Abstract: For a lower bound analysis, a quadrilateral element is more efficient and accurate, compared with a triangle element. The regional integral resulted from the weak form of equilibrium equations is reduced into a boundary integral through Green's theorem, and linear equilibrium equations are greatly simplified. The numerical integral scheme and coordinate transform in Gaussian integral are avoided. Furthermore, the stress continuity equations are also reduced. The presented method not only simplifies the calculation, but also makes it easy to program. The examples show that the method has higher accuracy.