TRUSS TOPOLOGY OPTIMIZATION INCLUDING UNILATERAL CONTACT

This work extends the ground structure approach of truss topology opti mization to include unilateral contact conditions. The traditional des ign objective of finding the stiffest truss among those of equal volum e is combined with a second objective of achieving a uniform contact f orce distribution, Design variables are the volume of bars and the gap s between potential contact nodes and rigid obstacles. The problem can be viewed as that of finding a saddle point of the equilibrium potent ial energy function (a convex problem) or as that of minimizing the ex ternal work among all trusses that exhibit a uniform contact force dis tribution (a nonconvex problem). These two formulations are related, a lthough not completely equivalent: they give the same design, but conc erning the associated displacement states, the solutions of the first formulation are included among those of the second but the opposite do es not necessarily hold. In the classical noncontact single-load case problem, it is known that an optimal truss can be found by solving a l inear programming (LP) limit design problem, where compatibility condi tions are not taken into account. This result is extended to include u nilateral contact and the second objective of obtaining a uniform cont act force distribution. The LP formulation is our vehicle for proving existence of an optimal design: by standard LP theory, we need only to show primal and dual feasibility; the primal one is obvious, and the dual one is shown by the Farkas lemma to be equivalent to a condition on the direction of the external load. This method of proof extends re sults in the classical noncontact case to structures that have a singu lar stiffness matrix for all designs, including a case with no prescri bed nodal displacements. Numerical solutions are also obtained by usin g the LP formulation. It is applied to two bridge-type structures, and trusses that are optimal in the above sense are obtained.