Optimal Design Of Elastic Beams Research Articles

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.

Optimal Design of Elastic Beams for Moving Loads

The problem of minimizing the volume (weight) of an elastic beam subjected to a compliance constraint is the subject of this paper. Specifically, the necessary conditions for the minimum volume are established for the case of beams, consisting of longitudinal segments of different constant cross‐sectional areas, subjected to moving loads. Two examples are considered, one for a moving concentrated load and the second for a distributed load. The results are the load location, the segment lengths and areas, and therefore the volume of the beam. For the case of the concentrated load, additional results are considered if the segment lengths are predetermined. That is, the load location and segment areas are established. The results are presented for beam cross sections that are sandwich type with different sheet thickness, solid cross sections of different widths and different depths.

Optimal design of elastic beams under multiple design constraints

This paper discusses the optimization of elastic beams under multiple load conditions and self-weight subject to stress and displacement constraints as well as limits on the cross-sectional area and its rate of spatial change (“Niordson constraint”). The general formulation allows for the effect of both bending moments and shear forces on the stresses and deflections. The proposed method is based on static-kinematic optimality criteria which have been successfully used in optimal plastic design. In the above approach, the Lagrangian of the equilibrium condition is regarded as an “associated” (or “Pragerian”) displacement field. The general theory is then illustrated with the example of a built-in beam subjected to stress and Niordson constraints; the statical redundancy of the beam provides a (zero) displacement constraint. Allowance is also made for the cost of clamping moments. It is found that, in general, some segments of the beams are “understressed” and the associated displacement field contains concentrated rotations (“curvature impulses”). Moreover, the solution of this example is found to take on a surprising number of different forms. A beam example with allowance for self-weight will be discussed in Part II of this study.

Optimal design of elastic beams with segmentwise constant height in the case of dynamic loading

A shape optimization method for elastic beams with segmentwise constant height is proposed. The ends of the beam are simply supported, clamped or free. Two problems are discussed; (i) a beam is loaded by lateral pressure, (ii) a beam is struck by an attached mass. External loads and volume of the beam are prescribed and the deflection in a given cross-section is to be minimized. For solving this problem, the deflections are expanded into a Fourier series by eigenfunctions. Numerical results are given for two-, three- and four-stepped beams.

Optimal Design of Elastic Beams with Unspecified Support Conditions

AbstractFormulation of the problem of optimal design for beams with unspecified support position and stiffness is presented. It is assumed that the beam is composed of segments of unspecified cross‐sections and lengths which are to be determined together with support positions and cross‐sections. Design for specified mean compliance, first frequency of free vibrations or buckling force is considered and proper optimality criteria are derived. Illustrative examples are presented and the iterative algorithm of numerical solution is discussed in detail.

Optimal Design of Elastic Beams for Minimum-Maximum Deflection

Optimal Design of Elastic Beams for Minimum-Maximum Deflection

Effect of shear deformation on optimal design of elastic beams

In order to avoid unbounded local stresses in the optimal structure, a realistic optimal design must include the effect of shear deformation. In this paper, such effect on the optimal design of elastic beams is illustrated by a problem of a circular ring. The cross section of the ring is assumed to be rectangular with given depth and variable width. Timoshenko beam theory is employed in the analysis. It is found that the optimal ring with shear deformation is more uniform in its cross section distribution than the optimal ring without shear deformation. Thus, by including the effect of shear deformation, the efficiency of optimal design is reduced.

Optimal design of elastic structures for multipurpose loading

The optimal design of elastic beams subjected to two alternative loading systems is considered for compliance constraints and a minimum-cross-section constraint. Sufficient conditions for optimality are established, and a technique for determining the optimal design is presented. Two examples are given. Generalizations to more than two loading systems and more complex structures are straightforward.