Prestress Loss and Transfer Length Prediction in Pretensioned Concrete Structures with Corrosive Cracking

Abstract

Concrete cracking caused by strand corrosion reduces bond strength and leads to loss of prestress and changes in transfer length. In this chapter, a model, considering the coupled effects of concrete cracking and bond degradation, is proposed to predict the corrosion-induced prestress loss in prestressed concrete structures at first. Then, an analytical model, considering the coupled effects of the Hoyer effect and corrosion-induced cracking, is proposed to predict the transfer length. The effects of strand corrosion on the effective prestress and transfer length are studied.

6.1 Introduction

Prestressed concrete is widely used in engineering structures because of its economical, durable and superior performance [8, 13]. Unfortunately, numerous of accidents in recent years have raised concerns about the safety of existing prestressed concrete structures [17, 18]. There are many factors that contribute to the deterioration of PC structures, and strand corrosion is one of the major factors [28, 39]. The assessment of prestress loss and transfer length caused by corrosive cracking is necessary for the serviceability and safety of existing concrete structures.

The assessment of prestress losses due to corrosion is a complex issue. Reduction in cross-section of corroded strands, concrete cracking, and bond degradation can all lead to loss of prestress. Compared with post-tensioned concrete members, corrosion cracking and bond degradation have greater influence on effective prestress in pretensioned concrete structures. How to evaluate the prestress loss in pretensioned concrete structures caused by corrosive cracking still needs to be studied further. Numerous researches have been carried out to clarify the effects of the strand diameter, the effective prestress, the concrete strength as well as the concrete cover on the transfer length [9, 23, 24, 33]. However, the above study did not examine the effect that corrosion has on the transfer length of PC structures. The transfer length prediction of PC beams with corrosion cracking also should be researched further.

Youakim et al. [37] evaluated the long-term prestress losses in prestressed concrete structures according to the principles of strain coordination and force balance. Kottari et al. [16] further improved the calculation method of prestress loss proposed by AASHTO-LRFD code and investigated the sensitivity of parameters, such as concrete age, the relative humidity, the number of steel, and concrete strength. In order to ensure the effective transfer of steel strand prestress, the effective bonding between steel strand and concrete interface is very important. When the PC beams are in a corrosive environment, the steel strands in the concrete will gradually corrode [17, 18, 20, 29, 41].

Corrosive cracking reduces the bond properties of the strands, adversely affects the transfer of effective prestress in the structure, and ultimately degrades the structural properties [3, 14, 19, 30, 38, 40]. Compared with other concrete structures, prestress transfer is quite important for PC structures. Corrosion cracking reduces the bond stress of the strand, which adversely affects the transfer length. The researchers conducted an experimental study on the bonding behavior of corroded prestressed strands. The results show that the bond stress first increases to a certain degree and then decreases with the increase of corrosion degree [7, 21, 22, 34, 35].

In this chapter, an analytical model is presented to evaluate the prestress loss in the corroded structure, which takes into account the coupling effects of concrete cracking and bond degradation. Then, an analytical model is proposed to predict the transfer length of PC beams by combining the coupled effects of Hoyer effect and corrosion-induced cracks.

6.2 Calculation of Corrosion-Induced Expansive Pressure

Corrosion of steel strands can cause cracking of the concrete structure and reduce the bond strength between the concrete and the steel strands, resulting in a prestress loss of the steel strands. It is important to consider the effects of concrete cracking and bond degradation in the prestress loss assessment.

When the tensile stress in the circumferential direction caused by corrosion expansion is large than the tensile strength of concrete, concrete is assumed to crack. The concrete cracking situation caused by the corrosion is shown in Fig. 6.1. Seven-wire strands are commonly used as prestressed tendon in PC structures. When the strand is corroded in an adverse environment, the corrosion first starts from the outside of the strand, as illustrated in Fig. 6.1. After the steel strand is corroded, the corrosion loss ρ can be denoted as

$$\rho = 6\Delta A/A_{p} ,$$

(6.1)

Fig. 6.1

Steel strand corrosion concrete cracking

where $\Delta A$ is the area loss of single wire, $\Delta A = \frac{2}{3}\pi \left( {R_{0}^{2} - R_{\rho }^{2} } \right)$, $A_{p}$ is the cross-sectional area of the corroded strand, $R_{o}$ is the radiuses of wire before corrosion, $R_{\rho } {\text{is }}$ after corrosion.

With the equivalent principle of volume, the volume reduction of strand $\Delta V_{w}$ per unit length can be written as

$$\Delta V_{w} = \frac{1}{\gamma - 1}(\Delta V_{c} + \Delta V_{e} ),$$

(6.2)

where ${ }\Delta V_{w} = 4{\uppi }\left( {R_{0}^{2} - R_{\rho }^{2} } \right)$; $\gamma$ is the rust expansion ratio; $\Delta V_{c}$ is the volume of corrosion products in pores and cracks per unit length, and $\Delta V_{e}$ is the volume change of concrete unit length, $\Delta V_{e} = 4{\uppi }\left( {R_{t}^{2} - R_{0}^{2} } \right)$, $R_{t}$ is the wire radius with corrosion products.

Corrosion products filled in the pores and cracks, the volume per unit length can be expressed as [2]

$$\Delta V_{c} = 4{\uppi }\left( {R_{t} - R_{o} } \right)\left( {R_{u} - R_{t} } \right),$$

(6.3)

where $R_{u}$ is the radius of cracked area.

The concrete displacement $u_{c}$ caused by expansive pressure can be calculated with Eqs. (6.1)–(6.3), which can be expressed as

$$u_{c} = R_{t} - R_{0} = \frac{{\left( {\gamma - 1} \right)A_{p} \rho }}{{4\pi \left( {R_{u} + R_{0} } \right)}}$$

(6.4)

The theory of elastic thick-walled cylinders is used to elaborate the concrete stress state. The hoop stress $\sigma_{\theta } \left( t \right)$ in uncracked concrete, radial displacement u(t) in uncracked concrete are [32]

$$\sigma_{\theta } \left( t \right) = \frac{{R_{u}^{2} P_{u} }}{{\left( {R_{c}^{2} - R_{u}^{2} } \right)}}\left( {1 + \frac{{R_{c}^{2} }}{{t^{2} }}} \right)$$

(6.5a)

$$u\left( t \right) = \frac{{R_{u}^{2} P_{u} }}{{E_{c} \left( {R_{c}^{2} - R_{u}^{2} } \right)}}\left[ {\left( {1 + v_{c} } \right)\frac{{R_{c}^{2} }}{t} + \left( {1 - v_{c} } \right)t} \right],$$

(6.5b)

where $t$ is the radius of the uncracked concrete zone, $R_{u} \le t \le R_{c}$, $R_{c} = R_{0} + C$, $C$ is the concrete cover; $P_{u}$ is the expansion pressure, and the force is located at the interface between the cracked area and the non-cracked area; $E_{c}$ is the elastic modulus of concrete, and $v_{c}$ is Poisson's ratio of the concrete.

According to the equivalence principle of stress distribution at the same location, the concrete tensile stress $\sigma_{\theta } \left( {R_{u} } \right)$ at the interface between uncracked region and cracked region should be equal to the concrete tensile strength $f_{t}$, i.e., $\sigma_{\theta } \left( {R_{u} } \right) = f_{t}$. The expansive pressure at the interface between uncracked region and cracked region can be obtained with Eq. (6.5a), which can be expressed as

$$P_{u} = f_{t} \frac{{R_{c}^{2} - R_{u}^{2} }}{{R_{c}^{2} + R_{u}^{2} }}.$$

(6.6)

The radial displacement $u\left( t \right)$ can be calculated in the uncracked concrete by substituting Eq. (6.6) into Eq. (6.5b). Assuming that the radial displacement $u\left( r \right)$ in the cracked concrete still satisfies the linear distribution principle, the radial displacement can be given as

$$u\left( r \right) = \frac{{f_{t} R_{u}^{2} }}{{E_{c} \left( {R_{c}^{2} + R_{u}^{2} } \right)}}\left[ {\left( {1 + v_{c} } \right)\frac{{R_{c}^{2} }}{r} + \left( {1 - v_{c} } \right)r} \right],$$

(6.7)

where $r$ is the radius of the concrete cracked region, $R_{0} \le r \le R_{u}$.

The hoop stress $\sigma _{\theta } \left( r \right)$ in the cracked concrete can be calculated, which can be denoted as [27]

$$\sigma _{\theta } \left( r \right) = \left\{ {\begin{array}{*{20}l} {E_{c} \varepsilon _{\theta } \left( r \right),} \hfill & {\varepsilon _{\theta } \left( r \right) \le \varepsilon _{{{\text{ct}}}} } \hfill \\ {f_{t} \left[ {1 - 0.85\frac{{\varepsilon _{\theta } \left( r \right) - \varepsilon _{{{\text{ct}}}} }}{{\varepsilon _{1} - \varepsilon _{{{\text{ct}}}} }}} \right],} \hfill & {\varepsilon _{{{\text{ct}}}} {\text{ < }}\varepsilon _{\theta } \left( r \right) \le \varepsilon _{1} ,} \hfill \\ {0.15f_{t} \left[ {\frac{{\varepsilon _{u} - \varepsilon _{\theta } \left( r \right)}}{{\varepsilon _{u} - \varepsilon _{1} }}} \right],} \hfill & {\varepsilon _{1} {\text{ < }}\;\varepsilon _{\theta } \left( r \right) \le \varepsilon _{u} ,} \hfill \\ \end{array} } \right.$$

(6.8)

where $\sigma_{\theta } \left( r \right)$ is the hoop stress of concrete and $\varepsilon_{\theta } \left( r \right)$ is the hoop strain of concrete; $\varepsilon_{\rm ct}$ and $\varepsilon_{u}$ are the strain corresponding to concrete tensile strength and the ultimate strain of concrete, respectively;$\varepsilon_{1}$ is the strain corresponding to 15% concrete tensile strength; $\varepsilon_{u}$ is the limiting strain of concrete.

Before cover cracking, the expansive pressure would be resisted by the confining stress in the uncracked concrete and the residual tensile stress in the cracked concrete. The expansion pressure $P_{c}$ at the interface between steel strand and concrete can be expressed as

$$P_{c} R_{0} = P_{u} R_{u} + \mathop \int \limits_{{R_{0} }}^{{R_{u} }} \sigma_{\theta } \left( r \right)dr.$$

(6.9)

After the concrete cover has cracked, the residual tensile strength in the concrete is mainly used to resist the expansion pressure generated by the corrosion products. The expansive pressure at the strand–concrete interface can be rewritten as [12, 32]

$$P_{c} R_{0} = \mathop \int \limits_{{R_{0} }}^{{R_{c} }} \sigma_{\theta } \left( r \right)dr$$

(6.10)

6.2.1 Prediction Model of Prestress Loss Under Corrosive Cracking

6.2.1.1 Stress Variation in Corroded Strand

The elements of half beam are numbered from 1 to n, as shown in Fig. 6.2. For any element $i$, the stress $f_{p,i}$ in the steel strand can be denoted as

$$f_{p,i} = f_{p,i + 1} - \Delta f_{p,i} ,$$

(6.11)

Fig. 6.2

Stress variation in corroded strand

where $\Delta f_{p,i}$ is the local stress change in element i of the corroded strand, $1 \le i \le n$.

The contact area $S$ between the outer wire and concrete is equal to two-thirds of the surface area, which can be written as $S = \frac{4}{3}\pi R_{\rho ,i} l_{i}$, $R_{\rho ,i}$ is the residual radius of corroded wire at the $i$ element$,l_{i}$ is the element length. The local stress variation in corroded strand can be written as

$$\Delta { }f_{p,i} = \frac{6S}{{A_{p,i} \left( \eta \right)}}\tau_{\eta } ,$$

(6.12)

where $A_{p,i} \left( \eta \right)$ is the residual cross-sectional area of corroded strand at the $i$ element.

6.2.2 Bond Degradation Due to Strand Corrosion

The geometrical shape of the strand can be reduced by corrosion. Corrosion cracking reduces the concrete restraint. All these factors affect the bonding performance of corroded strand. The bond strength can be evaluated in terms of restraint stress, expansive pressure, and bond stress. The stress in the bond of the strand $\tau_{\eta }$ that is corroded can be represented as

$$\tau_{\eta } = \tau_{a} + \tau_{b} + \tau_{c} ,$$

(6.13)

where $\tau_{a}$ is the bonding stress at the bonding interface; $\tau_{b}$ is the confinement stress of the concrete, and $\tau_{c}$ is the bonding stress induced by the expansion pressure.

The adhesive stress of the corroded strand can be represented as [5]

$$\tau_{a} = \frac{{kA_{r} \left[ {\cot \delta + \tan \left( {\delta + \theta } \right)} \right]}}{{\pi Ds_{r} }}f_{{{\text{coh}}}} ,$$

(6.14)

where $k$ is rib count on ribs in the transverse direction; $A_{r}$ is the area where the ribs are at right angles to the strand axis in the plane; $\delta$ is the rib orientation; $\theta$ is the friction angle between steel strands and concrete; $D$ and $s_{r}$ are the strand diameter and the rib spacing, respectively; $f_{{{\text{coh}}}}$ is a factor of the bonding stress.

The confinement stress around the concrete is denoted as [11]

$$\tau_{b} = \frac{{kC_{r} {\text{tan}}\left( {\delta + \theta } \right)}}{\pi }p_{x} ,$$

(6.15)

where $C_{r}$ is the shape factor constant; $p_{x}$ is the maximum pressure at bond failure.

The bond stress induced by expansive pressure can be expressed as

$$\tau_{c} = k_{c} P_{c} ,$$

(6.16)

where $k_{c}$ is the coefficient of friction between the corroded strand and the cracked concrete.

Substituting Eqs. (6.14)–(6.16) into Eq. (6.13), the bond stress of corroded steel strand can be obtained. For pretensioned concrete structures, the prestress transfers from strand to concrete through the adhesive stress. Since corrosive cracking and bond degradation have been estimated, a model for corrosion-induced prestress loss can be proposed. The effective prestress of the uncorroded strand minus the effective prestress of the corroded steel strand can be defined as the prestress loss caused by corrosion. The effective prestress in corroded strand could be evaluated based on the strain compatibility and force equilibrium equations.

6.2.3 Calculation Flow Chart of Prestress Loss

For corroded pretensioned concrete structures, the prestress of strand at the beam end is zero, i.e., $f_{p,1} = 0$. The tension force of corroded strand $T_{p,i}$ can be calculated as

$$T_{p,i} = f_{p,i} A_{p,i} \left( \eta \right).$$

(6.17)

After corrosion, the strain change in strand at the $i$ element $\Delta \varepsilon_{p,i}$ can be denoted as

$$\Delta \varepsilon_{p,i} = \frac{{T_{pi} }}{{E_{p} A_{p} }} - \frac{{T_{p,i} }}{{E_{p} A_{p,i} \left( \eta \right)}},$$

(6.18)

where $T_{pi}$ is the initial prestressing force of uncorroded strand at the $i$ element; $E_{p}$ is the elastic modulus of strand.

When the stress in the corroded strand equals the effective prestress, the strain change in concrete $\Delta \varepsilon_{c,i}$ should be equal to the change in strain of the corroded strand location $\Delta \varepsilon_{p,i}$ to maintain the strain compatibility, i.e., $\Delta \varepsilon_{c,i} = \Delta \varepsilon_{p,i}$. After corrosion, the concrete strain $\Delta \varepsilon_{p,i}$ for the ith element should be written as

$$\varepsilon_{\text{cp},i} = \frac{{T_{pi} }}{{E_{c} }}\left( {\frac{1}{A} + \frac{{e_{p}^{2} }}{I}} \right) - \Delta \varepsilon_{c,i} ,$$

(6.19)

where $e_{p}$ is the rate of eccentricity of the steel strand; A is the cross-sectional area of concrete; and I is the moment of inertia of the gross section of concrete.

This study primarily investigates the prestress loss caused by strand corrosion, and corrosion of the steel reinforcements is not considered. The distribution of strain in the beam section is shown in Fig. 6.3. The steel strain in the tension region is $\varepsilon_{s,i}$, the steel strain in the compression region is $\varepsilon_{s,i}^{\prime}$, which can be written as

$$\varepsilon_{s,i} = \frac{{h_{x} - a_{s} }}{{h_{x} - a_{p} }}\varepsilon_{\text{cp},i}$$

(6.20a)

$$\varepsilon_{s,i}^{\prime} = \frac{{h - h_{x} - a_{s}^{\prime} }}{{h_{x} - a_{p} }}\varepsilon_{\text{cp},i} ,$$

(6.20b)

Fig. 6.3

Strain distribution in the cross-section

where $h$ is the height of beam; $h_{x}$ is the length of the concrete beam from the center of gravity, $a_{p}$ and $a_{s}$ are the strand center and the tensile reinforcement center to the bottom of beam, respectively; $a_{s}^{\prime}$ is the height from the center of the stressed reinforcement to the top of the concrete beam.

The stress–strain behavior of steel bars can be described using an elastic–plastic constitutive model [11], which is given as

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

(6.21)

where $f_{s}$ is the stress of reinforcement$; \varepsilon_{s}$ is strain of reinforcement; $E_{s}$ is the elastic modulus of reinforcement and $E_{sp}$ is the hardening modulus of reinforcement; $f_{sy}$ is the yield strength of reinforcement; $\varepsilon_{sy}$ is yield strain of reinforcement.

$F_{s,i}$ is the reinforcement force in the tensile zones, $F_{s,i}^{\prime}$ is the forces of reinforcements in the compression zones, which can be written as

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

(6.22a)

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

(6.22b)

where $A_{s}$ is the section areas of reinforcement in tension zones, $A_{s}^{\prime}$ is the section areas of reinforcement in compression zones; $f_{s} (\varepsilon_{s,i} )$ is the stresses of reinforcement in the tension, and $f_{s} (\varepsilon_{s,i}^{\prime} )$ is the compression zones.

The mechanical behavior of concrete in tension can be simulated by a linear elastic constitutive law [25]. The nonlinear constitutive law of concrete proposed [6] by is utilized to describe the mechanical behavior of concrete in compression. The stress–strain curve of concrete is denoted as

$$f_{c} = \left\{ {\begin{array}{*{20}c} {f_{c}^{\prime} \left[ {2\left( {\frac{{\varepsilon_{c} }}{{\varepsilon_{0} }}} \right) - \left( {\frac{{\varepsilon_{c} }}{{\varepsilon_{0} }}} \right)^{2} } \right]} & {\text{in compression}} \\ {E_{c} \varepsilon_{c} } & {\text{in tension}} \\ \end{array} } \right.,$$

(6.23)

where $f_{c}$ is the stress of concrete, $\varepsilon_{c}$ is concrete strain, $f_{c}^{\prime}$ is the compressive strength of concrete, $\varepsilon_{0}$ is the strain corresponding to the concrete compressive strength and taken as 0.002.

The total force $C_{i}$ of the concrete is denoted as

$$C_{i} = \mathop \int \limits_{{A_{c} }}^{{}} f_{c} {\text{d}}A_{c} ,$$

(6.24)

where $A_{c}$ is the region of the damaged concrete section.

For corroded pretensioned concrete structures, the forces in prestressing strand, steel reinforcements, and concrete should satisfy the equilibrium equation, which can be written as

$$C_{i} + F_{s,i}^{\prime} - T_{p,i} - F_{s,i} = 0.$$

(6.25)

As mentioned above, we propose a new model incorporating the coupling effects of concrete cracking and bond degradation to evaluate the corrosion-induced prestress loss. The flow chart of prestress loss calculation is shown in Fig. 6.4.

Fig. 6.4

Calculation flowchart of prestress loss

6.2.4 Evaluation of Effective Prestress

6.2.4.1 Accelerated Corrosion and Data Measurement

The concrete was poured with ordinary silicate cement. The concrete mix contained: 676 kg/m³ fine aggregates, 417 kg/m³ cement, and 1026 kg/m³ coarse aggregates. The concrete water–cement ratio was 0.44. To catalyze the corrosion process, sodium chloride was added to the concrete. The uniaxial compressive strength measured after 28 days of curing of the concrete elements was 44.1 MPa.

The testing parameters are given in Table 6.1. In order to study the prestress loss under different stresses and corrosion levels, four stress levels of strand were designed as 0, $0.25f_{p}$, $0.5f_{p}$, and $0.75f_{p}$, respectively, where $f_{p}$ was 1860 MPa. The corrosion time of A and B groups was 15 days and 20 days, respectively.

Table 6.1 Concrete compressive strength

The test was designed with eight concrete beams with a cross-section of 130 × 150 mm and the length of 2000 mm. The test beam was strengthened with 7-wire steel strands of 15.2 mm in diameter. The yield strength of the steel strand selected in the test was 1830 MPa, and the maximum strength was 1910 MPa. Stirrups with a diameter of 6 mm and a spacing of 100 mm were used in the beam, two 6 mm steel bars are set at the bottom of the beam as longitudinal reinforcement. The yield strength of deformed reinforcement was 400 MPa, and the ultimate strength of deformed reinforcement was 540 MPa. The details of the specimen are shown in Fig. 6.5. The concrete cover of reinforcement was 30 mm, and the concrete cover of strand was 42.4 mm.

Fig. 6.5

Details of specimen

The compressive strength of concrete is listed in Table 6.2. In this research test, electrochemical methods were used to accelerate the corrosion of strands. Only the effect of corrosion on the prestress loss of steel strands is considered, the epoxy resin is applied to the surface of steel bars to prevent corrosion. The accelerated corrosion device consists of a DC power supply and a stainless-steel plate. The steel strand is connected to the anode and the stainless-steel plate is connected to the cathode, and 10% sodium chloride solution is used as the corrosion solution. A constant potential meter is used to apply a DC current to the strand. The operating current during accelerated corrosion is 0.1 A.

Table 6.2 Testing parameters

After the accelerated corrosion, the crack width in 10 cm intervals on the concrete surface was measured using a portable microscope with a resolution of 0.01 mm. The average mass loss of the strand in the longitudinal direction was measured to reflect the corrosion level of strand in the global region. The average mass loss of strand in 10 cm intervals was measured to reflect the variation of the corrosion degree in different zones.

In the current study, the prestress loss was predicted using the uniform corrosion model, which is reasonable from the predicted results. In this experimental study, in order to reflect the longitudinal corrosion loss of steel strands and the change of crack width of concrete members, the crack widths of concrete members and the mass loss of steel strands are given at intervals of 10 cm. Corrosion-induced cracking will be accelerated under the action of prestress. Study have shown that the mass loss of corroded steel bars is closely related to the width of concrete cracks under corrosion [15]. In summary, it is feasible to use the concrete crack width to assess the mass loss of the corroded strand.

Comparison of the experimental data in Table 6.3. By changing the stress level of the strand from 0 to 75%$f_{p}$, the average increase in the mass loss is 37.0%. The results show that prestress can accelerate the mass loss of the strand. The mass losses of PA0 and PA3 were 7.05% and 9.41%, respectively. The mass losses of PB0 and PB3 were 10.45% and 14.69%, respectively.

Table 6.3 Experiment results summary

6.2.5 Effective Prestress Evaluation

Four methods of estimating the effective prestress in prestressed concrete structures have been employed in the current research [4]: (1) Cut strands to representative lengths to measure the changes of strand strain; (2) cracking load determined by load test; (3) Measurement of the longitudinal strain in concrete at the prestressed strand's center of gravity as a function of time; (4) determine the lateral pressure used to close the cracks in the small cylindrical holes. All four methods require processing of the test results before calculating the effective prestress.

As illustrated in Fig. 6.6, a four-point flexural test was performed to obtain the load–deflection curves of corroded specimens. The test beams have a bending span of 600 mm and a net span of 1800 mm. The load is applied monotonically, and the loading value is measured by a pressure transducer. Electronic digital micrometers are used to measure vertical deflection in loading points, support points, and spans. The load–deflection curves of the test beams were reflected by the mid-span displacements in Fig. 6.7. The cracking and ultimate loads for the beams are given in Table 6.3. As the applied load increases, the deflection of the beam increases accordingly. When the applied load reaches the failure load, the applied load suddenly decreases and the test beam fails.

Fig. 6.6

Diagram of load testing

During load testing, once the tensile stress exceeds the concrete tensile strength, cracks appear at the bottom of the test beam. The critical condition of concrete cracking can be expressed as

$$f_{t} = f_{p,\eta } A_{p} \left( \eta \right)\left( {\frac{1}{{A_{c} }} + \frac{{e_{p} }}{{I_{c} }}y_{b} } \right) - \frac{{M_{s} }}{{I_{c} }}y_{b} - \frac{{M_{c} }}{{I_{c} }}y_{b} ,$$

(6.26)

where $f_{p,\eta }$ is the effective prestress in corroded PC beams; $A_{p} \left( \eta \right)$ is the remaining cross-sectional area of the corroded strand; $y_{b}$ is the length from the center axis to the beam bottom; $M_{s}$ is the bending moment due to the self-weight of the beam; $M_{c}$ is the cracking moment; $I_{c}$ is the moment of inertia of the total section the damaged concrete.

The effective prestressing force and prestressing loss of the corroded prestressed concrete structure can be estimated, as given in Table 6.3. Studies have found that the high stress of strand will accelerate the loss of prestress caused by corrosion. The prestress losses of PA1 and PA3 are 12.0% and 31.3%, respectively. The prestress losses of PB1 and PB3 are 34.4% and 55.3%, respectively. By changing the stress level of strand from 25%$f_{p}$ to 75%$f_{p}$, the corrosion-induced prestress loss increases by 20.1% in the current experimental study.

Fig. 6.7

Load–deflection curves

6.2.6 Validation on Prestress Loss Model

For estimating the corrosion-induced prestress loss, the bond degradation is the vital factor and should be clarified at first. In order to study the effect of strand corrosion on the bond strength, bond strengths under different corrosion levels are given in Fig. 6.8. The normalized bond strength in Fig. 6.8 is defined as the bond strength ratio of corroded strand to uncorroded strand.

Fig. 6.8

Adhesion strength at different corrosion losses

As shown in Fig. 6.8, when the strand corrosion is less than 6.6%, the bond stress would increase with the increase of corrosion, and the further corrosion of the strand would gradually cause the bond stress to degrade. When the corrosion degree of the steel strand is less than 6.6%, the corrosion products increase the friction force and gear force of the interface between the steel strand and the concrete. Consequently, slightly corroded strand will result in an increase of the bond stress. Corrosion levels above 6.6% could lead to severe concrete cracking, reducing concrete confinement, and resulting in deterioration of bond stress.

The predicted effective prestress and experimental results are shown in Fig. 6.9. Strands with a corrosion loss of less than 6.6% are considered to have similar bond stress to uncorroded strands. The prediction of the effective prestress was carried out by applying the bond strength model proposed in this study.

Fig. 6.9

Experimental and predicted prestress

The average error of prediction is 4.8%, and standard deviation of prediction is 0.02. The prediction error is defined as $\frac{{\left| {P_{t} - P_{p} } \right|}}{{P_{t} }}$, where $P_{t}$ is tested effective prestress, and $P_{p}$ is predicted effective prestress. Errors in results may result due to model simplification. In addition, the uncertainty of the measured experimental data can also lead to errors. However, due to the complexity of the corrosion process, the accuracy of the model predictions is acceptable.

Wang et al. [32] designed ten beams to investigate the bond degradation at the strand–concrete interface under different levels of corrosion. An empirical model for predicting bond strength of corroded strand is proposed, which can be expressed as

$$R_{\eta } = \left\{ {\begin{array}{*{20}l} {1.0,} & {\eta \le 6.0\% } \\ {2.03{\text{e}}^{ - 11.8\eta } } & {\eta > 6.0\% } \\ \end{array} } \right.,$$

(6.27)

where $R_{\eta }$ is the bond stress ratio of corroded strand to uncorroded strand.

As shown in Fig. 6.10, the critical corrosion levels for this model [33] are 6.6% and 6.0%, respectively, and when the corrosion levels are below the critical values, the bond strength and effective prestressing are not reduced. Strand corrosion can cause severe cracking of PC beams when corrosion levels are exceeds the critical values, thereby reducing bond strength and effective prestress. When the corrosion level exceeds 34.0%, the effective prestress would drop to zero. The two models predict the prestress loss towards a similar direction, as shown in Fig. 6.10. The results show that the model proposed in this study can accurately predict the corrosion-induced prestress loss in prestressed concrete structures.

Fig. 6.10

Prestress loss and corrosion loss

6.2.7 Prediction of Transfer Length Under Corrosive Cracking

6.2.7.1 Expansive Pressure Induced by Hoyer Effect and Corrosion

For predicting the transfer length of PC beams under corrosion cracking, the expansion pressure at the strand–concrete interface is an important parameter. During the corrosion of the test beam, the expansion pressure is influenced by the coupling effect of Hoyer effect and corrosion. As illustrated in Fig. 6.11, the expansion pressure caused by the strand corrosion products would influence the concrete cracking. In predicting the transfer length of corroded prestressed concrete structures, the Hoyer effect and the expansion pressure due to corrosion need to be considered. In this study, a prediction model of transfer length is proposed, which has the advantage of considering the coupling effect of Hoyer effect and steel strand corrosion. The theory of thick-walled cylinders was also applied to evaluate the expansion pressure [26], as shown in Fig. 6.12.

Fig. 6.11

Schematic diagram of the expansion pressure due to Hoyle effect and corrosion of steel strands

Fig. 6.12

Theory of thick-walled cylinders for concrete cracks

After tensioning the prestressed strand, the radius $R_{t}$ is denoted as

$$R_{t} = \left( {1 - \frac{{f_{pt} }}{{E_{p} }}v_{p} } \right)R_{0} ,$$

(6.28)

where $R_{0}$ is strand original radius, $v_{p} ,f_{pt} ,$ and $E_{p}$ are Poisson’s ratio, the tensile stress, and the elastic modulus of strand, respectively.

After prestress releasing, the radial displacement of the prestressed strand u is [26]

$$u = \frac{{ - P_{h} R_{t} }}{{E_{c} \left( {1/R_{c}^{2} - 1/R_{t}^{2} } \right)}}\left[ {\frac{{\left( {1 - v_{c} } \right)}}{{R_{c}^{2} }} + \frac{{\left( {1 + v_{c} } \right)}}{{R_{t}^{2} }}} \right] - \frac{{v_{c} f_{cz} R_{t} }}{{E_{c} }},$$

(6.29)

where $R_{c}$ is the distance from the center of the strand to the edge of the concrete cover, $E_{c}$ and $v_{c}$ are the elastic modulus and Poisson’s ratio of concrete, respectively, $f_{cz}$ and $P_{h}$ are the concrete compressive stress at the strand location and the expansion pressure under the Hoyer effect, respectively.

At the location of the prestressed strand, the concrete longitudinal compressive stress $f_{cz}$ can be denoted as

$$f_{cz} = f_{pz} A_{p} \left( {\frac{1}{A} + \frac{{e_{p} }}{{I_{c} }}y_{b} } \right),$$

(6.30)

where $f_{pz}$ is the axial stress of strand at the corresponding position, $A_{p}$ and A are the initial cross-sectional area of strand and the cross-sectional area of concrete, respectively, $I_{c}$ is the moment of inertia of concrete cross-section, $e_{p}$ is the eccentricity of prestressing strand, $y_{b}$ is the distance from the neutral axis of beam to the bottom of beam.

Considering the condition of Hoyer effect, the expansion pressure $P_{h}$ is expressed as [26]

$$P_{h} = \frac{{R_{0} \left( {1 - v_{p} f_{pz} /E_{p} } \right) - R_{t} \left( {1 - v_{c} f_{cz} /E_{c} } \right)}}{{\left( {1 - v_{p} } \right)R_{0} /E_{p} + \left[ {v_{c} - \left( {R_{t}^{2} + R_{c}^{2} } \right)/\left( {R_{t}^{2} - R_{c}^{2} } \right)} \right]R_{t} /E_{c} }}$$

(6.31)

The residual cross-sectional area of prestressing strand after corrosion $A_{p}^{\prime}$ can be obtained with the corrosion loss $\rho$ as

$$A_{p}^{\prime} = A_{p} \left( {1 - \rho } \right).$$

(6.32)

After the corrosion of steel strands in concrete, the concrete compressive stress can be rewritten as

$$f_{cz}^{\prime} = \left( {1 - \rho } \right)f_{cz} .$$

(6.33)

Combining Eqs. (6.29)–(6.33), the radial displacement of prestressing strand $u_{c}$ and the expansive pressure $P$ induced by Hoyer effect and strand corrosion can be calculated as

$$u_{c} = \frac{{ - PR_{t} }}{{E_{c} \left( {1/R_{c}^{2} - 1/R_{t}^{2} } \right)}}\left[ {\frac{{\left( {1 - v_{c} } \right)}}{{R_{c}^{2} }} + \frac{{\left( {1 + v_{c} } \right)}}{{R_{t}^{2} }}} \right] - \frac{{v_{c} f_{cz}^{\prime} R_{t} }}{{E_{c} }}$$

(6.34)

$$P = \frac{{R_{0} \left( {1 - v_{p} f_{pz} /E_{p} } \right) - R_{t} \left( {1 - v_{c} f_{cz}^{\prime} /E_{c} } \right)}}{{\left( {1 - v_{p} } \right)R_{0} /E_{p} + \left[ {v_{c} - \left( {R_{t}^{2} + R_{c}^{2} } \right)/\left( {R_{t}^{2} - R_{c}^{2} } \right)} \right]R_{t} /E_{c} }}$$

(6.35)

6.2.8 Calculation of Transfer Length

The transfer of prestress in PC beams can be obtained by the bond strength between strand and concrete. Corrosive cracking of the concrete affects the expansion pressure and reduces the bond strength between the concrete and the strand, thus changing the transfer length of the PC beam. The bond stress $\tau$ can be written as

$$\tau = \mu \cdot P,$$

(6.36)

where $\mu$ is the coefficient of friction.

One-half of the beam is discretized into the several elements with a length of $\Delta l$ to analyze the stress variation in prestressing strand, as shown in Fig. 6.13. The stress increment of strand $\Delta f_{p,i}$ at the ith element can be expressed as

$$\Delta f_{p,i} = \frac{{\pi d_{p}^{\prime} \tau_{i} }}{{A_{p}^{\prime} }}\Delta l,$$

(6.37)

Fig. 6.13

Schematic of element discretization in corroded PC beams

where $d_{p}^{\prime}$ is the remaining diameter of strand after corrosion, $\tau_{i}$ is the bond stress of the corroded strand at the ith element.

Stress change of prestressing strand $f_{p,i}$ and strain change of prestressing strand $\Delta \varepsilon_{p,i}$, at the ith element can be written as

$$f_{p,i} = f_{p,i - 1} + \Delta f_{p,i}$$

(6.38)

$$\Delta \varepsilon_{p,i} = \varepsilon_{p,0} - \frac{{f_{p,i} }}{{E_{p} }},$$

(6.39)

where $\varepsilon_{p,0}$ is the initial prestrain of strand.

The concrete strain at the ith element $\varepsilon_{c,i}$ can be given as

$$\varepsilon_{c,i} = \frac{{f_{p,i} A_{p}^{\prime} }}{{E_{c} }}\left( {\frac{1}{A} + \frac{{e_{p} }}{{I_{c} }}y_{b} } \right)$$

(6.40)

There is no stress in the strand at the end of beam. At the end of transfer length, the stress of strand is developed to the effective prestress. The transfer length can be determined when the strain change of prestressing strand at the ith element $\Delta \varepsilon_{p,i}$ is equal to the strain of concrete $\varepsilon_{c,i}$, i.e., $\Delta \varepsilon_{p,i}$ = $\varepsilon_{c,i} .$ The transfer length $l_{t}$ in corroded PC beams can be written as

$$l_{t} = i \cdot \Delta l$$

(6.41)

6.3 Evaluation of the Transfer Length in Corroded PC Beams

6.3.1 Specimen Design and Data Analysis

Ten specimens were designed to study the transfer length in the corroded PC beams. The dimensions of the specimen beams were all the same, and a hollow slot was designed in the mid-span with a length of 500 mm, a width of 60 mm, and a height of 100 mm. The width of the test beam was 200 mm, the height was 350 mm, and the length of the beam was 3800 mm. The prestressing strand had a diameter of 15.2 mm and a concrete cover thickness of 67.4 mm.

Compressive reinforcement was two deformed reinforcement bars with the diameter of 10 mm embedded in the top of the beam. Two deformed bars with a diameter of 16 mm were set at the bottom of the beam as tensile bars. The spacing of the stirrups with the diameter of 8 mm was 70 mm at the end of the beam and 100 mm in the middle of the span.

The elastic modulus, yield strength, and ultimate strength of deformed bars were 200 GPa, 400 MPa and 570 MPa, respectively. The elastic modulus, yield strength, and ultimate strength of strand were 195 GPa, 1830 MPa, and 1910 MPa, respectively. The average compressive strength of the concrete after 28 days of curing was 41.7 MPa. Figure 6.14 illustrates the details of the test beam.

Fig. 6.14

Specimen details (Unit: mm)

The electrochemical corrosion method was employed to achieve the corrosion of strand in this study. The reinforcement bars were coated with epoxy resin to avoid the corrosion. The uncorroded beam S0 was used as the control specimen. Nine specimens were classified into group A, group B, and group C based on the different corrosion positions, as shown in Fig. 6.15. Each group contained three specimens with different corrosion time, as given in Table 6.4.

Fig. 6.15

Local corrosion location of specimens (Unit: mm)

Table 6.4 Corrosion location and corrosion schedule

As illustrated in Fig. 6.15, a 200 mm long trough made of PVC material was installed in the area of localized corrosion. The corrosion device is composed of three parts: DC power supply, prestressed steel strand, and stainless-steel plate immersed in chloride solution. The anode and cathode are connected by steel strands and stainless-steel plates, respectively. During the corrosion process, the working current intensity is 0.5 A.

There is a correlation between the width of the crack caused by corrosion and the effect of corrosion on the transfer length. The corrosion loss, average, and maximum crack widths of specimens were measured. Corrosion-induced crack widths increase as corrosion extends from the mid-span region to the ends of the beam. Comparing to B3 and C3, the average crack width of A3 increases by 4.8% and 22.2%, respectively. Comparing to B3 and C3, the maximum crack width of A3 increases by 6.9% and 25.6%, respectively (Fig. 6.16).

Fig. 6.16

Corrosion-induced crack width: a average value; b maximum value

6.3.2 Evaluation of Transfer Length Under Corrosive Cracking

There is a relationship between the effective prestress and the transfer length, and the transfer length can be evaluated by reverse calculation of the cracking load. The design standard for the transfer length in PC beams is defined in ACI 318 ( ACI 318) as follows:

$$l_{t} = 0.048f_{{{\text{pe}}}} d_{P} ,$$

(6.42)

where $f_{{{\text{pe}}}}$ is the effective prestress, $d_{P}$ is the diameter of prestressing strand.

Under the applied load, the tensile stress of concrete at the bottom of the beam increases, and the effective prestress is expressed as

$$f_{{{\text{pe}}}} = \left( {f_{t} + \frac{{M_{s} }}{{I_{c} }}y_{b} + \frac{{M_{c} }}{{I_{c} }}y_{b} } \right)\left/\left[ {A_{p}^{\prime} \left( {\frac{1}{{A_{c} }} + \frac{{e_{p} }}{{I_{c} }}y_{b} } \right)} \right],\right.$$

(6.43)

where $M_{s}$ and $M_{c}$ are the moment due to the beam’s weight and the cracking moment, respectively.

As mentioned above, the effective prestress can be obtained from the cracking load calculated by Eq. (6.43). The effective prestress in Eq. (6.42) can be used to estimate the transfer length of corroded PC beams. In this study, the cracking loads of the specimens are obtained from static tests, and the transfer lengths are estimated by the cracking loads.

The four-point bending test was used to measure the cracking load of beams. The supporting span of the test beam was set to 3500 mm. Loading is monotonic and progressive until the structure eventually fails. As shown in Fig. 6.17, the dial gauges were used to measure the vertical displacement at the point of load support and mid-span. The vertical displacement and crack width were recorded under each loading phase.

Fig. 6.17

Schematic diagram of the load device (Unit: cm)

The vertical displacement of the mid-span is used to represent the deflection of the beam. The load–deflection curves of specimens are shown in Fig. 6.18. The whole loading process can be summarized in three stages: cracking point, yield point-, and damage point. When the applied load is less than the cracking load, the load–deflection curve of the test beam is approximately linear. Because the flexural stiffness mainly depends on the moment of inertia of the concrete section. Therefore, the strand corrosion has no significant effect on the bending stiffness of the test beams before concrete cracking.

Fig. 6.18

Load–deflection curves of the test beams

The flexural stiffness of the test beams was significantly reduced when the corrosion of the strands increased. The flexural stiffness in group A, group B, and group C decreases gradually under the same corrosion loss. This indicates that under the action of corrosion, the flexural stiffness degradation of the mid-span area after concrete cracking will be more serious than that of the end area. Finally, when the strand in the compression zone breaks or the concrete is crushed, the beam will fail.

As illustrated in Fig. 6.18, the cracking and ultimate load of the test beam also decrease as the corrosion degree increases. Compared with S0, the cracking loads are reduced by 60.0%, 77.8%, and 100.0% for A3, B3, and C3, respectively. The ultimate loads of all groups decreased to different degrees, with A3 decreased by 10.7%, B3 decreased by 14.4%, and C3 decreased by 35.0%.

Using S0 as a control for the other groups, the transfer length in the test beam can be calculated by taking the concrete cracking load and the strand corrosion loss. The transfer length of PC beams increases with the increase of corrosion degree. The transfer lengths of A3, B3, and C3 increase by 24.3%, 17.7%, and 14.4%, respectively, as compared to S0. Besides, the transfer length caused by strand corrosion will also extend when the local corrosion position moves from the mid-span to the beam end. The transfer lengths of A3 and B3 increase by 8.6% and 2.9%, respectively, as compared to C3.

6.4 Model Validation and Parameter Sensitivity Analysis

6.4.1 Verification of Proposed Model

The experimental results were consistent with the prediction of the proposed model. During the corrosion process, the corrosion product will diffuse from the local corrosion area to the end of the sample. Assuming that the corrosion of the strand reduces from localized corrosion areas to the end of the beam in a linear fashion; in this research model, the corrosion loss of the beam end strand is reduced to zero. The elastic modulus of the concrete used in the model is $3.25 \times 10^{4}$ MPa. Poisson's ratio of the chain is 0.3. [36] demonstrates that the friction coefficient in prestressing strand varies from 0.23 to 0.7, and the friction coefficient ${ }\mu$ is selected as 0.34 in the model. The theoretical transfer lengths in specimens are calculated and given in Table 6.5.

Table 6.5 Cracking load and transmission length of test beams

This study compares the theoretical transfer length of the test beam with the experimental transfer length, as shown in Fig. 6.19. The average error is 6.1%. The error in the data can be defined as ($l_{t,p}$ − $l_{t,e}$)/$l_{t,e}$ according to the experimental results, where $l_{t,p}$ is theoretical transfer length, and $l_{t,e}$ is experimental transfer length. The prediction error of the model is acceptable considering the uncertainty of the strand corrosion process, which means that the transfer length in PC beams under corrosive cracking can be accurately predicted by the proposed model.

Fig. 6.19

Theoretical and experimental transfer length

6.4.2 Effect of Material Parameters on Expansive Pressure

The expansion pressure at the interface between the strand and the concrete is an important factor in assessing the transfer length of prestressed concrete beams under corrosion. When the strand corrosion is 35%, the relationship between the expansion pressure and the normalized transfer length can be clarified, as shown in Fig. 6.20. The expansion pressure under the Hoyer effect coupled with corrosion is maximum at the end of the beam and decreases gradually along the longitudinal direction. At the end of the transfer length, the expansion pressure due to corrosion will be present, but the expansion pressure due to the Hoyer effect is zero. The expansion pressure at the end of the beam increases by 13.6% when the modulus of elasticity of the concrete increases from $3 \times 10^{4}$ MPa to $3.45 \times 10^{4}$ MPa. The expansion pressure at the interface between corroded steel strand and concrete has a more affected by Poisson's ratio than that by the elastic modulus of concrete.

Fig. 6.20

Influence of material parameters on expansion pressure

6.4.3 Effect of Material Parameters on Transfer Length

In this study, the transfer lengths at different levels of corrosion are discussed to evaluate the effect of corrosion on the transfer lengths, as shown in Fig. 6.21. The growth rate of the transfer length also increases accordingly as the corrosion degree of the strand increases. Under the corrosive cracking of PC beams, the transfer length decreases as Poisson's ratio of the strand and the modulus of elasticity of the concrete increase. By changing Poisson’s ratio of strand from 0.2 to 0.4, the transfer length with the corrosion loss of 35% reduces by 88.2%. By changing the elastic modulus of concrete from $3 \times 10^{4}$ MPa to$3.45 \times 10^{4}$ MPa, the transfer length is reduced by 8.6% when the corrosion loss reaches 35%. Compared to the modulus of elasticity, Poisson's ratio affects significantly the transfer length in beams with corrosive cracking.

Fig. 6.21

Effect of material parameters on transfer length after strand corrosion

6.5 Conclusions

• 1. A new model is proposed to predict the corrosion-induced prestress loss in prestressed concrete structures. The model takes into account the coupling effect of corrosion-induced concrete cracking and bond degradation.

• 2. Corrosion-induced cracking does not reduce the bond strength and effective prestress when the corrosion level does not exceed 6.6%. With the development of corrosion, the bond strength and effective prestress decrease, and the bond strength and effective prestress descend to zero when the corrosion level reaches 34.0%.

• 3. The high stress of strand will accelerate the corrosion-induced prestress loss. In the current experimental study, the prestress loss induced by corrosion increases by 20.1% by changing the strand stress level from 25 to 75% of the strand tensile strength.

• 4. An analytical model, incorporating the Hoyer effect and corrosion-induced cracking, is proposed to predict the transfer length in PC structures under corrosive cracking. The innovation of the proposed model is that it can consider the coupling effects of Hoyer effect and corrosion-induced cracking.

• 5. The transfer length depends on the corrosion positions and corrosion degrees. The transfer length with the corrosion loss of 35% will extend 8.6% when the corrosion position moves from the mid-span to the beam end. The transfer length increases by 24.3% once the corrosion loss of strand reaches 35%.

• 6. The transfer length of PC beams under corrosion cracking is influenced by Poisson's ratio of strand and modulus of elasticity of concrete. The transfer length under corrosion cracking increases accordingly with the increase of Poisson’s ratio of strand and modulus of elasticity of concrete.