Bearing Capacity Prediction of Corroded PT Beams Incorporating Grouting Defects and Bond Degradation

Abstract

Corrosion can reduce the sectional area of strand, induce concrete cracking, and degrade bond strength, which can further decrease the bearing capacity of prestressed concrete beams. First, an analytical model is proposed to predict the flexural bearing capacity of partially ungrouted post-tensioned concrete beams. Then, a prediction model of corroded prestressed concrete beams is established to calculate the incompatible deformation between strand and concrete under different load conditions.

9.1 Introduction

Grouting is a critical program in the manufacture prestressed concrete (PC) beams, which can avoid the effect of environment erosion and improve the bearing capacity of PC beams. Some adverse voids can be caused by the poor construction, water diversion evaporation, and retention of air pockets in the grouting duct [25, 26]. Grouting problems can be discovered in PC beams used for long time [19, 23, 28, 31].

The bond degradation is also detrimental to the bearing capacity of PC beams. It can cause corroded strands to slip out of the surrounding concrete, causing incompatible strains and reducing the bending capacity of PC beams. [12]. More is worse, the bond degradation could lead to prestress loss and cause the anchorage problem of prestressing strands, which could cause the early failure of PC beams [24].

The prediction of the bearing capacity of PC beams under corrosion is noted by experts [7, 20]. The bond performance of prestressed tendons and concrete in the defective grouting area is deteriorated, and the coordination is degraded. The strain deformation of beams no longer satisfies the plane section assumption, but it can be determined by the overall deformation of the entire defective grouting section [14]. Some scholars have studied the calculation of flexural performance of unbonded structural members by some finite element calculation methods and established the calculation model of strain and stress increment of unbonded prestressed tendon considering the applied force [1, 14, 29].

The calculation of defective grouting is more complex than that of uncoordinated structural parts, which is also affected by the different types, positions, and lengths of defective grouting. Cavell et al. [8] calculated the bearing capacity of members considering the effects of local non-grouting, and the corrosion fracture of prestressed tendons. However, the analytical model merely considers the case of corroded prestressed tendons, but the effect of performance degradation of corroded prestressed tendons is not considered. There is not still a calculation method of structural deformation considering local grouting defects and corrosion.

The strand corrosion is the key factor for the degradation of structural performance in concrete. Strand corrosion can cause reduction in the section zone of strand, concrete cover cracking and bond degradation, and the deterioration of mechanical properties of materials. These factors will weaken the bearing capacity of concrete beams. Furthermore, corrosion in PC beams often causes the brittle failure without warnings under high stress state, which is more dangerous than reinforced concrete (RC) members.

As the corrosion loss increases, the failure mode of the specimen changes from bending failure to shear failure [4, 9, 32]. Compared to the researches on the flexural strength of corroded RC beams, few researches focus on the bearing strength of PC beams under corrosion. Some experimental researches have researched the influence of corrosion on cracking, stiffness, ductility, ultimate strength, and failure mode of PC members [8]. It was found that the corrosion loss can reduce the number of bending cracks and increase the crack distance, and decrease the ultimate force [5]. When the corrosion loss reached 20%, the bearing capacity decreased by 67% [6]. Some investigations also studied the bearing capacity of pretensioned members in aging concrete bridges, such as twenty-eight 25-years old concrete panels [16] and two concrete girders serviced at least 45 years [15]. These girders were loaded to study the degradation of flexural behavior, which point out that the strand corrosion can reduce the ultimate capacity of strand and the ductility of strand, and the failure mode will also be changed.

The flexural capacity of RC beams after corrosion should be accurately predicted, which is important for making maintenance and reinforcement decisions. Currently, however, the existing prediction theories consider bond degradation by means of empirical factors [1, 18, 27]. These empirical factors [1, 18, 27] were derived from experimental results, therefore, the effectiveness of these factors is limited. The bond behavior loss between the corroded reinforcement and concrete is considered to predict the flexural capacity of beams by Eyre and Nokhasteh [34] and Cairns and Zhao [35]. Bhargava et al. [4] discussed the ultimate capacity caused by the bond failure of strand at the mid-span of specimens. El Maaddawy et al. [36] have researched the reinforcement strain between the two bending cracks for the prediction of flexural capacity considering the bond degradation effects, while the analytical model ignores the slip of reinforcements in the different condition of cracks.

Corrosion can reduce the sectional zone of strand, degrade bond strength, induce concrete cracking and deteriorate the material property, which can further decrease the bearing capacity of PC beams. The combined effects of these factors should be reasonably considered in the bending capacity prediction model. How to reasonably consider the effect of the above elements on the bearing capacity of PC beams considering corrosion still needs to investigate further.

This chapter is organized as follows. An analytical model is developed to predict the flexural capacity of locally ungrouted PT beams at first. Next, a model is built to assessment the flexural capability of PT beams considering corrosion-induced bond degradation. Finally, the conclusions are discussed.

9.2 Analytical Model for Flexural Capacity of PT Beams

9.2.1 Simplified Calculation Method

For locally ungrouted PC beams, an analytical model was employed to assessment the residual flexural capacity of PC beams with local grouting. Incompatible strains, the strand zone loss, and material degradation due to corrosion were considered in the analytical model. A new method is developed to predict local rigidity degradation for asymmetric deformation of beams.

The ungrouted duct is the unbonded zone. Within the unbonded zone, the strain of strand is incompatible with the strain of the concrete, but the total elongation of the strand in the ungrouted zone should be equivalent to the elongation of the corresponding concrete. Based on this law, the strain of strand in the ungrouted duct can be obtained from the calculations mentioned below.

To simplify the calculation of strain, fully grouted and ungrouted ducts can be considered as completely fully bonded and unbonded zones, respectively. In the bonded zone, local slipped between the strand and the concrete was not considered. Next, the PC structure is separated into some segments, as shown in Fig. 9.1. The fully grouted segments are numbered from 1 to ${ }e$, $f{ }$ to ${ }m$, and $n{ }$ to ${ }g$. The ungrouted segments are numbered from ${ }e{ }$ to ${ }f{ }$ and $m{ }$ to ${ }n$. These arbitrary segments in fully grouted and ungrouted zones are denoted by ${ }i{ }$ and ${ }j{ }$, respectively.

Fig. 9.1

Segment division of beams

The stress–strain distribution in the segment ${ }i{ }$ considering the applied force, numbered as $n\sim m$, is shown in Fig. 9.2. Combing the above theories, the strain increment ($\Delta \varepsilon_{{{\text{pc}},i}}$) of concrete at the strand location can be computed as

$$\Delta \varepsilon_{{{\text{pc}},i}} = \left( {\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{cbe}},i}} } \right) + \frac{{d_{{\text{p}}} }}{h}\left[ {\left( {\varepsilon_{{{\text{ct}},i}} - \varepsilon_{{{\text{cte}},i}} } \right) - \left( {\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{cbe}},i}} } \right)} \right],$$

(9.1)

Fig. 9.2

Stress–strain distribution

where $\varepsilon_{{{\text{ct}},i}}$ and $\varepsilon_{{{\text{cb}},i}}$ are the strains of concrete at the top and bottom fibers, respectively; $\varepsilon_{{{\text{cte}},i}}$ and $\varepsilon_{{{\text{cbe}},i}}$ are the compressive strains of concrete at the top and bottom fibers, respectively; $h$ is the beam height; d_p is the distance from the top fiber of concrete to the centroid of strand.

The elongation of the strand in the ungrouted duct is equivalent to the total elongation in the corresponding concrete, which is calculated as

$$\Delta L_{{\text{w}}} = \Delta L_{{\text{c}}} = \mathop \sum \limits_{i = n}^{m} \Delta \varepsilon_{{{\text{pc}},i}} l_{{{\text{w}},i}} ,$$

(9.2)

where $\Delta L_{{\text{w}}}$ and $\Delta L_{{\text{c}}}$ are the total elongation of the strand and corresponding concrete in the ungrouted duct, respectively; $l_{{{\text{w}},i}} { }$ is the length in the arbitrary segment i; $n$ and $m$ are the first and last segment in the ungrouted duct.

In the ungrouted zones, the friction between the strand and the concrete is usually ignored. Next, the average strain increment of strand in any ungrouted segment ${ }i$ ($\Delta \varepsilon_{{{\text{p}},i}}$) can be expressed as

$$\Delta \varepsilon_{{{\text{p}},i}} = \frac{{\Delta L_{{\text{w}}} }}{{L_{{\text{w}}} }},$$

(9.3)

where $L_{w}$ is the total length in the ungrouted duct.

At any segment j in the fully grouted zone m ~ e, it is usually assumed that the strand is well integrated with the concrete. The strain increment of the strand (${{\Delta \varepsilon }}_{{{\text{p}},j}}$) is equivalent to the strain increment of the corresponding concrete, which is calculated as

$${\Delta }\varepsilon_{{{\text{p}},j}} = \left( {\varepsilon_{{{\text{cb}},j}} - \varepsilon_{{{\text{cbe}},j}} } \right) + \frac{{d_{{\text{p}}} }}{h}\left[ {\left( {\varepsilon_{{{\text{ct}},j}} - \varepsilon_{{{\text{cte}},j}} } \right) - \left( {\varepsilon_{{{\text{cb}},j}} - \varepsilon_{{{\text{cbe}},j}} } \right)} \right],$$

(9.4)

where $\varepsilon_{{{\text{ct}},j}}$ and $\varepsilon_{{{\text{cb}},j}}$ are the concrete strains at the extreme top and bottom fibers, respectively; $\varepsilon_{{{\text{cte}},j}}$ and $\varepsilon_{{{\text{cbe}},j}}$ are the precompressive strains of concrete at the extreme top and bottom fibers, respectively.

The effective prestrain of strand will be induced by improving the strain rate, the strain of the strand ($\varepsilon_{p,k}$) in any segment $k$ is calculated as

$$\varepsilon_{p,k} = {\Delta }\varepsilon_{p,k} + \varepsilon_{pe,k} ,$$

(9.5)

where $k$ is the number of any section, $k = i$ in the ungrouted zone, and $k = j$ in the fully grouted zone; ${ }\varepsilon_{{{\text{p}},k}}$ is the strain of the strand; ${\Delta }\varepsilon_{{{\text{p}},k}}$ is the strain increment of the strand; $\varepsilon_{{{\text{pe}},k}} { }$ is the effective prestrain of the strand.

The tensile force of the strand under the applied load is associated with its cross-sectional zone, the strain of the strand, and intrinsic structural laws. The proposed constitutive law can be used for tensile testing of corroded strands. Furthermore, the maximum corrosion loss can usually be used to measure the remaining cross-sectional zone of the strand at the equivalent force of strand in the ungrouted zone. Next, the tension force of strand in any segment $k$ is calculated as

$$T_{{{\text{p}},k}} = \sigma \left( {\varepsilon_{{{\text{p}},k}} {}\rho } \right)A_{{\text{p}}} \left( {1 - \rho } \right),$$

(9.6)

where $T_{{{\text{p}},k}}$ is the tension force of strand; $\sigma \left( {\varepsilon_{{{\text{p}},k}} {}\rho } \right)$ is the stress of strand, which is associated with the strain of strand and the corrosion loss of strand which can be calculated by Eq. (9.1); $\varepsilon_{{{\text{p}},k}}$ is the strain of the strand; $A_{{\text{p}}}$ is the initial zone of the strand.

Based on the above-mentioned sectional assumption, the strain of longitudinal bars and hanger bars in the cross-sectional $k$ can be computed as

$$\varepsilon_{{{\text{s}},k}} = \varepsilon_{{{\text{ct}},k}} + \frac{{\varepsilon_{{{\text{cb}},k}} - \varepsilon_{{{\text{ct}},k}} }}{h}h_{0}$$

(9.7)

$$\varepsilon_{{{\text{s}},k}}^{^{\prime}} = \varepsilon_{{{\text{ct}},k}} + \frac{{\varepsilon_{{{\text{cb}},k}} - \varepsilon_{{{\text{ct}},k}} }}{h}a_{{\text{s}}}^{^{\prime}} ,$$

(9.8)

where $\varepsilon_{{{\text{s}},k}}$ and $\varepsilon_{{{\text{s}},k}}^{^{\prime}}$ are the strains of longitudinal bars and hanger bars, respectively; $h_{0}$ and $a_{{\text{s}}}^{^{\prime}}$ are the distances from the extreme top fiber of concrete to the centroids of longitudinal bars and hanger bars, respectively.

The elastic–plastic model can be employed to assessment the stress–strain relationship of steel bar. The force of longitudinal bars ($F_{{{\text{s}},k}}$) and the force of hanger bars ($F_{{{\text{s}},k}}^{^{\prime}}$) can be computed as

$$F_{{{\text{s}},k}} = A_{{\text{s}}} E_{{\text{s}}} \varepsilon_{{{\text{s}},k}} \le f_{{{\text{sy}}}} A_{{\text{s}}}$$

(9.9)

$$F_{s,k}^{^{\prime}} = A_{s}^{^{\prime}} E_{s}^{^{\prime}} \varepsilon_{s,k}^{^{\prime}} \le f_{sy}^{^{\prime}} A_{s}^{^{\prime}} ,$$

(9.10)

where A_s, E_s, f_sy are the segment zone, elastic modulus, yield strength of longitudinal bars, and $A_{{\text{s}}}^{^{\prime}} , E_{{\text{s}}}^{^{\prime}}$, $f_{sy}^{^{\prime}}$ are the segment zone, elastic modulus, yield strength of hanger bars, respectively.

In this study, the nonlinear composition method of concrete proposed by Zhang et al. [33] was used. The concrete composition law ignores the concrete tension, which can be expressed as

$$f_{{\text{c}}} = \left\{ {\begin{array}{*{20}l} {f_{{\text{c}}}^{^{\prime}} \left[ {2\left( {\varepsilon /\varepsilon_{0} } \right) - \left( {\varepsilon /\varepsilon_{0} } \right)^{2} } \right],} \hfill & { \varepsilon \ge 0} \hfill \\ {0,} \hfill & {\varepsilon < 0} \hfill \\ \end{array} } \right.,$$

(9.11)

where $f_{{\text{c}}}^{^{\prime}}$ is the specified compressive strength of concrete, and $\varepsilon_{0}$ is the corresponding strain of the steel strand.

Combining the ultimate stress of the fibers at the bottom of the concrete with the ultimate stress of the fibers at the top of the concrete is an effective method to calculate the compression force of the concrete, and it can be expressed as

$$F_{{{\text{c}},k}} = \mathop \int \limits_{0}^{h} f_{{\text{c}}} b{\text{d}}y,$$

(9.12)

where $F_{{{\text{c}},k}}$ is the compression force of the concrete; $b$ is the width of the beam.

$$Y_{k} = \frac{{\mathop \int \nolimits_{0}^{h} f_{{\text{c}}} by{\text{d}}y}}{{F_{{{\text{c}},k}} }},$$

(9.13)

where $Y_{k}$ is the distance from the centroid of the concrete compression force to the extreme top fiber of the concrete compression.

For the given force condition, the applied bending moment is known in each segment. In any segment $k$, the total forces of the strand, hanger bars, longitudinal bars, and concrete is equivalent to be a constant of zero. Moreover, the total moment of each segment is equivalent to the applied moment in the force condition. The moment of bending, the force of strand, the force of concrete, the force of longitudinal bar, and the force of hanger bar should satisfy the below calculation equations

$$F_{{{\text{c}},k}} + F_{{{\text{s}},k}}^{^{\prime}} + F_{{{\text{s}},k}} + T_{{{\text{p}},k}} = 0$$

(9.14)

$$M_{{{\text{s}},k}} = F_{{{\text{c}},k}} \left( {d_{{\text{p}}} - Y_{k} } \right) + F_{{{\text{s}},k}}^{^{\prime}} \left( {d_{{\text{p}}} - a_{{\text{s}}}^{^{\prime}} } \right) + F_{{{\text{s}},k}} \left( {h - d_{{\text{p}}} - a_{{\text{s}}} } \right),$$

(9.15)

where $M_{{{\text{s}},k}}$ is the bending moment; $a_{{\text{s}}}$ is the distance from the extreme bottom fiber of concrete to the centroid of longitudinal reinforcement bars.

9.2.2 Calculation Procedure

Resolving the ungrouted segment is not easy as it is determined by the total elongation of the concrete in the ungrouted zone. Therefore, a method of iterative is proposed to resolve the segment as per the following instructions. For fully grouted segments, the variables are independent of each other, and the strain of the concrete in Eqs. (9.14) and (9.15) can be resolved quickly by the plane section assumption. First, the flexural moment of the concrete ($M_{{{\text{s}},k}}$) can be calculated for the applied force ($P$), and the tension in each duct segment within the ungrouted zone ($T_{{{\text{p}},k}}$) is assumed to be similar. Then, the concrete strains ($\varepsilon_{{{\text{ct}},k}}$ and $\varepsilon_{{{\text{cb}},k}}$) can be calculated by using the above calculation equations (Eqs. 9.14 and 9.15). According to this above criterion, the strand force ($T_{{{\text{p}},k}}^{^{\prime}}$) can be computed by Eqs. 9.1–9.6. Revising $T_{{{\text{p}},k}}$ and repeating the procedure until $T_{{{\text{p}},k}}$ is equivalent to $T_{{{\text{p}},k}}^{^{\prime}}$. The corresponding of the strain of concrete, the force of strand, the force of longitudinal reinforcements, the force of hanger reinforcements, and the force of the concrete can be derived from the plane section assumption.

The cross-sectional curvature is a feasible way to tackle the deflection of PC beams. The existing models have been proposed to calculate the bearing capacity by cross-sectional curvature [16]. These works mainly study the symmetric deflection of the structure. The deformation of locally ungrouted beams considering corrosion is shown in Fig. 9.3. Therefore, the proposed model is suitable to assessment the deflection of ungrouted beams.

Fig. 9.3

Description of deflection prediction of specimens

This paper presents a two-step method for predicting the asymmetric deformation of corroded beams. First, the trial-and-error method is used to determine the maximum deformation and its location. Second, combining the section curvature is an effective method to assessment the deformation of the other section.

According to the strain distribution of the cross-section of beams in Fig. 9.2, the cross-section of curvature at any segment $k$ is calculated as

$$\phi_{k} = \frac{{\left( {\varepsilon_{{{\text{cb}},k}} - \varepsilon_{{{\text{cbe}},k}} } \right) - \left( {\varepsilon_{{{\text{ct}},k}} - \varepsilon_{{{\text{cte}},k}} } \right)}}{h},$$

(9.16)

where $\phi_{k}$ is the sectional curvature; $\varepsilon_{{{\text{cte}},k}}$ and $\varepsilon_{{{\text{cbe}},k}}$ are the precompressive strains of concrete at the extreme top and bottom fibers, respective

It can be assumed that the maximum deformation location occurs at segment $q$ (see Fig. 9.3), which can be computed as

$$\Delta_{{{\text{max}},ql}} = \mathop \sum \limits_{k = 1}^{q} \phi_{k} l_{w,k} x_{l,k}$$

(9.17)

$$\Delta_{{{\text{max}},qr}} = \mathop \sum \limits_{k = q + 1}^{g} \phi_{k} l_{{{\text{w}},k}} x_{{{\text{r}},k}} ,$$

(9.18)

where $\Delta_{{{\text{max}},ql}}$ and $\Delta_{{{\text{max}},qr}}$ are the maximum deflections calculated from the left and right support points, respectively; $x_{l,k}$ and $x_{{{\text{r}},k}}$ are the distances from left and right support points to segment $k$, respectively; $l_{{{\text{w}},k}}$ is the length of segment k; $g$ is the segment number.

If $q$ is the segment of the true maximum deflection, both deformations can be obtained by Eqs. (9.17) and (9.18), which must be equal. Otherwise, this relationship is invalid. According to the trial difference method, the maximum deformation of any segment can be computed by the above-mentioned law. First, the maximum deformation in any segment can be considered as a constant. Next, the deformations can be obtained by using the two formulas mentioned above. The above two calculated maximum deflections are calibrated and the segment is corrected until the maximum deflection in the left support points and the maximum deflection in the right support points to be equal. The maximum deformation can be expressed as

$$\Delta_{{{\text{max}},q}} = \Delta_{{{\text{max}},ql}} = \Delta_{{{\text{max}},qr}} ,$$

(9.19)

where $\Delta_{{{\text{max}},q}}$ is the maximum deformation of beams.

After the above calculation steps described, the maximum deflection and the position can be calculated. Then, the deformation of the segment $p$ (see Fig. 9.3) can be expressed as

$$\Delta_{{\text{p}}} = \Delta_{{{\text{max}},q}} - \mathop \sum \limits_{k = p}^{q} \phi_{k} l_{{{\text{w}},k}} X_{{{\text{p}},k}} ,$$

(9.20)

where $\Delta_{{\text{p}}}$ is the deflection at segment p; $X_{{{\text{p}},k}} { }$ is the distance from the segment p to the segment $k$.

If the calculation starts from a smaller loading condition, which can be corrected with the increase of the given loading. The strains in the concrete, the forces in the strand, the force of longitudinal beam, the force of the concrete as well as the deflection of the deformed segment can be obtained from the overall bending response of the beam. The ultimate state of PC beams can be computed as the following two conditions. For example, when (1) the concrete strain at the top fiber ($\varepsilon_{{\text{ct,max}}}$) exceeds the ultimate compressive strain ($\varepsilon_{{{\text{cu}}}}$); or (2) the strand strain ($\varepsilon_{{\text{p,max}}}$) reaches the ultimate strain ($\varepsilon_{{{\text{pu}}}}$). The calculation procedure is shown in Fig. 9.4.

Fig. 9.4

Calculation procedure of flexural capacity of PC beams

9.3 Model Validation

The proposed model can effectively assess the flexural performance of the test beam. The beams are segmented into 200 segments with 9 mm length each. The initial force can be set to be a constant of 1 kN and loaded with an increase of 1kN per step. The ultimate compressive strain of concrete ($\varepsilon_{{{\text{cu}}}}$) can be taken as a constant of 0.0035. The stress–strain relationship of corroded strands can usually be decided by the proposed constitutive law in Eq. (9.1). The yield of the initial strain and the final strain was 0.0094 and 0.028, respectively.

The results show that the prediction of the models is in good consistent with the experimental results for PC beams. The experimental errors in the prediction of ultimate deformation are due to uncertainty in the evaluation of corrosion loss of the strand and material degradation. As described above, for PC beams with severe corrosion, the ultimate strain of the strand could usually be taken to keep it constant.

Figure 9.5 shows the load–deformation curves of PC beams under the similar corrosion loss. Because of the existing condition, the four beams (PD1, PD4, PD7, PD8) can be selected by the following comparisons. The experimental and predicted rotation angles are shown in Fig. 9.6. The theoretical rotation angles in the ungrouted end and the fully grouted end are in well accordance with the experimental rotation angle in the ungrouted end and the fully grouted zone.

Fig. 9.5

Difference condition of load–deformation response: a PD8; b PD3; c PD4; d PD7; e PD6; f PD5; g PD1; h PD2

Fig. 9.6

Experimental and predicted load-rotation angle curves: a Load; b Corrosion loss

Figure 9.7 shows the prediction of compressive stress in concrete for four beams considering the ultimate force state. For comparison purposes, the figure also shows the existing cracking pattern. The ungrouted end has a different stress distribution as compared with the fully grouted end. The compressive stress at the fully grouted end is less than the compressive stress at the ungrouted end, and the pressure zone depth is greater than the compressive stress at the ungrouted end. These differences will become significant with the increase of corrosion losses. This phenomenon is in good agreement with cracking patterns. These cracks at the ungrouted end extend deeper into the pressure region than the cracks at the fully grouted end. Through the above-mentioned comparisons, the developed stress distribution model can reasonably predict the stress distribution of the material.

Fig. 9.7

Prediction of concrete stresses and crack patterns: a PD8; b PD4; c PD7; d PD1

Table 9.1 gives the experimental and predicted ultimate strengths of specimens. The ultimate strength prediction error in the experimental beams is less than 5%. The prediction error can be obtained by three aspects: variability of material properties, experimental error, and the error of model. Overall, the proposed model can reasonably assess the degradation of bending performance of beams after corrosion. The asymmetric deformation of the PC structure is computed considering the distribution of the stress and ultimate strength of specimens.

Table 9.1 Experimental data and verification results

9.4 Quantification of Corrosion-Induced Uncoordinated Deformation in Bond–Slip Zone

9.4.1 Quantification Principle of Bond–Slip Zone

The bond stress between strand and concrete is related to the associated slip value. Adhesion action is an effective way to provide the initial bond stress before bond–slip zone. As the slip increases, the adhesion stress can be supplied by frictional and gear forces. The friction force and gear force will increase until the excessive slip shows that the concrete gear has been sheared off. Then, the longitudinal friction force can provide the bond force only, but the value is small. Thus, the effective bond force between the strand and concrete can be primarily supplied by the bond force and gear force before over bonding slips, and the corresponding zone is named as effective bond region. On the other hand, the bond force after the excessive slip is named as the residual bond force which is supplied by the small friction force between strand and concrete, and its region is called as the slip region.

Figure 9.8 shows the changes of effective bond force in the longitudinal direction with the increase of the applied force. The adhesive force will increase with the increase of the applied force ($F_{{\text{p}}}$). In the effective bond zone, the corresponding slip value is small. The effective bond region starts to move when the force of the strand ($F_{p}$) exceeds the total of effective bond force and prestressed force of the tendon ($F_{{{\text{eb}}}} + F_{{{\text{pe}}}}$). An increase in the slip zone area results in an increase in residual adhesion force. The effective bond zone is moved until the total of the effective bond force, the residual bond force, and the prestressed force of the tendon ($F_{{{\text{eb}}}} + F_{{{\text{pe}}}} + F_{{{\text{rb}}}}$) is equivalent to the force of the strand ($F_{{\text{p}}}$). The above-mentioned theories can be used to define the extent of slip region in aging PC beams.

Fig. 9.8

Different shifting condition of bond force: a before shifting; b after shifting

The corrosion loss of strand and $F_{{\text{p}}}$ can determine the range of slip region, which can be found specifically in Fig. 9.9. The bond between prestressed strand and concrete in uncorroded and slightly corroded PC beams is better than that in PC beams with severe corrosion. Throughout the entire loading procedure, the total of effective bond force and prestressed force ($F_{{{\text{eb}}}} + F_{{{\text{pe}}}}$) exceeds the maximum tensile force ($F_{{\text{p}}}$). The effective bond zone remains unchanged without excessive slip, which can be shown without bond–slip in Fig. 9.9a. The strains in the strand and concrete are determined by the plane section assumption. With the force increases, the tension force in the strand ($F_{{\text{p}}}$) may be greater than the sum of effective bond force and prestressed load ($F_{{{\text{eb}}}} + F_{{{\text{pe}}}}$). In Fig. 9.9b, the effective bond zone moves toward the beam end. This movement decreases the tension in the prestressed steel, but increases the overall bond force, and it is stopped until the tension of the strand is equivalent to the effective bond force. Next, the range of the slip region is determined according to the above-mentioned theory. The strains in the strand and concrete within the slip zone satisfy the condition that the total elongation of the strand is equivalent to the elongation of the corresponding concrete in the slip zone, but it does not obey the plane section assumption. The strains in the strand and concrete can be evaluated according to the above-mentioned theory, which is shown in details in the next segment.

Fig. 9.9

Determination of the bond–slip region

It points out that the effective bond zone can move toward the beam end as the applied force increases, as shown in Fig. 9.9c. Under this condition, the post-tensioned segments in the slip regions were varied from the pretensioned segments in the slip regions. The pretensioned strand conveys the bond force directly to the surrounding concrete, which is stopped until the bond length exceeds a length from the end beam. When the effective bond zone moves to the end of the PC beam, the post-tensioned of the strand continues to be anchored in the beam segment. As a result, the extent of the slip zone can be determined by subtracting the two lengths in the effective zone from the total length of the member. The tension force near the effective bond zone of the strand increases as the force increases, and the strain on the strand and the concrete can also be obtained from their equivalent total elongation of the slip zone. And, the corresponding force of the strand should be named as the bearing capacity in the beam. For pretensioned prestressed members, the strand anchorage would fail instantly when the effective bond zone moves to the PC beams end [5, 21, 22], as shown in Fig. 9.9d.

9.4.2 A Quantitative Method for Uncoordinated Deformation

As mentioned before, the solution of deflection response of the beam relies on the applied force $\left( {F_{{\text{p}}} } \right)$ and corrosion loss. The uncorroded, slightly corroded, and severely corroded beams will not slip in a small provided force. According to these conditions, the method of plane section assumption will be used to calculate its flexural response. For the severely corroded beams in a very large force condition, which it will occur in the excessive slip. The slip zone can be decided before it is discussed. In this condition, the strand strains and the concrete strains will not obey the plane section assumption, but the total strand elongation is equivalent to the corresponding concrete in the slip zone. The flexural response of PC beams under corrosion can be obtained from the next section by the above-mentioned theories.

To optimize the flexural response of the calculation process, the beam is divided into several any segments to obtain the total elongations of the strand and concrete, and it can be shown in Fig. 9.10. These segments are numbered from 1 to g. The slip zone length is set to L_s, and these segments within the slip zone are marked from m to n. The cross-segments of the entire bending zone can be numbered from e to f. The cross-sectional length is defined as l_s,i. The average values of the cross-sections are used to express the strain–stress relationships in the arbitrary segment.

Fig. 9.10

Segment division of specimens

Zhang et al. [3] proposed the constitutive law for strand with the difference of the corrosion losses of PC beams, and it is employed to characterize the mechanical properties of the strand. All strands can be assumed to have the equivalent stress–strain curve before it has yielded. Then, the stress–strain curve will be changed with the increase of corrosion loss. When the strand corrosion loss is smaller than the critical value ($\eta_{{\text{c}}}$), the strand can enter the stiffening phase. When the corrosion loss is less than the critical value, the ultimate strains of them decreases linearly with the corrosion loss increases. The further corroded strand fails instantly after it has yielded.

According to the experimental results, the corrosion loss ($\eta_{{\text{c}}}$) was adopted as the value of 11%. The principal structure law for the strand considering the difference of corrosion losses, which is described as follows:

$$f_{{\text{p}}} = \left\{ {\begin{array}{*{20}l} {E_{{\text{p}}} \varepsilon ,~~} \hfill & {\varepsilon \le \varepsilon _{{{\text{py}}}} } \hfill & {} \hfill \\ {f_{{{\text{py}}}} + E_{{{\text{pp}}}} \left( {\varepsilon - \varepsilon _{{{\text{py}}}} } \right),} \hfill & {\varepsilon _{{{\text{py}}}}< \varepsilon \le \varepsilon _{{{\text{pu}}}} - \frac{\eta }{{\eta _{{\text{c}}} }}\left( {\varepsilon _{{{\text{pu}}}} - \varepsilon _{{{\text{py}}}} } \right)} \hfill & {\eta \le \eta _{{\text{c}}} } \hfill \\ {E_{{\text{p}}} \varepsilon ,} \hfill & {\varepsilon \le \varepsilon _{{{\text{py}}}} } \hfill & {\eta > \eta _{{\text{c}}} } \hfill \\ \end{array} } \right\},$$

(9.21)

where $f_{{\text{p}}}$ and $\varepsilon$ are the stress of strand and the strain of strand, respectively;$E_{{\text{p}}}$, $E_{{{\text{pp}}}}$, $\varepsilon_{{{\text{py}}}}$, $\varepsilon_{{{\text{pu}}}}$ are the modulus of the elastic value, the modulus of hardening value, the strain of the yielding, and the ultimate strain, respectively; $f_{{{\text{py}}}}$ is the yield strength of uncorroded strand.

The stress–strain relationship of concrete under compression can be characterized by a parabolic relationship [11], which can be calculated as follows:

$$f_{{\text{c}}} \left( {\varepsilon_{{\text{c}}} } \right) = f_{{\text{c}}}^{^{\prime}} \left[ {\frac{{2\varepsilon_{{\text{c}}} }}{{\varepsilon_{{{\text{c}}0}} }} - \left( {\frac{{\varepsilon_{{\text{c}}} }}{{\varepsilon_{{{\text{c}}0}} }}} \right)^{2} } \right],$$

(9.22)

where $f_{{\text{c}}}^{^{\prime}}$ is the specified compressive strength of concrete and $\varepsilon_{{{\text{c}}0}}$ is the corresponding strain in concrete.

The stress–strain relationship for the steel can be idealized as a linear elastic–plastic, considering strain hardening as a constant of 1% after it has yielded [33], which is expressed as follows

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

(9.23)

where $E_{{\text{s}}}$, $f_{{{\text{sy}}}}$ and $\varepsilon_{{{\text{sy}}}}$ are the modulus of the elastic, the strength of the yield and yield strain of the strand, respectively; $E_{{{\text{sp}}}}$ is hardening modulus of the strand.

For a given force condition, the slip zone can be identified as the effective adhesive force is equivalent to the tension increment near the effective bond–slip zone. In m or n segment, the increment of strand tension force should be equivalent to the effective adhesive force (F_eb). For the segment of m, the corresponding value is defined as

$$F_{{{\text{p}},m}} = F_{{{\text{pe}}}} + \Delta F_{{{\text{p}},m}} = F_{{{\text{pe}}}} + F_{{{\text{eb}}}} ,$$

(9.24)

where F_p,m is the strand tension force of segment m near the effective adhesive region; F_pe is the effective prestressed tension force, and ΔF_p,m is the increment tension force of segment m.

The strand tension force of any segment i in the slip zone is computed as

$$F_{{{\text{p}},i}} = \left\{ {\begin{array}{*{20}l} {F_{{{\text{p}},m}} + L_{{\text{p}}} R\left( \eta \right)\tau_{f} l_{im} } \hfill & {m \le i < e} \hfill \\ {F_{{{\text{p}},m}} + L_{{\text{p}}} R\left( \eta \right)\tau_{f} l_{em} } \hfill & {e \le i < f} \hfill \\ {F_{{{\text{p}},m}} + L_{{\text{p}}} R\left( \eta \right)\tau_{f} \left( {l_{nm} - l_{im} } \right)} \hfill & {f \le i < n} \hfill \\ \end{array} } \right.,$$

(9.25)

where F_p,i is the strand tension force of any segment i; $l_{im}$ is the distance from segment i to segment m; l_em is the distance from segment e to segment m, and l_nm is the distance from segment n to segment m.

According to the proposed constitutive law of strand given in Eq. (9.21), the strand strain in any segment i is defined as follows:

$$\varepsilon_{{{\text{p}},i}} = \left\{ {\begin{array}{*{20}l} {\frac{{F_{{{\text{p}},i}} }}{{\left( {1 - \eta } \right)A_{{\text{p}}} E_{{\text{p}}} }}} \hfill & {F_{{{\text{p}},i}} \le \left( {1 - \eta } \right)A_{{\text{p}}} f_{{{\text{py}}}} } \hfill \\ {\varepsilon_{py} + \frac{{F_{{{\text{p}},i}} }}{{\left( {1 - \eta } \right)A_{{\text{p}}} E_{{{\text{pp}}}} }} - \frac{{f_{{{\text{py}}}} }}{{E_{{{\text{pp}}}} }}} \hfill & {F_{{{\text{p}},i}} > \left( {1 - \eta } \right)A_{{\text{p}}} f_{{{\text{py}}}} } \hfill \\ \end{array} } \right.,$$

(9.26)

where ε_p,i is strand strain in segment i; $A_{{\text{p}}}$ is the segment zone of strand before corrosion.

The sum of the strand elongation in the slip zone, including the sum of the two effective adhesive zones and the elongation of the strand in all slip segments, and it can be calculated as follows

$$\Delta L_{{\text{p}}} = 2s_{3} + \mathop \sum \limits_{i = m}^{n} \left[ {l_{{{\text{s}},i}} \left( {\varepsilon_{{{\text{p}},i}} - \varepsilon_{{{\text{pe}},i}} } \right)} \right],$$

(9.27)

where ΔL_p is the total elongation of the steel strand within the slip zone; ε_pe,i is the prestrain of the strand; s₃ is the slip distance considering a relatively small value within the effective bond zone, as shown in Fig. 9.10. The value of s₃ is decreased by the increase of the corrosion, which is ignored in the current research [10].

All parts of the bending moment (M_s,i) can be quickly obtained with a given force condition. The tension force of the strand (F_p,i) in each segment is calculated by Eq. (9.25). Next, the strains of the concrete at the extreme top fiber and the bottom fiber (ε_ct,i and ε_cb,i) in each segment can be obtained according to the equilibrium equation as the following mentioned content.

It is assumed that the steel reinforcement is not corroded in this study. At this point, the strains of both reinforcement and concrete are in accordance with the plane section assumption, which is shown in Fig. 9.11. The strains in the longitudinal bars and the hanger bars can be calculated as follows:

$$\varepsilon_{{{\text{s}},i}} = \varepsilon_{{{\text{ct}},i}} + \frac{{\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{ct}},i}} }}{h}h_{0}$$

(9.28)

$$\varepsilon_{{{\text{s}},i}}^{^{\prime}} = \varepsilon_{{{\text{ct}},i}} + \frac{{\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{ct}},i}} }}{h}a_{{\text{s}}}^{^{\prime}} ,$$

(9.29)

Fig. 9.11

Strain and stress distribution

where $\varepsilon_{{{\text{s}},i}}$ is the strain in longitudinal bars and $\varepsilon_{{{\text{s}},i}}^{^{\prime}}$ is the hanger bars, respectively; h is the height of the segment; h₀ is the distances from the center of longitudinal bars to the extreme top fiber of beams; and $a_{{\text{s}}}^{^{\prime}}$ is the center of hanger bars to the extreme top fiber of beams.

The force in the longitudinal bars and the hanger bars is calculated as follows

$$F_{{{\text{s}},i}} = A_{{\text{s}}} f_{{\text{s}}} \left( {\varepsilon_{{{\text{s}},i}} } \right)$$

(9.30)

$$F_{{{\text{s}},i}}^{^{\prime}} = A_{{\text{s}}}^{^{\prime}} f_{{\text{s}}} \left( {\varepsilon_{{{\text{s}},i}}^{^{\prime}} } \right),$$

(9.31)

where $F_{{{\text{s}},i}}$ is the force of longitudinal bars and $F_{{{\text{s}},i}}^{^{\prime}}$ is the force of hanger bars; $A_{{\text{s}}}$ is the segment zones of longitudinal bars and $A_{{\text{s}}}^{^{\prime}}$ is the segment zones of hanger bars; $f_{{\text{s}}} \left( {\varepsilon_{{{\text{s}},i}} } \right)$ is the stress of longitudinal bars and $f_{{\text{s}}} \left( {\varepsilon_{{{\text{s}},i}}^{^{\prime}} } \right)$ is the stress of hanger bars. The above-mentioned parameters can be decided by the related strains according to the constitutive law given in Eq. (9.23).

The compression force in the concrete is obtained by integrating the ultimate stress from 0 to h, which can be calculated as the following formula

$$F_{{{\text{c}},i}} = \mathop \int \limits_{0}^{h} f_{{\text{c}}} \left( {\varepsilon_{{\text{c}}} } \right)b{\text{d}}y$$

(9.32)

$$\overline{y}_{i} = \frac{{\mathop \int \nolimits_{0}^{h} f_{{\text{c}}} \left( {\varepsilon_{{\text{c}}} } \right)by{\text{d}}y}}{{F_{{{\text{c}},i}} }},$$

(9.33)

where $F_{{{\text{c}},i}}$ is the compression force of concrete; $f_{{\text{c}}} \left( {\varepsilon_{{\text{c}}} } \right)$ is the concrete stress; $b$ is the beam width; $y$ is the distance from the extreme top fiber of concrete to the any segment of the height; $\overline{y}_{i}$ is the distance from the extreme top fiber to the centroid of concrete equivalent-stress-block.

The forces of the reinforcement, prestressing strands, bending moments, and concrete can meet the following mentioned equilibrium equations in all segments. The strains of concrete at the extreme top fiber (ε_ct,i) and the bottom fiber (ε_cb,i) in each segment are expressed as follows:

$$F_{{{\text{p}},i}} + F_{{{\text{s}},i}} + F_{{\text{s}}}^{^{\prime}} + F_{{{\text{c}},i}} = 0$$

(9.34)

$$F_{{{\text{s}},i}} \left( {h_{0} - d_{{\text{p}}} } \right) + F_{{{\text{s}},i}}^{^{\prime}} \left( {d_{{\text{p}}} - a_{{\text{s}}}^{^{\prime}} } \right) + F_{{{\text{c}},i}} \left( {d_{{\text{p}}} - \overline{y}_{i} } \right) = M_{{{\text{s}},i}} ,$$

(9.35)

where d_p is the distance from the center of the prestressed strand to the extreme top fiber of PC beams.

According to the above-mentioned theories, the increment of the strain in concrete at the position of the strand is calculated as follows:

$$\Delta \varepsilon_{{{\text{pc}},i}} = \left( {\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{cbe}},i}} } \right) + \frac{{d_{{\text{p}}} }}{h}\left[ {\left( {\varepsilon_{{{\text{ct}},i}} - \varepsilon_{{{\text{cte}},i}} } \right) - \left( {\varepsilon_{{{\text{cb}},i}} - \varepsilon_{{{\text{cbe}},i}} } \right)} \right],$$

(9.36)

where $\Delta \varepsilon_{{{\text{pc}},i}}$ is the strain increment of concrete at the position of the steel strand; $\varepsilon_{{{\text{cte}},i}}$ and $\varepsilon_{{{\text{cbe}},i}}$ are the prestrains in concrete at the extreme top fiber and the bottom fiber, respectively.

The sum of the elongation in concrete at the position of the steel strand is obtained as follows

$$\Delta L_{{{\text{pc}}}} = \mathop \sum \limits_{i = m}^{n} l_{{{\text{s}},i}} \Delta \varepsilon_{{{\text{pc}},i}} ,$$

(9.37)

where $\Delta L_{{{\text{pc}}}}$ is the sum of the elongation in concrete at the position of the strand.

Before the sum of the elongation of prestressed strand is equivalent to the sum of the corresponding elongation in concrete, and it can be expressed as

$$\Delta L_{{\text{p}}} = \Delta L_{{{\text{pc}}}}$$

(9.38)

If the above-mentioned strand elongation is not equivalent to the corresponding concrete elongation, the hypothetical length in the slip zone (L_s) could be corrected and the calculation process could be corrected by Eq. (9.37). The strains, stresses, and forces of concrete, reinforcement bars, and prestressed strands can be obtained at the given force condition. As the increase of the given force, the strains of concrete or prestressed strands exceed the corresponding values of the permissible is a crucial way to determine the flexural strength of beams.

9.5 Bearing Capacity Assessment Considering Bond Degradation

9.5.1 Bonding Degradation Model

In the current research, Wang et al. [30] proposed the proposed constitutive law to assess the bond–slip relationship between concrete and strands with corrosion. The proposed constitutive law considering the local bond characteristics is proposed to solve the problem of deformed bar by CEB Model Code [3], and it has been verified to be suitable for the concrete embedded strand [30]. As shown in Fig. 9.12, the bonded transfer zone can be divided into three different zones: the nonlinear zone will continue to increase until the maximum value of the adhesive stress, the linear decreasing region, and the linear invariant region, which can be calculated as follows:

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

(9.39)

Fig. 9.12

Bond stress-slip at the strand–concrete interface

where τ_max is the maximum value of the adhesive stress, and it is denoted as 1.25 $\sqrt {f_{ck} }$ in the good bond condition or denoted as 2.5 $\sqrt {f_{ck} }$ in other bond conditions; f_ck is the standard compressive stress of concrete; τ_f is the stress defined as 0.4τ_max considering residual friction; α is defined as a constant of 0.4; for the corresponding slip zone, s₂ is the maximum bond stress and s₃ is the residual friction stress.

In the current study, an equivalent bi-uniform bond stress model is proposed to simplify the above-mentioned calculation process. The bond stresses are obtained by equating the energy dissipation before the excessive slip and are defined as a constant, which is an equivalent dissipation of energy along the length of strand to satisfy the requirement of the non-uniform distribution, as shown in Fig. 9.12. The equivalent mean adhesive stress (τ_ave) is computed as follows:

$$\tau_{{{\text{ave}}}} = \frac{{\mathop \int \nolimits_{0}^{{s_{2} }} \left[ {\tau_{{{\text{max}}}} \left( {\frac{s}{{s_{2} }}} \right)^{\alpha } } \right]{\text{d}}s + \mathop \int \nolimits_{{s_{2} }}^{{s_{3} }} \left[ {\tau_{{{\text{max}}}} - \left( {\tau_{{{\text{max}}}} - \tau_{{\text{f}}} } \right)\left( {\frac{{s - s_{2} }}{{s_{3} - s_{2} }}} \right)} \right]{\text{d}}s}}{{s_{3} }}.$$

(9.40)

By substituting these parameters into Eq. (9.40), the equivalent mean adhesive stress (τ_ave) is probably computed as:

$$\tau_{{{\text{ave}}}} = \frac{{s_{2} }}{{\left( {\alpha + 1} \right)s_{3} }}\tau_{{{\text{max}}}} + \frac{{s_{3} - s_{2} }}{{2s_{3} }}\left( {\tau_{{{\text{max}}}} + \tau_{{\text{f}}} } \right) \approx 0.7\tau_{{{\text{max}}}} .$$

(9.41)

The increase of strand corrosion loss will reduce the bond strength and cause the early transfer of the effective bond force at a small given force condition. In the current study, Wang et al. [30] have studied the model to assess the degradation of the adhesive stress after the corrosion of the strand. The maximum bond stress is reduced after normalization, and it can be considered as a function of the strand considering corrosion loss, as shown in Fig. 9.13. When the corrosion of strand is lower than 6%, the maximum bond stress of strand will keep a constant. When the corrosion loss exceeds a constant of 6%, the bond stress is exponentially decreased. The function can be expressed as:

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

(9.42)

Fig. 9.13

Normalized bond stress and corrosion loss

where $R\left( \eta \right)$ is the maximum bond stress after normalization;$\eta$ is the corrosion loss of the strand.

The excessive slip normally appears after the strand has yielded. According to ACI 318 [13], the increment of effective bonded length under the strand tension can be expressed as

$$l_{{{\text{eb}}}} = \frac{{f_{{{\text{py}}}} - f_{{{\text{pe}}}} }}{7} d_{{\text{p}}} ,$$

(9.43)

where l_eb is the length in the effective bond zone; f_py is the yield strength of the strand; f_pe is the effective stress of the strand; d_p is the diameter of stranded wire.

The effective bond force in the strand under corrosion is computed as

$$F_{{{\text{eb}}}} = 0.7R\left( \eta \right) \tau_{{{\text{max}}}} L_{{\text{p}}} l_{{{\text{eb}}}} ,$$

(9.44)

where F_eb is the effective bond force of the strand; L_p is the circumference of the strand.

9.5.2 Calculation of Bearing Capacity

For post-tensioned members with poor anchorage or pretensioned members, the bearing capacity is decided by the different of anchorage failure in the strand. Figure 9.14 shows the calculation process of the bearing capacity as the following mentioned content.

Fig. 9.14

Flowchart of the calculation procedure

Stage I: calculate the parameter effective bonding force and divide the plane element.

Effective cohesive force (F_eb) and effective bond length (l_eb) between strand and concrete are calculated. Calculate the maximum length of the corrosion bond–slip zone (L_{S, max}), which is about the beam length minus twice the effective bond length. Then, the maximum bond–slip zone is divided into plane elements.

Stage II: analysis and calculation before effective bond zone slip.

Given a smaller calculated force (P), calculate the bending moment value (M_s,i) of each element. At this time, the force value is small and the concrete beam does not slip. Therefore, the maximum concrete strain, strand strains, and tensile force are calculated according to the plane section assumption. The strand tension will increase with the increase of the calculated load. When it exceeds the sum of the effective bonding force and the effective pretension of the strand (F_pe + F_eb), proceed to the next step of calculation. Otherwise, continue to increase the force and repeat this step. During the calculation, check whether the tensile strain of strands and the compressive strain of the concrete exceed their allowable values, and then determine their flexural capacity.

Stage III: analysis and calculation in the slip process of effective bonding zone.

At this time, the tensile force of the strand near the effective bonding segment is known (F_p,m = F_pe + F_eb), but the length of the bonding slip segment (L_S) is unknown. Therefore, first, assuming the length of the bond–slip segment (L_S) for the strand, calculate the corresponding stress deformation and the corresponding total elongation in the bond–slip segment according to formulas (9.25–9.27) (ΔL_p); then, calculate the concrete strain of each unit and the total elongation of concrete in the bond–slip segment according to formula (9.28–9.37) (ΔL_c).If the calculated total elongation of strand considering the effective bonding segment (ΔL_P) is not equivalent to the calculated total elongation of concrete (ΔL_C), the length in the bond–slip segment (L_c) needs to be assumed again and calculated again until it is equivalent. The value of ${L}_{c}$ will increase with the increase of the calculated force. Once it exceeds its maximum length (L_S,max), proceed to the next step of calculation, otherwise continue to increase the force and repeat this step. During the calculation, check whether the tensile strain of strand and the compressive strain of concrete exceed their allowable values, and then determine their flexural capacity.

Stage IV: analysis and calculation when the effective bonding zone slides to the end.

At this time, the length of bond–slip segment of concrete beam is known (L_S = L_s,max). However, the tensile force (F_p,m) of the strand near the effective bonding segment is unknown. Therefore, first, assuming the tensile force (F_p,m) of the strand at the end, calculate the corresponding stress deformation in the bonding slip segment and the corresponding total elongation according to formula (9.25–9.27) (ΔL_p); then, calculate the concrete strain of each unit and the total elongation of concrete in the bond–slip section according to formula (9.28–9.37) (ΔL_c). If the calculated total elongation of the strand in the effective bonding segment (ΔL_p) is not equal to total elongation of concrete(ΔL_c), it is necessary to re-assume the tensile force (F_{p, m}) of the end strand and recalculate until it is equal. The tensile force (F_p,m) of the steel strand will increase at the beam end with the increase of calculated force. Once the total effective bonding force and the anchorage force at the end of the reinforcement (F_eb + F_p,end) are smaller than F_p,m, the anchorage failure of the test piece occurs. Otherwise, continue to increase the force and repeat this step. During the calculation, check whether the tensile strain of strand and the compressive strain of concrete exceed their allowable values to determine their flexural capacity.

9.5.3 Model Verification

Eight post-tensioned PC beams were employed to verify the feasibility of models. The size of the beam was 150 mm × 220 mm × 2000 mm. The bottom of the beam as anchored by two 8 mm common reinforcement bars. The top of the beam was anchored by 8 mm stirrups with 90 mm spacing, and two 12 mm deformed bars. The bottom and top of beams were prestressed with a 15.2 mm seven-wire strand considering the initial prestressing of 1395 MPa and casted in the 32 mm concrete pipe. The duct was grouted after the prestressed strand has occurred. Acceleration corrosion of the prestressed strand under different time can cause the different corrosion loss. A third point loading test was performed on the beam to study its flexural strength, which is shown in Fig. 9.15.

Fig. 9.15

Beam details (Unit: mm)

The yield strength and ultimate strength were adopted as 1830 MPa and 1910 MPa for prestressed strand, 235 MPa and 310 MPa for the 8 mm plain bars, 335 MPa and 425 MPa for the 12 mm deform bars, respectively. The elasticity modulus of the prestressed strand was adopted as 195 GPa$\text{,}$ and the mild steel bars were adopted as 210 GPa. The strength (f_c), corrosion loss of the strand (η), and the bearing strength of each PC beams under experimental condition (M_exp) can be found in Table 9.2.

Table 9.2 Corresponding parameter values of beams

The flexural strength of tested beams can be decided by the above-mentioned model. The corroded beam is segmented into 200 segments with the length of 10 mm. With a given load of 1kN, each step is calculated in 1 kN increments. The entire shear-flexure span of the effective bond zone could be decided by the above-mentioned calculation method, and the effective bond zone would be displaced during the loading process. According to the above-mentioned condition, the maximum bond stress can be adopted as the large value of 2.5 $\sqrt {f_{ck} }$, and the length of the effective bond zone is corrected to a small value according to the strand transmission length of calculation method proposed by ACI 318 [13], which can be calculated as:

$$l_{{{\text{eb}}}} = \frac{{f_{{{\text{py}}}} - f_{{{\text{pe}}}} }}{21} d_{{\text{p}}} .$$

(9.45)

The predicted flexural strengths (M_cal,p) are shown in Fig. 9.16 and listed in Table 9.1. The prediction of the bearing capacity of beams by the above-mentioned method is a good match with test results, the maximum prediction error is taken as about $\text{15\%}$. The above-mentioned model can provide an effective method to reasonably predict the bearing capacity in corroded PC beams. A simplified theory that considers only the zone loss of the strand and material deterioration is presented in Table 9.2, which predicts the flexural capacity (M_cal,s). The error in the simplified theory slowly becomes larger due to the incompatible strain induced by the bond degradation between the corroded strand and concrete, and the value of the error has an important influence for the bearing capacity of the PC structure as the corrosion loss increases. And, the above-mentioned model gives a crucial method to calculate the incompatible strain of the PC structure. The coordination factor of beams under different force procedures is shown in Fig. 9.16.

Fig. 9.16

Comparison of test and predicted flexural capacity

9.5.4 Effect of Corrosion on Uncoordinated Deformation

As the above-mentioned theory, the bond degradation caused by the corrosion will lead to incompatible strains between strand and concrete, which have an impact on the bearing capacity. For this condition, a compatibility factor value is employed to assess the incompatible strain in these beams. And, the proposed model is employed to calculate the influence of the increase of corrosion loss on the incompatible strain.

Figure 9.12 schematically shows the strains in the strand, concrete, and reinforcement bars at a given force. The compatibility coefficient can be decided by the following mentioned calculation formula and it is employed to calculate the corresponding value as

$$\Omega = \frac{{\varepsilon_{{\text{p}}} }}{{\varepsilon_{{\text{p,cal}}} }},$$

(9.46)

where $\Omega$ is the compatibility coefficient; $\varepsilon_{p}$ is the strain of the strand at mid-span position of the PC structure; $\varepsilon_{{\text{p,cal}}} = \varepsilon_{{{\text{pe}}}} + \Delta \varepsilon_{{{\text{pc}}}}$ is the corresponding calculated strain of the steel strand according to the plane section assumption method; $\varepsilon_{{{\text{pe}}}}$ is the effective prestress in the strand; $\Delta \varepsilon_{{{\text{pc}}}}$ is the strain increment in the strand position of concrete.

The coordination factor of each beam during loading processes was calculated using this model. Figure 9.17 shows the calculated compatibility factors of all tested PC beams. The compatibility factors for the beam with 15 and 38% corrosion loss can also be shown in Fig. 9.17. The compatibility factors can be determined by the corrosion loss increases, and the force rate increases. For the uncorroded structure PE0 and lightly corroded structure PE5, no slip will occur due to the effective bond. The compatibility factors can usually be taken as a constant of 1.0 throughout the entire loading process. It shows that corrosion loss less than 13%, and the value of it can hardly have an impact on the corresponding incompatible strain.

Fig. 9.17

Different compatibility coefficient of corroded PC beams (CL: Corrosion loss)

As the force increases, the compatibility coefficient decreases in the further corroded beam. The more serious corrosion of PC beams, the earlier the coordination coefficient will decrease under the condition of small force. The compatibility coefficient of mildly corroded beams PE6 and PE7 will be decreased under a large given force condition. The beam fails more slowly with the decrease of the compatibility coefficient. For beams with severe corrosion, the coordination coefficient will be reduced in the small loading condition, but these PC beams are slow to destroy and essentially reach the equivalent coordination coefficient value.

The ultimate coordination coefficient is defined as the coordination factor of beams under the ultimate condition, which is an effective method to assess the bearing capacity as shown in Fig. 9.18 and listed in Table 9.2. The ultimate compatibility coefficient will remain a constant of 1.0 until the corrosion loss of the PC beam is more than 13%. Next, the ultimate compatibility coefficient decreases rapidly to be a small constant. Then, the ultimate coordination coefficient can be kept a constant of the small value for the further corrosion of the beam.

Fig. 9.18

Ultimate compatibility coefficient in PC structures under corrosion

The prediction of ultimate compatibility coefficient and the variation of the compatibility coefficient has a good match with the reduction of the bond capacity occurs. For slightly corroded beams and the uncorroded beams, the slip and coordination strain will not happen during the loading test process in which the condition of adhesive capacity can be defined as the best bonding capacity. The corresponding compatibility factor is usually kept as a constant of 1.0. For the modestly corrosion of PC beams, the strand starts the condition of slippage until the tension force exceeds the effective bond force, resulting in a reduced coordination factor. These beams are subject to failure under the movement procedure that occurs in the effective bond zone. The difference in the slip region leads to a decrease of ultimate compatibility coefficient in corroded PC beams. For beams with severe corrosion, the effective bond zone tends to shift quickly to the end of the PC beam with the poorer adhesive capacity. The above-mentioned method can accurately assess the degradation of the effective bond and reasonably predict the bearing capacity.

9.6 Conclusions

• 1. Corrosion can reduce the ultimate strain of the strand, leading to the brittle damage of the strand. Corrosion has a slight impact on the modulus of elasticity and yield strength of strand. Corrosion can reduce the ultimate capacity and change the failure mode of PC beams.

• 2. An analytical model for the residual bearing capacity of corroded PC beams is proposed in the present study. The model can consider the incompatible strain caused by bond deterioration and the different failure modes.

• 3. The accuracy of the prediction model of corroded PC beams anchored by straight, hooked, and welded bars is verified through experimental study. The proposed model can effectively quantify the bond deterioration caused by corrosion, and can reasonably predict the flexural capacity and the failure mode of corroded PC beams.

• 4. The bond degradation causes an incompatible strain. When the corrosion loss is lower than 13%, it has a slight impact on the corresponding incompatible strain. As the applied force increases, further corrosion rapidly reduces the coordination coefficient to a small value.