy of the system depends on the number of cables, types of connections (fixed or movable) of cables with the pylons, and the nature of the pylon connection at its base with the stayed girder or pier. The girder is usually made continuous over three spans. Figure 15.55 shows the order of redundancy for various single-plane systems of cables.
If the bridge has two planes of cables, two stayed girders, and double pylons, it usually also must be provided with a number of intermediate cross diaphragms in the floor system, each of which is capable of transmitting moment and shear. The bridge may also have cross girders across the top of the pylons. Each of these cross members adds two redundants, to which must be added twice the internal redundancy of the single-plane structure, and any additional reactions in excess of those needed for external equilibrium as a space structure.
The redundancy of the space structure is very high, usually of the order of 40 to 60. Therefore, the methods of plane statics are normally used, except for large structures.
For a cable-stayed structure such as that illustrated in Fig. 15.56a, it is convenient to select as redundants the bending moments in the stayed girder at those points where the cables and pylons support the girder. When these redundants are set equal to zero, an articulated, statically determinate base system is obtained, Fig. 15.56b. When the loads are applied to this choice of base system, the stresses in the cables do not differ greatly from their final values; so the cables may be dimensioned in a preliminary way.
Other approaches are also possible. One is to use the continuous girder itself as a statically indeterminate base system, with the cable forces as redundants. But computation is generally increased.
A third method involves imposition of hinges, for example at a and b (Fig. 15.57), so placed as to form two coupled symmetrical base systems, each statically indeterminate to the fourth degree. The influence lines for the four indeterminate cable forces of each partial base system are at the same time also the influence lines of the cable forces in the real system. The two redundant moments Xa and Xb are treated as symmetrical and antisymmetrical group loads, Y = Xa + Xb and Z = Xa + Xb , to calculate influence lines for the 10- degree indeterminate structure shown. Kern moments are plotted to determine maximum effects of combined bending and axial forces.
A similar concept is illustrated in Fig. 15.58, which shows the application of independent symmetric and antisymmetric group stress relationships to simplify calculations for an 8- degree indeterminate system. Thus, the first redundant group X1 is the self-stressing of the lowest cables in tension to produce M1= +1 at supports.
The above procedures also apply to influence-line determinations. Typical influence lines for two bridge types are shown in Fig. 15.59. These demonstrate that the fixed cables have a favorable effect on the girders but induce sizable bending moments in the pylons, as well as differential forces on the saddle bearings.
Note also that the radiating system in Fig. 15.55c and d generally has more favorable bending moments for long spans than does the harp system of Fig. 15.59. Cable stresses also are somewhat lower for the radiating system, because the steeper cables are more effective.
But the concentration of cable forces at the top of the pylon introduces detailing and construction difficulties. When viewed at an angle, the radiating system presents esthetic problems, because of the different intersection angles when the cables are in two planes.
Furthermore, fixity of the cables at pylons with the radiating system in Fig. 15.55c and d produces a wider range of stress than does a movable arrangement. This can adversely influence design for fatigue.
A typical maximum-minimum moment and axial-force diagram for a harp bridge is shown in Fig. 15.60.
The secondary effect of creep of cables (Art. 15.12) can be incorporated into the analysis.
The analogy of a beam on elastic supports is changed thereby to that of a beam on linear viscoelastic supports. Better stiffness against creep for cable-stayed bridges than for comparable suspension bridges has been reported.
Static Analysis Deflection Theory
Distortion of the structural geometry of a cable-stayed bridge under action of loads is considerably less than in comparable suspension bridges. The influence on stresses of distortion of stayed girders is relatively small. In any case, the effect of distortion is to increase stresses, as in arches, rather than the reverse, as in suspension bridges. This effect for the Severn Bridge is 6% for the stayed girder and less than 1% for the cables. Similarly, for the Du¨sseldorf North Bridge, stress increase due to distortion amounts to 12% for the girders.
The calculations, therefore, most expeditiously take the form of a series of successive corrections to results from first-order theory (Art. 15.19.1). The magnitude of vertical and horizontal displacements of the girder and pylons can be calculated from the first-order theory results. If the cable stress is assumed constant, the vertical and horizontal cable components V and H change by magnitudes V and H by virtue of the new deformed geometry. The first approximate correction determines the effects of these V and H forces on the deformed system, as well as the effects of V and H due to the changed geometry. This process is repeated until convergence, which is fairly rapid.