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

Authors: Yukui Wang; Zhefeng Liu; Weijun Yang; Jian Wang; Xiao Zheng
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

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. The 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. The 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. The 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. Therefore, 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.


---

## Body

## 1. Introduction

In performance-based design methods, the determination of the structural performance levels is based on the damage index [1–3]. 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 [4–7].In 1986, Park and Ang [5] proposed the Park-Ang damage index by linearly combining the deformation damage term with the energy damage term. The 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]. Therefore, 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 [12–14], period [15], and deformation energy [16]. These 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. The 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.


## 2. 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. The length of the test section was 1500 mm, and the cross-section size was 250 mm × 350 mm.Figure 1
Test specimen (dimensions in mm).The properties and imposed displacement patterns of the 24 specimens are summarized in Table1. The 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.Table 1
The properties and imposed displacement patterns of 24 specimens [17].
Specimen
V
y (kN)
u
y (mm)
ρ
w (%)
ρ (%)
Displacement pattern
Displacement amplitude of CC stage
Group
No.
1
R1
67.60
12.97
0.226
0.766
SV1
20 mm, 40 mm and 60 mm
R2
67.60
12.97
SV2
40 mm, 20 mm and 60 mm
R3
67.60
12.97
SV3
60 mm, 40 mm and 20 mm
2
R4
67.60
12.97
0.402
0.766
SV1
20 mm, 40 mm and 60 mm
R5
67.60
12.97
SV2
40 mm, 20 mm and 60 mm
R6
67.60
12.97
SV3
60 mm, 40 mm and 20 mm
3
R7
67.60
12.97
0.804
0.766
SV1
20 mm, 40 mm and 60 mm
R8
67.60
12.97
SV2
40 mm, 20 mm and 60 mm
R9
67.60
12.97
SV3
60 mm, 40 mm and 20 mm
4
R10
51.50
11.70
0.402
0.587
SV4
20 mm, 40 mm, 60 mm and 40 mm
R11
51.50
11.70
SV5
40 mm, 20 mm, 60 mm and 40 mm
R12
51.50
11.70
SV6
60 mm, 40 mm, 20 mm and 40 mm
5
R13
80.11
14.39
0.402
0.971
SV4
20 mm, 40 mm, 60 mm and 40 mm
R14
80.11
14.39
SV5
40 mm, 20 mm, 60 mm and 40 mm
R15
80.11
14.39
SV6
60 mm, 40 mm, 20 mm and 40 mm
6
R16
89.02
16.66
0.402
1.198
SV4
20 mm, 40 mm, 60 mm and 40 mm
R17
89.02
16.66
SV5
40 mm, 20 mm, 60 mm and 40 mm
R18
89.02
16.66
SV6
60 mm, 40 mm, 20 mm and 40 mm
7
R19
67.60
12.97
0.226
0.766
RV1
—
R20
67.60
12.97
0.402
0.766
RV1
—
R21
89.02
16.66
0.402
1.198
RV1
—
8
R22
89.02
16.66
0.804
1.198
RV2
—
R23
89.02
16.66
0.226
1.198
RV2
—
R24
51.50
11.70
0.226
0.587
RV2
—
Note. V
y is the yield load, uy is the yield displacement, ρw is the transverse reinforcement ratio at the plastic hinge region, and ρ is the longitudinal reinforcement ratio in the test section.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 Figure2, 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. The 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.Figure 2
The SV displacement pattern, which was composed of four CC stages.The 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. The RV displacement patterns imposed on the test specimens are shown in Figure3. Details of the test can be found in the companion paper [17].Figure 3
The RV displacement patterns imposed on the test specimens [17]: (a) RV1. (b) RV2.
(a)
(b)


## 3. The Stiffness Degradation Index of the Reinforced Concrete Members

### 3.1. The Stiffness Degradation Index under the SV Displacement Patterns

The hysteresis path between two consecutive zero-forces is defined as a half-cycle. As shown in Figure4, 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. uk is the displacement amplitude of the k-th half-cycle. The slope of the red line Kk is defined as the initial loading stiffness of the k-th half-cycle.Figure 4
The initial loading stiffness of thek-th half-cycle.The stiffness degradation index of the reinforced concrete members is defined as(1)DK,k=1−KkVy/uy,When the initial stiffnessKk is larger than Vy/uy, the value of DK,k is taken as 0. The 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).TheDK,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 DK,k monotonically increased from 0 to 1; (2) the difference in the displacement patterns has a significant influence on the development of DK,k. For the three specimens of the same group, when the displacement amplitude increases gradually from small to large (SV1 pattern), the development of DK,k is the slowest; when the displacement amplitude decreases gradually from large to small (SV3 pattern), the development of DK,k is the fastest. (3) The increase in the reinforcement ratio can slow down the development of DK,k, which is more effective under the SV1 displacement pattern (such as specimens R1, R4, and R7).Figure 5
D
K,k-k relations of specimens R1 to R24. (a) Group 1. (b) Group 2. (c) Group 3. (d) Group 4. (e) Group 5. (f) Group 6. (g) Group 7. (h) Group 8.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)To study the relationship betweenDK,k, the displacement history (deformation damage term), and the cumulative energy dissipation (cumulative damage term) and to obtain a simplified estimation method for DK,k, the normalized effective amplitude μk and the normalized cumulative energy dissipation nk at the k half-cycle are defined as [17](2)μk=.uk+uk−12uy,k>1,ukuy,k=1.(3)nk=∑i=1k−1EH,i0.5Vyuy.Here,∑i=1k−1EH,i is the sum of dissipated energy from the first to the k-1th half-cycle.TheDK,k-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12 are shown in Figure 6. The different half-cycle sequences (HS) are represented by different symbols in Figure 6. The number next to the symbol represents the value of μk of the HS.Figure 6
D
K,k-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


#### 3.1.1.DK,k-nk Relations for the Initial Half-Cycle Sequence (IHS)

Figure6 shows that the DK,k-nk relations for the IHS increase monotonically from 0 to 1, but the slope of the DK,k-nk relations for the IHS gradually decreases with increasing nk. According to these characteristics, a damage model of the stiffness degradation index for the IHS is proposed:(4)DK,k=α1−e−βnk.As shown in Figure7, parameter α is the peak value of DK,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 ∞. The larger the value of the parameter β is, the steeper the DK,k-nk relation is, which means that less energy is dissipated when the stiffness deterioration tends to stop.Figure 7
Damage model of the stiffness degradation index for the IHS.According to equation (4), the DK,k-nk relations for the IHS are fitted by a nonlinear regression analysis procedure, and the values of parameter α and parameter β are obtained. The values of parameter α and parameter β for specimens R1 to R18 are given in Table 2.Table 2
The values of parameterα and parameter β for specimens R1 to R18.
No.
α
β
R1
0.833
0.486
R2
0.910
0.148
R3
0.932
0.093
R4
0.641
0.174
R5
0.876
0.094
R6
0.924
0.074
R7
0.620
0.160
R8
0.838
0.080
R9
0.902
0.071
R10
0.660
0.234
R11
0.842
0.099
R12
0.894
0.108
R13
0.596
0.608
R14
0.864
0.121
R15
0.910
0.126
R16
0.551
0.263
R17
0.768
0.239
R18
0.901
0.153The relationship betweenμk and parameter α is shown in Figure 8. It can be seen that the value of parameter α increased with increasing μk. Figures 8(a) and 8(b) show the influence of ρ and ρw on the α-μk relations, respectively. The influence of ρ on the α-μk relations is not clear, but the difference of ρw causes the α-μk relations to be obviously stratified, and the increase in ρw leads to the decrease of parameter α. Based on the correlation of parameter α with ρw and μk, equation (5) is obtained by nonlinear curve fitting of the data points in Figure 8(b).(5)α=0.951−e−μk0.68+11.35e−14.24ρw.Figure 8
Relationship betweenμk and parameter α. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)As shown in Figure9, the relationship between μk and parameter β presents a decreasing function. Figures 9(a) and 9(b) show the influence of ρ and ρw on the β-μk relations, respectively. Similar to Figures 8(a) and 8(b), only ρw has a regular influence on the development of the β-μk relations, and the increase in ρw leads to the decrease in parameter β. Based on the correlation of the parameter β with ρw and μk, equation (6) is obtained by nonlinear curve fitting of the data points in Figure 9(b). (6)β=0.31ρw−1.561+μk1.04ρw−0.45.Figure 9
Relationship betweenμk and parameter β. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)


#### 3.1.2.DK,k-nk Relations of the Subsequent Half-Cycle Sequence (SHS)

As shown in Figure6, the DK,k-nk relation of the SSHS continues the trajectory of the previous CC stage. For the LSHS, the DK,k-nk relation increases rapidly on the basis of the previous CC stage and then tends to stabilize. The shape of the DK,k-nk relation for the LSHS is similar to that of the IHS.To study the correlation between theDK,k-nk relation of the LSHS and IHS, the DK,k-nk relations of the LSHS and IHS with the same normalized effective amplitude and reinforcement conditions are plotted together. The DEk-nk 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 DK,k-nk relation for the LSHS can be obtained by translating the DK,k-nk relation of the IHS to the right.Figure 10
DEk-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12 according to the principle of the same normalized effective amplitude.
(a)
(b)
(c)
(d)
(e)
(f)Based on the correlation of theDK,k-nk relation for the LSHS and IHS, the translation hypothesis of the DK,k-nk relation for the LSHS is proposed. The expression formula of the translation model is as follows:(7)DK,k=α1−e−βnk−xt.Here,xt is the translation value, Figure 11 shows the translation model of the LSHS, and nc is the normalized cumulative energy dissipation at the first half-cycle of the LSHS.Figure 11
The translation model of LSHS.According to equation (7), the DK,k-nk relations for the LSHS are fitted by a nonlinear regression analysis procedure, and the values of xt are obtained. Figure 12 shows the relationship between xt and nc, which shows a linear growth relationship:(8)xt=0.65nc−3.2.Figure 12
Relationship betweenxt and nc.Substituting equation (8) into equation (7), the estimation method of DK,k for the LSHS is obtained. Therefore, the estimation method of DK,k under the SV displacement patterns is as follows:(9)DK,k=α1−e−βnk,for IHS,α1−e−βnk−0.65nc+3.2,for LSHS,DK,I−1,for SSHS.Here,DK,I-1 represents the estimation method of DK,k for the SSHS, and the estimation method of the SSHS is the same as that of the previous CC stage.


### 3.2. The 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 amplitudeμk needs to be transformed into a quasi-stable normalized effective amplitude according to equation (10). Then, the stiffness degradation index must be calculated according to the estimation method of DK,k for the SV displacement patterns. The meaning of equation (10) is that the normalized effective amplitude μk 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 shows the process of the quasi-stable normalized effective amplitude for specimen R24. See companion paper [17] for the detailed calculation process:(10)μk=1b−a+2∑j=a−1bμj,k=a−1,b,(11)uj−uj−1<0.15umax,j=a,b,(12)∑j=abuj−uj−1<0.15umax.where umax is the maximum normalized effective amplitude.Figure 13
The process of the quasi-stable normalized effective amplitude for specimen R24. (a). The value ofμk of specimen R24. (b) Removal of the data points under the line of μk = 1. (c) Averaging of the similar values of μk.
(a)
(b)
(c)For the estimation method ofnk, the energy dissipation of the k-th half-cycle EH,k needs to be calculated first according to the companion paper [17], and then nk is estimated by equation (3). The companion paper [17] proposed computing the energy dissipation of the k-th half-cycle EH,k as follows:(13)EH,k=fρw,Vy,uy,μk.Figure14 shows the prediction procedure of DK,k. The previous step of the calculation of DK,k requires the calculation of nk 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 DK,k is dependent on both the cumulative energy dissipation (cumulative damage term) and displacement history (deformation damage term).Figure 14
Prediction procedure ofDK,k.The test data of 7 specimens (specimens R25 to R31 in Table3) 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. The yield loads and the yield displacements were calculated from the skeletal curves of the hysteresis curves according to the equal energy criterion. The properties of the 7 specimens in the PEER database are shown in Table 3.Table 3
The properties of 7 specimens in the PEER database.
No.
Data sources
V
y (kN)
u
y (mm)
B ×h (mm)
ρ (%)
ρ
w (%)
R25
ORC1 [18]
251.71
37.92
305 × 508
1.265
1.372
R26
C5-00N [19]
57.83
13.22
203 × 203
0.965
0.9
R27
C5-00S [19]
R28
U1 [20]
263.75
24.72
350 × 350
1.605
0.3
R29
A1 [21]
45.37
12.44
0.623
R30
B1 [21]
36.39
9.48
152.4 × 152.4
1.225
0.705
R31
C1 [21]
38.74
14.00To understand the accuracy of the estimation method, Figure15 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. The hollow point is DK,k for specimens R1 to R24, and the solid point is DK,k for specimens R25 to R31 in the PEER database. The results show that the proposed method provides promising prediction reliability and accuracy.Figure 15
Estimated values and experimental values ofDK,k for each half-cycle for 31 specimens.


## 3.1. The Stiffness Degradation Index under the SV Displacement Patterns

The hysteresis path between two consecutive zero-forces is defined as a half-cycle. As shown in Figure4, 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. uk is the displacement amplitude of the k-th half-cycle. The slope of the red line Kk is defined as the initial loading stiffness of the k-th half-cycle.Figure 4
The initial loading stiffness of thek-th half-cycle.The stiffness degradation index of the reinforced concrete members is defined as(1)DK,k=1−KkVy/uy,When the initial stiffnessKk is larger than Vy/uy, the value of DK,k is taken as 0. The 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).TheDK,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 DK,k monotonically increased from 0 to 1; (2) the difference in the displacement patterns has a significant influence on the development of DK,k. For the three specimens of the same group, when the displacement amplitude increases gradually from small to large (SV1 pattern), the development of DK,k is the slowest; when the displacement amplitude decreases gradually from large to small (SV3 pattern), the development of DK,k is the fastest. (3) The increase in the reinforcement ratio can slow down the development of DK,k, which is more effective under the SV1 displacement pattern (such as specimens R1, R4, and R7).Figure 5
D
K,k-k relations of specimens R1 to R24. (a) Group 1. (b) Group 2. (c) Group 3. (d) Group 4. (e) Group 5. (f) Group 6. (g) Group 7. (h) Group 8.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)To study the relationship betweenDK,k, the displacement history (deformation damage term), and the cumulative energy dissipation (cumulative damage term) and to obtain a simplified estimation method for DK,k, the normalized effective amplitude μk and the normalized cumulative energy dissipation nk at the k half-cycle are defined as [17](2)μk=.uk+uk−12uy,k>1,ukuy,k=1.(3)nk=∑i=1k−1EH,i0.5Vyuy.Here,∑i=1k−1EH,i is the sum of dissipated energy from the first to the k-1th half-cycle.TheDK,k-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12 are shown in Figure 6. The different half-cycle sequences (HS) are represented by different symbols in Figure 6. The number next to the symbol represents the value of μk of the HS.Figure 6
D
K,k-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


### 3.1.1.DK,k-nk Relations for the Initial Half-Cycle Sequence (IHS)

Figure6 shows that the DK,k-nk relations for the IHS increase monotonically from 0 to 1, but the slope of the DK,k-nk relations for the IHS gradually decreases with increasing nk. According to these characteristics, a damage model of the stiffness degradation index for the IHS is proposed:(4)DK,k=α1−e−βnk.As shown in Figure7, parameter α is the peak value of DK,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 ∞. The larger the value of the parameter β is, the steeper the DK,k-nk relation is, which means that less energy is dissipated when the stiffness deterioration tends to stop.Figure 7
Damage model of the stiffness degradation index for the IHS.According to equation (4), the DK,k-nk relations for the IHS are fitted by a nonlinear regression analysis procedure, and the values of parameter α and parameter β are obtained. The values of parameter α and parameter β for specimens R1 to R18 are given in Table 2.Table 2
The values of parameterα and parameter β for specimens R1 to R18.
No.
α
β
R1
0.833
0.486
R2
0.910
0.148
R3
0.932
0.093
R4
0.641
0.174
R5
0.876
0.094
R6
0.924
0.074
R7
0.620
0.160
R8
0.838
0.080
R9
0.902
0.071
R10
0.660
0.234
R11
0.842
0.099
R12
0.894
0.108
R13
0.596
0.608
R14
0.864
0.121
R15
0.910
0.126
R16
0.551
0.263
R17
0.768
0.239
R18
0.901
0.153The relationship betweenμk and parameter α is shown in Figure 8. It can be seen that the value of parameter α increased with increasing μk. Figures 8(a) and 8(b) show the influence of ρ and ρw on the α-μk relations, respectively. The influence of ρ on the α-μk relations is not clear, but the difference of ρw causes the α-μk relations to be obviously stratified, and the increase in ρw leads to the decrease of parameter α. Based on the correlation of parameter α with ρw and μk, equation (5) is obtained by nonlinear curve fitting of the data points in Figure 8(b).(5)α=0.951−e−μk0.68+11.35e−14.24ρw.Figure 8
Relationship betweenμk and parameter α. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)As shown in Figure9, the relationship between μk and parameter β presents a decreasing function. Figures 9(a) and 9(b) show the influence of ρ and ρw on the β-μk relations, respectively. Similar to Figures 8(a) and 8(b), only ρw has a regular influence on the development of the β-μk relations, and the increase in ρw leads to the decrease in parameter β. Based on the correlation of the parameter β with ρw and μk, equation (6) is obtained by nonlinear curve fitting of the data points in Figure 9(b). (6)β=0.31ρw−1.561+μk1.04ρw−0.45.Figure 9
Relationship betweenμk and parameter β. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)


### 3.1.2.DK,k-nk Relations of the Subsequent Half-Cycle Sequence (SHS)

As shown in Figure6, the DK,k-nk relation of the SSHS continues the trajectory of the previous CC stage. For the LSHS, the DK,k-nk relation increases rapidly on the basis of the previous CC stage and then tends to stabilize. The shape of the DK,k-nk relation for the LSHS is similar to that of the IHS.To study the correlation between theDK,k-nk relation of the LSHS and IHS, the DK,k-nk relations of the LSHS and IHS with the same normalized effective amplitude and reinforcement conditions are plotted together. The DEk-nk 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 DK,k-nk relation for the LSHS can be obtained by translating the DK,k-nk relation of the IHS to the right.Figure 10
DEk-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12 according to the principle of the same normalized effective amplitude.
(a)
(b)
(c)
(d)
(e)
(f)Based on the correlation of theDK,k-nk relation for the LSHS and IHS, the translation hypothesis of the DK,k-nk relation for the LSHS is proposed. The expression formula of the translation model is as follows:(7)DK,k=α1−e−βnk−xt.Here,xt is the translation value, Figure 11 shows the translation model of the LSHS, and nc is the normalized cumulative energy dissipation at the first half-cycle of the LSHS.Figure 11
The translation model of LSHS.According to equation (7), the DK,k-nk relations for the LSHS are fitted by a nonlinear regression analysis procedure, and the values of xt are obtained. Figure 12 shows the relationship between xt and nc, which shows a linear growth relationship:(8)xt=0.65nc−3.2.Figure 12
Relationship betweenxt and nc.Substituting equation (8) into equation (7), the estimation method of DK,k for the LSHS is obtained. Therefore, the estimation method of DK,k under the SV displacement patterns is as follows:(9)DK,k=α1−e−βnk,for IHS,α1−e−βnk−0.65nc+3.2,for LSHS,DK,I−1,for SSHS.Here,DK,I-1 represents the estimation method of DK,k for the SSHS, and the estimation method of the SSHS is the same as that of the previous CC stage.


## 3.1.1.DK,k-nk Relations for the Initial Half-Cycle Sequence (IHS)

Figure6 shows that the DK,k-nk relations for the IHS increase monotonically from 0 to 1, but the slope of the DK,k-nk relations for the IHS gradually decreases with increasing nk. According to these characteristics, a damage model of the stiffness degradation index for the IHS is proposed:(4)DK,k=α1−e−βnk.As shown in Figure7, parameter α is the peak value of DK,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 ∞. The larger the value of the parameter β is, the steeper the DK,k-nk relation is, which means that less energy is dissipated when the stiffness deterioration tends to stop.Figure 7
Damage model of the stiffness degradation index for the IHS.According to equation (4), the DK,k-nk relations for the IHS are fitted by a nonlinear regression analysis procedure, and the values of parameter α and parameter β are obtained. The values of parameter α and parameter β for specimens R1 to R18 are given in Table 2.Table 2
The values of parameterα and parameter β for specimens R1 to R18.
No.
α
β
R1
0.833
0.486
R2
0.910
0.148
R3
0.932
0.093
R4
0.641
0.174
R5
0.876
0.094
R6
0.924
0.074
R7
0.620
0.160
R8
0.838
0.080
R9
0.902
0.071
R10
0.660
0.234
R11
0.842
0.099
R12
0.894
0.108
R13
0.596
0.608
R14
0.864
0.121
R15
0.910
0.126
R16
0.551
0.263
R17
0.768
0.239
R18
0.901
0.153The relationship betweenμk and parameter α is shown in Figure 8. It can be seen that the value of parameter α increased with increasing μk. Figures 8(a) and 8(b) show the influence of ρ and ρw on the α-μk relations, respectively. The influence of ρ on the α-μk relations is not clear, but the difference of ρw causes the α-μk relations to be obviously stratified, and the increase in ρw leads to the decrease of parameter α. Based on the correlation of parameter α with ρw and μk, equation (5) is obtained by nonlinear curve fitting of the data points in Figure 8(b).(5)α=0.951−e−μk0.68+11.35e−14.24ρw.Figure 8
Relationship betweenμk and parameter α. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)As shown in Figure9, the relationship between μk and parameter β presents a decreasing function. Figures 9(a) and 9(b) show the influence of ρ and ρw on the β-μk relations, respectively. Similar to Figures 8(a) and 8(b), only ρw has a regular influence on the development of the β-μk relations, and the increase in ρw leads to the decrease in parameter β. Based on the correlation of the parameter β with ρw and μk, equation (6) is obtained by nonlinear curve fitting of the data points in Figure 9(b). (6)β=0.31ρw−1.561+μk1.04ρw−0.45.Figure 9
Relationship betweenμk and parameter β. (a) Different symbols represent the difference in ρ. (b) Different symbols represent the difference in ρw.
(a)
(b)


## 3.1.2.DK,k-nk Relations of the Subsequent Half-Cycle Sequence (SHS)

As shown in Figure6, the DK,k-nk relation of the SSHS continues the trajectory of the previous CC stage. For the LSHS, the DK,k-nk relation increases rapidly on the basis of the previous CC stage and then tends to stabilize. The shape of the DK,k-nk relation for the LSHS is similar to that of the IHS.To study the correlation between theDK,k-nk relation of the LSHS and IHS, the DK,k-nk relations of the LSHS and IHS with the same normalized effective amplitude and reinforcement conditions are plotted together. The DEk-nk 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 DK,k-nk relation for the LSHS can be obtained by translating the DK,k-nk relation of the IHS to the right.Figure 10
DEk-nk relations for specimens R4 to R6, R7 to R9, and R10 to R12 according to the principle of the same normalized effective amplitude.
(a)
(b)
(c)
(d)
(e)
(f)Based on the correlation of theDK,k-nk relation for the LSHS and IHS, the translation hypothesis of the DK,k-nk relation for the LSHS is proposed. The expression formula of the translation model is as follows:(7)DK,k=α1−e−βnk−xt.Here,xt is the translation value, Figure 11 shows the translation model of the LSHS, and nc is the normalized cumulative energy dissipation at the first half-cycle of the LSHS.Figure 11
The translation model of LSHS.According to equation (7), the DK,k-nk relations for the LSHS are fitted by a nonlinear regression analysis procedure, and the values of xt are obtained. Figure 12 shows the relationship between xt and nc, which shows a linear growth relationship:(8)xt=0.65nc−3.2.Figure 12
Relationship betweenxt and nc.Substituting equation (8) into equation (7), the estimation method of DK,k for the LSHS is obtained. Therefore, the estimation method of DK,k under the SV displacement patterns is as follows:(9)DK,k=α1−e−βnk,for IHS,α1−e−βnk−0.65nc+3.2,for LSHS,DK,I−1,for SSHS.Here,DK,I-1 represents the estimation method of DK,k for the SSHS, and the estimation method of the SSHS is the same as that of the previous CC stage.


## 3.2. The 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 amplitudeμk needs to be transformed into a quasi-stable normalized effective amplitude according to equation (10). Then, the stiffness degradation index must be calculated according to the estimation method of DK,k for the SV displacement patterns. The meaning of equation (10) is that the normalized effective amplitude μk 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 shows the process of the quasi-stable normalized effective amplitude for specimen R24. See companion paper [17] for the detailed calculation process:(10)μk=1b−a+2∑j=a−1bμj,k=a−1,b,(11)uj−uj−1<0.15umax,j=a,b,(12)∑j=abuj−uj−1<0.15umax.where umax is the maximum normalized effective amplitude.Figure 13
The process of the quasi-stable normalized effective amplitude for specimen R24. (a). The value ofμk of specimen R24. (b) Removal of the data points under the line of μk = 1. (c) Averaging of the similar values of μk.
(a)
(b)
(c)For the estimation method ofnk, the energy dissipation of the k-th half-cycle EH,k needs to be calculated first according to the companion paper [17], and then nk is estimated by equation (3). The companion paper [17] proposed computing the energy dissipation of the k-th half-cycle EH,k as follows:(13)EH,k=fρw,Vy,uy,μk.Figure14 shows the prediction procedure of DK,k. The previous step of the calculation of DK,k requires the calculation of nk 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 DK,k is dependent on both the cumulative energy dissipation (cumulative damage term) and displacement history (deformation damage term).Figure 14
Prediction procedure ofDK,k.The test data of 7 specimens (specimens R25 to R31 in Table3) 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. The yield loads and the yield displacements were calculated from the skeletal curves of the hysteresis curves according to the equal energy criterion. The properties of the 7 specimens in the PEER database are shown in Table 3.Table 3
The properties of 7 specimens in the PEER database.
No.
Data sources
V
y (kN)
u
y (mm)
B ×h (mm)
ρ (%)
ρ
w (%)
R25
ORC1 [18]
251.71
37.92
305 × 508
1.265
1.372
R26
C5-00N [19]
57.83
13.22
203 × 203
0.965
0.9
R27
C5-00S [19]
R28
U1 [20]
263.75
24.72
350 × 350
1.605
0.3
R29
A1 [21]
45.37
12.44
0.623
R30
B1 [21]
36.39
9.48
152.4 × 152.4
1.225
0.705
R31
C1 [21]
38.74
14.00To understand the accuracy of the estimation method, Figure15 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. The hollow point is DK,k for specimens R1 to R24, and the solid point is DK,k for specimens R25 to R31 in the PEER database. The results show that the proposed method provides promising prediction reliability and accuracy.Figure 15
Estimated values and experimental values ofDK,k for each half-cycle for 31 specimens.


## 4. Performance Classification Levels of the Stiffness Degeneration Index

The damage development process of specimens R1 to R24 was not recorded in detail during the test. Therefore, specimens R1 to R24 cannot be used to study the performance classification levels of the stiffness degradation index. The damage development process was recorded for specimens S1 to S9 from reference [22], and the damage stages faithfully corresponded to the number of half-cycles. The 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. The four performance levels corresponding to the four damage stages are mild damage, moderate damage, severe damage, and destruction. Therefore, specimens S1 to S9 from the reference [22] were used to discuss the performance classification levels of the stiffness degeneration index. The properties and the number of half-cycles corresponding to the damage of specimens S1 to S9 are shown in Table 4.Table 4
The properties and the number of half-cycles corresponding to the damage for specimens S1 to S9 [22].
Group
No.
V
y (kN)
u
y (mm)
ρ
w (%)
ρ (%)
Displacement pattern
The number of half-cycles corresponding to damage
Cracking
Spalling
Severe spalling
Buckling
1
S1
47.69
10.25
0.226
0.587
B1-X
5
22
46
54
S2
47.69
10.25
0.226
0.587
B1
6
21
48
51
S3
47.69
10.25
0.226
0.587
B2
2
12
100
185
2
S4
84.89
12.75
0.226
1.198
B1-X
8
30
47
54
S5
84.89
12.75
0.226
1.198
B1
6
20
47
54
S6
84.89
12.75
0.226
1.198
B1-D
-
3
45
54
3
S7
85.97
13.32
0.804
1.198
B1-X
6
30
48
-
S8
85.97
13.32
0.804
1.198
B1
5
29
52
-
S9
85.97
13.32
0.804
1.198
B2
3
14
180
-The 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. There 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 Figure16. 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 16
Four displacement patterns imposed on specimens S1 to S9: (a) B1. (b) B1-X. (c) B1-D. (d) B2.
(a)
(b)
(c)
(d)Figure 17
Hysteresis loops for specimens S1 to S9: (a) S1. (b) S2. (c) S3. (d) S4. (e) S5. (f) S6. (g) S7. (h) S8. (i) S9.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure18 shows the DK,k-k relations for specimens S1 to S9. It can be seen from Figure 18 that the DK,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 DK,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 DK,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 DK,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 DK,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 DK,k is the fastest.Figure 18
DEk-nk relations for specimens S1 to S9: (a) S1, S2, S7 and S8. (b) S4, S5 and S6. (c) S3 and S9.
(a)
(b)
(c)Figure18 shows that, regardless of the difference between the reinforcement condition and displacement pattern, the DK,k values for the same damage stage are similar. The 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. This result shows that the performance levels of the reinforced concrete members can be graded by using the value of DK,k. Table 5 shows the performance classification levels of the reinforced concrete members based on the value of DK,k.Table 5
Performance classification levels of the reinforced concrete members based on the value of DK,k.
No.
D
K,k
Damage stages
Performance levels
Performance status
1
D
K,k ≤ 0
No cracking
No damage
Intact
2
0 <DK,k ≤ 0.3
Cracking of the concrete
Mild damage
Repairable
3
0.3 <DK,k ≤ 0.7
Slight spalling of the concrete
Moderate damage
4
0.7 <DK,k ≤ 0.9
Severe spalling of the concrete
Severe damage
No collapse
5
0.9 <DK,k ≤ 1
Buckling of the longitudinal reinforcement
Destruction
CollapseSubstituting equations (3), (5) and (6) into equation (9) gives the following:(14)DK,k=gμk,∑i=1k−1EH,i,Vy,uy,ρw.According to equation (14), when DK,k is defined as the damage state of the reinforced concrete member at the end of the earthquake action (performance design objective), μk and ∑i=1k−1EH,i are the ductility demand and the energy demand, respectively. Vy and uy are related to the section size, concrete strength, and reinforcement ratio. Therefore, 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. Therefore, this method can be used for the performance design of reinforced concrete members.


## 5. Correlation betweenDK,k and Different Parameters

The correlation betweenDK,k and different parameters is discussed according to equation (9). Since the estimation method of DK,k for the SSHS can be replaced by the estimation method of the previous CC stage, equation (9) can be expressed in another form:(15)DK,k=αmax1−e−βmaxnk−xt,max.The values ofαmax, βmax and xt,max are calculated from(16)αmax=0.951−e−μk,max0.68+11.35e−14.24ρw,(17)βmax=0.31ρw−1.561+μk,max1.04ρw−0.45,(18)xt,max=0.65nc,max−3.2,where μk,max = max (μ1... μk), xt,max = max (xt,1... xt,k), and when xt,max is less than 0, xt,max is equal to 0, nc,max = max (nc,1... nc,k).Substituting equations (16) to (18) into equation (15) gives the following:(19)DK,k=gρw,μk,max,nk,nc,max.Figure19 shows the correlation between DK,k and different parameters. The x coordinate axis is ρw, which varies from 0.2% to 1%. The y coordinate axis is nk, which varies from 0 to 100, and the z coordinate axis is DK,k, which varies from 0 to 1. μk,max is 3 and 1.5, which represent the large ductility condition and the small ductility condition, respectively. nc,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.Figure 19
Correlation betweenDK,k and different parameters. (a) Peak displacement occurs at the front of the displacement history. (b) Peak displacement occurs at the end of the displacement history.
(a)
(b)As shown in Figure19, 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. The results show that the development of DK,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. Therefore, 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.


## 6. Conclusions

(1)
An estimation method for the stiffness degradation index was proposed, and the method provides promising prediction reliability and accuracy.
(2)
The 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 (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.
(3)
The 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. Therefore, 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.



---

*Source: 9464275-2020-02-29.xml*