# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 5 Numerical Examples
### Long Span Planar Skeletal Structure  
A large span truss model [44] with a length of 25 m and the height of 1m is shown in Figure 12. The elastic modulus is \(E=2.0\times 10^{5}\) MPa, the density is 7800 kg/m\({}^{3}\), and the cross-section area of each bar is 100 mm\({}^{2}\). In FEM, the large displacement Saint Venant Kirchhoff material model and complete Lagrange description [45] are adopted for the truss elements. There are 101 member bars in total, and each member bar is a single element with hinged constraints at both ends. As shown in Figure 9, Points \(A\) and \(B\) are located near the middle of the span and are set to gain an initial velocity \(v=100\) m/s. This example is used to test both the numerical stability and numerical dissipation of the algorithm, and the integration scheme in Equations (34)-(38) with \(\alpha\geq 0\). Firstly, setting the time step \(\Delta t=0.01\) s, this problem was solved with trapezoidal rule and the proposed method with \(\alpha=0\). The vertical responses at Point \(A\) are shown in Figure 13. It can be obviously seen that after about 2 s, the numerical errors of the acceleration start to accumulate significantly for the trapezoidal rule, while the proposed method with \(\alpha=0\) still performs well. Enlarging the time step size to be \(\Delta t=0.02\) s and \(\Delta t=0.04\) s, the responses resulted from the proposed method with \(\alpha=0\) remain stable, as shown in Figure 14, i.e., the proposed method performs better in numerical stability than the trapezoidal rule for this example. Figure 11: Convergence curves of displacements and velocities of the seven integration schemes: (**a**) Displacement/rotational angle. (**b**) Velocity. Figure 12: Long span planar skeletal structure. Secondly, in order to verify the efficiency in numerical dissipation of the proposed algorithm, we set the time step interval \(\Delta t=0.001\) s and solve the problem with \(\alpha=0\), 0.01, 0.05, 0.1, 0.2 respectively, the resulting responses are shown in Figures 15 and 16. It can be seen that, when \(\alpha=0\), the high-frequency oscillation of the system is very intense. With an increase of \(\alpha\), i.e., numerical damping is introduced, the high-order-frequency response of the system obviously decays. The higher  
Figure 14: Vertical displacement responses of Point \(A\) with \(\Delta t=0.02\) s and \(\Delta t=0.04\) s: (**a**) Displacement response with \(\Delta t=0.02\) s. (**b**) Displacement response with \(\Delta t=0.04\) s.  
Figure 13: Vertical response of Point \(A\) with the trapezoidal rule and the proposed method when \(\Delta t=0.01\) s: (**a**) Displacement response. (**b**) Velocity response. (**c**) Acceleration response. \(\alpha\) is, the faster the high frequency response decays. In addition, the low frequency response remains intact while no obvious amplitude decay and no period elongation appear. Therefore, the proposed method can effectively filter out the high frequency response of the system while retaining the low frequency response well through the controllable numerical damping. Thirdly, the vertical displacement responses at Point \(A\) calculated with the Wilson \(\theta\)-method, [PERSON] method, and the proposed method are shown and compared in Figures 17 and 18, where \(\Delta t=0.01\) s and the computational times are correspondingly listed in Table 3. It is found that, for the [PERSON] method the computational time is short, but the numerical damping is a little large, which results in excessive dissipation of low frequency modes and significant amplitude decay. For [PERSON] method, the low frequency modes are retained well and the high frequency modes of the response are quickly filtered out, but the computational time is longer. Nevertheless, with the setting of different \(\alpha\), the present method performs better than the [PERSON] method in numerical dissipation and better than Bathe method in computational efficiency. Figure 16: Vertical velocity responses of Point \(A\) with \(\Delta t=0.001\) s: (**a**) Velocity responses.