The present study focuses on a prediction of crack width and load-carrying capacity of flexural reinforced concrete (RC) elements strengthened with fibre-reinforced polymer (FRP) reinforcements. Most studies on cracking phenomena of FRP-strengthened RC structures are directed to empirical corrections of crack-spacing formula given by design norms. Contrary to the design norms, a crack model presented in this paper is based on fracture mechanics of solids and is applied for direct calculation of flexural crack parameters. At the ultimate stage of crack propagation, the load-carrying capacity of the element is achieved; therefore, it is assumed that the load-carrying capacity can be estimated according to the ultimate crack depth (directly measuring concrete’s compressive zone height). An experimental program is presented to verify the accuracy of the proposed model, taking into account anchorage and initial strain effects. The proposed analytical crack model can be used for more precise predictions of flexural crack propagation and load-carrying capacity.
Retrofitting of existing structures is one of the main challenges for civil engineers today. One of the most advantageous material types for strengthening is fibre-reinforced polymers (FRP) due to their corrosion resistance and high strength to low weight ratio [
In this chapter, the estimation methods of the crack width and the mean crack spacing proposed in design standards [
The mean value of crack spacing can be defined as follows [
Assuming stabilized cracking, the characteristic value of the crack width of FRP-strengthened RC structures is calculated according to fib bulletin 14 [
The mean crack spacing, taking into account the effect of both the internal and the external reinforcement, can be calculated as [
Neglecting the tension-stiffening effect and initial strain, the characteristic crack width is as follows [
Hence, a denser cracking and the smaller crack widths are obtained for RC beams strengthened with FRP; the crack widths estimated by the methodology proposed in [
In accordance with Jokūbaitis and Juknevičius and Jokūbaitis et al
The model for calculation of normal crack propagation: (a) crack model; (b) strain distribution.
The proposed crack model is presented in Figure
General expression of ultimate value of concrete’s tensile strain is used for analysis:
The mean value of concrete’s tensile strength and secant modulus of elasticity are estimated in accordance with EC2 [
The ratio of ultimate strain of concrete in tension and the strain of FRP reinforcement can be calculated as follows:
A wide range of experimental research was conducted by Jokūbaitis et al. [
Below is the same expression with some modifications which could be used for FRP-strengthened structures:
Factor
Stiffness of the interface between separate members:
Factor
The distance to tensile steel and FRP-reinforcement resultant (in mm) (see Figure
Effective tension area: (a) EBR; (b) NSM.
In EC2, the effective reinforcement ratio of tensile zone of concrete is expressed as follows:
As proposed in [
The equivalent factor of the tensile zone, taking into account both steel and FRP reinforcements, can be derived from (
Bond perimeter of FRP reinforcement: (a) FRP rods; (b) FRP strips; (c) EBR.
Subsequently, the crack width is derived from (
The same relation in (
Furthermore, there will always be a retained condition:
Therefore, the crack depth can be expressed as follows:
When the load of the RC element strengthened with FRP is close to its ultimate value, the strain in tensile steel reinforcement, in most cases, shall exceed the yield strength and large plastic deformations will occur in the element (
State of stress in RC beam strengthened with (a) EB FRP reinforcement and (b) NSM FRP reinforcement.
Although FRP stress is unknown, the equilibrium condition between the ultimate crack depth and the FRP stress can be reached iteratively. This way, the ultimate crack depth could be evaluated by the following:
Limit state of strain in the flexural member.
In accordance with EC2 [
Afterwards, the compressive reinforcement stress is calculated and the ultimate crack depth is revised, evaluating the impact of compressive reinforcement and reduced stiffness in the interface between the RC member and the FRP reinforcement.
Real reduction coefficients of a concrete compressive zone stress diagram can be determined using the modified technique proposed by Dulinskas et al. [
Stress distribution diagrams for concrete in compression: (a) curvilinear: 1—ascending part, 2—descending part; (b) strains in the cross-section; (c) curvilinear concrete’s compressive zone stress diagram and centroids of its parts:
The areas of separate parts and the whole curvilinear concrete’s compressive zone diagram [
When
If
Stress distribution diagrams for concrete in compression, when mean value of concrete compressive strength is not reached: (a) strains in cross-section; (b) curvilinear concrete compressive zone stress diagram; (c) rectangular stress diagram.
Next, the ascending part of the concrete stress diagram is divided into the simpler figures, and the parameters of equivalent stress diagram are calculated by (
Subsequently, the flexural strength of the strengthened member can be expressed as
A simplified methodology proposed by Slaitas et al
The analytical model proposed above could be used for more reliable prediction of concrete crack parameters and flexural strength for FRP-strengthened RC structures.
Four-point bending tests were carried out on seven full-scale beams (see Figure
Loading scheme of tested beams.
Properties of specimens and materials.
Beam ID | Anchored | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
CB | 0.15 | 0.3 | 2.7 | 308 | 569 | — | — | 226 | 0 | 25.40 | — |
B1-0 | 0.15 | 0.3 | 2.7 | 308 | 569 | 144 | 2334 | 226 | 0 | 25.40 | Yes |
B2-P |
0.15 | 0.3 | 2.7 | 308 | 569 | 144 | 2334 | 226 | 0 | 25.40 | Yes |
B3-0 | 0.1 | 0.2 | 1.2 | 226 | 318 | 16.7 | 4800 | 100 | 0 | 28.70 | No |
B4-0 | 0.1 | 0.2 | 1.2 | 226 | 318 | 16.7 | 4800 | 100 | 0 | 28.70 | No |
B5-7 | 0.1 | 0.2 | 1.2 | 226 | 318 | 16.7 | 4800 | 100 | 7 | 28.70 | No |
B6-7 | 0.1 | 0.2 | 1.2 | 226 | 318 | 16.7 | 4800 | 100 | 7 | 28.70 | No |
Additionally, an extended database of 27 beams total was used for comparison of numerical and experimental results. Additional 21 beams were taken from a research conducted in [
Properties of additional beams for serviceability stage [
Beam ID | Anchored | ||||||||
---|---|---|---|---|---|---|---|---|---|
1-A2 | 180 × 100 | 2000 | 157 | 456 | 16.70 | 3450 | 100 | 10.63 | No |
1-A3 | 180 × 100 | 2000 | 157 | 456 | 33.40 | 3450 | 100 | 11.90 | No |
1-A4 | 180 × 100 | 2000 | 157 | 456 | 33.40 | 3450 | 100 | 11.05 | No |
1-A5 | 180 × 100 | 2000 | 157 | 456 | 16.70 | 3450 | 100 | 12.33 | No |
1-A7 | 180 × 100 | 2000 | 157 | 456 | 16.70 | 3450 | 100 | 12.75 | No |
1-A8 | 180 × 100 | 2000 | 100 | 513 | 16.70 | 3450 | 157 | 7.86 | No |
1-B2 | 180 × 100 | 1800 | 226 | 432 | 16.70 | 3450 | 100 | 12.75 | No |
1-B3 | 180 × 100 | 1800 | 226 | 432 | 33.40 | 3450 | 100 | 12.00 | No |
1-B4 | 180 × 100 | 1800 | 226 | 432 | 33.40 | 3450 | 100 | 10.50 | No |
1-B5 | 180 × 100 | 1800 | 226 | 432 | 16.70 | 3450 | 100 | 12.75 | No |
1-B7 | 180 × 100 | 1800 | 226 | 432 | 16.70 | 3450 | 100 | 13.13 | No |
2-A2 | 150 × 100 | 1800 | 100 | 530 | 18.15 | 3450 | 100 | 4.61 | No |
2-A3 | 150 × 100 | 1800 | 100 | 530 | 18.15 | 3450 | 100 | 4.80 | No |
2-A4 | 150 × 100 | 1800 | 100 | 530 | 16.50 | 3450 | 100 | 4.91 | No |
2-B2 | 150 × 100 | 1800 | 157 | 570 | 18.15 | 3450 | 157 | 6.45 | No |
2-B3 | 150 × 100 | 1800 | 157 | 570 | 18.15 | 3450 | 157 | 6.45 | No |
2-B4 | 150 × 100 | 1800 | 157 | 570 | 33.00 | 3450 | 157 | 6.19 | No |
2-C3 | 100 × 150 | 1800 | 100 | 530 | 13.20 | 3450 | 100 | 4.46 | No |
2-C4 | 100 × 150 | 1800 | 100 | 530 | 13.20 | 3450 | 100 | 4.01 | No |
2-D3 | 100 × 150 | 1400 | 157 | 570 | 13.20 | 3450 | 157 | 7.34 | No |
2-E3 | 100 × 150 | 1800 | 314 | 570 | 33.00 | 3450 | 157 | 10.01 | No |
A number of additional RC beams, strengthened with carbon fibre-reinforced polymer (CFRP) and glass fibre-reinforced polymer (GFRP) sheets, plates, strips, and rods, tested by different researchers, were analysed in a comparison of numerical and experimental results of load-carrying capacity (sample size: 98 beams). The properties of extra 71 beams are listed in Table
Properties of additional beams for load-carrying capacity.
Ref. | EBR/NSM | ||||||
---|---|---|---|---|---|---|---|
[ |
0.85 | 400 | 0.11 | 3100 | 165 | 1000 | EBR |
[ |
0.85 | 400 | 0.13 ÷ 0.14 | 2068 | 131 | 0 ÷ 1000 | NSM |
[ |
0.40 | 426 | 0.04 ÷ 0.12 | 2453 ÷ 3479 | 165 ÷ 230 | 0 | EBR |
[ |
0.40 | 426 | 0.04 ÷ 0.11 | 1878 ÷ 2453 | 121 ÷ 165 | 0 | NSM |
[ |
0.45 | 436 | 0.04 ÷ 0.22 | 1500 ÷ 2483 | 100 ÷ 167 | 0 | NSM |
[ |
0.29 ÷ 1.19 | 466 ÷ 501 | 0.08 | 2850 | 165 | 0 ÷ 1323 | EBR |
[ |
0.50 ÷ 0.75 | 525 ÷ 531 | 0.11 | 3263 | 251 | 0 | EBR |
[ |
0.58 | 545 | 0.12 ÷ 0.26 | 1350 ÷ 2350 | 64 ÷ 170 | 0 | NSM |
[ |
0.58 | 540 | 0.13 ÷ 0.26 | 1350 ÷ 2500 | 64 ÷ 170 | 0 | NSM |
[ |
0.77 | 475 | 0.08 | 2167 | 130 | 0 ÷ 1241 | NSM |
[ |
0.54 ÷ 0.94 | 730 | 0.16 ÷ 0.24 | 2740 | 159 | 0 | NSM |
[ |
0.39 | 585 | 0.06 | 1922 | 164 | 0 ÷ 823 | NSM |
The crack pattern of the higher beams (B1-0 and B2-P) is presented in Figure
Experimental crack pattern of beams B1-0 and B2-P.
The crack pattern of smaller beams (B3-0, B4-0, B5-7, and B6-7) is presented in Figure
Experimental crack pattern of beams B3-0, B4-0, B5-7, and B6-7.
In beams without initial strain, the crack distribution was denser, because from the start the beams had higher reinforcement ratios. Until the strengthening moment, the beams B5-7 and B6-7 were acting as ordinary RC beams and the primary cracks had already developed; after strengthening, the crack development was slower, but the spacing remained the same as earlier; that is, the distribution and propagation of cracks are different if the stiffness of beams at the moment of strengthening is different. As a result, crack widths of the beams strengthened under external load action (B5-7, B6-7) were 2 times higher than those strengthened without it (B3-0, B4-0). This validates the evaluation of the initial crack width in (
It should be mentioned that in smaller beams, mainly primary cracks were developing and the absence of anchorage has led to the horizontal cracks in the contact zone of concrete and FRP, which appeared when the external load has reached about 80% of the load-carrying capacity. The failure result of these beams was concrete cover separation.
The function of cross-sectional parameters
The function of cross-sectional parameters
The constants
The comparison of experimental and numerical results of the crack widths is presented in Figure
Comparison of the crack width estimation results with experimental ones at service load: (a)
Statistical parameters of crack width estimation.
1.03 | 1.00 | 0.68 | 0.60 | |
0.53 | 0.22 | 0.70 | 0.64 | |
51.44 | 21.62 | 103.59 | 107.50 | |
Crack widths calculated by design provisions while evaluating the tension-stiffening effect and without it were overestimated with up to 85% (mean 32%) and 86% (mean 40%) errors (
Generally, the numerical calculations of the load-carrying capacity without reduction of FRP stress (due to the slippage between concrete and FRP,
Comparison of experimental and numerical results of load-carrying capacity.
Statistical parameters of load-carrying capacity.
1.00 | 1.14 | |
0.24 | 0.24 | |
24.07 | 20.84 | |
The statistical parameters in Table
The analysis of experimental and numerical results proves that this calculation method allows the accurate evaluation of the load-carrying capacity of the normal section of flexural RC beams strengthened with FRP.
The proposed calculation method could be treated as appropriate for practical application when choosing the most effective strengthening material and when determining the crack width and load-carrying capacity of the strengthened member.
A crack width propagation and load-carrying capacity prediction model was presented in this paper. The conclusions of the analysis of experimental and numerical results are presented below.
The crack width calculation techniques proposed in the design recommendations overestimate the crack width with up to 86% (average 32% and 40%) error and very high scatter (coefficient of variation more than 100%). The predicted crack widths by the proposed model agreed very well with experimental results (with 0% average error and a rather low coefficient of variation: 21.62%), even for beams without additional anchorage and with initial strain (common situation in practice). The crack propagation analysis revealed that prestressing of the strengthening material reduced the maximum crack width 2.67 times. This was evaluated in the proposed calculation method (calculated crack width of the non-prestressed beam was 2.55 times bigger than that of the prestressed one). The propagation of cracks differs if the stiffness of the beams at the moment of strengthening is different. In the beams strengthened under external load action (with initial strain), the cracks had developed as in RC beams. After strengthening, further development of the cracks has slowed down, but the spacing remained the same. As a result, the crack widths of the beams strengthened under external load action (B5-7, B6-7) were 2 times higher than those strengthened without it (B3-0, B4-0). This validates the evaluation of the initial crack width in ( The results of numerical calculations of the load-carrying capacity without reduction of FRP stress (due to the slippage between concrete and FRP,
All data and results of a research are provided in the manuscript.
The authors declare that there is no conflict of interest regarding the publication of this paper.