# Stiffness and Crack Behavior of Unbonded Posttensioned Concrete Beam Strengthened with Aluminum Alloy Plate

Authors: Hong Chang; Wei Zhou
Journal: Advances in Civil Engineering
 (2020)

---

## Abstract

Corrosion resistance of aluminum alloy plates externally bonded by magnesium phosphate cement provides the ability to strengthen inshore infrastructures in harsh environments subject to moisture and humidity. In this study, the aim is to study the stiffness and cracking behavior of concrete beams using this strengthening technique. Six damaged unbonded posttensioned concrete beams were repaired and strengthened and then subjected to monotonic load until failure. This technique improved the stiffness and limited the development of cracks. The formula of elastic-plastic stiffness coefficient related to the comprehensive reinforcement index was established. An influence coefficientδ considering the effect of aluminum alloy plates and unbonded tendons was introduced, and the crack expansion coefficient under short-term load was obtained by statistical analysis. Finally, some simplified methods were proposed to evaluate the stiffness and cracks of unbonded posttensioned concrete beams strengthened with aluminum alloy plates.


---

## Body

## 1. Introduction

Numerous studies have verified the mechanical properties of structures strengthened by thin steel plates and fiber reinforced polymer (FRP) sheets, and the research results have contributed to the development of specifications, codes, and standards. Meanwhile, these techniques have been used widely in various infrastructures [1–8]. However, the corrosion of steel in wet or humid conditions can decrease its strengthening effect. In addition, the brittle failure characteristics of FRP does not meet the ductile requirements of concrete structures. As a strengthening material, there are two advantages of aluminum alloy (AA) compared to steel and FRP: (1) it has a similar tensile strength to steel, and lower density and (2) it has a similar corrosion resistance to FRP, and higher ductility. In addition, epoxy used as an adhesive for structural strengthening behaves poorly under high temperature and aging, and has a pungent odor. Magnesium phosphate cement (MPC) has the advantages of high strength, low shrinkage, high temperature resistance, aging resistance, and nontoxicity, and therefore has the potential to replace epoxy as the adhesive for structural strengthening in harsh environments [9, 10].Previous studies [11–14] have shown experimentally that the technique of strengthening with AA plates can effectively improve the flexural capacity of reinforced concrete (RC) beams, with better ductility than that of beams strengthened with FRP. The RC beams strengthened by this technique have the behavior of debonding under the loading point, which is similar to the behavior shown by FRP sheet and steel plate and causes the premature failure of strengthened beams. The strengthening technique of near-surface mounting with AA plates and external bonding using AA plates with anchoring can effectively avoid early failure of strengthened beams caused by debonding of AA plates. The formula of ACI 440.2R-08 is found to be feasible to predict the debonding strain of the beams strengthened by AA plates. Epoxy was used as the adhesive for AA plates in most previous studies. Researchers have applied MPC to the interface between AA plates and concrete, studied the behavior of the interface through bilateral shear tests, and proposed the constitutive model of the bond-slip relationship [15]. The feasibility of using MPC as the adhesive bonding AA plates to strengthen unbonded posttensioned concrete (UPC) beams has been proved [16]. The ductility of the strengthening beam decreased while the flexural capacity was effectively improved. Studies on stiffness and cracking of UPC beams strengthened by AA plates bonded with MPC have not yet been carried out.In this study, an investigation was carried out on the stiffness and cracking of UPC beams strengthened with AA plates bonded by MPC. Six damaged UPC beams were repaired and strengthened, and then loaded to failure under a monotonic load, and attention was specifically paid to the deflection and cracks of the beams. Finally, a calculation method of stiffness and cracking of UPC beams strengthened with AA plates has been proposed based on experimental results and available theory.


## 2. Specimens and Methods

### 2.1. Specimens

A total of six damaged UPC beams were repaired and strengthened. Figure1 shows the dimensions, reinforcement details, cross-section details, and unbonded prestressed details of the specimens. Each beam has a length of 6000 mm and different reinforcement ratios. The properties of specimens are shown in Table 1.Figure 1
The reinforcement of specimens: (a) SS beams; (b) SL beams.
(a)(b)Table 1
Properties of specimens.
No.Cross-sectional area (mm × mm)Steel reinforcementsSteel strandSL1200 × 4002C121AS17.8SL2200 × 4002C181AS17.8SL3200 × 4002C251AS17.8SS1200 × 3002C121AS17.8SS2200 × 3002C181AS17.8SS3200 × 3002C221AS17.8The reinforcement index of the non-prestressed reinforcementsβs = fyAs/fcbhp, that of the unbonded tendons βp = σpeAp/fcbhp, that of the AA plate βa = σaAa/fcbh, that of the compression reinforcement βs′ = fy′As′/fcbhp, and the compositive reinforcement index β0 = βs + βp + βa−βs′, where b is the width of the UPC beams; hs, hp, and ha are the distance from the top face to the centroid of the non-prestressed reinforcement, unbonded tendons, and AA plate, respectively. fy, σpe, σa, and fy′ are the yield stress of non-prestressed reinforcement, unbonded tendons, AA plate, and compression reinforcement, respectively.All six UPC beams were damaged by static loads, followed by the crushing of the concrete in the compression zone. All the damaged beams exhibited relatively small residual deformation, and most of the tensile cracks were closed. The reinforcement yielded before the concrete in the compression zone crushing. The initial damage of each damaged UPC beam is shown in Table2.Table 2
Initial damage of each damaged beam.
No.Cracking load (kN)Ultimate load (kN)Maximum crack width (mm)Number of cracksInitial prestress (MPa)FrontBackSL180204513121570SL260180116161570SL3442000.318201570SS14084217161361SS2301040.219181570SS3341180.420201570The diameter of the stirrup of the specimens was 8 mm, which was distributed in the range of 2000 mm at the end of the beams with a spacing of 100 mm (see Figure1). The stainless-steel bolts with a diameter of 10 mm were anchored into the hole with a depth of 50 mm at the bottom of the beams with MPC. The uses of the stainless-steel bolts are as follows: (1) to bear the interfacial shear force when the bonding fails; (2) to avoid the debonding of the AA plates before the failure of strengthened beams; (3) to apply uniform pressure to the AA plates in the setting process of MPC to improve the adhesion effect. Considering the above factors, it was determined that the stainless-steel bolts would be staggered with a spacing of 330 mm (see Figure 2).Figure 2
The detailed description of bolts and AA plate.


### 2.2. Properties of Materials

Concrete with grade 40 was used to cast the beams. P.O 42.5 Portland cement and class F fly ash were used as binders, with a water-binder ratio (w/b) of 0.3. Sand particles in zone II were used as the fine aggregate, and crushed rocks were used as coarse aggregate. The axially compressive strength of the concrete was 45 MPa. The elastic modulus of the concrete Ec was 33600 MPa.Steel strands of grade 1860 with a diameter of 17.8 mm were used as unbonded tendons. Their tensile strength was 1915 MPa, and nominal yield strength was 1732 MPa. The mechanical properties of non-prestressed reinforcements are shown in Table3.Table 3
Properties of reinforcing bars.
Bar sizeGrade (MPa)Yielding stress (MPa)Ultimate stress (MPa)Elasticity modulusEs (MPa)C8400402575197000C12400402575197000C18400452648197000C22400405629197000C25400489672197000A unidirectional tensile test of 5083 AA plates was carried out on a universal electronic test machine (see Figure3), which showed that its yielding strength was 112 MPa, ultimate strength was 210 MPa, elastic modulus Ea was 278000 MPa, and ultimate elongation was 27%.Figure 3
Unidirectional tensile test.MPC is composed of ammonium dihydrogen phosphate or potassium dihydrogen phosphate, magnesium oxide, and a setting retarder. Borax was used as a retarder agent in this experiment, and the mix ratio of MPC by weight was ammonium dihydrogen phosphate (NH4H2PO4): borax (Na2B4O7 · 10H2O): magnesium oxide (MgO): water = 26 : 4:51 : 19. The compressive strength and flexural strength of the MPC were 23.5 MPa and 7.5 MPa, respectively.


### 2.3. Experimental Designs

Two-point loads were applied at the one-third span points of the beam (see Figure4). Three displacement sensors were set under the loading points and the midspan of the beam, two sensors were located at the end of the AA plate to measure the relative slip between the AA plate and the beam, and another two were set above the supports to measure the beam end movement. The AL beams were loaded with 5 kN for each level, and the AS beams with 3 kN for each level.Figure 4
Loading device.


### 2.4. Failure Modes

Among the six specimens, the failure mode of SS3 was the sudden fracture failure, and the failure of the other five beams can be attributed to concrete crushing in the compression zone. The details are as follows.


## 2.1. Specimens

A total of six damaged UPC beams were repaired and strengthened. Figure1 shows the dimensions, reinforcement details, cross-section details, and unbonded prestressed details of the specimens. Each beam has a length of 6000 mm and different reinforcement ratios. The properties of specimens are shown in Table 1.Figure 1
The reinforcement of specimens: (a) SS beams; (b) SL beams.
(a)(b)Table 1
Properties of specimens.
No.Cross-sectional area (mm × mm)Steel reinforcementsSteel strandSL1200 × 4002C121AS17.8SL2200 × 4002C181AS17.8SL3200 × 4002C251AS17.8SS1200 × 3002C121AS17.8SS2200 × 3002C181AS17.8SS3200 × 3002C221AS17.8The reinforcement index of the non-prestressed reinforcementsβs = fyAs/fcbhp, that of the unbonded tendons βp = σpeAp/fcbhp, that of the AA plate βa = σaAa/fcbh, that of the compression reinforcement βs′ = fy′As′/fcbhp, and the compositive reinforcement index β0 = βs + βp + βa−βs′, where b is the width of the UPC beams; hs, hp, and ha are the distance from the top face to the centroid of the non-prestressed reinforcement, unbonded tendons, and AA plate, respectively. fy, σpe, σa, and fy′ are the yield stress of non-prestressed reinforcement, unbonded tendons, AA plate, and compression reinforcement, respectively.All six UPC beams were damaged by static loads, followed by the crushing of the concrete in the compression zone. All the damaged beams exhibited relatively small residual deformation, and most of the tensile cracks were closed. The reinforcement yielded before the concrete in the compression zone crushing. The initial damage of each damaged UPC beam is shown in Table2.Table 2
Initial damage of each damaged beam.
No.Cracking load (kN)Ultimate load (kN)Maximum crack width (mm)Number of cracksInitial prestress (MPa)FrontBackSL180204513121570SL260180116161570SL3442000.318201570SS14084217161361SS2301040.219181570SS3341180.420201570The diameter of the stirrup of the specimens was 8 mm, which was distributed in the range of 2000 mm at the end of the beams with a spacing of 100 mm (see Figure1). The stainless-steel bolts with a diameter of 10 mm were anchored into the hole with a depth of 50 mm at the bottom of the beams with MPC. The uses of the stainless-steel bolts are as follows: (1) to bear the interfacial shear force when the bonding fails; (2) to avoid the debonding of the AA plates before the failure of strengthened beams; (3) to apply uniform pressure to the AA plates in the setting process of MPC to improve the adhesion effect. Considering the above factors, it was determined that the stainless-steel bolts would be staggered with a spacing of 330 mm (see Figure 2).Figure 2
The detailed description of bolts and AA plate.


## 2.2. Properties of Materials

Concrete with grade 40 was used to cast the beams. P.O 42.5 Portland cement and class F fly ash were used as binders, with a water-binder ratio (w/b) of 0.3. Sand particles in zone II were used as the fine aggregate, and crushed rocks were used as coarse aggregate. The axially compressive strength of the concrete was 45 MPa. The elastic modulus of the concrete Ec was 33600 MPa.Steel strands of grade 1860 with a diameter of 17.8 mm were used as unbonded tendons. Their tensile strength was 1915 MPa, and nominal yield strength was 1732 MPa. The mechanical properties of non-prestressed reinforcements are shown in Table3.Table 3
Properties of reinforcing bars.
Bar sizeGrade (MPa)Yielding stress (MPa)Ultimate stress (MPa)Elasticity modulusEs (MPa)C8400402575197000C12400402575197000C18400452648197000C22400405629197000C25400489672197000A unidirectional tensile test of 5083 AA plates was carried out on a universal electronic test machine (see Figure3), which showed that its yielding strength was 112 MPa, ultimate strength was 210 MPa, elastic modulus Ea was 278000 MPa, and ultimate elongation was 27%.Figure 3
Unidirectional tensile test.MPC is composed of ammonium dihydrogen phosphate or potassium dihydrogen phosphate, magnesium oxide, and a setting retarder. Borax was used as a retarder agent in this experiment, and the mix ratio of MPC by weight was ammonium dihydrogen phosphate (NH4H2PO4): borax (Na2B4O7 · 10H2O): magnesium oxide (MgO): water = 26 : 4:51 : 19. The compressive strength and flexural strength of the MPC were 23.5 MPa and 7.5 MPa, respectively.


## 2.3. Experimental Designs

Two-point loads were applied at the one-third span points of the beam (see Figure4). Three displacement sensors were set under the loading points and the midspan of the beam, two sensors were located at the end of the AA plate to measure the relative slip between the AA plate and the beam, and another two were set above the supports to measure the beam end movement. The AL beams were loaded with 5 kN for each level, and the AS beams with 3 kN for each level.Figure 4
Loading device.


## 2.4. Failure Modes

Among the six specimens, the failure mode of SS3 was the sudden fracture failure, and the failure of the other five beams can be attributed to concrete crushing in the compression zone. The details are as follows.


## 3. Stiffness Analysis

### 3.1. Relationship between Load and Deflection

The typical curve of load-deflection of prestressed concrete beams should have obvious turning at the points of cracking of beams and yielding of non-prestressed reinforcements. In this study, the test beams have been damaged and repaired, which has a certain impact on the stiffness of strengthened beams. As a result, the stiffness mutation of strengthened beams was not obvious, as shown in Figure5. The attached figures show the characteristics of a trilinear model with the inflection points of the crack load, yield load, and ultimate load of strengthened beams. It can be observed from Figure 5 that the curves still have similar characteristics to those of the prestressed concrete beams.Figure 5
Load-deflection curves: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
(a)(b)(c)(d)(e)(f)The three dotted lines were divided into three stages, with the slope of each segment being the stiffness of the corresponding stage.Stage I is the stage before the concrete cracks. The beam behaves mainly elastically. The relationship between stress and strain is basically linear. This stage continued until the tensile stresses exceeded the tensile strength of concrete.Stage II is the serviceability stage that occurs after the concrete has been cracked and non-prestressed reinforcements take up almost all the tension force, but the non-prestressed reinforcements have not yet yielded. The neutral axis shifts upward with the increase of the applied loads.Stage III starts with the yielding of non-prestressed reinforcements and ends with the failure of the beam. In this stage, the stiffness of the beam is weakened further. The load-deflection relationship becomes clearly nonlinear. Finally, the excessive developing of cracks and the crushing of concrete in the compression zone lead to the ultimate collapse of the beam.Figure5 shows that the stiffness changes of the SL1 and SS1 beams with the lowest reinforcement ratio are obvious at the crack load, while those of other beams are not obvious. The behavior at the yield load is opposite to that at the crack load; the stiffness of the SL1 and SS1 beams does not change significantly at the yield load, while the stiffness of other beams changes significantly at the yield load. It can be concluded that (1) the influence of cracking on the stiffness is inversely proportional to the reinforcement ratio and (2) the influence of the yielding of non-prestressed reinforcement on the stiffness is proportional to the reinforcement ratio.Figure6 shows the failure modes of the specimens.Figure 6
The failure modes picture of the specimens: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
(a)(b)(c)(d)(e)(f)


### 3.2. Stiffness Calculation

A formula for calculating the deflection of homogeneous elastic materials is proposed in [17], f=aMl2/EcI0, but the concrete is a heterogeneous and inelastic material, so the bending stiffness of the interface changes during bending. The bending stiffness not only decreases with the increase of load but also decreases with the increase of load duration, so the reduction coefficient, namely, the elastic-plastic stiffness coefficient, should be considered. According to the empirical values used in the Chinese design standards, the elastic-plastic stiffness coefficient of concrete beams before cracking is 0.85, and the elastic-plastic stiffness coefficient after cracking β′ needs further research.According to the bilinear stiffness model, the bending moment-deflection curve of prestressed concrete beams can be regarded as consisting of two straight sections OA and AB. With the cracking point as the turning point, the deflection under the cracking momentMcr is fcr, and the deflection under the serviceability moment is fs, as shown in Figure 7.Figure 7
Bending moment-deflection curve of the prestressed beam.According to Figure5, f=fcr+fs=al2/EcI0Mcr/0.85+Ms−Mcr/β′.The elastic-plastic stiffness coefficient under the increment of bending moment (Ms − Mcr) is then obtained as(1)β′=al2Ms−Mcrfs−fcrEcI0,where a is the coefficient of load and support conditions, which according to the material mechanics, in this case, is 0.106.In order to study the elastic-plastic stiffness coefficientβ′ after cracking, it is necessary to analyze the bending moment increment, and Ms is calculated as 1.67Mcr. It was found by calculation that when the Ms of the SL1 beam was 150 kN, the non-prestressed reinforcements have yielded, so no research was conducted on the SL1 beam. According to the experimental data of the other five strengthened beams, the elastic-plastic stiffness coefficient was as calculated in Table 4.Table 4
Calculation results of elastic-plastic stiffness coefficient.
No.Mcr (MPa)fcr (mm)Ms (MPa)fs (mm)β′SL26516108310.19SL36011100190.23SS1452875620.18SS2422070400.22SS3391965340.23By analysis of the experimental data, it was found that there was an approximate linear relationship between 1/β′ and 1/β0, and the fitting equation is as equation (2) with a correlation coefficient of 0.9588. Figure 8 shows the relationship between experimental data and the fit line:(2)1β′=2.3+0.52β0.Figure 8
Relationship between 1/β′ and 1/β0.Thus,(3)β′=β02.3β0+0.52.The stiffness of a UPC beam changes during bending; therefore, in order to accurately calculate the deformation of the beam under short-term load, an appropriate average stiffnessBs is proposed as the short-term stiffness. Thus, on the basis of f=aMl2/Bs, the short-term stiffness of UPC beams strengthened by AA plates can be expressed as(4)Bs=0.85β′MsEcI0β′Mcr+0.85Ms−Mcr.


### 3.3. Prediction and Analysis of Deflection

A UPC beam can be regarded as equivalent to an RC beam with a pair of prepressure in the anchorage position, which still show the characteristics of an RC beam after cracking. Therefore, the stiffness of UPC beams strengthened with AA plates can be measured using the same research method as the RC beams.The stiffness under short-term load of the strengthened beam was constant before cracking, so the stage before cracking has not been investigated, and the deflection under short-term load of five strengthened beams, apart from SL1, was analyzed.The value ofBs is substituted into equation (4) to calculate the deflection of the strengthened beam after cracking, and the calculation results are shown in Table 5. Although most of the deflection of UPC beams in the initial damage was restored by the prestress, there were still some residual deflection that was not restored, which has been taken into account.Table 5
Comparison of calculation results and experimental results of deflection.
No.β′Calculation results of deflection (mm)Initial residual deflection (mm)Experimental result of deflection (mm)Error (%)SL20.1979.51590.06.67SL30.2363.5566.83.29SS10.1895.420106.4−8.08SS20.2283.825119.08.4SS30.2372.01589.73.01The calculation errors by the method proposed in this study were all within 10%, so the calculation method is applicable to the stiffness calculation of UPC beams strengthened by AA plates.


## 3.1. Relationship between Load and Deflection

The typical curve of load-deflection of prestressed concrete beams should have obvious turning at the points of cracking of beams and yielding of non-prestressed reinforcements. In this study, the test beams have been damaged and repaired, which has a certain impact on the stiffness of strengthened beams. As a result, the stiffness mutation of strengthened beams was not obvious, as shown in Figure5. The attached figures show the characteristics of a trilinear model with the inflection points of the crack load, yield load, and ultimate load of strengthened beams. It can be observed from Figure 5 that the curves still have similar characteristics to those of the prestressed concrete beams.Figure 5
Load-deflection curves: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
(a)(b)(c)(d)(e)(f)The three dotted lines were divided into three stages, with the slope of each segment being the stiffness of the corresponding stage.Stage I is the stage before the concrete cracks. The beam behaves mainly elastically. The relationship between stress and strain is basically linear. This stage continued until the tensile stresses exceeded the tensile strength of concrete.Stage II is the serviceability stage that occurs after the concrete has been cracked and non-prestressed reinforcements take up almost all the tension force, but the non-prestressed reinforcements have not yet yielded. The neutral axis shifts upward with the increase of the applied loads.Stage III starts with the yielding of non-prestressed reinforcements and ends with the failure of the beam. In this stage, the stiffness of the beam is weakened further. The load-deflection relationship becomes clearly nonlinear. Finally, the excessive developing of cracks and the crushing of concrete in the compression zone lead to the ultimate collapse of the beam.Figure5 shows that the stiffness changes of the SL1 and SS1 beams with the lowest reinforcement ratio are obvious at the crack load, while those of other beams are not obvious. The behavior at the yield load is opposite to that at the crack load; the stiffness of the SL1 and SS1 beams does not change significantly at the yield load, while the stiffness of other beams changes significantly at the yield load. It can be concluded that (1) the influence of cracking on the stiffness is inversely proportional to the reinforcement ratio and (2) the influence of the yielding of non-prestressed reinforcement on the stiffness is proportional to the reinforcement ratio.Figure6 shows the failure modes of the specimens.Figure 6
The failure modes picture of the specimens: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
(a)(b)(c)(d)(e)(f)


## 3.2. Stiffness Calculation

A formula for calculating the deflection of homogeneous elastic materials is proposed in [17], f=aMl2/EcI0, but the concrete is a heterogeneous and inelastic material, so the bending stiffness of the interface changes during bending. The bending stiffness not only decreases with the increase of load but also decreases with the increase of load duration, so the reduction coefficient, namely, the elastic-plastic stiffness coefficient, should be considered. According to the empirical values used in the Chinese design standards, the elastic-plastic stiffness coefficient of concrete beams before cracking is 0.85, and the elastic-plastic stiffness coefficient after cracking β′ needs further research.According to the bilinear stiffness model, the bending moment-deflection curve of prestressed concrete beams can be regarded as consisting of two straight sections OA and AB. With the cracking point as the turning point, the deflection under the cracking momentMcr is fcr, and the deflection under the serviceability moment is fs, as shown in Figure 7.Figure 7
Bending moment-deflection curve of the prestressed beam.According to Figure5, f=fcr+fs=al2/EcI0Mcr/0.85+Ms−Mcr/β′.The elastic-plastic stiffness coefficient under the increment of bending moment (Ms − Mcr) is then obtained as(1)β′=al2Ms−Mcrfs−fcrEcI0,where a is the coefficient of load and support conditions, which according to the material mechanics, in this case, is 0.106.In order to study the elastic-plastic stiffness coefficientβ′ after cracking, it is necessary to analyze the bending moment increment, and Ms is calculated as 1.67Mcr. It was found by calculation that when the Ms of the SL1 beam was 150 kN, the non-prestressed reinforcements have yielded, so no research was conducted on the SL1 beam. According to the experimental data of the other five strengthened beams, the elastic-plastic stiffness coefficient was as calculated in Table 4.Table 4
Calculation results of elastic-plastic stiffness coefficient.
No.Mcr (MPa)fcr (mm)Ms (MPa)fs (mm)β′SL26516108310.19SL36011100190.23SS1452875620.18SS2422070400.22SS3391965340.23By analysis of the experimental data, it was found that there was an approximate linear relationship between 1/β′ and 1/β0, and the fitting equation is as equation (2) with a correlation coefficient of 0.9588. Figure 8 shows the relationship between experimental data and the fit line:(2)1β′=2.3+0.52β0.Figure 8
Relationship between 1/β′ and 1/β0.Thus,(3)β′=β02.3β0+0.52.The stiffness of a UPC beam changes during bending; therefore, in order to accurately calculate the deformation of the beam under short-term load, an appropriate average stiffnessBs is proposed as the short-term stiffness. Thus, on the basis of f=aMl2/Bs, the short-term stiffness of UPC beams strengthened by AA plates can be expressed as(4)Bs=0.85β′MsEcI0β′Mcr+0.85Ms−Mcr.


## 3.3. Prediction and Analysis of Deflection

A UPC beam can be regarded as equivalent to an RC beam with a pair of prepressure in the anchorage position, which still show the characteristics of an RC beam after cracking. Therefore, the stiffness of UPC beams strengthened with AA plates can be measured using the same research method as the RC beams.The stiffness under short-term load of the strengthened beam was constant before cracking, so the stage before cracking has not been investigated, and the deflection under short-term load of five strengthened beams, apart from SL1, was analyzed.The value ofBs is substituted into equation (4) to calculate the deflection of the strengthened beam after cracking, and the calculation results are shown in Table 5. Although most of the deflection of UPC beams in the initial damage was restored by the prestress, there were still some residual deflection that was not restored, which has been taken into account.Table 5
Comparison of calculation results and experimental results of deflection.
No.β′Calculation results of deflection (mm)Initial residual deflection (mm)Experimental result of deflection (mm)Error (%)SL20.1979.51590.06.67SL30.2363.5566.83.29SS10.1895.420106.4−8.08SS20.2283.825119.08.4SS30.2372.01589.73.01The calculation errors by the method proposed in this study were all within 10%, so the calculation method is applicable to the stiffness calculation of UPC beams strengthened by AA plates.


## 4. Crack Analysis

### 4.1. Mean Crack Spacing

The crack development mechanism of UPC beams is similar to that of RC beams. When the concrete in the tensile zone cracks, the concrete is no longer in tension, but the non-prestressed reinforcement continues to undergo tension. Due to the difference in the elastic modulus of concrete and steel, the tension causes a relative slip between the non-prestressed reinforcements and the concrete bonded with them, which causes the development of cracking. During this process, there is no bond force between the unbonded prestressed tendons and concrete, and the tensile stress of concrete in the tensile zone is offset by the prestress of unbonded prestressed tendons in the initial period of loading. Prestress can delay the occurrence of cracking in the concrete, but it cannot control the distribution of cracks, which are mainly controlled by the relationship of bond-slip between the non-prestressed reinforcement and the concrete.The experimental results show that the cracks in the tensile concrete of the strengthened beams are all the same as in the initial damage, and no new cracks appeared. The experimental data also show that there is a regular relationship between the mean crack spacinglcr and deq/ρs and ta/ρa, with the regression equation as follows:(5)lcr=0.042deqρs−0.033taρa+74.76.The correlation coefficient of the above equation is 0.9538, wheredeq is the equivalent diameter of the reinforcement, ta is the thickness of the AA plate, ρs and ρa are the reinforcement ratios of the non-prestressed reinforcement and AA plate, respectively.Table6 shows the results calculated by equation (5) and the experimental results. The calculated errors are all less than 5%, indicating that it is suitable for calculating the mean crack spacing of UPC beams strengthened by AA plate.Table 6
Calculation results and experimental data of mean crack spacing.
No.ρsρaExperimental data oflcr (mm)Calculation results oflcr (mm)Error (%)SL10.0060.0151541571.9SL20.0130.015133128−3.9SL30.0250.0151111110SS10.0080.02142137−3.7SS20.0170.021141140SS30.0250.021051060.9


### 4.2. Mean Crack Width

#### 4.2.1. Stress Increment of Unbonded Prestressed Tendons in Serviceability Stage

In the initial loading, the stress in the unbonded prestressed tendons grew slowly. After the concrete in the tension zone cracked, the stress in the unbonded prestressed tendons grew quickly. The cracks then gradually developed, the stiffness gradually declined, and the deformation increased more quickly. Due to the different reinforcement ratio, initial prestress, and other parameters of each strengthened beam, the stress increment of the unbonded prestressed tendons could not be compared directly. In order to have a unified comparison standard for the strengthened beams, the relative values of the stress increment and prestress of each strengthened beam were analyzed. Based on the experimental results, it was found that there was a linear relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr) in the serviceability stage, described as follows:(6)ΔσpΔσpy=0.583Mk−McrMy−Mcr+0.376.The correlation coefficient of the fit line was 0.94, where Δσp is the stress increment of unbonded tendons in the serviceability stage, Δσpy is the stress increment of the unbonded tendons when the non-prestressed reinforcements were yielding, Mk is the bending moment in the serviceability stage, and My is the bending moment when the non-prestressed reinforcements were yielding. The relationship between the fit line and the experimental data of each test beam is shown in Figure 9.Figure 9
The relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr).


#### 4.2.2. Stress Increment of Unbonded Tendons When the Non-Prestressed Reinforcement Is Yielding

In order to calculate Δσp using equation (6), the equation governing Δσpy needs to be established.Analysis of the experimental data shows that Δσpy has a regular relationship with β0 and βs, with the fitting equation as follows:(7)Δσpy=11.04β0−0.56βs.The fitting curve has a correlation coefficient of 0.9309. The relationship between experimental data and fit line is shown in Figure10.Figure 10
The relationship between Δσpyβs and β0.The results of Δσpy calculated by equation (7) and the experimental data are shown in Table 7. It can be seen that the error was within 10% except for the SS2 beam with large dispersion. Therefore, equation (7) is applicable to the calculation of the stress increment of unbonded tendons when the non-prestressed reinforcement is yielding.Table 7
The calculation error of Δσpy.
No.Experimental data of Δσpy (MPa)Calculated value of Δσpy (MPa)Error (%)SL118.0719.035.04SL231.8532.271.3SL314.0314.493.1SS139.5920.90−89.4SS235.5432.26−9.2SS320.4418.65−9.6Equation (6) can be translated into(8)Δσp=0.583Mk−McrMy−Mcr+0.376Δσpy.When equation (8) is substituted into equation (7), the formula for Δσpy can be obtained as follows:(9)Δσp=0.583Mk−McrMy−Mcr+0.37611.04β0−0.56βs.


#### 4.2.3. Bending Moment in Serviceability Stage

The serviceability stage of a UPC beam usually refers to the period between beam cracking and yielding of non-prestressed reinforcement. The cracking moment can be expressed as(10)Mcr=W0σpc+ftk,where W0 is the bending modulus of the transformed section, σpy is the prestress of the tensile edge of the concrete while deducting all losses, and ftk is the characteristic value of axial compressive strength.The distribution of stress and strain of the cross section when the non-prestressed reinforcement was yielding is shown in Figure11.Figure 11
The distribution of stress and strain of the section when the non-prestressed reinforcement was yielding.
(a)(b)(c)From Figure11, the tensile force is equal to the resultant compression force:(11)0.5Asfy+Apσp+σaAa=Ecεcyby.In equation (11),(12)σp=σpe+Δσpy.The experimental data show that the yielding in the non-prestressed reinforcement was earlier than that in the AA plate. Therefore, when the non-prestressed reinforcement was yielding, the distribution of stress and strain in the cross section conforms to the plane-section assumption. So,(13)εcy=yh0−y,εsy=fyyEsh0−y,(14)εay=h−yh0−y,εsy=fyh−yEsh0−y.Equation (14) can be translated as(15)σay=fyEah−yEsh0−y.According to equations (11), (12), (13), and (15), the depth of the compression zone when the non-prestressed reinforcement was yielding can be obtained, and the yielding moment can be expressed as(16)My=Asfy+Apσpe+Δσpyh0−0.5y+Aaσah−0.5y.


#### 4.2.4. Mean Crack Width

The distribution of stress and strain of the cracking section in the serviceability stage is shown in Figure12.Figure 12
The distribution of stress and strain of the section in serviceability stage.
(a)(b)For equilibrium, the following equation can be established:(17)Mk−Npez−ep=σsAsηsh0+αΔσpApηphp+σaAaηah,where Npe is the prepressure compression, ηsh0, ηphp, and ηah are, respectively, the internal force arms of the non-prestressed reinforcement, unbonded tendons, and AA plate which undergoes the bending moment.In the process of crack development of UPC beams, the contribution of unbonded prestressed tendons to the crack resistance of the UPC beam is less than that of the same quantity of bonded prestressed tendons. Therefore, a reduction factorα for the unbonded tendons has been introduced in equation (17), which is the ratio of the stress increment of unbonded prestressed tendons to that of non-prestressed reinforcement at the same location with service load. According to the existing research [18], the recommended value of α is 0.23.According to the plane-section assumption, the stress in non-prestressed reinforcement in the serviceability stage is as follows:(18)σsk=Mk−Npez−ep−0.23ΔσpApηphp+σaAaηahAsηsh0.This can be translated to(19)σsk=Mk−Npez−epAsηsh01+δ,δ=h−yEaAaηahh0−yEsAsηsh0+0.23hp−yEpApηphph0−yEsAsηsh0.In the above,z is the distance from the point of the resultant force of non-prestressed reinforcement and unbonded tendons to the centre of compression of the concrete, which can be calculated by the following equations:(20)z=0.87−0.12h0e2h0,(21)e=ep+MkNpe,where e is the distance between the action point of axial pressure and that of the resultant force of the non-prestressed reinforcement and ep is the distance between the action point of prepressure and that of the resultant force of the unbonded tendons, non-prestressed reinforcement, and AA plate.According to the development mechanism of cracks in reinforced concrete beams, the mean crack width of the beams is approximately the difference between the elongationεs¯lcr of non-prestressed reinforcement and the elongation εc¯lcr of concrete at the corresponding position within the crack spacing. The formula for the mean crack width wm of a UPC beam is as follows:(22)wm=0.85ψ⋅σsEs⋅lcr,where ψ is the coefficient of uniformity of non-prestressed reinforcement, which can be calculated according to the following formula:(23)ψ=1.1−0.65ftkρteσs,where ρte is the reinforcement ratio of non-prestressed reinforcement calculated according to the effective tensile area of concrete and considering the effect of the AA plate, and ρte = (As + EaAa/Es)/Ate, where Ate is the effective tensile area of concrete, and Ate = 0.5bh, ftk is the characteristic value of concrete axial compressive strength.


### 4.3. Maximum Crack Width

Previous studies showed that the crack width of the RC beams has a large dispersion, and the reasonable maximum crack width should be determined by statistical analysis. According to this study, the distribution of thei-th crack width wi to the mean crack width wm under short-term loading, it was found that the value of wi/wm was approximately a normal distribution, as shown in Figure 13.Figure 13
Frequency distribution graph.The fitting equation of the normal distribution curve is as follows:(24)f=0.014+10.2582πexp−wi/wm−1.05/2×0.2582,with a correlation coefficient of 0.9661.The maximum crack width is determined by the guarantee rate of 95%, and the characteristic value corresponding towi/wm is the crack expansion coefficient under short-term loading τs:(25)τs=1.05+1.645×0.258=1.474.Thus, the equation for calculating the maximum crack width under the short-term load is as follows:(26)wmax,s=τswm=1.474×0.85×ψσsEs0.042deqρs−0.033taρa+74.76.Even if the load remains unchanged under a long-term load, due to shrinkage, creep, and slippage of the concrete in the tension zone, the concrete in tension between the cracks will continuously break away, and the strain of the non-prestressed reinforcement near the cracks will gradually increase. The crack width of the beam will thus increase over time. Therefore, the effect of long-term loading should be considered in the calculation of cracks, and the expansion coefficienttl should be introduced. According to previous studies, tl = 1.5 [19]. Thus, the calculation equation for the maximum crack width of UPC beams strengthened with AA plates under long-term loading is as follows:(27)wmax=τlτswm=1.5×1.474×0.85×ψσsEs0.042deqρs−0.033taρa+74.76.


## 4.1. Mean Crack Spacing

The crack development mechanism of UPC beams is similar to that of RC beams. When the concrete in the tensile zone cracks, the concrete is no longer in tension, but the non-prestressed reinforcement continues to undergo tension. Due to the difference in the elastic modulus of concrete and steel, the tension causes a relative slip between the non-prestressed reinforcements and the concrete bonded with them, which causes the development of cracking. During this process, there is no bond force between the unbonded prestressed tendons and concrete, and the tensile stress of concrete in the tensile zone is offset by the prestress of unbonded prestressed tendons in the initial period of loading. Prestress can delay the occurrence of cracking in the concrete, but it cannot control the distribution of cracks, which are mainly controlled by the relationship of bond-slip between the non-prestressed reinforcement and the concrete.The experimental results show that the cracks in the tensile concrete of the strengthened beams are all the same as in the initial damage, and no new cracks appeared. The experimental data also show that there is a regular relationship between the mean crack spacinglcr and deq/ρs and ta/ρa, with the regression equation as follows:(5)lcr=0.042deqρs−0.033taρa+74.76.The correlation coefficient of the above equation is 0.9538, wheredeq is the equivalent diameter of the reinforcement, ta is the thickness of the AA plate, ρs and ρa are the reinforcement ratios of the non-prestressed reinforcement and AA plate, respectively.Table6 shows the results calculated by equation (5) and the experimental results. The calculated errors are all less than 5%, indicating that it is suitable for calculating the mean crack spacing of UPC beams strengthened by AA plate.Table 6
Calculation results and experimental data of mean crack spacing.
No.ρsρaExperimental data oflcr (mm)Calculation results oflcr (mm)Error (%)SL10.0060.0151541571.9SL20.0130.015133128−3.9SL30.0250.0151111110SS10.0080.02142137−3.7SS20.0170.021141140SS30.0250.021051060.9


## 4.2. Mean Crack Width

### 4.2.1. Stress Increment of Unbonded Prestressed Tendons in Serviceability Stage

In the initial loading, the stress in the unbonded prestressed tendons grew slowly. After the concrete in the tension zone cracked, the stress in the unbonded prestressed tendons grew quickly. The cracks then gradually developed, the stiffness gradually declined, and the deformation increased more quickly. Due to the different reinforcement ratio, initial prestress, and other parameters of each strengthened beam, the stress increment of the unbonded prestressed tendons could not be compared directly. In order to have a unified comparison standard for the strengthened beams, the relative values of the stress increment and prestress of each strengthened beam were analyzed. Based on the experimental results, it was found that there was a linear relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr) in the serviceability stage, described as follows:(6)ΔσpΔσpy=0.583Mk−McrMy−Mcr+0.376.The correlation coefficient of the fit line was 0.94, where Δσp is the stress increment of unbonded tendons in the serviceability stage, Δσpy is the stress increment of the unbonded tendons when the non-prestressed reinforcements were yielding, Mk is the bending moment in the serviceability stage, and My is the bending moment when the non-prestressed reinforcements were yielding. The relationship between the fit line and the experimental data of each test beam is shown in Figure 9.Figure 9
The relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr).


### 4.2.2. Stress Increment of Unbonded Tendons When the Non-Prestressed Reinforcement Is Yielding

In order to calculate Δσp using equation (6), the equation governing Δσpy needs to be established.Analysis of the experimental data shows that Δσpy has a regular relationship with β0 and βs, with the fitting equation as follows:(7)Δσpy=11.04β0−0.56βs.The fitting curve has a correlation coefficient of 0.9309. The relationship between experimental data and fit line is shown in Figure10.Figure 10
The relationship between Δσpyβs and β0.The results of Δσpy calculated by equation (7) and the experimental data are shown in Table 7. It can be seen that the error was within 10% except for the SS2 beam with large dispersion. Therefore, equation (7) is applicable to the calculation of the stress increment of unbonded tendons when the non-prestressed reinforcement is yielding.Table 7
The calculation error of Δσpy.
No.Experimental data of Δσpy (MPa)Calculated value of Δσpy (MPa)Error (%)SL118.0719.035.04SL231.8532.271.3SL314.0314.493.1SS139.5920.90−89.4SS235.5432.26−9.2SS320.4418.65−9.6Equation (6) can be translated into(8)Δσp=0.583Mk−McrMy−Mcr+0.376Δσpy.When equation (8) is substituted into equation (7), the formula for Δσpy can be obtained as follows:(9)Δσp=0.583Mk−McrMy−Mcr+0.37611.04β0−0.56βs.


### 4.2.3. Bending Moment in Serviceability Stage

The serviceability stage of a UPC beam usually refers to the period between beam cracking and yielding of non-prestressed reinforcement. The cracking moment can be expressed as(10)Mcr=W0σpc+ftk,where W0 is the bending modulus of the transformed section, σpy is the prestress of the tensile edge of the concrete while deducting all losses, and ftk is the characteristic value of axial compressive strength.The distribution of stress and strain of the cross section when the non-prestressed reinforcement was yielding is shown in Figure11.Figure 11
The distribution of stress and strain of the section when the non-prestressed reinforcement was yielding.
(a)(b)(c)From Figure11, the tensile force is equal to the resultant compression force:(11)0.5Asfy+Apσp+σaAa=Ecεcyby.In equation (11),(12)σp=σpe+Δσpy.The experimental data show that the yielding in the non-prestressed reinforcement was earlier than that in the AA plate. Therefore, when the non-prestressed reinforcement was yielding, the distribution of stress and strain in the cross section conforms to the plane-section assumption. So,(13)εcy=yh0−y,εsy=fyyEsh0−y,(14)εay=h−yh0−y,εsy=fyh−yEsh0−y.Equation (14) can be translated as(15)σay=fyEah−yEsh0−y.According to equations (11), (12), (13), and (15), the depth of the compression zone when the non-prestressed reinforcement was yielding can be obtained, and the yielding moment can be expressed as(16)My=Asfy+Apσpe+Δσpyh0−0.5y+Aaσah−0.5y.


### 4.2.4. Mean Crack Width

The distribution of stress and strain of the cracking section in the serviceability stage is shown in Figure12.Figure 12
The distribution of stress and strain of the section in serviceability stage.
(a)(b)For equilibrium, the following equation can be established:(17)Mk−Npez−ep=σsAsηsh0+αΔσpApηphp+σaAaηah,where Npe is the prepressure compression, ηsh0, ηphp, and ηah are, respectively, the internal force arms of the non-prestressed reinforcement, unbonded tendons, and AA plate which undergoes the bending moment.In the process of crack development of UPC beams, the contribution of unbonded prestressed tendons to the crack resistance of the UPC beam is less than that of the same quantity of bonded prestressed tendons. Therefore, a reduction factorα for the unbonded tendons has been introduced in equation (17), which is the ratio of the stress increment of unbonded prestressed tendons to that of non-prestressed reinforcement at the same location with service load. According to the existing research [18], the recommended value of α is 0.23.According to the plane-section assumption, the stress in non-prestressed reinforcement in the serviceability stage is as follows:(18)σsk=Mk−Npez−ep−0.23ΔσpApηphp+σaAaηahAsηsh0.This can be translated to(19)σsk=Mk−Npez−epAsηsh01+δ,δ=h−yEaAaηahh0−yEsAsηsh0+0.23hp−yEpApηphph0−yEsAsηsh0.In the above,z is the distance from the point of the resultant force of non-prestressed reinforcement and unbonded tendons to the centre of compression of the concrete, which can be calculated by the following equations:(20)z=0.87−0.12h0e2h0,(21)e=ep+MkNpe,where e is the distance between the action point of axial pressure and that of the resultant force of the non-prestressed reinforcement and ep is the distance between the action point of prepressure and that of the resultant force of the unbonded tendons, non-prestressed reinforcement, and AA plate.According to the development mechanism of cracks in reinforced concrete beams, the mean crack width of the beams is approximately the difference between the elongationεs¯lcr of non-prestressed reinforcement and the elongation εc¯lcr of concrete at the corresponding position within the crack spacing. The formula for the mean crack width wm of a UPC beam is as follows:(22)wm=0.85ψ⋅σsEs⋅lcr,where ψ is the coefficient of uniformity of non-prestressed reinforcement, which can be calculated according to the following formula:(23)ψ=1.1−0.65ftkρteσs,where ρte is the reinforcement ratio of non-prestressed reinforcement calculated according to the effective tensile area of concrete and considering the effect of the AA plate, and ρte = (As + EaAa/Es)/Ate, where Ate is the effective tensile area of concrete, and Ate = 0.5bh, ftk is the characteristic value of concrete axial compressive strength.


## 4.2.1. Stress Increment of Unbonded Prestressed Tendons in Serviceability Stage

In the initial loading, the stress in the unbonded prestressed tendons grew slowly. After the concrete in the tension zone cracked, the stress in the unbonded prestressed tendons grew quickly. The cracks then gradually developed, the stiffness gradually declined, and the deformation increased more quickly. Due to the different reinforcement ratio, initial prestress, and other parameters of each strengthened beam, the stress increment of the unbonded prestressed tendons could not be compared directly. In order to have a unified comparison standard for the strengthened beams, the relative values of the stress increment and prestress of each strengthened beam were analyzed. Based on the experimental results, it was found that there was a linear relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr) in the serviceability stage, described as follows:(6)ΔσpΔσpy=0.583Mk−McrMy−Mcr+0.376.The correlation coefficient of the fit line was 0.94, where Δσp is the stress increment of unbonded tendons in the serviceability stage, Δσpy is the stress increment of the unbonded tendons when the non-prestressed reinforcements were yielding, Mk is the bending moment in the serviceability stage, and My is the bending moment when the non-prestressed reinforcements were yielding. The relationship between the fit line and the experimental data of each test beam is shown in Figure 9.Figure 9
The relationship between Δσp/Δσpy and (Mk − Mcr)/(My − Mcr).


## 4.2.2. Stress Increment of Unbonded Tendons When the Non-Prestressed Reinforcement Is Yielding

In order to calculate Δσp using equation (6), the equation governing Δσpy needs to be established.Analysis of the experimental data shows that Δσpy has a regular relationship with β0 and βs, with the fitting equation as follows:(7)Δσpy=11.04β0−0.56βs.The fitting curve has a correlation coefficient of 0.9309. The relationship between experimental data and fit line is shown in Figure10.Figure 10
The relationship between Δσpyβs and β0.The results of Δσpy calculated by equation (7) and the experimental data are shown in Table 7. It can be seen that the error was within 10% except for the SS2 beam with large dispersion. Therefore, equation (7) is applicable to the calculation of the stress increment of unbonded tendons when the non-prestressed reinforcement is yielding.Table 7
The calculation error of Δσpy.
No.Experimental data of Δσpy (MPa)Calculated value of Δσpy (MPa)Error (%)SL118.0719.035.04SL231.8532.271.3SL314.0314.493.1SS139.5920.90−89.4SS235.5432.26−9.2SS320.4418.65−9.6Equation (6) can be translated into(8)Δσp=0.583Mk−McrMy−Mcr+0.376Δσpy.When equation (8) is substituted into equation (7), the formula for Δσpy can be obtained as follows:(9)Δσp=0.583Mk−McrMy−Mcr+0.37611.04β0−0.56βs.


## 4.2.3. Bending Moment in Serviceability Stage

The serviceability stage of a UPC beam usually refers to the period between beam cracking and yielding of non-prestressed reinforcement. The cracking moment can be expressed as(10)Mcr=W0σpc+ftk,where W0 is the bending modulus of the transformed section, σpy is the prestress of the tensile edge of the concrete while deducting all losses, and ftk is the characteristic value of axial compressive strength.The distribution of stress and strain of the cross section when the non-prestressed reinforcement was yielding is shown in Figure11.Figure 11
The distribution of stress and strain of the section when the non-prestressed reinforcement was yielding.
(a)(b)(c)From Figure11, the tensile force is equal to the resultant compression force:(11)0.5Asfy+Apσp+σaAa=Ecεcyby.In equation (11),(12)σp=σpe+Δσpy.The experimental data show that the yielding in the non-prestressed reinforcement was earlier than that in the AA plate. Therefore, when the non-prestressed reinforcement was yielding, the distribution of stress and strain in the cross section conforms to the plane-section assumption. So,(13)εcy=yh0−y,εsy=fyyEsh0−y,(14)εay=h−yh0−y,εsy=fyh−yEsh0−y.Equation (14) can be translated as(15)σay=fyEah−yEsh0−y.According to equations (11), (12), (13), and (15), the depth of the compression zone when the non-prestressed reinforcement was yielding can be obtained, and the yielding moment can be expressed as(16)My=Asfy+Apσpe+Δσpyh0−0.5y+Aaσah−0.5y.


## 4.2.4. Mean Crack Width

The distribution of stress and strain of the cracking section in the serviceability stage is shown in Figure12.Figure 12
The distribution of stress and strain of the section in serviceability stage.
(a)(b)For equilibrium, the following equation can be established:(17)Mk−Npez−ep=σsAsηsh0+αΔσpApηphp+σaAaηah,where Npe is the prepressure compression, ηsh0, ηphp, and ηah are, respectively, the internal force arms of the non-prestressed reinforcement, unbonded tendons, and AA plate which undergoes the bending moment.In the process of crack development of UPC beams, the contribution of unbonded prestressed tendons to the crack resistance of the UPC beam is less than that of the same quantity of bonded prestressed tendons. Therefore, a reduction factorα for the unbonded tendons has been introduced in equation (17), which is the ratio of the stress increment of unbonded prestressed tendons to that of non-prestressed reinforcement at the same location with service load. According to the existing research [18], the recommended value of α is 0.23.According to the plane-section assumption, the stress in non-prestressed reinforcement in the serviceability stage is as follows:(18)σsk=Mk−Npez−ep−0.23ΔσpApηphp+σaAaηahAsηsh0.This can be translated to(19)σsk=Mk−Npez−epAsηsh01+δ,δ=h−yEaAaηahh0−yEsAsηsh0+0.23hp−yEpApηphph0−yEsAsηsh0.In the above,z is the distance from the point of the resultant force of non-prestressed reinforcement and unbonded tendons to the centre of compression of the concrete, which can be calculated by the following equations:(20)z=0.87−0.12h0e2h0,(21)e=ep+MkNpe,where e is the distance between the action point of axial pressure and that of the resultant force of the non-prestressed reinforcement and ep is the distance between the action point of prepressure and that of the resultant force of the unbonded tendons, non-prestressed reinforcement, and AA plate.According to the development mechanism of cracks in reinforced concrete beams, the mean crack width of the beams is approximately the difference between the elongationεs¯lcr of non-prestressed reinforcement and the elongation εc¯lcr of concrete at the corresponding position within the crack spacing. The formula for the mean crack width wm of a UPC beam is as follows:(22)wm=0.85ψ⋅σsEs⋅lcr,where ψ is the coefficient of uniformity of non-prestressed reinforcement, which can be calculated according to the following formula:(23)ψ=1.1−0.65ftkρteσs,where ρte is the reinforcement ratio of non-prestressed reinforcement calculated according to the effective tensile area of concrete and considering the effect of the AA plate, and ρte = (As + EaAa/Es)/Ate, where Ate is the effective tensile area of concrete, and Ate = 0.5bh, ftk is the characteristic value of concrete axial compressive strength.


## 4.3. Maximum Crack Width

Previous studies showed that the crack width of the RC beams has a large dispersion, and the reasonable maximum crack width should be determined by statistical analysis. According to this study, the distribution of thei-th crack width wi to the mean crack width wm under short-term loading, it was found that the value of wi/wm was approximately a normal distribution, as shown in Figure 13.Figure 13
Frequency distribution graph.The fitting equation of the normal distribution curve is as follows:(24)f=0.014+10.2582πexp−wi/wm−1.05/2×0.2582,with a correlation coefficient of 0.9661.The maximum crack width is determined by the guarantee rate of 95%, and the characteristic value corresponding towi/wm is the crack expansion coefficient under short-term loading τs:(25)τs=1.05+1.645×0.258=1.474.Thus, the equation for calculating the maximum crack width under the short-term load is as follows:(26)wmax,s=τswm=1.474×0.85×ψσsEs0.042deqρs−0.033taρa+74.76.Even if the load remains unchanged under a long-term load, due to shrinkage, creep, and slippage of the concrete in the tension zone, the concrete in tension between the cracks will continuously break away, and the strain of the non-prestressed reinforcement near the cracks will gradually increase. The crack width of the beam will thus increase over time. Therefore, the effect of long-term loading should be considered in the calculation of cracks, and the expansion coefficienttl should be introduced. According to previous studies, tl = 1.5 [19]. Thus, the calculation equation for the maximum crack width of UPC beams strengthened with AA plates under long-term loading is as follows:(27)wmax=τlτswm=1.5×1.474×0.85×ψσsEs0.042deqρs−0.033taρa+74.76.


## 5. Conclusion

(1)
The technique of strengthening UPC beams with AA plates can improve stiffness and limit the development of cracks.(2)
The relationship of the load-deformation curve of damaged beams strengthened with AA plates is different from that of typical reinforced concrete beams, i.e., the characteristic trilinear model with cracking and yielding as inflection points of the strengthened beams was less obvious than that of undamaged reinforced concrete beams.(3)
Based on the double broken line model, an equation for the elastic-plastic stiffness coefficient in the serviceability stage with integrated reinforcement indexβ0 as parameter was proposed, and a calculation method for the stiffness of a UPC beam strengthened with AA plates was proposed.(4)
The relative relationship between average crack spacinglcr, deq/ρs, and ta/ρa was analyzed, and a calculation method of mean crack spacing for UPC beams strengthened with AA plates was proposed by regression.(5)
Based on the analysis of the stress increment of unbonded prestressed tendons in the serviceability stage, a calculation method was proposed. An influence coefficient considering the effect of unbonded prestressed tendons and AA plates was introduced, and a calculation method for the crack width in UPC beams strengthened with AA plates was proposed.The behavior of concrete structures strengthened with aluminum alloy plate in corrosive environments should be studied in future research.



---

*Source: 3824543-2020-09-30.xml*