## 7 Seismic Analysis
### Results of the Piers Models
#### 7.2.1 Pier 1: Shear-Diagonal Failure
Models with combined hourglass control dissipated more energy through plasticity and damage. Figure 15 illustrates the damage patterns at the NC state, expressed in terms of maximum strains. The type of hourglass control and the scaling factor value influence the distribution of damage. Nonetheless, the primary failure mode remains consistent: the initiation of a diagonal crack, which aligns with the outcomes from the implicit analyses. For the subsequent analyses, the relax stiffness control with a scaling factor of 0.01 was selected. It is noted that either stiffness or combined controls with the same scaling factor value could have been selected since, overall, their behaviour was similar. The subsequent phase involves three models to evaluate the influence of the damping parameter \(\beta\) on the solution and time increment. These models were created using the previously selected hourglass control. Three values for \(\beta\) were assigned as follows. Using the Rayleigh formulation, a first value for \(\beta\) was calculated as \(\beta\) = 2.596 \(\times\) 10\({}^{-4}\). The model is denoted \(\alpha+\beta\). The second value was set to \(\beta\) = 2.596 \(\times\) 10\({}^{-5}\), which is a reduction of 10% (denoted \(\alpha+0.1\beta\)). This value also aligns with the same order of magnitude of the initial time increment without damping (\(\Delta t\) = 3.70 \(\times\) 10\({}^{-5}\)) [87]. Lastly, \(\beta\) = 0 was applied, in line with common practice. This model is referred as \(\alpha\). The time history results of the three models are shown in Figure 16. For comparison, the results from the QF and LR are also included. The introduction of the damping parameter \(\beta\) delayed and controlled the increase in artificial energy. Nevertheless, when \(\beta\) is not reduced, the response in terms of displacements is significantly diminished and the initiation of damage is delayed. The models with \(\beta\) = 0 and 0.1\(\beta\) exhibited comparable levels of energy dissipation (plastic and damage energies) at the NC state. Figure 15: Damage pattern at the NC limit state (in terms of maximum principal strains) for different hourglass controls and different scaling factor values, obtained from explicit solutions. The primary failure mode in the model with 0.1\(\beta\) and \(\beta\) = 0 remains consistent with the previous analyses, consisting of the occurrence of a diagonal shear crack, as seen in Figure 17. It is noted that the image corresponding to the model \(\alpha+\beta\) belongs to the last step just before numerical problems arose. Based on these outcomes, the subsequent set of analyses was conducted using the value of 0.1\(\beta\). The time increment for \(\beta\) = 2.596 \(\times\) 10\({}^{-4}\) was considerably small (see Figure 18). Indeed, using this value not only increases computational time but also yields an unrealistic response (Figure 16). Regarding the model with 0.1\(\beta\), there was a minor reduction in the time increment, from \(\Delta t\) = 3.70 \(\times\) 10\({}^{-5}\) to \(\Delta t\) = 2.01 \(\times\) 10\({}^{-5}\), while the response became closer to the QF model of the implicit method. Figure 16: Comparison of the time history results for the three models varying \(\beta\). Figure 17: Damage pattern at the NC limit state (in terms of maximum principal strains) for different values of \(\beta\), obtained from explicit solutions. In the last series for diagonal shear failure, the effect of the bulk viscosity parameter was assessed, assigning five values: 0.01, 0.06 (by default in Abaqus), 0.50, 1.00, and 5.00. The models were generated using the previously selected relax stiffness equal to 0.01 and \(\beta=2.596\times 10^{-5}\). Since bulk viscosity is a type of damping, it has an impact on the time increment. The reduction became more evident as the value increased, as seen in Figure 18.  
In terms of displacements, models with bulk equal to 1.00 and 5.00 behaved similarly to the LR model of the implicit solution.