# Dynamic Damage Mechanism and Seismic Fragility Analysis of an Aqueduct Structure
## Dynamic Response Analysis  
### Concrete Damage Plasticity (CDP) Constitutive Model  
The CDP constitutive model assumes that material is damaged by isotropic compression and tension. The damage factor d (Damage) is presented based on the plastic model to represent the irrecoverability of materials produced by damage to simulate the actual mechanical behavior of concrete [30; 31]. It ranges from 0 (the material is not harmed) to 1 (the material is entirely destroyed). Based on the energy equivalent hypothesis, this study adapts [PERSON]'s [32] universal compression and tensile damage factor calculation approach and integrates it directly into the ABAQUS program. In Figure 1, taking C40 concrete as an example, the tensile-compression damage plastic parameters are calculated. According to the plastic damage model theory in ABAQUS, the total strain tensor \(\varepsilon\) is composed of elastic strain rate \(\varepsilon^{el}\), and equivalent plastic strain rate \(\varepsilon^{\rm pl}\) can be determined as follows  
\[\varepsilon=\varepsilon^{el}+\varepsilon^{\rm pl} \tag{1}\]  
When there is no damage to the concrete, the stress-strain relationship of the concrete is  
\[\sigma=D^{el}(\varepsilon-\varepsilon^{\rm pl}) \tag{2}\]  
in which \(\sigma\) is the total stress and \(D^{el}\) is the elastic stiffness matrix. When the concrete is damaged, the material damage factor is introduced to describe the degradation of stiffness,and the relationship of damaged concrete in a three-dimensional multiaxial state can be expressed by the damage elastic equation:  
\[\sigma=(1-d)\overline{\sigma}=(1-d)D^{d}(\varepsilon-\varepsilon^{\text{Pl}}) \tag{3}\]  
In this formula, \(\overline{\sigma}\) is the effective stress, and \(d\) is the damage factor, which varies from 0 (no damage) to 1 (complete damage). The complex damage mechanism of concrete under cyclic alternating stress is related to the cracking, consolidation, and changing of early cracks. When concrete is changed from tension to compression, the stiffness of the concrete recovers locally. To reflect the phenomenon, it is assumed that the relationship between the damage factor and tensile and compressive damage variables (\(d_{\text{t}}\) and \(d_{\text{c}}\)) under an alternating load is  
\[(1-d)=(1-s_{\text{t}}d_{\text{c}})(1-s_{\text{c}}d_{\text{t}}),\;0\leq s_{ \text{t}},\;s_{\text{c}}\leq 1 \tag{4}\]  
where \(s_{\text{t}}\) and \(s_{\text{c}}\) are the relationship of stress action under stiffness recovery, which can be determined as follows  
\[\left\{ \begin{array}{l}s_{\text{t}}=1-w_{\text{t}}r^{*}(\overline{\sigma}), \;0\leq w_{\text{t}}\leq 1\\ s_{\text{c}}=1-w_{\text{c}}(1-r^{*}(\overline{\sigma})),\;0\leq w_{\text{c}} \leq 1\end{array} \right. \tag{5}\]  
where \(r^{*}(\overline{\sigma})\) is the weight factor in the relationship of principal stress under a multiaxial state, which is  
\[r^{*}(\overline{\sigma})\overset{\text{def}}{=}\frac{\sum_{i=1}^{3}\langle \overline{\sigma}_{i}\rangle}{\sum_{i=1}^{3}|\overline{\sigma}_{i}|},\;0\leq r ^{*}(\overline{\sigma})\leq 1 \tag{6}\]  
in which \(w_{\text{t}}\) and \(w_{\text{c}}\) are stiffness recovery weighting factors related to material properties. \(\overline{\sigma}_{i}(i=1,2,3)\) is the main stress component; \(\langle\bullet\rangle\) can be expressed as  
\[\langle x\rangle=(|x|+x)/2\]  
Figure 1 shows the stiffness recovery curve of the concrete damage model under a uniaxial alternating load. The weight factors under tension and compression are \(\omega_{\text{t}}=0\) (compression \(\rightarrow\) tension) and \(\omega_{\text{c}}=1\) (tension \(\rightarrow\) compression), respectively.