By minimizing the total potential energy function and deploying the virtual work principle, a higher-order stiffness matrix is achieved. This new tangent stiffness matrix is used to solve the frame with geometric nonlinear behavior. Since authors’ formulation takes into account the higher-order terms of the strain vector, the convergence speed of the solution process will increase. In fact, both linear and nonlinear parts of the frame axial strains are included in the presented formulation. These higher-order terms affect the resulting unbalanced force and also frame tangent stiffness. Moreover, the finite element method, updated Lagrangian description, and arc length scheme are employed in this study. To check the efficiency of the proposed strategy, several numerical examples are solved. The findings indicate that the authors’ technique can accurately trace the structural equilibrium paths having the limit points.

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