Introduction

Corrosion of the strands has led to many instances of premature failure in existing prestressed concrete (PC) bridge decks since 1980 [1,2,3,4,5,6,7,8]. The ingress of moisture and chloride ions occurring through either concrete cover (pre-tensioned girders) or end blocks (post-tensioned girders) is a primary reason for corrosion. For post-tensioned girders, it gets aggravated in case of improper grouting of the ducts [9]. The main difference between corrosion in a reinforced concrete member and that in a PC member is that less extent of pitting corrosion in the latter can trigger brittle failure of the component or the structure as a whole [10].

The quantitative evaluation of corrosion [11] can be broadly classified as non-localized average cross-section loss (gravimetric mass loss [12,13,14,15]) and localized maximum cross-section loss (pit-configurations [16,17,18,19], contour gauge [20, 21], three-dimensional geometric modelling using laser scanning [22]). Several studies showed that the tensile deformability and ductility of prestressing wires as well as strands are highly sensitive to corrosion. The post-yield branch of the stress versus strain curve even disappeared at a less extent of corrosion [14, 15, 17,18,19]. In case of a multi-wire strand, the corrosion level of the maximum (critically) corroded wire is significant to cause the failure of the strand [19]. This is the main reason for the less accurate predictions of the tensile stress versus strain behaviour of strands with average section loss less than 5% [18, 19]. Hence, a new method was introduced in the present study to consider the localized pitting of critically corroded wire, as well as the distribution of corrosion in the strand, for predicting the tensile behaviour of a strand.

The objective of the study was to develop a methodology to predict the tensile behaviour of corroded prestressing strands. The paper is organized in the following manner. First, the test specimen details, artificial corrosion technique and the tensile test set-up are discussed under the section materials and methods. This is followed by the results from experiments, development of analytical models and generation of synthetic stress–strain curves. Finally, the proposed model is corroborated using data from recent literature.

Materials and Methods

Specimen Design and Fabrication

Four uncorroded strand specimens and five corroded strand specimens (as obtained from concrete prisms) were used in this study. Prisms with cross-section of 100 mm × 100 mm and length of 750 mm, were selected to simulate corrosion of the strands while embedded in concrete. Each prism specimen was reinforced with one 12.7 mm nominal diameter low-relaxation strand [23] placed at the centre. The concrete mix proportion of coarse aggregate, fine aggregate, water and cement by weight was 2.94:1.96:0.36:1.0. The 7-day average cube compressive strength was 41.7 MPa. The coarse aggregate and fine aggregate used were granite of maximum size 20 mm and natural river sand, respectively. Ordinary Portland Cement of Grade 53 S [24] was selected.

Accelerated Corrosion of Strands

The impressed current technique was adopted to accelerate the diffusion of chloride ions through the concrete cover of a prism specimen. A small reservoir of plan inner dimension 200 mm × 45 mm was constructed at the middle of the top surface of a prism specimen, to contain 5% sodium chloride (NaCl) solution. The strand (anode) and the Nickel chromium mesh (cathode) were connected to the positive and negative terminals, respectively, of a direct current (DC) galvanostatic Regulated Power Supply (RPS) of 64 V as shown in Fig. 1. Multimeter was used to monitor the current passing through the circuit. The duration of current was monitored to attain a target amount of corrosion (mass loss) in the strand based on the Faraday’s law (Table 1). Subsequently, the corroded strands were retrieved by breaking the prism specimens (P1 to P5), followed by cleaning procedure stipulated in ASTM G1-03 [11].

Fig. 1
figure 1

Schematic diagram of artificial corrosion set-up

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Table 1 Details of accelerated corrosion applied to prism specimens
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Tension Test of Strands

The uncorroded and corroded strand specimens of length 750 mm, were tested under tension to characterize the average stress versus average strain behaviour. The static tests were conducted based on the standard procedure of ASTM A1061/A1061M-2016 [25], at ambient temperature using a displacement-controlled test facility (Servo-hydraulic testing system MTS Model 311.32 of 1 MN capacity), as shown in Fig. 2. The stress can vary among the wires in a strand. Even within a wire, the stress can vary across the section due to the effect of residual stress [26]. Here, an average stress was considered across the section of a strand. The failure of a corroded strand is defined as the instant at which any one of the corroded wires ruptures [19]. Modified end grips were used for the strands to prevent localised failure at the end grip region. It consisted of hardened aluminium plates with smooth, semi-cylindrical grooves and coated with slurry (Grade 3-F aluminium oxide as abrasive and glycerine as carrier). In addition to this, aluminium foil was wrapped to protect the end portions of a strand specimen within the grip [19, 27].

Fig. 2
figure 2

Tension test set-up of a strand till failure

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The sudden rupture of corroded wires of a strand specimen would have damaged the external extensometers. Hence, the load and elongation were recorded using in-built load cell and extensometer of the testing machine. The extensometer recorded the total elongation between the grips. The slips occurring at the end grips were calibrated and deducted from the elongation, by conducting a parallel tension testing of an uncorroded strand specimen (Fig. 3). In the latter test, two LVDTs were fixed using magnetic base at the grip locations to obtain the slips till yielding of strand. It was assumed that additional slip after the yielding of a strand was negligible. The elongation between the grips was corrected by deducting the slips from the reading of the extensometer. An additional strain transducer was also attached at the middle region of the strand specimen to measure the strain in the test region. The strain calculated from the elongation was found to match with the strain obtained from the strain transducer. With this slip data as reference, slip excluded elongation was extracted for all the uncorroded and corroded strand specimens. For a corroded specimen, the net elongation of the corroded length was calculated by subtracting the elongations of the adjacent uncorroded lengths. The average strain in a corroded strand was calculated by dividing the net elongation by the measured length of the corroded region (ℓc). Here onwards, the average stress and average strain are simply referred to as stress and strain, respectively. The modulus of elasticity obtained in this paper is the apparent modulus of elasticity, as it is defined using average strain over a gage length.

Fig. 3
figure 3

Tension test set-up of a strand till elastic limit to capture slip at end grips

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Test Results and Analysis

Measure of Severity of Corrosion

The corrosion in strands was analysed after the tension tests of the specimens. The corrosion morphology of corroded strands showed severe pitting just beneath the reservoir region and slight pitting in the adjacent regions. The spatial distribution of section loss along the length of a strand was generated for all the corroded strands using X-ray Computed Tomography scanning technique (CT scan) (Table 2). This was done by analysing the transverse sectional images at every 1 mm interval along the length of a specimen. The length of a specimen had to be limited to 150 mm due to the space constraint of the scanning machine. The adopted technique was able to provide precise values of localised section loss along the length of a strand, as compared to the gravimetric method. It is evident from the corrosion morphology that with increasing corrosion, several uneven local pits (as in CS1 and CS2) get connected together to form a combination of smeared corrosion as well as pitting corrosion (as in CS3, CS4 and CS5). This observation is in good agreement with the findings of Liu et al. [14].

Table 2 Maximum area loss of strand, section loss profile of critical wire (CW) and corrosion morphology of corroded strand
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In corrosion of metals, pitting factor (Pf) for a planar surface is used as a measure of severity of corrosion in terms of local pitting depth (P) and depth of uniform corrosion (d). It can be approximately defined as Pf = P / d. ASTM: G1-03 [11] also defines the pitting factor in a similar way. In the present study, for cylindrical surfaces of wires and strands, this concept was modified and defined as the ratio of maximum area loss of critical wire (ΔAmax,cw) to average area loss of strand (ΔAavg,s) (Eq. (1) and Fig. 4). Thus, it considers both the maximum corrosion of the critically corroded wire, as well as the morphology of corrosion along the length of the strand. This concept is found to be similar to the cross-sectional area spatial heterogeneity factor defined by Zhang et al. [22]. Values of ΔAmax,cw and ΔAavg,s could be extracted from the section loss profiles of the corroded strand specimens (Table 3). The value of Pf decreases with amount of corrosion, having a lower bound of approximately 1.0, as there will be more average area loss of the strand as compared to maximum area loss of the critical wire.

Fig. 4
figure 4

Schematic representation of definition of pitting factor (Pf) of a corroded strand

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Table 3 Measured variables for obtaining pitting factor
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$$ {\text{P}}_{{\text{f}}} {\mkern 1mu} {\text{ = }}\frac{\Delta{{{\text{A}}_{{{\text{max,cw}}}} }}}{{\Delta {\text{A}}_{{{\text{avg,s}}}} }} $$
(1)

Analysis of Stress versus Strain Behaviour

The stress versus strain behaviour of uncorroded (US) strand of nominal diameter 12.7 mm was characterized for reference using four specimens (designated as US1 to US4). It was observed that the bi-linear stress–strain model is suitable for modelling the behaviour (Eq. (2)). This model can be conveniently modified to incorporate the effect of corrosion. The average mechanical properties of the uncorroded specimens are given in Table 4. As per IRC:112–2020 [28], the yield stress (fpy) was assumed to be 0.87fpu and the corresponding yield strain as εy = fpy / Es. The comparison of experimental results with the representative bi-linear model is shown in Fig. 5a.

Table 4 Mechanical properties of uncorroded strand (US) specimens
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Fig. 5
figure 5

Average stress–strain curves (a) Uncorroded strands (US) (b) Corroded strands (CS)

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For εp ≤ εy,

$$f_{p} = \, \varepsilon_{p} E_{s}$$
(2a)

For εy < εp ≤ εu,

$$f_{p} = f_{{py}} + {\text{ }}\frac{{f_{{pu}} - {\mkern 1mu} f_{{py}} }}{{\varepsilon_{u} - {\mkern 1mu} \varepsilon_{y} }} \times {\mkern 1mu} \left( {\varepsilon_{p} - {\mkern 1mu} \varepsilon_{y} } \right)$$
(2b)

In case of corroded strands, the elongation varied among the wires based on their residual areas. The average stress was calculated as the load divided by the average residual area of the corroded strand (Ares,s), which was obtained from ΔAavg,s. A sudden failure by rupture of the critically corroded wire was observed in all the specimens. Hence, the ultimate (breaking) load for a corroded strand was considered in this study to be governed by the critically corroded wire. εu_cs, fpu_cs, Ecs, Ares,s and ℓc are given in Table 5 for the corroded strand specimens. Comparison of the average stress versus strain curves for corroded strands with the bi-linear model for uncorroded strand is shown in Fig. 5b. For higher level of corrosion (ΔAavg,s > 12.3%), there was no yielding at all. The critical corrosion loss of 8% was considered to define no yield condition of corroded wires [14] and strands [18] in previous studies. Thus, the ductility of the strands was found to get reduced with increasing severity of corrosion. An analogous bi-linear stress–strain model is proposed for modelling the behaviour of a corroded strand using Ecs, εy_cs, and fpy_cs with (εu_cs, fpu_cs) as the failure point (Eq. (3)). The yield stress (fpy_cs) is defined as 0.87fpu_cs for fpu_cs ≥ fpy and the corresponding yield strain as εy_cs = fpy_cs / Ecs. No yielding was observed for specimens with fpu_cs < fpy (CS3, CS4 and CS5).

Table 5 Measured variables of corroded strand (CS) specimens
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For εp ≤ εy_cs

$$f_{p\_cs} = \, \varepsilon_{p} E_{cs}$$
(3a)

For εy_cs < εp ≤ εu_cs

$$f_{{p\_cs}} = f_{{py\_cs}} + {\text{ }}\frac{{f_{{pu\_cs}} - {\mkern 1mu} f_{{py\_cs}} }}{{\varepsilon_{{u\_cs}} - {\mkern 1mu} \varepsilon_{{y\_cs}} }} \times {\mkern 1mu} \left( {\varepsilon_{p} - {\mkern 1mu} \varepsilon_{{y\_cs}} } \right)$$
(3b)

Development of Analytical Models

Models for Mechanical Properties

It was evident from the experimental results that the degradation of mechanical properties (Ecs, εu_cs and fpu_cs) occur as a result of the corrosion of a strand. For a corroded strand with non-prismatic section, Ecs was calculated based on the average behaviour over the corroded length. To consider the localized pitting of critically corroded wire as well as the distribution of corrosion in the strand, εu_cs and fpu_cs were expressed in terms of Pf of the corroded strand. Each of these parameters were then normalised with respect to their corresponding values in uncorroded condition to get the reduction factors kE, kεu and kfpu as shown in Eqs. (4), (5) and (6).

$${\text{k}}_{\text{E}}= \text{ } \frac{{\text{E}}_{\text{cs}}}{{\mathrm{E}}_{\mathrm{s}}}$$
(4)
$$k_{{\varepsilon u}} = \frac{{\varepsilon _{{u\_cs}} }}{{\varepsilon _{u} }}$$
(5)
$${\text{k}}_{\text{fpu}}\text{=}\frac{{\text{f}}_{\text{pu\_cs}}}{{\text{f}}_{\text{pu}}}$$
(6)

The analytical models developed using experimental data points are shown in Fig. 6. They are expressed as follows.

Fig. 6
figure 6

Analytical models (a) kE versus average section loss of strand (b) \(k_{{\varepsilon u}}\) versus pitting factor and (c) \({\text{k}}_{{\text{f}}{\text{pu}}}\) versus pitting factor

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$${\text{k}}_{\text{E}} \, \text{=}\text{ a}{ \Delta \mathrm{A}}_{\mathrm{avg},\mathrm{s}} \, \text{+}\text{ b}$$
(7)
$$ k_{{\varepsilon u}} {\mkern 1mu} = {\text{c}}{\mkern 1mu} e^{{f{\text{P}}_{f} }} {\mkern 1mu} + {\text{g}} $$
(8)
$${\text{k}}_{\text{fpu}}\text{ = h}{\text{ e}}^{\text{m }{\text{P}}_{\text{f}}}\text{ + n}$$
(9)

The coefficients of Eqs. (7), (8) and (9) were derived from regression analysis and were obtained as: a = − 0.019, b = 1.0; c = − 1.306, f = − 0.092, g = 1.048; h = − 4.800, m = − 0.660 and n = 1.0. These models satisfied the bounding limits. The upper and lower limits used in each model are as follows. (i) kE = 1 for ΔAavg,s = 0% and kE = 0 for for ΔAavg,s = 53% (ii) kεu = 1 for Pf ≥ 40 and kεu = 0 for Pf ≤ 2.3 (iii) kfpu = 1.0 for  Pf ≥ 40 and kfpu = 0 for Pf ≤ 2.3.

Pitting Factor for a Corroded Strand

In absence of CT scan results, to get Pf of a corroded strand specimen, it needs to be related to the commonly computed data ΔAavg,s. It was observed that at a section, the residual areas of the individual wires can be different in strands with same ΔAavg,s [29]. Hence, the number of severely corroded wires (Ncc) and corroded wires (Nc) were both incorporated in relating the pitting factor for a given ΔAavg,s. The value of Ncc can be more than 1, based on the method of corrosion. The model relating Pf and ΔAavg,s (Eq. (10)) was developed based on Ncc and Nc data of the experimental corroded strand specimens (Table 6). Pf gets reduced if both Ncc and Nc are more for a given ΔAavg,s. The comparison of Pf obtained from proposed model and that from experiments is given in Fig. 7.

Table 6 Values of Ncc and Nc for corroded strands
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Fig. 7
figure 7

Comparison of pitting factor obtained from proposed model and experiment

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$$P_{f} {\text{ }} = {\text{ }}\frac{5}{{N_{{cc}} N_{c} }}\left( {\frac{p}{{\Delta A_{{avg,s}} }}{\text{ }} + {\text{ }}q} \right)$$
(10)

The coefficients of Eq. 10 were derived from regression analysis and were obtained as: p = 37.6 and q = 0.69.

In an existing bridge girder, the rate of corrosion of strand (mm/year) can be obtained from the instrument such as GECOR-9, which works based on linear polarization resistance (LPR) technique. The ΔAavg,s can be estimated from this data by adopting an appropriate area loss model for a corroding strand. Nc and Ncc can be obtained based on the diffusion of chloride front through the concrete cover and wire orientation of strand. Hence, Pf can be estimated using Eq. 10. The ΔAavg,s and Pf can be used for further analysis of existing girder. Thus, Pf can be used in field to get the severity of pitting corrosion of critical wire of each strand in existing structures from the measured rate of corrosion data.

Prediction of Stress Versus Strain Curves

The methodology adopted for developing the stress versus strain curve of any corroded strand is given in Fig. 8 as a flowchart. The results obtained from the tensile tests of the corroded strands were then compared with the corresponding predicted stress–strain curves, as shown in Fig. 9. Here, the pitting factor obtained from section loss profile was used. It was found that the theoretical prediction of stress–strain curve is in reasonable agreement with the tensile test results. The loss of ductility can be modelled by limiting the capacity of the strand corresponding to the breaking strain. Minor deviations of the predicted curves from the experimental results are due to the irregularity and variability of pitting corrosion along the lengths of the strands.

Fig. 8
figure 8

Flowchart for theoretical prediction of average stress versus strain curves of corroded strand

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Fig. 9
figure 9

Comparison of experimental data and theoretical prediction of average stress versus strain behaviour of corroded strands

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Validation of Developed Models

The developed models were used to corroborate the parameters for corroded strands with the data obtained from the test results in the available literature [15, 18, 19]. Values of ΔAavg,s were available for the specimens of Jeon et al. [19] and Belletti et al. [15], which were naturally corroded strands from existing structures. Values of ΔAmax,s obtained from the specimens of Lu et al. [18] were converted to ΔAavg,s using a relationship established from experiments. In the absence of details about the number of corroded wires which contributed to ΔAavg,s, Ncc and Nc were evaluated based on the best fit with the corresponding experimental data. The plots for correlation of theoretical and experimental values of reduction factors kE, kεu and kfpu are shown in Fig. 10, having coefficients of correlation equal to 0.89, 0.99 and 0.87, respectively. Additionally, the accuracy of proposed method was checked and compared with the available theoretical predictions of Jeon et al. [19] and Lu et al. [18], as shown in Fig. 11. The available section loss models for pitting corrosion such as Type 1, Type 2 and Type 3 pit configurations were used in Jeon et al. [19], whereas a hemispherical pit model was adopted in Lu et al. [18]. It can be seen that the proposed models are giving reasonably good predictions for kE and kεu when compared to the experimental data. Further studies are required to refine the proposed model for better prediction of kfpu.

Fig. 10
figure 10

Comparisons of proposed models for reduction factors with available experimental data (a) Modulus of elasticity, kE, (b) Breaking strain, kεu, (c) Ultimate stress, kfpu

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Fig. 11
figure 11

Comparison of proposed models for reduction factors with available theoretical predictions (a) kE,pred/kE,exp (b) kεu,pred/kεu,exp (c) kfpu,pred/kfpu,exp

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Influence of Steel Micro-Cracking

The development of steel micro-cracking emanating from the pits can be attributed to the presence of synergic effects of stress, corrosive medium, diffusion of corrosive hydrogen and material properties. Figure 12 shows the presence of micro-cracks in one of the corroded strand specimens captured using Scanning Electron Microscopy (SEM). The local effects caused by such damage would have led to the reduction of apparent elastic modulus and brittle failure [30]. The degradation due to corrosion in existing PC bridge girders can be similar to that discussed above.

Fig. 12
figure 12

Scanning electron microscope image of the pit region of a corroded wire of strand (a) \(100\mu m\) magnification (b) Micro-cracks of Region A at \(20\mu m\) magnification

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Conclusions

This study primarily investigated the effect of sectional area loss on average tensile stress versus strain behaviour of corroded 12.7 mm diameter 7-wire strands. The major conclusions of the present study are as follows:

  1. 1)

    The level of corrosion of a single wire with respect to the average corrosion of the wires is significant to cause the failure of the strand. This aspect is considered in the proposed analytical models.

  2. 2)

    The severity of corrosion in a strand was defined in terms of the pitting factor (Eq. (1)). The values of the factor for the corroded strand specimens were obtained from the CT scan results. The values become smaller as the corrosion increases. Analytical models were developed to predict the pitting factor from an estimable quantity, such as the average section loss of the strand (ΔAavg,s, Eq. (10)).

  3. 3)

    The three mechanical properties of the tensile behaviour of a corroded strand, modulus of elasticity (Ecs), ultimate strength (fpu_cs), and breaking strain (εu_cs), were obtained from tensile tests. It was observed that with increasing corrosion, the ductility of the strands gets reduced significantly. For ΔAavg,s > 12.3%, there was no yielding in the average stress versus strain curve of a strand. Models were developed for predicting the reductions of Ecs, εu_cs and fpu_cs (Eqs. (7), (8) and (9)).

  4. 4)

    The developed models in this study were substantiated by comparing with the results of the tension tests. They were also corroborated using test results and theoretical predictions available in the literature.

More number of corroded samples can be used to generate a database for refining the analytical models developed in this study. The study can be extended to corroded strand specimens subjected to prestress and fatigue loading behaviour.