SEGMENTAL PRESTRESSED CONCRETE BOX GIRDER BRIDGES

Abstract

Segmental prestressed concrete box girder bridges are widely popular as they combine the advantages of prestressing with the structural efficiency of a box section and speed of erection by segmental construction to provide an ideal solution to intermediate span bridge structures. The procedure adopted for this type of construction is usually the well-known cantilever construction technique. The cross-section is generally a single cell, rectangular in shape. As the box crosssection is deformable, the structural behaviour is more complex than the primary behaviour as a beam in the longitudinal direction. Secondary effects such as shear lag and cross-sectional distortion affect stresses substantially in these box girders. Structural analysis and design are highly inter active processes and, moreover, for prestressed segmental box girder bridges, the detailed design must follow the construction sequence. Nevertheless, analysis of the completed structure usually governs the design with regard to cross-sectional proportioning and the amount of prestress. Because of the complexity of analysis of the completed bridge, the finite element method using flat shell type elements is considered most appropriate. The live loads specified by the Indian Roads Congress (IRC; need to be considered as patch loads occupying only a portion ii of the element in which they occur. The order of the elements used should be high enough to eschew problems of numerical instabilities while determining design sensitivities for optimization purposes. At the same time, the chosen elements should remain computationally viable. Optimization of such a class of structure using the finite element method ( PEM) has hardly been reported in the literature. This is inspite of the fact that the FEM is a very powerful analytical tool and that the use of crude analytical techniques have been the source of major errors in a sophisticated optimization procedure. Optimum structural design problems, being highly non linear, can only be solved iterativaly. It is normally possible to regard the non-linear problem as a sequential linear programming problem ( SLP). The powerful simplex algorithm of optimization is then used with advantage. Such an integrated approach combining the FEM technique with the SLP may be called the FEM-SLP procedure. Research workers in the area of shape optimization discourage the grouping together of size and geometry variables. Optimizing with respect to geometry variables, though desirable, is extremely complex as their behaviour is non-linear to varying degrees and the structural topology is affected. Finally, the number of the design variables chosen cannot be large as the computational costs would then become prohibitive. Including the above Ill constraints and considerations, an optimization procedure for prestressed segmental box girder bridges has been formulated and described in the present work. The procedure uses a parabolic shell finite element which has been derived on the lines of a flat shell element. The derived shell element has six degrees of freedom per node so that consistency is maintained at the global level when individual stiffnesses of elements lying in different planes are assembled. The computer program SHELL written for analyzing shell structures in general and box girder bridges in particular is shown to give excellent results. Because of the isoparametric formulation, curved shapes are quite accurately represented. The program's performance has been checked through test problems reported in the literature. A numerical procedure to evaluate equivalent nodal loads for patch loads acting on a patch of general geometry within an element, is described. This procedure is written as a general purpose subroutine, IRC, and incorporated in the program SHELL. The accuracy of this subroutine has also been tested with a test problem. An iterative design procedure consists of an analysis of the current design, a sensitivity analysis corresponding to changes in the design variables and a decision to redesign the structure. In this work, the decision to redesign has been formulated as a linear programming problem and the sensitivity analysis has been integrated within the FEM procedure. As the information in respect of stresses,generated iv by the FEM analysis, is quite large, it has been found essential to use a failure criterion which involves all components of the stress at a point. For this purpose, an 'effective stress' has b^n devised to represent the stress-state at any point in the continuum. The stress constraints arc generated at a few critical points only. The location of these critical points is necessarily to be specified by the user, based on his judgement, so that the entire continuum gets considered. In an attempt to reduce the number of design variables, the web and soffit thicknesses have been prescribed a parabolic variation from a minimum at the free end of the cantilever to a maximum at the pier. The size variables selected for optimization purposes arc, thus, the maximum web and soffit thicknesses. The geometry variables defining the cross-sectional shape have been kept as the box width/ deck width ratio and the ratio of the maximum girder depth at the pier to the cantilever span. The girder depth has been assumed to vary parabolically from a minimum at the free end of the cantilever to a maximum at the pier. The program SHELL has been drastically altered to yield the computer program BROPT which computerizes the derived FEM-SLP procedure. For a particular cross-sectional shape, the program BROPT yields the optimum web and soffit thicknesses while minimizing the weight of the bridge. This represents a local optimum design point for a particular cross-sectional shape. The optimization in respect of thicknesses is essential as only through studying the local optima can the global optimum be achieved. Accordingly, the shape optimization has been carried out in a parametric manner. Varying the shape parameters one at a time, local optimum solutions have been obtained through the program BROPT. Astudy of the local optima leads to the global optimum in respect of both the shape parameters and the thicknesses. While optimizing with respect to the shape arameters, the need for an iterative procedure has been examined. Finally, the effect of varying the amount of prestress forces, for the considered prestressing profile, viz., straight tendons anchored in the deck slab, on the characteristic of the optimum design point has been studied. The study has been done on two bridge spans in order to verify and illustrate the performance of the formulated optimization procedure. The devised effective stress has been found to represent satisfactorily the stress-state at critical points. The algorithms for computing design sensitivities is 'exact'. The convergence of the program BROPT is fast and only two design iterations have been found generally to be sufficient to locate the local optima. The variation of the bridge weight with respect to the shape parameters has been found to be highly non-linear. While optimizing with respect to these parameters, the use of an iterative procedure seems unnecessary. T^e gain in the objective is linearly dependent on the amount of prestress in the given prestressing profile. p