This paper proposes a method to modify the strut effectiveness factor in the strutandtie model for CFRPstrengthened reinforced concrete deep beams. Two groups of deep beams comprising six ordinary reinforced concrete deep beams and six CFRPstrengthened reinforced concrete deep beams were experimentally tested under the fourpoint bending configuration. The shear spantoeffective depth ratio of the beams in each group was 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00. The theoretical principal tensile strain in CFRPstrengthened struts was modified based on a proposed empirical relationship, based on two ratios: the experimental to the theoretical value of principal tensile strain and the shear spantoeffective depth of deep beams.
According to ACI 31811, a deep beam has a clear span less than or equal to four times the overall depth. Regions with concentrated loads spanning twice the member depth from the support are also considered deep beams (
The strengthening of concrete structures with carbon fiberreinforced polymer (CFRP) has become a topic of interest among researchers in the last decade because CRFP is lightweight and corrosion resistant. CFRPs are easy to install and have high tensile strength, making these materials a useful tool for strengthening concrete structures.
Numerous studies explored the effects of CFRP in three forms (sheet, plate, and bar) on the behavior of RC beams (
Several equations and models to predict concreteCFRP bond strength (
According to various codes and standards, the strutandtie model (STM) is a rational approach to analyze deep beams (
Equations [
The strut effectiveness factor can be calculated as follows from Equation [
According to AASHTO LRFD, εs is calculated as below.
The average principal tensile stress for cracked concrete in concrete struts in tension proposed in the above research is presented in Equation [
The use of CFRP sheets to strengthen concrete structural elements continues to increase worldwide. Despite the wide application of STM in structural member design (
The reinforced concrete deep beams consisted of two groups: ordinary and CFRPstrengthened deep beams. Each group comprised six deep beams with shear span to effective depth ratios of 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00. CFRP sheets are usually installed on two or three sides, or fully wrapped around the beam. Anchorage is achieved with the threeside and fullwrap systems. Twosided installation is more common in strengthening, retrofitting, and even repair because of its ease of installation and costeffectiveness compared to the other two installation systems. This study was therefore confined to twosided CFRP installation, for it aims to investigate the effect of installing a CFRP sheet without anchorage on the tensile strength of an inclined RC strut. The effect of CFRP anchorage in threesided and fullywrapped CFRP installation systems should be explored in future research.
The 140 mm × 350 mm deep beams were essentially identical; they measured 1840 mm.
long and had a rectangular cross section, as illustrated in
Crosssection of beam.
Typical reinforcement.
The beams were cast with a single supply of readymixed concrete. One layer of unidirectionally woven carbonfiber fabric with a thickness of 0.111 mm/ply was wetlaid on the deep beams with a twopart epoxy resin. The direction of the fiber in the installed CFRP sheet was vertical.
Typical properties of CFRP sheets and epoxy
Materials  Tensile strength ( 
Tensile modulus of elasticity ( 
Elongation at failure  Bond strength ( 
Thickness ( 

CFRP sheet  3900  230  1.5% (7days at + 23 °C)  –  0.111 
Epoxy resin  30  4.5  0.9% (7days at + 23 °C)  >4  – 
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 ASTME8 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 fourpoint bending configuration. The load was increased to failure with a 5000kN 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 Dregions of the beams. The DEMEC discs were then properly positioned on the beam surface using a DEMEC invar bar at 200mm 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 CFRPstrengthened 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.
Experimental test set up.
The results were carefully measured using calibrated tools since there only one test was available for each a/d ratio in this study. This section presents the relationship between the shear strength of CFRP deep beams and the shear spantoeffective depth ratio with a view to modifying the STM for CFRPstrengthened deep beams. Then, an empirical relationship was established to modify the value of the strut effectiveness factor of the CFRPstrengthened deep beams.
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 CFRPstrengthened RC deep beams than in ordinary RC deep beams. The ultimate shear strength of ordinary RC deep beams and CFRPstrengthened deep beams and their respective midspan deflection values are shown in
Empirical values for the ultimate shear strength of deep beams and respective midspan deflection
a/d  P_{uordinary} ( 
P_{uFRP} ( 
Δ _{ordinary} ( 
Δ _{CFRP}
_{strengthening} ( 

0.75  756.95  905.31  3.29  3.99 
1.00  709.01  857.89  3.40  4.13 
1.25  604.08  740.02  3.54  4.53 
1.50  555.91  691.04  3.59  4.66 
1.75  403.02  510.01  3.64  5.00 
2.00  360.02  468.05  3.74  5.17 
However, the ductility and energy absorption of ordinary and CFRPstrengthened RC deep beams should be explored in further research. The partial rupturing of the CFRP sheet was the dominant failure mode in the twosided CFRPstrengthened 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 spantoeffective depth ratios.
Ultimate shear strength of deep beams
0.75  756.95  905.31  19.60 
1.00  709.01  857.89  21.00 
1.25  604.08  740.02  22.51 
1.50  555.91  691.04  24.31 
1.75  403.02  510.01  26.55 
2.00  360.02  468.05  30.02 
While part of the IR shown in
Empirical relationship for predicting the shear strength of CFRPstrengthened deep beams.
In this study, the relationship between two significant ratios (IR and a/d) was considered in evaluating the behavior of CFRPstrengthened deep beams.
STM was developed with an empirical equation to predict the ultimate shear strength of CFRPstrengthened 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 CFRPstrengthened struts was the factor recommended by AASHTO LRFD over the ACI 31811 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 CFRPstrengthened 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 CFRPstrengthened deep beams.
The behavior of the CFRPstrengthened Dregion was evaluated based on the principal tensile strain for bottleshaped struts in STM. Based on Equation [5], the ε_{1} for ordinary concrete struts was calculated from Equation [
Equation [
Therefore, the principal tensile strain in CFRPstrengthened concrete struts in which the contribution of CFRP bonding stress is taken into consideration is proposed in Equation [
The contribution of CFRP bonding stress in Equation [
Principal tensile strain in CFRPstrengthened concrete struts, empirical relationship.
Calculation of the modification ratio based on ε_{1−FRP} and ε_{1−FRP−test}

ε_{1−FRP} 
ε_{1−FRPtest} 


0.75  4.85  25.87  5.33 
1.00  6.50  35.25  5.41 
1.25  8.27  64.29  7.77 
1.50  10.07  84.33  8.37 
1.75  11.86  142.97  12.05 
2.00  13.60  169.82  12.48 
Using the a/d ratio, the modification ratio (R) was calculated from Equation [
Ultimate shear strength of CFRPstrengthened deep beams,recommended method and empirical findings

ε_{1−FRPrecommended
}





0.75  22.09  944.72  905.31  1.04 
1.00  40.08  801.66  857.89  0.93 
1.25  64.28  732.18  740.02  0.99 
1.50  94.47  634.65  691.04  0.92 
1.75  130.28  540.66  510.01  1.06 
2.00  171.26  459.95  468.05  0.98 
This study investigated the application of the strutandtie model for CFRPstrengthened deep beams. It sought to establish an empirical relationship to modify the calculated value of the principal tensile strain on CFRPstrengthened struts. The following conclusions may be drawn.
An empirical relationship was established to modify the value of the strut effectiveness factor for CFRPstrengthened struts and to predict the value of principal tensile strain in struts for CFRPstrengthened deep beams.
The modified STM, which utilized the proposed empirical relationship, can be employed to predict the shear strength of CFRPstrengthened deep beams.
The experimental results showed that CFRPstrengthening increases the ultimate shear strength of deep beams from 19.60 to 30.02 with shear spantoeffective depth ratios of 0.75 to 2.00, respectively.
The partial rupturing of the CFRP sheet is the dominant failure mode in twosided CFRPstrengthened deep beams.
The authors acknowledge the support from the Housing Research Center (HRC) and Dura Technology Company for facilitating the experimental work. Author a is deeply indebted to Taw Ly Wen, the English Instructor from University Putra Malaysia for her comments.
E_{s}: Young's modulus for steel bars (
υ: strut effectiveness factor
τ: average bond strength of concreteCFRP (
α, β: reduction factors
Δ _{ordinary}: midspan deflection of ordinary RC deep beams (
Δ _{CFRP}
_{strengthening}: midspan deflection of CFRPstrengthened RC deep beams (