M.Sc Student Voronov Ariel Generalization of Fully Stressed Design Optimization of Beams Department of Mechanical Engineering Professor Emeritus Eli Altus Abstract

Much progress has been made in recent years in the field of structural optimization. Since the 1970s analytical approaches have been mostly abandoned and current studies focus on taking advantage of the advancement of computational abilities to solve structural optimization problems mostly numerically. In this research, an Euler Bernoulli beam - a one dimensional, elastic and linear structure, is studied instead of a complicated 2D or 3D structure. This simplification, along with a mathematical tool recently developed for the analytical approximation of differential equation solutions for beams, allow for analytical development of the Fully Stressed Design (FSD) approach to structural optimization.

An implicit expression for the optimal stiffness is the obtained from the functional differentiation of a target functional, which is defined based on the core concept of FSD. Next, an expression for the moment is obtained from the differential equation of the beam. These two expressions yield an integral equation for the FSD stiffness function, which is solved using a numerical algorithm.

The cases of static determinate and indeterminate beams with a single predetermined load distribution are solved with a functional constraint on the stiffness, which results in an altered implicit expression for the optimal stiffness. Next, an FSD is obtained for infinite possible loading conditions, which satisfy a functional constraint. This case is relevant to many design problems where, for current solutions, the vague nature of the loading conditions requires large safety factors which dramatically decrease the efficiency of the design.

In the dynamic case, an analytical approximation for the moment is obtained using a method developed by Nachum and Altus (2007) based on the Functional Perturbation Method (FPM). The target functional is generalized to consider the additional time dimension and an FSD is obtained. An FSD for infinite loading conditions is obtained for the dynamic case as well.

Lastly, a different approach is developed for static indeterminate beams. A completely analytical solution is found in specific cases; in other cases, the numerical calculation is reduced to the relatively simple solution of a set of algebraic equations. A general solution is shown for a clamped beam with n supports. In addition, a proof for the uniqueness of the solution is presented for a clamped beam with a single support. One of the main advantages of this method is that it is not iterative, and therefore it is significantly more efficient in many cases.