Here, we calculate the maximum failure load that the reinforced composite beam can withstand by the conversion section method or ultimate stress method of material mechanics and compare the difference between the calculated maximum failure load theoretical value and the experimental value. See
Appendix A for the theoretical formulas of material mechanics.
3.3.1. Comparison of Experimental and Theoretical Values of PC Beams Reinforced with RPC
Reinforce the 10 × 10 × 35 cm PC beam with 1 cm thick RPC1, and the span length L = 30 cm. Calculate the maximum failure load P when the PC beam is subjected to a three-point bending test. The basic properties of concrete substrates and reinforcing materials are as follows:
PC beam: elastic modulus E1 = 2.41 × 105 kg/cm2, compressive strength σ1 = 347 kgf/cm2, flexural strength σ2 = 53 kgf/cm2, oblique shear strength τ = 117 kgf/cm2
RPC1: elastic modulus E2 = 3.01 × 105 kg/cm2, flexural strength σ3 = 201 kgf/cm2
The shear force diagram and bending moment diagram of a simply supported beam subjected to a P force are as shown in
Figure 6.
The converted section method (shown in
Figure 7) is used to calculate the internal stress of the composite beam as follows:
PC concrete subjected to maximum compressive stress:
PC concrete subjected to maximum tensile stress:
RPC1 concrete subjected to maximum tensile stress:
Shear stress at the bonding interface:
The maximum failure load is controlled when the PC concrete is subjected to the maximum tensile stress.
Theoretical value
The experimental value P = 1658.94 kg.
RPC1 reinforced 1 cm increases the strength by 34.9% compared with the unreinforced theoretical value, however, the strength of the experimental value increased by 39.5%.
The above results show that the theoretical maximum breaking load of the reinforced bending test body is P = 1603.9 kgf, which is 34.9% higher than that of the unreinforced bending strength. The bending strength increases by 39.5%, and the maximum failure load of the experimental value is very close to the calculated theoretical value.
In addition, the calculation results are shown in
Table 5 for the increase ratio of the maximum failure load theoretical value of the 2 cm thick RPC1 reinforced PC beam, the 1 cm thick RPC2 reinforced PC beam, and the 2 cm thick RPC2 reinforced PC beam. Since the RPC-reinforced concrete is analyzed by the internal stress of the beam in material mechanics, the theoretical value and the experimental value of the maximum failure load can obtain similar results. Therefore, this method is suitable for evaluating the PC beams reinforced with RPC.
3.3.2. Comparison of Experimental and Theoretical Values of PC Beams Reinforced with CFRP
The theoretical value of the maximum failure load of the CFRP-reinforced PC beam was calculated using the conversion section method and the ultimate stress method of material mechanics. A 10 × 10 × 35 cm PC beam is reinforced with a layer of CFRP patch, and the span length L = 30 cm. The maximum failure load P is calculated when the PC beam is subjected to a three-point bending test. The basic properties of concrete substrates and reinforcing materials are as follows:
PC beam: elastic modulus E1 = 2.41 × 105 kg/cm2, compressive strength σ1 = 347 kgf/cm2, flexural strength σ2 = 53 kgf/cm2, oblique shear strength τ = 117 kgf/cm2
CFRP1: elastic modulus E2 = 7.4 × 105 kg/cm2, flexural strength σ3 = 7800 kgf/cm2, oblique shear strength τ = 100 kgf/cm2, thickness t = 0.3 mm.
The shear force diagram and bending moment diagram of a simply supported beam subjected to a P force are as shown in
Figure 6.
- (1)
Using the converted section method (shown in
Figure 8) to calculate the internal stress of the composite beam is as follows:
PC concrete subjected to maximum compressive stress:
PC concrete subjected to maximum tensile stress:
CFRP1 subjected to maximum tensile stress:
Shear stress at the bonding interface:
The maximum failure load is controlled when the PC concrete is subjected to the maximum tensile stress.
Theoretical value .
Experimental value P = 1990.3 kg.
CFRP1 reinforced 1 layer increased the strength by 2.8% compared to the unreinforced theoretical value, however, the strength of the experimental value increased by 67.3%.
- (2)
The ultimate stress method (shown in
Figure 9) calculates the internal stress of the composite beam:
Assume that the concrete cannot withstand tensile stress. Concrete strain .
Find the neutral axis position:
Theoretical value .
Experimental value P = 1990 kgf.
CFRP1 reinforced 1 layer increased the strength by 104.9% compared to the unreinforced theoretical value, however, the strength of the experimental value increased by 67.3%. The theoretical calculation and experimental results of CFRP reinforcement are shown in
Table 5. It is obvious that the theoretical values of one-layer, two-layer, and three-layer CFRPs are all overestimated. The analysis of the internal stress of the beam by the conversion section method and the ultimate stress method cannot correctly evaluate the effect of CFRP reinforcement. There is such a large difference between the theoretical value and the experimental value, which may be due to: (a) The CFRP patch has a higher tensile strength than the bonded interface. (b) When the concrete has cracks at the loading point, the concrete can hardly bear the tensile force at this time, and the tensile force is borne by the bonding interface between the CFRP patch and the concrete. (c) The test load is jointly borne by the CFRP and the concrete, and the CFRP can withstand larger deformation than the concrete. (d) When the tensile force is greater than the bond strength that the interface can withstand, the bonding interface between the concrete and the CFRP will fall off and be damaged, but the CFRP will not be torn [
15,
25]. The interface bonding between CFRP and concrete plays an important role in the reinforcement of concrete members using CFRP [
25]. In addition, factors such as bending capacity, ductility, and maximum deflection should be considered. An important parameter in these analyses is the value of the elongation at failure of the CFRP composite, which can lead to premature failure of the “beam-bond-reinforcement” system [
26].
The theoretical value obtained by the conversion section method is because the concrete is damaged by tension, and the strength provided by the bonding interface between the CFRP and the concrete is not considered after the concrete is damaged. Therefore, the calculated theoretical value will be significantly lower than the experimental value. The theoretical value obtained by the ultimate stress method is based on the assumption that the concrete cannot bear the tensile force and the CFRP patch has reached a subdued state, which is different from the phenomenon observed in the experiment, so the theoretical value obtained by the ultimate stress method will be higher [
13]. Xiang et al. [
27] related study on the calculation method of flexural strength of damaged reinforced concrete beams strengthened by CFRP sheets, and the section analysis method was used to analyse the flexural strength of damaged beams. After the effective strain equation of the CFRP sheet is recommended, the experimental results can be verified.
ACI PRC 440.2 [
28] provides guidance for the selection, design and installation of FRP systems for external reinforcement of concrete structures. Ross et al. [
29] also proposed a practical design method, which agrees well with the experimental results. To evaluate the strength enhancement provided by the FRP panels, an inelastic section analysis procedure was developed that can accurately predict the load–displacement response of the modified beam.