# Failure Analysis of a Concrete Anchor under Severe Seismic Action
## 4 Dynamic Simulation under DS-0050  
In this section, a simplified electrical power transformer (single lumped mass system) with dimensions of 2.2 \(\times\) 1.5 \(\times\) 1.2 m is anchored using anchors differing in diameter (M24, M36, and M48) and embedded length (150-450 mm). The weight of the transformer is 1300 kN (132.52 tonnes); this is applied at the center of gravity Figure 14a. The \"anchor supplement\" is beam-linked to the center of gravity. Surface interactions occur between the bottom face of the anchor supplement and top surface of the concrete. The friction coefficient is set to 0.4. The circumferential surface of the embedded anchor and corresponding inner concrete hole are multiply constrained. The concrete bed has dimensions of 3650 \(\times\) 3020 \(\times\) 500 (-700) mm and the anchor embedments are \(h_{ef}\) = 150, 300, and 450 mm for M24, M36, and M48, respectively. The anchor steel supplements are a single vertical PL (300 \(\times\) 140 \(\times\) 20 mm), a single horizontal PL (140 \(\times\) 115 \(\times\) 20 mm), and two triangular PL diaphragms (172 \(\times\) 115 \(\times\) 16 mm) Figure 14b. Figure 12: Pushover simulations. Response modification factor \(R\), overstrength factor \(\Omega_{o}\), and inelastic energy factor \(R_{\mu}\). Figure 13: Cyclic pushover simulations (**a**) for different failure mechanisms. (**b**) Normal and shear stresses vs. plastic strain. In order to consider material damage, a nonlinear material model was considered. The Drucker-Prager Cap model was used as a concrete damage model, and its yield surface is shown in Figure 15. This failure surface consists of a shear fracture surface and an equivalent compression fracture surface. This is the shear failure surface (\(F_{s}\)), the compression failure surface (\(F_{c}\)) using the end-cap, and the transition surface (\(F_{t}\)) to supplement the occlusion continuity by connecting them. The nonlinear strain hardening section represents plastic compression and can control volume expansion. \[F_{s}(\sigma,\beta,d)=q-p\cdot tan\beta-d=0 \tag{11}\]  
Here, \(p=-\left(\frac{1}{\beta}\right)\text{trace}(\sigma)\), \(\text{q}=\sqrt{\frac{3}{2}(s\cdot s)}\), \(s=\sigma+pI\), and \(p\) is an isotropic confining stress, \(q\) is a von-Mises equivalent stress, and \(s\) is a deviatoric. The shear failure surface (\(F_{s}\)) is determined as shown in Equation (11) by the internal friction angle \(\beta\), cohesion, \(d\) and resistance functions for the current deriving stress tensor \(\sigma\). As shown in Equation (12), the Cap compression failure surface provides an inelastic mechanism representing compression softening by limiting the fracture surface during isotropic compression, and simulates shear hardening behavior as a function of an inelastic volume increase occurring when concrete materials are fractured. Plastic volume compression (when compressed and yielded from the Cap surface) causes softening, and plastic volume expansion (when yielded from the Drucker-Prager shear failure surface) causes hardening. \[F_{c}=\sqrt{\left(p-p_{o}\right)^{2}+\left[\frac{R\cdot p}{(1+\alpha-\alpha/ cos\beta)}\right]^{2}}-R(d+p_{a}tan\beta)=0 \tag{12}\]  
Figure 14: (a) Finite element model used for dynamic time history analysis under an earthquake load. (b) Model for the anchor and steel supplements. Figure 15: Failure criterion of Cap plasticity for concrete materials. Here, \(R\) is the material coefficient that controls the cap shape, \(\alpha\) is the value used to define the smooth transition surface between the shear failure surface and the cap, and \(p_{a}\) is the stress representing hardening/softening that causes plastic volume deformation. The hardening/softening law is a function that defines the relationship between isotropic compression yield stress, \(p_{b}\) and volumetric plastic strain, \(\varepsilon_{v}^{p}\). For nonlinear steel material, the perfectly plasticity was assigned.