Search

A method is given for the optimum design of linearly elastic trusses which can be partitioned into a series of statically determinate substructures. In the case of such structures the design problem is formulated as a problem in optimal control theory. The member areas of substructures are treated as control variables and the interacting forces between substructures are taken as the state variables. Transformation equations for state variables are derived using equilibrium and displacement compatibility conditions. The design problem is envisaged as a problem of determining optimal controls to minimize the weight of the truss. This necessitates the solution of a number of smaller problems in sequence, instead of a single problem of larger dimension. Solution of the resulting optimal control problem can be obtained using the method of local variations.

This note presents the solution to the problem of the absolute minimum volume of reinforcement for strength criterion. It is for the case of clamped square reinforced concrete slabs, using Hillerborg’s strip method and the necessary and sufficient conditions for an absolute minimum to exist as proposed by Rozvany and Charret based on the Prager and Shield hypothesis, for uncurtailed and curtailed reinforcement layouts.

A finite-difference method is developed for obtaining the natural frequencies and mode shapes for rectangular plates, of abruptly varying stiffnesses causing re-entrant corners, by assuming average curvatures at junctions. This eliminates the use of fictitious points. It is shown that the discrete model analogue proposed by Witteveen 1 in 1966 for flexural analysis of such plates is a degenerate case of this method. The method is applied to the case of plates with simply supported and clamped boundary conditions. Numerical examples are presented to show the convergence and the versatility of the method. Checks are made for the limiting cases of plates of constant stiffnesses.

By replacing the continuous system of the plate with a discrete system of beams and torsional panels with finite degrees-of-freedom, a grid framework model is formulated for obtaining the critical buckling load of plates whose stiffness changes abruptly in two directions. This approach is then generalized to consider the effect of lateral deformations by a mathematically consistent first order finite-difference method. However, it is found that the effect of Poisson's ratio on the critical buckling load is negligible. The advantage in the discrete formulation is that no fictitious points need be considered, thereby making it possible to easily deal with cases of nonhomogeneity, complex boundary conditions, etc. An advantage over the finite element method is that the number of equations are far less, since there is only one unknown at a grid point. Numerical examples for different boundary conditions are presented to show the convergence and versatility of the method.

A general problem in the optimal design of structures for loading and other environmental conditions wherein structures material-interplay is to be given due recognition falls into the category of multifactor optimization problems. This can be visualized by noticing that a single objective function cannot be identified in such aforementioned design situation, since the design is to meet optimally several structural and environmental objectives. Also different candidate materials and configurations, have different degrees of suitabilities to each of the design requirements. The present algorithm is an attempt to generate a suitable design procedure to choose the best material out of an available set of materials to meet several design criteria optimally by seeking a trade off between the conflicting design objectives. In the algorithm that is generated, the candidate material properties, the field equations relating the material properties are the input information. The weightage of design requirements with the associated utility measures for each material for each criterion (either on a deterministic or probability basis) according to the subjectivity of the designer, serve as a basis to rate different materials with reference to their suitability to the design situation. A material is rated by its total expected utility value. This method is compared to existing methods available so far and it is shown that it obliviates some of their limitations. An example worked out and the general flow chart is discussed m detail in the paper.

A grid frame work model is extended to obtain the natural frequencies of square plates with stiffened square openings. This technique eliminates the use of fictitious points at the re-entrant corner point, junction point, etc. on the stiffened opening. Numerical results are presented for the first time for the cases of plates with simply supported and clamped boundary conditions in order to show the convergence and versatility of the method. Also, this investigation provides a general feeling about the changes in fundamental frequencies that occur when stiffeners are introduced.

A method is given for the analysis of nonlinear trusses which can be partitioned into a series of statically determinate substructures such that each substructure is coupled to two adjacent substructures. In the case of such structures, the number of independent variables to be determined in order to evaluate the forces in all the members is considerably reduced. The method is the same as the state space analysis herein used for controlled systems. The formulation given results in a set of operator equations with the interacting forces between the substructures as dependent variables and the properties of truss elements as independent variables. The method is applied to trusses having conservative, nonlinear elements undergoing small deformations. The resulting set of equations are solved using the Newton-Raphson technique.