## 7 Seismic Analysis
### Results of the Piers Models
#### 7.2.1 Pier 1: Shear-Diagonal Failure  
Based on preliminary analyses, an amplitude of 75% was determined as a suitable dynamic input for conducting the analyses. In accordance with Eurocode 8 guidelines, two critical limit states based on structural drift are recommended: significant damage (SD) and near collapse (NC). For primary structural elements, the SD limit corresponds to a drift of 0.40%, while the NC limit is set at 0.53% [97]. The drifts were calculated at the midpoint of the top of the pier. In the first set of analyses, the four element types were tested through the implicit solver. For linear elements with reduced integration, the three hourglass controls were implemented. The fundamental frequency of the model was 7.58 Hz. The time increment was set to 0.005 s and the calculated Rayleigh damping parameters were \(\alpha=2.26981\) and \(\beta=2.596\times 10^{-4}\). All the models exhibited similar drift responses before the onset of large plastic deformations at approximately 4.5 s (see drifts in Figure 12). Notably, linear elements seemed to be stiffer, particularly the LF model (full integration), which never reached the limit states. The mass on top of the wall also imposed some degree of flexural behaviour on the pier. Therefore, the LF model underestimated the displacements, since linear elements with full integration are stiffer against this deformation type. The employment of a greater number of elements did not alleviate the shear locking effect. Regarding the three models with reduced integration (LR), their behaviour was identical irrespective of the hourglass control applied. In the CDP constitutive law, the concept of damage energy represents the dissipated energy due to the reduction of material strength, whereas plastic energy refers to the energy required to induce plastic deformations. As seen in Figure 12, when the NC limit was attained, LR models dissipated more energy through plasticity but less through damage compared to QF and QR models. The damage patterns at the NC state, expressed in terms of maximum strains, are depicted in Figure 13. The damage in all the models started at the corners of the base, indicating an initial flexion. However, the predominant failure mode was the abrupt initiation of a diagonal crack, as expected. The next series of analyses were carried out using the explicit solver. Nine models were created to assess the three selected hourglass controls: relax stiffness, (pure) stiffness, and combined, each with three different scaling factor values: 0.01, 0.02, and 0.03. It is noted that only the \(\alpha\) damping parameter was used. The initial time increment automatically calculated by Abaqus was 3.697682 \(\times\)\(10^{-5}\) s.  
Figure 14 presents the time history results for these nine models, and for comparison, the responses of the LR and QF models are also included. In this work, it is assumed that the results from quadratic elements with full integration are the more accurate and thus the reference. Models with a scaling factor of 0.03 generated more artificial energy, while the models with 0.01 were more efficient. In nearly all the cases, the NC limit was reached before the artificial energy achieved 5%. Figure 14: Comparison of the time history results for all the nine models, obtained from explicit solutions, varying the hourglass control implemented and the scaling factor value. Figure 12: Comparison of the time history results for all the models, obtained from implicit solutions: quadratic elements with full integration (QF) and reduced integration (QR), and linear elements with full integration (LF) and reduced integration (LR). Figure 13: Damage pattern at the NC limit state (in terms of maximum principal strains) obtained from implicit solutions. Independently of the hourglass control, the response of the nine models is conservative compared to the implicit solution. Before the material nonlinearities increased considerably at 4.2 s, the drifts were similar to the implicit solution, in particular to the LR model (see drifts in Figure 14).