Optimal Design Of Beams Research Articles

Optimal Designs of I‐Beams against Lateral Buckling

This paper concerns the optimal distribution of a given volume of material in I‐beams so as to maximize the elastic flexural‐torsional buckling capacities. The material distribution has been restricted to different top‐to‐bottom flange‐width ratios, linear tapering of flange width, or linear tapering of web depth. Based on the Rayleigh‐Timoshenko energy method, a canonical form of the Ray‐leigh quotient is derived for the three types of design considered. For the maximum buckling capacity, the quotient is first minimized with respect to the displacement function and then maximized with respect to the design parameter. To avoid inelastic behavior and a small cross‐sectional area in the optimal beam designs, a maximum permissible normal‐stress constraint is imposed. Optimal designs of simply supported I‐beams under general moment gradient are presented. A comparison study is made to determine which of the three design types is the most effective way of distributing material for maximum buckling capacities.

An optimal control approach to the design of vibrating elastic-viscoelastic sandwich beams

This work features the application of an optimal control algorithm to a new class of continuous one-dimensional structural design problems. A sandwich beam of rectangular cross-section is considered. It has a variable-thickness core and two equal variable-thickness cover layers and is subjected to harmonic forced vibrations. The objective is to distribute both the core and layer mass so as to minimize a measure of dynamic compliance for forced steady-state vibration and fixed material volumes. Either or both materials may be viscoelastic. Any constitutive relation may be used provided it is linear and time-invariant. The design problem is formulated as an optimal control problem. The resulting problem contains ten state variables, two control functions, four control parameters, and six terminal state constraints. Simple transformations are used to treat the minimum-gage constraints. A conjugate gradient/gradient projection optimal control algorithm is then used to obtain numerical solutions. Several optimal beam designs are presented and compared for a variety of problem parameter values.

Optimal design of beams discretized by elastic plastic finite element

The optimum design of beams discretized by elastic plastic finite elements is formulated accounting for behavioural constraints. The novelty of the present work derives from adopting a beam model within which plasticity may spread. The corresponding optimality criterion is discussed, and an iterative procedure is proposed and applied to some meaningful examples.

Optimal design of beams in torsion under periodic loading

The optimal design of beams in torsion under harmonically varying torques is discussed. The analysis covers the cases when the excitation frequency is either less than or greater than the fundamental frequency of the beam. The beams analyzed are in the main assumed to have rectangular cross-section but the theory is easily extended to other section shapes. In each case the problem is stated in variational form with the introduction of constraints through Lagrange multipliers. The mathematical analysis of the various problems presented results in a system of non-linear differential equations with associated boundary conditions. The solutions given for some of the cases provide expressions for the design variable and the response, along the length of the beam, in terms of the forcing frequency and some constants which can be determined for the particular problem. The computed results and data are given in tabular form and some optimum profiles are shown graphically.

Optimal design of beams under flexural vibration

The minimum weight design of a cantilever beam in flexural vibration is considered. The aim is the maximization of a given natural bending frequency (usually the first) for a given beam weight or equivalently the minimization of beam weight for a specified value of a natural frequency. The beams considered are of rectangular section and are subject, in a range of cases presented, to a variety of constraints on lower and upper bounds on the cross-section dimensions or to the specification of a point mass at the end of the beam. Simple bending theory is regarded as applicable to the problem. A variational statement of the problem is made and the necessary conditions for a minimum are obtained as a system of non-linear equations which are solved numerically. Results are given in the form of tables and of figures showing computed optimum profiles. Some experiments on a sample set of beams of equal mass are described briefly. The optimum profile beam was found to have the greatest fundamental frequency, in support of the theoretical predictions.

Pareto optimal design of beams subjected to support motions

The optimal design of a clamped-hinged beam is determined with the objectives of minimizing the maximum deflection, maximum normal and shearing stresses and of maximizing the fundamental eigenfrequency. The design variable is the area function of the beam which is approximated by linear splines in order to convert the continuous problem into a finite dimensional one. The multi-objective design problem is formulated by using the concept of Pareto optimality. The design algorithm consists of successive stages of analysis and optimization. Each analysis is carried out by iteratively solving an equivalent integral equation formulation of the original problem. Mathematical programming techniques are employed for optimization. The numerical results for a number of design examples are presented in the form of graphs of optimal trade-off curves which give the optimum performance of the structure with respect to specified objectives.

An Optimal Design of Beam with Variable Cross Section : Design of Cantilever by Dynamic Programming

The problem of determining the shape of an elastic cantilever with a cross section varied stepwise is discussed. The criterion is the minimization of the strain energy of the cantilever under the constraints of given length and volume. The beam is subject to the load distributing proportionally to the power of the distance from the free end. The length of each beam segment as well as the size of its cross section is taken into consideration as the design variable. The dynamic programming procedure is used to obtain a recursive equation which is modified into an appropriate non-dimensional one to decide optimal values independently of the magnitudes of constraints (or, total volume and length) at every decision stage. Beams with a rectangular cross section of a constant width and beams with a circular cross section are treated in numerical examples. Moreover, from a viewpoint of optimization through discretization the results are compared with the analytically optimized beams with a continuously variable cross section.

An Optimal Design of Beam : Design of Cantilever by Dynamic Programming

An Optimal Design of Beam : Design of Cantilever by Dynamic Programming

Optimal design of beams with minimum compliance

An optimization problem of non-linear elastic or viscous beams is discussed. To the beam an additional support is introduced whose location must be selected so as to minimize the compliance of the beam. The problem is solved with the aid of optimal control theory. Both rigid and flexible supports are considered. Some new optimization conditions, which are valid for arbitral compliance criterions, are deduced. A few illustrating examples are given.

Optimal Design of Beams against Buckling: A Potential Energy Approach

Abstract A rather general development of the optimum design of structures against buckling is extended to include a broader class of structures and loadings. The development is specialized to the problem of optimal design of thin-walled beams against lateral buckling. The optimum design for simply supported, cantilever, and fixed-fixed rectangular beams in pure bending is found to be a uniform one. For simply supported rectangular beams of variable height the savings in weight compared to a uniform beam is insignificant. Significant savings can be realized for a constrained design of a class of I-beams of variable depth.

Optimal design of beams for alternative loads and constraints on generalized compliance and stiffness

The paper presents an analytic approach to the optimal design of beam-type structures subject to alternative loads and constraints on generalized compliance and bending stiffness. Necessary and sufficient optimality conditions are established. The determination of optimal designs is illustrated by examples and general features of optimal design are discussed.

The optimal design of beams and frames with compliance constraints

The optimal design of simple structures composed of a material which satisfies a viscous power law relation is considered. The structure must be designed for minimum weight subject to a specified rate of work of the external loads. It is shown that the optimal design problem can be treated as a structural design problem with the constitutive relation modified by requirements which are necessary and sufficient to establish optimality. Linear elastic and rigid-plastic problems can be recovered from this treatment.

Optimal design of beams for prescribed compliance under alternative loads

A global optimality condition is established for minimum-weight design of sandwich beams with elementwise constant cross section for prescribed compliances in alternative states of loading. This condition requires a nonnegative linear combination of the mean-square curvatures of an element in the considered states of loading to have the same value for all elements. The use of the condition in the determination of the minimum-weight design is illustrated by examples.