W. Gutkowski

Discrete structural optimization

Abstract The paper deals with minimum weight design of elastic structures under constraints imposed upon stresses, displacements and maximum and minimum cross-section dimensions. The structures may be designed from rolled profiles and sheets. The proposed method is based on the simultaneous solution of nonlinear equations and inequalities resulting from the Kuhn-Tucker necessary conditions for an optimal problem. These conditions constitute a set of bilinear forms and inequalities based on a standard stiffness matrix. Two numerical examples comparing results with nonlinear programming technique are given.

A DISCRETE METHOD FOR LATTICE STRUCTURES OPTIMIZATION

The paper presents a semi-analytical approach to the discrete optimization of large space-truss structures. The optimization problem is stated as a cost (weight) minimization subject to constraints both on stresses and displacements. Some of the variables such as the cross-sectional areas of bars are chosen from a discrete set (catalogue) of available sections. The mined nonlinear programming problem is transformed to discrete nonlinear programming using the Galerkin procedure in solving the partial difference equations which describe displacements of nodes of the structure. Finally, introducing Boolean variables the problem is solved using algorithms of logical programming. Numerical results are presented for a “Unistrut” type space-truss structure.

Minimum Weight Design of Structures under Nonconservative Forces

The paper deals with a method of determining minimum weight design of a structure subjected to nonconservative forces with stability constraints. The relatively simple solution algorithm based on Kuhn-Tucker necessary conditions is proposed. The procedure of scaling design variables for required critical loading and normalization condition for eigenvectors are introduced at each iterative step to overcome difficulties due to the high sensitivity fo the problem. The numerical example of Beck’s column shows efficiency of the proposed method which in this case gives the highest so far known value of the critical force with no additional constraints added on equality of two first critical loads.

An effective method for discrete structural optimization

Deals with an approximate, but very simple, method of finding the minimum of structural weight under static loads. The design consists of assigning an appropriate rolled profile, from a given catalogue, to each structural member. The design is then formulated as a discrete structural optimization problem. The structure may be subjected to an arbitrary number of constraints imposed on stresses and structural material from members with the least stress. Presented examples are showing that the problems with k0 catalogue elements, and j0 structural members, including k0 to the power j0 combinations, can be solved with k0j0 analyses only. The knowledge needed to solve the problem is limited to structural analysis.

2D shape optimization with static and dynamic constraints

The paper presents an approach that allows us to consider in the shape optimization several static loading conditions together with constraints imposed on eigenfrequencies. The idea of the method is based upon simultaneous solutions of equations and inequalities arising from the Kuhn-Tucker necessary conditions for an optimum problem. The paper is illustrated with four examples in which stress and eigenfrequency are active constraints.

Optimal design of a truss configuration under multiloading conditions

The paper deals with the optimum structural design of a truss for which coordinates of structural nodes as well as member sizes constitute a set of design variables. The truss may be loaded by as many loading conditions as needed and is subjected to constraints imposed on stress displacements and complementary energy. An important relation between the cost function, Lagrange multipliers and limit values of constraints is derived. The paper presents the outline of an algorithm for the solution of a system of equations and inequality arising from the Kuhn-Tucker necessary condition for an optimum problem. Five numerical examples of 2D trusses are presented.

Discrete structural optimization by removing redundant material

A very simple method for finding the minimum weight of a structure designed from a list of available parameters is presented. The structure can be subjected to multiple loading conditions with constraints imposed on displacements, stresses and eigenfrequency. The method consists of a recursive removal of redundant material, starting from the heaviest structure. The number of analyses required is a factor of 102 less than for most stochastic methods. The knowledge needed for application of the method is limited to the finite-element method.

Manufacturing tolerances of fiber orientations in optimization of laminated plates

The paper deals with the effect of manufacturing imperfections in fiber orientations on the optimum design of laminate plates exposed to compression load. It is assumed that each dimensional imperfection cannot exceed one of two accepted values, called tolerances. In general, these are the largest allowed deviations of the design variable from its nominal value. The incorporation of tolerances is achieved by diminishing the limit values of state variables by the product of assumed tolerances and appropriate sensitivities. Therefore, the given method allows the introduction of tolerances into design in a relatively simple way and ensures safe results. The presented approach is a deterministic approximation of an exact stochastic solution of the problem. The paper is illustrated with two examples of the rectangular laminated plate design. The first one is maximization of buckling load for a given plate thickness. The second one is minimum weight (thickness) design for a given load. Numerical results show the reliability-based design to be important for structural safety when compared to the approach where tolerances are not considered

Mechanical problems of elastic lattice structures

Abstract A number of modern structures such as parabolic antennae, various aerial masts, hangers, domes, towers, etc., are composed of a very large number of similar bars. These two features, that is, large number and similarity of elements, imply a special approach to the theoretical and numerical investigation of the structures in question. The aim of the present paper is to present and illustrate by several examples the methods used for solving problems of regular elastic lattice systems, widely applied in modern structural engineering.

On the analysis of polar lattice plates

Abstract The paper deals with regular, polar, elastic, plane frames, loaded transversely (see Fig. 1). The equilibrium equations of structural joint and adjacent members, together with force-deformation relations, lead to a set of two difference equations of second order with variable coefficients. The equations are with respect to displacements and rotations of structural nodes. The solution of the problem, in the form of a finite series, is obtained by means of Laplaces method. At the end of the paper two numerical examples are given. Although the final solutions seem complex, on closer examination they are relatively simple and reduce to the calculation of factorials.