# Hysteretic Behavior of Steel Reinforced Concrete Columns Based on Damage Analysis
## 5 Damage-Based Hysteretic Analysis
### Cyclic Strength and Stiffness Degradation in the Hysteretic Model
#### 5.3.2 Degradation Analysis of Strength and Stiffness  
(1) Degradation of yield load  
With the increase of the number of load cycles, the yield load of the member decreases continuously after each reverse loading and reloading, and the degradation rule of the yield load is:  
\[P_{yi}^{\pm}=(1-\beta_{i})P_{y(i-1)}^{\pm}. \tag{24}\]  
where \(P_{yi}^{\pm}\) is the yield load of the member after the \(i\)-th cycle loading, and \(P_{y(i-1)}^{\pm}\) is the yield load of the member before the \(i\)-th cycle loading. The superscript \"\(+\)\" means positive loading, \"\(-\)\" means reverse loading, and the same is shown below. (2) Degradation of hardening stiffness  
With the increase of the number of load cycles, the hardening stiffness of the members is degenerated, and the degeneration rule is:  
\[K_{\text{si}}^{\pm}=(1-\beta_{i})K_{\text{s}(i-1)}^{\pm}. \tag{25}\]  
where \(K_{\text{si}}^{\pm}\) is the hardening stiffness of the member after the \(i\)-th cycle loading, and \(K_{\text{s}(i-1)}^{\pm}\) is the hardening stiffness of the member before the \(i\)-th cycle loading. (3) Degradation of softening stiffness  
With the increase of the number of load cycles and the displacement, the softening stiffness of the member gradually degrades and approaches to the origin, the degradation rule of softening stiffness can be determined by the following equation:  
\[K_{\text{ni}}^{\pm}=(1-\beta_{i})K_{\text{n}(i-1)}^{\pm} \tag{26}\]  
where \(K_{\text{ni}}^{\pm}\) is the softening stiffness of the member after the \(i\)-th cycle loading and \(K_{\text{n}(i-1)}^{\pm}\) is the softening stiffness of the member before the \(i\)-th cycle loading. Figure 12 shows the degradation rule of member yield load, hardening stiffness, and softening stiffness. The member is loaded from point 0 to point 2 along the loading path 0-1-2. The initial loading stiffness is \(K_{\text{e}}\), the hardening stiffness is \(K_{\text{s0}}^{+}\), the softening stiffness is \(K_{\text{n}}^{+}\). Then along point 2 unloaded to point 3, the unloading path is 2-3. A half cycle is completed from point 0 to point 3 according to Equation (18) to calculate a damage value. Then the degradation index \(\beta_{i}\) is calculated through Equation (22), and the respectively corresponding strength and stiffness values are calculated by Equation (24), Equation (25), and Equation (26). In the reverse loading, the yield load of the members decreases from \(P_{y}^{-}\) to \(P_{y1}^{-}\), the slope of the hardening stiffness decreases from \(K_{\text{s0}}\) to \(K_{\text{s1}}^{-}\), and the slope of the negative stiffness decreases from \(K_{\text{n}}^{-}\) to \(K_{\text{n1}}^{-}\). From point 6 unloads to point 7, the  
damage value is calculated by Equation (18) again. Next the half-cycle damage value increment \(\Delta D_{i}\) is calculated, and then the degradation index \(\beta_{i}\) is recalculated by Equation (22). When the member is reloaded along the load path 7-8-9, the yield load of the structure reduces from \(P_{y}^{+}\) to \(P_{y1}^{+}\), the hardening stiffness decreases from \(K_{\text{s0}}\) to \(K_{\text{s1}}^{+}\), and the softening stiffness decreases from \(K_{\text{n}}^{+}\) to \(K_{\text{n1}}^{+}\). (4) Degradation of unloading stiffness  
The existing research results show that the unloading stiffness of the member does not significantly degrade during the elastic phase and the elastoplastic stage, which is almost the same as the initial stiffness \(K_{\text{e}}\).When the horizontal load reaches the peak load and the structure is in the plastic force state, the unloading stiffness of the members degenerates and can be described as follows:  
\[K_{\text{u}i}=(1-\beta_{i})K_{\text{u}(i-1)} \tag{27}\]  
where \(K_{\text{u}i}\) is the unloading stiffness of the member after the \(i\)-th cycle loading, and \(K_{\text{u}(i-1)}\) is the unloading stiffness of the component before the \(i\)-th cycle loading.