Elevated temperature creep model of parallel wire strands

Abstract

Parallel wire strands (PWSs), which are widely used in prestressed steel structures, are typically in high-stress states. Under fire conditions, significant creep effects occur, reducing the prestress and influencing the mechanical behavior of PWSs. As there is no existing approach to analyze their creep behavior, this study experimentally investigated the elevated temperature creep model of PWSs. A charge-coupled camera system was incorporated to accurately obtain the deformation of the specimen during the elevated temperature creep test. It was concluded that the temperature level had a more significant effect on the creep strain than the stress level, and 450 °C was the key segment point where the creep rate varied significantly. By comparing the elevated temperature creep test results for PWSs and steel strands, it was found that the creep strain of PWSs was lower than that of steel strands at the same temperature and stress levels. The parameters in the general empirical formula, the Bailey–Norton model, and the composite time-hardening model were fitted based on the experimental results. By evaluating the accuracy and form of the models, the composite time-hardening model, which can simultaneously consider temperature, stress, and time, is recommended for use in the fire-resistance design of pre-tensioned structures with PWSs.

Graphical abstract

Keywords

parallel wire strands / experimental study / elevated temperature creep model

1 Introduction

Parallel wire strands (PWSs), as a type of steel cable, are a major component of pretensioned structures such as membrane structures and suspension bridges, as they are used to apply pretension and are always in a high-stress state. Numerous studies have investigated the mechanical properties of PWSs at ambient temperature [1−4]. Nonetheless, fire, which is one of the most frequent extreme conditions, can occur throughout the service life of buildings and bridges. Existing research [5–7] has highlighted that the degradation of the mechanical properties of building materials under fire conditions is severe and cannot be ignored. The effects of creep and degraded material properties in prestressed steel cables at elevated temperatures may lead to stress relaxation and pretension loss [8], thereby influencing the stiffness and strength of cable-supported structures and causing structural resistance degradation or further collapse.

Establishing a mathematical model of a material, structural component, or structure is essential for accurately evaluating their mechanical performance [9–14]. To establish the creep model, a series of experimental and theoretical analyses of the elevated temperature mechanical behavior of steel cables manufactured using various technologies have been conducted. Harmathy and Stanzak [15] conducted elevated temperature creep tests on 1725 MPa PWSs to explore the creep strain–temperature curves. The concept of temperature-compensated time was proposed, and a strain-hardening criterion was introduced to improve the Dorn creep model [16] and to propose the Harmathy creep model. However, the Harmathy creep model is more applicable to constant stress states, and large errors may occur in various stress states [17]. John et al. [18] performed steady-state and transient-state creep tests on steel strands using the digital image correlation method and calibrated the key parameters in the Harmathy creep model. Zhang and Zheng [19] tested the creep strain of 1770 MPa steel strands with diameters of 5 mm at elevated temperatures. A corresponding elevated temperature creep model was also proposed based on the experimental results of the thermal strain–time curves.

Concerning the test method, elevated temperature material property tests of steel cables can be classified into external and internal measurement methods. External measurement methods measure the deformation of a nonuniform temperature range within the specimen, and the actual strain is difficult to obtain accurately; therefore, they have been gradually abandoned by researchers. On the other hand, internal measurement methods can be further divided into direct and indirect internal measurement methods. Direct internal measurement methods generally use elevated temperature extensometers that extend the rod into the furnace to contact the surface of the specimen directly to obtain axial deformation. Indirect internal measurement methods improve the external measurements by transferring the deformation of the uniformly elevated temperature part of the specimen outside the furnace, in which the clamping part is transferred from the outside to the inside, and the accuracy can be improved. However, elevated temperature extensometers used in direct measurement methods are highly prone to relative slipping between the surfaces of the specimen, resulting in large errors in the test data, which prevents accurate elevated temperature mechanical property indicators. In addition, to avoid damaging the extensometer, it must be removed before the specimen fails, and the entire stress–strain curve cannot be measured. In indirect measurement methods, the axis of the rod for transferring the deformation must be parallel to the axis of the specimen, and errors can occur because the test equipment inevitably has slightly different axial degrees, resulting in bending or twisting of the specimen. Moreover, the rod expands slightly owing to the elevated temperature, which can cause a certain degree of amplification or reduction in the transferred deformation. In summary, the reliability and accuracy of the test results of elevated temperature material property tests of steel cables are highly dependent on the accuracy of the deformation measurement. It is worth noting that the twisting deformation of steel strands influences their elevated temperature material properties, whereas studies [20,21] ignored the influence of conducting tests only on the middle wire.

To ensure the accuracy of deformation measurements, John et al. [18] and Du and Gou [22] used a noncontact video strain measurement system to perform elevated temperature creep tests. They precisely captured the full range of the stress–strain relationship. In Ref. [23], steady-state creep tests were conducted under different target temperatures and stress ratios to investigate the full-range creep strain–time relationship of 2013.5 MPa [24] steel strands. The authors proposed an improved composite time-hardening model using calibrated parameters. The results indicate that the twisting deformation of the steel strands resulted in different creep rates with respect to the existing parameters recommended in AS/NZS 4672 [25], ASTM A416 [26], ASTM A421 [27], and BS 5896 [2] for the Harmathy creep model. Moreover, it is worth noting that if elevated temperature creeping is ignored, the steel strands will fail because of strength failure. However, the actual elevated temperature creep strengthened the relaxation deformation, and the steel strands failed to reach the ultimate strain. Therefore, it is not only essential to consider the effect of elevated temperature creep, but also to accurately quantify the creeping effect using advanced techniques when conducting the fire-resistance design of structures with prestressed steel cables.

In the tensile test at ambient temperature, the failure mode of the PWSs was consistent with that of single wires. However, the above studies showed that the elevated temperature mechanical properties of steel strands, which are formed by uniformly twisting multiple high-strength steel wires in a spiral shape in one direction by 30°−45°, can be different from those of single steel wires. The forming process of PWSs differs from that of steel strands because PWSs tie multiple high-strength steel wires tightly and twist them by only 2°−4°. However, the influence of twisting deformation on the elevated temperature mechanical properties of PWSs, particularly their creeping behavior, is still unclear.

To explore the creeping behavior and mechanical properties of PWSs at elevated temperatures, experimental research was conducted using an advanced noncontact video strain measurement system. The remainder of this paper is organized as follows. Section 2 introduces the specimens, test setup, test parameters, and test scheme for the elevated temperature creep tests on 36 PWS specimens. Section 3 presents the test results, including the test phenomenon and elevated temperature creep strain–time curves of the specimens. The results were analyzed and discussed by comparing them with steel strands in the existing literature. Section 4 proposes elevated temperature creep models based on test data, a general empirical formula, the Bailey−Norton model, and a composite time-hardening model. The accuracy and practicability of the proposed model were evaluated.

2 Creep tests

2.1 Specimen configuration

Each PWS specimen consisted of seven parallel steel wires with diameters of 7 mm. The configurations of the PWS specimens are presented in Fig.1. The geometric characteristics and design values of the ultimate strength at ambient temperature f_pk are listed in Tab.1, where n_w indicates the number of wires, d_w is the diameter of a single wire, A and d indicate the cross-sectional area and diameter of the PWS specimen, respectively, and L is the effective length of the PWS specimen. A white-elevated-temperature-resistant painting was sprayed on the middle of the PWS specimens to form the speckle belt shown in Fig.2(a), which was used for the deformation measurement conducted using a noncontact video strain measurement system.

It should be noted that the clamping device of the testing machine can damage the specimen, making the clamping part a weak area. Therefore, both ends of the PWS specimen were cast into an anchor head to prevent the failure of the specimen at the clamping part. In addition, the anchor heads can also relieve the relative slipping between the specimen and test machine; thus, the accuracy of the test results can be further improved.

2.2 Test equipment and instrumentation

The test setup, including the loading system, noncontact video strain measurement system, and electric furnace, is shown in Fig.3. The details of the connection section between the specimen and material testing machine in Fig.3 are shown in Fig.4.

The loading system was an Instron hydraulic servo universal test machine that could provide a maximum load of 500 kN and a maximum displacement of 75 mm. It should be noted that the anchor heads on both ends of the specimens had internal threads and were connected to the loading machine using a double-edged screw, as shown in Fig.4. Thus, the deviation of the axes caused by manual installation can be avoided, and the accuracy of the test results can be further improved.

The electric furnace, with a size of 30 cm × 10.5 cm (height × diameter), had three heating units and a side-view window where the three units could be heated simultaneously, facilitating a uniform temperature distribution within the furnace. The maximum temperature inside the furnace was 1250 °C. A thermocouple was placed at the center of each unit to measure the gas temperature.

Through the side-view window, the noncontact video strain measurement system, which adopts a charge-coupled device (CCD) camera, can capture any adjacent speckles within the speckle belt on the PWS specimen to form a gauge length smaller than 20 mm, as shown in Fig.2(b). The components of the system are shown in Fig.5. The variation in the speckle belt, that is, the deformation of the specimen, was recorded using a CCD camera and sent to the data acquisition system. After processing, the data were sent to a computer in the form of a gray level matrix.

2.3 Effective yield strength at elevated temperature

Day et al. [28] highlighted that the elevated temperature creep of a metal is dependent on the stress ratio, which is defined as the ratio of the target tensile stress to the effective yield strength at the ambient temperature. A series of yield strengths of PWSs under different strain levels at elevated temperatures was reported using tensile tests conducted by Du and Xiao [29], as tabulated in Tab.2. In Tab.2, θ is the temperature; E_p,θ is the elastic modulus at temperature θ; f_pp,θ and f_pt,θ are the proportional limit strength and ultimate strength at temperature θ, respectively; f_ε,θ is the nominal yield strength with a strain level of ε at temperature θ; ε_pp,θ, ε_pt,θ, and ε_pu,θ are the proportional limit strain, ultimate strain, and fracture strain at temperature θ, respectively. The data in Tab.2 can help determine the target tensile strain for the effective yield strength of PWSs at elevated temperatures.

Tab.2 shows that f_2.0%,θ was close to f_pt,θ, as the largest deviation, was only 7.8% at 200 °C. Meanwhile, as the values of f_2.0%,θ are all smaller than f_pt,θ, f_2.0%,θ can conservatively represent the yield strength of steel strands at elevated temperatures. Thus, f_2.0%,θ, with a strain level of 2.0%, was also defined as the effective yield strength for PWSs at elevated temperatures.

2.4 Target temperature and stress level

According to the Chinese standard GB/T 2039-2012 [30], the target temperature and stress level affect the elevated temperature creep of steel cables. According to the principles suggested in Ref. [31], the target temperature of creep tests is usually 0.3−0.5 times the melting point of the material (absolute temperature). As the melting point of galvanized steel is 1545 °C, the target temperatures for the elevated temperature creep tests were determined to be 350, 400, 450, and 500 °C.

As elevated temperature creep occurs when the stress level is lower than the yield strength of the metal, the stress level in the creep test should be determined based on the actual yield strength. According to the concept of effective yield strength proposed in Subsection 2.3, the stress levels are taken as at least 0.5f_2.0%,θ referring to the data in Ref. [32], and are carefully selected according to the strain development in preliminary tests. The actual stress levels selected for the creep tests and their corresponding stress ratios (with respect to f_2.0%,20) are listed in Tab.3, where three stress levels are selected for each target temperature.

In summary, there were four target temperatures, each with three stress levels; thus, there were 12 test conditions. Three parallel specimens were tested under each test condition, and the total number of specimens was 36.

2.5 Test scheme

It is notable that although transient-state tests (i.e., heating the specimen under a constant stress level) are closer to the actual fire scenario, the strain obtained in transient-state tests includes both thermal strain and creep strain, which are difficult to distinguish accurately. As we focus on the elevated temperature creep behavior of PWSs, the steady-state test method described in EN 10291-2000 [33] was adopted to perform creep tests on PWSs at elevated temperatures. The creep test scheme can be divided into the following steps.

S1. A pretension of 1 kN was added to the specimen to eliminate gaps within the specimen and loading machine.

S2. The specimen was heated to the target temperature at a rate of 10 °C/min.

S3. The target temperature was maintained for 1 h to achieve a uniform temperature distribution within the specimen, allowing the thermal strain to develop fully.

S4. The target temperature was kept constant, and a tensile load was applied to the PWS specimen up to the stress level specified in Subsection 2.4 within 10 min. The loading process should not last too long because creep is time-dependent, and a relatively short loading process can avoid excessive creep strain development during the loading process.

S5. The creep strains of the specimens were reset to zero. The strain of the specimen was monitored using a CCD camera with a frequency of 5 Hz, as long as the stress level was reached. The stress level was maintained for 2 h or until the specimen failed. Notably, a duration of 2 h satisfied the requirements for the analysis of common building fires.

During the entire test process, the loading machine was set to load control such that the jack could move to accommodate the creep strain of the specimens, thereby ensuring a constant stress level.

3 Creep test results and discussion

3.1 Test phenomenon

Fig.6 shows the appearance of typical PWS specimens when they were naturally cooled to normal temperature after the 2 h creep test at various temperatures and stress levels. It can be observed from Fig.6 that there was no obvious necking within the specimen when the target temperature was not above 500 °C.

3.2 Creep strain–time curves

The elevated temperature creep strain–time curves of the PWS specimens (average curve of the three parallel specimens) under different target temperatures and stress levels are shown in Fig.7 and Fig.8, respectively. It is noteworthy that the curves in Fig.7 and Fig.8 are the same; however, they are plotted under different control parameters. Note that all creep strain–time curves fluctuate with a root mean square of approximately 0.001% owing to the slight vibration of the loading machine during operation. Therefore, the test data collected by the noncontact video measurement system were smoothed using mean filtering to reduce noise interference.

Generally, creep strain–time curves can be divided into two stages: primary and secondary (steady phase). The rate of increase in strain gradually decreased in the primary stage. When the rate of increase decreased to its minimum value, it remained almost constant in the secondary stage. The demarcation points of the two stages are marked by a red cross in Fig.7, which is determined by the variation in the slope of the curves, as indicated by the blue dotted lines in Fig.7(a).

The following conclusions regarding the influence of the stress levels on the creep strain–time curve can be drawn from Fig.7.

1) After exposure to the target temperature of 350 °C for 2 h, the elevated temperature creep strain of the PWS specimen reached 0.0863%, 0.1689%, and 0.2692% under stress levels of 511, 617, and 734 MPa, respectively. Although the creep strain increased with the stress level, the overall creep strain was still relatively small, indicating that creep was not significant for PWSs at the target temperatures below 350 °C. Similarly, the creep resistance of PWSs was still acceptable when the target temperature was 400 °C and the stress level was below 442 MPa. However, when the stress level reached 511 MPa, the creep strain reached 0.4861% after 2 h, which is considerable and cannot be neglected when conducting a structural fire-resistance analysis. As shown in Fig.7(c) and Fig.7(d), large creep strains still existed when the target temperature was 450 or 500 °C, even when subjected to lower stress levels. For example, at 500 °C, the creep strain after 2 h reaches 0.9332% under a stress level of 238 MPa. Therefore, the effect of stress level on the creep effect becomes more significant with an increase in the target temperature.

2) At 350 °C, it took approximately 40, 55, and 70 min to reach the demarcation point under stress levels of 511, 617, and 734 MPa, respectively. In addition, the slope at the demarcation point increases with increasing stress levels. These characteristics can also be observed in Fig.7(b)−Fig.7(d), i.e., under target temperatures 400–500 °C, it can be concluded that as the stress level increases, the increase rate of creep strain in the secondary stage increases, and the duration of the primary stage is extended.

3) As shown in Fig.7(d), when the target temperature was 500 °C, the lower stress level of 238 MPa induced a higher creep strain than those under lower target temperatures and higher stress levels, as shown in Fig.7(b) and Fig.7(c). Hence, the influence of the stress level on elevated temperature creep was more significant when the target temperature was higher.

The following conclusions regarding the influence of the target temperature on the creep strain–time curve are drawn in Fig.8.

1) When the stress level was 442 MPa, the ultimate creep strain of PWS specimens after 2 h achieved 0.1572% and 0.9026% at 400 and 450 °C, respectively, indicating that the ultimate creep strain increased by 5.74 times due to an increase of 50 °C. Similarly, the ultimate creep strain increased by 2.73 times and 2.87 times under stress levels 511 and 617 MPa, respectively, when the temperature increased from 350 to 400 °C. Therefore, it can be concluded that, under the same stress level, the creep strain significantly increases with temperature.

2) Under the same absolute stress, the rate of increase in the creep strain in the secondary stage and the duration of the primary stage increased with the target temperature. Nonetheless, it should be noted that when the target temperature increased from 400 to 450 °C, the increase rate of the creep strain in the secondary stage increased more rapidly than at other temperatures under the same stress level. Therefore, the target temperature of 450 °C can be regarded as a key value with respect to the elevated temperature creep behavior of PWSs.

3) At a stress level of 442 MPa, the creep strain increased by 474% and the target temperature increased by 12.5% (400–450 °C), as shown in Fig.8(a). At 400 °C, the creep strain increased by only 29.04% when the stress level increased by 16.01%, as shown in Fig.7(b). Therefore, it can be concluded that the influence of the target temperature on the creep strain of PWSs is more significant than that of the stress level.

Moreover, the maximum stress level considered in this study was 734 MPa, and the corresponding creep strain ε_cr,θ only reached 6.25% of the ultimate strain, as listed in Tab.4, where σ is the stress level. As the creep strain increases with temperature, even though the stress level decreases, it is conservative to take the creep strain of ε_cr,350 with a stress level of 734 MPa in the practical application of pretensioned steel structures with PWSs when the stress level is lower than 734 MPa and the temperature is lower than 350 °C.

3.3 Comparison of creep behavior

A comparison of the elevated temperature creep strain–time curves between steel strands [23] and PWSs is shown in Fig.9. It can be observed from Fig.9 that the elevated temperature creep strain developed more significantly in the steel strands than in the PWSs. Specifically, under a stress level of 441 MPa, the ultimate creep strain of the steel strands was 29.8% larger than that of the PWSs at 450 °C after heating for 2 h, as shown in Fig.9(a). As shown in Fig.9(b), the ultimate creep strain of the steel strands was almost ten times that of the PWSs. The twisting angle of the steel strands is much larger than that of the PWSs, and the stress relaxation is more dramatic, as described in the introduction. Thus, the creep model for steel strands proposed in Ref. [23] cannot be directly applied to describe the creep behavior of PWSs, and an accurate and reliable creep model should be proposed based on the experimental results obtained in this study.

4 Creep model at elevated temperatures

A typical elevated temperature creep curve of the steel cables is shown in Fig.10, which can be divided into three stages: primary (I), secondary (II), and tertiary (III) creep stages. The experimental results obtained in Subsection 3.2 indicate that the PWSs will not enter Stage III under common building fire durations (2 h). Therefore, this section explores a model that describes the elevated temperature creep behavior of PWSs before rupture, that is, the primary and secondary stages.

4.1 Creep model based on general empirical formula

Taira et al. [34] proposed the following relationship between creep strain and time under constant temperature and stress levels:

(1){\epsilon }_{\text{cr}}=b{t}^n+kt,

where n is a positive number less than 1, and b and k are parameters related to the different stages of elevated temperature creeping. The derivative of Eq. (1) can be obtained as

(2){\dot{\epsilon }}_{\mathrm{c}\mathrm{t}}=bn{t}^{n-1}+k.

In Stage I, t tends to be 0, and bntⁿ⁻¹ is very large, as n− 1 is a negative number, and it plays a leading role. When entering Stage II, the first term tends to zero, the creep rate tends to k, and the second term plays a dominant role. Therefore, Eq. (1) can be expected to reflect the elevated temperature creep behavior of Stages I and II.

The experimental data of each creep curve were used to fit parameters b, n, and k using optimization software “1stOpt” [35] according to Eq. (1). The fitted values of the parameters and the goodness-of-fit R² of the creep curves under different temperatures and stress levels are listed in Tab.5.

Because the parameters b, n, and k change with the temperature and stress levels, the surfaces of the parameter values with respect to the temperature and stress levels are plotted in Fig.11−Fig.12 where the black dots indicate the fitted values listed in Tab.5. Based on the form of the surfaces, the practical formulas for the parameters are as follows:

(3)\begin{array}{rl}b=& \exp (-9224.666+1045.15\sigma -436.997\sigma \ln \sigma \\ & +168.422{\sigma }^{1.5}-12.969{\sigma }^2+1.51{\sigma }^2\ln \sigma \\ & -0.025{\sigma }^{2.5}+0.146\theta -0.018\theta \ln \theta ),\end{array}

(4)\begin{array}{rl}n=& -5.432+\frac{23321.919}{\sigma }-\frac{1.873\times {10}^7}{{\sigma }^2}+\frac{7.31\times {10}^9}{{\sigma }^3}\\ & -\frac{1.386\times {10}^{12}}{{\sigma }^4}+\frac{1.019\times {10}^{14}}{{\sigma }^5}-\frac{4093.39}{\theta }+\frac{772576}{{\theta }^2},\end{array}

(5)k=\exp \left(-190.049-\frac{1428.34}{\sigma }+6.15{\theta }^{0.5}+\frac{3889.995\ln \theta }{\theta }\right).

Based on the fitted parameters, the creep strain–time curves at different temperatures and stress levels calculated using the general empirical formula were compared with the test results, as shown in Fig.13. It can be seen that, except for cases of θ = 400 °C and σ = 442 MPa, θ = 450 °C, σ = 316 MPa and θ = 450 °C and σ = 381 MPa, where the fitting curve deviates from the test curve, the general empirical formula is in good agreement with the test results.

4.2 Creep model based on Bailey–Norton model

The Bailey–Norton model [36] is the most widely used model for describing Stage II of elevated temperature creep behavior, whose expression is given as

(6){\epsilon }_{\text{cr}}=B{t}^m{\sigma }^n,

where B, m, and n are parameters that depend on the temperature and can be determined by tests. It should be noted that when m = 1, Eq. (6) is the Norton model that only considers Stage II, whereas the Bailey–Norton model can consider Stages I and II by assigning m < 1 and m > 1, respectively.

To describe both Stages I and II, we used the experimental data to fit the Bailey–Norton model expressed as

(7){\epsilon }_{\text{cr}}=B_0{t}^m{\sigma }^{n_0}+Bt{\sigma }^n,

where additional parameters B₀ and n₀ are introduced to describe Stage I. The fitted parameters are listed in Tab.6.

A comparison between the Bailey–Norton model and experimental creep strain–time curves is also plotted in Fig.13. It can be found that the Bailey–Norton model is in good agreement overall, as R² is 0.99.

4.3 Creep model based on composite time-hardening model

The creep tests conducted in this study revealed that the target temperature, stress level, and fire exposure time governed the elevated temperature creep strain. A previous numerical study on the fire behavior of twisted strands conducted by Zienkiewicz and Cormeau [37] indicated that the following composite time-hardening model could be applied to describe the elevated temperature creep behavior of steel cables:

(8){\epsilon }_{\text{cr}}\left(t\right)=\frac{a_1}{a_3+1}{\sigma }^{a_2}\cdot t^{a_3+1}\cdot {\text{e}}^{-\frac{a_4}{\theta }}+a_5{\sigma }^{a_6}\cdot t\cdot {\text{e}}^{-\frac{a_7}{\theta }},

where ε_cr(t) is the elevated temperature creep strain at time t, ɑ₁−ɑ₇ are the parameters to be fitted.

With respect to Eq. (8), it is difficult to fit one set of coefficients to cover all combinations of influencing parameters. According to Subsection 3.2, when the target temperature reached 450 °C, the rate of increase in the creep strain suddenly increased. Therefore, 450 °C was considered the segment point, that is, the creep model was described using two sets of coefficients.

Regression was also performed using the optimization software 1stOpt [35], and the values of coefficients a₁−a₇ are tabulated in Tab.7. As R² is 0.997 and 0.995 for the two segments, it can be concluded that the proposed creep model based on the composite time-hardening model can accurately describe the elevated temperature creep behavior of PWSs. A comparison of the creep strain–time curves obtained using the proposed model and experimental results is shown in Fig.13.

4.4 Creep model comparison

As shown in Fig.13, by comparing the creep curves fitted by the PWSs based on the general empirical formula, the Bailey–Norton model, and the composite time-hardening model, it can be found that the accuracy of the general empirical formula is the worst. The behaviors of the Bailey–Norton model and composite time-hardening models are relatively similar, and they are in good agreement with the test results, indicating that the two models can effectively predict the elevated temperature creep strain of the PWSs.

Nonetheless, from the perspective of the formula, because the Bailey–Norton model is limited to a specific temperature level, it can only predict the elevated temperature creep response of PWSs when the thermal-structural coupled analysis is performed at a specific temperature. On the other hand, the composite time-hardening model considers the temperature, stress, and time simultaneously, which can well simulate Stages I and II of PWSs under different cases and has strong practicability for predicting the elevated temperature creep strain. Therefore, it is recommended to use the composite time-hardening model (Eq. (8)), with the fitted parameters listed in Tab.7 when conducting a thermal-structural coupled analysis of pretensioned structures with PWSs. The scope of application of the creep models proposed in this study is 20 °C ≤ θ≤ 500 °C and σ≤ 1780 – 2.90θ, which is determined by the experimental conditions specified in Section 2.

5 Conclusions

In this study, elevated temperature creep tests of 1670 MPa PWSs were conducted. The main conclusions are summarized as follows.

1) The ultimate creep strain of the PWSs increased with temperature and stress level. The temperature level had a more significant effect on creep strain than the stress level.

2) At the same temperature and stress level, the creep strain of the PWSs was lower than that of the steel strands.

3) The rate of creep strain increase of PWSs suddenly increased at 450 °C, which was regarded as a key segment point because the creep behavior below and above this temperature can be different.

4) Creep models for describing the elevated temperature behavior of PWSs are proposed based on the general empirical formula, Bailey–Norton model, and composite time-hardening model, in order to serve the fire-resistance design of pretensioned structures with PWSs. The composite time-hardening model is recommended for practical engineering applications based on its goodness of fit and simplicity. The scope of application of the creep models proposed in this study was 20 °C ≤ θ≤ 500 °C and σ≤ 1780 – 2.90θ.

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