The shear-tensile strength of closed-type winding GFRP stirrups

Abstract

Closed-type winding GFRP (CW-GFRP) stirrups are obtained by winding continuous fibers fully impregnated with resin on the mould layer by layer. Compared to pultruded stirrups, the CW-GFRP stirrups have a higher bend strength and no overlap sections. For concrete beams with CW-GFRP stirrups subjected to shear, due to the angle between the stirrup and the diagonal crack, the main tensile stresses acting on the stirrup can be resolved into a tensile force along the stirrup and a shear force perpendicular to the stirrup. The strength of the stirrup straight section in this composite state was called the shear-tensile strength. So far, there are few studies about the shear-tensile strength of CW-GFRP stirrups. In this paper, a new shear-tensile test method was proposed and tested for 23 specimens, including twenty CW-GFRP stirrups and three pultruded GFRP stirrups. The main parameters included the cross-sectional dimensions and diagonal crack angle. The test results showed that the bend strength of CW-GFRP stirrups was improved by 81% ∼ 115% and the shear-tensile strength was enhanced by 4% ∼ 19% compared with those of pultruded stirrups. Besides, as the angle of diagonal cracks increased from 35° to 55°, the shear-tensile strength decreased by 10% ∼ 28%, and the decrease was related to the cross-sectional dimensions of CW-GFRP stirrups. The greater the width-to-thickness ratio, the more significant the decrease. Furthermore, the shear-tensile strength overall increased with increasing thickness and decreased with increasing width. When the cross-sectional area was certain, the shear-tensile strength decreased with increasing width-to-thickness ratio and the bend strength exhibited opposite trends. Finally, a shear-tensile strength prediction model for CW-GFRP stirrups was proposed. For the application of CW-GFRP stirrups in beams, width-to-thickness ratios ranging from 1.5 to 2.67 and taking 60% of the tensile strength as the shear-tensile strength are recommended.

Keywords

FRP Stirrups shear-tensile strength diagonal cracks prediction model

Introduction

Reinforced concrete (RC) structures are a key component of marine and offshore infrastructure, but they face significant challenges due to steel corrosion caused by the high concentration of chloride ions in marine environments. This corrosion not only weakens structural integrity but also increases maintenance costs and reduces the service life of these critical infrastructures. In response, the use of non-corrosive materials as alternatives to traditional steel reinforcement has emerged as a promising solution. Among these materials, fiber-reinforced polymer (FRP) has gained considerable attention due to its exceptional properties, including lightweight, high tensile strength, and excellent resistance to corrosion (Cheung and Tsang, 2010; Dong et al., 2018; Tahir et al., 2019b, 2019c, 2021). The replacement of steel with FRP bars in RC structures provides a long-term solution to the corrosion problem, enhancing the durability and sustainability of marine and offshore infrastructure.

Like RC structures, FRP-reinforced concrete (FRP-RC) structures rely on stirrups to resist shear forces. The widely used FRP pultruded stirrup is as shown in Figure 1(a), which is formed by bending the pultruded FRP rod prior to resin polymerization. During stirrup fabrication, the bending process can lead to the kinking of the inner fibers in the stirrup bent section, resulting in non-uniform fiber stress distribution. This issue results in a considerable reduction in the bend strength of FRP stirrups, which is typically only 30%∼60% of their tensile strength at straight part (Ahmed et al., 2009; Dong et al., 2018; El-Sayed et al., 2007; Lee et al., 2013; Spadea et al., 2017; Tahir et al., 2019b, 2022). During shear tests on concrete beams reinforced with pultruded FRP stirrups, it was consistently observed that the stirrups fractured in the bent section (Lee et al., 2016; Tahir et al., 2019c; Yuan and Wang, 2019). This failure mode highlights a critical problem: the low bend strength of the FRP stirrups significantly limits the ability to utilize the high tensile strength of the material fully. As a result, the potential benefits of FRP such as corrosion resistance and high strength are not fully realized in shear-critical regions, undermining the overall performance of FRP-RC structures. To address the above problem, Dong et al. (2018) proposed the closed-type winding glass-FRP (CW-GFRP) stirrup as shown in Figure 1(b), which is traditionally employed in the production of FRP tubes. In this method, resin-impregnated glass fibers are continuously wound around a mold, forming a composite structure that is then cut along the tube’s cross-section to produce rectangular CW-GFRP stirrups. These stirrups feature continuous fibers and a completely closed loop design. This manufacturing approach significantly mitigates the kinking and stress concentration that typically occurs during the bending process, thereby enhancing the overall bend strength of the stirrups. As a result, the bend strength of the CW-GFRP stirrup produced using this method improves dramatically, reaching 62% ∼ 81% of the tensile strength (Yuan et al., 2022).

Figure 1.

The comparison of pultruded stirrups and closed-type winding stirrups. (a) Pultruded FRP stirrup. (b) CW-GFRP stirrup production process.

As shown in Figure 2, there are two common failure locations of the stirrups when the concrete beams are subjected to shear forces, which are the straight section and the bent section of the stirrups intersecting with the critical diagonal crack. The stirrup failure at different locations depends on the strength of the corresponding location. In the study of the shear performance of concrete beams with FRP stirrups, it is commonly assumed that the stirrup failure is determined by the minimum of tensile strength and bend strength. For the beams with pultruded stirrups, the bend strength is much smaller than the tensile strength, which makes the stirrup fracture mainly occur in the bent section (Lee et al., 2016; Tahir et al., 2019a; Yuan and Wang, 2019). For the beams with CW-GFRP stirrups, due to the greatly improved bend strength, the probability of the fracture of stirrup straight section is greatly increased. Yuan et al. (2022) tested the GFRP-RC beams with CW-GFRP stirrups. The results showed that both the straight section failure and bent section failure of CW-GFRP stirrups were common in concrete beams subjected to shear forces. Therefore, it is necessary to study the strength of the bent section and the strength of the straight section of CW-GFRP stirrups. For the stirrup bent section, there have been some primary studies on pultruded stirrups and CW-GFRP stirrups. Lee et al.(2014) made closed stirrups by winding CFRP strips and tested the bend strength. The results showed that the bend strength could reach 60%∼78% of the tensile strength, which greatly improved the material utilization efficiency. Spadea et al.(2017) made novel closed rectangular stirrups by wounding filaments around the bar and tested the bend strength. The results showed that the bend strength was significantly improved compared to the pultruded stirrups, and it was also pointed out that the width-to-thickness ratio has a large effect on the bend strength. Gong et al., 2025 investigated the effect of cross-sectional dimensions on the bend strength of CW-GFRP stirrups and proposed a bend strength prediction model based on the width-to-thickness ratio and the bending radius.

Figure 2.

Schematic drawing of critical diagonal crack intersecting with stirrups.

The stirrup straight section intersecting with the diagonal crack, not only undergoes tensile stress but also undergoes the shear stress at the diagonal crack (El-Sayed et al., 2007), which is in the state of shear-tensile complex stress, hereafter called shear-tensile strength. FRP is an anisotropic material in which the transverse strength in the perpendicular fiber direction is much lower than the longitudinal strength in the parallel fiber direction (Aldajah and Haik, 2012), which makes the shear-tensile strength less than the tensile strength of the stirrup straight section. Shehata et al.(2000) tested the shear-tensile strength of pultruded stirrups by presetting artificial cracks in concrete blocks (the test setup is shown in Figure 3). The results showed that the shear-tensile strength of pultruded stirrups was more than 60% of the tensile strength. As the increase of the angle of the critical diagonal crack, the shear-tensile strength decreased and then increased, with the lowest strength at an angle of 45°. However, the test setup required two identical FRP stirrups to be crossed and symmetrically placed, which was difficult to achieve for the completely closed CW-GFRP stirrups. Nakamura and Higai (1995) showed that the shear-tensile strengths of FRP stirrups were about 50% ∼ 90% of the tensile strength and proposed a shear-tensile strength calculation equation based on elasticity theory. The equation only considers the effect of normal stress, ignoring the adverse effect of shear on the strength of the stirrup straight section. If the crack inclination is large, the accuracy of calculation results will be low. Since CW-GFRP stirrups have rectangular cross-sections and higher bend strength compared to pultruded stirrups, the effect on the failure mode of the stirrups during the shear of concrete beams will be significant. Therefore, it is necessary to test the shear-tensile strength of the straight section of the CW-GFRP stirrups to provide a basis for the shear design of concrete beams.

Figure 3.

Shear-tensile strength test method by Shehata et al.(2000).

There are few studies on the shear-tensile strengths of CW-GFRP stirrups. In this paper, the shear-tensile strengths of CW-GFRP stirrups were investigated. A new shear-tensile test method was proposed, and 23 specimens were tested, including twenty CW-GFRP stirrups and three pultruded GFRP stirrups. The effects of the form of the stirrup, cross-sectional dimensions, and angle of diagonal cracks on the shear-tensile strength were mainly investigated. Furthermore, a prediction model for the shear-tensile strength of CW-GFRP stirrups was proposed.

Test methodology

Tensile test and bent test

In this paper, the reinforcement was ECT (Electrical & Chemical Treatment) glass fiber and the matrix was thermosetting epoxy resin for all GFRP stirrups. Moreover, the tensile strength and bend strength of the pultruded GFRP stirrups and CW-GFRP stirrups were tested initially to characterize the shear-tensile strengths better. For the tensile test, the pultruded GFRP stirrups were determined by testing the 8 mm pultruded bars according to ASTM D7205-06 (2011) as shown in Figure 4(a). The average tensile strength was 1064.4 MPa and the elastic modulus was 50.4 GPa. The tensile test method in ASTM D3039-14 (2014) was used to test the tensile strength of the straight section for CW-GFRP stirrups, as shown in Figure 4(b). The tensile strengths and the modulus of elasticity with different cross-sectional dimensions of CW-GFRP stirrups were similar, so the average tensile strength with four cross-sectional dimensions was taken as the tensile strength of CW-GFRP stirrups. The average tensile strength of the CW-GFRP stirrup was 1098.1 MPa, and the modulus of elasticity was 55 GPa. For the bent test, all stirrups were tested by using test method B.5 recommended by ACI 440.3R-12 (2012), which is shown in Figure 4(c). The results of the bent test are shown in Table 1.

Figure 4.

Test setup and instrumentation details for the tensile strength of FRP bars. (a) Tensile test of pultruded bars. (b) Tensile test of CW-GFRP bars. (c) Bent test of all stirrups.

Table 1.

Specimen Design and Test Results of the Shear-Tensile Test.

Specimen	Numbers	α (°)	w_s (mm)	t_s (mm)	A (mm²)	F (kN)	f_fs (MPa)	f_fs,avg (MPa)	$\frac{f_{fs}}{f_{fu}}$	f_fb (MPa)	$\frac{f_{fb}}{f_{fu}}$	$\frac{f_{fs}}{f_{fb}}$
C-9 × 3-45	3	45	9	3	27	27.6	722.6	686.3	0.63	892.2	0.81	0.78
		45	9	3	27	24.7	646.9
		45	9	3	27	26.3	689.3
C-9 × 3-35	1	35	9	3	27	34.7	784.2	784.2	0.71			0.88
C-9 × 3-55	1	55	9	3	27	18.1	597.3	597.3	0.54			0.67
C-18 × 3-45	3	45	18	3	54	51.6	675.7	624.5	0.57	784.6	0.71	0.79
		45	18	3	54	46.1	603.4
		45	18	3	54	45.4	594.3
C-18 × 3-35	1	35	18	3	54	62.1	796.8	796.8	0.68			0.94
C-18 × 3-55	1	55	18	3	54	38.5	571.0	571.0	0.52			0.72
C-12 × 4.5-45	3	45	12	4.5	54	56.1	734.3	725.6	0.66	760.6	0.69	0.96
		45	12	4.5	54	53.9	706.2
		45	12	4.5	54	56.2	736.2
C-12 × 4.5-35	1	35	12	4.5	54	70.9	801.7	801.7	0.73			1.06
C-12 × 4.5-55	1	55	12	4.5	54	42.5	685.6	685.6	0.62			0.89
C-9 × 6-45	3	45	9	6	54	62.1	813.6	774.6	0.70	726.5	0.66	1.06
		45	9	6	54	59.2	774.6
		45	9	6	54	56.2	735.7
C-9 × 6-35	1	35	9	6	54	73.0	824.6	824.6	0.75			1.14
C-9 × 6-55	1	55	9	6	54	44.0	746.6	746.6	0.68			1.03
P-8-45	3	45	d = 8		50.2	41.2	580.3	598.4	0.71	415.2	0.39	1.78
		45	d = 8		50.2	48.1	677.1
		45	d = 8		50.2	38.2	537.8

Note: α is the angle between the stirrup and the vertical direction; w_s is the width of CW-GFRP stirrups; t_s is the thickness of CW-GFRP stirrups; d is the diameter of reinforcement; A is the cross-sectional area of the reinforcement; F is the average of the peak loads of the two jacks when the stirrup fractured; f_fs is the shear-tensile strength of the stirrup straight section at the intersection with the diagonal crack; f_fu is the average tensile strength of the straight section of the stirrups, f_fb is the bend strength of stirrups.

Shear-tensile test

Test method

For the shear-tensile strength of stirrup straight sections, this paper simplifies the changes of the stirrup straight section in the crack opening to facilitate the test. The forces acting on the stirrup straight section at its intersection with a critical diagonal crack are illustrated in Figure 2. During shear tests on concrete beams, when the diagonal crack intersects with the stirrup, the principal tensile stress perpendicular to the diagonal crack’s direction can be decomposed into the tensile force along the stirrup and the shear force perpendicular to the stirrup, owing to the angle between the stirrup and the diagonal crack. As the principal tensile stress continuously increases, the diagonal crack passing through the stirrup gradually opens up as the load increases. The crack opening, which occurs perpendicular to the diagonal crack’s direction, induces first deformation in the stirrup. F₁ is the principal tensile force at the first deformation; T₁ is the tensile force along the stirrup at the first deformation and P₁ is the shear force perpendicular to the stirrup. When the crack opening reaches a certain width, the concrete above the crack shifts upward, while the concrete below shifts downward along the diagonal crack’s direction, resulting in the crack misalignment. F₂ is the principal tensile force at the first deformation; T₂ is the tensile force along the stirrup at the first deformation and P₂ is the shear force perpendicular to the stirrup. This crack misalignment, which occurs parallel to the diagonal crack’s direction, induces second deformation in the stirrup. Therefore, the relative misalignment of the cracks is ignored to account for the most unfavorable case. Also, for the convenience of the test and analysis, the edges of the cracks where the stirrup intersects the critical diagonal cracks are treated as parallel straight lines.

Based on the above-mentioned., a new shear-tensile test method was developed to simulate the forces on stirrup straight sections at diagonal cracks, as shown in Figure 5. The test method focuses on the zone where the stirrup straight section intersects the critical diagonal crack. For the simulation of the critical diagonal crack, a steel plate was embedded in the concrete block to make a weak cross-section, dividing the concrete block into two sections, and guiding the cracks to appear at a predefined location. In the real concrete beam, the critical diagonal crack intersects the stirrup diagonally at an angle (α) to the horizontal axis, as shown in Figure 2. Therefore, the effect of α was considered in the shear-tensile test. To simulate the real state of stirrups in concrete beams, CW-GFRP stirrups were placed in concrete blocks, and the tests used simultaneous loading by jacks installed on the left and right sides of the concrete block to simulate crack opening and using force sensors to monitor the load in real-time. During the test, the test setup was placed on a level ground and rollers were placed between the ground and the test setup to minimize the effect of friction. To avoid the relative misalignment of concrete blocks in the horizontal direction in the test, a side-restraining device was provided to the left and right of the concrete blocks. The side-restraining device consists of a steel plate, rollers, and high-strength bolts. The high-strength bolts fixed the steel plates on both sides of the concrete block, thus restraining the lateral displacement of the concrete block. The rollers were used to eliminate friction between the side limiters and the concrete block.

Figure 5.

Test setup and instrumentation details for the shear-tensile strength of stirrups.

The size and reinforcement of the specimens are shown in Figure 6. To ensure that specimen failure occurs at the pre-cracked location, steel bars were configured in the concrete block. To ensure the stirrup angles were accurate, holes were pre-punched in the formwork at the designed angles, and the CW-GFRP stirrups were suspended in their intended position by means of a steel wire rope passing through the holes. Wood blocks were placed between the base mould and the stirrups to ensure the thickness of the protective layer met the design requirements.

Figure 6.

Dimensions and reinforcement details of the specimen (Units: mm).

Test specimens

The details of specimens for the shear-tensile test of stirrup straight sections are shown in Table 1. The inner width of the stirrups was 60 mm and the inner length was 300 mm. The bend radius of all stirrups was 30 mm. To investigate the effect of CW-GFRP stirrup cross-sectional dimensions on the shear-tensile strength, four different cross-sectional dimensions, 9 mm × 3 mm, 18 mm × 3 mm, 12 mm × 4.5 mm, and 9 mm × 6 mm, were set up. Meanwhile, to investigate the effect of the angle of diagonal crack, three different angles (α) between the stirrup and the vertical direction, 35°, 45°, and 55°, were set for each cross-sectional dimension. In addition, 8 mm diameter pultruded GFRP stirrup was set up as a control specimen with α as 45°. The specimens are named with a three-part code. The initial character represented the form of stirrups (C: CW-GFRP stirrup, P: pultruded GFRP stirrup); the second part represented the cross-sectional dimension; the third character represented α. For instance, C-18 × 3-35 represents a specimen with CW-GFRP stirrups having a cross-sectional dimension of 18 × 3 mm and a diagonal crack angle of 35°. Three specimens were cast for tests with a 45° crack angle, while for others one specimen was made for each test variable, in total 23 specimens.

Test results and discussion

The effect of the stirrup’s form

The failure of all specimens occurred in the stirrup straight section, and no cracks were found in the concrete block at the end of the test. The failure modes of the shear-tensile test of the stirrup straight sections are shown in Figure 7. In the final stage, the crack between the concrete blocks suddenly opened up, and the straight section of all stirrups split and pulled off at the intersection with the crack. After separating the concrete blocks, local spalling of the concrete layer adjacent to the stirrups was observed.

Figure 7.

Failure modes of the shear-tensile test of stirrups. (a) Specimen after destruction. (b) The broken stirrups. (c) Failure mode of CW-GFRP stirrups. (d) Failure mode of pultruded stirrups.

Table 1 summarises the shear-tensile strength (f_fs), the bend strength (f_fb), the ratio of the shear-tensile strength to the tensile strength (f_fs/f_fu), and the ratio of the shear-tensile strength to the bend strength (f_fs/f_fb). To reduce the test error, this paper used the average value of the shear-tensile strength (f_{fs, avg}) instead of the shear-tensile strength (f_fs). The results showed that the f_fs of the CW-GFRP stirrups was 4% ∼ 29% higher than that of the pultruded stirrups. The shear-tensile strength of the straight section of all stirrups was 0.52 ∼ 0.73 of the tensile strength, which demonstrated that it was inaccurate to take the tensile strength as one of the standards for evaluating the stirrup failure in the concrete beam. Moreover, the f_fb of the CW-GFRP stirrups was improved by 81% ∼ 115% than that of the pultruded stirrups. The bend strength of CW-GFRP stirrups was significantly enhanced.

When shear failure of FRP-RC beams led to the fracture of their stirrups, it is noteworthy that the location of the stirrup fracture is closely related to the position of the stirrup relative to the critical diagonal crack. Stirrup can fail either in its straight or bent sections, and the failure strength depends on the minimum values of the bend strength (f_fb) and the shear-tensile strength (f_fs). The f_fs/f_fb of CW-GFRP stirrups was about 0.79 ∼ 1.06 and the f_fs/f_fb of pultruded stirrups was about 1.78. In pultruded stirrups, the f_fs was significantly higher than f_fb, so the failure of the pultruded stirrup in the shear test of concrete beams mostly occurred in the bent section. As a result, the shear-tensile strength of the pultruded stirrup could not be effectively utilized in the straight section. However, for CW-GFRP stirrups, f_fs was closer to f_fb, which was more conducive to the full use of the shear-tensile strength.

The effect of the diagonal crack angle

In concrete beams subjected to shear, the angle (α) of the critical diagonal crack and horizontal direction affects the failure mode and shear performance. Shear tests performed on FRP-RC beams showed that the α may be distributed from about 30° to 60° (Razaqpur and Spadea, 2015). The stirrup straight section is subjected to both tensile and shear forces at the intersection with the diagonal crack, and the shear force increases with the increase of α. Because FRP is an anisotropic material, the action of shear force causes a significant reduction in its shear-tensile strength. Therefore, it is necessary to investigate the effect of α on shear-tensile strength.

The shear-tensile strength tended to decrease continuously with increasing angle of diagonal cracks. Figure 8 shows the effect of α on the f_fs/f_fu of CW-GFRP stirrups. When α was increased from 35° to 55°, the f_fs/f_fu with the cross-sections of 9 mm × 6 mm and 12 mm × 4.5 mm for CW-GFRP stirrups only decreased by about 10% and 15%. However, for CW-GFRP stirrups with cross-sections of 18 mm × 3 mm and 9 mm × 3 mm, the decrease in f_fs/f_fu reached 23% and 28%. This decrease in f_fs/f_fu is attributed to increasing shear force due to higher stirrup deformation with increasing crack angle. As FRP is weak in the transverse direction, the increase in shear force causes early failure of the stirrups. It is worth noting that the difference in this variation is related to the cross-sectional dimensions of the stirrups. The greater the width-to-thickness ratio, the higher the decrease in shear-tensile strength as the diagonal crack angle increased.

Figure 8.

The variation curve of f_fs/f_fu of specimens with the angle of diagonal cracks.

The effect of cross-sectional dimensions

Figure 9 shows the effect of the cross-sectional dimensions of CW-GFRP stirrups on the ratio of shear-tensile strength to tensile strength (f_fs/f_fu) at different diagonal crack angles. The effect of the thickness (t_s) on f_fs/f_fu for CW-GFRP stirrups with a width of 9 mm is shown in Figure 9(a). When t_s increased from 3 mm to 6 mm, due to the increase in the number of fiber layers in the stirrup, the f_fs/f_fu tended to increase with the increase in t_s, and the greater the angle of the diagonal crack, the more significant is the effect of the thickness. When the angle of the diagonal crack was 35°, the thickness increased from 3 mm to 6 mm and the f_fs/f_fu only improved by 6%. When the angles of the diagonal crack were 45° and 55°, the f_fs/f_fu improved by 11% and 28%, respectively.

Figure 9.

The effect of the cross-sectional dimensions at different diagonal crack angles. (a) The effect of the thickness. (b) The effect of the width. (c) The effect of the width-to-thickness ratio.

Figure 9(b) shows the effect of the CW-GFRP stirrup width (w_s) on f_fs/f_fu when the t_s was 3 mm. The f_fs/f_fu decreased with increasing width, but not significantly at small angles of inclination of diagonal cracks. When α was 35°, the f_fs/f_fu of the CW-GFRP stirrups with a cross-section of 9 mm × 3 mm was similar to that of the stirrups with a cross-section of 18 mm × 3 mm. When α was 45° and 55°, the f_fs/f_fu with a cross-section of 9 mm × 3 mm increased by 11 % and 5 %, respectively, compared to f_fs/f_fu with a cross-section of 9 mm × 3 mm. This may be attributed to non-uniform stress distribution in the width direction which is caused by deformation of the edges of the stirrups. As the width of the stirrup increases, the non-uniformity of stress distribution of the stirrup may be aggravated, thus causing the tearing of the edge of the stirrup.

The effect of the w_s/t_s on f_fs/f_fu and f_fb/f_fu for CW-GFRP stirrups is shown in Figure 9(c). The increase in w_s/t_s means that the stirrup section becomes wider and thinner. Based on the combined effect of width and thickness, the f_fs/f_fu of stirrup straight section generally tended to decrease as the w_s/t_s increased and the effect of the width-to-thickness ratio was more noticeable at large diagonal crack angles. When the angle of diagonal crack was 35°, the width-to-thickness ratio increased from 1.5 to 6, and f_fs/f_fu decreased by only 3%. When the diagonal crack angle was 45° and 55°, w_s/t_s increased from 1.5 to 6, and f_fs/f_fu decreased by 19% and 24%, respectively. However, the bend strength increased with the width-to-thickness ratio. When increasing the width-to-thickness ratio from 1.5 to 6, the f_fb/f_fu increased by 8%. The width-to-thickness ratio had the opposite effect on shear-tensile strength and bend strength. Therefore, in the shear test of beams, it is necessary to determine the width-to-thickness ratio of the stirrup reasonably to ensure that both shear-tensile strength and bend strength can be effectively utilized. Based on the test results, the width-to-thickness ratio in the range of 1.5 ∼ 2.67 was the reasonable recommendation with a cross-sectional area of 54 mm² (ϕ 8).

The prediction model for the shear-tensile strength of CW-GFRP stirrups

There are few studies on calculation models for the shear-tensile strength of stirrup straight section (Nakamura and Higai, 1995). To provide a basis for the shear design of FRP-RC beams with CW-GFRP stirrups, a model for calculating the shear-tensile strength of CW-GFRP stirrups has been developed by drawing on the model proposed by Friberg (1940) for dowels through transverse joints in concrete pavements. The schematic of the model proposed by Friberg is shown in Figure 10.

Figure 10.

Schematic drawing of the dowel model proposed by Friberg (Friberg, 1940). (a) Schematic drawing of dowel forces between concrete joints. (b) Simplified model. (c) Schematic of the deformation of the dowel.

As shown in Figure 10(a), the intersection of the dowels and the concrete joint is subjected to a shear force (P) and a moment (M₀). By using the solution for semi-infinite beams on elastic foundations, Friberg developed equations for determining the slope and deflection of the dowel through joints in concrete pavements as follows:

y = \frac{e^{- β x}}{2 β^{3} E I} (P \cos β x - β M_{0} (\cos β x - \sin β x))

(1)

β = \sqrt[4]{\frac{K b}{4 E I}}

(2)

where x is the distance from the left end of the beam; y is the vertical displacement; y₀ is the vertical displacement of the left end of the beam; P is the shear acting at the left end of the elastic beam; M₀ is the moment acting at the left end of the elastic beam; E is the elastic modulus of the beam; I is the moment of inertia of the beam; b is the width of the elastic beam; and K is the modulus of dowel support. Once the dowels were subjected to shear forces, the concrete cracked and spalled. Therefore, it is necessary to modify K according to the test results.

From equation (1), the displacement of the left end of the beam in the y-direction when x is 0 is obtained:

y_{0} = \frac{P - β M_{0}}{2 β^{3} E I}

(3)

By taking the derivative of equation (1) and substituting x equal to 0, the angle of deflection at the left end of the beam is obtained:

\frac{d y}{d x} = \frac{2 β M_{0} - P}{2 β^{2} E I}

(4)

Based on the Friberg model, this paper further considers the tensile force on the stirrup in the shear-tensile test. When the stirrup stress ( $σ_{x}$ ) reaches the shear-tensile strength (f_fs), the deformation of the stirrup straight section at the intersection with the diagonal crack is shown in Figure 11. Where the midpoint O is the midpoint of the stirrup straight section; a is the crack width; The point A is the intersection of the stirrup and the edge of the concrete crack; α is the angle between the stirrup and the vertical direction; $ϕ$ is the angle between T and the vertical direction at the point O, increases with the increasing deformation of the stirrup; y_A is the displacement of point A perpendicular to the direction of stirrup; x₁ is the displacement of point O in the horizontal direction caused by y_A; x₂ is the displacement of point O in the horizontal direction caused by the deflection angle ( $φ$ ) of point A; x₃ is the projection in the x-direction of the bending deformation at point O under the action of P; x₄ is the projection of the horizontal direction of the shear deformation at point O under the action of P. The predictive model for the shear-tensile strength of CW-GFRP stirrup is based on the deformation of the stirrup during the test. When the load applied by the jack is small, the stirrup does not deform, The F₁ is the combined force of the tensile force (T₁) and the shear force (P₁). As the load increases, the stirrup undergoes deformation, causing both the magnitude and direction of the tensile and shear forces to continuously change. The combined force of the tensile and shear forces remains equal in magnitude and opposite in direction to the load applied by the jack. When the stirrup deforms to the position shown in Figure 11, it undergoes shear-tensile failure. At this stage, the combined force (F₂) of the shear force (P₂) and tensile force (T₂) represents the failure load of the stirrup.

Figure 11.

Schematic of a straight section of stirrups crossing a crack in the shear-tensile test.

When the stirrup stress ( $σ_{x}$ ) reaches the shear-tensile strength (f_fs), T is:

T = w_{s} t_{s} f_{fs}

(5)

P and T are the shear and tensile forces on the stirrup at the point O; Since P is perpendicular to T, to facilitate the solution, it is assumed that the relationship between P and T is:

P = η T

(6)

where

η

\tan ϕ

Since the surface of the CW-GFRP stirrup is smooth, the bond between the stirrup straight section and the concrete can be ignored. Thus, the relationship between a and f_fs when the shear-tensile failure occurs can be obtained as:

a = \frac{f_{fs}}{E} L \cos α

(7)

where L is the length of the straight section of the CW-GFRP stirrup in the shear-tensile test, which is 234 mm.

From equation (4), y_A can be obtained as:

y_{A} = \frac{P - β M_{A}}{2 β^{3} E I}

(8)

where M_A is the moment applied in section A.

The x₁ caused by y_A is given by:

x_{1} = \frac{y_{A}}{\cos α}

(9)

After the deformation of the stirrup, because of the change in slope at point A, the deflection angle ( $φ$ ) is generated. From equation (4) and the geometrical relationship in the Figure 11, the equation for x₂ can be obtained as:

x_{2} = \frac{φ l}{\cos α}

(10)

where l is the length of OA in mm.

The x₃ under the action of the P is given by:

x_{3} = \frac{P}{3 E I} l^{3} \cos ϕ

(11)

Assuming that the shear stress ( $τ$ ) is uniformly distributed within the section, the x₄ can be obtained as:

x_{4} = \frac{τ}{G} l \cos ϕ = \frac{P l}{w_{s} t_{s} G} \cos ϕ

(12)

Where G is the shear modulus of the stirrup. Since the glass fibers are unidirectionally wound in the CW-GFRP stirrup fabrication, G takes the shear modulus of the epoxy resin, which is 3.5 GPa.

The M_A is as follows:

M_{A} = P l

(13)

l = \frac{a}{2 \cos α}

(14)

Neglecting the relative deflections of points A and O, the equation for a can be obtained as

ϕ = α - φ

(15)

φ = \frac{2 β M_{A} - P}{2 β^{2} E I}

(16)

In the test, the stirrups were always symmetric about the center of point O. Thus, the following relationship can be obtained:

x_{1} + x_{2} + x_{3} + x_{4} = \frac{a}{2} \tan α

(17)

Since P is unknown, the relationship between P and T should be assumed based on equation (6), after which the iterative calculations based on equations (5) to (17) are used to solve for x₁, x₂, x₃, and x₄, respectively. When equation (17) holds, the value of P can be obtained. However, equation (17) does not determine whether the stirrup failure has occurred.

In this paper, the Tsai-Hill failure criterion is applied to the critical section of the stirrup to determine whether the stirrup failure occurs or not. The point A is simultaneously subjected to T, P, and M_A. Therefore, section A is a critical section for shear-tensile failure. By assuming a parabolic distribution of the shear stress ( $τ_{A}$ ) and a linear distribution of positive stress ( $σ_{A}$ ) in this section as shown in Figure 12, the distribution equations of $σ_{A}$ and $τ_{A}$ can be obtained as follows:

σ_{A} (w) = \frac{T}{w_{s} t_{s}} - \frac{M_{A}}{I} w

(18)

τ_{A} (x) = (- \frac{6 w^{2}}{{w_{s}}^{3}} + \frac{3}{2 w_{s}}) \frac{P}{t_{s}}

(19)

Where w is the coordinate on the coordinate axis established with the point A.

τ_{A, \max} = τ_{A} (0) = \frac{3}{2} \cdot \frac{P}{w_{s} t_{s}}

(20)

σ_{A, \max} = σ_{A} (- 0.5 w_{s}) = \frac{T}{w_{s} t_{s}} + \frac{M_{A}}{I} \cdot 0.5 w_{s}

(21)

σ_{A, \min} = σ_{A} (0.5 w_{s}) = \frac{T}{w_{s} t_{s}} - \frac{M_{A}}{I} \cdot 0.5 w_{s}

(22)

Figure 12.

Distribution of normal and shear stress along the cross-section at point A.

The equation for determining whether the failure has occurred in the section A is presented as follows:

\frac{σ_{1}^{2}}{S_{L}^{2}} - \frac{σ_{1} σ_{2}}{S_{L}^{2}} + \frac{σ_{2}^{2}}{S_{T}^{2}} + \frac{τ_{12}^{2}}{S_{LT}^{2}} = \frac{σ_{A}^{2} (w)}{S_{L}^{2}} + \frac{τ_{A}^{2} (w)}{S_{LT}^{2}} = 1

(23)

Where

σ_{1}

is the longitudinal stress in the fiber direction;

σ_{2}

is the transverse stress perpendicular to the fiber direction;

σ_{3}

is the shear stress perpendicular to the fiber direction; S_L, S_T, and S_LT are the material strengths under longitudinal, transverse, and shear failure, respectively. Based on the shear-tensile test results of CW-GFRP stirrups, the S_L was determined to be 1098 MPa. The S_LT was determined to be 82.7 MPa based on the mechanical property parameters of GFRP epoxy composites in the literature (Gibson, 2007).

The calculation flow of the model is shown in Figure 13, and the calculation process is as follows:

(1) Assuming a small $σ_{x}$ and η, then iteratively solve to determine the true value coefficient η, which determines the relationship between P and T.

(2) From equation (23), determine whether the failure has occurred in section A. If it has not, increase the $σ_{x}$ .

(3) By cycling through step (1) until section A fails, the shear-tensile strength (f_fs) of CW-GFRP stirrups can be found when the shear-tensile failure occurs.

Figure 13.

Flow chart for shear-tensile calculation.

The equation for the coefficient K on CW-GFRP stirrups was obtained by fitting the K values calculated for the specimens based on the test results as:

K = 1.46 {w_{s}}^{0.439} {t_{s}}^{- 0.419} α^{0.819}

(24)

Figure 14 compares the experimental values (f_fs,exp) and the predicted values (f_fs,pre) of shear-tensile strength. The difference between the f_fs,exp and the f_fs,pre is within 10% and the model proposed by this paper is accurate.

Figure 14.

Comparison of the experimental and calculated shear-tensile strength.

Figure 15 shows the relationship between the angle (α) of the diagonal crack on the ratio of shear-tensile strength to tensile strength of CW-GFRP stirrups (f_fs/f_fu) based on the model of this paper. When α is in the range of 0°∼10°, the shear-tensile strength decreases slowly with the growth of α. This is because the diagonal crack is approximately perpendicular to the stirrup leg at small α, and the shear-tensile strength of the stirrup leg is approximately equal to the tensile strength. As α increases, the shear-tensile strength decreases until it reaches a minimum at a value equal to 55°, and then increases slightly with α. The model proposed by this paper considers the effects of normal and shear stresses in the stirrup failure. As the angle α changes, the magnitude and direction of the normal and shear stresses change, which is the reason for the trend of the shear-tensile strength. The test results of Shehata et al. (2000) also showed a law of first decrease and then an increase in shear-tensile strength with an increase in α, but the shear-tensile strength is lowest when α is 45°.

Figure 15.

The angle of the diagonal crack on the ratio of shear-tensile strength to tensile strength.

When the beams with FRP stirrups were subjected to shear failure, the angle of the diagonal cracks was mostly within 50°. (Bentz et al., 2010; Razaqpur and Spadea, 2015; Yuan et al., 2022). The values in which the diagonal crack angles were 50° shown in Figure 15 can be adopted as the recommended values for the shear-tensile strength of CW-GFRP stirrups of different cross-section dimensions. Besides, the shear-tensile strength decreases with increasing width-to-thickness ratio, which is compatible with the test results. In FRP-RC beams subjected to shear, it is necessary to limit the width-to-thickness ratio to effectively utilize the bend strength and shear-tensile strength of the stirrups. Figure 16 represents the experimental values of bend strength and recommended values of shear-tensile strength for specimens with different width-thickness ratios. At the width-to-thickness ratio in the range of 1.5 ∼ 2.67, it is obvious that there is not much gap between the shear-tensile strength and bend strength of the stirrups. Therefore, this paper suggests using CW-GFRP stirrups with a width-to-thickness ratio in the range of 1.5 ∼ 2.67, and the shear-tensile strength is taken as 60% of the tensile strength.

Figure 16.

Experimental values of bend strength and recommended values of shear-tensile strength for each specimen.

Conclusions

This paper investigates the shear-tensile strengths of closed-type winding GFRP (CW-GFRP) stirrups, respectively. A new test method was proposed to test 23 specimens, including twenty CW-GFRP stirrups and three pultruded GFRP stirrups. The main test variables included the stirrup’s form, cross-sectional dimensions and the angle of critical diagonal crack. A prediction model applicable to the shear-tensile strength of CW-GFRP stirrups was proposed. The following conclusions were drawn in this paper:

(1) For pultruded GFRP stirrups, the shear-tensile strength and bend strength were 71% and 39% of the tensile strength, respectively. Due to the low bend strength of the pultruded stirrup, the ratio of the shear strength to the bend strength reached 1.78, which will make the stirrup failure tend to occur in the bent section, and the stirrup straight section strength can not be effectively utilized.

(2) For CW-GFRP stirrups, the bend strength was improved by 81% ∼ 115% and the shear-tensile strength was enhanced by 4% ∼ 19% compared with those of pultruded stirrups. The ratio of shear-tensile strength to bend strength of CW-GFRP stirrups was about 0.79 ∼ 1.06. Compared to pultruded GFRP stirrups, CW-GFRP stirrups had a smaller gap between the shear-tensile strength and bend strength, allowing better utilization of the strength of the bent and straight sections during stirrup failure.

(3) As the angle of diagonal cracks increased from 35° to 55°, the shear-tensile strength decreased by 10% ∼ 28%, and the decrease was related to the cross-sectional dimensions of CW-GFRP stirrups. The greater the width-to-thickness ratio, the more significant the decrease. Besides, the shear-tensile strength increased by 6% ∼ 28% with increasing thickness from 3 mm to 6 mm and decreased by 5% ∼ 11% with increasing width from 9 mm to 18 mm. When the cross-sectional area was certain, the shear-tensile strength decreased by 3% ∼ 24% with an increasing width-to-thickness ratio from 1.5 to 6.

(4) A relatively accurate prediction model of shear-tensile strength for CW-GFRP stirrup was proposed. For the application of CW-GFRP stirrups in beams, width-to-thickness ratios ranging from 1.5 to 2.67 and taking 60% of the tensile strength as the shear-tensile strength are recommended to better utilize the bend strength and the shear-tensile strength of CW-GFRP stirrups.

Footnotes

Author contributions

Zhenyu Wang, Yongbo Shao and Ye Yuan designed the study and made significant revisions to the manuscript; Yanbin Gong and Tianyou Wang drafted and wrote the manuscript, acquired, analyzed, or interpreted data for the study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (52278222) and project ZR2022ME007 supported by Shandong Provincial Natural Science Foundation.

ORCID iD

Zhenyu Wang

Data Availability Statement

The data will be available upon request.*

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