# 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  
#### 5.3.1 Damage-Based Cyclic Degradation Index  
The cycle degradation index \(\beta_{i}\) can be used to describe the degradation of the performance of the structures and members under an earthquake. The previous research results have shown that the loading path has a significant influence on the energy dissipation capacity of the members, therefore, the cyclic degradation index that was established by hysteretic energy dissipation cannot satisfactorily consider the influence of the loading path. Meanwhile, it is difficult to give the unified expression of component energy dissipation capacity under an earthquake. Therefore, in this paper, the cycle degradation index, which was based on the damage index obtained above, was proposed to better consider the effect of the loading path and cycle number on the seismic performance of the frame column, and it is expressed as:  
\[\beta_{i}=[\Delta D_{i}/(1-D_{i-1})]^{\varphi} \tag{22}\]  
where \(\Delta D_{i}\) is the increment of the component damage value for the \(i\)-th loading cycle; \(D_{i-1}\) is the cumulative damage value of the member before the \(i\)-th loading cycle; and \(\varphi\) is the correlation coefficient, based on the test results, and is equal to 1.2 in this research. Based on the degree of damage of the member under cyclic loading, the cyclic degradation index can describe the degradation of member performance. Meanwhile, the impact of the loading path on the cyclic degradation index is also considered. The value of the cyclic degradation index \(\beta_{i}\) is between 0 and 1. When the value is closer to 1, it indicates that the degradation of the member performance is more serious. If \(\beta_{i}<0\) or \(\beta_{i}>1\), it means that the damage value increment of the member exceeds the residual damage value of the member under a certain cyclic loading, which indicates that the structural member is invalid. Therefore, the structure members failure criterion can be expressed as:\[\Delta D_{i}>1-D_{i-1}. \tag{23}\]  
In the following, the degradation rule of the member performance is described by using the damage-based cyclic degradation index \(\beta_{i}\).  
**Remark:** The damage index (Equation (18)), the cyclic degradation index (Equation (22)), and the associate failure criterion (Equation (23)) are hysteresis based. Both are obtained based on the hysteresis curves through destructive tests. However, currently there is a lack of studies to link the sensor-based damage index with the hysteresis-based damage index through destructive tests. It would be interesting to conduct tests of SRC specimens integrated with embedded piezoceramic smart aggregates and to correlate these two indices. If these two indices correlate well, the health status of an in-service structure can be monitored in real-time through embedded sensors based on the sensor enabled damage index. In addition, the failure criterion (Equation (23)) can help to establish the corresponding criterion for the sensor-based damage index.