Local bond-slip model between steel bars and concrete after exposure to high temperatures

Abstract

The impact of high-temperature exposure on the residual bond behavior between deformed steel bars and concrete is a critical aspect that affects their composite interaction and the load-bearing capacity of reinforced concrete (RC) members after fire events. This paper presents an analytical method to predict the residual bond behavior and failure modes of steel bars embedded in concrete after exposure to high temperatures. The concrete cover is modeled as a thick-walled cylinder with the inner surface under uniform pressure. During the loading process, the concrete cover is divided into two portions: an intact outer portion and a cracked inner portion. The material properties of both steel and concrete, along with the tension-softening characteristics of the cracked inner portion, are well-defined in the analytical method. Simplified equations are developed to estimate the maximum radial stresses, residual bond strength and interfacial slip parameters, which together define the bond-slip relationship at the steel bar-to-concrete interface after high-temperature exposure. The accuracy and validity of these simplified equations are extensively verified by comparing the predictions obtained from these simplified equations with corresponding analytical results, experimental data from the literature and the load-displacement curves predicted by the proposed finite element model.

Keywords

steel bar concrete cover analytical method residual bond strength local bond-slip relationship high temperatures simplified expressions

Introduction

Reinforced concrete (RC) structures are widely used in civil infrastructure construction, where fire poses a severe risk particularly in enclosed space. Observations from fire incidents have indicated that while total collapse of RC structures under fire exposure is rare, post-fire damages—such as concrete cracking, spalling, material property deterioration and reduced bond strength between steel reinforcement and concrete—can significantly influence the residual load-carrying capacity of RC members (Chang et al., 2006; Gao et al., 2017; Liu et al., 2016; Wu et al., 2019). Among these effects, bond strength deterioration is crucial, as it governs the load transfer and stress interactions between the steel reinforcement and concrete. Numerous experimental studies have explored the residual bond strengths of steel bars in concrete following exposure to high temperatures (Deshpande et al., 2020; Hlavička, 2017; Li et al., 2023; Liu et al., 2023; Yang et al., 2018; Yin et al., 2011; Zhang et al., 2019). As the exposure temperatures increase, the bond strength gradually decreases. Once the temperature exceeds 600°C, the bond strength decreases to only 10% to 50% of its original strength at room temperature. Such a substantial decline in bond strength can significantly influence the post-fire load-bearing capacity and stiffness of RC members. Hence, it is imperative to formulate a precise and robust local bond-slip model to describe the bond behavior between steel bars and concrete after exposure to high temperatures. This aspect is critical for accurately evaluating the post-fire performance of RC elements and offering appropriate strategies for post-fire repair and strengthening.

In the field of structural engineering, pullout tests (Huang et al., 2022; Li et al., 2021, 2023; Liu et al., 2023; Wang et al., 2022a; Yang et al., 2018; Zhang et al., 2019, 2020) are commonly employed to investigate the bond performance between deformed steel bars and concrete under varying environmental conditions. Previous studies have indicated that several design parameters, including concrete type and strength, steel bar diameter, concrete cover thickness, embedment length, presence of stirrups and high temperatures, can significantly influence the bond behavior (Gao et al., 2013; Harajli et al., 2004; Li et al., 2021; Liu et al., 2023; Xu et al., 2022; Yin et al., 2011; Wang et al., 2022a). Pullout failures are frequently observed in cases where the concrete cover layer is thick or the steel bars are heavily confined, especially in specimens with a high steel stirrup reinforcement ratio. Inadequate concrete cover may lead to splitting failure, causing cracks to propagate from the bond interface to the outer surface of the specimen. Tensile load-displacement curves are typically measured in these tests to establish the bond stress and interfacial slip relationship, commonly referred to as the “bond-slip model” in the existing literature (Gao et al., 2019; Khalaf et al., 2016; Li et al., 2021; Wang et al., 2022a; Xu et al., 2022; Zhang et al., 2019). The local bond-slip model, initially introduced by Eligehausen et al. (1983) and subsequently incorporated into Mode Code 2010 (2010), is widely used for evaluating and designing the local bond-slip behavior under standard environmental conditions. Furthermore, various analytical models have been proposed to predict the bond behavior and failure mechanisms at the steel bar-to-concrete interface under sustained pullout loading (Chang et al., 2021; Den Uijl and Bigaj, 1996; Gao et al., 2019; Khalaf et al., 2016; Nielsen and Bićanić, 2002; Pothisiri and Panedpojaman, 2013; Wang and Liu, 2003; Xu et al., 2022). Recent research has focused on predicting the residual bond strengths of steel bars after exposure to high temperatures. A thick-wall cylinder model has been employed to elucidate the bond interactions between internal steel bars and the surrounding concrete cover (Wang et al., 2022b, 2023). However, there is a lack of simplified design equations for predicting the residual bond strengths of steel bars following high-temperature exposure. Additionally, there is a shortage of reliable interfacial constitutive laws that describe the local bond-slip model governing the residual bond behavior between steel bars and concrete after exposure to high temperatures. This model is essential for accurately predicting the post-fire load-bearing capacity and deformation characteristics of RC members.

This paper presents an analytical method for predicting the bond behavior and failure mechanisms of pullout specimens after exposure to high temperatures. The study primarily focuses on the load transfer mechanism between steel bars and the concrete cover. The analytical model defines the concrete cover as a thick-walled cylinder subjected to uniform pressure on its inner surface (Tepfers, 1979). Cracking of the concrete within the cover occurs during the pullout loading process due to increased internal pressure, requiring accurate consideration of the tensile softening behavior of the internally cracked concrete. Furthermore, the residual bond strength of the steel bar-to-concrete interface after exposure to high temperatures is analyzed and determined based on the theoretical analytical method that considers two different failure modes of the interface—pullout failure and concrete splitting failure—while accounting for temperature-induced degradation of material properties. A local bond-slip model is then developed based on the analytical results and numerical parametric investigations, in which simplified expressions are developed for estimating the maximum radial stresses, residual bond strength and interfacial slip parameters that are used to define the local bond-slip relationship at the steel bar-to-concrete interface after exposure to high temperatures. The validation of the simplified design expressions for residual bond strength and interfacial slip parameters in estimating the bond-slip relationship for steel bars embedded in concrete after exposure to high temperatures involves comparisons between analytical results, existing pullout test results and the corresponding predictions derived from the simplified expressions. Additionally, a finite element (FE) model is proposed to simulate pullout tests of steel bars following exposure to high temperatures. The FE model has incorporated the local bond-slip model of the steel bar-to-concrete interface using the proposed simplified design expressions. The accuracy and reliability of the simplified expressions and the proposed bond-slip model are further confirmed through comparisons of load-displacement curves from FE predictions with pullout test results.

Analytical method for bond strength

The relationship between bond stress and radial stress at the steel bar-to-concrete interface

The interaction between deformed steel bars and concrete involves chemical adhesion, friction and mechanical interlocking. Among these mechanisms, chemical adhesion exerts the least significant influence on bond stress. When the deformed steel bars and concrete interface experiences initial sliding, the chemical adhesion weakens and the mechanical interlocking emerges as the primary factor affecting the bond stress (Wang et al., 2022b). Figure 1 illustrates the bond stress and radial pressure generated at the interface between the steel bar and the concrete. As the interfacial slip progresses, the concrete adjacent to the deformed steel bar ribs undergoes compressed, leading to the development of a conical failure surface. The compressive stress $p_{x, T}$ and the friction force $f p_{x, T}$ acting on the conical failure surface are counterbalanced by the pullout force. The angle between the failure surface and the longitudinal axis of the steel bar is denoted by $φ$ , while $f$ represents the friction coefficient between the cracking concrete and the ordinary concrete on the failure surface. Consequently, equations (1) and (2) are derived from the force equilibrium equations at the interface between the steel bar and the concrete:

p_{c, T} = p_{x, T} \cos φ - f p_{x, T} \sin φ

(1)

τ_{T} = p_{x, T} \sin φ + f p_{x, T} \cos φ

(2)

Figure 1.

Interaction between the deformed steel bar and the surrounding concrete.

In the aforementioned equations, $p_{c, T}$ represents the radial pressure acting on the steel bar after exposure to high temperatures, and $τ_{T}$ denotes the bond stress between the steel bar and the surrounding concrete. The value of $φ$ is influenced by various design parameters, such as the steel bar diameter, the concrete strength and the concrete cover thickness. Xu (1990) complied pullout test data from existing literature, where the recorded values of $φ$ ranged between 10° and 40°. This study adopted the average angle of 24° as recommended by Guo and Shi (2003). The friction coefficient $f$ was assumed to be 0.5 according to Zhao and Xiao (2011). By combining equations (1) and (2), the relationship between bond stress $τ_{T}$ and radial pressure $p_{c, T}$ can be formulated as:

τ_{T} = \frac{\sin φ + f \cos φ}{\cos φ - f \sin φ} \cdot p_{c, T} = 1.216 p_{c, T}

(3)

Radial stress during the elastic stage

The thick-walled cylinder model proposed by Tepfers (1979) is used as a basic framework to describe the bond interactions between deformed steel bars and the surrounding concrete. In this model, the steel bar is represented as a cylinder with a radius of $R_{b}$ , while the concrete cover is represented as an outer cylinder with a radius of $R_{c}$ . The stress distribution within the concrete cylinder following exposure to high temperatures is depicted in Figure 2. During the elastic stage, the steel bar experiences radial pressure $p_{c, T}$ from the adjacent concrete, and the resultant reaction force $p_{c, T}^{'}$ is counterbalanced by the tensile stress generated within the concrete. In the absence of cracks in the concrete cover, both the steel bar and concrete are deemed elastic materials at the initiation of slippage. According to the elasticity theory elucidated by Timoshenko and Goodier (1982), the hoop tensile stress in the concrete at the radius $r$ can be mathematically expressed as:

σ_{t, T} (r) = \frac{R_{b}^{2} p_{c, T}}{R_{c}^{2} - R_{b}^{2}} (1 + \frac{R_{c}^{2}}{r^{2}})

(4)

where

r

represents the distance from a point within the concrete cover to the center of the steel bar,

R_{b}

represents the radius of the steel bar, and

R_{c}

denotes the concrete cover depth which is defined as the distance from the center of the steel bar to the outermost edge of the surrounding concrete.

Figure 2.

Stress distribution in the elastic stage.

Within the uncracked concrete area, the tensile stress increases gradually as the radius decreases, as depicted in Figure 2. In situations where the pullout force is relatively low, the tensile stress at the interface between the steel bar and concrete (denoted as $σ_{t, T} (R_{b})$ ) may be lower than the residual tensile strength of the concrete after exposure to high temperatures, $f_{t, T}$ . As the pullout force increases, the elastic stage ends when $σ_{t, T} (R_{b})$ is equal to $f_{t, T}$ , as shown in the following equation:

σ_{t, T} (R_{b}) = \frac{R_{b}^{2} p_{c, T}}{R_{c}^{2} - R_{b}^{2}} (1 + \frac{R_{c}^{2}}{R_{b}^{2}}) = f_{t, T}

(5)

Radial stress during the partial cracking stage

After concrete cracking, the concrete cover can be divided into two distinct parts: the internal cracked portion and the outer uncracked portion. The outer uncracked part of the concrete exhibits linear elastic behavior. It is essential to carefully analyze the tensile softening behavior of cracked concrete in the internal portion. Typically, the relationship between tensile stress and crack opening displacement (COD) is utilized to describe the tensile behavior in this situation (Hillerborg et al., 1976). By establishing the force equilibrium equations for these two portions, the radial stress expression during the partial cracking stage can be determined.

Tensile stress-COD relationship of concrete after high-temperature exposure

The tensile softening behavior of concrete is often characterized by a curve that depicts the relationship between tensile stresses and COD. This study employs the nonlinear tensile softening model proposed by Wang et al. (2022a):

{\begin{cases} \frac{σ_{t, T}}{f_{t, T}} = 1 - {(\frac{ω_{T}}{ω_{u, T}})}^{λ} & 0 \leq \frac{ω_{T}}{ω_{u, T}} \leq 1 \\ σ_{t, T} = 0 & \frac{ω_{T}}{ω_{u, T}} > 1 \end{cases}

(6)

where

σ_{t, T}

is the tensile stress of concrete after exposure to high temperatures,

ω_{T}

is the COD,

ω_{u, T}

is the COD corresponding to

σ_{t, T}

= 0, and

λ

is a coefficient.

ω_{u, T}

is affected by many factors, including the strength of the concrete, the residual fracture energy of the concrete after exposure to high temperatures and the maximum aggregate size (Pantazopoulou and Papoulia, 2001; Zhao and Xu, 2001). In this study, the equation introduced by Pantazopoulou and Papoulia (2001) is used in this investigation for computation of

ω_{u, T}

ω_{u, T} = \frac{k G_{F, T}}{f_{t, T}}

(7)

The value of $k$ depends on the maximum aggregate size $d_{a}$ . When the value of $d_{a}$ is 8 mm, 16 mm or 32 mm, the corresponding value of $k$ is 8, 7 or 5, respectively (Pantazopoulou and Papoulia, 2001). For $d_{a}$ with values other than those specified, a linear interpolation method is used to calculate the value of $k$ . $G_{F, T}$ represents the energy absorbed during the entire concrete cracking process, which can be determined by the area enclosed by the $σ_{t, T} - ω_{u, T}$ curve, as illustrated in Figure 3(a):

G_{F, T} = \int_{0}^{ω_{u, T}} σ_{t, T} d ω_{T}

(8)

Figure 3.

The relationship between tensile stress and crack opening displacement (COD) of concrete after exposure to high temperatures: (a) tensile stress-COD relationship; and (b) normalized tensile stress-COD relationship with temperatures.

Limited research has been conducted on the residual fracture energy of concrete following exposure to high temperatures. Figure 4 shows the collected experimental results, which have been normalized against the corresponding results obtained at room temperature (Baker, 1996; Dabbaghi et al., 2021; Menou et al., 2006; Nielsen and Biéanić, 2003; Tang and Lo, 2009; Yu et al., 2013; Zhang and Bicanic, 2006; Zhang et al., 2000). The figure illustrates that there is no discernible trend in the relationship between concrete fracture energy and temperature variation. Consequently, the current analytical method disregards the alternations in concrete fracture energy following exposure to high temperatures. In this context, $G_{F, T} = G_{F 0}$ , where $G_{F 0}$ denotes the fracture energy of concrete at room temperature and is calculated using the formula specified in Mode Code 2010 (2010):

G_{F, T} = G_{F 0} = 73 f_{c m}^{0.18}

(9)

where

f_{c m}

is the average compressive strength of concrete, and its conversion relationship with the characteristic compressive strength

f_{c k}

is given by Mode Code 2010 (2010) as follows:

f_{c m} = f_{c k} + Δ f

(10)

where

Δ f

= 8 MPa.

Figure 4.

Normalized fracture energy of concrete after exposure to high temperatures.

By substituting equations (6) and (7) into (8), the fracture energy $G_{F}$ can be expressed as:

G_{F} = \int_{0}^{ω_{u, T}} σ_{t, T} d ω_{T} = \int_{0}^{ω_{u, T}} f_{t, T} [1 - {(\frac{ω_{T}}{ω_{u, T}})}^{λ}] d ω_{T} = \frac{ω_{u, T} f_{t, T}}{k}

(11)

Solving equation (11) results in determining the coefficient $λ$ as a function of $k$ :

λ = \frac{1}{k - 1}

(12)

By substituting equations (9) into (11), the ratio of $ω_{u, T} / ω_{u, 0}$ can be expressed as:

\frac{ω_{u, T}}{ω_{u, 0}} = \frac{f_{t, 0}}{f_{t, T}}

(13)

where

ω_{u, 0}

is the COD corresponding to

σ_{t, 0} = 0

at room temperature, and

f_{t, 0}

is the tensile strength of concrete under ambient conditions.

By substituting equations (7), (9), and (11) into (6), the tensile softening behavior of concrete after exposure to high temperatures can be formulated as:

{\begin{cases} \frac{σ_{t, T}}{f_{t, T}} = 1 - {(\frac{ω_{T} f_{t, T}}{k G_{F}})}^{\frac{1}{k - 1}} & 0 \leq \frac{ω_{T}}{ω_{u, T}} \leq 1 \\ σ_{t, T} = 0 & \frac{ω_{T}}{ω_{u, T}} > 1 \end{cases}

(14)

Figure 3(b) depicts the relationship between the normalized tensile stress $σ_{t, T} / f_{t, 0}$ and the normalized COD (i.e., $ω_{T} / ω_{u, 0}$ ). The graph shows that as the exposure temperature rises, the rate of decline in the curve becomes more gradual. This phenomenon is associated with the decrease in the tensile strength of concrete after exposure to high temperatures.

Determination of the maximum radial stress

Once the tensile stress in the concrete at the steel bar-to-concrete interface $σ_{t, T} (R_{b})$ reaches the residual tensile strength $f_{t, T}$ , cracking in the concrete begins at the interface. As the pullout force applied increases, the cracks in the concrete propagate from the interface towards the outer region of the concrete cover. The distance from the crack tips to the interface is defined as the crack radius $R_{i, T}$ , where the tensile stress is equal to the tensile strength $f_{t, T}$ . Consequently, the thick-walled cylinder can be divided into two portions by this radius $R_{i, T}$ : an outer portion without cracks and an inner one with cracks, as depicted in Figure 5.

Figure 5.

Stress distribution in the partial cracking stage: (a) thick-walled cylinder; (b) uncracked outer portion; and (c) cracked inner portion.

The uncracked external portion ( $R_{i, T} < r \leq R_{c}$ ) is regarded as maintaining its elasticity. The state of equilibrium is upheld by the combined effects of the hoop tensile stress $σ_{t, T}$ distributed within the concrete and the compressive stress $p_{i, T}^{'}$ applied by the internal part on its inner surface (see Figure 5(b) for further information).

The tensile stress within the uncracked outer portion $σ_{t, T} (r)$ can be determined by applying the analytical method of the elastic stage, where the variables of $R_{b}$ and $p_{c, T}$ in equation (4) are substituted with their respective values of $R_{i, T}$ and $p_{i, T}$ :

σ_{t, T} (r) = \frac{R_{i, T}^{2} p_{i, T}}{R_{c}^{2} - R_{i, T}^{2}} (1 + \frac{R_{c}^{2}}{r^{2}}); R_{i, T} < r \leq R_{c}

(15)

where

p_{i, T}

represents the compressive stress exerted by the uncracked concrete on the outer wall of the cracked inner portion, acting as the reaction force to

p_{i, T}^{'}

, and the two have equal values.

The fictitious crack model proposed by Hillerborg et al. (1976) is employed to elucidate the distribution of tensile stress within the cracked inner cylinder ( $R_{b} \leq r \leq R_{i, T}$ ). At the crack radius $R_{i, T}$ , the hoop tensile stress in concrete $σ_{t, T} (R_{i, T})$ is equivalent to the residual tensile strength $f_{t, T}$ . As the radius $r$ decreases, the tensile stress $σ_{t, T} (r)$ gradually reduces. In the cracked portion of the concrete, the proposed model adopts the assumption that the crack opening displacement is linearly distributed along the crack length (Chang et al., 2021; Gao et al., 2019). The total circumferential elongation $Δ_{r, T}$ of the concrete at the radius $r$ after exposure to high temperatures can be expressed as the sum of the elastic deformation $2 π ε_{r, T}$ occurring between cracks and the accumulative COD $n ω_{r, T}$ of the cracks:

Δ_{r, T} = 2 π r ε_{r, T} + n ω_{r, T}; R_{b} \leq r \leq R_{i, T}

(16)

where

ε_{r, T}

is the hoop strain at the radius

r

ω_{r, T}

is the COD at the radius

r

n

is the number of cracks in the concrete, typically ranging from 2 to 4, with an average value of 3 used in this paper (Cattaneo and Rosati, 2009; Nielsen and Bićanić, 2002).

By substituting equations (14) into (16), the total hoop elongation $Δ_{r, T}$ at the radius $r$ can be expressed as:

Δ_{r, T} = 2 π r ε_{r, T} + \frac{n k G_{F}}{f_{t, T}} {(\frac{ε_{r, T}}{ε_{c t, T}} - \frac{σ_{t, T} (r)}{f_{t, T}})}^{k - 1}; R_{b} \leq r \leq R_{i, T}

(17)

At the crack tips ( $r = R_{i, T}$ ), $ω_{r, T}$ is zero, resulting in a hoop strain equal to the tensile cracking strain $ε_{c t, T}$ ( $ε_{c t, T} = f_{t, T} / E_{T}$ , $E_{T}$ is the residual elastic modulus of the concrete after exposure to high temperatures). Therefore, the hoop elongation $Δ_{r, T} (R_{i, T})$ at the radius $R_{i, T}$ can be described as follows:

Δ_{r, T} (R_{i, T}) = 2 π R_{i, T} ε_{c t, T}

(18)

In previous studies, it was assumed that the total hoop elongation inside the cracked inner portion ( $R_{b} \leq r \leq R_{i, T}$ ) was not affected by the radius $r$ (Gao et al., 2019; Pothisiri and Panedpojaman, 2013; Wang and Liu, 2003). Substituting equations (18) into (17) yields the following equation:

2 π R_{i, T} ε_{c t, T} = 2 π r ε_{r, T} + \frac{n k G_{F}}{f_{t, T}} {(1 - \frac{σ_{t, T} (r)}{f_{t, T}})}^{k - 1}

(19)

The above formula can be reformulated as:

σ_{t, T} (r) = {1 - {(\frac{2 π f_{t, T} ε_{c t, T}}{n k G_{F}})}^{\frac{1}{k - 1}} {(R_{i, T} - \frac{ε_{r, T}}{ε_{c t, T}} r)}^{\frac{1}{k - 1}}} f_{t, T}; R_{b} \leq r \leq R_{i, T}

(20)

Figure 6 shows the stress distribution of concrete in the partial cracking stage. At this stage, the inner concrete region is subjected to the tensile stress $σ_{t, T}$ distributed over a radius $r$ ( $R_{b} \leq r \leq R_{c}$ ) and the compressive stress $p_{c, T}^{'}$ (= $p_{c, T}$ ) exerted by the steel bar on its inner surface. The radial pressure can be determined by establishing an equilibrium of forces within the concrete, and it is expressed as follows:

2 π \cdot R_{b} p_{c, T} = 2 π \cdot \int_{R_{b}}^{R_{c}} σ_{t, T} (r) d r

(21)

Figure 6.

Schematic diagram of concrete stresses in the partial cracking stage.

By substituting equations (15) and (20) into equation (21), the radial pressure $p_{c, T}$ can be expressed as:

p_{c, T} = f_{t, T} [\frac{R_{i, T}}{R_{b}} \cdot \frac{R_{c}^{2} - R_{i, T}^{2}}{R_{c}^{2} + R_{i, T}^{2}} + (\frac{R_{i, T}}{R_{b}} - 1) - \frac{{(C \cdot R_{b})}^{\frac{1}{k - 1}} (k - 1)}{k} {(\frac{R_{i, T}}{R_{b}} - 1)}^{\frac{k}{k - 1}}]

(22)

where

C = \frac{2 π ε_{c t, T}}{n ω_{u, T}}

(23)

Then, the maximum value of the radial pressure $p_{c, T}^{\max}$ can be obtained by taking the derivative of $p_{c, T}$ with respect to $R_{i, T}$ , and setting the resulting differential equation to zero as follows:

\frac{d p_{c, T}}{d R_{i, T}} = \frac{1}{R_{b}} \cdot \frac{R_{c}^{4} - 4 R_{c}^{2} R_{i, T}^{2} - R_{i, T}^{4}}{{(R_{i, T}^{2} + R_{c}^{2})}^{2}} + \frac{1}{R_{b}} - \frac{{(C \cdot R_{b})}^{\frac{1}{k - 1}}}{R_{b}} {(\frac{R_{i, T}}{R_{b}} - 1)}^{\frac{1}{k - 1}} = 0

(24)

The solution of equation (24) is called the critical cracking radius $R_{i, T}^{c r}$ , which corresponds to the maximum value of the radial pressure, $p_{c, T}^{\max}$ . The residual bond strength $τ_{T, \max}$ of the steel bar embedded in concrete can be calculated by substituting the corresponding value of $p_{c, T}^{\max}$ into equation (3).

Impact of steel stirrups

Steel stirrups play a crucial role in improving the bond properties between the deformed steel bars and the surrounding concrete (Harajli et al., 2004; Xu et al., 2022), thereby enhancing bond strength. In a study by Xu (1990), the confinement effect of steel stirrups was investigated by varying parameters such as the diameter and spacing of stirrups and the concrete cover depth. Furthermore, Xu (1990) proposed an enhancement coefficient $K_{s v}$ to quantify the confinement effect:

τ_{T, \max}^{s v} = K_{s v} \cdot τ_{T, \max}

(25)

where

τ_{T, \max}^{s v}

is the bond strength considering the confinement effect of steel stirrups, and the enhancement coefficient

K_{s v}

can be calculated using the following formula:

K_{s v} = 1 + 8.5 ρ_{s v} = 1 + 8.5 \frac{π d_{s v}^{2}}{4 c S_{s v}}

(26)

where

ρ_{s v}

is the volume ratio of steel stirrups,

c

is the concrete cover depth,

d_{s v}

and

S_{s v}

are the diameter and spacing of the stirrups, respectively.

Analytical method of the local bond-slip model

As mentioned in the introduction section, local bond-slip models are commonly used to describe the constitutive behavior of the interface between deformed steel bars and the surrounding concrete. In this study, the four-stage local bond-slip model outlined in Mode Code 2010 (2010) is employed as the foundational constitutive relationship to represent the influence of high temperatures, as depicted in the following equation:

{\begin{cases} τ_{T} = τ_{T, \max} {(s / s_{1, T})}^{m} & 0 \leq s < s_{1, T} \\ τ_{T} = τ_{T, \max} & s_{1, T} \leq s < s_{2, T} \\ τ_{T} = τ_{T, \max} - (τ_{T, \max} - τ_{T, f}) (\frac{s - s_{2, T}}{s_{3, T} - s_{2, T}}) & s_{2, T} \leq s < s_{3, T} \\ τ_{T} = τ_{T, f} & s \geq s_{3, T} \end{cases}

(27)

where

τ_{T}

and

s

are the bond stress and interfacial slip between the deformed steel bar and the surrounding concrete after exposure to high temperatures, in which

m

is the coefficient controlling the ascending branch of the bond-slip curve, assigned a value of 0.4 according to Mode Code 2010 (2010). The bond strength,

τ_{T, \max}

, can be determined by substituting the maximum radial stress

p_{c, T}^{\max}

into equation (3):

τ_{T, \max} = \frac{\sin φ + f \cos φ}{\cos φ - f \sin φ} \cdot p_{c, T}^{\max} = 1.216 p_{c, T}^{\max}

(28)

where

p_{c, T}^{\max}

is derived by substituting the critical cracking radius

R_{i, T}^{c r}

into equation (22).

Previous studies on pullout tests of steel bars have identified two distinct failure modes (Deshpande et al., 2020; Li et al., 2023; Liu et al., 2023; Yang et al., 2018; Yin et al., 2011; Zhang et al., 2019). The primary failure mode is pullout failure, which is particularly common in situations where the concrete cover is thick or when the steel bars are subjected to high levels of restraint (e.g., confinement by a high ratio of steel stirrups). Essentially, pullout failure occurs before the radial propagation of concrete cracks reaches the surface of the specimen. Conversely, in cases with a thin concrete cover, radial concrete cracks propagate towards the concrete surface of the specimen, resulting in a failure mode known as splitting failure. Various analytical solutions for these distinct failure modes are currently being considered to accurately characterize the bond-slip behavior between steel bars and concrete after exposure to high temperatures. To establish the bond-slip relationship for the steel bar-to-concrete interface after high-temperature exposure, the failure mode is typically determined based on the test observations. In cases where the failure mode is not explicitly noted, it is generally assumed that the pullout failure occurs if the ratio $R_{c}$ / $R_{b}$ is higher than 10; otherwise, the splitting failure mode is presumed.

Pullout failure

As the interfacial slip increases, the concrete cover layer between adjacent steel bar ribs experiences radial extension (Gao et al., 2019), as depicted in Figure 7. The concrete region between two neighboring ribs can be approximated as a truncated cone with a specific angle $θ$ . Therefore, it can be calculated using the following equation:

\tan θ = \frac{v_{T}}{s_{1, T}}

(29)

where

v_{T}

represents the radial displacement at the steel bar-to-concrete interface in the failure state, which can be determined through the following calculation:

v_{T} = R_{i, T}^{c r} \cdot ε_{c t, T}

(30)

Figure 7.

Equivalent cone of the deformed steel bar.

For the sake of simplicity, the effects of exposure to high temperatures on the values of $θ$ are disregarded (Wang et al., 2022b). The typical slip $s_{1}$ of the bond interface during pullout failure at room temperature is set at 1.0 mm according to Mode Code 2010 (2010). Therefore, $\tan θ$ can be formulated as $\tan θ = v_{0} / 1.0$ , where $v_{0}$ denotes the radial displacement at room temperature which can be calculated using equation (30). $s_{1, T}$ can be evaluated using the subsequent equation:

s_{1, T} = \frac{v_{T}}{\tan θ} = \frac{v_{T}}{v_{0} / s_{1}} = \frac{R_{i, T}^{c r} \cdot ε_{c t, T} \cdot s_{1}}{R_{i, 0}^{c r} \cdot ε_{c t, 0}}

(31)

where

R_{i, 0}^{c r}

and

ε_{c t, 0}

are the critical cracking radius and the ultimate tensile strain of concrete at room temperature, respectively.

The characteristic slip $s_{2}$ at ambient temperature is determined to be 2.0 mm according to Mode Code 2010 (2010), which is 1.0 mm larger than $s_{1}$ . This relationship remains unchanged in the analytical method, and thus, $s_{2, T}$ = $s_{1, T}$ +1.0.

Once the interface slip $s$ equals the spacing between adjacent ribs, the mechanical interlocking contribution to the bond force ceases. Consequently, the residual bond stress is attributed solely to friction. The residual bond stress, $τ_{T, f}$ , is reported as 0.2 $τ_{T, \max}$ following the suggestion by Gao et al. (2019). The characteristic slip $s_{3, T}$ is described as the distance between the ribs of the steel bar, $l_{s}$ . In cases where not explicitly stated, the expression provided in the current design guideline GB/T 1499.2-2018 (2018) is employed for assessment:

s_{3, T} = l_{s} = - 5.56 d^{2} \times 10^{- 3} + 0.564 d + 1.575

(32)

where

d

represents the diameter of the deformed steel bar.

By substituting the acquired residual bond strength $τ_{T, \max}$ , the residual bond strength $τ_{T, f}$ and the characteristic slips $s_{1, T}$ , $s_{2, T}$ and $s_{3, T}$ into equation (27), the local bond-slip curve under pullout failure (as illustrated in Figure 8(a)) can be determined.

Figure 8.

Local bond-slip curves associated with distinct failure modes: (a) local bond-slip relationship; (b) normalized local bond-slip relationships of splitting failure at different temperatures.

Splitting failure

When the splitting failure occurs, the absence of a “plastic plateau” characterized by consistent bond strength $τ_{T, \max}$ leads to a bond-slip curve with only three stages (Figure 8(a)). The prescribed characteristic slip $s_{1}$ of the bond interface for pullout failure at room temperature is set to 0.6 mm according to Mode Code 2010 (2010). Thus, $\tan θ$ can be expressed as $\tan θ$ = $v_{0} / 0.6$ , and $s_{1, T}$ can be obtained by substituting $s_{1}$ = 0.6 mm into equation (31).

The pullout specimens that are confined with steel stirrups exhibit residual bond properties after splitting failure, which can be attributed to the supplementary confinement provided by the stirrups. Therefore, the residual bond stress $τ_{T, f}$ associated with splitting failure in the presence of stirrups is assumed to be 0.2 times the bond strength $τ_{T, \max}$ , and the characteristic slip $s_{3, T}$ is defined as 0.5 times the stirrup spacing $l_{s}$ . In contrast, the unconfined pullout specimens under splitting failure show a significant decrease in bond stress after the peak, eventually reaching zero with no residual bond stress. In this case, $τ_{T, f}$ is treated as zero and $s_{3, T}$ is assigned a value of 1.2 times $s_{1, T}$ . By providing precise parameter definitions, the local bond-slip curve of the deformed steel bar-to-concrete interface during splitting failure can be established.

Figure 8(b) depicts a set of normalized local bond-slip curves that show the behavior of the steel bar-to-concrete interface after exposure to different high temperatures. These curves illustrate the splitting failure mode of the steel bar under the influence of confinement by stirrups. The residual bond strength gradually decreases as the exposure temperature increases, with the values of $s_{1, T}$ initially increasing and then decreasing as the temperature approaches 800°C.

Simplified expressions of the local bond-slip relationship

The previous section has presented detailed analytical solutions for determining the parameters of the local bond-slip relationship at the interface between steel bars and concrete after exposure to high temperatures. Given the complex nature of the analytical method, computational methods are needed to calculate the critical cracking radius and other relevant parameters. As a result, the direct application of the local bond-slip model is considered impractical. The following section aims to develop simplified formulations for the parameters that define the local bond-slip relationship by employing data fitting methods. This study will rely on numerical results obtained from parametric investigations carried out using the proposed analytical method.

Properties of concrete materials after exposure to high temperatures

A comprehensive investigation on the degradation of material properties in concrete after exposure to high temperatures, such as tensile strength and elastic modulus, is crucial for defining the parameters required to formulate the local bond-slip model. Liang (2008) carried out an in-depth analysis of experimental data and formulated a design equation to elucidate the reduction rate of the elastic modulus of concrete after exposure to high temperatures:

\frac{E_{T}}{E_{0}} = \frac{1}{1 + 18.5 \times {(T / 1000)}^{2.47}} 20 ° C < T \leq 800 ° C

(33)

where

E_{0}

is the elastic stiffness of concrete at room temperature, which can be calculated using the formula outlined in Mode Code 2010 (2010):

E_{0} = E_{c 0} \cdot α_{E} \cdot {(\frac{f_{c m}}{10})}^{\frac{1}{3}}

(34)

where

α_{E}

= 1.0 for concrete made with quartzite aggregates, and

E_{c 0}

= 21500 MPa as specified by Mode Code 2010 (2010).

f_{c m}

can be calculated using equation (10).

The decrease in the tensile strength of concrete after exposure to high temperatures is described by the following formula presented by Qin et al. (2004):

\frac{f_{t, T}}{f_{t, 0}} = 1.01 - 0.00111 T 20 ° C < T \leq 800 ° C

(35)

where

f_{t, 0}

is determined in accordance with Mode Code 2010 (2010):

f_{t, 0} = {\begin{cases} 0.3 {(f_{c m} - 8)}^{2 / 3} & for f_{c m} < 50 MPa \\ 2.12 \cdot \ln (1 + 0.1 f_{c m}) & for f_{c m} \geq 50 MPa \end{cases}

(36)

Simplified expressions for critical cracking radius and maximum radial stress

Once the critical cracking radius $R_{i, T}^{c r}$ is determined, the residual bond strength can be calculated using equations (28) and (39). The value of $R_{i, T}^{c r}$ can be computed using equation (24), which in influenced by parameters such as the steel bar radius $R_{b}$ , the concrete cove depth $R_{c}$ , the compressive strength $f_{c m}$ , the coefficient $k$ and the exposure temperature $T$ . The current simplified process does not account for the impact of high-temperature exposure and concrete strength on the critical cracking radius $R_{i, T}^{c r}$ . Instead, a constant value is assumed for $R_{i, T}^{c r}$ , which is determined based on a concrete strength of 45 MPa at room temperature ( $R_{i, 0}^{c r}$ ). This assumed value is then utilized in the analytical method.

The numerical results suggest that the critical cracking radius $R_{i, T}^{c r}$ values exhibit an increase with the steel bar radius $R_{b}$ and the ratio of $R_{b} / R_{c}$ . An empirical equation has been formulated using data fitting to describe the relationship between the ratio of $R_{i, T}^{c r}$ to $R_{c}$ (denoted as $q$ ):

q = \frac{R_{i, T}^{c r}}{R_{c}} = 0.735 + 1.21 \times 10^{- 3} R_{b} - 8.10 \times 10^{- 4} R_{c}

(37)

To validate the precision of the aforementioned design equation, within the framework of the particular scenario where

R_{b}

= 8 mm,

R_{c}

= 56 mm, and

f_{c m}

= 45 MPa, the detailed values of the critical cracking radius

R_{i, T}^{c r}

derived from the analytical method and the simplified expressions are outlined in Table 1. The proposed equation has proven its efficacy in predicting the

R_{i, T}^{c r}

values, with a maximum deviation of less than 10% noted at 800°C.

Table 1.

Comparisons of analytical method and simplified expressions.

$T$ (^oC)	$R_{i, T}^{c r}$ (mm)			$τ_{T, \max}$ (MPa)			$s_{1, T}$ (mm)
$T$ (^oC)	Analytical	Simplified	$\| e \|$ (%)	Analytical	Simplified	$\| e \|$ (%)	Analytical	Simplified	$\| e \|$ (%)
20	38.9	39.2	0.77	16.2	16.2	0.00	0.60	0.60	0.00
100	39.2	39.2	0.00	14.9	14.7	1.34	0.58	0.58	0.00
200	39.3	39.2	0.25	13.1	13.0	0.76	0.65	0.65	0.00
300	39.1	39.2	0.26	11.2	11.2	0.00	0.80	0.80	0.00
400	39.0	39.2	0.51	9.3	9.4	1.08	1.01	1.00	0.99
500	39.1	39.2	0.26	7.5	7.6	1.33	1.20	1.20	0.00
600	39.6	39.2	1.01	5.8	5.8	0.00	1.33	1.30	2.26
700	40.5	39.2	3.21	4.0	4.0	0.00	1.28	1.23	3.91
800	42.6	39.2	7.98	2.2	2.3	4.54	0.95	0.86	9.47

The maximum residual radial pressure ( $p_{c, T}^{\max}$ ) can be calculated by substituting the critical cracking radius $R_{i, T}^{c r}$ into equation (22), resulting in:

p_{c, T}^{\max} = f_{t, T} [\frac{R_{i, T}^{c r}}{R_{b}} \cdot \frac{R_{c}^{2} - {(R_{i, T}^{c r})}^{2}}{R_{c}^{2} + {(R_{i, T}^{c r})}^{2}} + (\frac{R_{i, T}^{c r}}{R_{b}} - 1) - \frac{{(C \cdot R_{b})}^{\frac{1}{k - 1}} (k - 1)}{k} {(\frac{R_{i, T}^{c r}}{R_{b}} - 1)}^{\frac{k}{k - 1}}]

(38)

Equation (38) can be reformulated as the sum of $p_{1}$ , $p_{2}$ and $p_{3}$ , where $p_{1}$ , $p_{2}$ and $p_{3}$ are expressed as follows:

{\begin{cases} p_{1} = f_{t, T} \frac{R_{i, T}^{c r}}{R_{b}} \cdot \frac{R_{c}^{2} - {(R_{i, T}^{c r})}^{2}}{R_{c}^{2} + {(R_{i, T}^{c r})}^{2}} \\ p_{2} = f_{t, T} (\frac{R_{i, T}^{c r}}{R_{b}} - 1) \\ p_{3} = - f_{t, T} \frac{{(C \cdot R_{b})}^{\frac{1}{k - 1}} (k - 1)}{k} {(\frac{R_{i, T}^{c r}}{R_{b}} - 1)}^{\frac{k}{k - 1}} \end{cases}

(39)

By substituting equation (37) into equation (39), the exact values of $p_{1}$ and $p_{2}$ can be obtained. A more simplified expression can be derived using data fitting techniques:

p_{1} + p_{2} = f_{t, 0} (1.01 - 1.09 \times 10^{- 3} T) (0.912 \frac{R_{c}}{R_{b}} - 0.827)

(40)

The expression of $p_{3}$ in equation (39) is associated with the radius of the steel bar, the ratio of $R_{c} / R_{b}$ , and the tensile strength and elastic modulus of concrete after exposure to high temperatures. By substituting equation (23) into equation (39), $p_{3}$ can be expressed as:

p_{3} = - f_{t, T} {(\frac{2 π ε_{c t, T}}{n ω_{u, T}})}^{\frac{1}{k - 1}} [\frac{{R_{b}}^{\frac{1}{k - 1}} (k - 1)}{k} {(\frac{R_{i, T}^{c r}}{R_{b}} - 1)}^{\frac{k}{k - 1}}]

(41)

Hence, the value of $p_{3}$ is associated with the factors of $X_{1}$ and $X_{2}$ :

{\begin{cases} X_{1} = {(\frac{2 π ε_{c t, T}}{n ω_{u, T}})}^{\frac{1}{k - 1}} \\ X_{2} = \frac{{R_{b}}^{\frac{1}{k - 1}} (k - 1)}{k} {(\frac{R_{i, T}^{c r}}{R_{b}} - 1)}^{\frac{k}{k - 1}} \end{cases}

(42)

It is evident that the value of $X_{1}$ is significantly influenced by the concrete strength and the exposure temperature, whereas the value of $X_{2}$ is correlated with the steel bar radius and the ratio of $R_{c} / R_{b}$ . Through the application of data fitting techniques, an equation is formulated to elucidate the relationship between $X_{1}$ values with the concrete tensile strength $f_{t, 0}$ and the exposure temperature $T$ :

X_{1} = 0.167 f_{t, 0}^{0.280} (0.997 - 1.72 \times 10^{- 4} e^{9.28 \times 10^{- 3} T})

(43)

By considering the case of $k = 6$ (i.e., $d_{a} = 24$ mm) and utilizing data fitting techniques, the following simplified expression for $X_{2}$ can be derived:

X_{2} = 0.761 R_{b}^{0.123} (1.30 \frac{R_{c}}{R_{b}} - 2.53)

(44)

By substituting equations (43) and (44) into equations (41) and (42), $p_{3}$ can be calculated by the following formula:

p_{3} = - f_{t, T} \cdot 0.761 R_{b}^{0.123} (1.30 \frac{R_{c}}{R_{b}} - 2.53) \cdot 0.167 f_{t, 0}^{0.280} (0.997 - 1.72 \times 10^{- 4} e^{9.28 \times 10^{- 3} T})

(45)

Nevertheless, the aforementioned simplified expression of $p_{3}$ is applicable solely when $k$ = 6, representing the maximum aggregate size in concrete ( $d_{a}$ ) as 24 mm. Consequently, numerical analysis is carried out to ascertain the values of $p_{3}$ for varying $d_{a}$ , and data fitting is executed to derive the correction coefficient $K_{d}$ for $k$ in the subsequent manner:

K_{d} = 0.203 k^{0.890}

(46)

By substituting equations (40), (45), and (46) into equations (38) and (39), a more simplified formula of the maximum radial pressure $p_{c, T}^{\max}$ can be expressed as:

p_{c, T}^{\max} = (1.01 - 1.09 \times 10^{- 3} T) f_{t, 0} (0.912 \frac{R_{c}}{R_{b}} - 0.827 - k^{0.890} \cdot f_{t, 0}^{0.280} \cdot \frac{0.0337 R_{c} - 0.0654 R_{b}}{R_{b}^{0.877}})

(47)

Consequently, the bond strength $τ_{T, \max}$ can be determined by substituting equation (47) into equation (28) or equation (25) for the steel bar, with or without the presence of steel stirrups for confinement.

Simplified expression of local interfacial slip

As previously stated, the critical cracking radius $R_{i, T}^{c r}$ is regarded as being unaffected by the exposure temperature $T$ and the compressive strength $f_{c m}$ of the concrete, i.e., $R_{i, T}^{c r}$ = $R_{i, 0}^{c r}$ . Consequently, equation (31) can be rephrased as follows to determine the value of $s_{1, T}$ :

s_{1, T} = \frac{R_{i, T}^{c r} \cdot ε_{c t, T} \cdot s_{1}}{R_{i, 0}^{c r} \cdot ε_{c t, 0}} = s_{1} \frac{f_{t, T} E_{0}}{f_{t, 0} E_{T}} = s_{1} (1.01 - 1.11 \times 10^{- 3} T) [1 + 18.5 \times {(\frac{T}{1000})}^{2.47}]

(48)

In addition, the precise values of the local interfacial slip parameters $s_{2, T}$ and $s_{3, T}$ can be ascertained by employing the definitions described in the above sections.

Validation of the simplified expressions

Validation using analytical method

A parametric study was conducted to assess the accuracy of the simplified expressions using the proposed analytical method to determine the bond strength $τ_{T, \max}$ and the characteristic slip $s_{1, T}$ . Various design parameters were taken into account in this research, such as variations in the ratio of $R_{c} / R_{b}$ , concrete compressive strength $f_{c m}$ , and exposure temperature $T$ .

In the specific case where $R_{b}$ = 8 mm, $R_{c}$ = 56 mm and $f_{c m}$ = 45 MPa, Table 1 presents a comparative evaluation between the analytical results of $R_{i, T}^{c r}, τ_{T, \max} and s_{1, T}$ and the corresponding estimations obtained from the simplified expressions. This comparative assessment is conducted under the assumption that splitting failure is the governing criterion in the comparison dataset.

The disparity between the two predictions is assessed using a relative deficit ratio, as defined in equation (49):

| e | = | \frac{V_{s} - V_{a}}{V_{a}} | \times 100 %

(49)

where

V_{s}

represents the value determined by the simplified expressions, and

V_{a}

denotes the value obtained by the analytical method.

After being subjected to temperatures exceeding 600°C, the discrepancies between the critical radius $R_{i, T}^{c r}$ and characteristic slip $s_{1, T}$ as predicted by the simplified expressions and the analytical method become more pronounced compared to those observed in the specimens tested at temperatures below 500°C. However, the relative deviation in $s_{1, T}$ remains below 10%, a threshold deemed acceptable for simplified design assessments. These disparities in the predictive results of $τ_{T, \max}$ and $s_{1, T}$ from the two methods have validated the reliability of the proposed simplified expressions.

Validation using existing pullout tests

To further validate the simplified expressions, pullout test results on the bond strengths of steel bars in concrete after exposure to various high temperatures were collected from the existing literature. Table 2 presents a summary of the collected test data. Each specimen in the table is identified by a name selected from the original pullout tests,

T_{\max}

is the maximum exposure temperature, and

l_{b}

is the bond length of the steel bar. The pullout tests involve standard concrete cylinders or cubes to assess the concrete compressive strength at room temperature. Once cubic concrete specimens are tested, the concrete compressive strength is determined using the design equation specified in the current design guideline of concrete structures GB 50010-2010 (2015), where

f_{c m}

= 0.6688

f_{c u, k}

+8 (

f_{c u, k}

represents the average cubic compressive strength of concrete). Additionally, the failure modes of all the specimens are provided based on observations from the original studies.

Table 2.

Details of the existing pullout and beam-end tests.

Ref.	Specimen name	$T_{\max}$ (^oC)	$R_{c}$ (mm)	$R_{b}$ (mm)	$d_{a}$ (mm)	$l_{b}$ (mm)	$f_{c m}$ / $f_{c u, k}$ (MPa)	$d_{s v} / S_{s v}$	Failure mode^c
Deshpande et al. (2020)	Con-35	800	76	8	19	48	34^a	—	SP, PO
	Con-60	800	76	8	19	48	57^a	—	SP, PO
Hlavička (2017)	M0	800	60	6	16	40	61.4^b	—	/
Zhang et al. (2019)	C0	800	100	10	25	40	47.6^b	6/50	PO
Li et al. (2023)	F0	800	75	7	20	70	53.5^b	—	SP, PO
Liu et al. (2023)	P30-18	800	100	9	20	90	34.9^b	—	PO
	P50-18	800	100	9	20	90	53.4^b	—	SP, PO
Yang et al. (2018)	C00	800	100	10	15	40	47.6^b	6/50	PO
Yin et al. (2011)	C25	700	75	8	20	80	27.5^b	—	PO
	C40	700	75	8	20	80	43.5^b	—	PO
Asghari Ghajari and Yousefpour (2023)	A-22-M	700	61	11	19	200	30.0^a	10/100	PO
Sureshbabu and Mathew (2021)	OPCC	800	50	4	/	50	17.9^b	—	/
Rashid et al. (2019)	/	500	50	5	19	140	30.69^a	—	SP
Yuan et al. (2006)	C0-6	650	75	8	25	100	35^b	6/50	PO
Yang et al. (2021)	ZL	500	75	10	31.5	100	50.6^b	—	SP, PO

^aCylinder compressive strength of concrete.

^bCubic compressive strength of concrete.

^cPO: pullout failure, SP: splitting failure.

^d—: the specimen was tested without steel stirrups.

^e/: the information was not reported in the original study.

Figure 9 presents a comprehensive comparison of the bond strengths estimated through the analytical method, the simplified expressions and the results from pullout tests conducted by various researchers (Asghari Ghajari and Yousefpour, 2023; Deshpande et al., 2020; Hlavička, 2017; Li et al., 2023; Rashid et al., 2019; Yang et al., 2018; Yuan et al., 2006; Zhang et al., 2019). The left column of the figures shows specific bond strength values, while the right column depicts the normalized bond strengths based on room temperature values to facilitate comparison. In Figure 9(h), the bond strengths from the specimens subjected to air-cooling and water-cooling methods are compared with the predictions. In Figure 9(i), a comparison between the bond strengths tested by Rashid et al. (2019) and the predictions has indicated that all the experimental results are lower than the predicted values. However, both the experimental and analytical results exhibit similar trends with increasing exposure temperature. As shown in Figure 9, the residual bond strengths between steel bars and concrete decrease to approximately 20% of the room temperature value when the exposure temperature reaches 800°C, highlighting the significance of considering bond degradation of steel bars after exposure to high temperatures. The good agreement among the analytical results, the predictions from the simplified expressions and the experimental data has validated the use of the proposed simplified expressions for predicting the bond strengths of steel bars following exposure to high temperatures.

Figure 9.

Comparisons among the bond strengths predicted by the analytical method and the simplified expressions as well as the experimental results.

Figure 10 illustrates a comparison between the residual bond strengths predicted by the simplified expressions and the experimental results obtained from 85 pullout tests, as detailed in Table 2. Additioal information regarding these experiments and the variables under study can be found in Table 2. The figure shows a close grouping of data points around the identity line $y$ = $x$ , indicating that the simplified expressions proposed are capable of accurately predicting the residual bond strengths between steel bars and concrete after exposure to high temperatures.

Figure 10.

Predicted residual bond strengths versus the experimental data.

Validation using finite element model

In order to evaluate the accuracy of the proposed simplified expressions in predicting the load-displacement behavior of the pullout specimens after exposure to high temperatures, a finite element (FE) model is implemented in the ABAQUS software. In this FE model, both steel bars and concrete are modeled as isotropic linear elastic materials to improve computational efficiency. Consequently, the tension-softening properties of concrete are not accounted for. In other words, it is assumed that the load-displacement curves of the tested specimens are primarily influenced by the interfacial local bond-slip behavior at the steel bar-to-concrete interface. This modeling approach has been utilized in prior research to simulate the load-displacement response of the bonded joints involving dissimilar materials, such as steel bars and concrete (Gao et al., 2019; Wang et al., 2022b) or fiber-reinforced polymer laminates and concrete (Gao et al., 2012, 2015; Guo et al., 2021; Jia et al., 2021; Lv et al., 2024; Zhou et al., 2022). The variations in the strength and stiffness of concrete post exposure to high temperatures are detailed in Section 4.1. The concrete cover is represented using solid three-dimensional (3D) elements (C3D8R), while the steel bar is modeled using two-node link elements (T3D2). The interaction between the steel bars and the surrounding concrete is simulated using user-defined nonlinear spring elements (as illustrated in Figure 11) to assess the validity of the local bond-slip relationship defined by the simplified expressions. Consequently, the local bond stress-interfacial slip relationship of the nonlinear spring elements in the longitudinal direction is established based on the bond-slip model defined by the simplified expressions, facilitating the computation of the force sustained by each spring element using the following equation:

F_{e} = π d l_{e} τ_{T}

(50)

where

F_{e}

represents the force sustained by the spring element in the longitudinal direction, and

l_{e}

is the length of each spring element. In the perpendicular directions, linear spring elements with an elastic stiffness ten times greater than the elastic modulus of the concrete are utilized. As depicted in Figure 11, the mesh size in the bond area involves an element length of 4 mm to accurately capture the local bond-slip behavior. Conversely, other regions have coarser mesh sizes with lengths of 10 mm to improve computational efficiency. Throughout the pullout loading process, a displacement-controlled loading method is applied at the end of the steel bar, commonly known as the loaded end.

Figure 11.

The FE model incorporating nonlinear spring elements.

Figure 12 illustrates a comparison between the load-displacement curves generated by the FE model and the measured curves obtained from pullout tests on the specimens exposed to high temperatures, as reported in previous studies (Asghari Ghajari and Yousefpour, 2023; Rashid et al., 2019; Yuan et al., 2006; Zhang et al., 2019). The applied loads in all cases are normalized by the maximum force recorded during the pullout tests to facilitate a clear comparison. In the FE model predictions, the tensile force ( $F_{T}$ ) is calculated as the sum of the bond stresses distributed longitudinally in the bond area, while the displacement $Δ$ represents the relative displacement between the steel bar and the surrounding concrete at the loaded end. Figure 12(c) shows the load-displacement curves of the specimens treated with air-cooling and water-cooling methods for comparison with the FE model predictions. Figure 12(d) compares the load-displacement curves obtained by Rashid et al. (2019) with those predicted by the FE model, and the results of this test series indicate a splitting failure mode occurring both at room temperature and after exposure to high temperatures. The loading point was positioned 550 mm away from the embedded region, and the load-displacement curves were generated by monitoring the load-time and displacement-time curves at the loading point. Since the pullout tests only provided the load-displacement curves for the ascending segment, Figure 12(d) only presents the comparison of the load-displacement curves up to the peak load of each curve. It is noteworthy that the measured displacements corresponding to the peak loads in each specimen are larger than 2 mm, which are much greater than those observed in Figures 12(a)–12(c). This difference is primarily attributed to the fact that the measured displacement in Figure 12(d) includes the elastic deformation of the steel bar in the unboned area. Overall, the comparisons shown in Figures 12(a)–(d) illustrate that the proposed local bond-slip model, derived from the simplified expressions, has demonstrated excellent accuracy in predicting the load-displacement curves of steel bar specimens embedded in concrete after exposure to high temperatures.

Figure 12.

Comparisons of the predicted load-displacement curves with the experimental results.

Conclusions

This paper presents an analytical method for predicting the bond behavior and failure modes of the steel bar-to-concrete interface after exposure to high temperatures. The analytical method conceptualizes the concrete cover in a pullout specimen as a thick-walled cylinder under uniform pressure on its inner surface. During the pullout loading process, the concrete cover is segmented into two portions: the cracked inner portion and the uncracked outer portion. The analytical method adequately addresses the changes in concrete material properties after exposure to high temperatures, including the tension-softening behavior of the cracked internal portion. It also accounts for the confinement effect of steel stirrups on the bond strength. Through the analytical method and numerical parametric investigations, simplified expressions are developed to estimate the maximum radial stresses, residual bond strength and interfacial slip parameters that define the local bond-slip relationship at the steel bar-to-concrete interface after exposure to high temperatures. These simplified expressions effectively consider the effects of various design parameters, including exposure temperature, steel bar diameter, concrete cover thickness, material strengths and stiffnesses, the maximum aggregate size of the concrete and the presence of stirrups.

The accuracy and reliability of the proposed simplified expressions for predicting the residual bond strengths of steel bars in concrete after exposure to high temperatures have been validated through the comparisons of their predictions with the analytical parametric results and the test results collected from existing literature sources. Furthermore, the local bond-slip relationship derived from the simplified expressions has been evaluated by comparing the load-displacement curves reported in previous experimental studies with those predicted by a finite element (FE) model that incorporates the proposed local bond-slip model. The excellent agreement between the predicted and observed load-displacement curves has demonstrated the robustness and accuracy of these simplified expressions for assessing post-fire bond-slip behavior at the steel-to-concrete interface after exposure to high temperatures.

The proposed simplified expressions and the corresponding local bond-slip model offer a practical tool specifically tailored for deformed steel bars in normal-weight concrete after exposure to high temperatures. The analytical method can also be adapted to different concrete types, including high-strength concrete, recycled aggregate concrete and lightweight aggregate concrete, by modifying properties to suit each material. However, this method may not be applicable to engineered cementitious composites (ECC) and ultra-high-performance concrete (UHPC), as their deformation characteristics are significantly different from conventional concrete, potentially leading to inaccurate predictions of interface slip parameters and bond strength. Overall, the proposed local bond-slip model aims to establish a foundational constitutive law for evaluating the residual bond performance of steel bars in concrete after high-temperature exposure and for estimating the post-fire load-carrying capacity of RC members.

Footnotes

Acknowledgments

The authors wish to acknowledge the financial support received from the National Natural Science Foundation of China (NSFC) (No. 51978398) and the Natural Science Foundation of Shanghai (No. 23ZR1429200). Furthermore, the authors are grateful for the funding provided by the Open Foundation of Yunnan Key Laboratory of Building Structure and New Materials (No. 2023-JKKF-01).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: this work was supported by the National Natural Science Foundation of China (NSFC) (No. 51978398) and the Natural Science Foundation of Shanghai (No. 23ZR1429200). Furthermore, the authors are grateful for the funding provided by the Open Foundation of Yunnan Key Laboratory of Building Structure and New Materials (No. 2023-JKKF-01).

ORCID iD

Wan-Yang Gao

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