# EM-Based Monitoring and Probabilistic Analysis of Prestress Loss of Bonded Tendons in PSC Beams

Authors: Zheheng Chen; Shanwen Zhang
Journal: Advances in Civil Engineering
 (2018)

---

## Abstract

The prestress level is a key factor of prestressed concrete (PSC) beams, affecting their long-term serviceability and safety. Existing monitoring methods, however, are not effective in obtaining the force or stress of embedded tendons. This paper investigates the feasibility of elastomagnetic (EM) sensors, which have been used for external tendons, in monitoring the long-term prestress loss of bonded tendons. The influence of ambient temperature, water, eccentricity ratio, plastic duct, and cement grouts on the test results of EM sensors is experimentally examined. Based on the calibrated EM sensors, prestress loss of a group of PSC beams was monitored for one year. In order to further consider the high randomness in material, environment, and construction, probabilistic analysis of prestress loss is conducted. Finally, the variation range of prestress loss with a certain confidence level is obtained and is compared with the monitored data, which provides a basis for the determination of prestress level in the design of PSC beams.


---

## Body

## 1. Introduction

Prestressed concrete (PSC) beams have been widely used in highway and railway bridges, in which the effective prestress is one key factor influencing the bridge serviceability and safety. However, it is well known that the time-dependent prestress loss is a very complex process, influenced by inherent material properties, external loads or environmental effects, and construction details. [1]. Moreover, there are abundant uncertainties in such a process [2], resulting in difficulties in accurate prediction on prestress loss. As a result, excessive deflections and cracks are often observed in practice [3–6], which highlights the importance of improved design of prestress level of PSC beams.In order to learn the authentic status of prestress loss in existing PSC beams, various long-term monitoring measures have been investigated. The vibrating wire strain gauges (VWSGs) can obtain the prestress loss due to shrinkage and creep of concrete [7–10], while they would not monitor the loss of stress relaxation. Similar problem exists when the optical fiber sensing is used [11, 12]. Due to the friction or bond of concrete, the stress along the tendon usually changes, and load cell at the beam ends can only obtain the total loss of prestress [13]. As to traditional resistance strain gauges, they are usually adhered to the surface of PT steels in the plastic corrugated pipes, while the narrow working space and the moisture may seriously affect the survival of gauges and their working performance. The elastomagnetic (EM) sensing is a nondestructive and noncontact monitoring technique that is based on the magnetic-elastic effects of ferromagnetic material [14]. At present, it has been usually used in stress monitoring of external tendons (i.e., unbonded tendons), such as the stayed cables, suspenders, and external tendons of PSC bridges, as shown in Figure 1. However, the application of EM sensing in bonded tendons requires further investigation due to the possible influence of plastic duct and cement grouts on measurement results.Figure 1
EM sensors used for the stress monitoring of external cables. (a) For stayed cables. (b) For suspenders. (c) For external tendons.
(a)
(b)
(c)In this paper, the feasibility of elastomagnetic (EM) sensors in monitoring the long-term prestress loss in bonded tendons is experimentally investigated, and the influence of ambient temperature, water, eccentricity ratio, plastic duct, and cement grouts on the test results is discussed. Based on the calibrated EM sensors, prestress loss of two PSC beams was monitored for one year. In order to further consider the high randomness in material, environment, and construction, probabilistic analysis of prestress loss is conducted based on the prediction model that can take into account the influence of nonprestressed steels and interaction among concrete creep, shrinkage, and steel relaxation. Finally, the variation range of prestress loss with a certain confidence level is obtained and is compared with the monitored data, which provides a basis for rational determination of prestress level in the design of PSC beams.


## 2. Elasto-Magnetic (EM) Sensing

### 2.1. Principle of EM Sensing

The EM sensor is a nondestructive and noncontact test method, based on the inherent magnetic-elastic effect in the ferromagnetic material. The change in tension force can be obtained by measuring variations in magnetic permeability of prestressed tendons. As shown in Figure2, the EM sensor consists of two layers of coils (i.e., the excitation and measurement coils) and the temperature sensor, and Figure 3 shows the measuring principle of EM sensor. The steel tendon will be magnetized when the pulse current flows into excitation coil, and pulsed magnetic field can be generated along the steel tendon. Due to electromagnetic induction, voltage is induced in the measurement coil. Magnitude of the induced voltage depends on the magnetic permeability of steel tendon, which is a function of the stress of the tendon, as shown in Equation (1).Figure 2
Configuration of an EM sensor.Figure 3
Measuring principle.The increment of magnetic permeability of the test specimen with respect to its zero stress state is expressed as follows:(1)μ=1+S0SfVoutV0−1,where V0 represents the integral value of voltage when there is no test specimen in the EM sensor, Vout denotes the integral value of voltage when there is a test specimen, S0 is the area of section surrounded by measurement coil, and Sf represents the net area of test specimen. The relationship between tension force and the increment of magnetic permeability can be expressed by the following equation:(2)f=C0+C1μ+C2μ2+C3μ3,where C0,C1, C2, and C3 are the coefficients of the cubic polynomials, which can be obtained from the calibration tests of EM sensors. Note that the effect of temperature on the magnetic permeability of ferromagnetic materials should be eliminated during the test, and the variation of magnetic permeability due to temperature effect can be modified by the following formula:(3)μf,T=μf,T0+αT−T0,where T and T0 are the test and the reference temperatures, respectively; μf,T and μf,T0 are the increments of magnetic permeability corresponding to the two temperatures; and α is the slope of the magnetic permeability vs. temperature relationship.


### 2.2. Tests on Influencing Factors of EM Sensor

A group of tests were designed to investigate the factors that may influence the performance of EM sensors. First, the influence of ambient temperature on the tests is experimentally investigated. It is known that the permeability of steel specimens changes with the temperature, while the slope of the relationship between the permeability and stress of steel specimens does not change within −30°C to 60°C [15]; therefore, the temperature calibration is carried out under the zero stress condition. In this study, a steel tendon with zero stress and the diameter of 15.24 mm was inserted in one EM sensor (type CCT43B) with the inner diameter of 43 mm, and they were put in a constant temperature box. The temperature started from 20°C and gradually rose to 50°C with an increment of 5°C. Under each temperature, the specimen was kept inside a temperature box for 1 hour before reading the relative magnetic permeability. The function curve of temperature and permeability is obtained, as shown in Figure 4, where linear relationship between temperature and the relative magnetic permeability is observed with the slope of 0.013.Figure 4
Temperature vs. permeability relationship.Secondly, the influence of ambient media on the test results was analyzed, where there were four tests. In the test No. 1, the EM sensor was put in air while in the test No. 2, the EM sensor was immerged in water. No specimens were put inside the sensor in the test Nos. 1 and 2. In the test Nos. 3 and 4, the EM sensors were put in air and immerged in water, respectively, with a steel tendon inside the sensor. In each test, the integral voltages were measured for six times, and the ambient temperature was 16°C. According to the test results in Table1, the integral voltages in water are almost the same as those in air, no matter there are steel specimens inside or not, showing that the water has no influence on the measurement results.Table 1
Integral voltages of specimen with various ambient media (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
1
526.7
526.3
526.7
527.0
526.9
527.0
526.8
2
526.6
526.7
526.3
526.7
526.9
526.5
526.6
3
641.0
640.7
640.8
640.6
641.2
640.6
640.8
4
640.9
640.4
641.1
640.5
640.7
640.8
640.7In addition, the inner diameter of EM sensor is usually significantly larger than the specimen; therefore, in some cases the specimen is difficult to be completely located in the centroid of EM sensor. To investigate the influence of eccentricity, another three tests were conducted, as shown in Figure5. In test No. 5, the steel tendon with the diameter of 15.24 mm was located in the centroid of the sensor; in test No. 6 the tendon was located at the lowest position in the sensor (having the largest vertical eccentricity) and in test No. 7 the tendon was inclined. The test results presented in Table 2 indicate that the value in test No. 6 is lower than those in tests No. 5 and 7, showing that the eccentricity has certain influence on the measurement results; therefore, the sensor should avoid excessive eccentricity when used.Figure 5
Positions of tendon in the sensor.Table 2
Integral voltages of specimen with various eccentricities (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
5
640.9
641.2
640.5
640.4
640.8
640.5
640.7
6
637.6
637.4
637.4
637.1
637.2
636.9
637.3
7
640.4
640.6
640.8
640.1
640.5
639.9
640.4In practice, the steel tendons are usually embedded in corrugated ducts with or without grouts. In order to study the influence of tendon ducts and grouts, the test No. 8 and test No. 9 were conducted, where in the test No. 8, there is a corrugated plastic duct outside the tendon and in the test No. 9, the duct was grouted. As compared with the results in the test No. 3, the two tests in Table3 are almost the same. Therefore, the corrugated ducts and cement grouts, being nonmetal materials, do not have any influence on the measurement results of EM sensor.Table 3
Integral voltages of specimens with duct and grout (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
8
640.4
640.9
640.7
640.5
640.2
640.1
640.5
9
640.2
640.4
640.5
640.0
640.4
640.0
640.3Finally, the relative magnetic permeability of the EM sensor was measured under various levels of loads, as illustrated in Figure6, where the relative magnetic permeability vs. tension force relationship is basically linear, and the cubic polynomial is used to describe the relation.Figure 6
Relative magnetic permeability vs. tension force relationship.


## 2.1. Principle of EM Sensing

The EM sensor is a nondestructive and noncontact test method, based on the inherent magnetic-elastic effect in the ferromagnetic material. The change in tension force can be obtained by measuring variations in magnetic permeability of prestressed tendons. As shown in Figure2, the EM sensor consists of two layers of coils (i.e., the excitation and measurement coils) and the temperature sensor, and Figure 3 shows the measuring principle of EM sensor. The steel tendon will be magnetized when the pulse current flows into excitation coil, and pulsed magnetic field can be generated along the steel tendon. Due to electromagnetic induction, voltage is induced in the measurement coil. Magnitude of the induced voltage depends on the magnetic permeability of steel tendon, which is a function of the stress of the tendon, as shown in Equation (1).Figure 2
Configuration of an EM sensor.Figure 3
Measuring principle.The increment of magnetic permeability of the test specimen with respect to its zero stress state is expressed as follows:(1)μ=1+S0SfVoutV0−1,where V0 represents the integral value of voltage when there is no test specimen in the EM sensor, Vout denotes the integral value of voltage when there is a test specimen, S0 is the area of section surrounded by measurement coil, and Sf represents the net area of test specimen. The relationship between tension force and the increment of magnetic permeability can be expressed by the following equation:(2)f=C0+C1μ+C2μ2+C3μ3,where C0,C1, C2, and C3 are the coefficients of the cubic polynomials, which can be obtained from the calibration tests of EM sensors. Note that the effect of temperature on the magnetic permeability of ferromagnetic materials should be eliminated during the test, and the variation of magnetic permeability due to temperature effect can be modified by the following formula:(3)μf,T=μf,T0+αT−T0,where T and T0 are the test and the reference temperatures, respectively; μf,T and μf,T0 are the increments of magnetic permeability corresponding to the two temperatures; and α is the slope of the magnetic permeability vs. temperature relationship.


## 2.2. Tests on Influencing Factors of EM Sensor

A group of tests were designed to investigate the factors that may influence the performance of EM sensors. First, the influence of ambient temperature on the tests is experimentally investigated. It is known that the permeability of steel specimens changes with the temperature, while the slope of the relationship between the permeability and stress of steel specimens does not change within −30°C to 60°C [15]; therefore, the temperature calibration is carried out under the zero stress condition. In this study, a steel tendon with zero stress and the diameter of 15.24 mm was inserted in one EM sensor (type CCT43B) with the inner diameter of 43 mm, and they were put in a constant temperature box. The temperature started from 20°C and gradually rose to 50°C with an increment of 5°C. Under each temperature, the specimen was kept inside a temperature box for 1 hour before reading the relative magnetic permeability. The function curve of temperature and permeability is obtained, as shown in Figure 4, where linear relationship between temperature and the relative magnetic permeability is observed with the slope of 0.013.Figure 4
Temperature vs. permeability relationship.Secondly, the influence of ambient media on the test results was analyzed, where there were four tests. In the test No. 1, the EM sensor was put in air while in the test No. 2, the EM sensor was immerged in water. No specimens were put inside the sensor in the test Nos. 1 and 2. In the test Nos. 3 and 4, the EM sensors were put in air and immerged in water, respectively, with a steel tendon inside the sensor. In each test, the integral voltages were measured for six times, and the ambient temperature was 16°C. According to the test results in Table1, the integral voltages in water are almost the same as those in air, no matter there are steel specimens inside or not, showing that the water has no influence on the measurement results.Table 1
Integral voltages of specimen with various ambient media (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
1
526.7
526.3
526.7
527.0
526.9
527.0
526.8
2
526.6
526.7
526.3
526.7
526.9
526.5
526.6
3
641.0
640.7
640.8
640.6
641.2
640.6
640.8
4
640.9
640.4
641.1
640.5
640.7
640.8
640.7In addition, the inner diameter of EM sensor is usually significantly larger than the specimen; therefore, in some cases the specimen is difficult to be completely located in the centroid of EM sensor. To investigate the influence of eccentricity, another three tests were conducted, as shown in Figure5. In test No. 5, the steel tendon with the diameter of 15.24 mm was located in the centroid of the sensor; in test No. 6 the tendon was located at the lowest position in the sensor (having the largest vertical eccentricity) and in test No. 7 the tendon was inclined. The test results presented in Table 2 indicate that the value in test No. 6 is lower than those in tests No. 5 and 7, showing that the eccentricity has certain influence on the measurement results; therefore, the sensor should avoid excessive eccentricity when used.Figure 5
Positions of tendon in the sensor.Table 2
Integral voltages of specimen with various eccentricities (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
5
640.9
641.2
640.5
640.4
640.8
640.5
640.7
6
637.6
637.4
637.4
637.1
637.2
636.9
637.3
7
640.4
640.6
640.8
640.1
640.5
639.9
640.4In practice, the steel tendons are usually embedded in corrugated ducts with or without grouts. In order to study the influence of tendon ducts and grouts, the test No. 8 and test No. 9 were conducted, where in the test No. 8, there is a corrugated plastic duct outside the tendon and in the test No. 9, the duct was grouted. As compared with the results in the test No. 3, the two tests in Table3 are almost the same. Therefore, the corrugated ducts and cement grouts, being nonmetal materials, do not have any influence on the measurement results of EM sensor.Table 3
Integral voltages of specimens with duct and grout (unit: voltage).
Test
Serial number of measurement
Mean
1
2
3
4
5
6
8
640.4
640.9
640.7
640.5
640.2
640.1
640.5
9
640.2
640.4
640.5
640.0
640.4
640.0
640.3Finally, the relative magnetic permeability of the EM sensor was measured under various levels of loads, as illustrated in Figure6, where the relative magnetic permeability vs. tension force relationship is basically linear, and the cubic polynomial is used to describe the relation.Figure 6
Relative magnetic permeability vs. tension force relationship.


## 3. Long-Term Monitoring of Prestress Loss of Bonded Tendons in PSC Beams

Based on the feasibility of EM sensors in monitoring the long-term prestress loss in bonded tendons, as depicted in Section2, a series of tests were conducted, and two of them are taken for analysis in this paper. The simply supported concrete beams were made using the C55 concrete, which has the 28-day cubic compressive strength of 55 MPa. The beams have a clear span of 4.2 m and a rectangular cross section of 200 × 300 mm, as shown in Figure 7. High-strength and low-relaxation 1860 grade steel strand was adopted as prestressed reinforcement, and the diameter of the reserved duct was 35.4 mm. The HRB335 grade twisted steel was used as longitudinal main reinforcement, the HPB235 grade round steel with diameter of 8 mm was adopted as transverse stirrups for the rest of the beams, and the stirrup spacing was 150 mm in all beams. The prestressing forces were applied directly to the target values at the age of 30 days. The initial tension forces of PC1 and PC4 are 100.1 kN and 135.9 kN, respectively, which were applied at one end of the beams with load cell installed to control the jacking force. Plain concrete blocks were applied as the secondary dead load at the two 1/3 beam spans immediately after the jacking, each with the weight of 3.125 kN. The detailed design parameters of the beams are shown in [16].Figure 7
Structural dimensions and sensor layout of beams PC1 and PC4. (a) Profile. (b) Cross section (at midspan).
(a)
(b)Figure8 illustrates the time-varying prestress loss of beams PC1 and PC4, monitored using EM sensors, where the development of prestress loss with time is similar in pattern to concrete creep and shrinkage. The long-term loss developed rapidly in the early monitoring period, and the growth rate of time-varying loss becomes gradually gentle after about 250 days. Due to the higher initial tension of the tendon in PC4, the long-term loss at 1/3 section of PC4 is 62.9 MPa after one year of monitoring, which is about 1.42 times of that of PC1. In addition, the monitored long-term loss exhibits some fluctuations during the whole monitoring period, which is due to the variation of environmental factors such as humidity.Figure 8
Long-term prestress losses of beams PC1 and PC4.


## 4. Probabilistic Analysis of Long-Term Prestress Loss of Bonded Tendons

### 4.1. Probabilistic Model of Long-Term Prestress Loss

Based on the method of age-adjusted effective modulus [17], the random prediction model for long-term prestress loss which takes into account nonprestressed steels and interaction among concrete creep, shrinkage, and steel relaxation is expressed in Equation (4), of which the detailed derivation can also be found in [16]:(4)σpt=Apσc,pτφt,τ+BpEpεsh+Cpσ¯prtDp,where n=σc,sτ/σc,pτ, α=Esp/Ecτ, Esp=Es+Ep/2, φt,τ=ψ1φ0βct,τ, and εsh=ψ2εsh0βst,τ; σc,pτ and σc,sτ represent the instantaneous elastic stresses of concrete at the centroids of PT tendons and nonprestressed steels at first loading time, respectively; Es, Ep, and Ecτ are the elastic moduli of nonprestressed steels, PT tendons, and concrete, respectively; φt,τ and εsh are the creep coefficient and shrinkage strain and aging coefficient at time t, respectively.In order to consider the uncertainties in creep model and shrinkage model, two coefficients,ψ1 and ψ2, are adapted in this study. For the CEB-FIP90 model, the statistical characteristic values of ψ1 and ψ2 can be taken as Eψ1=1, Vψ1=0.339 and Eψ2=1, Vψ2=0.451 [18], respectively; φ0 is the nominal creep coefficient; βct,τ is the coefficient to describe the development of creep with time; εsh0 is the nominal shrinkage strain; βst,τ is the coefficient to describe the development of shrinkage with time; σ¯prt is the reduced relaxation loss and equals to λrσprt; λr denotes the reduction factor, and a value of 0.75 [19] is adopted in this study, and the calculation method of intrinsic relaxation, σprt by Youakim et al. [20], is adopted in this paper.The coefficientsAp, Bp, CP, and DP are as shown in the following equation:(5)Ap=α1+αρsrs,s1+χφ−nαρsrp,s1+χφ,Bp=1+αρs1+χφrs,s−rp,s,Cp=1+αρsrs,s1+χφ,Dp=1+α1+χφρsrs,s+ρprp,p,where ρs and ρp represent the ratio of nonprestressed steels and PT tendons, respectively; ri,j=1+eiej/rc2, where i or j represents the nonprestressed steels (s) or PT tendons (p), and es and ep are the eccentricity of nonprestressed steels and the PT tendons with respect to the centroid of concrete, respectively, and rc is the radius of gyration of the concrete section; χ is the aging coefficient, which varies between 0.6 and 1.0 [21], and the value of 0.8 is adopted in this study for simplicity.


### 4.2. Statistical Properties of Random Parameters

The mechanism and influence factors of concrete shrinkage and creep are very complicated, and there are usually large deviations between the predicted and measured values of effects of concrete creep and shrinkage. There are many reasons leading to this phenomenon, and one of the main reasons is that the selected prediction model of shrinkage and creep itself is not perfect. At present, most of the shrinkage and creep models are the empirical formulas obtained through statistical regression based on experimental data, and the models themselves have the fitting errors; therefore, model uncertainty factors are used to describe the deviation. Some scholars have conducted extensive research on uncertainty of shrinkage and creep models [22–24]. For the CEB-FIP90 model, the statistical characteristic values of the model uncertainty factor are shown in Table 4.Table 4
Statistical properties of random variables.
Random variables
Unit
Mean
COV
Distribution
Sources
Uncertainty factor for creep model,ψ1
—
1
0.339
Normal
[18]
Uncertainty factor for shrinkage model,ψ2
—
1
0.451
Normal
[18]
Concrete strength,fcu
MPa
63.9
0.089
Normal
[25]
Density of concrete,DW
kN/m3
25.5
0.046
Normal
[26]
Initial stress in steel,σp
MPa
723 (PC1)
0.020
Normal
[27]
977 (PC4)
Annual relative humidity,RH
%
75.0
0.161
Normal
Observed
Initial loading age,t0
d
30
0.110
Uniform
Assumed
Elastic modulus of prestressing steel,Ep
MPa
1.95 × 105
0.060
Normal
[28]
Area of nonprestressed steel,As
mm2
565.5
0.035
Normal
[29]
Area of prestressed steel,Ap
mm2
140
0.0125
Normal
[30]Meanwhile, it is necessary to consider the randomness of parameters in the prediction model during the analysis of long-term loss of prestress, where the concrete compressive strength, ambient relative humidity, and loading age are treated as random variables. According to [25], the concrete strength used in this study follows the normal distribution with a mean value of 63.9 MPa and a coefficient of variation (COV) of 0.089. The ambient relative humidity is another significant factor which influences the time-varying effects of concrete structure, and the relative humidity in the laboratory exhibits seasonal fluctuations. According to the meteorological data of Nanjing, the probability density function (PDF) of ambient relative humidity is obtained, which follows a normal distribution with the mean value of 75% and the COV of 0.161. Other random factors include the density of concrete, initial loading age, elastic modulus of prestressing steel, areas of nonprestressed, and prestressed steel, which are described in Table 4. The geometric sizes of specimens are treated as design constants due to its minor variation. In order to facilitate stochastic analysis, random variables are assumed to be independent of each other.


### 4.3. Probabilistic Analysis and Discussion

According to the probabilistic model in Section4.1 and the random factors in Table 4, probabilistic analysis of long-term prestress loss in beams PC1 and PC4 is carried out using the Latin hypercube sampling technique (5000 samples), and the time-varying PDFs are obtained, as shown in Figure 9. It can be seen that the PDFs of long-term prestress loss approximately follow the normal distribution at different ages, and the probability density curves gradually become flat as time goes by, indicating that at the same confidence level, the width of confidence interval of long-term prestress loss becomes larger. As presented in Figure 10, the standard deviation (SD) of long-term loss increases continuously in each calculation condition from 1 month to 30 years after jacking, which shows that the discreteness of prestress loss due to the variation of influencing parameters gradually becomes larger with time, and the increase is most significant in the first ten years.Figure 9
Time-varying PDFs of long-term loss. (a) PC1. (b) PC4.
(a)
(b)Figure 10
Time-varying SDs of long-term loss.As shown in Figure11, the measured long-term loss exhibits some fluctuations during the whole monitoring period, due to variations of ambient temperature and humidity. Although the test results of PC1 are basically consistent with the mean values of stochastic analysis, for PC2, the differences between test results and mean analyzed results become larger, which is due to the uncertainties in analysis of time-varying effect. Note that the mean values of prestress loss are usually adopted in structural design, which might underestimate their actual values and lead to unsafe design. Considering the prestress loss approximately obeying normal distribution at each time, the 95% confidence interval can be supposed as [f¯lt−1.96SDt,f¯lt+1.96SDt], where f¯lt represents the mean value of probability analysis of time-varying loss at time t based on Equation (4), and SDt represents the SD of time-varying loss at time t. As shown in Figure 10, during the monitored period, the measured long-term prestress loss for beams PC1 and PC4 are located within the 95% confidence interval of probabilistic analyses. Taking PC4 as an example, the measured value of long-term loss is 70.6 MPa after about 1 year of monitoring, and the upper bound of 95% confidence interval is 97.1 MPa, which is about 37.5% larger than the measured value. Meanwhile, the width of the confidence interval of long-term loss gradually increased with time, varying from [4.3, 34.1] at one month of monitoring to [13.2, 97.1] at one year of monitoring, which indicates that the discreteness of time-varying loss becomes more obvious due to the larger variability of concrete shrinkage and creep as time goes by. Therefore, it is suggested that the upper bound of 95% confidence interval, instead of the mean value, should be used as the maximum long-term loss which may occur in the actual PSC structures.Figure 11
Comparison of probability results of long-term loss with measured values. (a) PC1. (b) PC4.
(a)
(b)


## 4.1. Probabilistic Model of Long-Term Prestress Loss

Based on the method of age-adjusted effective modulus [17], the random prediction model for long-term prestress loss which takes into account nonprestressed steels and interaction among concrete creep, shrinkage, and steel relaxation is expressed in Equation (4), of which the detailed derivation can also be found in [16]:(4)σpt=Apσc,pτφt,τ+BpEpεsh+Cpσ¯prtDp,where n=σc,sτ/σc,pτ, α=Esp/Ecτ, Esp=Es+Ep/2, φt,τ=ψ1φ0βct,τ, and εsh=ψ2εsh0βst,τ; σc,pτ and σc,sτ represent the instantaneous elastic stresses of concrete at the centroids of PT tendons and nonprestressed steels at first loading time, respectively; Es, Ep, and Ecτ are the elastic moduli of nonprestressed steels, PT tendons, and concrete, respectively; φt,τ and εsh are the creep coefficient and shrinkage strain and aging coefficient at time t, respectively.In order to consider the uncertainties in creep model and shrinkage model, two coefficients,ψ1 and ψ2, are adapted in this study. For the CEB-FIP90 model, the statistical characteristic values of ψ1 and ψ2 can be taken as Eψ1=1, Vψ1=0.339 and Eψ2=1, Vψ2=0.451 [18], respectively; φ0 is the nominal creep coefficient; βct,τ is the coefficient to describe the development of creep with time; εsh0 is the nominal shrinkage strain; βst,τ is the coefficient to describe the development of shrinkage with time; σ¯prt is the reduced relaxation loss and equals to λrσprt; λr denotes the reduction factor, and a value of 0.75 [19] is adopted in this study, and the calculation method of intrinsic relaxation, σprt by Youakim et al. [20], is adopted in this paper.The coefficientsAp, Bp, CP, and DP are as shown in the following equation:(5)Ap=α1+αρsrs,s1+χφ−nαρsrp,s1+χφ,Bp=1+αρs1+χφrs,s−rp,s,Cp=1+αρsrs,s1+χφ,Dp=1+α1+χφρsrs,s+ρprp,p,where ρs and ρp represent the ratio of nonprestressed steels and PT tendons, respectively; ri,j=1+eiej/rc2, where i or j represents the nonprestressed steels (s) or PT tendons (p), and es and ep are the eccentricity of nonprestressed steels and the PT tendons with respect to the centroid of concrete, respectively, and rc is the radius of gyration of the concrete section; χ is the aging coefficient, which varies between 0.6 and 1.0 [21], and the value of 0.8 is adopted in this study for simplicity.


## 4.2. Statistical Properties of Random Parameters

The mechanism and influence factors of concrete shrinkage and creep are very complicated, and there are usually large deviations between the predicted and measured values of effects of concrete creep and shrinkage. There are many reasons leading to this phenomenon, and one of the main reasons is that the selected prediction model of shrinkage and creep itself is not perfect. At present, most of the shrinkage and creep models are the empirical formulas obtained through statistical regression based on experimental data, and the models themselves have the fitting errors; therefore, model uncertainty factors are used to describe the deviation. Some scholars have conducted extensive research on uncertainty of shrinkage and creep models [22–24]. For the CEB-FIP90 model, the statistical characteristic values of the model uncertainty factor are shown in Table 4.Table 4
Statistical properties of random variables.
Random variables
Unit
Mean
COV
Distribution
Sources
Uncertainty factor for creep model,ψ1
—
1
0.339
Normal
[18]
Uncertainty factor for shrinkage model,ψ2
—
1
0.451
Normal
[18]
Concrete strength,fcu
MPa
63.9
0.089
Normal
[25]
Density of concrete,DW
kN/m3
25.5
0.046
Normal
[26]
Initial stress in steel,σp
MPa
723 (PC1)
0.020
Normal
[27]
977 (PC4)
Annual relative humidity,RH
%
75.0
0.161
Normal
Observed
Initial loading age,t0
d
30
0.110
Uniform
Assumed
Elastic modulus of prestressing steel,Ep
MPa
1.95 × 105
0.060
Normal
[28]
Area of nonprestressed steel,As
mm2
565.5
0.035
Normal
[29]
Area of prestressed steel,Ap
mm2
140
0.0125
Normal
[30]Meanwhile, it is necessary to consider the randomness of parameters in the prediction model during the analysis of long-term loss of prestress, where the concrete compressive strength, ambient relative humidity, and loading age are treated as random variables. According to [25], the concrete strength used in this study follows the normal distribution with a mean value of 63.9 MPa and a coefficient of variation (COV) of 0.089. The ambient relative humidity is another significant factor which influences the time-varying effects of concrete structure, and the relative humidity in the laboratory exhibits seasonal fluctuations. According to the meteorological data of Nanjing, the probability density function (PDF) of ambient relative humidity is obtained, which follows a normal distribution with the mean value of 75% and the COV of 0.161. Other random factors include the density of concrete, initial loading age, elastic modulus of prestressing steel, areas of nonprestressed, and prestressed steel, which are described in Table 4. The geometric sizes of specimens are treated as design constants due to its minor variation. In order to facilitate stochastic analysis, random variables are assumed to be independent of each other.


## 4.3. Probabilistic Analysis and Discussion

According to the probabilistic model in Section4.1 and the random factors in Table 4, probabilistic analysis of long-term prestress loss in beams PC1 and PC4 is carried out using the Latin hypercube sampling technique (5000 samples), and the time-varying PDFs are obtained, as shown in Figure 9. It can be seen that the PDFs of long-term prestress loss approximately follow the normal distribution at different ages, and the probability density curves gradually become flat as time goes by, indicating that at the same confidence level, the width of confidence interval of long-term prestress loss becomes larger. As presented in Figure 10, the standard deviation (SD) of long-term loss increases continuously in each calculation condition from 1 month to 30 years after jacking, which shows that the discreteness of prestress loss due to the variation of influencing parameters gradually becomes larger with time, and the increase is most significant in the first ten years.Figure 9
Time-varying PDFs of long-term loss. (a) PC1. (b) PC4.
(a)
(b)Figure 10
Time-varying SDs of long-term loss.As shown in Figure11, the measured long-term loss exhibits some fluctuations during the whole monitoring period, due to variations of ambient temperature and humidity. Although the test results of PC1 are basically consistent with the mean values of stochastic analysis, for PC2, the differences between test results and mean analyzed results become larger, which is due to the uncertainties in analysis of time-varying effect. Note that the mean values of prestress loss are usually adopted in structural design, which might underestimate their actual values and lead to unsafe design. Considering the prestress loss approximately obeying normal distribution at each time, the 95% confidence interval can be supposed as [f¯lt−1.96SDt,f¯lt+1.96SDt], where f¯lt represents the mean value of probability analysis of time-varying loss at time t based on Equation (4), and SDt represents the SD of time-varying loss at time t. As shown in Figure 10, during the monitored period, the measured long-term prestress loss for beams PC1 and PC4 are located within the 95% confidence interval of probabilistic analyses. Taking PC4 as an example, the measured value of long-term loss is 70.6 MPa after about 1 year of monitoring, and the upper bound of 95% confidence interval is 97.1 MPa, which is about 37.5% larger than the measured value. Meanwhile, the width of the confidence interval of long-term loss gradually increased with time, varying from [4.3, 34.1] at one month of monitoring to [13.2, 97.1] at one year of monitoring, which indicates that the discreteness of time-varying loss becomes more obvious due to the larger variability of concrete shrinkage and creep as time goes by. Therefore, it is suggested that the upper bound of 95% confidence interval, instead of the mean value, should be used as the maximum long-term loss which may occur in the actual PSC structures.Figure 11
Comparison of probability results of long-term loss with measured values. (a) PC1. (b) PC4.
(a)
(b)


## 5. Conclusions

This paper provides a preliminary investigation on the feasibility of EM sensors in monitoring the long-term prestress loss of bonded tendons. According to the presented study, conclusions are drawn as follows:(1)
The performance tests of EM sensor show that the external media such as water, cement grouting, and plastic duct have no influence on the measurement results, showing that the EM sensor can be well used in steel members bonded in concrete. The temperature has certain influence on the magnetic permeability of the tested specimens, and the correction coefficient can be obtained through temperature calibration at the zero stress state. The eccentricity of the steel specimens relative to the EM sensor has some impact on the test results, and therefore, the steel specimens should be located in the centroid of the EM sensor when it is installed.(2)
The long-term prestress loss due to shrinkage, creep of concrete, and stress relaxation is complicated, and there are significant uncertainties. Therefore, the uncertainty analysis of long-term loss should be carried out in order to make the design of bridges more safe and realistic. Based on the probabilistic analysis model adapted in this study, the probability analyses were conducted, which indicate that the long-term loss of prestress does not reject the normal distribution at each time, and its discreteness gradually become larger with the extension of time. The bandwidth surrounded by the upper and lower bound of 95% confidence interval also gradually increase with time. In addition, the predicted prestress loss in bridge design from deterministic analysis may be significantly different from the authentic monitored values. Therefore, to conduct a safe design, it is suggested that some suitable confidence limits such as 95% should be adopted instead of the mean value.(3)
For the internal prestressed tendons of widespread PSC box girder bridges, the EM sensors installed in the construction stage can be used for the real-time monitoring of effective prestress force in the service process of bridges, which would provide the basis for the long-term performance assessment of the structures. These will be done in the next study.



---

*Source: 4064362-2018-09-23.xml*