Lower bound limit analysis and design of reinforced concrete rectangular slabs

Abstract

It is becoming increasingly necessary to use reinforced concrete slabs with different end conditions - fixed (partially or fully), continuous, simply supported, or even free. The understanding of their strength, structural behaviour, and failure mechanisms is, however, inadequate. Until very recent times, most analyses of slabs were based on the theory of elasticity in classical functional forms and thereby restricted to simple shapes and boundary conditions. The present trend is towards greater use of plastic theory, such as limit analysis, in view of better prediction of safety and the resulting economy in their designs. Establishing the need and scope for the research study in Chapter 1, the basic principles of limit analysis, normal moment criterion, and associated flow rule are outlined in Chapter 2. An in-depth study (Chapter 3) of scope, requirements, limitations, and difficulties inherent in obtaining a rigorous limit analysis solution for fixed, non-circular slab systems provides valuable insight into this difficult problem and highlights, in particular, the realistic boundary conditions in terms of moments.

Chapter 4 reports the detailed development of the lower bound solutions for rectangular slabs supported all around, with all possible combinations of fixed and simply supported edges (a total of nine cases). The method, in essence, consists of apportioning the load at a point to the three constituents of the plate equilibrium equation, the apportionment being a polynomial of the coordinates of the point. This approach has been extended in Chapter 5 to include slabs with free edges. Solutions have been obtained for slabs with one or two free edges and having all possible combinations of edge conditions - fixed, simply supported, or free. Both Poisson’s and Kirchhoff’s boundary conditions have been employed along a free edge, and the corresponding moment fields for all the thirteen cases have been presented.

In order to serve as a comparison, upper-bound solutions for all the above-cited twenty-two cases have been obtained using simple failure modes (Appendix I). Evaluation of the results reported in Chapter 8 establishes that the solution deduced from the generalized procedure is more efficient in general and in no case less efficient than those available in the literature. A limit design method is developed in Chapters 6 and 7 for rectangular slabs without and with free edges. The slabs are divided into end and middle zones (with and without twist, respectively). The moment field (bending and twisting) of the lower bound solution developed in Chapters 4 and 5 has been transformed into a moment capacity field in the end and middle zones, in conformity with the flow rule of the possible failure mechanism. This facilitates the direct deduction of the steel to be provided in the x- and y-directions. The present design method is shown to be more rational than Hillerborg’s modified strip method.

The results of the entire study are reviewed in Chapter 8, with a view to highlighting their significance and importance. Useful and relevant conclusions, deduced from the study, are summarised in Chapter 9.