Keywords: topology optimization, probabilistic loading, multiply load case, stochastic loading, uncertain point of application, optimal layout, optimality criteria method, optimal design, robust design.

Loading uncertainties can be expressed by the stochastic nature of any part of the load determination. If any information is probabilistic amongst the three independent data items specifying the loadings (the magnitude, the line of action and the point application) a more precise design requires a probabilistic method to complete the design procedure. Here a new type of probabilistic optimal topology design method is elaborated where the points of application of the loads are given randomly. As a result of the probabilistic nature of the multiple loading the compliance expressions are probabilistic ones. Two equivalent topology optimization algorithms are elaborated: minimization of the maximum structural compliance with respect to a given volume or minimization of the volume of the structure subjected to compliance constraints. By the use of an appropriate approximation technique, the original stochastic mathematical programming problem is substituted by a deterministic one. The numerical procedure is based on an iterative formula which is formed by the use of the first order optimality condition of the Lagrangian function. In addition to the numerical modelling a parametric study is given where the influence of the multiply loading is investigated to determine the optimal layout in function of the initial layout. The application is illustrated using numerical examples.

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