Research Article  Open Access
Yukui Wang, Zhefeng Liu, Weijun Yang, Jian Wang, Xiao Zheng, "Research on the Estimation Method of the Stiffness Degeneration Index of Reinforced Concrete Members", Advances in Civil Engineering, vol. 2020, Article ID 9464275, 16 pages, 2020. https://doi.org/10.1155/2020/9464275
Research on the Estimation Method of the Stiffness Degeneration Index of Reinforced Concrete Members
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 (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. 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.
1. Introduction
In performancebased design methods, the determination of the structural performance levels is based on the damage index [1–3]. Since the deformationbased 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 ParkAng 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
Twentyfour 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 crosssection size was 250 mm × 350 mm.
The properties and imposed displacement patterns of the 24 specimens are summarized in Table 1. 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.
 
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. 
Stable variable (SV) displacement patterns were imposed on specimens R1 to R18. An SV displacement pattern was composed of several constantamplitude 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. The first stage of the CC was defined as the initial halfcycle 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 halfcycle sequence (SSHS); when the displacement amplitude was larger than the maximum historical displacement amplitude, the CC stage was defined as the larger subsequent halfcycle sequence (LSHS). Six SV displacement patterns were designed in the test.
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 Figure 3. Details of the test can be found in the companion paper [17].
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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 zeroforces is defined as a halfcycle. 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 halfcycles by the abscissa axis. u_{k} is the displacement amplitude of the kth halfcycle. The slope of the red line K_{k} is defined as the initial loading stiffness of the kth halfcycle.
The 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. The initial loading stiffness of each halfcycle of the 24 specimens was extracted, and the stiffness degradation index of each halfcycle was calculated according to equation (1).
The 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) The 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).
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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 halfcycle are defined as [17]
Here, is the sum of dissipated energy from the first to the k1th halfcycle.
The D_{K,k}n_{k} relations for specimens R4 to R6, R7 to R9, and R10 to R12 are shown in Figure 6. The different halfcycle sequences (HS) are represented by different symbols in Figure 6. The number next to the symbol represents the value of μ_{k} of the HS.
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3.1.1. D_{K,k}n_{k} Relations for the Initial HalfCycle Sequence (IHS)
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:
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 ∞. The 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. The values of parameter α and parameter β for specimens R1 to R18 are given in Table 2.

The 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 αμ_{k} relations, respectively. The influence of ρ on the αμ_{k} relations is not clear, but the difference of ρ_{w} causes the αμ_{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(b).
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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 βμ_{k} relations, respectively. Similar to Figures 8(a) and 8(b), only ρ_{w} has a regular influence on the development of the βμ_{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).
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3.1.2. D_{K,k}n_{k} Relations of the Subsequent HalfCycle 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. The 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. The 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.
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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. The 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 halfcycle 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. Therefore, the estimation method of D_{K,k} under the SV displacement patterns is as follows:
Here, D_{K,I1} 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.
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 amplitude μ_{k} needs to be transformed into a quasistable normalized effective amplitude according to equation (10). Then, the stiffness degradation index must be calculated according to the estimation method of D_{K,k} for the SV displacement patterns. The meaning of equation (10) is that the normalized effective amplitude μ_{k} of the similar halfcycles is replaced by the average value. To identify the continuous halfcycles with similar normalized effective amplitudes, equations (11) and (12) are applied. Figure 13 shows the process of the quasistable normalized effective amplitude for specimen R24. See companion paper [17] for the detailed calculation process:where u_{max} is the maximum normalized effective amplitude.
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For the estimation method of n_{k}, the energy dissipation of the kth halfcycle E_{H,k} needs to be calculated first according to the companion paper [17], and then n_{k} is estimated by equation (3). The companion paper [17] proposed computing the energy dissipation of the kth halfcycle E_{H,k} as follows:
Figure 14 shows the prediction procedure of D_{K,k}. The 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 halfcycle 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).
The 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. 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.
To understand the accuracy of the estimation method, Figure 15 shows the estimated values and experimental values of each halfcycle for 31 specimens, where the abscissa is the experimental value and the ordinate is the estimated value. The 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. The results show that the proposed method provides promising prediction reliability and accuracy.
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 halfcycles. 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 halfcycles corresponding to the damage of specimens S1 to S9 are shown in Table 4.

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, B1X, B1D, 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.
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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 halfcycles, 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 B1X 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 B1D 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.
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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. 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 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 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. 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 between D_{K,k} and Different Parameters
The 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:
The values of α_{max}, β_{max} and x_{t,max} are calculated fromwhere 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. The x coordinate axis is ρ_{w}, which varies from 0.2% to 1%. The 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.
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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. The 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. 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 (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)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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Research Promotion Project of Changsha University of Science and Technology (Grant no. 2019QJCZ060), the Research and Innovation Project of Hunan Graduate (Grant no. CX2018B537), and the Bridge Engineering Open Fund Project of Changsha University of Science and Technology (Grant no. 18KD03).
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Copyright © 2020 Yukui Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.