Research on the Estimation Method of the Stiffness Degeneration Index of Reinforced Concrete Members

Based on variable amplitude displacement cycle tests of 24 reinforced concrete members with different reinforcement conditions, the stiffness degradation index was proposed to describe the damage..e relationship between the stiffness degradation index, the displacement history, and the cumulative energy dissipation was studied; on this basis, an estimation method for the stiffness degradation index was proposed. By comparing the experimental values and estimated values of the stiffness degradation index, the proposed method provides promising prediction reliability and accuracy. .e stiffness degradation index has an effective relationship with the structural design parameters. Based on the stiffness degradation index, the reinforced concrete members can be divided into five performance levels: no damage (DK,k< 0), mild damage (0<DK,k≤ 0.3), moderate damage (0.3<DK,k≤ 0.7), severe damage (0.7<DK,k≤ 0.9), and destruction (0.9<DK,k≤ 1), which can provide a good reference for the seismic design of reinforced concrete members..e increase in the transverse reinforcement ratio can significantly reduce the stiffness damage, and the effect is more obvious under the conditions of small ductility. Under the same conditions, the smaller the ductility condition is, the smaller the stiffness damage of the reinforced concrete members will be. .erefore, the control of the ductility condition and the increase in the transverse reinforcement ratio are stable and effective methods for controlling the stiffness damage of reinforced concrete members.


Introduction
In performance-based design methods, the determination of the structural performance levels is based on the damage index [1][2][3]. Since the deformation-based damage index is not convenient for considering damage accumulation, it is the development direction of the research on the damage index to consider the deformation and cumulative energy dissipation [4][5][6][7].
In 1986, Park and Ang [5] proposed the Park-Ang damage index by linearly combining the deformation damage term with the energy damage term. e damage index has an effective relationship with the structural design parameters (such as the ductility, yielding load, and reinforcement), which can provide a good reference for seismic design [6]. erefore, an increasing number of applications have been obtained [7]. However, the simplified calculation method for the damage index cannot fully reflect the damage mechanism. On the one hand, the influence of the difference displacement path is not considered in the damage index, while the deformation capacity and energy dissipation capacity of the structure are related to the displacement history [8,9]. On the other hand, to maintain the linear expression of the damage index, the nonlinear relationship between the deformation damage term and the energy damage term is completely borne by the influence of the factor beta, which results in a large dispersion of the statistical results of the beta value. To accommodate for these shortcomings, scholars have performed many revisions on the determination of on the damage index [10,11], which are of great significance for perfecting the damage index. However, these revisions do not touch on the influence of the variable amplitude displacement history on the damage development, which causes the verification of the revised damage index to lack sufficient theoretical support. Another type of damage index describes the damage according to the change in the structural characteristics before and after damage, such as the stiffness [12][13][14], period [15], and deformation energy [16]. ese damage indexes do not directly contain the deformation damage term and energy dissipation term but reflect the results caused by two kinds of damage terms. Since this type of damage index corresponds to concrete physical indicators, the physical meaning of the damage index is clear, and the verification is more convenient. However, this kind of damage index is not directly linked to the design parameters of the structure, so it is not convenient for guiding the performance design of the structure.
In this paper, based on the variable amplitude displacement cycle tests of 24 reinforced concrete members [17], the stiffness degradation index was proposed to describe the damage. e relationship between the stiffness degradation index and the energy dissipation capacity, displacement path, and structural design parameters was studied. On this basis, an estimation method of the stiffness degradation index was proposed, which is intended to provide a reference for seismic design and damage assessment of reinforced concrete members in the future.

Experiments
Twenty-four specimens in the companion paper [17] were selected for this paper. As shown in Figure 1, the dimensions and details of the 24 specimens were identical except for the longitudinal reinforcement ratio in the test section and the transverse reinforcement ratio at the plastic hinge region. e length of the test section was 1500 mm, and the crosssection size was 250 mm × 350 mm. e properties and imposed displacement patterns of the 24 specimens are summarized in Table 1. e yield loads and the yield displacements of the 24 specimens were calculated from the skeletal curves according to the equal energy criterion. Specimens R1 to R18 were divided into six groups according to reinforcement type, and three specimens in each group had the same reinforcement type.
Stable variable (SV) displacement patterns were imposed on specimens R1 to R18. An SV displacement pattern was composed of several constant-amplitude cycle (CC) stages. As shown in Figure 2, the SV displacement pattern was composed of four CC stages, and the displacement amplitudes of the four CC stages were 20 mm, 40 mm, 60 mm, and 40 mm. e first stage of the CC was defined as the initial half-cycle sequence (IHS). For the subsequent stages of the CC, when the displacement amplitude was smaller than the maximum historical displacement amplitude, the CC stage was defined as the smaller subsequent half-cycle sequence (SSHS); when the displacement amplitude was larger than the maximum historical displacement amplitude, the CC stage was defined as the larger subsequent half-cycle sequence (LSHS). Six SV displacement patterns were designed in the test. e random variable (RV) displacement patterns were imposed on specimens R19 to R24, and the displacement patterns RV1 and RV2 were imposed on test specimens R19 to R21 and R22 to R24, respectively. e RV displacement patterns imposed on the test specimens are shown in Figure 3. Details of the test can be found in the companion paper [17].

e Stiffness Degradation
Index under the SV Displacement Patterns. e hysteresis path between two consecutive zero-forces is defined as a half-cycle. As shown in Figure 4, the abscissa axis is the displacement amplitude, the ordinate axis is the force, and one hysteresis loop is divided into upper and lower half-cycles by the abscissa axis. u k is the displacement amplitude of the k-th half-cycle. e slope of the red line K k is defined as the initial loading stiffness of the k-th half-cycle. e stiffness degradation index of the reinforced concrete members is defined as When the initial stiffness K k is larger than V y /u y , the value of D K,k is taken as 0. e initial loading stiffness of each half-cycle of the 24 specimens was extracted, and the stiffness degradation index of each half-cycle was calculated according to equation (1). e D K,k -k relations of specimens R1 to R24 are shown in Figure 5. It can be seen that (1) whether the displacement pattern is an SV or RV, the value of D K,k monotonically increased from 0 to 1; (2) the difference in the displacement patterns has a significant influence on the development of D K,k . For the three specimens of the same group, when the displacement amplitude increases gradually from small to large (SV1 pattern), the development of D K,k is the slowest; when the displacement amplitude decreases gradually from large to small (SV3 pattern), the development of D K,k is the fastest. (3) e increase in the reinforcement ratio can slow down the development of D K,k , which is more effective under the SV1 displacement pattern (such as specimens R1, R4, and R7).
To study the relationship between D K,k , the displacement history (deformation damage term), and the cumulative energy dissipation (cumulative damage term) and to obtain a simplified estimation method for D K,k , the normalized effective amplitude μ k and the normalized cumulative energy dissipation n k at the k half-cycle are defined as [17] 2 Advances in Civil Engineering Note. V y is the yield load, u y is the yield displacement, ρ w is the transverse reinforcement ratio at the plastic hinge region, and ρ is the longitudinal reinforcement ratio in the test section.

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Here, k− 1 i�1 E H,i is the sum of dissipated energy from the first to the k-1th half-cycle.
e D K,k -n k relations for specimens R4 to R6, R7 to R9, and R10 to R12 are shown in Figure 6. e different halfcycle sequences (HS) are represented by different symbols in Figure 6. e number next to the symbol represents the value of μ k of the HS.  Figure 6 shows that the D K,k -n k relations for the IHS increase monotonically from 0 to 1, but the slope of the D K,k -n k relations for the IHS gradually decreases with increasing n k . According to these characteristics, a damage model of the stiffness degradation index for the IHS is proposed:

D K,k -n k Relations for the Initial Half-Cycle Sequence (IHS).
As shown in Figure 7, parameter α is the peak value of D K,k when the stiffness deterioration tends to stop, the value of parameter α varies from 0 to 1, and the larger the value of parameter α is, the more severe the stiffness degradation is. Parameter β reflects the energy dissipation requirements of the stiffness degradation, and the value of the parameter β varies from 0 to ∞. e larger the value of the parameter β is, the steeper the D K,k -n k relation is, which means that less energy is dissipated when the stiffness deterioration tends to stop. According to equation (4), the D K,k -n k relations for the IHS are fitted by a nonlinear regression analysis procedure, and the values of parameter α and parameter β are obtained. e values of parameter α and parameter β for specimens R1 to R18 are given in Table 2.
e relationship between μ k and parameter α is shown in Figure 8. It can be seen that the value of parameter α increased with increasing μ k . Figures 8(a) and 8(b) show the influence of ρ and ρ w on the α-μ k relations, respectively. e influence of ρ on the α-μ k relations is not clear, but the difference of ρ w causes the α-μ k relations to be obviously stratified, and the increase in ρ w leads to the decrease of parameter α. Based on the correlation of parameter α with ρ w and μ k , equation (5) is obtained by nonlinear curve fitting of the data points in Figure 8 Figure 6: D K,k -n k relations for specimens R4 to R6, R7 to R9, and R10 to R12.

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As shown in Figure 9, the relationship between μ k and parameter β presents a decreasing function. Figures 9(a) and 9(b) show the influence of ρ and ρ w on the β-μ k relations, respectively. Similar to Figures 8(a) and 8(b), only ρ w has a regular influence on the development of the β-μ k relations, and the increase in ρ w leads to the decrease in parameter β. Based on the correlation of the parameter β with ρ w and μ k , equation (6) is obtained by nonlinear curve fitting of the data points in Figure 9(b).
3.1.2. D K,k -n k Relations of the Subsequent Half-Cycle Sequence (SHS). As shown in Figure 6, the D K,k -n k relation of the SSHS continues the trajectory of the previous CC stage.
For the LSHS, the D K,k -n k relation increases rapidly on the basis of the previous CC stage and then tends to stabilize. e shape of the D K,k -n k relation for the LSHS is similar to that of the IHS.
To study the correlation between the D K,k -n k relation of the LSHS and IHS, the D K,k -n k relations of the LSHS and IHS with the same normalized effective amplitude and reinforcement conditions are plotted together.
e D Ek -n k relations of specimens R4 to R6, R7 to R9, and R10 to R12, according to the principle of the same normalized effective amplitude, are shown in Figure 10. It is not difficult to find that the D K,k -n k relation for the LSHS can be obtained by translating the D K,k -n k relation of the IHS to the right.
Based on the correlation of the D K,k -n k relation for the LSHS and IHS, the translation hypothesis of the D K,k -n k relation for the LSHS is proposed. e expression formula of the translation model is as follows: Here, x t is the translation value, Figure 11 shows the translation model of the LSHS, and n c is the normalized cumulative energy dissipation at the first half-cycle of the LSHS.
According to equation (7), the D K,k -n k relations for the LSHS are fitted by a nonlinear regression analysis procedure, and the values of x t are obtained. Figure 12 shows the relationship between x t and n c , which shows a linear growth relationship: Substituting equation (8) into equation (7), the estimation method of D K,k for the LSHS is obtained. erefore, the estimation method of D K,k under the SV displacement patterns is as follows: Here, D K,I-1 represents the estimation method of D K,k for the SSHS, and the estimation method of the SSHS is the same as that of the previous CC stage.

e Stiffness Degradation Index under the RV Displacement Patterns.
For the estimation method of the stiffness degradation index under the RV displacement patterns, the normalized effective amplitude μ k needs to be transformed into a quasi-stable normalized effective amplitude according to equation (10). en, the stiffness degradation index must    Advances in Civil Engineering be calculated according to the estimation method of D K,k for the SV displacement patterns. e meaning of equation (10) is that the normalized effective amplitude μ k of the similar half-cycles is replaced by the average value. To identify the continuous half-cycles with similar normalized effective amplitudes, equations (11) and (12) are applied. Figure 13

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shows the process of the quasi-stable normalized effective amplitude for specimen R24. See companion paper [17] for the detailed calculation process: where u max is the maximum normalized effective amplitude. For the estimation method of n k , the energy dissipation of the k-th half-cycle E H,k needs to be calculated first according to the companion paper [17], and then n k is estimated by equation (3). e companion paper [17] proposed computing the energy dissipation of the k-th halfcycle E H,k as follows: (13) Figure 14 shows the prediction procedure of D K,k . e previous step of the calculation of D K,k requires the calculation of n k by equation (3). In the calculation of equation (9), it is necessary to judge the subsequent half-cycle sequence as the LSHS or SSHS according to the loading displacement history and then select the corresponding estimation method. In summary, the value of D K,k is dependent on both the cumulative energy dissipation (cumulative damage term) and displacement history (deformation damage term). e test data of 7 specimens (specimens R25 to R31 in Table 3) with the following conditions were selected from the Pacific Earthquake Engineering Research Center (PEER) database: (a) the failure type is flexural failure; (b) the axial load is zero; and (c) the section is rectangular. e yield loads and the yield displacements were calculated from the skeletal curves of the hysteresis curves according to the equal energy criterion. e properties of the 7 specimens in the PEER database are shown in Table 3.
To understand the accuracy of the estimation method, Figure 15 shows the estimated values and experimental values of each half-cycle for 31 specimens, where the abscissa is the experimental value and the ordinate is the estimated value. e hollow point is D K,k for specimens R1 to R24, and the solid point is D K,k for specimens R25 to R31 in the PEER database. e results show that the proposed method provides promising prediction reliability and accuracy.

Performance Classification Levels of the Stiffness Degeneration Index
e damage development process of specimens R1 to R24 was not recorded in detail during the test. erefore, specimens R1 to R24 cannot be used to study the performance classification levels of the stiffness degradation index. e damage development process was recorded for  10 Advances in Civil Engineering specimens S1 to S9 from reference [22], and the damage stages faithfully corresponded to the number of half-cycles. e damage stages of specimens S1 to S9 were divided as concrete cracking, slight spalling of the concrete, severe spalling of the concrete, and buckling of the longitudinal reinforcement. e four performance levels corresponding to the four damage stages are mild damage, moderate damage, severe damage, and destruction. erefore, specimens S1 to S9 from the reference [22] were used to discuss the performance classification levels of the stiffness degeneration index. e properties and the number of halfcycles corresponding to the damage of specimens S1 to S9 are shown in Table 4. e dimensions of specimens S1 to S9 are the same as those of specimens R1 to R24. Specimens S1 to S9 were divided into three groups according to the reinforcement type. ere were three specimens in each group with the same reinforcement type. Group 1 and group 2 have the same transverse reinforcement ratio at the plastic hinge region but different longitudinal reinforcement ratios in the test section. Group 2 and group 3 have the same longitudinal reinforcement ratio in the test section but different transverse reinforcement ratios. Four RV displacement patterns were imposed on specimens S1 to S9 : B1, B1-X, B1-D, and B2. Each displacement pattern contained the main part (black solid line) and the additional part (blue solid line) in Figure 16. Figure 17 shows hysteresis of the loops for specimens S1 to S9, and the different symbols are used to represent the different damage stages. Figure 18 shows the D K,k -k relations for specimens S1 to S9. It can be seen from Figure 18 that the D K,k -k relations exhibit a zigzag pattern due to the asymmetry of the stiffness degradation of the positive and negative half-cycles, but it does not affect the tendency of D K,k to increase monotonically from 0 to 1. Comparing specimens S1 and S2 with specimens S7 and S8 in Figure 18(a), it can be found that, under the same displacement pattern, the increase in the reinforcement ratio can significantly slow down the development of D K,k , and similar phenomena can be observed in Figure 18(c). Comparing specimens S4, S5, and S6 in Figure 18(b), it can be seen that, under the same reinforcement conditions, the difference in the displacement pattern has a significant impact on the development of D K,k . For the three specimens, when the B1-X displacement pattern (the displacement amplitude of the main part increases from small to large) is adopted, the development of D K,k is the slowest. When the B1-D displacement pattern (the displacement amplitude of the main part increases from large to small) is adopted, the development of D K,k is the fastest. Figure 18 shows that, regardless of the difference between the reinforcement condition and displacement pattern, the D K,k values for the same damage stage are similar. e cracking of the concrete is distributed in the range of 0∼0.3, the slight spalling of the concrete is distributed in the range of 0.3∼0.7, the severe spalling of the concrete is in the range of 0.7∼0.9, and the buckling of the longitudinal reinforcement is distributed in the range of 0.9∼0.1. is result shows that the performance levels of the reinforced concrete members can be graded by using the value of D K,k . Table 5 shows the performance classification levels of the reinforced concrete members based on the value of D K,k .
Substituting equations (3), (5) and (6) into equation (9) gives the following: According to equation (14), when D K,k is defined as the damage state of the reinforced concrete member at the end of the earthquake action (performance design objective), μ k and k− 1 i�1 E H,i are the ductility demand and the energy demand, respectively. V y and u y are related to the section size, concrete strength, and reinforcement ratio.
erefore, when the ductility demand, the energy demand, and the performance design objective of reinforced concrete members are determined, the geometrical dimensions of the member, the concrete strength, and the transverse reinforcement ratio can be obtained. erefore, this method can be used for the performance design of reinforced concrete members.

Correlation between D K,k and
Different Parameters e correlation between D K,k and different parameters is discussed according to equation (9). Since the estimation method of D K,k for the SSHS can be replaced by the estimation method of the previous CC stage, equation (9) can be expressed in another form: e values of α max , β max and x t,max are calculated from where μ k,max � max (μ 1 ... μ k ), x t,max � max (x t,1 ... x t,k ), and when x t,max is less than 0, x t,max is equal to 0, n c,max � max (n c,1 ... n c,k ). Substituting equations (16) to (18) into equation (15) gives the following: Figure 19 shows the correlation between D K,k and different parameters. e x coordinate axis is ρ w , which varies from 0.2% to 1%. e y coordinate axis is n k , which varies from 0 to 100, and the z coordinate axis is D K,k , which varies from 0 to 1. μ k,max is 3 and 1.5, which represent the large ductility condition and the small ductility condition, respectively. n c,max is 10 and 90, which represent the peak displacement occurring at the front of the loading displacement history and the peak displacement occurring at the end of the loading displacement history, respectively.
As shown in Figure 19, the increase in the cumulative energy dissipation will lead to the aggravation of the stiffness damage, but the effect will slow down after the cumulative energy dissipation exceeds a threshold value. Comparing  Figures 19(a) and 19(b), when the peak displacement occurs at the front of the displacement history, the stiffness damage of the reinforced concrete members will be more serious than that of the peak displacement at the end of the displacement history. e results show that the development of D K,k is controlled by both the cumulative energy dissipation (cumulative damage term) and displacement history (deformation damage term). At the level of structural design, the increase in the transverse reinforcement ratio can reduce the damage, and the effect is more obvious under the conditions of small ductility. However, the effect will disappear after the transverse reinforcement ratio exceeds a certain threshold value. Under the same conditions, the smaller the ductility condition is, the smaller the damage of reinforced concrete members will be. erefore, the increase in the transverse reinforcement ratio within a certain threshold range and the control of the ductility condition are stable and effective ways to control the seismic damage of reinforced concrete members.  Figure 18: D Ek -n k relations for specimens S1 to S9: (a) S1, S2, S7 and S8. (b) S4, S5 and S6. (c) S3 and S9.

Conclusions
(1) An estimation method for the stiffness degradation index was proposed, and the method provides promising prediction reliability and accuracy. (2) e stiffness degradation index has an effective relationship with structural design parameters. Based on the stiffness degradation index, the reinforced concrete members can be divided into five performance levels, i.e., no damage (D K,k < 0), mild damage (0 < D K,k ≤ 0.3), moderate damage (0.3 < D K,k ≤ 0.7), severe damage (0.7 < D K,k ≤ 0.9), and destruction (0.9 < D K,k ≤ 1), which can provide a good reference for the seismic design of reinforced concrete members. (3) e increase in the transverse reinforcement ratio can significantly reduce the stiffness damage, and the effect is more obvious under the conditions of small ductility. Under the same conditions, the smaller the ductility condition is, the smaller the stiffness damage of the reinforced concrete members will be. erefore, the control of the ductility condition and the increase in the transverse reinforcement ratio are stable and effective methods for controlling the stiffness damage of reinforced concrete members.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.