CROSS SECTIONAL OPTIMIZATION OF PRESTRESSED CONCRETE BOX GIRDER BRIDGE SUPERSTRUCTURES

Abstract

The continuing expansion of the highway network throughout the world is largely the result of the great increase in traffic, population and the extensive growth of metropolitan urban areas. This expansion has led to many changes in the use and development of various kinds of bridges. Single or multicell prestressed concrete box girder bridges have been proposed and widely used as economic and aesthetic solutions for the overcrossings, undercrossings, separation structures and viaducts. They take full advantage of the modern methods of construction. For long span bridges, where the segmental method of construction is preferred, prestressed concrete box girder bridges have proved to be economical. The very large torsional rigidity of the box girder's closed cellular section provides structural efficiency, while its broad, unbroken soffit, viewed from beneath is moce aesthetically pleasing than the open-web type construction. In the past, box girder bridges were designed in a conservative manner by adopting thick box walls in which secondary stresses arising from distortional and warping effects were negligible and hence simple beam . theory and St. Venant's torsion theory adequately predicted the behaviour. But increase in span lengths necessitated the reduction in weight of the superstructure by thinning down the sizes of various components forming the box girder cross section. The developments in the methods of analysis, easy availability of the computational facilities and better understanding of the material behaviour made it possible to analyse such large and complicated structures accurately. The confidence thus gained has led to the use of box girders for larger span, thinning the wall thicknesses further and increasing the size of the box cell. Since box girder bridges are being preferred over other bridge forms increas ingly, it has become necessary to optimize both the box dimensions and the (viii) wall thicknesses to achieve an economical solution. This has been the primary objective of the present thesis. For an accurate assessment of the stress-resultants, finite element analysis has been chosen for its. versatility in the present study. Eight noded isoparametric parabolic shell elements have been used to discretise the struc tures for their accuracy and ability to idealise curved surfaces realistically. For numerical computations, live loads, as specified by the Indian Roads Congress, have been considered. Since wheel loads are spread over sizeable areas on the deck, these have been treated as patch loads. However it has been observed that, as the size of the patch area becomes small compared to the size of the element, a wheel load may be considered as a point load located at the centroid of the patch without affecting the accuracy of the analysis. The tedium of preparation of voluminous data has been minimized and feeding of input data automated by writing a subroutine 'BOX'. The sub routine performs the preliminary design of the section and calculates the sectional properties viz. area, moment of inertia, self weight, prestressing forces required at different sections for both cantilever and simply supported bridges. The prestressing forces are transferred to the relevant nodes as equivalent nodal loads. Magnitude of each wheel load and its location has, however, to be specified externally. The subroutine converts these loads as equivalent nodal loads and transfers them to the relevant nodes. It has been reported that the cost of the superstructure including that of the formwork is nearly 55 percent of the total cost of a bridge. A noteworthy saving can be achieved if some reduction in element thicknesses can be achieved by the application of sophisticated finite element analysis and a suitable optimization technique. It is therefore, proposed to achieve an optimal design (ix) with the objective of minimizing the total weight of the structure, by minimi zing the thicknesses of the component plates by integrating the analysis procedure with the optimization method. Optimum design of a prestressed concrete box girder bridge is a non linear problem because of the interactive behaviour of various component plates constituting the section, and as such, optimization has to be achieved iteratively." Based on initial values of the design variables, analysis of the structure is first carried out. The sensitivity analysis corresponding to the change in each of the design variables is then done. On the basis of this sensitivity analysis, a decision is taken as regards the design variable to be changed and magnitude of this change to yield an optimal weight. The design variables involved in a problem of optimal design of a pre stressed concrete box girder bridge may be grouped as the size variables which include the thicknesses of the web, the soffit slab, the deck slab and the cantilever slab, and the shape or geometry variables such as the ratios of box cell width (B) to the total deck width (C) and depth of the bridge at the pier (D) to the span length (L). Prestressing force and grade of concrete are other design variables. Research workers in the area of shape optimization are not in favour of grouping the size and shape variables together because of different degrees of non-linearity introduced in the problem and changes in topology of the structure in every iteration requiring redefinition of element mesh and recalcu lation of structural stiffness. In the present study, the optimization has been carried out in three phases : first with respect to the size variables for the given geometry and in the second phase varying the shape or geometry variables. On achieving the global optimum with respect to these size and geometry variables, the prestressing force has then been varied in the third (x) step to study its effect on the optimum value. With the variation in the amount of prestressing force, the values of thicknesses and hence the weights are further optimized. The effect of variation in concrete grade on the optimum weight so obtained is then studied for a simply supported bridge. The computer program prepared for the purpose determines, for a particular cross-sectional shape the optimum thicknesses of the web, the soffit slab, the deck slab and the cantilever slab subject to the condition that stresses everywhere arc within the specified limits. The state of stress at a point is represented by an 'effective' stress which has been devised on the basis of the failure envelops suggested by Kupfer et.al. for different conditions of biaxial stress in concrete. The 'effective' stress is calculated only at critical locations, which are preselected and specified based on judgement, so that these points represent the state of stress in the continuum in an over-all manner. The computer program analyses the bridge, calculates displacements and their derivatives with respect to the changes in each of the design variables as well as 'effective' stresses and their gradients at each of the critical points for both transfer and service conditions. With this data, the equations of Linear Programming Problem (LPP) are set up which are processed by the Simplex algorithm. The values of the thicknesses and the corresponding weight represent the local optimum for the chosen shape. The shape parameters arc then varied one at a time and local optima obtained for the various values of these shape parameters. From the plot of these local optima, a global optimum is determined. The prestressing force considered for parametric study in the first instance is the full design prestressing force which balances the total live load and self weight. The effect of prestressing force on the optimum value is then studied by varying its magnitude by different degrees. A limited study with variation of concrete grade is also done. (xi) The present computer program is applicable to both cantilever and simply supported type prestressed concrete box girder bridges and accordingly, the cables have been assumed to be located in the deck slab or in webs respectively. In all, six problems of two-lane prestressed concrete box girder bridges of different spans, three each of cantilever and simply supported types, have been investigated. One bridge from each type is without foot paths. The results of investigation are presented in the form of graphs. It has been generally observed that the reduction in weight achieved by integrating finite element analysis with optimization technique, the FEM-SLP technique, is significant compared to conventional designs with merely 30% to 50% extra computational effort over the conventional FEM analysis.