A deep beam is a structural member whose behavior is dominated by shear deformations. In practice, engineers typically encounter deepbeams when designing transfer girders, pile supported foundations, or bridge bents. Until recently, the design of deepbeams 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- tiemodeling (STM) for the design of deepbeams 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.
The strut efficiency factor ( 𝛽 𝑠 ) is an important for the strength of concrete for the analysis and design of reinforceddeepbeams 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 reinforceddeepbeams 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
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 reinforcedconcretedeepbeams has been weel determined by the effect of strut and tie models . Estimation of the localized compressive failure zone of concrete by AE method is well determined for modelling of RC failure zone behaviour . Shear strength of reinforcedconcretebeams under uniformly distributed loads in accordance with the strength design method is well determined with the approach method of reinforcement detailing [9-11].
Reinforcedconcrete (RC) deepbeams are structural members characterized by relatively small shear span to depth (a/d) ratios. Sectional analysis as well as design procedures are not valid for these members due to the complex interaction of flexure and shear. The strut-and-tie method (STM) has been widely accepted and used as a rational approach for the design of such disturbed regions (D regions) of reinforcedconcrete members, where traditional flexure theory cannot be used. The flow of stress is idealized as a truss consisting of compressive struts (concrete) and tension ties (reinforcing steel) transmitting the loads to the supports. Usually, STM considers only equilibrium. Hence, there is no unique solution for a given system, as one can find more than a single truss geometry admissible for a given force field. Therefore, the model which gives the maximum capacity can be considered as the most appropriate one. This paper attempts to predict the ultimate strength of deepbeams failing in diagonal compression as well as tension, from the experimental database available in literature based on STM. A modified approach has been used, considering the crushing and splitting failures of the diagonal strut separately. Crushing failure of the diagonal strut has been predicted using a plastic Strut-and-tie model with varying compression zone depth. A localized STM has been considered to predict the splitting failure of the diagonal strut.
The effectiveness of the Strut and Tie Model of reinforcedconcretedeepbeams as provided in the design codes from Canada (CSA A23.3-04 ), USA (ACI 318-08 ) and Europe (EN 1992-1-1-2004E ) 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 deepbeams 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 deepbeams, 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.
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 deepbeams 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 concretebeams, 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.
Deepbeams usually fail through shearing action which tends to be more sudden and brittle [17, 25]. However, the accurate prediction of the shear strength of reinforcedconcretedeepbeams is vital in engineering design and management . Nonetheless, several of the developed shear strength models for RC deepbeams either overestimate  or under estimate [26, 27] experimental observations. More so, studies that have evaluated the adequacy of code- based shear provisions have either reported relatively lower correlation coefficient  or larger coefficient of variation  in comparison with developed models. The adequacy of these models is highly reliant on the range of variables considered as well as the size of the database employed . Hence it is becomes necessary to develop explicit shear strength model for PKSC deepbeams. This study investigated the shear strength of 12 reinforced PKSC and Normal Concrete (NC) deepbeams. The main variables considered in the study were shear span-to-effective depth ratio, vertical shear reinforcement ratio, and type of concrete. The shear capacity predictability of 3 codes of practice; ACI 318-99 (empirical formulae based), and ACI 318-05 and EC 2 (both based on strut-and-tie models) and one kinematic model were evaluated for such RC deepbeams. A calibration procedure is proposed to provide reliable and consistent shear strength estimate for engineering design.
The topology optimization of strut-and-tie models in non-flexural reinforcedconcrete 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 reinforcedconcrete members such as deepbeams 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 reinforcedconcrete 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 reinforcedconcrete structures.
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 deepbeams. Strut- and-tiemodeling 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-tiemodeling 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 reinforcedconcrete 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
The strut-and-tie method (STM) is a simple and conservative method for designing concrete structures, especially deepbeams. 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 concretedesign 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 designprovisions conservatively estimate failure loads for all members.
building design. American Concrete Institute Building Code Requirements for Structural Concrete (ACI) 318-08 provides two methods for the design of deepbeams, Strut-and-TieModeling (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 reinforcedconcrete 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 deepbeams 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 reinforcedconcrete 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).
Deepbeams are commonly used in tall buildings, offshore structures, and foundations. According to many codes and standards, strut-and-tie model (STM) is recommended as a rational approach for deep beam analyses. This research focuses on the STM recom- mended by ACI 318-11 and AASHTO LRFD and uses experimental results to modify the strut effectiveness factor in STM for reinforcedconcrete (RC) deepbeams. This study aims to refine STM through the strut effectiveness factor and increase result accuracy. Six RC deepbeams with different shear span to effective-depth ratios (a/d) of 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00 were experimentally tested under a four-point bending set-up. The ulti- mate shear strength of deepbeams obtained from non-linear finite element modeling and STM recommended by ACI 318-11 as well as AASHTO LRFD (2012) were compared with the experimental results. An empirical equation was proposed to modify the principal tensile strain value in the bottle-shaped strut of deepbeams. The equation of the strut effectiveness factor from AASHTTO LRFD was then modified through the aforementioned empirical equation. An investigation on the failure mode and crack propagation in RC deepbeams subjected to load was also conducted.
Strut-and-tie models are often used for the design of shear critical deep members since they can rationalise the shear transfer within discontinuous or disturbed regions in RC structural elements. Most current codes of practice adopt the strut-and-tie method but provide very little guidance on how to select appropriate strut-and-tie layout and dimensions. Furthermore, the effectiveness factors used to account for the biaxial state of stresses in struts of deepbeams are not reliable. This paper reviews the application of strut-and-tie models for the design of RC deepbeams and evaluates current formulations of the effectiveness factor. Experimental and numerical studies are used to assess how the effectiveness factor is influenced by different parameters including concrete compressive strength, shear span to depth ratio and shear reinforcement ratio and to arrive at a more reliable strain based effectiveness factor. Various effectiveness factors are examined against an extensive database of experimental results on RC deepbeams with and without shear reinforcement. The results show that the proposed effectiveness factor yields the most reliable and accurate predictions and can lead to more economic and safe design guidelines.
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.
beams with concentrated loads within twice the member depth from the support that are loaded on one face and supported on the opposite face so that compression struts can develop between the loads and supports. The shear behavior of deep beam is considerable different from that of slender beams that exceed the limits noted above. The assumptions of plane sections in analysis of normal beams can no longer be used for deepbeams. The behavior of deepbeams is dominated by shear deformation, thus in design of deep beam one needs a design procedure based on mechanisms of failure of deepbeams. In practice, one typically encounters deepbeams when designing transfer girders in tall buildings, pile supported foundation, or bent caps in bridges. Prior to 1999, ACI Code 318 included a design equation for designing reinforcedconcretedeepbeams, but since 2002, the strut and tie model and nonlinear analysis have been specified. The strut and tie model is a powerful tool for designing reinforcedconcrete structures, but it is a multiple-step, complex procedure. Using Strut-and-Tie models requires some structural design and analysis experience. For nonlinear analysis, one needs a computer and a nonlinear program. In addition, designers need some finite element method knowledge, basic theory for algorithm, and an understanding of the primary variables. It is the reason that many practical designers are not confident using either procedure.
Concretereinforced with steel bars and randomly distributed fibers is named in this study as reinforced fibrous concrete (RFC). Basic purpose of this experimental research study is to obtain experimental data on the flexural strength of RFC beams containing metallic fibers in mono and hybrid forms at different loading stages. The experimental results are to be compared with analytical solution based on engineering practices in reinforcedconcrete calculation, in order to render fiber addition in concrete elements more attractive for practical applications.
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.
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.
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.