The objective of this study is to develop a topology-optimal design method that enables structural robustness and stiffness maximization for severe loading conditions where the direction of the design load varies uncertainly in three-dimensional space. Here, a hyperelastic material model is applied to take into account the geometric nonlinear properties of the structure, and the sum of the expected value of the end compliance and the standard deviation is formulated as the objective function. By introducing the proposed quadratic approximation of the objective function, it is shown that a highly accurate design sensitivity can be obtained at a low computational cost, and the validity of the method is verified through several numerical examples.