In this thesis, the tangent stiffness matrix for the plane frame structure with geometrical nonlinearity is derived by Lagrange method. The tangent stiffness matrix is composed of constant part, force part and displacement part. Newton's method is used to solve the static cases. The mode vector method and the direct method are used to obtain the equation of motions. Each of them has its own advantage and disadvantage, and Runge-Kutta method is applied to obtain both the linear and nonlinear steady state response of the plane frames. The nonlinearity of Plane frames becomes obviously as the external forces increase. The numerical results have shown that the displacement part is dominated in the stiffness matrix, and the increasing of member's E and I can cause a significant decreasing for displacement for both static and forced cases. Response histories of linear and nonlinear structures about the exciting force's angular speed are also investigated. Key words: Plane frame, Lagrange method, geometrical nonlinearity, steady state response, multi-degree-of-freedom system.