We present a second-order approximation for the robust counterpart of general uncertain NLP with state equation that is given by a discretized PDE. This approach is an extension of the first-order robust approximations that have been proposed by M. Diehl, H. G. Bock and E. Konstina and by Y. Zhang. We show how the approximated worst-case functions, which are the essential part of the approximated robust counterpart, can be formulated as trust-region problems that can be solved efficiently. Also, the gradients of the approximated worst-case functions can be computed efficiently combining a sensitivity and an adjoint approach. However, there might be points where these functions are nondifferentiable. Hence, we introduce an equivalent formulation of the approximated robust counterpart as an MPEC, in which the objective and all constraints are differentiable. We present numerical results that show the efficiency of the described method when applied to shape optimization in structural mechanics in order to obtain optimal solutions that are robust with respect to uncertainty in acting forces. The robust formulation can further be extended to model the presence of actuators that are capable of applying forces to a structure in order to counteract the effects of uncertainty. Also the robust optimization problem with additional actuators can be solved efficiently.