Geometry and Topology Optimization of Statically Determinate Beams under Fixed and Most Unfavorably Distributed Load

Abstract

The paper concerns topology and geometry optimization of statically determinate beams with an arbitrary number of pin supports. The beams are simultaneously exposed to uniform dead load and arbitrarily distributed live load and optimized for the absolute maximum bending moment. First, all the beams with fixed topology are subjected to geometrical optimization by genetic algorithm. Strict mathematical formulas for calculation of optimal geometrical parameters are found for all topologies and any ratio of dead to live load. Then beams with the same minimal values of the objective function and different topologies are classified into groups called topological classes. The detailed characteristics of these classes are described.

Shape optimization of 2D beam-structures to obtain the uniform strength

“…It is worth pointing out that the equations (23)(24)(25), due to the law of ℎ given in equation (20), involve a non-linear relationship among forces and displacements.…”

“…For a statically determinate structure the nodal forces are known only by the balance law and thus solved in a one-shot; hence, the optimal shape is already known without any extra efforts by the equations (20,(23)(24)(25). The resulting shape, namely the weight of the structure, remains a function of the chosen iso-stress value 𝜎 𝑖𝑠𝑜 .…”

“…The searches for an optimum involving Frame structures (generally beams that englobe truss structures) differ in terms of purpose, weights [15,16], minimal bending moments [17,18], stress minimizations [5,19], maximal stiffnesses. Analytical solutions may be involved for statically determined beams [19][20][21][22][23], on the other hand, numerical approaches are recurrent for statically undetermined structures [24][25].…”