تجزیه و تحلیل حساسیت و طراحی بهینه ساختارهای چارچوب 3D برای محدودیت استرس و فرکانس

The present paper deals with the problem of determining the optimal joint positions and cross-sectional parameters of linearly elastic space frames with imposed stress and free frequency constraints. The frame is assumed to be acted on by different load systems, including temperature and self-weight loads. The stress state analysis includes tension, bending, shear, and torsion of beam elements. By a sequence of quadratic programming problems, the optimal design is attained. The sensitivity analysis of distinct as well as multiple frequencies is performed through analytic differentiation with respect to design parameters. Illustrative examples of optimal design of simple and medium complexity frames are presented, and the particular case of bimodal optimal solution is considered in detail.

In this paper, we shall discuss the problem of sensitivity analysis and optimal design of frame structures for which both cross-sectional and configuration design parameters are to be determined from the solution. We shall impose the stress and free frequency constraints on the optimal design. The first constraint provides proper stress levels under specified loads, the other constraint is aimed to assume proper structure response under dynamic excitation for which the resonant conditions are avoided and proper frequency spectrum is obtained. The optimal design problems with free frequency constraints were treated in numerous papers (cf. [5], [9] and [10] and references cited therein). Similarly, the sensitivity analysis for single and multiple eigenfrequencies was considered by numerous authors (cf. Wittrick [22], Masur and Mroz [11] and [12], Haug and Rousselet [4], Pedersen [17] and [16], Mills-Curran [14], Mc Gee and Phan [13], Olhoff et al. [15], Krog and Olhoff [8]). The aim of this paper is to extend the previous analyses and consider variation of both cross-sectional and configurational parameters. It turns out that structures are much more sensitive to configuration changes, so search for optimal structure joint positions provides much more efficient designs. The use of stress and frequency constraints assures practical designs for which both static and dynamic responses are controlled. One of the characteristic features of free frequency constrained designs is occurrence of multiple or nearly equal eigenvalues. Such coincidence of free frequencies is associated with structural symmetry or is induced by the evolution of eigenvalue spectrum due to redesign process toward an optimum with constraint set on the fundamental frequency. It is well known that multiple frequencies are not differentiable in the common sense (that is are not Frechet differentiable) and only directional sensitivity derivatives can be calculated. This fact creates some difficulties in finding sensitivities of multiple frequencies with respect to design parameters and in applying the effective gradient optimization techniques. The present paper is devoted to development of an efficient method of sensitivity analysis and optimization of space frame structures with single and multiple eigenfrequencies. The sequential quadratic programming scheme is used with direct application of sensitivity derivatives. It was found that the applied method is reliable and accurate. The specific examples are concerned with a two-beam space frame, frame dome with 52 beam elements, and finally with a frame of mobile crane.

A general optimization formulation is presented for three-dimensional frames for stress and multiple frequency constraints. The optimization can be performed using ordinary methods of design sensitivity analysis and effective gradient optimization methods. The efficiency and accuracy of the optimization method has been demonstrated by a thorough numerical study of space frames. The present work extends the previous treatments of optimal design of space frames as both size and configuration design parameters are treated in the uniform way using the analytically derived sensitivity derivatives. It turns out that the structure response is much more sensitive with respect to joint positions variation, and more effective designs can be generated by optimizing both shape and size parameters. The phenomenon of veering at the bimodal state is one of the interesting observations following from this paper. When curve veering occurs, there is an exchange of eigenmodes at the bimodal point and the frequency lines do not intersect. On the other hand, when intersection of frequency lines occurs, the eigenmodes are preserved along the same lines.