به‌كارگيري روش نيرو و روش مركزها در بهينه‌سازي وزن سازه‌ي خرپا

COMBINATION OF FORCE FORMULATION AND METHOD OF CENTERS FOR MINIMUM WEIGHT DESIGN OF TRUSS STRUCTURES

در اين تحقيق بهينه‌سازي وزن سازه‌ي خرپا تحت قيود تنش، سطح مقطع، و ارتباط بين سطح مقاطع اعضا تحت يك يا چند حالت بارگذاري استاتيكي، بررسي مي‌شود. روش نقاط مياني كه از تكنيك كره‌هاي محاطي بهره مي‌جويد و از گراديان توابع در آن استفاده مي‌‌شود، به‌عنوان ابزار بهينه‌سازي در نظر گرفته شده، و از روش نيروها در تحليل سازه به‌عنوان ابزار فرمول‌بندي استفاده مي‌شود. در اينجا با تعميم روش مركزها براي در نظر‌گرفتن قيود تساوي، امكان استفاده‌ي هم‌زمان از اين دو روش به‌منظور بهره‌گيري هر‌چه بيشتر از خصوصيات ذاتي آنها فراهم مي‌شود. چند نمونه مثال عددي براي نشان‌دادن كارايي روش پيشنهادي ارايه شده است. مقايسه‌ي نتايج به دست آمده با آنچه كه پيش‌تر در مراجع گزارش شده است، حاكي از قدرت و كارايي اين روش است.

In this article, minimum weight design of trusses under limitations of stresses, size limitations, and the inter-relationship between the cross sectional areas, under several loading conditions, has been considered. Normally, a displacement method of analysis is employed by most researchers for the analysis part of the optimization of truss structures. However, using the force method makes the stress constraints for the elements become linear, which facilitates the use of most optimization procedures and improves efficiency. Therefore, in this article, the force method of analysis is employed as the analysis section, in order to utilize these advantages. The optimization method selected for this research is based on the idea of moving towards the optimum in the feasible part of design space, as defined by the constraints. Whereupon, the method of center points, as defined by the center of inscribed hyper-spheres to the feasible-usable design space, previously developed by researchers for inequality constraints, is adopted. This method is extended, herein, to be capable of taking into account the nonlinear equality constraints arising in the analysis part of the structures also. The method of inscribed hyper-spheres as a robust procedure for solving engineering optimization problems has important features, which increase its efficiency in comparison to other methods. Firstly, the convergence to the optimum through the feasible design space is important from an engineering point of view. Secondly, its capability of considering only the near active constraints that adequately define the interior surface of the feasible-useable design space, is important, in that it enhances the computational efficiency in practical applications where a large amount of constraints should be satisfied. Thirdly, a uniform margin of safety is observed in the sequence of design points generated in the optimization process, creating a series of near optimum acceptable designs. Fourthly, as the movement towards optimum point is always inside the feasible space and uniformly distant from near active constraints, it shows low sensitivity, with respect to its exact definition. In the extension of this algorithm for handling nonlinear equality constraints, attempts are made to maintain the main useful features of the method as much as possible. Therefore, in inscribing the hyper-sphere in feasible-usable design space, it is always forced to be tangent to the linearized surface of the nonlinear equalities. As the optimization steps proceed, the radius of the sphere shrinks to zero and its center finds an optimum location on the joint equality constraint surfaces. In this article, cross-sectional areas of elements and forces of redundant members in a truss structure in each loading are chosen as design variables. This selection alleviates the necessity of separate analyses steps in the optimization procedure. Thus, the combination of the method of hyper-sphere and the forced method of analysis provides higher computational efficiency and rapid convergence ability. Most of the well known examples that have appeared in the literature so far have been solved by the proposed method. The procedure developed proves to be simple and efficient, as evidenced by the results of several classical problems. Observation of the results shows that rapid decrease of weight is achieved in the early steps and the trend in the reduction of weight is monotonic and relatively uniform.