THEORY AND DESIGN AIDS FOR SUSPENSION BRIDGES WITH REVERSE CABLES

Abstract

This thesis presents the nonlinear deflection equation of suspension bridges, which was originally developed by Joseph Melan in 1888 and is still current, in a non-dimensional form. This equation is linearized so that the true solution can be bracketed within close-enough bounds and an influence line approach can be used to calculate the envelopes of load effects in terms of non-dimensional parameters due to a moving point load and multiple moving patch loads placed anywhere on the span. The code-specified live loads can be approximated as a linear superposition of these load types. For these analyses, uniform properties along the span have been assumed. Consequently, the overall maximum values in the envelopes govern the design. These governing values for point and patch loads are used for the development of design charts to serve as design aids. The influence lines corresponding to these governing values are also presented to give an idea of the critical loading patterns along the span. The first six-mode shapes and their frequencies have also been determined for vertical, transverse and torsional vibrations. It was found that the fundamental vertical mode shape and the governing influence lines are quite similar. These envelopes, design aids, governing influence lines, mode shapes and frequencies are developed for single and for three span suspension bridges without and with reverse cables in terms of non-dimensional parameters so that they are usable for any span and structural properties. Vehicular suspension bridges are not currently provided with reverse cables, but pedestrian suspension bridges are routinely provided with reverse cables to prevent windinduced uplift of the very light deck provided. For single-span bridges a two-hinged stiffening girder; whereas for three-span bridges a stiffening girder continuous over piers as-well-as independent two-hinged stiffening girders for each span have been considered. Borgaonkar’s Ph.D. thesis (2012) on single span suspension bridges is the only research work till date to study the role of multiple reverse cables, and it found that they reduced the load effects quite considerably. Reverse cables that are funicular for a uniform patch load and together apply a uniform downward load on the entire span are considered. Thus, for 2-reverse cables, 1st cable shall be parabolic in the 1st segment (half) of the stiffening girder and linear elsewhere, whereas the 2nd cable shall be parabolic in the 2nd segment (half) and linear elsewhere. Borgaonkar has developed all these design aids for single span suspension bridges without and with 1 to 10-reverse cables considering equal-length segments of the span. (ii) The main purpose of this study is to develop similar design aids for three span suspension bridges without and with reverse cables. To achieve this objective, the theory of suspension bridges is developed in a systematic manner from the very beginning, and then the algorithms for numerical solution of the non-dimensional form of the deflection equation of suspension bridges are developed. Results for single-span suspension bridges are then computed to verify Borgaonkar’s results. Finally, the results for three-span suspension bridges are computed. A suspension bridge is the only bridge type in which the gravity stiffness (i.e., geometric stiffness due to effects) arising due to unmoving outward horizontal thrusts, , provided by the anchor blocks to resist the horizontal component, , of cable tensions adds to the flexural stiffness, , of the stiffening girder to increase the stiffness of the bridge. It is therefore not surprising that the longest pedestrian bridges as-well-as the longest vehicular bridges are suspension bridges. Pedestrian suspension bridges are designed where other types cannot be economically constructed, and their design is largely strength-based, and no understanding of gravity stiffness is typically required. On the other hand, the design of modern vehicular suspension bridges relies on gravity stiffness and is carried out by using the nonlinear deflection theory. Because of the relatively few bridge designers being involved in the design of vehicular suspension bridges, gravity stiffness is largely unknown among bridge designers. Consequently, suspension bridges are deemed suitable only for long spans. An arch bridge is the only other type of bridge wherein gravity stiffness plays a role. But, in arch bridges the abutments provide an unmoving inward horizontal thrusts, , resulting in negative gravity stiffness. Consequently, arch bridges are susceptible to buckling for longer spans in the absence of sufficiently large flexural stiffness, , of the arch rib. Because of the rigid behavior of short-span stone masonry arch bridges in which arches with huge, , have customarily been provided, arch bridges are perceived to be inherently very stiff. On the other hand, because of the flexible behavior of long-span suspension bridges in which stiffening girders with very low, , have customarily been provided, suspension bridges are perceived to be inherently very flexible. However, if the steel stiffening girder of a suspension bridge and the rib of a steel arch bridge having same span are provided with equal flexural stiffnesses, , then the deflections, bending moments and shear forces in the suspension bridge due to live loads shall be less than those in the arch bridge. Further, with the provision of reverse cables in the suspension bridge, much lower deflections, bending moments and shear forces shall occur. (iii) Besides positive gravity stiffness, suspension bridges have other advantages also. They are constructed on whole-to-part principle. The anchorages and pylons are first constructed, followed by the spinning of the main cables to span from anchorage to anchorage. Afterwards portions of the stiffening girder are attached to the main cables through hangers. Lastly the decking system is constructed on the stiffening girder. There is no subsidiary launching system required for their construction. Because the construction sequence of suspension bridges follows the whole-to-part principle, and because the bridge is statically determinate at all stages of its construction, there are no locked-in unknown-stresses to worry about. Other bridge types often require subsidiary launching systems. Slab bridges, beam bridges, box-bridges, arch bridges, etc. require shuttering and centering. Steel truss bridges are either launched as cantilevers with the help of a much lighter launching-nose truss attached in front of the bridge truss or through a cable suspension system. Arch bridges also require counter-pulling cable systems during construction. In cable-stayed bridges the pylons are first constructed, and then stiffening girder portions are added as cantilevers on both sides of the pylon along with stays to support them. Thus, the construction of cable-stayed bridges follows part-to-whole principle and there are unknown locked-in stresses that need to be meticulously figured-out to keep them within bounds. Thus, a lot of construction-stage backward and forward analyses involving effects of shrinkage and creep of concrete, relaxation of steel, slips at anchorages and temperature changes are required for the design and construction of cable-stayed bridges. Owing to these advantages, it should be possible to economically design and construct suspension bridges with reverse cables with steel truss stiffening girders for single and multispan configurations for smaller spans also. Such bridges require space for provision of anchor blocks and adequate free-board for reverse cables. If such space and suitable foundation soils are available, then suspension bridges with reverse cables can be considered. This study has shown that in multi-span configurations, suspension bridges with multiple reverse cables and hinged stiffening girders behave like independent single span suspension bridges with no interaction between adjacent spans. Therefore, the design of multi-span suspension bridges with reverse cables and hinged stiffening girders can be carried out using the design aids for single span suspension bridges with reverse cables. If in such bridges, equal spans are provided then the pylons can be shared between adjacent spans, and only two anchorages at the ends shall be required.