Bond–Slip Model of Corroded Strand Considering Rotation Effect

Abstract

Corrosion leads to the cracking of concrete cover and degrades the bond strength of strand. This chapter theoretically analyzed the bond strength model of corroded strands by considering the rotation characteristics. First, an analytical model is developed to predict the bond strength of uncorroded strands by considering strand rotation. Then, the effect of corrosion-induced concrete cracking on the ultimate bond strength of corroded strand is analyzed theoretically. Finally, the theoretical model of bond strength of corroded strands is obtained.

5.1 Introduction

The bond performance between prestressed steel strands and concrete is very important for PC structures. For convenience of calculation, the bond between steel strands and concrete can be idealized as the shear stress along the bond interface [23]. The strands are subject to corrosion due to the penetration of chloride ions in an aggressive environment. Damage to the contact surface and corrosion-induced concrete cracking destroys the strand bond.

Many researches have been undertaken to investigate the bond behavior of corroded steels in RC structures, which can be concluded as: experimental research, theoretical analysis, and numerical simulation. Most of the existing research on the bond strength under the influence of corrosion is carried out based on experiments [6, 31, 33]. And most experiments are based on the central pull-out test. Existing tests show that the bond strength of steel strands increases at the beginning of corrosion, decreasing beyond a certain critical corrosion state, and finally reaches a steady state. Based on the existing test results [38, 40] have established empirical formulas to evaluate the degradation of the bond strength of corroded strands. However, these test prediction results are highly dependent on test conditions and have limited application in practical engineering.

Some scholars experimentally explored the bond–slip behavior between corroded rebars and concrete with stirrup constraints. Almusallam et al. [3] studied the bond strength and failure modes of corroded RC beams with the pull-out tests. The effects of concrete cover thickness, steel diameter, concrete strength, type of stirrups, and crack widths on bond performance of corroded steels have been discussed. Al-Hammoud et al. [2] compared the difference of bond properties of corroded RC beams under static and dynamic loads. Choi et al. [13] discussed the difference of the bond properties of concrete structures under natural corrosion and accelerated corrosion. The accelerated corrosion method may underestimate the effect of corrosion on the bond degradation. Additionally, some scholars had also discussed the degradation of bond behavior of corroded RC beams with common and recycled aggregates.

Some analytical models have also been proposed to predict the bond performance of corroded RC beams. Wang et al. [38] pointed out that the bond strength between corroded steels strands and concrete was affected by many factors, such as the corrosion degree, bar type, the adjacent spacing, and the number of stirrups. Among these factors, the corrosion degree is most significant to affect the bond strength. Some scholars proposed some models to predict the bond strength of corroded steel strands based on the thick-walled theory, which considering the relationship between corrosion depth of steel and the radial displacement of concrete. Chen et al. [12] proposed a model to predict the bond strength of corroded RC beams incorporating the softening behavior of cracked concrete. Abrishami et al. [1] established a computational model for the peak bond stress in the transfer and anchorage regions of pretensioned members.

Some scholars have employed the finite element software to numerically simulate the bond behavior of corroded RC beams. Lee et al. [23] used the bond plane unit and bond interface unit to model the bond behavior of corroded RC beams. Some researchers simulated the bond properties of corroded RC beams with two numerical methods, i.e., the friction and damage type method. Bolmsvik and Lundgren [9] discussed the influences of corrosion on the bond properties of ribbed and plain bars, respectively. Some scholars used the Lattice approaches to simulate the bond properties of corroded RC beams, which can reasonably model the influence of filling extent of corrosion products in cracks. These study are all based on deformed steel bars, and there are very few study on prestressed steel strands.

Based on the above research background, this chapter theoretically deduces and analyzes the bond strength model of corroded strands by considering the rotation characteristics caused by the twisting structure of steel strands. First, an analytical model is developed to predict the bond strength of uncorroded strands by considering strand rotation. Then, the influence of corrosion-induced concrete cracking on the ultimate bond strength of corroded strand is analyzed theoretically. Finally, the theoretical model of bond strength of corroded strands was acquired.

5.2 Bond Strength of Strand Considering Rotation Effect

The seven-wire prestressed steel strand is often used in bridges. This seven-wire steel strand is made by twisting and rotating the six outer steel wires around the central steel wire. Due to the twisted structure, the steel strand is often accompanied by concrete splitting or strand rotation when it reaches the ultimate bond strength. For the pull-out beams with seven-wire steel strands, two typical bond failure modes can be identified: concrete splitting failure and pull-out failure. The confinement of concrete on the strand determines these two failure modes, leading to different bond strengths case. Concrete splitting failure can be explained as the bond failure when the surrounding concrete completely splits, as seen in Fig. 5.1a. Concrete splitting failure occurs when steel strand is not sufficiently constrained by concrete cover. On the contrary, the type of pull-out failure occurs under a well-constrained condition. When the constraints are sufficient, due to the twisted structure of the steel strands, the steel strands have rotational slip before the maximum confinement stress of the concrete is reached, and the concrete is not completely split at this time, as shown in Fig. 5.1b. Various parameters determine the failure mode, including surface condition of strand, thickness of concrete cover, and tensile strength of concrete. Besides these parameters, the aggregate size may also play a role in determining the failure mode. The aggregate size has been shown to influence the interlock of ribbed bars.

Fig. 5.1

Bond failure modes: a concrete splitting failure; b pull-out failure

In general, adequate concrete cover and additional reinforcements were designed around the steel strands to prevent splitting failure of the PC structures. This leads to the most common type of pull-out failure. However, the type of pull-out failure of steel strand differs from that of the deformed bars. When the pull-out failure occurs, the strands are not pulled out directly from concrete block. The steel strand tends to be pulled out by a helical movement around the tunnel formed by the concrete. The bond mechanism of strand determined the rotation behavior. This part theoretically analyzes the ultimate bond strength in the case of pull-out failure of the helical strand.

5.2.1 Theoretical Expressions for Bond Strength

In the part, a prediction model of bond strength is proposed according to the pull-out failure mode with strand rotation. In general, the bond stress between steel reinforcement and concrete can be regarded as a uniform shear stress along the bond interface. For a single seven-wire steel strand, the average bond stress $\tau_{b}$ is written as:

$$\tau_{b} = {{F_{p} }}\left/{{\left( {i \cdot \pi \cdot d \cdot l_{t} } \right)}}\right.,$$

(5.1)

where, $F_{p}$, $l_{t}$, and $d$ are the tensile force, the bond length, and the nominal diameter of the strand, respectively; $i$ is the perimeter factor, and its value can be taken as 4/3 for the seven-wire strand [23].

Steel strand consists of six outer helical wires wound on a straight wire in the center. Due to the helical shape of the outer wires, the projections of any cross-section of steel strand in its longitudinal direction do not entirely overlap within a lay length. Lay length is calculated as the distance for the outer wires turn around the center wire. It is supposed that the steel strand is longitudinally divided into several equal segments. The height of each segment is $dz$. A schematic diagram of a strand segment is shown in Fig. 5.2.

Fig. 5.2

Schematic diagram of a strand segment with the height of $dz$

As can be seen in Fig. 5.2, the longitudinal extension of the length $dz$ is related to a rotation of dα angle on the projection plane of strand. It is supposed that the six outer wires wrap uniformly around the straight wire in the center. The relationship between da and $dz$ is written as:

$$\frac{d\alpha }{{2\pi }} = \frac{dz}{{l_{\text{lay}} }},$$

(5.2)

where $l_{\text{lay}}$ is the lay length of steel strand. According to the ASTM A416 standard, $l_{\text{lay}}$ can be approximately considered as 14d.

Additionally, the rotation of outer wires along the core wire produces an inclined angle δ (see Fig. 5.2), which can be written as:

$$\tan \delta = \frac{{\left( {r_{c} + r_{e} } \right) \cdot d\alpha }}{dz}.$$

(5.3)

For strands with exact diameter and lay length, $\delta$ can be computed by combining Eqs. (5.2) and (5.3). Therefore, the lay length is $14d$, $\delta$ can be determined to be 8.7° for different diameters of strands.

The mechanical interlock is produced on the helical ribs because of the inclined planes corresponding to the longitudinal direction. One segment projection along the longitudinal direction of the steel strand is shown in Fig. 5.3a.

Fig. 5.3

a Projection of one segment along the longitudinal direction of the steel strand; b effective bearing face on the rib of an outer wire

The shaded part in Fig. 5.3a is the stress area where the rib of steel strand provides the interlocking force. The shaded part can be approximately regarded as six incomplete crescent shapes. It is assumed that the outer wire of steel strand has the same force. One of the outer wires with the crescent shape was taken for analysis, as shown in Fig. 5.3b. This simplified crescent-shaped rib is similar to the rib of the deformed bar. Due to the helical feature of the outer wires, the bearing faces are helical along the strand length, resulting in complex boundary conditions. It is assumed that the ribs providing mechanical interlocking are generated only by tangential movement of the outer wire and not by rotation of the central wire. Therefore, the helical surface can be simplified as a flat surface. The crescent-shaped rib is comparable to simulate the rib of deformed bars. The force analysis on the ribs of deformed bars is related to the force analysis of strand [11]. For a central embedded strand, it is assumed that the boundary conditions of the six outer wires are the same at a certain cross-section. Therefore, a steel wire is arbitrarily chosen for mechanical analysis. It is worth mentioned that the steel strand is considered as a whole, so the effect of the force between the steel wires is ignored.

Consider the forces acting on a rib section subtended by an angle $d\theta$ (see Fig. 5.3b). The corresponding area $dA$ on the rib face is written as:

$$dA = h_{w} /\sin \delta \cdot r_{e} \cdot d\theta ,$$

(5.4)

where $h_{w}$ is the rib height; θ is the effective coverage angle of the incomplete crescent, the value range of θ is [−π/6, π/2].

Figure 5.4a shows a force analysis of the mechanical interlock force on a single rib. It is assumed that pull-out failure happens when the shear stress on the bond interface exceeds the adhesion capacity and friction between the helical ribs and concrete. The mechanical interlocking forces acting on rib surface can be decomposed into $dF_{g}$ inclined at an angle $\emptyset$ to the rib surface and $dF_{s}$ parallel to the rib surface. $dF_{s}$ and $dF_{g}$ can be written as:

$$dF_{s} = \sigma_{c} \cdot dA,$$

(5.5)

$$dF_{g} = \sigma_{n} \cdot dA/\cos \phi ,$$

(5.6)

where $\sigma_{c}$, is unit cohesion between bearing face of rib and concrete, which can be estimated as 0.11f_c [11], $\phi$ and $\sigma_{n}$ are the friction angle between strand and concrete and the normal stress on the shear failure plane, respectively.

The forces $dF_{g}$ and $dF_{s}$ can be separated into forces $dF_{\text{pu}}$ and $dF_{\text{bu}}$. $dF_{\text{pu}}$ is parallel to the axal direction of steel strand and $dF_{\text{bu}}$ is perpendicular to the axial direction of steel strand.

$$dF_{\text{pu}} = \left( {\sigma_{n} \cdot \frac{dA}{{\cos \phi }}} \right) \cdot \sin \left( {\delta + \phi } \right) + \sigma_{c} \cdot dA \cdot \cos \delta ,$$

(5.7)

$$dF_{\text{bu}} = \left( {\sigma_{n} \cdot \frac{dA}{{\cos \phi }}} \right) \cdot \cos \left( {\delta + \phi } \right) - \sigma_{c} \cdot dA \cdot \sin \delta .$$

(5.8)

Integration of Eq. (5.7) along the rib bearing face results in the contribution to the total resisting force $F_{p}$

$$F_{p} = 6\mathop \int \limits_{ - \pi /6}^{\pi /2} dF_{\text{pu}} d\theta = 6\left[ {\frac{{\sigma_{n} }}{\cos \phi }\frac{{\sin \left( {\delta + \phi } \right)}}{\sin \delta } + \sigma_{c} \cdot \cot \delta } \right] \cdot A_{r} ,$$

(5.9)

where $d_{e}$ is the diameter of outer wire; $A_{r}$ is the area of one incomplete crescent shape, which can be written as:

$$A_{r} = \frac{2}{3}\left\{ {\pi \frac{{d_{e}^{2} }}{4} - \pi \frac{{d_{e} }}{2}\left[ {\frac{{d_{e} }}{2} - \left( {\frac{{d_{c} + d_{e} }}{2}} \right)d\alpha } \right]} \right\},$$

(5.10)

where $d_{c}$ is the diameter of the central wire.

Ignoring the difference in diameter between the center wire and the outer wires, by combining Eqs. (5.9) and (5.10) result in $F_{p}$ within the length of $dz$.

$$F_{p} = 2\left[ {\frac{{\sigma_{n} }}{\cos \phi } \cdot \frac{{\sin \left( {\delta + \phi } \right)}}{\sin \delta } + \sigma_{a} \cot \delta } \right]\pi d_{e}^{2} d\alpha .$$

(5.11)

Substitution of $F_{p}$ from Eq. (5.11) into Eq. (5.1) results in the bond stress:

$$\tau_{b} = \frac{3}{14}\pi \left( {\frac{{d_{e} }}{d}} \right)^{2} \left[ {\frac{{\sigma_{n} }}{\cos \phi } \cdot \frac{{\sin \left( {\delta + \phi } \right)}}{\sin \delta } + \sigma_{c} \cdot \cot \delta } \right].$$

(5.12)

From Eq. (5.12), it can be seen that $\tau_{b}$ is relates to parameters, for instance, the ratio of outer wire diameter to nominal diameter of strand $d_{e} /d$, friction angle, $\emptyset$, angle of inclination of strand ribs, $\delta$, unit bond between concrete and steel strand, $\sigma_{c}$, and normal stress at the shear plane, $\sigma_{n}$. The following part explains to calculate normal confining stress, $\sigma_{n}$, by considering the strand rotation.

For pull-out failure of beams accompanied with the rotation of steel strand, by integrating of the radial component of the mechanical interlock, $dF_{\text{bu}}$ produces a bursting force, $F_{\text{rib}}$, around the outer wires (see Fig. 5.4b). Therefore, the torque, $M_{\text{rib}}$, is applied on the steel strand because of the helical shape of outer wires. When $M_{\text{rib}}$ applied to the strand overcomes the resistance provided by friction, strand begins to rotate in a helical way. Once the strand rotates, both bond strength and confining stress of steel strand reach the peak values, respectively. Therefore, the normal stress, $\sigma_{n}$, corresponding to pull-out failure of the beams accompanied by the strand rotation. $\sigma_{n}$ is considered to be a critical confining stress, $\sigma_{n,\text{crit}}$, which can be acquired on the basis of the moment balance about the strand’s center.

Fig. 5.4

a Force analysis on a single rib; b torque generated by bursting force

First, $F_{\text{rib}}$ from the outer wire needs to be calculated. And $F_{\text{rib}}$ is consisted of a vertical force, $F_{\text{rib}v}$, and a horizontal force, $F_{\text{rib}h}$, which are written as:

$$F_{\text{rib}v} = \mathop \int \limits_{{ - \frac{\pi }{6}}}^{{\frac{\pi }{2}}} dF_{\text{bu}} \cdot \cos \theta \cdot d\theta = \frac{3}{4}\left[ {\frac{{\sigma_{n} \cos \left( {\delta + \phi } \right)}}{\cos \phi \sin \delta } - \sigma_{c} } \right]h_{w} d_{e} ,$$

(5.13)

$$F_{\text{rib}h} = \mathop \int \limits_{{ - \frac{\pi }{6}}}^{{\frac{\pi }{2}}} dF_{\text{bu}} \cdot \sin \theta \cdot d\theta = \frac{\sqrt 3 }{4}\left[ {\frac{{\sigma_{n} \cos \left( {\delta + \phi } \right)}}{\cos \phi \sin \delta } - \sigma_{c} } \right]h_{w} d_{e} .$$

(5.14)

Therefore, $F_{\text{rib}}$, for a single outer wire is written as:

$$F_{\text{rib}} = \sqrt {F_{\text{rib}v}^{2} + F_{\text{rib}h}^{2} } = \frac{\sqrt 3 }{2}\left[ {\frac{{\sigma_{n} \cos \left( {\delta + \phi } \right)}}{\cos \phi \sin \delta } - \sigma_{c} } \right]h_{w} d_{e} .$$

(5.15)

The torque, $M_{\text{rib}}$, caused by the six outer ribs within the length of $dz$ is written in terms of $\sigma_{n}$ as:

$$M_{\text{rib}} = \frac{9}{2}\left[ {\frac{{\sigma_{n} \cos \left( {\delta + \phi } \right)}}{\cos \phi \sin \delta } - \sigma_{c} } \right]d_{e}^{3} d\alpha .$$

(5.16)

The torque, $M_{\text{fri}}$, was generated by the friction force around the steel strand. The friction force is considered to be evenly distributed throughout the rib face (see Fig. 5.5). In this instance, the torque, $M_{\text{fri}}$, supplied by the friction force can be written as

$$\frac{{M_{\text{fri}} }}{12} = \mathop \int \limits_{{ - \pi {/6}}}^{{\pi {/2}}} \mu \sigma_{n} dA\frac{{d_{e} }}{2}\left( {\cos \theta } \right)^{2} + \mathop \int \limits_{ - \pi /6}^{\pi /2} \mu \sigma_{n} dA\sin \theta \left( {\frac{{d_{e} + d_{c} }}{2} + \frac{{d_{e} }}{2}\sin \theta } \right),$$

(5.17)

Fig. 5.5

Torque generated by friction force

where $\mu = \tan \emptyset$, and $\mu$ is the friction coefficient.

The diameter of the central wire, d_c, is supposed to be same as the diameter of the outer wires, d_e. The torque $M_{\text{fri}}$ supplied by the force of friction can be written as:

$$M_{\text{fri}} = \left( {3\sqrt 3 + 2\pi } \right)\frac{{\mu \sigma_{n} }}{\sin \delta }d_{e}^{3} d\alpha .$$

(5.18)

The stress $\sigma_{n,\text{crit}}$ in Eq. (5.12) for pull-out failure of the beams accompanied with strand rotation can be acquired by equating Eqs. (5.16) and (5.18).

$$\sigma_{n,\text{crit}} = k \cdot \sigma_{c} ,$$

(5.19)

where $k = 1/\left[ {\frac{{\cos \left( {\delta + \phi } \right)}}{\cos \phi \cdot \sin \delta } - \left( {\frac{2\sqrt 3 }{3} + \frac{4\pi }{9}} \right)\frac{\tan \phi }{{\sin \delta }}} \right]$.

Replacing $\sigma_{n,\text{crit}}$ from Eq. (5.19) into Eq. (5.12) results in the bond stress:

$$\tau_{b} = \frac{3}{14}\pi \left( {\frac{{d_{e} }}{d}} \right)^{2} \left[ {\frac{{k \cdot \sigma_{c} }}{\cos \phi } \cdot \frac{{\sin \left( {\delta + \phi } \right)}}{\sin \delta } + \sigma_{c} \cdot \cot \delta } \right].$$

(5.20)

It is worth noting that $\sigma_{n,\text{crit}}$ in Eq. (5.19) is computed for the situation of pull-out failure accompanied with the rotation of steel strand, which only appears under a well-constrained condition. That is, if surrounding concrete supplied sufficient confinement on the strand, $\sigma_{n,\text{crit}}$ in Eq. (5.19) may be lower than the maximum confining stress. In this instance, the maximum bond stress, $\sigma_{n,\text{crit}}$ should be used instead of the maximum confining stress to avoid overestimating the bond stress.

The compressive stress $\sigma_{n}$ on the concrete is defined as the normal stress on the shear failure plane. $\sigma_{n}$ is almost perpendicular to the longitudinal direction of strand. The generation of $\sigma_{n}$ is closely relates to the helical characteristics of steel strand. Due to the shape of the helical ribs, the strand is forced to move perpendicular to the bond surface, resulting in radial confining stress around the steel strand. Consequently, the radial confining stress is assumed to be equivalent to the normal stress, $\sigma_{n}$.

5.2.2 Model Verification

In this part, some test results related to the pull-out test are collected. There is a total of 63 specimens, including 50 strands with a diameter of 12.7 mm and 13 strands with a diameter of 15.2 mm. The test data was compared with the results of the model analysis. Other details of beams such as the minimum thickness of concrete cover and bond length are shown in Table 5.1. Additionally, in the selected tests, all steel strands were tension-free before the pull-out test, which is consistent with the stress state in Eq. (5.12).

Table 5.1 Details of test beams from previous research and prediction results

Before applying Eq. (5.20) in the calculation of ultimate bond stress for the pull-out failure with strand rotation, the maximum confining stress provided by concrete $\sigma_{n,\max }$, for all the selected tests also needs to be confirmed to avoid overestimation of bond stress. For a center-embedded strand with a certain thickness of concrete cover and concrete compressive strength, $\sigma_{n,\max }$ provided by the surrounding concrete to the steel strands can be considered as a constant value.

The concrete confining model proposed by Den [16] is introduced to determine the maximum confining stress $\sigma_{n,\max }$. The confining pressure model is closely related to the thickness of concrete cover and concrete strength. According to this study, the concrete surrounding steel strands can be regarded as a thick-walled cylinder. Among them, the inner radius of the cylinder is the nominal diameter of steel strand, the outer radius of the cylinder is minimum cover of concrete. The radial confining stress of concrete at different stages, such as uncracked, partially cracked stages and entirely cracked stages, was determined on the basis of the radial displacement at the interface of concrete. In partially cracked stage, when the splitting crack penetrates roughly 70% of the thickness of the cylinder wall, the maximum radial stress $\sigma_{n,\max }$ is reached (Den [16]). Further crack penetration results in rapid reduction of radial stress. The radial confining stress $\sigma_{n}$ at partially cracked stage is written as:

$$\sigma_{n} = \frac{{r_{k} }}{{r_{s} }}f_{\text{ct}} \left( {\frac{{c^{2} - r_{k}^{2} }}{{c^{2} + r_{k}^{2} }}} \right) + f_{\text{ct}} \left[ {\frac{{a \cdot 2\pi \varepsilon_{c} \cdot r_{s} }}{{2n \cdot w_{0} }}\left( {\frac{{r_{k} }}{{r_{s} }} - 1} \right)^{2} + b\left( {\frac{{r_{k} }}{{r_{s} }} - 1} \right)} \right],$$

(5.21)

where $r_{s} = d/2$ is the nominal radius of steel strand; c is the minimum cover of concrete; $r_{k}$ is the radius of crack front, its value is from $r_{s}$ to c; $f_{\text{ct}}$ is the tensile strength of concrete; a and b are constants, and whose values related to the softening behavior of cracked concrete; $\varepsilon_{c} = f_{\text{ct}} /E_{c}$, where $E_{c}$ is the elastic modulus of concrete; n and w₀ are the number of fictitious splitting cracks (n = 3) and the minimum crack width at concrete failure (w₀ = 0.2 mm), respectively [32].

The stress $\sigma_{n}$ in Eq. (5.21) changes as the crack front $r_{k}$ develops. However, $\sigma_{n,\max }$ is a constant value, which relates to a specific thickness of cover and concrete tensile strength. When $r_{k}$ of beams penetrates into about 70% of the cylinder wall thickness, the radial confining stress reaches its peak value, that is, a constant value.

Before determining the maximum confining stresses and the critical confining stresses in Eqs. (5.19) and (5.21), it is necessary to determine the parameters, for example, tensile strength of concrete, $f_{\text{ct}}$, the unit adhesion between strand and concrete, $\sigma_{c}$, the inclination angle of rib, $\delta$, and friction coefficient, $\mu$. The tensile strength $f_{\text{ct}}$, can be correlate with the concrete compressive strength, $f_{c}$, as $f_{\text{ct}} = 0.56\,\, \cdot \,f_{c}^{1/2}$ on the basis of ACI 318-14. $\sigma_{c}$, is considered as $0.11f_{c}$ [11]. The strands in all data had not been coated; the angle of friction, $\emptyset$, is estimated to be 20° on the basis of earlier study, which corresponds to the coefficient of friction, $\mu$, of 0.36. The value of the inclination angle, $\delta$, was taken as 8.7° in all the tests. It is assumed that there is no excess concrete is adhered to the bearing face to increase the angle of inclination in the case of pull-out failure.

Table 5.1 gives the ratio of the maximum confining stress $\sigma_{n,\max }$ and the critical confining stress corresponding to pull-out failure with strand rotation $\sigma_{n,\text{crit}}$ for the 12.7 mm and 15.2 mm steel strands, respectively.

The maximum confining stresses in Table 5.1 are larger than that of the confining stresses corresponding to pull-out failure accompanied with the strand rotation for all of the selected references. Therefore, the radial confining stress in Eq. (5.19) can be used in Eq. (5.20) to calculate the ultimate bond strength in the case of the pull-out failure of the strand. Table 5.1 and Fig. 5.6 illustrate the comparison between the experimental and predicted bond strengths.

Fig. 5.6

Prediction and experimental bond strengths: a 12.7 mm strand; b 15.2 mm strand

The results in Fig. 5.6 and Table 5.1 show that the proposed model combines the pull-out failure and strand rotation to predict the ultimate bond strength between strand and concrete. The prediction accuracy can be improved by considering the strand rotation. By considering strand rotation, the average ratio of the test results to the predicted results for strands with a diameter of 12.7 mm is 0.92–1.04 with a minimum standard deviation of 0.15. For the strands with a diameter of 15.2 mm, this average ratio is 0.99–1.01 with a minimum standard deviation of 0.17.

5.3 Model for Bond Strength of Corroded Strand

5.3.1 Ultimate Bond Strength of Corroded Strand

In Sect. 5.2, a theoretical analysis of the bond strength of uncorroded seven-wire strands at concrete splitting failure is first carried out. Then, the mechanical interlocking forces on the loading bearing surfaces of steel strands are studied. The bond stress was expressed by the normal stress of the strand–concrete interface. In this section, by considering additional pressure induced by corrosion at the interface, the proposed bond model of an uncorroded strand is revised to predict the bond strength of corroded strand.

For Eq. (5.12) in Sect. 5.2, the bond stress of an uncorroded strand $\tau_{b}$ relates to parameters, for example, the ratio of the outer wire diameter to the nominal strand diameter $d_{e} /d$, angle of friction $\phi$, inclined angle of the rib to the center line of the steel strand $\delta$, unit adhesion strength $\sigma_{c}$, and normal stress along the shear face $\sigma_{n}$.

In order to supply a model for the corroded strand, some other parameters need to be added on the basis of the bond model for corroded deformed bars. The corrosion pressure at the interface is needed to be considered. The second thing is to modify the influence parameters in Eq. (5.12). Thus, the ultimate bond strength of steel strand with corrosion $\tau_{b} \left( {\rho_{p} } \right)$ can be written as:

$$\tau_{b} \left( {\rho_{p} } \right) = \tau_{\rm CP} \left( {\rho_{p} } \right) + \tau_{\rm AD} \left( {\rho_{p} } \right) + \tau_{\text{COR}} \left( {\rho_{p} } \right),$$

(5.22)

where $\rho_{p}$ is the strand corrosion loss; $\tau_{\text{CP}} \left( {\rho_{p} } \right)$ is the contributions of maximum confining stress; $\tau_{\text{AD}} \left( {\rho_{p} } \right)$ is the adhesion stress; $\tau_{\text{COR}} \left( {\rho_{p} } \right)$ is the pressure induced by corrosion.

Assuming the same diameter reduction for all the seven corroded wires, and ignoring the original radius difference between the central and the outer wires, the radius of a random corroded wire is calculated as, $r_{\rm es} = r_{e} - x$, where $r_{e}$ is the radius of an uncorroded outer wire.

In the current investigation, it is considered that the corrosion effect was uniformly distributed. The corrosion loss $\rho_{p}$ can be written as a function of corrosion penetration depth x as:

$$\rho_{p} = \left( {1 - \frac{{\left( {r_{e} - x} \right)^{2} }}{{r_{e}^{2} }}} \right) \times 100\% .$$

(5.23)

On the basis of the bond strength model of strand from Eq. (5.12) in Sect. 5.2, $\tau_{\text{CP}} \left( {\rho_{p} } \right)$, $\tau_{\text{AD}} \left( {\rho_{p} } \right),$ and $\tau_{\text{COR}} \left( {\rho_{p} } \right)$ can be written as

$$\tau_{\text{CP}} \left( {\rho_{p} } \right) = \frac{3}{14}\pi \left( {\frac{{d_{e} \left( {\rho_{p} } \right)}}{{d\left( {\rho_{p} } \right)}}} \right)^{2} \frac{{\sin \left( {\delta + \phi } \right)\left( {\rho_{p} } \right)}}{{\cos \phi \left( {\rho_{p} } \right) \cdot \sin \delta \left( {\rho_{p} } \right)}}\sigma_{{{\text{ n}},{\text{max}}}} \left( {\rho_{p} } \right),$$

(5.24a)

$$\tau_{\text{AD}} \left( {\rho_{p} } \right) = \frac{3}{14}\pi \left( {\frac{{d_{e} \left( {\rho_{p} } \right)}}{{d\left( {\rho_{p} } \right)}}} \right)^{2} \cot \delta \left( {\rho_{p} } \right)\sigma_{c} \left( {\rho_{p} } \right),$$

(5.24b)

$$\tau_{\text{COR}} \left( {\rho_{p} } \right) = \mu \left( {\rho_{p} } \right) \cdot p_{r} \left( {\rho_{p} } \right),$$

(5.24c)

where $\sigma_{{{\text{n}},{\text{max}}}} \left( {\rho_{p} } \right)$ and $p_{r} \left( {\rho_{p} } \right)$ are the maximum radial confining stress on the interface supplied by surrounding concrete at bond failure and the corrosion-induced pressure, respectively.

For the quantitative evolution of other influence parameters of corrosion damage, it is assumed that the coefficient of friction μ exhibits a linear change with the corrosion penetration depth x. The coefficient of friction $\mu$ between corroded strand and cracked concrete is calculated as $\mu = 0.345 - 0.26\left( {x - x_{\text{cr}} } \right)$ [7]. The chemical adhesion conditions at strand–concrete interface were altered because of the oxidation of steel strand and the accumulation of corrosion products. The adhesive stresses have been acted on the rib surfaces, which were quantitatively degraded. Some authors have assumed that the adhesion of deformed bars varies linearly with the level of corrosion [12, 15, 39]. Based on the similarity of the bond mechanism between deformed bars and steel strand, it is assumed that the adhesion of corroded strand decreases linearly with the increasing corrosion loss x. And $\sigma_{c}$ is calculated as $\sigma_{c} = 3 - 22.08\left( {x - x_{\text{cr}} } \right)$,where $x_{\text{cr}}$ is the critical penetration depth at concrete cover [7]. Apart from these, the corrosion pressure $p_{r} \left( {\rho_{p} } \right)$ and $\sigma_{{{\text{n}},{\text{max}}}} \left( {\rho_{p} } \right)$ need to be computed.

5.3.2 Corrosion-Induced Pressure at Bond Interface

Corrosive pressure at the interface is measured according to the processes below. Firstly, the radial displacement at the interface due to volumetric expansion of corrosion products is acquired by considering the section characteristics of the steel strand. Secondly, the confinement model proposed by Den [16] describes the functional relationship between radial pressure and radial displacement at the interface and is used to infer the variation of corrosion pressure, $p_{r} \left( {\rho_{p} } \right)$, with corrosion propagation.

On the basis of the thick-walled cylinder theory, the corrosive radial displacement at interface is inferred. The radius of the cylinder is the minimum concrete cover on the strand. In general, the propagation of corrosion products caused the development of multiple cracks from the strand–concrete interface to the outer surface of the concrete. Assuming that the direction of these cracks is to develop toward the outer surface of the concrete, radius of the crack front is written as $r_{i}$. The value of concrete strain at crack front $r_{i}$ is consistent with the nominal tensile strain of concrete $\varepsilon_{\text{ct}}$, where $\varepsilon_{\text{ct}} = f_{\text{ct}} /E_{0} .$ $f_{\text{ct}}$ is the tensile strength of concrete; $E_{0}$ is the elastic modulus of concrete.

The volume loss of steel strand with corrosion per unit length $\Delta V_{s}$ is written as

$$\Delta V_{s} = 6\left( {\pi r_{e}^{2} - \pi r_{\rm es}^{2} } \right) = 6\left( {2\pi r_{e} x - \pi x^{2} } \right).$$

(5.25)

In order to simplify the calculation process, the corrosion products of the six outer wires are assumed to be evenly distributed along the nominal diameter of the strand. The outer radius of the corrosion products is $r_{r}$. Schematic diagram of the development of the corrosion products is depicted in Fig. 5.7.

Fig. 5.7

Thick-walled cylinder model

In the corrosion process, the products formed by corrosion of strand not only accumulate around the steel strand but also penetrate into the corrosion-induced cracks. The volume of the products formed by corrosion $\Delta V_{r}$ is expressed as:

$$\Delta V_{r} = m \cdot \Delta V_{s} = \pi \left( {r_{r}^{2} - r_{0}^{2} } \right) + 6\pi \left( {r_{e}^{2} - r_{\rm es}^{2} } \right) + \sum w \cdot \left( {r_{i} - r_{r} } \right)/2,$$

(5.26)

where ∑w is the sum widths of corrosion-induced cracks at the radius of $r_{r}$, which can be calculated as:

$$\sum w = 2\pi u_{r} \left| {r = r_{0} = 2\pi } \right.\left( {r_{r} - r_{0} } \right),r_{r} = r_{0} + u_{r} .$$

(5.27)

The corrosion-induced radial displacement $\left. {u_{r} } \right|_{{r = r_{0} }}$ at interface is written as:

$$\left. {u_{r} } \right|_{{r = r_{0} }} = r_{r} - r_{0} = \frac{{\left( {m - 1} \right)\left( {12r_{c} x - 6x^{2} } \right)}}{{r_{0} + r_{i} }},$$

(5.28)

where m is the volume expansion ratio of products formed by corrosion to its original steel strand. According to the type of products formed by corrosion, m ranges from 1.7–6.15 [30].

The concrete cover in the corrosion process can go through three stages according to the degree of cracking: namely uncracked stages, partially cracked stages, and complete cracked stages [12]. The radial displacement $u_{r,r0}$ gradually increases at radius $r_{0}$. The analysis shows that the corrosion pressure was closely related to the radial displacement and the corrosion pressure in the uncracked stages and partially cracked stages. When concrete is completely cracked, the corrosion pressure gradually decreases. Since $u_{r,r0}$ has been represented by x in Eq. (5.28), the radial pressure model proposed by Den [16] is used to calculate $p_{r}$ as the cracking stages develops. The radial pressure model associates $u_{r,r0}$ with the radial pressure.

As for the uncracked stage of concrete, the functional relationship between $u_{r,r0}$ and $p_{r}$ can be written as:

$$u_{{r,r_{0} }} = \frac{{r_{0} p_{r} }}{{E_{0} }}\left( {\frac{{r_{c}^{2} + r_{0}^{2} }}{{r_{c}^{2} - r_{0}^{2} }} + v_{c} } \right),$$

(5.29)

where $v_{c}$ is the Poisson’s ratio of concrete, and $r_{c}$ is the radius of cylinder, which equals to the minimum concrete cover.

Equations (5.26) and (5.28) can be combined to acquire $p_{r}$ at the uncracked stage, which can be expressed as:

$$p_{r} = \frac{{\left( {m - 1} \right)E_{0} \left( {r_{c}^{2} - r_{0}^{2} } \right)\left( {12r_{b} x - 6x^{2} } \right)}}{{2r_{0}^{2} \left[ {r_{c}^{2} \left( {1 + v_{c} } \right) + r_{0}^{2} \left( {1 - v_{c} } \right)} \right]}}.$$

(5.30)

The maximum corrosion pressure $p_{r,1}$ at the uncracked stage of concrete appears when the value of circumferential tensile stress at the strand–concrete interface is equals to $f_{\text{ct}}$.

$$p_{r,1} = f_{\text{ct}} \frac{{r_{c}^{2} - r_{0}^{2} }}{{r_{c}^{2} + r_{0}^{2} }}.$$

(5.31)

The corrosion depth $x_{1}$ is obtained by substitution of Eq. (5.31) into Eq. (5.30). The corrosion depth $x_{1}$ corresponds to the initial cracking of concrete.

At partially cracked stage, both the uncracked cylinders and the cracked cylinders contribute to $p_{r}$ and $u_{r,r0}$, respectively. On the basis of the fictitious crack model, the contribution of the cracked cylinder to the radial displacement is calculated [32], which takes into account the bilinear softening properties of the cracked concrete. The bilinear softening model of cracked concrete is depicted in Fig. 5.8.

Fig. 5.8

Bilinear model for softening concrete in tension [30]

As shown in Fig. 5.8, the bilinear curve changes at this point (α, β). $\alpha$ and $\beta$ are coefficients in the bilinear softening model, which are closely relevant to the performances of cracked concrete including fracture energy $G_{f}$, concrete tensile strength $\sigma_{\text{ct}}$, etc. On the basis of the CEB-FIB code, $\alpha$ and $\beta$ are taken as 0.15, where W is the normalized crack width, and W is written as $W\left( r \right) = f_{\text{ct}} \,\, \cdot \,w\left( r \right)/G_{f}$, where $w\left( r \right)$ is the actual width of the crack at radius r. $G_{f} = 0.5\cdot\left( {\alpha + \beta } \right) \cdot W_{u} \cdot f_{\text{ct}}$, where $W_{u}$ is the localized deformation at failure.

The displacement $u_{{r,r_{0} }}$ at partially cracked stage is written as:

$$\begin{aligned} u_{{r,r_{0} }} & = r_{i} \varepsilon_{\text{cr}} \left( {1 + v_{c} C_{1} } \right) + \varepsilon_{\text{cr}} C_{1} r_{i} \ln \frac{{r_{i} }}{{r_{0} }} + \varepsilon_{\text{cr}} b\left( {r_{i} \ln (r_{i} /r_{0} ) - \left( {r_{i} - r_{0} } \right)} \right) \\ & + \frac{{\varepsilon_{\text{cr}} aC_{2} }}{4}\left( {2r_{i}^{2} \ln (r_{i} /r_{0} ) + \left( {r_{i} - r_{0} } \right)\left( {r_{0} - 3r_{i} } \right)} \right), \\ \end{aligned}$$

(5.32)

where $C_{1} = \frac{{r_{c}^{2} - r_{i}^{2} }}{{r_{c}^{2} + r_{i}^{2} }}$; $C_{2} = \frac{{2\pi \varepsilon_{\text{ct}} }}{{nW_{u} }}$; n is the number of fictitious cracks. The influences of crack width on the tensile stress of concrete $\sigma_{\text{ct},r}$ are represented by a and b. Table 5.2 show the values of coefficients a and b in different situations.

Table 5.2 a and b corresponding to softening behavior of cracked concrete

The stress $p_{r}$ at partially cracked stage can be written as:

$$p_{r} = \frac{{r_{i} }}{{r_{0} }}f_{\text{ct}} C_{1} + f_{\text{ct}} \left( {\frac{{aC_{2} r_{0} }}{2}\left( {\frac{{r_{i} }}{{r_{0} }} - 1} \right)^{2} + b\left( {\frac{{r_{i} }}{{r_{0} }} - 1} \right)} \right)$$

(5.33)

The functional relationship between the radius of crack front $r_{i}$ and the penetration depth x can be acquired by equating Eqs. (5.28) and (5.32). The penetration depth, $x_{2}$, corresponds to the completely cracked concrete cover, which can be acquired by substituting $r_{c}$ into $r_{i}$. In addition, the stress p_r for different penetration depth, x, in the partially cracked stage can be obtained by substituting $r_{i}$ into Eq. (5.33).

The contribution of the cracked cylinder to $p_{r}$ in the complete cracked stage is comparable to that of the partially cracked stage. It is assumed that the total elongations $\vartriangle_{\text{tot}}$ of the cylinder at any radius r are the same, where $\vartriangle_{\text{tot}} = 2\pi r\cdot\varepsilon_{\text{ct}} + nW_{u} \left( {\sigma_{\text{ct},r} /f_{\text{ct}} - b} \right)/a$. The displacement $u_{r,r0}$ at the complete cracked stage of concrete is written as:

$$u_{{r,r_{0} }} = C_{3} \frac{{nW_{0} }}{2\pi } + \varepsilon_{\text{cr}} \left( {aC_{3} + b} \right)\left( {r_{c} \ln \frac{{r_{c} }}{{r_{0} }} - r_{c} + r_{0} } \right) - \frac{{aC_{2} \varepsilon_{\text{cr}} }}{4}\left( {2r_{c}^{2} \ln \frac{{r_{c} }}{{r_{0} }} - r_{c}^{2} + r_{0}^{2} } \right)$$

(5.34)

where $C_{3} = \Delta_{\text{tot}} /\left( {nW_{u} } \right)$.

The stress p_r can be written as

$$p_{r} = f_{\text{ct}} \left( {aC_{3} + b} \right)\left( {\frac{{r_{c} }}{{r_{0} }} - 1} \right) - \frac{{f_{\text{ct}} aC_{2} r_{0} }}{2}\left( {\left( {\frac{{r_{c} }}{{r_{0} }}} \right)^{2} - 1} \right).$$

(5.35)

According to the Eqs. (5.30), (5.33) and (5.35), $p_{r}$ can be calculated.

The corrosion pressure model mentioned before requires the iterative analysis during concrete cracking. Therefore, the expressions at different stages may be complicated in the actual structural evaluation. Therefore, a simplified equation is needed for effective evaluation on the basis of the existing equations. The original graph of the corrosion pressure during the corrosion process is shown in Fig. 5.9.

Fig. 5.9

Comparison of the corrosion pressure between computed value and fitting curve

The general trend of the curve is almost parabolic until the concrete cracks completely. Then, the stress $p_{r}$ drops sharply and slowly approaches an approximate stable value, which is consistent with the nature of an exponential function. Hence, a piecewise simplified function, which is divided by the critical corrosion loss $\rho_{\text{crit}}$, is acquired base on the data fitting of the computed values. Some parameters have been proven to influence the stress $p_{r}$, for example, thickness of concrete cover $r_{c}$, strength of concrete tensile $f_{\text{ct}}$, and diameter of steel strand $r_{0}$, these parameters are also considered in the equation.

$$p_{r} = \left[ { - 56.7\rho^{2} + \left( {94.2 + r_{0} } \right)\rho + r_{0}^{2} - 43.29} \right] \cdot f_{\text{ct}} /r_{0} ;\rho \le \rho_{\text{crit}}$$

(5.36a)

$$p_{r} = \left( {0.067 + 32.8 \cdot e^{ - 3\rho } } \right) \cdot f_{\text{ct}} \cdot \left( {r_{c} /r_{0} - 1} \right);\rho > \rho_{\text{crit}} .$$

(5.36b)

The existing model and the simplified equation predicted the comparison of corrosion pressure are shown in Fig. 5.9. There is a general satisfactory agreement.

5.3.3 Confining Stress at Bond Failure

In order to calculate the ultimate bond strength of corroded steel strand, it is necessary to determine the maximum confining stress $\sigma_{n,\max }$, at bond failure in Eq. (5.24a). For beams without additional reinforcement, only the concrete acts as a constraint on the steel strands. For beams with additional reinforcement, $\sigma_{n,\max }$ is supplied by the combined action of surrounding concrete and stirrups. Giuriani et al. [19] modified the constraint models of cracked concrete and stirrups for uncorroded bars to incorporate the influence of corrosion.

$$\sigma_{n,\max } \left( {\rho_{p} } \right) = \sigma_{n,\max c} \left( {\rho_{p} } \right) + \sigma_{n,\max s} \left( {\rho_{p} } \right),$$

(5.37a)

$$\sigma_{n,\max c} \left( {\rho_{p} } \right) = \left\{ {\frac{{b_{p} }}{{\left[ {d_{0} \left( {\rho_{p} } \right) + 2t_{r} \left( {\rho_{p} } \right)} \right]}} - 1} \right\}\left( {a\frac{{W_{t} }}{{W_{0} }} + b} \right)f_{\text{ct}} ,$$

(5.37b)

$$\sigma_{n,\max s} \left( {\rho_{p} } \right) = \left\{ {\frac{{n_{s} A_{s} }}{{\left[ {d_{0} \left( {\rho_{p} } \right) + 2t_{r} \left( {\rho_{p} } \right)} \right]S_{v} }}} \right\} \times E_{\text{st}} \sqrt {\frac{{a_{2} W_{t}^{2} }}{{\alpha_{\text{st}}^{2} d_{\text{st}}^{2} }} + \frac{{a_{1} W_{t} }}{{\alpha_{\text{st}} d_{\text{st}} }} + a_{0} ,}$$

(5.37c)

where $\sigma_{n,\max c}$ and $\sigma_{n,\max s}$ are the maximum confining stresses at bond failure supplied by the cracked concrete and stirrups, respectively; $t_{r} \left( {\rho_{p} } \right)$ is the thickness of corrosion layer; b_p is specimen width; $n_{s}$ is the number of stirrups within the range of $b_{p}$; $A_{s}$ is the cross-sectional area of stirrups in the specimen; $W_{t}$ is fictitious crack width; $S_{v}$ is the spacing of stirrups; $E_{\text{st}}$ is the elastic modulus of stirrups, $d_{\text{st}}$ is the diameter of stirrups. $a_{0}$, $a_{1}$, $a_{2}$ are coefficients, which are obviously related to the local bond–slip law of stirrups. The values of these coefficients can be acquired from the study [19]. $\alpha_{\text{st}}$ is the shape factor for stirrups, and its value is 2.

It should be explained that for the case of pull-out failure, the confining stresses based on Eq. (5.37) does not consider the influence of strand rotation. As mentioned before, when the confining stresses provided by the concrete or transversal reinforcement is enough, the strand slip with rotation due to its twisting structure and then the pull-out failure occurs. Under the cases, beams fail as the strand rotates before the confinements of concrete and steel reinforcement are maximized, respectively. This means that the maximum confinement on Eq. (5.37) may be overestimated during the uncorroded stage and initial corrosion stage. In this case, the maximum confinement at pull-out failure accompanies with strand rotation, which needs to be considered.

For pull-out failure case, when the steel strand rotates, the integration of the radial component for the mechanical interlock, $dF_{\text{bu}} \left( {\rho_{p} } \right)$, leads to a bursting force, $F_{\text{rib}} \left( {\rho_{p} } \right)$, around the strand (see Fig. 5.10). Due to the helical structure of the outer wires, a torque, $M_{\text{rib}} \left( {\rho_{p} } \right)$, generates on the strand. Friction around the steel strand resisted this torque during the initial pull-out stage. As the pull-out force increases further, when the frictional force is no longer able to resist the torque, the steel strand begins to rotate in the concrete groove. When the steel strand rotates, the bond strength and the confining stress reach their maximums value, respectively. Therefore, the confining stress,$\sigma_{n} \left( {\rho_{p} } \right)$, corresponding to pull-out failure can be computed according to the balance of the central moment of the steel strand.

Fig. 5.10

Moment balance between bursting forces and friction forces

Firstly, the force $F_{\text{rib}} \left( {\rho_{p} } \right)$ from a single outer wire needs to be determined. $F_{\text{rib}} \left( {\rho_{p} } \right)$ is composed of a vertical component, $F_{\text{rib}v} \left( {\rho_{p} } \right)$, and a parallel component, $F_{\text{rib}h} \left( {\rho_{p} } \right)$, which can be expressed as:

$$\begin{aligned} F_{\text{rib}v} \left( {\rho_{p} } \right) &= \mathop \int \limits_{{ - \frac{\pi }{6}}}^{{\frac{\pi }{2}}} dF_{\text{bu}} \left( {\rho_{p} } \right) \cdot \cos \theta \cdot d\theta \hfill \\ &= \frac{3}{4}\left[ {\frac{{\sigma_{n} \left( {\rho_{p} } \right)\cos \left( {\delta + \phi \left( {\rho_{p} } \right)} \right)}}{{\cos \phi \left( {\rho_{p} } \right)\sin \delta }} - \sigma_{c} \left( {\rho_{p} } \right)} \right]h_{r} \left( {\rho_{p} } \right)d_{e} \left( {\rho_{p} } \right) \hfill \\ \end{aligned}$$

(5.38a)

$$\begin{aligned} F_{\text{rib}h} \left( {\rho_{p} } \right) & = \mathop \int \limits_{{ - \frac{\pi }{6}}}^{{\frac{\pi }{2}}} dF_{\text{bu}} \left( {\rho_{p} } \right) \cdot \sin \theta \cdot d\theta \\ & = \frac{\sqrt 3 }{4}\left[ {\frac{{\sigma_{n} \left( {\rho_{p} } \right)\cos \left( {\delta + \phi \left( {\rho_{p} } \right)} \right)}}{{\cos \phi \left( {\rho_{p} } \right)\sin \delta }} - \sigma_{c} \left( {\rho_{p} } \right)} \right]h_{r} \left( {\rho_{p} } \right)d_{e} \left( {\rho_{p} } \right) \\ \end{aligned}$$

(5.38b)

Thus, the bursting force, $F_{\text{rib}} \left( {\rho_{p} } \right)$, for an outer wire can be acquired as:

$$\begin{aligned} F_{\text{rib}} \left( {\rho_{p} } \right) & = \sqrt {F_{\text{rib}v}^{2} \left( {\rho_{p} } \right) + F_{\text{rib}h}^{2} \left( {\rho_{p} } \right)} \\ & = \frac{\sqrt 3 }{2}\left[ {\frac{{\sigma_{n} \left( {\rho_{p} } \right)\cos \left( {\delta + \phi \left( {\rho_{p} } \right)} \right)}}{{\cos \phi \left( {\rho_{p} } \right)\sin \delta }} - \sigma_{c} \left( {\rho_{p} } \right)} \right]h_{r} \left( {\rho_{p} } \right)d_{e} \left( {\rho_{p} } \right) \\ \end{aligned}$$

(5.39)

The torque, $M_{\text{rib}} \left( {\rho_{p} } \right)$, induced by the ribs of the six outer wires within the length of $dz$ can be written by $\sigma_{n} \left( {\rho_{p} } \right)$ as:

$$M_{\text{rib}} \left( {\rho_{p} } \right) = \frac{9}{2}\left[ {\frac{{\sigma_{n} \left( {\rho_{p} } \right)\cos \left( {\delta + \phi \left( {\rho_{p} } \right)} \right)}}{{\cos \phi \left( {\rho_{p} } \right)\sin \delta }} - \sigma_{c} \left( {\rho_{p} } \right)} \right]d_{e}^{3} \left( {\rho_{p} } \right)d\alpha .$$

(5.40)

For the torque, $M_{\text{fri}} \left( {\rho_{p} } \right)$, is supplied by the frictional force around the strand. The frictional force is supposed to be evenly distributed along the rib surface (see Fig. 5.10). The maximum torque, $M_{\max } \left( {\rho_{p} } \right)$, supplied by the friction can be acquired as:

$$\begin{aligned} \frac{{M_{\max } \left( {\rho_{p} } \right)}}{12} & = \mathop \int \limits_{{ - \frac{\pi }{6}}}^{{\frac{\pi }{2}}} \mu \left( {\rho_{p} } \right)\sigma_{n} \left( {\rho_{p} } \right)dA\frac{{d_{e} \left( {\rho_{p} } \right)}}{2}\left( {\cos \theta } \right)^{2} \\ & + \mathop \int \limits_{ - \pi /6}^{\pi /2} \mu \left( {\rho_{p} } \right)\sigma_{n} \left( {\rho_{p} } \right)dA\sin \theta \left( {\frac{{d_{f} \left( {\rho_{p} } \right) + d_{e} \left( {\rho_{p} } \right)}}{2} + \frac{{d_{e} \left( {\rho_{p} } \right)}}{2}\sin \theta } \right). \\ \end{aligned}$$

(5.41)

Assuming $d_{f} \left( {\rho_{p} } \right) = d_{e} \left( {\rho_{p} } \right)$, the torque, $M_{\max } \left( {\rho_{p} } \right)$, can be simplified as:

$$M_{\max } \left( {\rho_{p} } \right) = \left( {3\sqrt 3 + 2\pi } \right)\frac{{\mu \left( {\rho_{p} } \right)\sigma_{n} \left( {\rho_{p} } \right)}}{\sin \delta }d_{e}^{3} \left( {\rho_{p} } \right)d\alpha .$$

(5.42)

The critical confining stress, $\sigma_{n,\text{crit}} \left( {\rho_{p} } \right)$, can be acquired by equating equations. (5.38) and (5.40).

$$\sigma_{n,\text{crit}} \left( {\rho_{p} } \right) = \sigma_{c} \left( {\rho_{p} } \right)/\left[ {\frac{{\cos \left( {\delta + \phi \left( {\rho_{p} } \right)} \right)}}{{\cos \phi \left( {\rho_{p} } \right) \cdot \sin \delta }} - \left( {\frac{2\sqrt 3 }{3} + \frac{4\pi }{9}} \right)\frac{{\tan \phi \left( {\rho_{p} } \right)}}{\sin \delta }} \right].$$

(5.43)

The critical confining stress in Eq. (5.43) is computed for the pull-out failure accompanied with the strand rotation. Under the cases, the beams fail at the critical confining stress. The confining stress in Eq. (5.37) and cannot reach the maximum. For the failure mode in this case, the critical confining stress should be substituted by the maximum confining stress to avoid overestimating the ultimate bond strength. When the critical confining stress of the corroded beams is greater than the maximum confining stress, the splitting failure of concrete occurs before pull-out failure. In this condition, $\sigma_{n,\max } \left( {\rho_{p} } \right)$ for the ultimate bond strength can be obtained directly from Eq. (5.37).

5.3.4 Model Validation

The validity of the proposed model for predicting the ultimate bond strength between the corroded steel strand and cracked concrete is verified by comparing the experimental results with the predicted results. For comparison, the pull-out tests results of 15.2 mm corroded strand specimens are used [37]. In the experimental study, 20 pull-out beams without stirrups and with stirrups were designed. Corrosion of steel strand is accelerated to cause the cracking of concrete cover. All beams have the same dimension of 150 mm × 150 mm × 1200 mm. The effective bond length is 1100 mm and the thickness of concrete cover is 67.4 mm. For the beams with stirrups, there are nine smooth steel bars with the spacing of 150 mm for additional confinement. All tests were carried out by pulling the steel strand out directly from the concrete block to investigate the influences of corrosion-induced concrete crack on the strand bond. More details can be found in the Sect. 4.2 in Chap. 4.

Before calculating the ultimate bond strength, it is necessary to determine the parameters in the proposed model. In this test, the concrete’s compressive strength $f_{c}$ was 35 MPa; the tensile strength of concrete, $f_{\text{ct}} = 0.56\left( {f_{c} } \right)^{1/2} = 3.29{\text{ MPa}}$; the elastic modulus of concrete, $E_{0} = 4735\cdot\left( {f_{c} } \right)^{1/2} = 2.8 \times 104{\text{ MPa}}$; the concrete tensile strain during cracking, $\varepsilon_{\text{ct}} = 0.000117$; and Poisson’s ratio of concrete, $v_{c} = 0.2$. In addition, based on the known concrete properties, other parameters related to the softening behavior of cracked concrete are also estimated. The localized deformation at failure, $W_{u} = 0.2 \,{\text{mm}}$. The volume expansion ratio of products formed by corrosion m because of expansive products formed by corrosion is taken as m = 4.25. For the number of fictitious cracks n, based on the existing study, it was assumed that n = 3 or n = 4. On the basis of experimental evidence of concrete cracking caused by strand corrosion, this paper finally determined n = 4 [36].

The predicted results of the beams without and with stirrups are compared with the experimental results as shown in Figs. 5.11 and 5.14, respectively. The variation of three components $\tau_{\text{CP}} \left( {\rho_{p} } \right)$, $\tau_{\text{AD}} \left( {\rho_{p} } \right),$ and $\tau_{\text{COR}} \left( {\rho_{p} } \right)$, which are caused by the maximum confining stress, adhesion stress, and pressure, respectively, can also be expressed as functions of the corrosion loss.

Fig. 5.11

Comparison between predicted and experimental results

From the comparison results in Fig. 5.11, the prediction of the ultimate bond strength of the beams without stirrups is in good agreement with the experimental results, especially when the corrosion loss is greater than 2%. It can be seen from the predicted results change with the development of the strand corrosion. The bond strength first increases in the early stage of concrete cracking. And then the bond strength drops when the cracks penetrate reach roughly 2/3 of concrete cover or the loss of corrosion exceeds 0.8%. In the initial stage of ascent, the corrosion pressure contributes most to the improvement of bond strength because both the interfacial bursting stress and the friction coefficient increase. This result is comparable to that of deformed bars with corrosion [7]. In the subsequent stage of descent, the concrete cover is completely cracked, the bond strength drops rapidly with a corrosion loss of 1.3%, and the contributions of both confining stress and corrosion pressure drop rapidly. When the corrosion loss is further increased to more than 2%, the bond strength is affected by a combination of the confining stress and the corrosion pressure. At this stage, the proposed theoretical model can predict well the bond strength and the decrease tendency when corrosion loss increases further.

The predicted ultimate bond strengths for the beams with stirrups are shown in Fig. 5.12. The proposed model in this part can reasonably predict the evolution of the ultimate bond strength of the corroded strand, especially for the completely cracked concrete. In particular, the influence of the stirrups in delaying the deterioration of bond induced by corrosion can be well modeled by considering the additional contribution of the stirrups in the confinement model (see Fig. 5.13).

Fig. 5.12

Comparison between predicted and experimental results

Fig. 5.13

Effect of stirrups in delaying bond deterioration

Fig. 5.14

Effect of concrete tensile strength on bond strength evolution

The effect of tensile strength of concrete on the evolution of bond strength is shown in Fig. 5.14. As predicted, the increased tensile strength can lead to the increase of the bond strength. This is because the contributions of corrosion pressure and confining stress are proportional to the tensile strength of concrete. Additionally, comparing the results of the two random curves shows that the discrepancy in bond strength gradually decreases as the corrosion progresses further. This phenomenon happens because of the softening behavior of concrete tensile strength. When the concrete cover is completely cracked, the degradation of tensile strength reduces its influence on the bond strength.

Additionally, this section also studies the effect of the number of fictious cracks n on the bond strength, as shown in Fig. 5.15. As mentioned earlier. n can be taken as 3 or 4, and the two values are compared in this section. According to the curve comparison results in Fig. 5.15, it can be known that the bond strength between strand and concrete of n = 3 is slightly lower than that of n = 4.

Fig. 5.15

Effect of number of fictious cracks on bond strength

5.4 Model for Bond–Slip Between Corroded Strand and Concrete

5.4.1 Method for the Local Bond Characteristics

In this study, the numerical model proposed by Haskett et al. [20] is improved to acquire an appropriate local bond–slip relationship between the corroded strand and concrete. The experimental force–displacement curves are also used to calibrate the model parameters. Details are shown below.

With the assumed local bond–slip relationship, the expression of slip ($s_{p}$) at the loaded end under the applied force $F_{p}$ is shown in Fig. 5.17. In order to simplify the calculation, the pull-out beam is divided into several sections. These sections were numbered from 1 to n, and any section $i$ has length $l_{i}$. The values of force and slip on each section are taken from their average. The specific calculation steps are shown in Fig. 5.16.

Fig. 5.16

Calculation flow chart

Fig. 5.17

Schematic diagram of the numerical analysis

$F_{2}$ and $s_{2}$ at the second section have been acquired. The numerical procedure continues to be repeated for calculating $F_{i}$ and $s_{i}$ of the remaining section ($i$) until the known boundary conditions are met the requirements. If the boundary conditions do not meet the requirements, the assumed loaded end slip ($s_{p}$) should be modified and the above process should be repeated.

According to the relative relationship between the bond length and the effective bond length, there are two boundary conditions. Firstly, for specimens with long bond lengths or when the pull-out load is small, the transmission of the pull-out force along the steel strand is limited to the inner section of the specimen and is not transmitted to the free end. The boundary condition currently is that the tension force ($F_{i}$) and slip ($s_{i}$) of the steel strand at one section ($i$) are both equal to 0. Secondly, for specimens with short bond length or when the pull-out load is large, the pull-out force is transmitted to the entire bond area, and the boundary condition at this time is that the tension force ($F_{n}$) of the steel strand at the free end is equal to 0.

The steel strand may yield during pull-out test. In that case, the influence of plastic deformation should be considered when the strain ($\varepsilon_{s}$) of strand is calculated. This study adopts the elastic–plastic constitutive model of steel strand proposed by Zhang et al. [40], which is written as:

$$\sigma = \left\{ {\begin{array}{*{20}l} {\varepsilon E_{s} } & {\varepsilon \le \varepsilon_{\text{sy}} } \\ {f_{\text{sy}} + E_{\text{sp}} \left( {\varepsilon - \varepsilon_{\text{sy}} } \right)} & {\varepsilon > \varepsilon_{\text{sy}} } \\ \end{array} } \right.,$$

(5.44)

where $f_{\text{sy}}$ is the yield strength of strand; $\varepsilon_{\text{sy}}$ is the yield strain of strand; and $E_{\text{sp}}$ is the steel hardening modulus.

The strand strain considering the influence of plastic deformation, and the strand strain can be calculated as:

$$\varepsilon_{si} = \left\{ {\begin{array}{*{20}l} {\frac{{F_{i} }}{{A_{s} E_{s} }}} & {F_{i} \le A_{s} f_{\text{sy}} } \\ {\varepsilon_{\text{sy}} + \frac{{F_{i} }}{{A_{s} E_{\text{sp}} }} - \frac{{f_{\text{sy}} }}{{E_{\text{sp}} }}} & {F_{i} > A_{s} f_{\text{sy}} } \\ \end{array} } \right..$$

(5.45)

Some scholars pointed out that when the nonlinear deformation of concrete occurs, the concrete stress exceeds 33% of its compressive strength. Before that, the concrete is basically in the linear elastic deformation stage [21]. In this section, the maximum compressive stress of concrete is about 12 MPa. The measured compressive strength of each beam is between 34.1 and 35.6 MPa. Therefore, in order to simplify the calculation, the nonlinear deformation of concrete is not considered in the calculation.

5.4.2 Local Bond–Slip Between Corroded Strand and Concrete

In existing research, many models have been come up with to express the local bond stress–slip relationship. Figure 5.18a, b shows the piecewise uniform function and the multi-variate linear function [4, 26, 35]. However, these two models are too simplistic for the current study. Figure 5.18c represents the logarithmic function [34]. The bond stress increases gradually with the increase of the load in this model. There is no descending section, which may lead to abnormally large bond stress. The CEB Model Code divides the function into four different zones. A plateau with the maximum bond stress is recommended to describe the bond behavior between the deformed bar and the concrete, as shown in Fig. 5.18d. But no plateau was observed experimentally by Eligehausen et al. [17].

Fig. 5.18

Common local bond stress–slip models: a piecewise uniform distribution; b multi-linear distribution; c logarithmic distribution; d CEB Model Code suggested distribution

In order to reflect the local bond behavior between deformed steel bars and concrete [20, 25], the CEB Model Code recommends an improved distribution of Fig. 5.18d, as shown in Fig. 5.19. The bond transfer is divided into three different zones: a nonlinear increase zone until the bond stress reaches a maximum, a linear decrease zone of the bond stress, and the zone of constant residual strength. The plateau with the maximum bond stress is removed from the improved model. As mentioned earlier, the bond mechanism is similar to that of deformed steel bars. Therefore, the improved local bond stress–slip model between deformed steel bars and concrete is also applicable to describe the bond behavior between steel strands and concrete.

Fig. 5.19

Improved local bond stress–slip model

The bond mechanism and the bond failure mode can be described by this model. The bond force and the bond stiffness are provided by the adhesive force. With the increase of slip, the adhesive force gradually disappears. However, the slip and rotation of steel strands are resisted by the concrete, resulting in the frictional and mechanical interlock forces at the steel–concrete interface. The bond stress increases with the increase of slip and gradually increases to a maximum value. Subsequently, the local crushing and micro-cracks appear on the confining surface of the concrete. The bond stress gradually decreases until the slip is too large, and the concrete is sheared or crushed. At this time, the bond force is only provided by the longitudinal friction force, its value is small and basically constant.

On the basis of CEB Model Code, the improved bond stress–slip model can be written as:

$$\tau = \left\{ {\begin{array}{*{20}l} {\tau_{\max } \left( {s/s_{2} } \right)^{\alpha } } & {0 \le s \le s_{2} } \\ {\tau_{\max } - \left( {\tau_{\max } - \tau_{f} } \right)\left( {\frac{{s - s_{2} }}{{s_{3} - s_{2} }}} \right)} & {s_{2} \le s \le s_{3} } \\ {\tau_{f} } & {s_{3} \le s} \\ \end{array} } \right.,$$

(5.46)

where $\tau_{\max }$ and $\tau_{f}$ are the maximum bond stress and the residual friction stress, respectively; $\tau_{f} = 0.4\tau_{\max }$, $\alpha$, $s_{2}$ and $s_{3}$ are the constants.

For the deformed steel bar, the maximum bond stress $\tau_{\max}$ is calculated as 1.25 $\sqrt {f_{\text{ck}} }$ and 2.5 $\sqrt {f_{\text{ck}} }$ corresponding to the good bond conditions and other bond conditions, respectively. $f_{\text{ck}}$ is the characteristic compression stress of the concrete. $\tau_{f}$ is calculated as $0.4\tau_{\max }$. $\alpha$ and $s_{2}$ can be taken as 0.4 and 3 mm, respectively. $s_{3}$ represents the clear spacing between the two ribs of the deformed bar. As mentioned earlier, the bond mechanism of strands embedded in concrete is very similar to that of the deformed bars. Therefore, this section approximately adopts parameters, which are similar to that of the deformed steel bars to characterize the local bond characteristics of steel strands. Since there is no ribs in steel strand, it is assumed that $s_{3}$ is represented as the half of distance between the adjacent wires and concrete gear, $s_{3} = 0.5s_{l}$, $s_{l} = s_{t} /\sin \alpha$, as shown in Fig. 5.20.

Fig. 5.20

Bond mechanism: a strand embedded in concrete; b deformed bar embedded in concrete

The parameters were applied to the local bond stress–slip model and calibrated using the beam PS0. The maximum bond stress $\tau_{\max }$ is taken as $1.25\sqrt {f_{\text{ck}} }$ and $s_{3}$ is taken as 12.0 mm. It can be seen from Fig. 5.21 that the computed load–displacement curve at the loading end is in good agreement with the experimentally obtained load–displacement curve at the loading end. The predicted failure mode of the beam PS0 is also consistent with the experimental result (i.e., the strand broken). It illustrates that the local bond stress–slip model and the assumed parameters are applicable for the uncorroded strand embedded in concrete.

Fig. 5.21

Experimental and computed load–displacement curves for specimen without corrosion

For the corroded beams, these parameters need to be recalibrated. As mentioned earlier, strand corrosion not only influences the bond strength, but corrosion also influences the bond stiffness. Based on the Eq. (5.46), the bond stiffness is related to the maximum bond stress $\tau_{\max}$, and the expression of the local bond stress–slip model is modified for corroded beams.

The maximum bond stress $\tau_{\max }$ for the corroded beam is calibrated using the experimental data. Figure 5.22 shows a comparison between the tested and calibrated load–displacement curves for all the corroded beams. The computed curve and the experimental curve coincide well. This illustrates that the optimized local bond stress–slip model is also suitable for the corroded beams by modifying $\tau_{\max }$.

Fig. 5.22

Experimental and computed force–displacement curves for beams with corrosion: a Beams PS1, PS3, PS5, and PS7; b Beams PS2, PS4, PS6, and PS9

Figure 5.22 shows the maximum bond stress $\tau_{\max }$ for the corroded beams. $\tau_{\max }^{^{\prime}}$ is the maximum bond stress for the uncorroded beams and its value is 1.25 $\sqrt {f_{\text{ck}} }$. $\tau_{\max }$ for the beams PS1 and PS2 with the less corrosion rate are larger than $\tau_{\max }^{^{\prime}}$ for uncorroded beams. When the corrosion is severe, $\tau_{\max}$ for corroded beams is less than $\tau_{\max }^{^{\prime}}$ for uncorroded beams, and $\tau_{\max }$ decreases with the increasing of corrosion loss.

The influence of strand corrosion on bond behavior was investigated by using the normalized maximum bond stress R, which is defined as the ratio of the maximum bond stress of corroded beam to the corresponding original beam, as shown in Fig. 5.23. Figure 5.23 also shows a fitting curve of R versus η. It can be observed from the figure that when the corrosion loss is less than 6%, R can be increased. However, the normalized maximum bond stress decreases significantly as corrosion propagates further. The bond behavior of the corroded PC and RC specimens subjected to the pull-out test have a similar variation law. [5, 8, 14, 18].

Fig. 5.23

Maximum normalized bond strength as a function of corrosion

Based on the existing experimental data in Fig. 5.23, an empirical model is introduced to depict the gradual degradation of bond between corroded strands and concrete:

$$R = \left\{ {\begin{array}{*{20}l} {1.0} & {\eta \le 6\% } \\ {2.03e^{ - 0.118\eta } } & {\eta > 6\% } \\ \end{array} } \right..$$

(5.47)

In order to confirm the accuracy of the empirical model, the predicted load–displacement curves of the beam PS8 were compared with the experimental curves as shown in Fig. 5.24. The predicted curve is relatively close to the experimental curve before the excessive slip occurs in the steel strand. After the strand slip is too large, the increase of the pull-out force is influenced by the uncorroded strand at the free end. This cannot be predicted by the empirical model and thus leads to some prediction errors. Generally, the empirical model has high prediction accuracy for the bond degradation between a corroded strand and concrete.

Fig. 5.24

Predicted and experimental load–displacement curves for beam SP8

The study found an increase in bond strength for slightly corroded strands. The increase in bond strength is conducive for concrete members. The empirical model does not account for the increase in bond strength. When the corrosion loss is smaller than 6%, R for the steel strand is assumed to remain at the value of 1.0. When the corrosion loss exceeds 6%, R decreases exponentially. The empirical model can be used to predict the residual bond strength of the corroded strand embedded in concrete.

5.5 Conclusions

• 1. The proposed bond strength model in this paper is derived on the basis of the helical-shaped surfaces of the steel strand. Through theoretical analysis, the factors such as the interfacial confining stress, compressive strength of concrete, diameter of strand, and friction coefficient are considered and applied into the model.

• 2. A comparison between the predicted results and the experimental results from the literature shows that the empirical model can reasonably predict the ultimate bond strength. The prediction accuracy can be improved by considering the strand rotation.

• 3. On the basis of the helical-shaped characteristics of corroded strand, the bond strength model between corroded strand and cracked concrete is proposed. The contributions of corrosion pressure, adhesion, and confinement to the ultimate bond strength of corroded strand are considered.

• 4. The ultimate bond strength is in an increasing state at the initial stage of concrete cracking. The ultimate bond strength decreases when the crack penetrates into about 2/3 of the concrete cover or the corrosion loss of beams exceeds 0.8%. With the further expansion of the concrete cracking, the ultimate bond strength gradually decays to the residual value.

• 5. A simplified empirical model has been proposed to predict residual bond strength between corroded strand and concrete. When the corrosion loss is less than 6%, the normalized maximum bond stress is assumed to remain at the value of 1.0. When the corrosion loss exceeds 6%, the normalized maximum bond stress decreases exponentially.