The digital design of thin-walled lightweight components made of carbon-fiber-reinforced plastics (CFRP) needs accurately calibrated material models on so-called coupon tests. Due to the high in-plane stiffness of the CFRP, tensile measurements turn out be very difficult, especially the clamping method plays a fundamental role. Therefore, the effective mechanical material parameters are often derived with three-point or four-point bending tests. The same holds true for materials like concrete which have a poor tensile strength. The measured plate bending stiffness can be predicted for laminate structures made of unidirectional fiber-reinforced laminae very well by using the classical laminate theory in a two-step approach. At first, the effective lamina stiffness is calculated with analytical formulas taking as input the elastic parameters of the fibers and matrix material and the volume fraction and orientation of the fibers. Secondly, based on the stacking of the laminae the effective plate bending stiffness is obtained by using the plane stress assumption. If the lamina exhibits a more complex geometry, e.g. due to fiber waviness, the first step can be replaced by numerical approaches. Especially, the FFT-based homogenization method of Moulinec and Suquet has proven to be a powerful tool for the computation of effective mechanical properties of micro-heterogeneous materials. When nonlinear effects enter the stage, however, a direct simulation of the bending is unavoidable. In our talk we will extend the Lippmann-Schwinger equation for nonlinear elasticity at small-strains by mixed strain/stress gradient loadings. To control all components of the effective strain/stress gradient the periodic boundary conditions are combined with constraints that enforce the periodically deformed boundary to approximate the kinematically fully prescribed boundary in an average sense. Afterwards, we will validate the resulting fixed point and Fletcher-Reeves algorithms by comparing its predictions with the laminate theory. Finally, we will showcase the low memory consumption and extraordinary computational speed of the algorithms by using them to predict effective nonlinear properties under bending loadings for a laminate.