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 coefficient
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 [
Previous studies [
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.
A total of six damaged UPC beams were repaired and strengthened. Figure
The reinforcement of specimens: (a) SS beams; (b) SL beams.
Properties of specimens.
No. | Cross-sectional area (mm × mm) | Steel reinforcements | Steel strand |
---|---|---|---|
SL1 | 200 × 400 | 2C12 | 1AS17.8 |
SL2 | 200 × 400 | 2C18 | 1AS17.8 |
SL3 | 200 × 400 | 2C25 | 1AS17.8 |
SS1 | 200 × 300 | 2C12 | 1AS17.8 |
SS2 | 200 × 300 | 2C18 | 1AS17.8 |
SS3 | 200 × 300 | 2C22 | 1AS17.8 |
The reinforcement index of the non-prestressed reinforcements
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 Table
Initial damage of each damaged beam.
No. | Cracking load (kN) | Ultimate load (kN) | Maximum crack width (mm) | Number of cracks | Initial prestress (MPa) | |
---|---|---|---|---|---|---|
Front | Back | |||||
SL1 | 80 | 204 | 5 | 13 | 12 | 1570 |
SL2 | 60 | 180 | 1 | 16 | 16 | 1570 |
SL3 | 44 | 200 | 0.3 | 18 | 20 | 1570 |
SS1 | 40 | 84 | 2 | 17 | 16 | 1361 |
SS2 | 30 | 104 | 0.2 | 19 | 18 | 1570 |
SS3 | 34 | 118 | 0.4 | 20 | 20 | 1570 |
The 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 Figure
The detailed description of bolts and AA plate.
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 (
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 Table
Properties of reinforcing bars.
Bar size | Grade (MPa) | Yielding stress (MPa) | Ultimate stress (MPa) | Elasticity modulus |
---|---|---|---|---|
C8 | 400 | 402 | 575 | 197000 |
C12 | 400 | 402 | 575 | 197000 |
C18 | 400 | 452 | 648 | 197000 |
C22 | 400 | 405 | 629 | 197000 |
C25 | 400 | 489 | 672 | 197000 |
A unidirectional tensile test of 5083 AA plates was carried out on a universal electronic test machine (see Figure
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.
Two-point loads were applied at the one-third span points of the beam (see Figure
Loading device.
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.
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 Figure
Load-deflection curves: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
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.
Figure
Figure
The failure modes picture of the specimens: (a) SL1; (b) SL2; (c) SL3; (d) SS1; (e) SS2; (f) SS3.
A formula for calculating the deflection of homogeneous elastic materials is proposed in [
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 moment
Bending moment-deflection curve of the prestressed beam.
According to Figure
The elastic-plastic stiffness coefficient under the increment of bending moment (
In order to study the elastic-plastic stiffness coefficient
Calculation results of elastic-plastic stiffness coefficient.
No. | |||||
---|---|---|---|---|---|
SL2 | 65 | 16 | 108 | 31 | 0.19 |
SL3 | 60 | 11 | 100 | 19 | 0.23 |
SS1 | 45 | 28 | 75 | 62 | 0.18 |
SS2 | 42 | 20 | 70 | 40 | 0.22 |
SS3 | 39 | 19 | 65 | 34 | 0.23 |
By analysis of the experimental data, it was found that there was an approximate linear relationship between 1/
Relationship between 1/
Thus,
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 stiffness
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 of
Comparison of calculation results and experimental results of deflection.
No. | Calculation results of deflection (mm) | Initial residual deflection (mm) | Experimental result of deflection (mm) | Error (%) | |
---|---|---|---|---|---|
SL2 | 0.19 | 79.5 | 15 | 90.0 | 6.67 |
SL3 | 0.23 | 63.5 | 5 | 66.8 | 3.29 |
SS1 | 0.18 | 95.4 | 20 | 106.4 | −8.08 |
SS2 | 0.22 | 83.8 | 25 | 119.0 | 8.4 |
SS3 | 0.23 | 72.0 | 15 | 89.7 | 3.01 |
The 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.
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 spacing
The correlation coefficient of the above equation is 0.9538, where
Table
Calculation results and experimental data of mean crack spacing.
No. | Experimental data of | Calculation results of | Error (%) | ||
---|---|---|---|---|---|
SL1 | 0.006 | 0.015 | 154 | 157 | 1.9 |
SL2 | 0.013 | 0.015 | 133 | 128 | −3.9 |
SL3 | 0.025 | 0.015 | 111 | 111 | 0 |
SS1 | 0.008 | 0.02 | 142 | 137 | −3.7 |
SS2 | 0.017 | 0.02 | 114 | 114 | 0 |
SS3 | 0.025 | 0.02 | 105 | 106 | 0.9 |
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 Δ
The correlation coefficient of the fit line was 0.94, where Δ
The relationship between Δ
In order to calculate Δ
Analysis of the experimental data shows that Δ
The fitting curve has a correlation coefficient of 0.9309. The relationship between experimental data and fit line is shown in Figure
The relationship between Δ
The results of Δ
The calculation error of Δ
No. | Experimental data of Δ | Calculated value of Δ | Error (%) |
---|---|---|---|
SL1 | 18.07 | 19.03 | 5.04 |
SL2 | 31.85 | 32.27 | 1.3 |
SL3 | 14.03 | 14.49 | 3.1 |
SS1 | 39.59 | 20.90 | −89.4 |
SS2 | 35.54 | 32.26 | −9.2 |
SS3 | 20.44 | 18.65 | −9.6 |
Equation (
When equation (
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
The distribution of stress and strain of the cross section when the non-prestressed reinforcement was yielding is shown in Figure
The distribution of stress and strain of the section when the non-prestressed reinforcement was yielding.
From Figure
In equation (
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,
Equation (
According to equations (
The distribution of stress and strain of the cracking section in the serviceability stage is shown in Figure
The distribution of stress and strain of the section in serviceability stage.
For equilibrium, the following equation can be established:
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 factor
According to the plane-section assumption, the stress in non-prestressed reinforcement in the serviceability stage is as follows:
This can be translated to
In the above,
According to the development mechanism of cracks in reinforced concrete beams, the mean crack width of the beams is approximately the difference between the elongation
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 the
Frequency distribution graph.
The fitting equation of the normal distribution curve is as follows:
The maximum crack width is determined by the guarantee rate of 95%, and the characteristic value corresponding to
Thus, the equation for calculating the maximum crack width under the short-term load is as follows:
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 coefficient
The technique of strengthening UPC beams with AA plates can improve stiffness and limit the development of cracks. 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. Based on the double broken line model, an equation for the elastic-plastic stiffness coefficient in the serviceability stage with integrated reinforcement index The relative relationship between average crack spacing 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.
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
The authors declare no conflicts of interest.
The authors are grateful to the members of their research group in Civil of HIT, and they are appreciated for their effort in the investigation. This research has been supported by the National Natural Science Foundation of China (NSFC) under grant no. 51778186.