A **deep** beam is a structural member whose behavior is dominated by shear deformations. In practice, engineers typically encounter **deep** **beams** when designing transfer girders, pile supported foundations, or bridge bents. Until recently, the **design** of **deep** **beams** per U.S. **design** standards was based on empirically derived expressions and rules of thumb. The structural **design** standards, AASHTO LRFD (2008) and ACI 318-08, adopted the use of **strut**-and- **tie** **modeling** (STM) for the **design** of **deep** **beams** or other regions of discontinuity in 1994 and 2002, respectively. Based on the theory of plasticity, STM is a **design** method that idealizes stress fields as axial members of a truss. The primary advantage of STM is its versatility. In other words, it is valid for any given loading or geometry. However, the primary weakness of STM is also its versatility. The freedom associated with the method results in a vague and inconsistently defined set of guidelines. Because of the lack of a well-ordered **design** process, many practitioners are reluctant to use STM. A goal of the current research program is overcome this ambiguity through the development of consistent and safe STM **provisions**.

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The **strut** efficiency factor ( 𝛽 𝑠 ) is an important for the strength of **concrete** for the analysis and **design** of **reinforced** **deep** **beams** based on the **strut** and **tie** model. Because of ACI 318M-14 code uses constant values for **strut** efficiency factor 𝛽 𝑠 , the proposed empirical formulas used to evaluate the **strut** efficiency factor 𝐵 𝑠 will be based on the effect of manyparameters ( 𝑓 𝑐 ′ ), the shear span to effective depth ratio of **beams** ( 𝑎 𝑣 /𝑑 ), longitudinal reinforcement percentage ( 𝜌 𝑠 ), horizontal reinforcement percentage ( 𝜌 ℎ ), vertical reinforcement percentage ( 𝜌 𝑣 ), yield strength of reinforcement ( 𝑓 𝑦 ), and effective depth ( 𝑑). A 121 **reinforced** **deep** **beams** from the literature are used in this study to predict the proposed equation that have minimize the mean absolute error (MAE), root mean square error (RMSE) and maximize the coefficient of multiple determinations (R

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There are many methods for **modeling** the behavior of **concrete** structures through analytical and numerical approaches with three dimensional non-linear models. Shear strength model and **design** formula of **reinforced** **concrete** **deep** **beams** has been weel determined by the effect of **strut** and **tie** models [7]. Estimation of the localized compressive failure zone of **concrete** by AE method is well determined for modelling of RC failure zone behaviour [8]. Shear strength of **reinforced** **concrete** **beams** under uniformly distributed loads in accordance with the strength **design** method is well determined with the approach method of reinforcement detailing [9-11].

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The effectiveness of the **Strut** and **Tie** Model of **reinforced** **concrete** **deep** **beams** as provided in the **design** codes from Canada (CSA A23.3-04 [2004]), USA (ACI 318-08 [2008]) and Europe (EN 1992-1-1-2004E [2004]) has been evaluated based on the experimental results of 397 test samples compiled from the literature. The influence of certain variables on the codes’ ability to predict the ultimate strength of **deep** **beams** is also studied. The investigation confirms that the **Strut** and **Tie** model is in general an appropriate method for the **design** and evaluation of **beams** with shear span-to-depth ratio less than or equal to two. It has been found that the code **provisions** are more accurate for **beams** with web reinforcement. The CSA code **provisions** appear to be very robust in estimating the capacity of **deep** **beams**, as compared to the other two codes. However, the **provisions** of all the selected codes do not have the ability to predict the failure mode and location accurately and reliably. The STM procedure in ACI code in bottle shaped **strut** is found to be more suitable than the uniform **strut** in predicting the ultimate capacity.

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The maximum performance indexes obtained for Cases (a) to (d) are 1.88, 1.3, 1.23, and 1.21, respectively. The optimal topol- ogy and corresponding **strut**-and-**tie** idealization for each case are presented in Fig. 11. It can be observed from Fig. 11 that the truss model that ideally represents the load transfer mechanism is changed from **deep** **beams** to slender **beams**. For **beams** with a span-depth ratio L/D ≥ 3, inclined tensile ties connecting the compressive **concrete** struts are necessary to form the truss mod- el, as shown in Fig. 11(b) to (d). For very slender **concrete** **beams**, optimal topologies obtained by the continuum topology optimization method are continuum-like structures, in which **strut**-and-**tie** actions are difficult to be identified, such as that shown in Fig. 11(d). For such cases, the flexural beam theory may be applied. These optimal **strut**-and-**tie** models indicate that the angles between compressive **concrete** struts and longitudinal ties are equal to or larger than 45 degrees. In detail **design**, some of the bottom steel bars may be bent up to resist the inclined ten- sile stresses or the shear in the shear spans.

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The topology optimization of **strut**-and-**tie** models in non-flexural **reinforced** **concrete** members using the Evolutionary Structural Optimization (ESO) procedure has been presented in this paper. The basic features of the ESO approach have been described in terms of the sensitivity numbers that identify inefficient materials and the performance index, which monitors the optimization process and measures the material efficiency. It is shown that the proposed procedure can effectively generate optimal topologies of **strut**-and-**tie** models in non-flexural **reinforced** **concrete** members such as **deep** **beams** and corbels. By means of systematically removing inefficient materials from the **concrete** member, the **strut**-and-**tie** model within the member is gradually evolved towards an optimum. The results obtained by the ESO method are supported by analytical solutions and experimental observations. The proposed method is useful for automatically tracing the actual load paths in non-flexural **reinforced** **concrete** members with complex geometry and loading conditions and is a valuable tool for structural designers in selecting the best **strut**-and-**tie** models in the **design** and detailing of **reinforced** **concrete** structures.

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behave differently from shallow **beams** and generally their ultimate capacity is controlled by shear strength. The conventional **design** formulas not be useable for this type of RC **beams**. Some semi rational methods such as **Strut**-and-**tie** method have proposed to analysis and **design** of **deep** **beams**. **Strut**- and-**tie** **modeling** is the most rational and simple method for designing nonflexural members currently available. Specific **strut**- and-**tie** models need to be developed, whereas shallow **beams** are characterized by linear strain distribution and most of the applied load is transferred through a fairly uniform diagonal compression field. **Design** of nonflexural members using **strut**-and-**tie** **modeling** incorporates lower bound theory of plasticity assuming that both the **concrete** and the steel are perfectly plastic. The behavior and dimensional properties of steel are well known and the strength of members failing in tension can be predicted with some degree of certainty. The foundation of the method was laid by Ritter in 1899. Ritter’s original goal was to explain that stirrups in **reinforced** **concrete** members provided more than dowel action in resisting shear. Mörsch (1909) expanded on Ritter’s model by proposing that the diagonal compressive stresses in the **concrete** need not be discrete zones, but could be a continuous field. Foster, S.J et al

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The **strut**-and-**tie** method (STM) is a simple and conservative method for designing **concrete** structures, especially **deep** **beams**. This method expresses complicated stress patterns as a simple truss or kinematic model made up of compression elements (struts), tension elements (ties), and the joints between elements (nodes). STM is based on lower- bound plasticity theorem, so using it properly will lead to a conservative **design**. Although the concepts of STM have been around in **concrete** **design** since the late 19 th century, STM was first introduced in AASHTO LRFD in 1994 and ACI 318-02 in 2002. ACI 318 defines two different types of struts (prismatic and bottle-shaped) based on whether compression stress can spread transversely along the length of the **strut**. Recent work has brought into question whether these two types of struts do exist and whether current **design** **provisions** conservatively estimate failure loads for all members.

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building **design**. American **Concrete** Institute Building Code Requirements for Structural **Concrete** (ACI) 318-08 provides two methods for the **design** of **deep** **beams**, **Strut**-and-**Tie** **Modeling** (STM) or **Deep** Beam Method (DBM). A **deep** beam is defined by ACI 318-08 as having a clear span equal to or less than four times the overall depth of the beam or the regions with concentrated loads within twice the member depth from the face of the support. The truss analogy was first introduced during the late 1890’s and early 1900’s by W. Ritter and E. Morsch (Schlaich, Schafer, & Jennewein, 1987). This method was introduced as the appropriate and rational way to **design** cracked **reinforced** **concrete** through testing data by researchers. The STM is a modified version of the truss analogy which includes the **concrete** contribution through the concept of equivalent stirrup reinforcement. Once the **concrete** has cracked, the stresses are transferred to the horizontal and vertical steel across the crack and back into the **concrete**. This method, however, cannot be applied where geometrical or statical discontinuity occurred. In 1987, the Pre-stressed **Concrete** Institute Journal, PCI Journal, published a four part article on the truss analogy, “Towards a Consistent **Design** of Structural **Concrete**” by Jorg Schlaich, Kurt Schafer, and Mattias Jennewein, which generalized the truss analogy by proposing an analysis method in the form of STMs that are valid in all regions of the structure (Schlaich, Schafer, & Jennewein, 1987). The STM is included in the ACI code, ACI 318-08, found in Appendix A. The more widely used approach by **design** professionals in the **design** of **deep** **beams** is through a nonlinear distribution of the strain, DBM, which is covered in ACI 318, Sections 10.2.2, 10.2.6, 10.7 and 11.7. Actual stresses of a **deep** beam are non-linear. Typically, a **reinforced** **concrete** beam is designed by a linear-elastic method of calculating the redistributed stresses after cracking. Applying the linear-elastic method to a **deep** beam revealed that the stresses determined were less than the actual stresses near the center of the span (Task Committee 426, 1973).

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524 Figure 3-Effect of concrete effectiveness factor on shear strength prediction of RC deep beams with 525 shear reinforcement by STM 526 Figure 4-Effect of concrete effectiveness facto[r]

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The maximum values of the tensile strains in the reinforcement at the mid-span of the lower edge of the member are: 0.27‰ (at the measuring point S8 of the specimen W1, Figure 10), 1.35‰ (at the measuring point S4 of the member W2, Figure 11) and 1.1‰ (at the measuring point S6 of the member W3, Figure 12). At other measuring points, in the member W1, the measured maximum value of the tensile strain did not exceed 1.8‰, the maximum value of the tensile strain measured in the member W2 was not greater than 1.25‰, while for the specimen W3, the measured maximum tensile strain value did not exceed 0.16‰. The maximum values of the tensile strains in the reinforcement at the measuring points S8, S4 and S6 of specimens W1, W2 and W3 respectively, did not reach the ultimate limit strain values. The reason is in the fact that the steel frame with its rigidity reduced stresses and deformations in the member. The member is designed in the software "ST method" under the assumption of a static simple beam system. However, during the test, because of the interaction between the frame and the specimen, the spreading between the points at the contact of the frame and the specimen (Figure 13, zones rounded in red) was partially prevented due to the stiffness of the frame. This phenomenon will be explained for specimen W2, and the other samples behaved analogously. Based on experimentally measured strains at half the height of the "U" profile (Figure 13, zones rounded in red colour, and Figure 27), the values of normal force, bending moment and transverse force were determined. At designed loads of 2x100 kN, the normal force, bending moment and transverse force were - 129.91 kN, 7.36 kNm and 17.24 kN, respectively. By experimental measurement of strains in the reinforcement along the lower edge of the member (measuring point S3/S4, Figure 11), the axial force in the reinforcement was calculated, which at a load value of 2x100 kN is 11.77 kN. The axial force in the **tie**, along the lower edge of the support, of the **Strut**-and-**Tie** model for **design**, determined in the program "ST method", is 33 kN. The difference between this axial force and the axial force calculated on the basis of experimental results is 33 - 11.77 = 21.23 kN. One part of this difference in the amount of 17.24 kN is the consequence of the stiffness of the steel frame (transverse force in the "U" profile), and the rest of 21.23 - 17.24 = 3.99 kN is the force due to the effective area of **concrete** in tension.

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The specimens divided to three sets, Set A, consists of sixteen specimens divided as four groups with same opening dimension (60 mm x 60 mm) and different location directions (Horizontal, Vertical, Diagonal and Perpendicular to the diagonal), four specimens are in each direction. Set B, examines fifteen specimens consists of three Groups, all of them have same opening location and different opening shape, such that square or rectangle. Finally set C examines eight specimens divided to two groups considering the behavior of the characteristic strength of **concrete**, one group contains four **beams** specimens of normal strength **concrete** while, the other group contains four **beams** of high strength **concrete**.

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Fig. 7 shows the strain in shear reinforcement against the diagonal crack width in H-series **beams**: Fig. 7 (a) for vertical shear reinforcement in **beams** having either vertical or orthogonal shear reinforcement, and Fig. 7 (b) for horizontal shear reinforcement in **beams** having either horizontal or orthogonal shear reinforcement. The relation between stains in shear reinforcement and the diagonal crack width in L-series **beams** was similar to that in H-series **beams**; therefore, not presented here. The strains of shear reinforcement were recorded by ERS gages at different locations as shown in Fig. 1. Shear reinforcement was not generally strained at initial stages of loading. However, strains suddenly increased with the occurrence of the first diagonal crack. In **beams** with a / h =0.6, only horizontal reinforcing bars reached its yield strength, whereas in **beams** with a / h =1.0, only vertical reinforcing bars yielded. This indicates that the reinforcement ability to transfer tension across cracks, which is a function of the crack width, strongly depends on the angle between the reinforcement and the axis of the **strut**.

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However, few studies based on ﬁnite-element (FE) models have reported that the presence of FRP in shear-strengthened RC **beams** limits strain in the transverse-steel (e.g., Chen et al. 2010). The experimental results of the current study contradict the results of those FE studies. This discrepancy between the experimental results and those of the FE studies might occur since the mentioned FE studies consider a single crack pattern in the **concrete** beam web which does not comply with the multi-crack pattern observed in the shear- strengthened RC beam strengthened with internal transverse steel reinforcement (See Moﬁdi and Chaallal 2011c). Nev- ertheless, the matter related to yielding of transversal-steel reinforcement is still a subject of debate among the researchers in this area.

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