Showing papers on "Minimum weight published in 1983"

Minimum weight design of truss structures with geometric nonlinear behavior

Abstract: The paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior. The nonlinear critical load is determined by finding the load level at which the Hessian of the potential energy ceases to be positive definite. A recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm. Sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables. NUMBER of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements, element stresses, system stability, etc.1'2 The optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables. The optimality criterion was derived by, differentiating the Lagrangian with respect to the design variables. The displacements were assumed to be small and, in the finite element analysis, linear equilibrium equations were solved to determine the response of the structure to the ap- plied loads. In the case of system stability the constraints were defined (see Refs. 3-6) by the associated linear eigenvalue problem. This definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections. The correct procedure for these structures would be to analyze the structure* by using nonlinear equilibrium equations. This will be particularly true for large space structures (LSS). In this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior. In the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode. This design tends to become im- perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub- stantially. The imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal (see Ref. 7). A real structure has geometric im- perfections and these structures tend to have nonlinear

Nonlinear analysis of optimized structure with constraints on systemstability

Abstract: An optimization algorithm based on an optimality criterion was used to design a minimum weight space truss with different constraint requirements on system stability. The constraints were specified so that the eigenvalues associated with all the critical buckling modes are either equal or separated by a specified factor. For the second case the critical buckling mode was preselected from all the possible critical modes. The designs obtained for the various constraint conditions were analyzed with and without specified geometric imperfections using a nonlinear finite element program which accounts for geometric nonlinearity. The results obtained for various designs were compared for their imperfection sensitivity.

Structural optimization with dynamic behavior constraints

Abstract: The minimum weight optimum design of damped linearly elastic structural systems subjected to periodic loading with behavior constraints on maximum deflections and side constraints on design variables is addressed. Attention is focused on the two major impediments to an optimal solution: (1) the time parametric nature of the behavior constraints; and (2) the severe nonconvexity of the design space. A solution method based on upper bound approximations for the behavior constraints and an innovative mathematical programming scheme for seeking the optimal frequency subspace is set forth. Numerical results for several test problems illustrate the effectiveness of the method reported.

Heuristics for planar minimum‐weight perfect metchings

Abstract: Several linear-time approximation algorithms for the minimum-weight perfect matching in a plane are proposed, and their worst- and average-case behaviors are analyzed theoretically as well as experimentally. A linear-time approximation algorithm, named the “spiral-rack algorithm (with preprocess and with tour),” is recommended for practical purposes. This algorithm is successfully applied to the drawing of road maps such as that of the Tokyo city area. I. INTRODUCTION Consider n (an even number) points in a plane. The problem of finding the minimumweight perfect matching, i.e., determining how to match the n points in pairs so as to minimize the sum of the distances between the matched points, as well as Euler’s problem of unicursal traversing on a graph, is of fundamental importance for optimizing the sequence of drawing lines by a mechanical plotter ([2-5, 81; details are discussed in Sec. V). The algorithm which exactly solves this problem in 0(n3) time [6] seems to be too complicated from the practical point of view. Even approximation algorithms of O(n2) or O(n log n) [lo] would not be satisfactory or need some improvement for the application to real-world problems of a size, say, n greater than lo4. In contrast with the matching problem, an Eulerian path can be found in linear time in the’number of edges. In this paper, linear-time* approximation algorithms are proposed for the matching problem in a unit square; their worst-case performances are analyzed theoretically; their average-case performances are investigated both theoretically and experimentally for the case where n points are uniformly distributed on the unit square; and an application to the drawing of a road map is shown. The quality of an approximate solution is measured by the absolute cost of the matching, i.e., the sum of the distances *We adopt the RAM model of computation which executes an arithmetic operation such as addition, multiplication, or integer division (hence, the “floor” operation) in a unit time [ 11.

A segmental method for the discrete optimum design of structures

Abstract: A simple method based upon linear programming is described for the design of minimum weight structures under the restrictions that member sizes and/or material properties may be chosen only from discrete sets. The types of structures considered are those composed of axial force bars, membrane plates and shear panels. The method avoids the combinatorial nature of discrete optimization by introducing the concept of segmental members. The segmental optimum design is found by linear programming. Its weight is a lower bound to the weight of the discrete optimum design. A simple method for achieving a discrete optimum design from a segmental optimum design is described. Several examples of discrete optimum truss designs are presented.

Minimum-Weight Design of an Orthotropic Shear Panel with Fixed Flutter Speed

Abstract: THIS paper deals with the weight minimization of rectangular flat panels placed in a high supersonic flowfield and subject to a flutter speed constraint. In the establishing of the structural operator a pure transverse shear plate model was used, which may, be considered as a complement of the Love-Kirchhoff type model. By using the theory of optimal control of distributed parameter systems, necessary conditions for the minimum-weight panel are derived. These are supplemented with a condition ensuring that the flutter speed of the optimal panel coincide with the prescribed one. It is shown that the optimal thickness distribution is symmetrical with respect to the panel midpoint. Numerical rough estimates obtained via Galerkin's method are presented. Contents The field of weight minimization of panels subjected to aeroelastic constraints has been investigated thoroughly during the past decade, as it may be inferred from the specialized literature. Throughout these investigations, whether dealing with one-dimensional (see Refs. 1-4) or twodimensional aeroelastic optimization problems,5-8 the appropriate structural operator was established on the basis of the Love-Kirchhoff type model. As it is known, this model involves the ab-initio disregard of transverse shear effects. In contrast to this approach, a somewhat opposite structural type model is used here, in which the rigidities in transverse shear are considered as finite, and in bending as negligible (such a panel will be termed a pure transverse shear panel). This model—first introduced by Armand 9—is practically motivated by the advent of new composite materials that enjoy exotic properties. A generalized form of this structural model, including transverse shear orthotropicity effects, will be used here for approaching the present aeroelastic optimization problem. The structure to be analyzed consists of an elastic, rectangular flat thin panel (axb) of nonuniform thickness h = h(x]yx2), where OxjX2 denotes the in-plane coordinate system (Ox2 is the stream wise coordinate, while Ox2—the span wise one—coincides with the panel leading edge). The panel is exposed to a high supersonic gas flow over its upper face. The aeroelastic optimization problem dealt with here consists of finding the thickness distribution which minimizes the panel weight, while maintaining the same flutter speed as that of a uniform-thickness reference panel. As usual, 8 the reference panel is defined as the panel of uniform thickness H0 having the same ofthbtropy characteristics and boundary conditions as its counterpart of nonuniform thickness.

Damage tolerant design using collapse techniques

Abstract: A new approach to the design of structures for improved global damage tolerance is presented. In its undamaged condition the structure is designed subject to strength, displacement and buckling constraints. In the damaged condition the only constraint is that the structure will not collapse. The collapse load calculation is formulated as a maximization problem and solved by an interior extended penalty function. The design for minimum weight subject to constraints on the undamaged structure and a specified level of the collapse load is a minimization problem which is also solved by a penalty function formulation. Thus the overall problem is of a nested or multilevel optimization. Examples are presented to demonstrate the difference between the present and more traditional approaches.

A square root bound on the minimum weight in quasi-cyclic codes

Abstract: We establish a square root bound on the minimum weight in the quasi-cyclic binary codes constructed by Bhargava, Tavares, and Shiva. The proof rests on viewing the codes as ideals in a group algebra over GF (4). Theorem 6 answers a question raised by F. J. MacWilliams and N. J. A. Sloane in {\em The Theory of Error-Correcting Codes.} Theorems 3, 4, and 5 provide information about the way the nonzero entries of a codeword of minimum weight are distributed among the coordinate positions.

Shape Optimization of a Solid of Revolution

Abstract: Shape optimal design of an elastic solid of revolution under multiple constraints is treated. As a specific example, a device that seals a gun bore and transmits high in‐bore pressure to shear loading on the projectile, is considered. The design objective is minimum weight, with constraints on stress throughout the body, tractions on one surface of the boundary, and dimensions of the body. Methods of the calculus of variations and functional analysis are used to transform the variation of a functional over a variable region as a functional over a fixed region. An adjoint variable method of operator theory is then used to reduce this variation to an explicit function of only design variations. The resulting sensitivity coefficients are used in an iterative optimization algorithm. Numerical results are presented and show that the algorithm is stable and efficient.

Optimal strength and stiffness design of beams

Abstract: Minimum weight designs of statically indeterminate elastic beams subject to given loading are provided. Constraints are prescribed on flexural and shear stresses, and on the maximum deflection. The resulting mathematical optimization problems are solved by using concepts from differential game theory and variational calculus. The game problem under stress constraints is reduced to a minimax one. The solution procedure is illustrated by several examples of statically indeterminate beams. The optimal designs are compared with prismatic and linearly tapered beams. Suggestions are made as to the extension of the procedure to other indeterminate structures, such as beam columns (a).

Optimum Design of Radial Drilling Machine Structure to Satisfy Static Rigidity and Natural Frequency Requirements

Abstract: A computational capability is developed for the optimum design of radial drilling machine structure to satisfy static rigidity and natural frequency requirements using finite element idealization. The radial drilling machine structure is idealized with frame elements and is analyzed by using different combinations of cross sectional shapes for the radial arm and the column. From the results obtained, the best combination of cross sectional shapes is suggested for the structure. With this combination of cross sectional shapes, mathematical programming techniques are used to find the minimum weight design of the radial drilling machine structure. A sensitivity analysis is conducted about the optimum point to find the effects of changes in design variables on the structural weight and the response quantities.

Minimum weight design of tapered steel i-columns

Abstract: As I-shapes are widely used in design, the optimum proportioning of a tapered I-column from the viewpoint of practical minimum weight design is presented. While the web is tapered by a linear variation in depth, the flange area necessary to prevent buckling in the plane perpendicular to the web has been kept unchanged throughout the length of the column. The column is assumed to be slender enough so as to permit an optimum proportioning based on elastic buckling. Using a direct search technique, the optimum tapered I-column is determined within the feasible design space.

Minimum weight design using finite elements

Abstract: The authors' aim was the development of a suitable procedure for the structural design of railway vehicles within an existing general purpose finite element package. A previous application to railway vehicles was based on the use of a sequence of linear programming steps incorporating move limits on the design variables. This approach was slow and in other fields mathematical programming techniques tended to give way to methods based on optimality criteria. Nevertheless because of its simplicity of concept, and because some apparently successful alternatives to the move limit idea were available, the use of linear programming was returned to and the method is now incorporated in the British Rail finite element program NEWPAC. Couched in terms of inverse variables and using linearized forms of the stress constraints, the linear programming approach converges with the optimality criteria method. Details of the implementation and of some standard examples are given.

Minimum Weight Design for Structural Eigenvalue Problems by Optimal Control Theory

Abstract: The application of optimal control theory to minimum weight design of continuous one-dimensional structural elements subject to eigenvalue constraints is discussed. If not only the value of an eigenvalue is prescribed but also its position in the sequence of the ordered eigenvalues—for example, the critical buckling load of a column—the corresponding optimal control problem is shown to include necessarily all eigenvalues. Considering the unspecified eigenvalues as free parameters, necessary conditions for minimum weight design are derived. These conditions are compared with those obtained by use of variational methods. Attention is focused on the special case of multimodal solutions.

Optimum design of welded steel columns

Abstract: A method for the discrete minimum weight design of welded steel columns is developed. The design problem is formulated as a discrete nonlinear programming problem. The problem is solved using an interior penalty function method combined with a variable metric discrete search technique. Starting with an initial feasible solution, the first unconstrained problem is solved by searching along the direction of the integer gradient of the modified objective function. For the next unconstrained problems the direction of search is modified by the inverse of the Hessian matrix of the objective function. This way the slow convergence of the gradient method near the optimum is avoided. An example is given to show the efficiency and rapid convergence of the method. The results obtained show that more efficient use of materials can be achieved by using welded steel sections. The method presented can also be used to find the minimum cost design of welded steel sections if the cost factors are known.

An optimality criterion method for structures with stress, displacement and frequency constraints

Abstract: Presented in this paper is a procedure, based on optimality criterion methods, for the minimum weight design of structures subjected to stress, displacement and natural frequency constraints. The technique presented is a combination of a previously developed method for stress and displacement constraints alone and one for frequency limited structures. The method is applied to a delta-wing example, and the optimal designs obtained are compared to previously published results. The new method is capable of obtaining the optimal design in a small number of iterations, without significant calculations beyond a standard analysis. No approximate analyses or determination of large numbers of Lagrange multipliers are involved.

Dynamic optimization of timoshenko beam

Abstract: The present paper discusses the minimum weight design problem for Timoshenko and Euler beams subjected to multi-frequency constraints. Taking the simply-supported symmetric beam as an example, we reveal the abnormal characteristics of optimal Timoshenko beams, i.e., the frequency corresponding to the first symmetric vibration mode could be higher than the frequency of antisymmetric vibration mode if a very thin and high strip is suitably formed at the middle of the beam, and, optimal Timoshenko beams subjected to two different sets of frequency constraints could have the same minimum weight. The above abnormal characteristics demonstrate the need for including maximum cross sectional area constraint in the problem formulation in order to have a well-posed problem.