In this paper, we propose a robust shape optimization method for a linear elastic structure with unknown loadings. The concept of principal compliance for minimizing the maximal compliance in the unknown loadings is applied to a shape optimization problem of a linear elastic structure. The principal compliance minimization problem is transformed to the equivalent maximization problem of the fundamental eigenvalue, and the problem is formulated as the distributed-parameter shape optimization problem based on the variational method. The derived shape gradient function is applied to the H¹ gradient method to determine the optimal shape variation, or the optimal free-form of the linear elastic structure. With this method, the optimal shape can be obtained without shape parameterization, while maintaining the surface smoothness. It is confirmed that the obtained shape has high and uniform stiffness in all directions. We confirm the proposed method is effective for designing the robust shape with high stiffness of a linear elastic structure with unknown loadings.