J. Cent. South Univ. (2020) 27: 3807-3820

DOI: https://doi.org/10.1007/s11771-020-4563-z

Unified strength model based on the Hoek-Brown failure criterion for fibre-reinforced polymer-confined pre-damaged concrete columns with circular and square cross sections

ZHANG Yang(张扬), LU Zhi-fang(卢志芳), CAO Yu-gui(曹玉贵)

Hubei Key Laboratory of Roadway Bridge & Structure Engineering, Wuhan University of Technology,Wuhan 430070, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Fibre-reinforced polymer (FRP) has the advantages of high strength, light weight, corrosion resistance and convenient construction and is widely used in repairing and strengthening damaged concrete columns. Most of the existing strength models were built by regression analysis of experimental data; however, in this article, a new unified strength model is proposed using the Hoek-Brown failure criterion. To study the strength of FRP-confined damaged and undamaged concrete columns, 900 test data were collected from the published literature and a large database that contains the cross-sectional shape of each specimen, the damage type, the damage level and the FRP-confined stiffness was established. A new strength model using the Hoek-Brown failure criterion was established and is suitable for both circular and square columns that are undamaged, load-damaged and fire-damaged. Based on the database, most of the existing strength models from the published literature and the model proposed in this paper were evaluated. The evaluation shows that the proposed model can predict the compressive strength for FRP-confined pre-damaged and undamaged concrete columns with good accuracy.

Key words: FRP-confined concrete; load-damaged; fire-damaged; unified strength model; Hoek-Brown failure criterion

Cite this article as: ZHANG Yang, LU Zhi-fang, CAO Yu-gui. Unified strength model based on the Hoek-Brown failure criterion for FRP-confined pre-damaged concrete columns with circular and square cross sections [J]. Journal of Central South University, 2020, 27(12): 3807-3820. DOI: https://doi.org/10.1007/s11771-020-4563-z.

1 Introduction

At present, a large number of concrete structures have produced varying degrees of damage for various reasons [1-10]. The damage of a concrete structure leads to the reduction in the bearing capacity. To ensure the safety of the structure, it is necessary to repair the damaged concrete structure. As a kind of reinforcement material, increasingly concrete structures are strengthened with FRP. For FRP-confined undamaged concrete columns, many studies have focused on the mechanical behaviour [11]. A large number of compressive strength models for FRP-confined undamaged concrete columns have been proposed. However, there are few relevant models for FRP-confined pre-damaged concrete.

In general, FRP-confined pre-damaged columns fall into two categories: I) concrete columns confined by FRP after being heated and II) concrete columns confined by FRP after unloading.

For damage type I, the compressive strength of unconfined concrete after being heated is smaller than the concrete strength f_co during the test. BISBY et al [12] experimentally investigated the effects of target exposure temperature on the compressive strength of FRP-confined circular columns. The experimental results show that the compressive strength value decreased with increasing exposure temperature. GUO et al [13] also obtained the same conclusion. The test results show that carbon FRP (CFRP) greatly improves the compressive strength and deformation capacity of fire-damaged high strength concrete (HSC) columns. LENWARI et al [14] found that the exposure temperature has a significant influence on the compressive strength of concrete after a fire. Wrapping with CFRP can enhance the compressive strength of the concrete at high temperatures. LIU et al [15] experimentally concluded that CFRP confinement significantly affects the compression failure of concrete after high temperature and greatly improves the strength of concrete after high temperature, indicating that CFRP confinement can be used as an effective repair and reinforcement measure for concrete structures after a fire. In addition, based on the existing model, a simplified compressive model for evaluating the mechanical behaviour of CFRP- confined concrete after high temperature was proposed. OUYANG et al [16, 17] also obtained the same conclusion. They further developed two compressive strength models for FRP-confined fire-damaged concrete columns with circular and square sections by using two formulas. For damage type II, the damage load applied on the unconfined concrete usually exceeds the unconfined concrete strength. WU et al [6] experimentally studied the mechanical properties of CFRP-confined load- damaged circular concrete and proposed a compressive strength model for FRP-confined load-damaged circular concrete columns. GUO et al [11] used test data of CFRP-confined load-damaged circular concrete columns to evaluate existing compression strength models for FRP-confined intact concrete. The existing compression strength models could not provide sufficiently accurate calculation results. MA et al [18] proposed a compressive strength model for FRP-confined load-damaged concrete with circular cross sections. LI et al [3] experimentally studied the mechanical behaviour of FRP-confined load-damaged square concrete columns with different corner radii and proposed a compressive strength model. Some literature studies [1, 19, 20] stated that FRP could effectively improve the compressive strength of load-damaged concrete cylinders, and the load damage had a large effect on the compressive strength of the FRP-confined load-damaged concrete. However, other studies [3, 6, 11] concluded that the load damage had a slight influence on the compressive strength for FRP- confined load-damaged concrete. It has been concluded that there are not uniform theoretical opinions on the effect of load-damaged levels on the compressive strength of FRP-confined concrete. Researchers have developed different compressive strength models for FRP-confined load-damaged concrete columns with different cross-sections. Furthermore, as far as the author knows, no theoretical unified model can predict the compressive strength of FRP-confined load- damaged concrete columns with different cross- sections.

This paper aims to propose a new unified compressive strength model by using the Hoek- Brown failure criterion, which considers different damage types, damage levels and cross-section types. First, the load-damaged degree and fire-damaged degree of concrete are defined. Then, applying the Hoek-Brown failure criterion to FRP- confined concrete, the new compressive strength model for FRP-confined concrete is established. Finally, the experimental data of the published literature are used to evaluate the proposed model in this paper and the existing model from the published literature. The evaluation results show that the proposed model can predict the compressive strength of FRP-confined pre-damaged concrete columns with good accuracy.

2 Pre-damaged degree of concrete

2.1 Damage infliction

In this paper, the concrete pre-damage has two types: I) load-damaged and II) fire-damaged. The former type loads the concrete to the design load to achieve pre-damage. Similarly, the latter type exposes the concrete to the design temperature to create pre-damage. The test procedure for the two types of damage will be specifically described below.

A typical stress-strain relationship for FRP-confined load-damaged concrete at room temperature is illustrated by path R-D in Figure 1(a). The test process is divided into three steps: Step one is the pre-damage phase. Suppose that the concrete columns are loaded to point C along the path O-A-B, and the column is completely unloaded to point R following the path B-R. Step two is wrapping with the FRP jacket. After cleaning and repairing the surface, FRP is used to wrap the load-damaged concrete from step one. Step three is the reloading test phase. After the maintenance meets the requirements, the FRP-confined pre-damaged concrete is reloaded along path R-D. The concrete loading process is shown in Figure 1(a).

Figure 1(b) also shows the effects of exposing FRP-confined circular concrete columns to room temperature and high temperature. The test process for FRP-confined fire-damaged concrete is also divided into three steps: step one is also a pre-damage phase. The concrete is exposed to the design temperature. After heating, the concrete was cooled. Step two is wrapping with the FRP jacket. The wrapping method is similar to that used for the FRP-confined load-damaged concrete. Step three is the reloading test phase. The reload path of the FRP repaired concrete column will follow path O-G. Comparing intact plain concrete and fire-damaged plain concrete, the difference between the peak point A in Figure 1(b) and the peak point L in Figure 1(b) is relatively large, the compressive strength (residual compressive strength) is smaller and the peak strain is large. The concrete loading process is shown in Figure 1(b).

Figure 1 Typical stress-strain relationship for FRP-confined damaged concrete:

It can be seen in Figure 1 that the forms of the stress-strain relationship between the FRP-confined load-damaged concrete columns and the FRP- confined fire-damaged concrete columns are similar, and the published literatures [12-17] also revealed that the high exposure temperature had an insignificant influence on the shape of the concrete stress-strain relationship; only the ultimate strain, compressive strength and elastic modulus of the concrete were different. This conclusion is similar to that of the effect of load damage on concrete. OUYANG et al [16, 17] introduced the residual strength f_cd of concrete and proposed the FRP-confined fire-damaged concrete compressive strength model. From another point of view, it explains that the effects of fire damage and load damage are the same.

Based on the above discussion, in this paper, WU et al’s definition [6] of the concrete damage degree δ is still used, which can be expressed as follows:

                       (1)

where f_co and f_cd are the concrete compressive strength (point A) and damaged concrete compressive strength (point C or point L), respectively.

2.2 Load-damaged degree

For the FRP-confined pre-damaged concrete columns, the residual strength f_cd is not available, so the damage degree δ cannot be directly obtained by Eq. (1), only the unloading stress f_ci (point B in Figure 1(a)) can be directly obtained. In this paper, the damage degree δ given by Eq. (2b) is related to the damage level p (%) given by Eq. (2a). To distinguish loading on the ascending branch or descending branch, we assume that the positive sign indicates unloading in the ascending branch, whereas a negative sign indicates unloading in the descending branch (Figure 1(a)). As shown in Figure 1(a), the damage level p (%) to the columns increases as the unloading stress f_ci in the ascending branch increases; on the contrary, the damage level p (%) of the columns increases when the unloading stress f_ci decreases in the descending branch. WU et al [6] obtained the relationship between the load-damage degree δ₁ and damage level p (%) by testing unconfined undamaged concrete. In this study, the same damage degree δ₁ calculation formula is used as follows:

                      (2a)

 (2b)

where f_ci is the stress at unloading point B in Figure 1 and ε_c is the axial strain at unloading point B.

2.3 Fire-damaged degree

In the FRP-confined process of repairing fire- damaged concrete, the residual strength f_cd (point L in Figure 1(b)) was not recorded, and the damage degree δ could only be quantified by the residual strength of the unconfined fire-damaged concrete columns. At the same time, OUYANG et al [16, 17] experimentally investigated whether the fire- damaged degree is related to the type of section by testing circular and square fire-damaged concrete columns. Figure 2(a) shows the relationships among the damage degree δ of the unconfined fire- damaged columns, the exposure temperature T and the corner radius ratio 2r/b. It is found from Figure 2 that the fire-damaged degree δ₂ has a linear relationship with the exposure temperature T under the same corner radius ratio 2r/b. By regression analysis of the data points in Figure 2(a), the fire-damaged degree δ₂ is identified as Eq. (3), where R² is 0.74, as shown in Figure 2(b).

           (3)

The temperature in the laboratory is between 20 °C and 50 °C, δ₂ is almost zero, and there is a good connection between the mathematical sense and the physical sense.

Figure 2 Fire-damaged degree δ₂:

3 Hoek-Brown failure criterion for FRP-confined pre-damaged concrete columns

3.1 Hoek-Brown failure criterion derivation for undamaged concrete

By analyzing a large number of experimental data, HOEK et al [21] obtained a failure criterion for rocks as follows:

                     (4)

where σ₁ is the major principal stress, σ₃ is the minor principal stress, σ_c is the uniaxial compressive strength of the intact rock and m and s are related to the rock type. The value of m changes from 0.001 to 25. The value of s ranges from 0 to 1. For intact rock, s is identified as 1.

Concrete is also a type of rock; therefore,Eq. (4) is suitable. For FRP-confined concrete columns, σ₁ is the FRP-confined compressive strength f_cc, σ₃ is the confinement pressure f_l, and σ_c is the compressive strength f_co of the unconfined concrete. For plain concrete, Eq. (4) reduces to Based on Eq. (1), f_cd/f_co=1-δ. When there is no confinement, f_cd=f_cc, s can be confirmed as follows:

                                (5)

For columns without pre-damaged, Eq. (5) reduces to 1. Substituting Eq. (5) into Eq. (4) yields the following:

                   (6)

When concrete is subjected to bi-directional tensile stress, its failure strength is basically unchanged regardless of stress ratio σ₁/σ₃. The biaxial tensile strength is close to the uniaxial tensile strength f_t [22]. In this case, it is reasonable to assume σ₁=σ₃=f_t, and Eq. (6) takes the following form:

                          (7)

For concrete columns without pre-damage, δ=0, and since the tensile strength f_t of the concrete is related to the compressive strength f_co, we can reasonably assume that  [23, 24]. The value of m is obtained by solving Eq. (7) to give the following:

                                (8)

Similarly,

                (9)

                           (10)

where E_frp is the elastic modulus of the FRP, t_frp is the thickness of the wrapped FRP, ε_frp is the ultimate strain obtained from the tensile coupon tests and b is the the diameter of a circular column or the side length of a square column.

The coefficients a and c can be determined by regression analysing the FRP-confined undamaged circular concrete test database in Table 1. Therefore, the compressive strength model for FRP-confined undamaged concrete can be written as follows:

           (11)

It is well known that the mechanical behaviour of FRP confined concrete is affected by cross section shape. For concrete specimens wrapped with the same number layers of FRP, the compressive strength of the specimen with circular section is greater than that of the specimen with square section. The reason is that square sections of FRP-confined concrete specimens are less effective in confining concrete than that of circular specimen [25].

In order to analyze the effect of cross section shape on the FRP confinement stress, WU et al [26] developed a relationship between the circular section and the square section by using corner radius ratio (ρ=2r/b), as shown in Figure 3. In Figure 3, r is the corner radius for square section or radius for circular section. When ρ is equal to 1, the cross section shape is circular, otherwise the cross section shape is square. Based on the discussion above, the confinement pressure f_l in Eq. (11) can be replaced by f_l·f(ρ). The Eq. (11) can be written as:

   (12)

Through regression analyses using FRP- confined undamaged square column data in Table 1, Eq. (12) becomes the following:

 (13)

Based on the Hoek-Brown failure criterion derivation, Eq. (13) can be calculated as the compressive strength for FRP-confined undamaged columns. Therefore, when 2r/b=1, Eq. (13) is simplified to the model for FRP-confined circular columns, or Eq. (11); when 2r/b≠1, Eq. (13) can be calculated as the strength for FRP-confined square columns. For columns without confinement or f_l=0,Eq. (13) will be simplified to f_cc=f_co, which is in line with the physical and mathematical meanings.

Figure 3 Unification of column shapes

3.2 Hoek-Brown failure criterion derivation for pre-damaged concrete

3.2.1 Load-damaged model

Equation (13) can predict the compressive strength for FRP-confined undamaged concrete with circular or square cross sections. For concrete columns with load damage, replacing 1 in Eq. (13) with (1-δ₁)² gives the following:

      (14)

It should be noted that the load-damaged degree δ₁ is calculated from Eq. (2b). For concrete columns without load damage, Eq. (14) can be simplified to Eq. (13).

3.2.2 Fire-damaged model

For concrete columns with fire-damaged, replacing δ₁ in Eq. (14) with δ₂ gives the following:

      (15)

4 Assessment of models

4.1 Database for FRP-confined concrete

The database for FRP-confined concrete includes 900 experimental tests collected from the published literatures [3, 6, 11-19, 27-39], which contains 218 FRP-confined undamaged circular columns, 138 FRP-confined undamaged square columns, 140 FRP-confined load-damaged circular columns, 63 FRP-confined load-damaged square columns, 291 FRP-confined fire-damaged circular columns and 50 FRP-confined fire-damaged square columns. The concrete strength, including normal strength concrete (NSC) and HSC, varied from 19 to 112 MPa. The specimens in the database were confined by three types of FRP that included basalt FRP (BFRP), CFRP and glass FRP (GFRP). The elastic modules of BFRP, CFRP, and GFRP varied from 70 to 105 GPa, 22 to 436 GPa, and 22 to 27 GPa, respectively. For the circular columns, the diameters D of the specimens varied from 94 to 300 mm; for the square columns, the side lengths were between 100 and 400 mm. The damage level p (%) for the FRP-confined load-damaged concrete columns ranges from 0% to -50%. The exposure temperature T for the FRP-confined fire-damaged concrete columns varied from 20 to 800 °C. In addition to the regression analysis described above, the database is also used in the evaluation of existing models in the following sections. As shown in Table 1, the database contains five parameter ranges.

4.2 Existing strength models

Based on experimental tests and theoretical analysis, RICHART et al [40] obtained the compressive strength model, as shown in Eq. (16):

                           (16)

where k is the confinement effectiveness coefficient; RICHART et al [40] recommended k=4.1.

Most existing compressive strength models for FRP-confined undamaged concrete columns are developed by Eq. (16). The existing compressive strength models for FRP-confined undamaged concrete columns are summarized in Tables 2 and 3. The differences in the compressive strength models in Tables 2 and 3 are the confinement effectiveness coefficient k and a function of f_l/f_co.

4.3 Assessment model for FRP-confined undamaged concrete

In this paper, the integral absolute error (IAE) is used to evaluate the performance of the model [1, 2, 55, 56]. The IAE is an accuracy indicator that is very sensitive to prediction accuracy. The expression of the integral absolute error IAE is as follows:

                    (17)

where Theo_i, Expe_i and n are the experimental values, theoretical values, and the number of experimental data, respectively. The lower the IAE value is, the more accurate the model is.

The performance of Eq. (13) is assessed by using the experimental data in Table 1. The performance of the integral absolute error IAE of the circular model in Table 2 and the square model in Table 3 are summarized in Figures 4(a) and (b), respectively. Figure 4(a) shows that the integral absolute error IAE of the proposed model is 0.100. The proposed model is more accurate than the existing models. Figure 4(b) shows the performance of the existing models for FRP-confined undamaged square columns and the proposed model. WU et al’s model [24] and the proposed model are better and more accurate than the existing models. Therefore, the model proposed by WU et al [24] was chosen for comparison with the proposed model. We can clearly see that the integral absolute errors of the proposed models are 0.088 and 0.074. Similarly, the proposed model is more accurate than the other existing models.

Table 1 Parameters of database

The proposed unified strength model can predict the compressive strength for both FRP- confined circular concrete columns and FRP- confined square concrete columns, and it is necessary to conduct a comprehensive evaluation of circular and square columns. To better describe the distribution of the integral absolute error IAE, a typical model is further compared to the proposed model. The integral absolute error IAE ranges from 0.092 to 0.297. In the existing models, KUMUTHA et al’s model [43] has a relatively large IAE of 0.297, whereas WU et al’s model [24] has a smaller IAE of 0.134, and the IAE of the proposed model is 0.092. Compared with the existing unified compressive strength models, the proposed model has higher prediction accuracy. At the same time, it can be seen that the error volatility is also minimal. In other words, the model can accurately predict the strength of FRP-confined circular and square concrete columns and the integral absolute error IAE of the proposed model is smaller than that of the other existing models. Figure 5 shows that the proposed model for both circular and square concrete columns performs better than other existing unified strength models. The reason may be that existing compressive strength models are developed with small amount of experimental data.

Table 2 Strength models for circular columns

Table 3 Strength models for square columns

Figure 4 IAE of the model for FRP-confined undamaged columns:

Figure 5 Performance of strength model for FRP-confined undamaged circular and square columns:

When a large amount of experimental data with different FRP types and different concrete strengths is used to evaluate existing models, the prediction accuracy of the existing models will be reduced, and LIM et al [57] also obtained similar conclusions.

4.4 Assessment model for FRP-confined load- damaged concrete

Based on the strength model for FRP-confined undamaged concrete columns or Eq. (13) and considering the load-damaged degree δ₁ or Eq. (2b), the unified strength model for FRP-confined load-damaged circular and square concrete columns or Eq. (14) is obtained. Therefore, Eq. (14) can predict the strength for FRP-confined undamaged circular and square concrete columns and predict the strength for FRP-confined load-damaged circular and square concrete columns. Based on the above discussion, the strength model accuracy for FRP-confined undamaged circular and square concrete columns has been fully described.

The experimental data in Table 1, which contains FRP-confined undamaged concrete columns and FRP-confined load-damaged concrete columns, are used to evaluate the proposed and other existing strength models. Figure 6 shows that the IAE of LI et al’s model [3] and the proposed model are 0.103 and 0.129, respectively. Figure 6 shows that LI et al’s model [3] performs slightly better than the proposed model. However, the theoretical values proposed by the proposed method are smaller than those in the experimental tests. This can explain why the actual predicted strength of the proposed model is safe. Similarly, the proposed model can predict the strength for FRP-confined undamaged and load-damaged concrete columns. As shown in Figure 7, the IAE of LI et al’s model [3] and the proposed model are 0.193 and 0.106, respectively. Compared with LI et al’s model [3], the proposed model performs better for FRP-confined undamaged and load-damaged concrete columns.

Figure 6 Performance of strength model for FRP-confined load-damaged circular and square columns:

Figure 7 Performance of strength model for FRP- confined undamaged and load-damaged circular and square columns:

4.5 Assessment model for FRP-confined fire- damaged concrete

The data in Table 1, which consist of FRP- confined fire-damaged circular and square was used to evaluate the proposed model and existing models. The evaluation results are shown in Figure 8. It can be seen from Figure 8 that the IAE of the proposed model is smaller than that of the other models. In other words, the proposed model has better performance, and can predict the strength model for FRP-confined fire-damaged circular and square concrete columns.

Figure 8 Performance of strength model for FRP- confined fire-damaged columns

5 Conclusions

In this paper, considering the damage degree, the proposed model is developed by the Hoek-Brown failure criterion. The proposed model has two advantages compared with the other existing models. 1) The proposed model is a unified strength model for FRP-confined undamaged circular and square concrete columns. 2) The proposed model is suitable for FRP-confined load-damaged and fire-damaged concrete columns. The proposed model is in line with physical and mathematical meanings. A comprehensive and updated database was established by collecting experimental results from all available FRP- confined circular and square columns from the published literature, and the model was evaluated. The proposed model performs better than the other models, and the measures of dispersion are also minimal.

Contributors

CAO Yu-gui provided the concept and edited the draft of manuscript. ZHANG Yang conducted the literature review and wrote the first draft of the manuscript. LU Zhi-fang edited the draft of manuscript.

Conflict of interest

ZHANG Yang, LU Zhi-fang, and CAO Yu-gui declare that they have no conflict of interest.

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(Edited by HE Yun-bin)

中文导读

基于Hoek-Brown破坏准则的FRP约束损伤混凝土圆柱与方柱统一强度模型

摘要：纤维增强聚合物(FRP)具有强度高、重量轻、耐腐蚀、施工方便等优点，广泛应用于受损混凝土柱的修复和加固。现有的强度模型大多是通过实验数据的回归分析建立的。本文利用Hoek-Brown破坏准则，提出了一种新的强度统一模型。为了研究FRP约束混凝土损伤和未损伤柱的强度，作者从已发表的文献中收集了900个试验数据，建立了包含各试件截面形状、损伤类型、损伤程度和FRP约束刚度的大型数据库。利用Hoek-Brown破坏准则建立了一种新的强度模型，该模型适用于未损伤、荷载损伤和火灾损伤的圆柱和方柱。在数据库的基础上，对发表文献中已有的强度模型和本文提出的强度模型进行了评估。评估结果表明，该模型能较好地预测FRP约束损伤和未损伤混凝土柱的抗压强度。

关键词：FRP约束混凝土；荷载损伤；火灾损伤；统一强度模型；Hoek-Brown破坏准则

Foundation item: Project(2017M622540) supported by the China Postdoctoral Science Foundation; Project(51808419) supported by the National Natural Science Foundation of China; Project(2019CFB217) supported by the National Natural Science Foundation of Hubei Province, China; Project(201623) supported by the Science and Technology Project of Wuhan Urban and Rural Construction Committee, China

Received date: 2020-05-15; Accepted date: 2020-10-22

Corresponding author: CAO Yu-gui, PhD, Associate Professor; Tel: +86-15327263275; E-mail: caoyugui@163.com; ORCID: https:// orcid.org/0000-0003-1038-0572