## 7 Seismic Analysis
### Results of the Piers Models
#### 7.2.2 Pier 2: Flexural Failure  
In this work, the drift values recommended by Eurocode 8 were considered, in which the damage limit states for flexural failure are determined based on the slenderness of the wall. For primary structural elements, the drift is defined as 0.8(\(H_{0}/D\)) for the SD state, and as 1.06(\(H_{0}/D\)) for the NC state, where \(H_{0}\) is the height of the wall and \(D\) denotes its  
Figure 19: Comparison of the time history results for the models varying the bulk viscosity values.  
Figure 20: Damage pattern at the NC limit state (in terms of maximum principal strains) for different values of bulk viscosity, obtained from explicit solutions.  
horizontal dimension. The values for the analysed pier are 1.32% and 1.75% for SD and NC states, respectively.  
To streamline the analysis and reduce the number of studied parameters, the created models only included the features with the best performance in the previous analyses in both implicit and explicit solvers. In the implicit method, linear elements with full integrations (LF) were excluded due to their inaccuracy in simulating bending problems. Thus, both quadratic elements (QF and QR) and linear elements with reduced integration (LR) were employed. The three hourglass controls (relax stiffness, stiffness, and combined) were applied using a scaling factor of 0.01. The fundamental frequency of the wall was 14.16 Hz. The time increment was set to 0.005 s, and the calculated Rayleigh damping parameters were \(\alpha\) = 4.79667 and \(\beta\) = 6.826 \(\times\) 10\({}^{-5}\).  
For the explicit solver, the hourglass modes of the three created models were controlled by the relax stiffness method with a scaling factor of 0.01. The first model featured a damping factor of \(\beta\) = 0 and a bulk value of 0.06, as commonly used in practice. The second model included the damping of 0.1\(\beta\) and a bulk viscosity value of 0.06. For the third model, the damping was set at 0.1\(\beta\) with a bulk viscosity value of 1.00, which was the combination with better performance in the shear failure analyses.  
Figure 21 shows the time history response of all the models. A key observation is that LR models (implicit solver) exhibited identical behaviour in terms of artificial energy, displacements, and damage distribution, irrespective of the applied hourglass control. These models exceeded the NC limit earlier in the analysis. Although the displacements of QF and QR models were similar during almost all seismic loading, the QR model never reached the NC state. Regarding the explicit solution, the artificial energy generated by the three models was negligible. Nevertheless, the hysteretic energy (plastic and damage) at the NC state was considerably larger compared to the implicit models. In terms of displacements, the model with 0.1\(\beta\) and a bulk viscosity value of 1.00 behaved more similarly to the LR models in the implicit solution.  
The damage pattern of all the models is depicted in Figure 22. A uniform and unique crack propagated throughout the base of the wall. Due to the low pre-compression level and the aspect ratio of Pier 2, the failure modes correspond to an IP rocking failure without crushing in the corners. As is evident from both the time history and damage distribution, the models with reduced integration elements reached the NC state in the opposite direction of the models with quadratic elements.  
Figure 21: Comparison of the time history results for flexural failure.  
Finally, six additional models were created by varying the hourglass control (stiffness and combined with a scaling factor of 0.01) and using the same variations for \(\beta\) and bulk viscosity. The results showed insignificant differences in terms of artificial energy, with the same response for displacements. Therefore, it can be concluded that, for this case, any of the three hourglass controls can be used to obtain consistent outputs.