Results in Table 5 show the effect of SF ratio on ultimate loads for deep beams tested under monotonic loading.

The hybrid beam with SF ratio (2%) failed with crushing compression zone which differs from the other two beams with (B-HS-M-HSC 0-ρ_w 0 and B-HS-M-Hy1-ρ_w 0) with SF ratio (0%and 1%) that are failed compression strut as shown previously in Fig. (3c) because of the presence of steel fiber in the shear zone with 2%ratio, which led to the strengthening of the zone and prevent the emergence of cracks and therefore failure occurred in the shear zone, however, in the bending zone steel fiber is not added. This explains the role of steel fiber in increasing tensile strength and prevent or reduce the appearance of cracks.

The ultimate load for hybrid beam with SF ratio of 2% (B-HS-M-Hy2-ρ_w 0) and SF ratio of 1% (B-HS-M-Hy1-ρ_w0) increased by 28.85% and 9.62%, respectively with respect to which beam B-HS-M-HSC 0-ρ_w0 is cast without SF. The presence of SF in shear span regions for hybrid deep beams contributes in fostering resistance of hybrid deep beams in the absence of web reinforcement. Also, the percentage increase in ultimate load is 17.54% as the SF ratio increased from 1% (B-HS-M-Hy1-ρ_w 0) to 2% (B-HS-M-Hy2-ρ_w 0).

Table 6 shows the effect of SF ratio on ultimate load of hybrid deep beams tested under repeated loading level of 75% of their corresponding control monotonic loading.

From the observation of results in Table 6 it was found that the extension of SF in hybrid beams under repeated loading is significant. The capacity of hybrid deep beams with SF ratio of 2% and 1% are increased by 26.19% and 4.76%, respectively with respect to beam B-HS-R-Hy1-ρ_w 0. Also, the percentage increase in ultimate load is 20.45% as the SF ratio increases from 1%( B-HS-R-Hy1-ρ_w 0) to 2% (B-HS-R-Hy2-ρ_w 0). From Tables 5 and 6 it can be concluded that addition of SF with a ratio of 2% is significant as compared to 1% for both types of loading (monotonic and repeated) in the absence of web reinforcement.

Fig. (4). Effect of SF ratio on ultimate load of beams under monotonic and repeated loading.

Fig. (4) shows the effect of SF ratio on ultimate load of hybrid deep beams under monotonic and repeated loading of 75% of control beam failure monotonic loadings.

Fig. (6) shows the effect of the extension of SF for hybrid deep beams on load-deflection response under monotonic loading. Three volumetric ratios of SF were used. Hybrid beam with SF ratio of 2% have the smaller deflection values at each stage of loading as compared to other beams that have SF ratios of 1% and 0%.

In the present work, ultimate load for three simply supported deep beams which were tested under monotonic loading are calculated according to ACI 318 R-14 Code [1]. From the results it can be noticed that the STM mentioned in ACI 318R-14 Code underestimates the load capacity of deep beams for some beams and overestimates capacities for the other beams. The mean value (X') of the analytical/test ratio results of ultimate loads (P_An/P_Exp) is 1.25 where P_An refers to ultimate loads obtained using analytical methods and P_Exp refers to ultimate loads obtained from experimental test, the Standard Deviation (SD) is 0.07 and the Coefficient of Variation (CV) is 0.073.

Zhang and Tan in March (2007) [5], submitted a modified STM. Based on a preceding investigation reported by Tan and Cheng [6] can be calculated as shear strength of reinforced concrete deep beams. In this investigation, the shear capacity of 233 simply supported deep beams has been estimated using the modified STM in order to test its adequacy. For these deep beams, concrete strength varied between 16 to 120 MPa, which involved high and normal strength concrete. The deep beams had a total depth h reaching from 200mm to 1750mm and the a/d ratio reaching from 0.28 to 2.0. The ratio of longitudinal reinforcement reached from 0.90% to 4.07%, and web reinforcement ratios (vertical and horizontal) reached from 0.0% to 2.86% and from 0.0% to 3.17%, respectively.

For simply supported reinforced concrete beams which were analyzed constructively dependent on symmetric two point loads, it is well known that the ultimate load (P) is equal to twice the shear force at the support (Eqs. 1 and 2).

(1)

The expression for calculation shear strength V_n according to Zhang and Tan [5], is as follows:

(2)

where;

V_n: deep beams shear strength (N).

A_c: is the beam active cross sectional area (mm²), equals to b_w d_c.

d_c: effective beam depth (mm).

A_str: the concrete diagonal strut cross sectional area (mm²), equal to w_s b_w.

w_s: active width of the inclined strut (mm).

b_w: width of deep beam (mm).

f_e: collective tensile strength of concrete and reinforcement (MPa).

θ_s: angle amid the axis of the strut and the straight axis of the member.

It can be noted that the expression is the composite resistance of tension and included assistances from concrete and reinforcement (main with the web bars), where;

(3)

f_ct : symbolizes the contribution of concrete tensile strength. (Eq. 3)

f_st: symbolizes the contribution of steel reinforcement which contains of 2 parts, fsw from the web reinforcement and fss from the longitudinal reinforcement as explain in equation Eq. 4.

(4)

Zhang and Tan submitted that the existence of web reinforcement in the strut restricts the inclined cracks of readily increase into every the strut end. Eq. 5 shows the tensile contribution of web reinforcement at the boundary of the nodal zone.

(5)

For conformity horizontal web reinforcement and cases of vertical, Eq. 6 is reduced to:

(6)

Where:

A_sv: overall areas of vertical web reinforcement in the shear span (mm²).

A_sh: overall areas of horizontal web reinforcement in the shear span (mm²).

f_yv: tensile yield strength of vertical web reinforcement (MPa).

f_yh: tensile yield strength of horizontal web reinforcement (MPa).

θ_s: angle amid the horizontal axis of the member and the axis of the strut.

θ_w: angle amid the horizontal axis of beams and the web reinforcement at the crossing of the reinforcement and the diagonal strut.

The expression f_ss refers to the contribution of bottom longitudinal steel, it can be obtained according to the following Eq. 7:

(7)

Where:

A_s total areas of bottom longitudinal main reinforcement (mm²).

f_y: tensile yield strength of main reinforcement (MPa).

According to Zhang and Tan in November 2007 [7] the size influence on shear strength of reinforced concrete deep beams can be investigated, carried out an experimental program containing of 3 collections of 11 samples, they noticed that an increasing of deep beams depth led to decreases in shear strength. The depth effect was not included in their previous expression for calculating shear strength Article (5.1.2). They stated that the causes of size effect in deep beams need then development, so they submitted that the size effect is influenced by strut geometry and boundary conditions.

Zhang and Tan submitted the following modification to Eq. (2) calculate the size effect for ultimate shear strength.

(8)

The term v refers to the efficiency factor calculates for the influence of strut boundary conditions and strut geometry, influence by web reinforcement (Eqs. 8 and 9). The term ν is expressed as follows:

(9)

where;

ζ: efficiency factor for the influence of strut geometry.

ξ :efficiency factor for the influence of strut boundary conditions effected by web reinforcement (Eqs 10 and 11). These variables are expressed as follows:

(10)

(11)

Where,

l: length of strut in mm, as shown in Fig. (8).

Fig. (8). Strut geometry and strut boundary conditions.

d_s: web steel bar diameter, when web steel is not providing, d_s is taken as the smallest diameter of bottom longitudinal steel bars.

l_s: maximum spacing of web steel stopped by the inclined strut, when not provided web steel, l_s is equal to l.

is a material factor, when not providing web steel, it is taken as half of the overhead value.

These results for three methods are shown in Table 7 and Fig. (9).

Fig. (9). Ultimate loads obtained using experimental and different analytical methods.