Plastic analysis of plates subjected to multiple loads

Abstract

A simple analysis for the determination of the carrying capacities of the axisymmetric plate under combined loading has been presented in this chapter. Some brief remarks are appropriate on the interpretation to be given to the loads obtained by the present analysis. The loads are correctly those required to initiate collapse due to plastic deformation when the material is rigid plastic. As the influence of the membrane forces on the bending forces is ignored in the theory, the analysis can best be applied to problems involving moderately thick plates where elastic deformations are small and bending action is predominant. Simple load interaction curves are presented for engineering estimates. In these design charts, the relations between the non dimensional load parameter and the geometrical parameter are arranged so that the permissible load can be found from the given geometrical parameters. This approach may be applicable to a large number of limit analysis problems that have not been solved satisfactorily to date. The degree of safety and economy inherent in the design of ductile metal structures depends on the accuracy with which the real strength of structures can be estimated. The plastic methods of analysis provide the means for estimating the real strength of a structure under a given set of loads. For plates of simple geometry and simple loading, plastic analysis has been fairly well established; whereas for plates of complex geometry and multiple loading, the analysis is quite difficult. It requires the manipulation of nonlinear differential equations at various stages when sections of the plate are elastic, plastic, etc. There are other complications as well, such as the yield condition with increased numbers of dimensions of the abstract space in which the limit surfaces are to be constructed, and the infinite number of combinations of external loads that cause plastic collapse. This thesis has attempted to provide a fairly satisfactory solution for such complex problems. The solutions are provided in the form of load interaction curves for plate structures subjected to several independent loads. These are expected to be useful for a designer to make reasonably accurate and rapid assessments of the load carrying capacity of plate structures. For structures governed by axisymmetry, analytical solutions have been obtained. Several complicating factors such as material orthotropy, combinations of in plane and lateral load components, and different boundary conditions have been accounted for. Different versions of generalising the yield conditions for the structure have also been examined. In a non axisymmetric situation, it is impossible to obtain complete analytical solutions; they may, at best, be bounded through the use of limit theorems. The solution for a rectangular “membrane plate” has therefore been restricted to providing lower bounds only. Solutions are presented for many different combinations of in plane and lateral load distributions. Solutions for more general cases, like plates of arbitrary shape, are possible only through computer oriented analytical procedures. Accordingly, computer programs have been developed along the lines of the finite element approach. The accuracy and the potential of these programs have been amply demonstrated in the examples provided. The results include the load deflection characteristics and the progression of the plastic regions in the plate at various load levels. The analyses presented in this thesis are based on the small deflection theory of thin plates. Further, the material behaviour is idealised as rigid perfectly plastic in the analyses of Chapters IV and V. However, the finite element analysis of Chapter VI accounts for elastic deformations as well. The influence of membrane forces on the bending action has been ignored for simplicity. The effects of shear and instability are also neglected. For the rectangular plate problem under combined loading, discussed in Chapter V, it may be noted that the determination of a statically admissible stress field is a matter of intelligent guesswork rather than a well defined procedure. There is less information available for finding the exact admissible stress distribution. Even for such a seemingly simple problem as a simply supported square plate under uniform load, the closest bounds are distressingly far apart. The difficulties in obtaining the upper bound solutions are also explained in Chapter V. The finite element formulation in Chapter VI has the advantage of predicting collapse loads based on several failure criteria (Section 6.6). For example, in practical applications a structure may become unserviceable due to excessive deflection before the theoretical load carrying capacity is achieved. If the design specification includes limitations on allowable deflections under working loads, plastic methods can be supplemented by a deflection check. The computer program developed is capable of handling this. Although the stiffness formulation is attractive, it has disadvantages. Because the displacement field in each element must be at least cubic to meet inter element compatibility conditions, the yield condition varies within elements, so elements can contain both plastic and elastic zones. Moreover, displacement methods require iterative assumptions about elastic/plastic states, since they cannot directly handle internal unloading. The significance of some basic assumptions is as follows: • Geometry changes may be important, as large deflections prior to yield could render a structure unserviceable. • Changes in geometry may alter load application points. • Post yield displacements introduce membrane stresses even when bending dominates. The present analysis should therefore be applied to moderately thick plates where bending dominates and elastic deformations remain small. As described in Chapter II, many approximations to actual yield conditions exist. It is worthwhile to investigate whether simple, accurate yield conditions can be developed for complex stress states. Appendix C generalises the linearised Tresca yield condition for orthotropic circular membrane plates with axi symmetry. Better upper and lower bounds for plate problems under combined loading could also be constructed using mathematical programming methods. Appendix D outlines a possible approach. Time dependent effects-including inertia under rapid loading and viscoelasticity, which increases short term strength-are also suggested for future research. Analyses could also consider anisotropic materials, different yield stresses in tension and compression, and direction dependent yield behaviour. Experimental validation is essential, particularly to assess the post yield behaviour of structures. Further work is needed to develop computational methods suitable for small capacity computers. Appendix A proposes a modified variable stiffness method for plasticity problems; its computational economy remains to be tested. In conclusion, many combined loading problems remain open, including shakedown, optimum design, elastoplastic stability, thick plate behaviour, dynamic loading, thermal effects, finite deformations and material property variations.