Swelling and shrinkage of wood provides challenges when used as construction material. On the positive side, it allows for manufacturing curved wood elements by self-shaping upon drying. We present a two-dimensional model for simulating the moisture-induced bending of wooden bilayers and more general composite plates with multiple, periodically distributed material phases. The model is derived in a mathematically rigorous way from a fully three-dimensional nonlinear hyperelasticity model of the plate and is computationally cheap to handle. We focus specifically on the scenario of free deformations. While the derived model is capable to describe general isometric bending deformations, it has the property that free deformations in equilibrium are given by uniaxial bending deformations that can be computed without the need for solving the plate equations via finite element simulations. The hereby gained computational efficiency enables extensive parameter studies to be conducted regarding the design of wooden composite plates with complex composite geometry. We validate the model for homogeneous wooden bilayers and obtain good agreement between simulations and experimental data. Then we use the model to predict the bending behavior of periodically perforated bilayers. The comparison to other existing models and implications for practical use are discussed.