The steel molds were designed for casting four hybrid deep beams for each batch as shown in Fig. (2). The inside dimensions for each mold were 1500mm length,350mm height and 150mm width. The molds were designed to cast the beams vertically due to the difficulty of casting layers in a horizontal state. The front cover of the mold face of dimensions 1500mm×350mm consists of three plates. The lower plate was fixed to cast the first layer while the two other plates above were movable (doors) to cast the two other layers. Each door was closed before casting the layer of beam in it.

Fig. (2). Steel molds used for casting hybrid deep beams.

In order to test the beam, the faces of the samples were coated in white color for observing cracks easily. The support and load point locations were positioned and thereafter, the points were mounted on the beams. The beams were categorized and the site of support point, loading point, and the digital gage places were noticeable on the beams to ease the accurate setup of testing machine. Thereafter, the beam was elevated by electrical crane and depressed on to supports. Beam specimens were located at the testing machine and checked so that the centerline supports, load arms and dial gages were fixed at their right and suitable positions. Loading process commences by application of two-point load from the testing machine to the higher face of beam.

All beam specimens have been loaded until failure for monotonic test and five cycles in repeated loading test. The beam samples have been loaded in increases of 10kN, the rate of load increase was around 1.5kN/sec.

The locations and expansion of cracks for each cycle were noticeable on the surface of the beam. The failure happened, while the beam failed abruptly at simultaneity with the load index stopped in record or reoccurrence back and the deflection increased very quickly. The ultimate load has been noted, and the load has been removed to permit taking some photographs of the crack pattern and the mode of failure.

From the observation of Table 3, the following notes can be noticed:

1. The decreased percentages in the ultimate load according to the repeated loading for non-hybrid deep beams of HSC and FHSC are convergent which are 22.95% and 20.98%, respectively.
2. The decreased percentages in the ultimate load according to the repeated loading of hybrid deep beams which have web reinforcement (0.004 and 0.006) as variable are convergent which are 23.97% and 22.22%, respectively.
3. The decreased percentages in the ultimate load according to the repeated loading of hybrid deep beams with compressive strength concrete (high and normal) as variable are convergent which are 23.97% and24.32%, respectively.
4. The average value of the decreased percentages of beams subjected to monotonic and 75% of the repeated loading is 22.88%.

Three arrangements of web reinforcement were considered (0,0.004,0.006). The ratio 0.004 represents minimum web reinforcement (minimum horizontal 0.0025 and minimum vertical 0.0025).

4.4.1. under Monotonic Loading

Effect of amount of web reinforcement is calculated under the constant ratio of SF and a/h ratio for two types of loading monotonic and repeated of 75% control ultimate load. The results of the variable ρ_w are shown in Fig. (8) and Table 8.

Fig. (8). Effect of ρw on ultimate load of beams under monotonic loading.

Table 8. Effect of ρ_w on ultimate load of beams under monotonic loading.

Beam Designation	Type of Beam	SF (%)	ρw	Ultimate Load (Pu) (kN)	% Increase Ultimate Load *
B-HS-M-Hy1-ρw 0	Hybrid	1	0.0	570	-
B-HS-M-Hy1-ρw4	Hybrid	1	0.004	730	28.07
B- HS-M-Hy1-ρw 6	Hybrid	1	0.006	900	**57.89

* The percentage of increase is measured with respect to beam B-HS-M-Hy1%-ρ_w =0.

4.4.2. Under Repeated Loading

The effect of an ultimate load for hybrid deep beams under repeated loading of level 75% of control monotonic loading is shown in Fig. (9) and Table 9.

Fig. (9). Effect of ρw on ultimate load of beams under repeated loading.

Table 9. Effect of ρ_w on ultimate load of beams under repeated loading.

Beam Designation	Type of Beam	SF (%)	ρw	Ultimate Load (Pu) (kN)	% Increase Ultimate Load *
B-HS-R-Hy1-ρw 0	Hybrid	1	0.0	440	-
B-HS-R-Hy1-ρw 4	Hybrid	1	0.004	555	26.14
B-HS-R-Hy1-ρw 6	Hybrid	1	0.006	700	**59.09

*The percentage of increase is measured with respect to beam B-HS-R-Hy1%-ρ_w =0. ** the increased percentages of load of the beam B-HS-R-Hy1-ρ_w 6 is 26.13% with respect B-HS-R-Hy1-ρ_w 4.

It can be seen that there is an increase in the value of ultimate load value of 26.14% as p_w increases from (0 to 0.004) while there is an increase in ultimate load value of 59.1% as the p_w increased from (0 to 0.006). It can be concluded that presence of web reinforcement contributes to enhance shear capacities of hybrid deep beams with significant effected under monotonic and repeated loading. Also, the increase in web reinforcement ratio from 0.004 to 0.006 increases the capacity by 26.13%.

1. Remodeling buildings often imply a modification of the mechanical behavior of the structural system, which may include the application of concentrated loads to beams that before have only been subjected to distributed loads. In the case of reinforced concrete beams, the new condition causes the beam to support a concentrated load in the cracked condition determined by the distributed loads that had been acting in the past. If the concentrated load is applied at or near the mid-span of the beam, consequently, the shear demand reaches its maximum where the shear capacity is low. At and near the mid-span, in fact, the cracks in the tension zone are vertical like the stirrups. Therefore, the truss mechanism provides the shear capacity with a nil contribution, since both the struts and ties are vertical. Fiber-reinforced polymer composites allow these beams to increase their shear strength, up to guaranteeing adequate safety. A method for analyzing the mechanical behavior of beams with vertical cracks, strengthened with FRP reinforcement, subjected to concentrated loads. The method defines the direction of the fibers that the reinforcement has to be composed of and computes the shear strength of the beam [4].

2. A multiscale analytical model that predicts the lifetime of concrete members with externally bonded FRP reinforcement. The lifetime is dictated by the de-bonding of external reinforcement, which takes place within the concrete cover, where micro-cracks (initial flaws) propagate due to the shear stresses that the bond subjects the concrete cover too. The lifetime is estimated from the propagation of such cracks until a critical crack length is eventually reached, which causes the external FRP reinforcement to lose the bond (delayed de-bonding). The model provides a closed form-solution for the life-through estimation of the external reinforcement, which consists of the interaction between bond shear stresses and lifetime (maximum bond shear stress versus delayed time, i.e. the ultimate domain). Crack growth is modeled at the mesoscale, where the velocity of the cracks depends on the model stress intensity factor, but not on the microstructure. The model assumes that the carbonation process has reduced the plasticity and cohesion of the concrete cover to zero; thus, the predictions are slightly conservative (lower bound model). Some experimental results on real scale beams are presented to corroborate the theoretical findings. A practical application of the model shows that delayed de-bonding significantly reduces the service life of concrete members with externally bonded FRP reinforcement [5].

3. The investigation relates to flexural concrete members strengthened by means of external reinforcement bonded adhesively onto the surface, in particular, by fiber-reinforced polymeric strips, sheets, or laminates. Investigation specifically devoted to external reinforcement being already in tension under a dead load or a low fraction of live load. The concrete cover exhibits initial flaws, which were unimportant when a member was not externally reinforced, but that may give rise to slow crack propagation up to delayed de-bonding when the member is externally reinforced. This paper presents a model for predicting the delayed de-bonding. The common de-bonding models, including code provisions, focus mainly on the structural and material scales, and thus ignore delayed failure. On the contrary, this new model focusses on the mesoscale, which considers the velocity of crack growth that leads to de-bonding. While on the nanoscale and microscale, the crack velocity depends on the microstructure. On the mesoscale, the crack velocity depends on the interfacial bond shear stresses and crack length. This dependence can be synthetized by the ratio between the two models. The model describes the delayed de-bonding in terms of interaction between bond shear stress and time (ultimate domain maximum bond shear stress versus delayed time) [6].

STM in deep beams is represented by a structural truss. Each type of the elements in an STM serves a unique purpose but must act in conformity to describe accurately the behavior of a structure.

STM consists of the following members and parts:

In the present work, the ultimate load for eight simply supported deep beams which were tested under monotonic loading is calculated according to ACI 318 R-14 Code [1]. The comparison between test results and the expected values of the ultimate load is shown in Table 10. From Table 11, 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') for the ratio of analytical/test results of ultimate loads (P_An/P_Exp) is 0.94 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 0.14 and the coefficient of variation (CV) is 0.15.

Zhang and Tan in March (2007) [8], submitted a modified STM for calculation of shear strength of reinforced concrete deep beams based on a previous investigation reported by Tan and Cheng [9]. From the structural analysis for simply supported reinforced concrete beams dependent to symmetric two point loads, it is well known that the ultimate load (P) is equal to twice the shear force at the support as per Eq. (1).

(1)

The expression for calculation shear strength V_n according to Zhang and Tan [5], is as follows in Eq. (2)

(2)

where;

V_n: shear strength of deep beams (N).

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

d_c: effective beam depth (mm).

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

W_s: effective width of the inclined strut (mm).

b_w: width of deep beam (mm).

f_t: combined tensile strength of reinforcement and concrete (MPa).

0_s: angle between the axis of the strut and the horizontal axis of the member.

It can be noted that the expression is the composite tensile strength included contributions from concrete and reinforcement (web and main bars), in Eq. (3)

(3)

where; f_ct: represents the contribution of concrete tensile strength. f_st : represents the contribution of steel reinforcement which consists of two parts, f_sw from the web reinforcement and f_ss from the longitudinal reinforcement as explain in Eq (4).

(4)

Zhang and Tan submitted that the presence of web reinforcement in the strut restricts the inclined cracks from readily increase to every end of the strut. Eq. (5) shows the tensile contribution of web reinforcement at the interface of the nodal zone.

(5)

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

(6)

Where:

A_sv: total areas of vertical web reinforcement within the shear span (mm²).

A_sh: total areas of horizontal web reinforcement within the shear span (mm²).

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

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

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

θ_w: angle between the web reinforcement and the horizontal axis of beams at the intersection 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).

Table 11 summarized the strength of the deep beams of the present investigation.

From the results, it can be seen overrates ultimate loads compared to test results expect B- HS-M-Hy 1- ρ_w 6 which was cast with ρ_w more than the minimum value (0.006) gives ultimate load in experimental higher than modified. The expression gives capacities which are convergent to experimental values for non-hybrid (FHSC) deep beam B-HS-M-FHSC 1 - ρ_w 4 with SF ratio of 1%. The X' for P_An/P_Exp ratio is 1.24, S.D. is 0.33 and the C.O.V. is 0.27. These results are shown in Fig. (13).

To investigate the size effect on shear strength of reinforced concrete deep beams, Zhang and Tan in November (2007) [10], carried out an experimental program consisting of three groups of 11 specimens, they noticed that increasing deep beams depth led to decrease in shear strength. They stated that the causes of size effect in deep beams need 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. (8) for ultimate shear strength, taking into account the size effect.

(8)

The term v refers to the efficiency factor accounts for the effect of strut geometry, and the effect of strut boundary conditions influenced by web reinforcement. The term ν is expressed as follows in Eq. (9)

(9)

where;

ξ: efficiency factor for the effect of strut geometry.

ζ: efficiency factor for the effect of strut boundary conditions influenced by web reinforcement. These parameters are expressed as follows in Eq. (10 and 11)

(10)

(11)

Where,

l: length of strut in mm, as shown in Fig. (14).

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

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

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

Table 12 summarized the strength of some deep beams of the present investigation which were tested under monotonic loading.

The modified STM overrate ultimate loads as compared to test results expect B- HS-M-Hy 1- ρ_w 6 gives ultimate load in test result higher than modified STM. The expression gives capacity which is convergent value for non-hybrid (FHSC)deep beam B-HS-M-FHSC 1- ρ_w 4 only. The X' for P_An/P_Exp ratio is 1.29, SD is 0.35 and the CV is 0.27. These results are shown in Fig. (14).

1. They assumed that for beams in which bearing and anchorage failures at the nodes are prevented, shear capacity is governed by the compressive capacity of the strut and the mean stress in the strut depends on its dimensions. The strut is assumed to have a uniform width (prismatic strut) defined by geometry of the node at the support.
2. A simple truss model and the resulting failure are resultant of crushing failure of concrete in diagonal strut and tensile failure by splitting of concrete along diagonal strut and yielding of reinforcement.
3. The arch action after the formation of diagonal cracks where it adversely affects the shear strength as the depth becomes larger.

Fig. (14). Comparison between ultimate loads obtained using different experimental and analytical methods.