Research on a Damage Assessment Method for Concrete Components Based on Material Damage

Abstract

With the popularization of the concept of seismic performance-based design, the correct and quantitative evaluation of post-earthquake damage to structural components has become a research focus. Referring to the concrete constitutive relationship mentioned in the Chinese national standard GB/T 50010-2010, this study proposes a damage assessment method for concrete components based on material damage. According to the value of the uniaxial damage evolution parameter of concrete (d_c(t)), the damage grades of concrete components are defined. It is specified that, when the value of d_c(t) is less than the d_c(t),r value corresponding to the peak concrete strain (ε_c(t),r), the concrete component is in a non-damaged state (Level L1). When the value of d_c(t) is greater than the d_c(t)u value corresponding to the concrete strain (ε_c(t)u), the concrete component is in a severely damaged state (Level L6). When the value of d_c(t) is between these two values, the damage grade of the concrete component (levels L2 to L5) is determined using linear interpolation. To promote its engineering application, this study also proposes a quantitative expression for the damage assessment method for concrete components based on d_c(t). To verify the rationality of the damage assessment method for concrete components based on d_c(t), a refined model of rectangular, T-shaped, and L-shaped concrete shear wall components was established using ABAQUS software, and a nonlinear finite element analysis was carried out. The simulation results show that (a) the damage assessment method for concrete components based on d_c(t) can better characterize damage to concrete shear wall components; (b) when defining the damage grades of concrete shear wall components, using d_c is more reasonable than using d_t; and (c), from a macroscopic perspective, the damage assessment method for concrete components based on d_c(t) is more in line with actual expectations and has a higher safety factor compared with the damage assessment method for concrete components based on the concrete compressive strain (ε_c) mentioned in the Chinese association standard T/CECA 20024-2022.

1. Introduction

With the popularization of the concept of seismic performance-based design, the correct and quantitative evaluation of post-earthquake damage to structural components has become a research focus [1,2,3,4]. For engineering applications, most of the current mainstream seismic design methods for building structures adopt the seismic design approach based on performance requirements [5,6]. Its essence lies in taking corresponding measures for different seismic performance levels of structures, ensuring that the damage grades of different types of components meet the expected requirements and that the seismic performance objectives of the structures are achieved. Therefore, the correct and quantitative evaluation of the damage grades of components after an earthquake is the core of the seismic design method based on performance requirements. Regarding damage assessments of components, most of the current research focuses on establishing a quantitative expression that reflects the damage grades of components. The Park–Ang model [7] is recognized as a relatively typical one. The internal essence of component damage is the result of the continuous development and accumulation of material damage. Unfortunately, the current quantitative expressions fail to reflect the degree of material damage. With the continuous development of computational technology and material constitutive relationships [8,9,10,11,12], it has become possible to establish a quantitative expression that reflects component damage at the material level. Based on this, referring to the concrete constitutive relationship mentioned in the Chinese national standard GB/T 50010-2010, this study proposes a damage assessment method for concrete components based on material damage. According to the value of the uniaxial damage evolution parameter of concrete (d_c(t)), the damage grades of concrete components are defined. This allows the damage assessment method to effectively reflect the degree of material damage and establishes a direct correlation between the mesoscopic damage evolution of materials and the macroscopic damage evaluation of components.

In recent years, there have been frequent seismic activities around the world. Typical examples include the 1994 Northridge earthquake [13], the 1995 Kobe earthquake [14], the 1999 Kocaeli earthquake [15], the 2008 Wenchuan earthquake [16], the 2010 Maule earthquake [17], the 2011 Tohoku earthquake [18], and the 2023 Antakya earthquake [19]. With the increase in the urban population and the reduction in land resources, shear wall structures with relatively high lateral stiffness have been widely used in high-rise buildings [20]. Once the vertical components of high-rise buildings have suffered serious seismic damage, the losses will be immeasurable. Therefore, it is of great research value to evaluate the damage grades of shear wall components after an earthquake. To verify the rationality of the proposed damage assessment method for concrete components based on d_c(t), a refined model of rectangular, T-shaped, and L-shaped concrete shear wall components was established using ABAQUS 2024 software, and a nonlinear finite element analysis was carried out. The value of d_c(t) was derived according to the simulation outputs, and the damage grade of the concrete shear wall component was defined. Finally, the damage assessment method for concrete components based on d_c(t) was compared with the damage assessment method for concrete components based on a compressive strain mentioned in the Chinese association standard T/CECA 20024-2022 [21], and the rationality of the damage assessment method for concrete components based on d_c(t) was determined from a macroscopic perspective.

2. Constitutive Relationship of Concrete

In the long process of studying the constitutive relationship of concrete, various theories such as classical elastic mechanics, plastic mechanics, and fracture mechanics have been successively introduced. Unfortunately, based on in-depth research, these theories all have certain limitations in reflecting the mechanical behavior of concrete under stress. It was not until Dougill [22] applied the basic concept of damage mechanics [23] to describe the nonlinearity of concrete that research on the constitutive relationship of concrete entered a new stage of development. Subsequent studies have shown that the nonlinearity of macroscopic damage evolution stems from the randomness of the distribution of fracture strains at the mesoscopic level. The macroscopic damage evolution law should seek its internal mechanism and modeling approach in a physical analysis at the mesoscopic level. Based on this, Li et al. [10] introduced the concept of energy equivalent strain, combined the elastoplastic damage model with the uniaxial random damage model, and then extended the one-dimensional damage evolution law to the multi-dimensional stress–strain space, establishing a multi-dimensional elastoplastic random damage constitutive model for concrete. In this way, the randomness of microscopic fracture strains is combined with the nonlinearity of macroscopic mechanical behavior, achieving a comprehensive reflection of the nonlinearity and randomness of concrete materials. This model can not only ideally reflect the special behaviors unique to concrete materials but also reflect the random variation range of the strength of concrete under multi-dimensional stress conditions, providing a basis for a nonlinear analysis at the structural level. To accurately describe the mechanical behavior of concrete, the constitutive relationship of concrete mentioned in GB/T 50010-2010 [24] also introduced the concept of damage, using d_t and d_c to characterize the damage of concrete. The uniaxial stress–strain curve of concrete mentioned in GB/T 50010-2010 is shown in Figure 1. The values of d_t can be determined according to Equations (1)–(4), and the d_t values corresponding to different concrete strength grades are shown in

Table 1. The values of d_t corresponding to different concrete strength grades.

Concrete Strength Grades	ft,r (MPa)	εt,r (10−6)	αt	εtu/εt,r	dt,r	dtu
C30	2.01	95	1.25	2.51	0.30	0.86
C35	2.20	100	1.53	2.26	0.30	0.85
C40	2.39	104	1.80	2.10	0.30	0.83
C45	2.51	107	1.95	2.02	0.30	0.83
C50	2.64	110	2.19	1.93	0.30	0.82
C55	2.74	112	2.36	1.87	0.31	0.82
C60	2.85	115	2.55	1.82	0.31	0.81
C65	2.93	116	2.69	1.79	0.31	0.81
C70	2.99	118	2.81	1.76	0.32	0.81
C75	3.05	119	2.91	1.74	0.32	0.80
C80	3.11	120	3.03	1.72	0.32	0.80

Note: ε_tu is the concrete tensile strain corresponding to stress that is equal to 0.5f_t,r in the tensile descending segment of Figure 1. d_t,r and d_tu are the uniaxial tensile damage evolution parameters of concrete corresponding to ε_t,r and ε_tu, respectively.

. When x ≤ 1, Equation (2) is directly fitted based on the test results, and the calculation results of the damage evolution parameter may be negative in some cases; thus, Equation (2) can be corrected according to Equations (5) and (6) [25].

$$\sigma =(1-d_{\mathrm{t}})E_{\mathrm{c}}\mathsf{\epsilon }$$

(1)

$$d_{\mathrm{t}}=\left\{\begin{array}{l}1-{\rho }_{\mathrm{t}}(1.2-0.2x^5)\hspace{0.17em}(x\le 1)\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{t}}}{{\alpha }_{\mathrm{t}}{(x-1)}^{1.7}+x}}(x>1)\end{array}\right.$$

(2)

$$x={\displaystyle \frac{\mathsf{\epsilon }}{{\mathsf{\epsilon }}_{\mathrm{t},\mathrm{r}}}}$$

(3)

$${\rho }_{\mathrm{t}}={\displaystyle \frac{f_{\mathrm{t},\mathrm{r}}}{E_{\mathrm{c}}{\mathsf{\epsilon }}_{\mathrm{t},\mathrm{r}}}}$$

(4)

$$d_{\mathrm{t}}=\left\{\begin{array}{l}1-{\displaystyle \frac{{\rho }_{\mathrm{t}}{n}_{\mathrm{t}}}{n_{\mathrm{t}}-1+x^{n_{\mathrm{t}}}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(x\le 1)\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{t}}}{{\alpha }_{\mathrm{t}}{(x-1)}^{1.7}+x}}(x>1)\end{array}\right.$$

(5)

$$n_{\mathrm{t}}=1/(1-{\rho }_{\mathrm{t}})$$

(6)

where α_t is a parameter related to the tensile descending segment of Figure 1, which is detailed in reference [24]. The Chinese national standard JGJ 3-2010 [26] states that, when carrying out the seismic performance design of structures, the standard values of material strength should be used for checking the calculation of the components. Therefore, in this article, f_t,r is the standard value of concrete tensile strength; ε_t,r is the concrete peak tensile strain corresponding to f_t,r, which is detailed in reference [24]; and d_t is the uniaxial tensile damage evolution parameter of concrete.

The values of d_c can be determined according to Equations (7)–(11). The d_c values corresponding to different concrete strength grades are shown in

Table 2. The values of d_c corresponding to different concrete strength grades.

Concrete Strength Grades	fc,r (MPa)	εc,r (10−6)	αc	εc,u/εc,r	dc,r	dcu
C30	20.1	1472	0.75	2.86	0.54	0.92
C35	23.4	1531	0.96	2.73	0.51	0.91
C40	26.8	1589	1.17	2.49	0.48	0.90
C45	29.6	1634	1.34	2.32	0.46	0.88
C50	32.4	1678	1.50	2.20	0.44	0.87
C55	35.5	1727	1.68	2.09	0.42	0.86
C60	38.5	1769	1.85	2.03	0.40	0.85
C65	41.5	1808	2.02	1.97	0.37	0.84
C70	44.5	1844	2.18	1.91	0.35	0.82
C75	47.4	1884	2.34	1.90	0.33	0.82
C80	50.2	1922	2.49	1.90	0.31	0.82

Note: ε_cu is the concrete compressive strain corresponding to stress that is equal to 0.5f_c,r in the compressive descending segment of Figure 1. d_c,r and d_cu are the uniaxial compressive damage evolution parameters of concrete corresponding to ε_c,r and ε_cu, respectively.

.

$$\sigma =(1-d_{\mathrm{c}})E_{\mathrm{c}}\mathsf{\epsilon }$$

(7)

$$d_{\mathrm{c}}=\left\{\begin{array}{l}1-{\displaystyle \frac{{\rho }_{\mathrm{c}}n}{n-1+x^n}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(x\le 1)\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{c}}}{{\alpha }_{\mathrm{c}}{(x-1)}^2+x}}(x>1)\end{array}\right.$$

(8)

$${\rho }_{\mathrm{c}}={\displaystyle \frac{f_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\mathsf{\epsilon }}_{\mathrm{c},\mathrm{r}}}}$$

(9)

$$n={\displaystyle \frac{E_{\mathrm{c}}{\mathsf{\epsilon }}_{\mathrm{c},\mathrm{r}}}{E_{\mathrm{c}}{\mathsf{\epsilon }}_{\mathrm{c},\mathrm{r}}-f_{\mathrm{c},\mathrm{r}}}}$$

(10)

$$x={\displaystyle \frac{\mathsf{\epsilon }}{{\mathsf{\epsilon }}_{\mathrm{c},\mathrm{r}}}}$$

(11)

where α_c is a parameter related to the compressive descending segment of Figure 1, which is detailed in reference [24]. JGJ 3-2010 [26] states that, when carrying out the seismic performance design of structures, the standard values of material strength should be used for checking the calculation of the components. Therefore, in this study, f_c,r is the standard value of concrete compressive strength; ε_c,r is the concrete peak compressive strain corresponding to f_c,r, which is detailed in reference [24]; and d_c is the uniaxial compressive damage evolution parameter of concrete.

3. Damage Assessment Method for Concrete Components

The Chinese association standard T/CECA 20024-2022, issued by the China Engineering and Consulting Association, mentions several methods for evaluating the damage of concrete components from the material level. The damage grades of concrete components can be classified from Level L1 (non-damaged state) to Level L6 (severely damaged state) according to the compressive strain of concrete or the compressive damage variable of concrete, as shown in

Table 3. Damage assessment method for concrete components based on concrete compressive strain/compressive damage variable [21].

Type	Damage Grade
	L1	L2	L3	L4	L5	L6
Compressive damage variable	≤0.01	≤0.2	≤0.5	≤0.65	≤0.8	>0.8
Compressive strain	≤0.5εc,r	≤1.0εc,r	≤1.5εc,r	≤1.0εcu	≤1.5εcu	>1.5εcu

. However, the compressive damage variable of concrete mentioned in the Chinese association standard T/CECA 20024-2022 is different from the compressive damage variable of concrete (d_c) mentioned in GB/T 50010-2010. It takes into account the residual strain of concrete under a repeated load, and the specific differences are detailed in references [21,24]. Since the Chinese association standard is not a Chinese national standard, for the convenience of engineering application and for the quantitative evaluation of the damage grade of concrete components, this study refers to the concrete constitutive relationship mentioned in GB/T 50010-2010 and proposes a damage assessment method for concrete components based on material damage. The damage grade of each concrete component was defined according to the value of the uniaxial damage evolution parameter of concrete (d_c(t)).

It is specified that, when the value of d_c(t) is less than the value of d_c(t),r corresponding to the concrete peak strain ε_c(t),r, the concrete component is in a non-damaged state (Level L1). When the value of d_c(t) is greater than the value of d_c(t)u corresponding to the concrete strain ε_c(t)u, the concrete component is in a severely damaged state (Level L6). When the value of d_c(t) is between these two values, the damage grade of the concrete component (Levels L2 to L5) is determined using the method of linear interpolation, as shown in

Table 4. Damage assessment method for concrete components based on d_c(t).

Damage Grade	Macro Description	dc	dt
L1	Non-damaged state	≤dc,r	≤dt,r
L2	Slightly damaged state	≤0.25dcu + 0.75dc,r	≤0.25dtu + 0.75dt,r
L3	Mildly damaged state	≤0.50dcu + 0.50dc,r	≤0.50dtu + 0.50dt,r
L4	Moderately damaged state	≤0.75dcu + 0.25dc,r	≤0.75dtu + 0.25dt,r
L5	Relatively severely damaged state	≤dcu	≤dtu
L6	Severely damaged state	>dcu	>dtu

and Figure 2. Substituting the data in Table 1, Table 2 and Table 4, the damage grades of concrete components corresponding to the d_c(t) values under different concrete strength grades are shown in

Table 5. Damage grades of concrete components corresponding to the uniaxial tensile damage evolution parameter of concrete (d_t).

Concrete Strength Grades	Damage Grade
	L1	L2	L3	L4	L5	L6
C30	≤0.30	≤0.44	≤0.58	≤0.72	≤0.86	>0.86
C35	≤0.30	≤0.44	≤0.58	≤0.71	≤0.85	>0.85
C40	≤0.30	≤0.43	≤0.57	≤0.70	≤0.83	>0.83
C45	≤0.30	≤0.43	≤0.57	≤0.70	≤0.83	>0.83
C50	≤0.30	≤0.43	≤0.56	≤0.69	≤0.82	>0.82
C55	≤0.31	≤0.44	≤0.57	≤0.69	≤0.82	>0.82
C60	≤0.31	≤0.44	≤0.56	≤0.69	≤0.81	>0.81
C65	≤0.31	≤0.44	≤0.56	≤0.69	≤0.81	>0.81
C70	≤0.32	≤0.44	≤0.57	≤0.69	≤0.81	>0.81
C75	≤0.32	≤0.44	≤0.56	≤0.68	≤0.80	>0.80
C80	≤0.32	≤0.44	≤0.56	≤0.68	≤0.80	>0.80

and

Table 6. Damage grades of concrete components corresponding to the uniaxial compressive damage evolution parameter of concrete (d_c).

Concrete Strength Grades	Damage Grade
	L1	L2	L3	L4	L5	L6
C30	≤0.54	≤0.64	≤0.73	≤0.83	≤0.92	>0.92
C35	≤0.51	≤0.61	≤0.71	≤0.81	≤0.91	>0.91
C40	≤0.48	≤0.59	0.69	≤0.80	≤0.90	>0.90
C45	≤0.46	≤0.57	≤0.67	≤0.78	≤0.88	>0.88
C50	≤0.44	≤0.55	≤0.66	≤0.76	≤0.87	>0.87
C55	≤0.42	≤0.53	≤0.64	≤0.75	≤0.86	>0.86
C60	≤0.40	≤0.51	≤0.63	≤0.74	≤0.85	>0.85
C65	≤0.37	≤0.49	≤0.61	≤0.72	≤0.84	>0.84
C70	≤0.35	≤0.47	≤0.59	≤0.70	≤0.82	>0.82
C75	≤0.33	≤0.45	≤0.58	≤0.70	≤0.82	>0.82
C80	≤0.31	≤0.44	≤0.57	≤0.69	≤0.82	>0.82

.

For engineering applications, an ideal damage index should meet at least two conditions: (a) when the component is in a non-damaged state, the value should converge to 0; (b) when the component is in a severely damaged state, the value should converge to 1. Obviously, it is not ideal to define the damage grades of concrete components solely by using d_c(t). As mentioned in the Introduction, regarding the damage assessment of components, most of the current research focuses on establishing a quantitative expression that reflects the damage grade of components. To meet the above two conditions and be consistent with the mainstream research, this study refers to the widely recognized Park–Ang model (Equation (12)), which considers the effects of elastoplastic deformation and low-cycle fatigue on damage [7], introduces a damage index, D_c(t), which is closely related to d_c(t), and establishes its quantitative expression, as shown in Equations (13) and (14). Referring to reference [27], the relationship between the tensile and compressive damage indices and the damage grades of concrete components is shown in

Table 7. Relationship between the tensile and compressive damage indices and the damage grades of concrete components.

Damage Grade	Macro Description	Damage Index Dc(t)0
L1	Non-damaged state	Dc(t)0 = 0
L2	Slightly damaged state	0 < Dc(t)0 ≤ 0.2
L3	Mildly damaged state	0.2 < Dc(t)0 ≤ 0.4
L4	Moderately damaged state	0.4 < Dc(t)0 ≤ 0.6
L5	Relatively severely damaged state	0.6 < Dc(t)0 ≤ 0.8
L6	Severely damaged state	Dc(t)0 > 0.8

.

$$D={\displaystyle \frac{x_{\mathrm{m}}}{x_{\mathrm{cu}}}}+\beta {\displaystyle \frac{E_{\mathrm{h}}}{F_{\mathrm{y}}{x}_{\mathrm{cu}}}}$$

(12)

$$D_{\mathrm{c}0}=\alpha {\displaystyle \frac{d_{\mathrm{c}}-d_{\mathrm{c},\mathrm{r}}}{d_{\mathrm{cu}}-d_{\mathrm{c},\mathrm{r}}}}+\beta {\displaystyle \frac{E_{\mathrm{h}}}{M_{\mathrm{y}}{\phi }_{\mathrm{cu}}}}$$

(13)

$$D_{\mathrm{t}0}=\alpha {\displaystyle \frac{d_{\mathrm{t}}-d_{\mathrm{t},\mathrm{r}}}{d_{\mathrm{tu}}-d_{\mathrm{t},\mathrm{r}}}}+\beta {\displaystyle \frac{E_{\mathrm{h}}}{M_{\mathrm{y}}{\phi }_{\mathrm{tu}}}}$$

(14)

where x_cu is the ultimate displacement under monotonic loading. x_m is the maximum deformation under actual seismic action. F_y is the yield shear force. E_h is the cumulative energy dissipation. α is a regulation coefficient, whose value is taken as 0.8, referring to reference [27]. D_c0 and D_t0 are, respectively, the compressive and tensile damage indices of the damage assessment method for concrete components based on material damage. β is the energy dissipation factor of the component. For Equations (13) and (14), since d_c(t) itself has taken into account the influence of cumulative energy consumption, the value of β is taken as 0. M_y is the yield-bending moment. φ_cu is the ultimate curvature at the maximum deformation.

4. Numerical Simulation

4.1. Application of Uniaxial Damage Evolution Parameter of Concrete d_c(t) in ABAQUS

ABAQUS provides three options for the constitutive model of concrete. The concrete damage plasticity model was first proposed by Lubliner et al. [28] and revised by Lee and Fenves [29]. Since it takes into account the differences in the tensile and compressive properties of the material, etc., it can describe the nonlinear behavior of concrete well and is widely used in the numerical simulation of concrete components [30,31]. Unfortunately, the ABAQUS user manual [32] lacks an explanation of the value assignment of the plastic damage factor D in the concrete damage plasticity model. If the value of d_c(t) provided by GB/T 50010-2010 is directly used, it cannot meet the requirements of data inspection in ABAQUS, and it is not easy to converge the calculation results. At present, there are many studies on the application of d_c(t) in ABAQUS. Equation (15) refers to reference [33]; that is, it is determined according to the assumption of energy equivalence. The conversion between the values of D and d of concrete is shown in Equation (15):

$$D=1-{(1-d)}^{\gamma }$$

(15)

where γ is the damage adjustment coefficient. For compressive damage, its value is 0.5; for tensile damage, its value is 0.35.

4.2. Test Results

To verify the rationality of the damage assessment method for concrete components based on d_c(t), a refined model of rectangular, T-shaped, and L-shaped concrete shear wall components was established using ABAQUS software. This research mainly focused on the damage of shear wall components under low cyclic repeated loads. The data cited the pseudo-static tests of cast-in-place shear wall specimens carried out by Gu et al. [34,35,36]. The test results of the specimens at the characteristic points are shown in

Table 8. Test results.

Specimen	Loading Direction	Crack		Yield		Peak		Ultimate
		Pcr (kN)	Δcr (mm)	Py (kN)	Δy (mm)	Pp (kN)	Δp (mm)	Pu (kN)	Δu (mm)
RCW-1	positive	410	6.58	573	14.36	686	28.86	650	45.70
	negative	387	2.25	597	7.21	746	30.33	580	50.03
RCW-L	positive	326	2.70	1009	17.58	1130	23.84	751	45.78
	negative	314	3.05	862	13.55	1011	46.10	840	60.60
RCW-T	positive	284	1.04	1246	13.68	1415	22.05	1010	31.81
	negative	276	0.80	942	9.84	1072	29.84	923	38.88

Note: P_cr, P_y, P_p, and P_u are the crack load, yield load, peak load, and ultimate load, respectively. Δ_cr, Δ_y, Δ_p, and Δ_u are the crack displacement, yield displacement, peak displacement, and ultimate displacement, respectively. P_y and Δ_y are determined using the equivalent energy method [37].

.

The cross-sectional dimensions and reinforcement arrangements of the shear wall specimens are shown in Figure 3. For a rectangular shear wall, the height, width, and thickness are 2700 mm, 1800 mm, and 200 mm, respectively. For the T-shaped shear wall, the height, web width, flange width, and thickness are 2700 mm, 1800 mm, 1650 mm, and 200 mm, respectively. For the L-shaped shear wall, the height, web width, flange width, and thickness are 2700 mm, 1800 mm, 925 mm, and 200 mm, respectively. The compressive strengths of the rectangular, T-shaped, and L-shaped shear wall specimens are 60.1 MPa, 37.2 MPa, and 37.2 MPa, respectively, and the strength grades of all the reinforcing bars are HRB400E.

The loading devices were all composed of vertical and horizontal loading devices. Vertical loads were applied using a vertical actuator fixed on the reaction frame, and horizontal loads were applied with a horizontal actuator through a loading beam. After the specimen was hoisted to the testing platform, it was fixed to the laboratory floor with anchor bolts. The load/displacement hybrid control loading system was applied. Before the specimen yielded, the load control was used. After the specimen yielded, the displacement control was employed. The cyclic loading history is shown in Figure 4. The details of the specimen design principles, mechanical properties of materials, loading schemes, etc., are described in references [34,35,36].

4.3. Development of FE Model

The concrete and steel bars were modeled separately. In terms of material models, the built-in concrete damage plasticity model was adopted for the concrete material, as shown in Figure 5a,b. The relevant physical meanings are detailed in reference [38]. According to reference [39], the values of the CDP model parameters are shown in

Table 9. Values of CDP model parameters.

Parameter	Value
Dilation angle ψ	30°
Eccentricity e	0.1
Ratio of the initial biaxial compressive yield stress to the initial uniaxial compressive yield stress fb0/fc0	1.16
Ratio of the second stress invariant on the tensile meridian Kc	0.667
Viscosity coefficient ν	0.005

. For the steel bar material, a bilinear elastic/plastic model was used, as shown in Figure 5c. f_y, f_u, ε_y, ε_u, and E_s are, respectively, the yield strength, ultimate strength, yield strain, ultimate strain, and elastic modulus of the steel bar.

The dimensions of the FE model are entirely identical to those of the test specimen. The C3D8R was adopted for the concrete, and the T3D2 was adopted for the steel bars. All the steel bars were embedded in the concrete. After a mesh sensitivity analysis, the mesh size of the concrete was selected as 50 mm, and the mesh size of the steel bars was selected as 25 mm. Further reducing the mesh size would not significantly improve the accuracy but would greatly increase the computational cost. The foundations of the FE models RCW-1, RCW-T, and RCW-L all employ fixed-end constraints, which are consistent with the experimental boundary conditions. The loading process was categorized into two distinct phases. First, vertical axial compression was applied via a concentrated force. Second, low-cycle repeated loads were applied via displacement, with the displacement being consistent with the experimental results. For T-shaped and L-shaped shear walls, the loading direction is specified as “positive” when the flange is in tension and “negative” when the flange is in compression.

4.4. Verification of FE Model

Figure 6 presents a comparison of the load/displacement curves between the tests and simulations of the rectangular, T-shaped, and L-shaped shear walls.

Table 10. Comparison between the test and simulation results.

Specimen	Loading Direction	Load			Displacement
		PpE (kN)	PpS (kN)	PpE/PpS	ΔuE (mm)	ΔuS (mm)	ΔuE/ΔuS
RCW-1	positive	686	737	1.07	28.86	25.53	0.88
	negative	746	769	1.03	30.33	26.27	0.87
RCW-L	positive	1130	1085	0.96	23.84	25.23	1.06
	negative	1011	1122	1.11	46.10	37.82	0.82
RCW-T	positive	1415	1521	1.07	22.05	24.20	1.10
	negative	1072	1140	1.06	29.84	27.56	0.92

Note: P_p^E and P_p^S are the peak loads of the test and simulation, respectively. Δ_u^E and Δ_u^S are the peak displacements of the test and simulation, respectively.

provides a comparison of the loads and displacements at the peak points between the tests and simulations.

It can be seen that, for rectangular, L-shaped, and T-shaped shear walls, the differences between the simulation’s and experiment’s peak loads (the average of absolute values of positive and negative differences) are 5.0%, 7.5%, and 6.5%, respectively. These results are relatively reliable and overall meet engineering accuracy requirements. For rectangular, L-shaped, and T-shaped shear walls, the differences between peak displacements corresponding to peak loads from the simulation and experiment (the average of absolute values of positive and negative differences) are 12.5%, 12.0%, and 9.0%, respectively. These differences can be mainly attributed to three factors: (1) the boundary conditions in the simulation are completely idealized, while there are gaps between the loading devices and the test specimens in the experiment; (2) bond slip between steel bars and concrete is ignored in the simulation; and (3) the viscous coefficient v was calibrated to minimize the difference in peak loads between simulation and experiment as much as possible. Overall, the differences between the simulation and test results are within an acceptable range, and the development trend of the simulation curve is consistent with that of the test.

4.5. Damage Analysis

For the rectangular shear wall, the concrete damage of RCW-1 at the characteristic points is shown in Figure 7. As the load was applied, the maximum value of the tensile plastic damage factor D_t reached 0.81 quickly. According to Equation (14), the d_t value at this time was 0.99, which is consistent with the appearance of horizontal cracks in the test shear wall. When the load was increased to the yield displacement, the maximum value of the compressive plastic damage factor D_c at the bottom of RCW-1 reached 0.90. According to Equation (14), the d_c value at this time was 0.99. According to Table 6, the concrete component was at the L6 level at this time, which is consistent with the crushing and spalling of the concrete at the corner of the test shear wall. When the load was increased to the peak displacement, the d_c value in about one-third of the area of RCW-1 was 0.99, and the d_t value in about half of the area was 0.99, which is basically consistent with the failure phenomenon shown by the test shear wall. It should be noted that, due to the poor tensile performance of the concrete and the early development of cracks, the tensile plastic damage factor reached the peak value quickly. Compared with the d_t value, it is more reasonable to define the damage grade of the concrete shear wall component by using the d_c value. As the damage at the bottom of RCW-1 was the most serious when the test was terminated, the bottom unit of the shear wall was selected as the analysis object, and its M-d_c curve is shown in Figure 8. In the initial stage of loading, the section bending moment increased, but the value of d_c was 0; this is mainly because, in the initial stage of loading, the concrete had good compressive performance, and there was no compressive damage at the bottom of RCW-1. As the applied load increased, the d_c value began to increase, and at this time, the section bending moment increased rapidly. After the load was increased to the yield displacement, the increase rate of the section bending moment slowed down, but the uniaxial compressive damage evolution parameter of the concrete d_c increased rapidly to 1.0. After unloading, the uniaxial compressive damage evolution parameter of the concrete d_c was still 1.0, as represented by the vertical section at the end of the curve.

Since the T-shaped and L-shaped shear walls are not completely symmetrical structures, the compressive and tensile damages on both sides of the web developed asymmetrically, and the damage degree of the web on the side close to the flange is lighter than that of the web on the other side. For the T-shaped shear wall, the concrete damage of RCW-T at the characteristic points is shown in Figure 9, and the M-d_c curve of its bottom unit is shown in Figure 10. For the L-shaped shear wall, the concrete damage of RCW-L at the characteristic points is shown in Figure 11, and the M-d_c curve of its bottom unit is shown in Figure 12. When loading in the positive direction, the curve development trend of the T-shaped and L-shaped shear walls was similar to that of the rectangular shear wall. When loading in the reverse direction, the initial value of the compressive damage evolution parameter of concrete d_c was already close to 0.3. As the applied displacement increased, the compressive stress of the unit at the junction of the flange and the web kept increasing. However, due to the relatively large stiffness of the flange, only the slope of the curve changed: the compressive damage evolution parameter of concrete d_c increased linearly to 1.0, which is consistent with the phenomenon that only slight damage occurs at the bottom of the flange when the test fails.

4.6. Damage Assessment

The step-by-step process for evaluating the damage grade using the proposed method is shown in Figure 13.

Table 11. Damage assessment of concrete shear wall components.

Damage Grade			εc (10−3)	dc	dt	Dc0
RCW-1	crack	value	0.3	0.37	0.99	0
		damage grade	L1	L1	L6	L1
	yield	value	5.16	0.81	0.99	0.62
		damage grade	L5	L5	L6	L5
	peak	value	90	0.98	0.99	0.96
		damage grade	L6	L6	L6	L6
RCW-L	crack	value	1.1/0.93	0.88/0.83	0.99	0.76/0.66
		damage grade	L2	L5	L6	L5
	yield	value	3.6/2.1	0.98	0.99	0.96
		damage grade	L4/L3	L6	L6	L6
	peak	value	27/20	0.98	0.99	0.96
		damage grade	L6	L6	L6	L6
RCW-T	crack	value	0.8/0.2	0.82/0.43	0.99	0.64/L1
		damage grade	L1	L4/L2	L6	L5/L3
	yield	value	18/0.8	0.98/0.65	0.99	0.96/0.3
		damage grade	L6/L1	L6/L4	L6	L6/L3
	peak	value	57/6	0.98	0.99	0.96
		damage grade	L6	L6	L6	L6

Note: If the data shown in the table are in the form of A/B, A represents the data of the bottom unit of the web plate, and B represents the data of the bottom unit of the flange. D_c0 indicates the compressive damage index.

provides the results of a structural comparison that quantitatively evaluated the rectangular, T-shaped, and L-shaped concrete shear wall components between the damage assessment method for concrete components based on d_c(t) and the damage assessment method for concrete components based on concrete compressive strain ε_c in the Chinese association standard T/CECA 20024-2022. The results are as follows: (1) Due to the poor tensile performance of concrete and the early development of cracks, the tensile plastic damage factor D_t reaches the peak value quickly. Compared with d_t, it is more reasonable to define the damage grade of the concrete shear wall component by using d_c. (2) When using d_c and the compressive damage index of concrete D_c0 to define the damage grade of the concrete shear wall component, the evaluation results are basically consistent. (3) From a macroscopic perspective, the damage assessment method for concrete components proposed in this study is more in line with actual expectations than the damage assessment method for concrete components based on d_c(t) and the damage assessment method for concrete components based on ε_c. It also has a higher safety factor. This is because, as shown in the literature [34,35,36], when a specimen yields, a large number of cross-diagonal cracks appears on its surface, accompanied by local concrete spalling and steel bar yielding. In engineering applications, this exceeds the serviceability limit state of the shear wall and constitutes irreversible damage. However, according to the damage assessment method for concrete components based on ε_c, the damage grade of the component is lower than that determined with the damage assessment method based on d_c(t). For the flange of model RCW-T, it even remains in the L1 non-damaged state, which is inconsistent with the actual damage of the component and poses significant safety risks. Therefore, the method proposed in this article has a higher safety factor.

5. Conclusions

Referring to the concrete constitutive relationship mentioned in GB/T 50010-2010, this study proposed a damage assessment method for concrete components based on material damage. According to the values of d_c(t), the damage grades of concrete components were defined. To verify the rationality of the damage assessment method for concrete components proposed in this study, a refined model of rectangular, T-shaped, and L-shaped concrete shear wall components was established using ABAQUS software, and a nonlinear finite element analysis was carried out. Finally, the damage assessment method for concrete components proposed in this study was used to assess the damage grades of concrete shear wall components, and the main conclusions are as follows:

(1)

The damage assessment method for concrete components proposed in this study can well represent the damage grades of concrete shear wall components. Due to the poor tensile performance of concrete and the early development of cracks, the tensile plastic damage factor D_t reaches the peak value quickly; it is more scientific and reasonable to define the damage grade of concrete shear wall components using the value of d_c than using the value of d_t, as it can more accurately reflect the actual damage status of the components.

(2)

Using d_c and the compressive damage index of concrete D_c0 to define the damage grade of the concrete shear wall component, the evaluation results are basically consistent.

(3)

From a macroscopic perspective, the damage assessment method for concrete components based on d_c(t) is more in line with actual expectations and has a higher safety factor compared with the damage assessment method for concrete components based on concrete compressive strain ε_c mentioned in the Chinese association standard T/CECA 20024-2022.