Nonlinear Structural Dynamics Analysis Using Weighted Residual Integration

A new time integration method is proposed for solving a differential equation of motion of structural dynamics problems with nonlinear stiffness. In this method, it is assumed that order of variation of acceleration is quadratic in each time step and, therefore, variation of displacement in each time step is a fourth-order polynomial that has five coefficients. Other than implementing initial conditions and satisfying equation of motions at both ends, weighted residual integral is also used in order to evaluate the unknown coefficients. By increasing order of acceleration, more terms of Taylor series are used, which were expected to have better responses than the other classical methods. The proposed method is non-dissipative and its numerical dispersion showed to be clearly less than the other methods. By studying stability of the method, it reveals that critical time step duration for the proposed method can be larger than the critical time step duration obtained from other classical methods; also, computational time for analysis with critical time step duration for the proposed method becomes less in comparison with linear acceleration and central difference methods. The order of accuracy of the proposed method is four, which showed to be very accurate in almost all practical problems.