This paper discusses a class of state constrained optimal control problems, for which it is possible to formulate second-order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints. This kind of result is classical in the case of polyhedric control constraints. Our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric, in the presence of state constraints satisfying some specific hypotheses. The analysis is based on the assumption that some strict semilinearized qualification condition is satisfied. We apply the theory to some optimal control problems of elliptic equations with state and control constraints.