The 140 mm × 350 mm deep beams were essentially identical; they measured 1840 mm.

long and had a rectangular cross section, as illustrated in Figure 1. The flexural reinforcement consisted of nine 16-mm diameter deformed steel bars placed in three layers. The steel bars were welded to 10-mm thick steel plates at both ends of the beams to provide adequate anchorage. The 120-mm high anchorage steel plates fully covered the width of the beams. The longitudinal bars should be anchored as in standard practice using development length or embedded length. However, the end steel plates were utilized in the experiment to provide additional anchorage because of laboratory size limitations. Steel mesh reinforcement with a mesh size of 100 mm was provided as transverse reinforcement. The mesh satisfied the required minimum amount of web reinforcement and the recommended maximum spacing of orthogonal grid reinforcement (300 mm) specified by ACI 318-11 and AASHTO LRFD (30, 1). To prevent premature local failure, additional steel reinforcements were provided under the load plates and on top of the support plates as illustrated in Figure 2. The orthogonal grid reinforcement, used along the length of the beams, is shown only on one side of the beam in Figure 2.

The beams were cast with a single supply of ready-mixed concrete. One layer of uni-directionally woven carbon-fiber fabric with a thickness of 0.111 mm/ply was wet-laid on the deep beams with a two-part epoxy resin. The direction of the fiber in the installed CFRP sheet was vertical. Table 1 lists the typical properties of the CFRP sheet and epoxy resin provided by the manufacturer. Both the CFRP sheet and epoxy resin were supplied by Sika Company with Sikadur-330 and Sikawrap-230 product data sheets. Strengthening with the CFRP sheet was performed only on the surface of the beams between the load and support plate to cover the shear span of the deep beams. The CFRP-strengthening was cured for at least two days at ambient temperature following the manufacturer's recommendation. The support and load plates, which are 70 mm wide and 10 mm thick, fully covered the bottom and top of the beam. The deep beams were tested 28 d after casting.

A universal tensile strength testing machine was used to measure the tensile strength of steel bars. Three samples were chosen from each size of steel bars and the average taken as the final tensile strength. The test was carried out according to standard ASTM-E8 with a strain rate of 0.005 in/in/min to measure the ultimate tensile strength of the bars. The tensile strength of the reinforcing steel bars determined as described (T16) was 440 MPa, while the compressive and splitting tensile strengths of concrete were 37.02 and 3.31 MPa, respectively.

The beams were tested to failure with a four-point bending configuration. The load was increased to failure with a 5000-kN hydraulic actuator. The load increment was 25 kN during the loading process. The positions of the DEMEC discs were carefully drawn on the surface of the D-regions of the beams. The DEMEC discs were then properly positioned on the beam surface using a DEMEC invar bar at 200-mm intervals. DEMEC disc spacing was accurately measured in each step of loading since the DEMEC resolution was 0.001 mm. As the ultimate shear strength of CFRP-strengthened RC deep beams was not predictable, the DEMEC measurement process for all beams was continued following the load increment steps of 25 kN till the beams failed. Figure 3 shows the experimental test set-up with the DEMEC discs positioned along and perpendicular to the strut centerline. The strain perpendicular to the strut centerline at mid-height of the beam section was the focus of calculation in this study.

According to the experimental observation, the tendency of having brittle failure perceptibly increases among the ordinary RC deep beams as the shear span to the effective depth ratio decreases. Nonetheless, the foregoing tendency was observed to be weaker in CFRP-strengthened RC deep beams than in ordinary RC deep beams. The ultimate shear strength of ordinary RC deep beams and CFRP-strengthened deep beams and their respective mid-span deflection values are shown in Table 2. According to Table 2, the mid-span deflection of ordinary and CFRP-strengthened RC deep beams corresponding to the ultimate load slightly increased with the shear span-to-effective depth ratio.

However, the ductility and energy absorption of ordinary and CFRP-strengthened RC deep beams should be explored in further research. The partial rupturing of the CFRP sheet was the dominant failure mode in the two-sided CFRP-strengthened deep beams in this experiment. In other words, only part of the beam section failed without CFRP sheet rupture, while the failure of the remaining beam section occurred simultaneously and involved rupture of the CFRP sheet.

Among the studies conducted on deep beams hitherto, no attention has been paid to CFRP strengthening of deep beams with various shear span-to-effective depth ratios. Table 3 shows the ultimate shear strength of ordinary and CFRP-strengthened deep beams based on the experimental results of this study. The last column in Table 3 shows the increase in ultimate shear strength of deep beams bearing CFRP sheets. Two crucial trends were observed from the experimental results. First, as the shear span-to-effective depth ratio (a/d) increased, the ultimate shear strength of the ordinary and CFRP-strengthened deep beams decreases. This trend is consistent with the findings of previous studies (2, 47). Second, the shear strength of the CFRP-strengthened deep beams increases faster than that of ordinary deep beams with the increase in a/d.

While part of the IR shown in Figure 4 and Table 3 may stem from the increasing efficiency of FRP with increasing a/d, the reduced strength of the RC beams with increasing a/d was another reason for the rise in IR. As shown in Table 3, the strength of non-strengthened RC beams declined significantly with increasing a/d, prompting an increase in IR values after strengthening. Therefore, part of the increase in IR was due to the increased efficiency provided by the FRP sheets.

In this study, the relationship between two significant ratios (IR and a/d) was considered in evaluating the behavior of CFRP-strengthened deep beams. Figure 4 illustrates the empirical relationship between the two ratios. This relationship served as the basis for STM modification to predict the shear strength of CFRP-strengthened deep beams. This issue is discussed in a separate section in this paper.

STM was developed with an empirical equation to predict the ultimate shear strength of CFRP-strengthened deep beams in terms of the main CFRP properties such as thickness and modulus of elasticity. In this research, the strut effectiveness factor chosen for modification to accommodate CFRP-strengthened struts was the factor recommended by AASHTO LRFD over the ACI 318-11 proposal. This selection was considered because the strut effectiveness factor recommended by AASHTO LRFD is calculated from the value of the principal tensile strain on the strut, which is measurable for the CFRP-strengthened struts used in the experiment. The principal tensile strain on struts was also measured to provide experimental support for verification, aside from the development of the strut effectiveness equation recommended by AASHTO LRFD for CFRP-strengthened deep beams.

The behavior of the CFRP-strengthened D-region was evaluated based on the principal tensile strain for bottle-shaped struts in STM. Based on Equation [5], the ε₁ for ordinary concrete struts was calculated from Equation [6] below. ε 1 = 0.005 ( f c r f c 1 - 1 ) 2

Equation [6] was derived from Equation [5] and developed for CFRP-strengthened concrete struts based on the contribution of the CFRP sheet. The compressive strength of concrete struts declined due to of concrete softening in response to the strut effectiveness factor in STM. The diagonal cracks along the strut centerline widened with the increase in the applied load. CFRP sheets prevented the diagonal cracks from widening compared to those diagonal cracks without CFRP-strengthening. This was because of the bond stress transferred to the concrete - CFRP interface.

Therefore, the principal tensile strain in CFRP-strengthened concrete struts in which the contribution of CFRP bonding stress is taken into consideration is proposed in Equation [7]. ε 1 - F R P = 0.005 ( f c r + α τ f c 1 + β τ - 1 ) 2

The contribution of CFRP bonding stress in Equation [7] varied for different shear span-to-effective depth ratios because of the non-linear behaviour of the D-region. This is the reason why reduction coefficients α and β were used in Equation [7]. In order to calibrate Equation [7] with the experimental results, the values of reduction coefficients α and β were assumed to be equal to 1.00. Consequently, an empirical relationship was established in which reduction coefficients were used to attain the value of ε_1−FRP as shown in Figure 5. This relationship was established as the ratio of ε_1−FRP to ε_{1−FRP−test} as indicated in Table 4. The principal tensile strain on both ordinary and CFRP-strengthened struts was measured with the DEMEC discs at the mid-height of the strut perpendicular to its centerline. ε_{1−ordinary–AASHTO} and utilized as the basis of the calculation to attain ε_1−FRP. Based on the following empirical equation, ε_{1−FRPrecommended} was obtained with the shear span-to-effective depth ratio and the R value. R = 6.4335 ( a d ) - 0.2765 Therefore, ε 1 - F R P r e c o m m e n d e d = R × ε 1 - F R P .

Using the a/d ratio, the modification ratio (R) was calculated from Equation [8]. The value of ε_1−FRP was calculated from Equation [7] since α and β were equal to 1.00. Subsequently, ε_{1−FRPrecommended} was calculated from Equation [9]. The strut effectiveness factor for CFRP-strengthened RC deep beams was obtained from the value of ε_{1−FRPrecommended}.

Table 5 provides a summary of the calculations and the recommended method and experimental ultimate shear strengths of CFRP-strengthened deep beams. Empirical Equation [9] gives the value of the principal tensile strain on CFRP-strengthened concrete struts.