The compression rod is an important stress member of house building and bridge structure. When the load on the compression rod reaches the critical load, the entire structure will lose its stability. We use the fractional-order differential equation of the curvature of the member to bend and apply the fourth-order differential equation’s general solution to establish the compression rod’s stability model in construction engineering. In this paper, the discrete boundary conditions are applied to the algebraic equation system by the substitution method to obtain the characteristic equation about the buckling load of the compression rod. The research found that the method proposed in the paper is simple. The critical load relation deduced in this paper is reasonable and efficient.