# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 5 Numerical Simulation
### 4-Span Rigid Frame Bridge  
In this part, a 4-span bridge with a total length of 140 m is used to show the influence of MSE on the virtual damping error and the damping truncation error. The data of the bridge can be found in the reference [2]. Some of the data, including the span and height of the bridge, some of the node number and element number is shown in Figure 10. The frequency response function of the bridge beyond the 8 th mode decreases rapidly and the energy of the ground motion is usually small after 10 Hz. Thus, in the analysis, the 1 st mode and the 8 th mode is used to calculate the Rayleigh damping matrix, and the damping ratio is assumed to be 0.04. A two-dimensional beam element is used as the element of this bridge. Spectral analysis is used in this paper to show the statistical feature of the damping formulation problem. The spatially variate ground motions are simulated with stationary random processes. Figure 10: Simplified computational diagram of a 4-span rigid frame bridge. Figure 9: Comparison of estimated error ratio and numerical results for different size of structures when different modeling methods are used, (**a**) \(r^{Ar}\) and \(r^{Rr}\), (**b**) \(r^{Am}\) and \(r^{Ar}\). As is shown in the previous section, LMM-M approaches RMM-M when the added large masses approach infinity. However, in LMM, a limited large number is practically assigned to the large masses instead of infinity. This process induces rigid body modes and modes with very low natural frequencies into the model. These modes do not actually exist in the real structure. If these modes were calculated in the responses, the responses will be overestimated. To study the error in LMM, a trick is indispensable. In this paper, the limit of FRF when the large masses approach infinity is used instead of the practical LMM. And ILMM is used to refer to this ideal LMM. In this study, the soil conditions of the supports are assumed to be the same. The modified Kanai-Tajimi spectrum [1], which is also called the Clough-Penzien spectrum, is used as the spectral density function of the ground motion. The expression of this spectrum is shown in (41). \[S_{e}=S_{0}\frac{\omega_{g}^{4}+4\xi_{g}^{2}\omega_{g}^{2}\omega^{2}}{\left( \omega_{g}^{2}-\omega^{2}\right)^{2}+4\xi_{g}^{2}\omega_{g}^{2}\omega^{2}} \frac{1}{\left(\omega_{f}^{2}-\omega^{2}\right)^{2}+4\xi_{f}^{2}\omega_{f}^{2 }\omega^{2}} \tag{41}\]  
Firm soil conditions are considered and the parameters are \(\omega_{g}=\)15 rad, \(\omega_{f}=1.5\) rad, \(\xi_{f}=0.6\), \(\xi_{g}=0.6\) [PERSON] et al. [18]. \(S_{0}\) represents the energy intensity of the ground motion. Peak ground acceleration of 0.5 g is used for determining the intensity of the ground motions. The mean peak value of the ground motion and the structural responses are calculated with the method introduced by [PERSON] [19]. The duration of the earthquake is assumed to be 20 s.  
The model derived by [PERSON] et al. [20] is used to describe the incoherence effect in Equation (37), the expression is shown in Equation (42). \[\gamma_{ij}^{l}=\exp[-\left(\frac{\lambda_{x}\omega d_{ij}}{v_{s}}\right)^{2}] \tag{42}\]  
where \(v_{s}\) is an estimation for the elastic S-wave velocity in the random medium, \(\lambda_{x}=\mu\sqrt{R/r_{0}}\), \(R\) is the distance in the random medium traveled by the wave; \(r_{0}\) is the scale length of the random inhomogeneities along the path; \(\mu^{2}\) is the measure of the relative variation of the elastic properties in the medium. The value \(\lambda_{x}/v_{s}=2.5\times 10^{-4}\) is recommended in the reference. [15]. The apparent propagation velocity of the ground motion is assumed to be 800 m/s. Existing researches have shown that the apparent propagation velocity is usually larger than 800 m/s. The response PSD derived from different modelling methods are compared. The mean peak responses are used to measure the amplitude of responses. A new error ratio is defined to measure the error of mean peak responses.