An established approach to reduce the effort for the numerical time integration of Finite Element (FE) structures is given by model order reduction (MOR) via subspace projection. These techniques lead to a significant decrease of the necessary degrees of freedom (DOF) by using proper deformation trial vectors. However, if nonlinear loads are applied on distributed regions of the FE structures surface, the computation of these forces is based on physical state-information of all involved nodes. To avoid this dependency, Hyper-Reduction (HR) methods provide a suitable framework to compute the nonlinearity with a reduced number of DOF too. In this contribution, the HR of the nonlinear surface load is based on stress trial vectors, which can be either determined in conjunction with the deformation trial vectors for the MOR (a priori) or as a result of given solution snapshots of the nonlinearity under consideration (a posteriori). In both cases, the stress trial vectors span a subspace, which is combined with a problem formulation via the calculus of variations and a procedure for a reduced selection of integration points (e.g., empirical cubature method). As a result, an HR approach is obtained that allows a more efficient evaluation of the acting nonlinear loads. A numerical comparison of an a priori and an a posteriori subspace is made by using a planar crank drive mechanism, where an elastohydrodynamic (EHD) contact is considered between the piston and the cylinder liner.