Abstract: This paper examines various first order approximation methods commonly used in structural optimization. It considers several attempts at improving the approximation by using previous analytical results and introduces an adaptation of a first order approximation method using an exponent adjusted to better fit the constraints and reduce the overall number of iterations needed to attain the optimum.

Abstract: Elastic energy is the measure for the overall stiffness/flexibility of a solid. For orthotropic materials, a combined finite element/optimization procedure is described, by which the orientation field of the most stiff/flexible solid can be determined. A study of these bounds for possible materials is performed and more general insight obtained. Recent analytical results from sensitivity analysis are applied and the paper also adds to theoretical aspects, such as proving coinciding directions for principal strains and principal stresses, even with different (but optimal) principal directions of the material.

Abstract: The aim of this paper is to demonstrate how the traditional design process can be imitated mathematically to arrive at designs of reinforced concrete structures which conform to the requirement of the new Australian Standard AS3600-1988 and are the least expensive to construct. The total cost includes the costs of concrete, reinforcing steel and formwork. The minimum cost problem is formulated as a non-linear programming problem whose solution is attempted by two techniques. Several numerical examples of multi-span beams and of columns are given using a computer package developed in standard FORTRAN 77. The sensitivity of the minimum cost designs to variations in the relative cost of formwork is also studied.

Abstract: The objective of this paper is to provide a method of optimizing areas of the members as well as the shape of both two-hinged and fixed arches. The design process includes satisfaction of combined stress constraints under the assumption that the arch ribs can be approximated by a finite number of straight members. In order to reduce the number of detailed finite element analyses, the Force Approximization Method is used. A finite element analysis of the initial structure is performed and the gradients of the member end forces (axial, bending moment) are calculated with respect to the areas and nodal coordinates. The gradients are used to form an approximate structural analysis based on first order Taylor series expansions of the member end forces. Using move limits, a numerical optimizer minimizes the volume of the arch with information from the approximate structural analysis. Numerical examples are presented to demonstrate the efficiency and reliablity of the proposed method for shape optimization. It is shown that the number of finite element analysis is minimal and the procedure provides a highly efficient method of arch shape optimization.

Abstract: The object of the study was to optimize composite plates concerning free vibration frequencies, buckling loads, and deflections under constant pressure. Layups without coupling between bending and extension but otherwise arbitrary selection of the ply angle variation through the thickness of the laminate were included. For these plates, four different boundary conditions were studied. The number of relevant parameters was successively reduced from the initial six bending stiffnesses that any laminate has. Bending-twisting coupling has only negative influence on fundamental eigenfrequency, buckling load and average deflection under a constant pressure, so the number of parameters could be reduced by two. The remaining four parameters are not independent, but are functions of only two independent parameters, the flexural lamination parameters. It was further seen that the optimal designs always were found on the boundary of the allowable region of the flexural lamination parameters, i.e. there is only one relevant parameter for the optimization problems. This parameter can be interpreted as the layup angle (Θ) in an orthotropic (+/ − Θ) laminate.

Abstract: The design of most engineering systems is a complex and time-consuming process. In addition, the need to optimize such systems where multidisciplinary analysis and design procedures are required can cost additional human and computational resources if proper software and numerical algorithms are not used. Several computational aspects of optimization algorithms and the associated software must be considered while making comparative studies and selecting a suitable algorithm for practical applications. Several parameters, such asaccuracy, generality, robustness, efficiency and ease of use, must be considered while deciding the superiority of an optimization approach. Approximate algorithms without sound mathematical basis can be sometimes more efficient for a specific problem, but fail to satisfy other requirements. They are, therefore, not suitable for general applications. An objective of the paper is to emphasize the critical importance of the above-mentioned parameters in large scalestructural optimization and other applications. Theoretical foundations of two promising approaches, thesequential quadratic programming (SQP) andoptimality criteria (OC), are presented and analysed. Recent numerical experiments and experiences with the SQP algorithm satisfying these requirements are described by solving a variety of structural design problems. An important conclusion of the paper is that the SQP method with a potential constraint strategy is a better choice as compared to the currently prevalent mathematical programming (MP) and OC approaches.

Abstract: A procedure is presented for the shape optimization of truss structures based on the reliability concept. Nodal coordinates are taken as the shape design variables together with the sizing design variables such as the cross-sectional areas of the members. These variables are determined to minimize the structural volume under the constraint on the structural failure probability.

Abstract: Optimization problems often depend on parameters that define constraints or objective functions. It is often necessary to know the effect of a change in a parameter on the optimum solution. An algorithm is presented here for tracking paths of optimal solutions of inequality constrained nonlinear programming problems as a function of a parameter. The proposed algorithm employs homotopy zero-curve tracing tecnniques to track segments where the set of active constraints is unchanged. The transition between segments is handled by considering all possible sets of active constraints and eliminating nonoptimal ones based on the signs of the Lagrange multipliers and the derivatives of the optimal solutions with respect to the parameter.

Abstract: The paper deals with discrete optimization of elastic trusses with geometrical nonlinear behaviour and constraints on stability. The problem consists of minimizing the weight and determining the optimal member distribution so that the external load does not cause a loss of stability of the structure. Member cross-sections are selected from a catalogue of available sections. Element stresses, elment stability and global structural stability constraints are considered. A controlled enumeration method according to the increasing value of the objective function is applied. Shallow space trusses are numerically analysed.

Abstract: The research work presented here is based on the concept of the integration of optimization techniques and numerical analysis with the finite element method (FEM) and computer-aided design (CAD). A microcomputer aided optimum design system, MCADS, has been developed for general structures. Certain techniques to be discussed in the paper, e.g. the semi-analytical method for design sensitivity analysis, optimization analysis modelling for shape design, application oriented user interfaces and the coupling of automated optimization and user intervention have rendered MCADS pratical and versatile in applications for engineering structures. The above techniques and an application are presented in this paper.

Abstract: The concept of structural optimization was introduced in the early sixties. It was at that time that Professor Schmit, from Los Angeles, California University, suggested a combination of structural analysis by finite element and optimization methods (Schmit 1960). At present, the use of optimization methods is generally well understood. The capabilities (and limitations) of these methods are known and some industrial softwares exist. One of these is SAMCEF which can be applied to design of thin-walled structures or to shape optimization. The techniques used for the solution of large scale problems are presented in this paper. A summary of the main difficulties involved in shape optimization is also given. Several industrial problems are solved to illustrate the proposed concepts.

Abstract: An integrated analysis and design approach for stochastic optimization of structures is presented. The proposed procedure employs a first order second moment (FOSM) method for evaluating the constraints associated with structural safety.

Abstract: This paper deals with the classical structural optimization problem of minimizing the weight of a truss structure, subject to constraints on displacements and stresses. Design variables are the cross-section areas of the elements. The main theoretical result presented is that the complete Taylor series expansions of all constraint functions can be explicitly expressed, once the displacements corresponding ton specific load cases have been calculated, wheren = the number of elements. Based on this theoretical result, new explicit approximations of the constraint functions are suggested. These approximations are shown to have some interesting theoretical properties. They also seem to work very well in practice, when included in an optimization scheme.

Abstract: The boundary element approach to shape sensitivity analysis of eigenvalues of free vibrating elastic structures is presented. The eigenvalue problem is described in terms of the boundary integral equation method. Using the variational approach for variable regions, first-order sensitivities of simple frequencies are derived. Dependence of eigenvalues with respect to the stochastic shape of the boundary is considered. The numerical procedure of discretization of the problem is characterized. Numerical examples for two-dimensional problems are presented.

Abstract: Research in optimum structural design has shown that mathematical programming techniques can be employed efficiently only in conjunction with explicit approximate constraints In the course of time a well-established approximation for homogeneous functions (scalable structures) has emerged based on the linear Taylor expansion of the displacement functions in the compliance design space (Reciprocal approximation) It has been shown that the quality of this approximation is based on the property that homogeneity of the constraints is maintained and consequently the approximation is exact along the scaling line The present paper presents a new family of approximations of homogenous functions which have the same properties as the Reciprocal approximation and which produce more accurate models in most of the tested cases The approximations are obtained by mapping the direct linear Taylor expansion of the constraints unto a space spanned by intervening variables (original design variables to a powerm) Taking the envelope of these constraints along the scaling line yields a new family of approximations governed by the parameterm It is shown that the Reciprocal approximation is a particular member of this family of approximations (m = −1) The new technique is illustrated with classical examples of truss optimization An optimal plate design is also reported A parametric study of the results for various values of the exponentm is presented It is shown that for special values of the exponentm the new approximations usually yield better models than those based on the Reciprocal approximation

Abstract: A procedure for the minimum weight design of helicopter rotor blades with constraints on multiple coupled flap-lag natural frequencies, autorotational inertia, and centrifugal stress is presented. Optimum designs are obtained for blades with both rectangular and tapered planforms and are compared within a reference blade. The effects of higher-frequency constraints and stress constraints on the optimum blade designs are assessed. The results indicate that there is an increase in blade weight and a significant change in the design variable distributions with an increase in the number of frequency constraints. The inclusion of stress constraints has different effects on the wall thickness distributions of rectangular and tapered blades, but tends to increase the magnitude of the nonstructural segment weight distributions for both blade types.

Abstract: Beams and circular plates on elastic foundations are considered. In some cases, additional elastic supports are present. The stiffness distribution of the foundation is designed so that the pressure on the foundation is uniform. Sometimes the depth of the beam or plate is also varied, with either a piecewise-constant sandwich or solid cross-section, and a global measure of the deflection is minimized. The total stiffness of the foundation and supports is specified, as well as the volume of the structure. In one type of problem, the edges of the structure are displaced downwards; in the other examples, a downward load is applied. Types of loads include a concentrated central load, a uniform load and a parabolic load. The uniform foundation pressure for the resulting design is often substantially lower than the maximum pressure for a corresponding uniform beam or plate on an elastic foundation with uniform stiffness.

Abstract: The paper presents an alternative solution methodology for the optimization of continuous structures described in terms of one spatial coordinate. Using a control systems state variable formulation for the structural model, Pontryagin's principle reduces the optimization problem to simultaneously satisfying the state, costate and optimality condition together with any constraints on the problem. System dynamics methodology is then used to obtain the numerical solution. This is achieved by constructing graphically a flow diagram representing the necessary optimality conditions; contraints are included through the use of logical relationships. The associated (DYNAMO) equations are then solved numerically as part of the standard system dynamics package.

Abstract: An optimization method is presented to design a minimum weight structure with constraints imposed on the closed-loop frequency distribution and damping parameters. The control approach used here is linear quadratic regulator theory. The control model reduction is achieved by using Model Error Sensitivity Suppresssion. The application of the method is illustrated by designing the structure for different order of control models with the same constraints. The different designs obtained by these approaches are compared. The optimization problem is solved by using a nonlinear mathematical approach.

Abstract: Thin-walled box-section circular arches loaded by a radial pressure are optimized with respect to their stability. Beside the overall both in-plane and out-of-plane buckling local web and flange instability is considered. The optimization aim is to determine either optimal values of cross-section dimensions — parametrical formulation or optimal functions of these dimensions — variational design. For the latter case suitable optimality conditions are derived by use of Pontryagin's maximum principle. The required modality of the problem formulation is discussed and selected numerical results are presented.

Abstract: The relationship between two optimal design problems is investigated: (a) The fixed geometry problem, where the topology is optimized for a predetermined geometry. (b) The fixed topology problem, where the geometry is optimized for a given topology. Assuming approximate linear programming formulations, conditions of optimality are derived and geometries of multiple optimal topologies are studied. Some considerations related to a general design procedure for optimization of topology, geometry and cross-sections are discussed.

Abstract: The stress-ratio algorithm associated with fully stressed design philosophy has been used as a convenient tool to achieve minimum weight design of strength-limited structures. The algorithm is effective and converges quickly for many cases. However, it presents extremely slow oscillatory iteration histories for plate-thickness design problems that involve transverse bending loads. Modification of the basic algorithm presented in this paper provides an effective remedy to this problem when both membrane and bending loads are present. The modified resizing algorithm requires numerical solutions of a fourth-order algebraic equation. No additional data, beyond the ordinary static analysis results, are required.

Abstract: The output error and equation error methods of system identification are compared for their effectiveness in assessing damage in structural systems Damage is modeled on an element-by-element basis as changes in sectional properties, which then contribute to variations in the terms of the structural stiffness matrix Both static displacements and eigenmodes of the structure are used in the damage detection process A rational basis for the proper selection of eigenmodes and loading conditions for the identification process is also presented Characteristics of the unconstrained optimization design space for the two approaches are discussed in context of their ability to yield the location and extent of damage