FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE WIDE …

Journal of Engineering Science and Technology Vol. 15, No. 6 (2020) 4134 - 4156 © School of Engineering, Taylor’s University

4134

FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE WIDE BEAMS USING INTERNAL STEEL PLATES

AS SHEAR REINFORCEMENT

IMAN MAJID ABDUL AMEER1,*, ABASS HATIF NAJI1, AMER MOHAMMED IBRAHIM2

1Middle Technical University, Institute of Technical Baqubah , Iraq 2Department of Civil Engineering, Diyila University College of Engineering, Iraq

*Corresponding author: [email protected]

Abstract

In the current study the finite element software ANSYS (version 16) was used for modeling wide reinforced concrete beams with steel plates as shear reinforcement. The results obtained by the ANSYS program were matched with the data of experiment of four wide beams having steel plates of different thickness (3 mm, 4 mm and 5 mm). The samples in this side were specified. The compression test was conducted for load-deflection curves at mid span, yield as well as ultimate load and their relation deflection, strain in longitudinal reinforcement, shear reinforcement and concrete and crack patterns. The compression results were positive. The results from the finite element analysis calculated the same outcome as that of experimental test of the beams. The load-deflection curves resulted from the analysis of the finite element comply with the findings of the experiment in the linear ranges, but the findings of finite elements were stiffer than these obtained from the experimental findings.

Keywords: Finite element modeling, Reinforced concrete, Shear reinforcement, Shear steel plates, Wide beam.

Finite Element Analysis of Reinforced Concrete Wide Beams . . . . 4135

Journal of Engineering Science and Technology December 2020, Vol. 15(6)

1. Introduction The implementation of large-scale concrete beams in structural frame system has gone through recent developments. Wide beams can offer appropriate transversal sections in order to achieve the required ability at a shallow depth more than the slender beam system with an equivalent distance in the plan. Stirrups are the most commonly used shear reinforcement for wide beams. In order to resist higher shear stresses, the number of stirrups is increased having their spacing reduced and/or their diameter is increased. Due to some difficulties in stirrups erection with higher cost and time entailed, some attempts have been made to find out new techniques for shear reinforcement. One of these new techniques is the use of swimmer bars. Swimmer bars are small inclined bars with both ends bent horizontally for a short distance, welded, bolted, or spliced to both top and bottom longitudinal reinforcement [1].

A pilot study was conducted by Adam et al. [2] to examine the shear behaviour of thick boards and wide beams in addition to the effect of the width member. In their research, they examined five samples of concrete of normal strength with a normally thickness of 470 mm. However, this varies in widths ranging from 250 to 3005 mm and a length of 2900 mm. On this basis, the research shows that all shear pressures resulting from wide beams, narrow beams and tiles are identical.

In another study, Adam et al. [3] investigated the shear reinforcements spacing impact on the shear capacity of wide reinforced concrete members. In their study, they did a test on 13 specimens of normal strength concrete specimens. The spacing of shear reinforcement is a major test variable. The specimens constitute shear reinforcement ratios close to ACI 318-11 minimum requirements [4]. Their conclusion that there was a decrease of the effect of the shear reinforcement, while simultaneously an increase of the web reinforcement legs spacing that occurred across the member’s width. Utilizing little shear reinforcing leg, even after extensively spaced up to a space of nearly 2d (d: spacing from the centre to the centre of stirrups), shows a decrease in the failure mode brittleness in relation to a geometrically identical member lacking a web reinforcement. To prove that the shears capability of all members with shear reinforcement were appropriate once of designing in step with ACI 318-11, the investigators prompted that the cross spacing of net reinforcement ought to be restricted to less depth of the particular member and 600 millimetres in addition.

Mohamed et al. [5] investigates to what extent the web shearing reinforced contributes to the wide beams shear strength. The results plainly explain the role of web reinforcement to improve the shear ability, the wide beams elasticity that was compatible with the familiar standards and codes of universal.

Amer et al. [6] investigated the effect of steel plates on shear strength of wide reinforced concrete beams. These Beams were designed to fail in shear. The ascent in the shear strength ranges from 9.52% to 47.62% as matched with the control sample.

Al et al. [7] study the effect of steel plates on shear strength of wide reinforced concrete beams. The results showed that the steel plate improves shear capacity of wide beam a little in comparison to other ways of reinforcement and the failure comes to be sudden failure under this kind of reinforcement. Using steel plate causes increased shear capacity of wide beams significantly that in combination with stirrups there is increase in load capacity as well as ductility.

4136 I. M. A. Ameer et al.


Amer et al. [8] investigate the effect of using internal steel plates for shear reinforcement on flexural behaviour of SCC beams instead of using traditional reinforcement bars (stirrups) and to study the effect of their spacing and thickness on strength.

In this study, modeling of wide reinforced concrete beam with steel plate using ANSYS program was conducted for the first time. The results obtained by the ANSYS program were matched with the data of the experiment of four wide beams having steel plates of different thickness 3 mm, 4 mm and 5 mm. The samples in this side were specified, and taken from Ahmed [9].

2. Geometry of The Beam This paper calculates the geometry of the beam as reported by Ahmed [9]. The dimension of the beams and reinforced details are shown in Table 1 and Figs . 1 to 5.

Table 1. Details of tested wide beams. Specimens Description

As = 2010 𝐦𝐦𝐦𝐦𝟐𝟐 Stirrup spacing =

125 mm

WBS Double stirrups

WBP3-1 Transverse Plates with Circle shape of voids 3 mm thickness.

WBP4 Transverse Plates with Circle shape of voids 4 mm thickness.

WBP5 Transverse Plates with Circle shape of voids 5 mm thickness.

Fig. 1. Beam details.

Fig. 2. Section A-A of wide beam WBS.



Fig. 3. Section A-A of wide beam WBP3-1.

Fig. 4. Section A-A of wide beam WBP4.

Fig. 5. Section A-A of wide beam WBP5.



3. Finite Element Modelling

3.1. Concrete

To model concrete using ANSYS, three-dimensional brick element with eight nodes of the type SOLID65 was used. SOLID65 was used for 3D modeling of solids with or without reinforcing bars. Solid element was subjected to cracking due to tension and crushing due to compression so it is suitable to model the concrete. SOLID65 was defined by eight nodes having three degrees of freedom at each node translations of the nodes in x, y, and z-directions and by the material properties. Figure 6 illustrates the geometry, nodes locations, and the coordinate system for this element [10]. Meshing the concrete media volume was carried out with rectangular shape by dividing the concrete prism volumes into small hexahedron brick elements having orthogonal side dimensions of 50 mm.

Fig. 6. Geometry of SOLID 65.

For concrete, system analysis program needs input data for the material properties which are ultimate uniaxial compressive strength, elastic modulus, poison’s ratio, modulus of rupture, shear transfer coefficient for opened and closed cracks and compression stress strain diagram for concretes. The properties of concrete are shown in Table 2. The modulus of elasticity and flexural strength are based on the Eqs. (1) and (2) respectively.

𝐸𝐸𝑐𝑐 = 4700�𝑓𝑓′𝑐𝑐 MPa (1)

𝑓𝑓𝑡𝑡 = 0.62√𝑓𝑓′𝑐𝑐 MPa (2)

Table 2. Parameters and values of the concrete. Parameter Definition Value

f'c Ultimate compressive strength (MPa) 36.6 𝒇𝒇𝒕𝒕 Ultimate tensile strength (MPa) 3.66 𝜷𝜷° Shear transfer coefficient for opened crack 0.55 𝜷𝜷𝒄𝒄 Shear transfer coefficient for closed crack 0.6 𝑬𝑬𝒄𝒄 Young’s modulus of elasticity (MPa) 28434.29 ʋ Poisson’s ratio 0.2



The analysis program requests the relationship of uniaxial stress strain for‖ the concrete in compression. For the element of SOLID 65, linear and multi linear isotropic material properties are required t‖o properly model the concrete. The simplified relationship of stress strain for the concrete in compression is obtained, which was presented in Fig .7.

Fig. 7. Stress strain relation model for concrete.

A typical uniaxial stress strain behaviour of concrete under uniaxial compression is shown in Fig. 8. It is nearly linear up to 0.3 to 0.5 fć. For stress above this point, the curve shows a gradual downward curvature up to 0.75 to 0.9 fć then the stress strain curve starts to descend at fć until failure occurs due to the crushing of concrete at the ultimate strain [11].

Fig. 8. Typical uniaxial stress strain curve for concrete in compression.

Under uniaxial tensile stress, the stress strain curves for concrete are nearly straight up to about 0.6 𝑓𝑓𝑡𝑡. At that point, it will in generally deviate from linearity because of the process of micro cracking propagation prior to form a continuous crack system at peak stress. This is followed by a formation of an unstable micro



crack system, which grows rapidly under the tensile stress bringing the material to its post peak section [12]. Typical stress strain curve of concrete under uniaxial tensile stress is shown in Fig. 9.

Fig. 9. Typical uniaxial stress strain curve for concrete in tension.

3.2. Steel reinforcement In this study, LINK 180׀ was used to represented longitudinal and shear reinforcement. The elements were described by two nodes with three degrees of translations, i.e., node translations in the three directions are x, y, and z. The nodes location, geometries and the coordinating system of these elements represented in Fig .10 [10].

Fig. 10. Geometry of LINK180.

In contrast with concrete steel, it is more homogenous material and its strain-stress behaviour may have similar tension and compression. Figure 11 illustrates a typical uniaxial stress-strain curve for steel samples loaded monotonically ‖ in tension [13].

Fig. 11. Typical stress strain curve for steel.

While using computer, potential computational difficulties may occur. As a result, these difficulties may have avoided by the relationship of alternative bilinear



stress strain as represented in Fig. 12. Table 3 shows a list of steel reinforcement properties of ANSYS model.

Fig. 12. Idealization of computer calculations for steel.

Table 3. Parameters and values of steel reinforcement. Parameter Definition Value

𝑨𝑨𝒃𝒃 Cross sectional area (mm2) Longitudinal reinforcement 202 Shear reinforcement 78.5

𝒇𝒇𝒚𝒚 Yield tensile stress (MPa) Longitudinal reinforcement 415 Shear reinforcement 397

𝑬𝑬𝒔𝒔 Modulus of elasticity (MPa) Longitudinal reinforcement 200000 Shear reinforcement 200000

𝑬𝑬𝒕𝒕 Steel hardening (MPa) Longitudinal reinforcement 5000 Shear reinforcement 5000

ʋ Poisson’s ratio Longitudinal reinforcement 0.3 Shear reinforcement 0.3

3.3. Steel plate

The steel plates are represented in the ANSYS program by the element SHELL281. At each node, there are 6 degrees of freedom in element. The rotations about the nodal on x, y and z-axes and the translation of the nodes in the directions x, y, and z. SHELL281 is appropriate to linear, great rotation and/or great strain nonlinear applications. The locations of nodes, geometry, and the coordinate system of this element are shown in Fig. 13 [10].

Fig. 13. Geometry of SHELL281.



Different steel plates are used in this research, which are 3 mm, 4 mm, and 5 mm. Steel plate’s properties are shown in Table 4.

Table 4. Properties of shear steel plates. Parameter Definition Value

T Thickness of SHELL281 (mm) 3,4, 5 fy Yield tensile stress (MPa) 210,241,400 Es Modulus of elasticity (MPa) 200000 Et Steel hardening (MPa) 5000 ʋ Poisson’s ratio 0.3

3.4. Bearing steel plates The bearing plates are represented in the ANSYS program by SOLID185 which was established by use of eight nodes. Each node has three freedom degrees in the directions x, y, and z. Steel plates used to support locations and loading points in the finite element models were added to supply a more stress distribution. Poisson’s ratio of 0.3 and elastic modulus of equivalent with 200000 MPa are used for the plates. Steel plates are expected to be linear elastic materials. Figure 14 shows the geometry of this element [10].

Fig. 14. Geometry of SOLID185.

4. Boundary Conditions and Applied Load Considering displacement boundary conditions to force the model serves to get a novel response. Two simple supports of the beam were modelled by confining the centre node for every bearing plate 100 mm from the edge of the beam. One as a roller Uy = 0 and also the different as a hinge Uy = 0 and Uz = 0 as shown in Fig. 15, each is dispensed across the width of the lowest surface of the beam. The applied load is equally divided to two loads. Every load is applied at 725 mm from the corresponding fringe of the beam leaving a distance of 350 mm between them. Figure 15 shows how every load is dispensed on the centre nodes of the corresponding bearing plate across the width of the highest surface of the beam.



Fig. 15. Boundary conditions and external loads.

5. Numerical Analysis and Comparison of Results

5.1. Load deflection behaviour In the central of mid span of the beams bottom, face deflections were measured. Figures 16 to 19 show the comparative between the analytical and experimental result. The analysis reveals an agreement between the load deflection curves of the beams from the finite element analyses and the experimental data.

The load deflection curves of all wide beams in the linear range were quite stiffer than the experimental plots. After the first cracking, the stiffness of the finite element models becomes again higher than that of the experimental beams. Many factors might lead to increase stiffness in the finite element analysis of reinforced concrete wide beams.

Firstly, micro cracks are found in the concrete of the experimental beams and these cracks occur due to drying shrinkage in the concrete and/or handling of the beams. Secondly, these minor cracks are not included in finite element models as they decrease the toughness of the experimental beams.

Finally, the assumption that there is a perfect bond between steel reinforcement and concrete in the analyses of finite element, however, the presumption does not be true for the experimental beams. When a bond lip happens, the complex actions between the steel reinforcement and concrete are lost [8]. Thus, the overall stiffness of the experimental beams is predicting to be lower than that of the finite element models.



Fig. 16. Load deflection curve of wide beam (WBS).

Fig. 17. Load deflection curve of wide beam (WBPL3-1).

Fig. 18. Load deflection curve of wide beam (WBP4).



Fig. 19. Load deflection curve of wide beam (WBP5).

5.2. Load and deflection at yield Table 5 displays comparisons between the loads and deflections at the yield of the experimental beams [5] and the loads and deflection at yield from finite element model. As in Table 5, the yield load gained from the experiment beams are lower than that of the finite elements by 1.15% on average. The deflections at yield load from experiment beams are larger than that of the finite element model by 14.31% on average. This is due to the restriction on the degree of freedom which increases the stiffness and subsequently leads to load increase and deflection decrease.

Table 5. Experimental and FE results of yield loads and related deflections.

Modelled Beams

Yield load 𝑷𝑷𝒚𝒚 (kN) Mid span deflection ∆𝒚𝒚 (mm) Exp. ANSYS %Diff Exp. ANSYS %Diff.

WBS 400 405 -1.25 10.83 9.75 +9.97 WBP3-1 420 424 -0.95 13.45 11.34 +15.69 WBP4 420 424 -0.95 14.80 12.60 +14.86 WBP5 410 416 -1.46 13.75 12.60 +16.73

Avg. of Diff.% - 1.15 Avg. of Diff.% +14.31

• %Diff. Indicates to the percentage difference.

5.3. Load and deflection at failure Table 6 displays the final load of the experiment beams as reported by Ahmed [9] and the final loads from the finite element models. The final loads of the experiment beams are smaller than that of the finite elements by 1.05% on average and the ultimate deflection from experiment beams are larger than that of the finite elements by 10.65% on average. The differences between experimental and theoretical results existed due the reason already mentioned in 5.2. Figures 20 to 23 show the variance of deflection in y-direction of wide beams at failure load.



Table 6. Comparison between experimental and FE results of ultimate loads and related deflections.

Modelled Beams

Yield load 𝑷𝑷𝒖𝒖 (kN) Mid span deflection ∆𝒖𝒖 (mm) Exp. ANSYS %Diff Exp. ANSYS %Diff.

WBS 440 445 -1.14 18.93 16.60 +12.31 WBP3-1 431 435 -0.93 36.45 32.65 +10.42 WBP4 431 435 -0.93 30.35 27.48 +9.46 WBP5 419 424 -1.20 17.75 15.90 +10.42

Avg. of % Diff - 1.05 Avg. of % Diff +10.65

Fig. 20. Variation of deflection of y-direction of WBS at failure load

Fig. 21. Variation of deflection of y-direction of WBPL3-1 at failure load.



Fig. 22. Variation of deflection of y-direction of WBP4 at failure load.

Fig. 23. Variation of deflection of y-direction of WBP5 at failure load.

5.4. Strain in longitudinal reinforcement Table 7 shows a comparison between the strain in longitudinal reinforcement of the experimental wide beams as reported by Ahmed [9] and the strain in longitudinal reinforcement as of the finite element models. It shows that the strain from experimental beams was higher than the strain from finite element models by 7.65%, 10.36% and 8.14% on average at crack, yield, and ultimate load respectively. It also shows the correspondence between experimental and finite element results of longitudinal reinforcement strain. The differences between experimental and theoretical results existed due the reason already mentioned in 5.2. Figures 24 to 27 show the variation of stress in steel reinforcing bars of wide beams at the ultimate load.



Table 7. Experimental and FE results of strain in longitudinal reinforcement.

Modelled Beams

Strain at crack load x 10-3

Strain at yield load x 10-3

Strain at ultimate load x 10-3

ANSYS %Diff Exp. ANSYS %Diff Exp. ANSYS %Diff WBS 0.99 +7.48 2.330 2.01 +12.61 2.32 1.98 +14.66

WBP3-1 0.79 +8.14 2.62 2.33 +11.07 2.95 2.76 +6.44 WBP4 0.69 +8.00 2.27 2.06 +9.25 2.82 2.63 +6.74 WBP5 1.33 +6.99 2.47 2.26 +8.50 2.97 2.83 +4.71

Avg of Diff% +7.65 Avg of Diff% +10.36 Avg of Diff% +8.14

Fig. 24. Variation in stress of steel reinforcing

bars of wide beam (WBS) at ultimate load.

Fig. 25. Variation in stress of steel reinforcing

bars of wide beam (WBPL3-1) at ultimate load



Fig. 26. Variation in stress of steel reinforcing bars of wide beam (WBPL4) at ultimate load

Fig. 27. Variation in stress of steel reinforcing bars of wide beam (WBP5) at ultimate load.

5.5. Strain in shear reinforcement (stirrups and steel plate) Table 8 shows a comparison between the strains in stirrups and steel plates of the experimental wide beams as reported by Ahmed [9] and the strain in stirrups and steel plates from the element of finite models. It shows that the strain from experimental beams is slightly higher than that of the finite element models by 11.07%, 4.26%, and 8.14% on average at crack, yield and ultimate load respectively. It also shows that



experimental and finite element findings of shear reinforcement strain have a good contract with others. The differences between experimental and theoretical results existed due the reason already mentioned in 5.2. Figures 28 to 30 show variation of stress at the ultimate load, in the shear reinforcement of wide beams at the final loads.

Table 8. Experimental and FE results of strain in shear reinforcement.

Modelled Beams

Strain at crack load×10-3

Strain at yield load×10-3

Strain at ultimate load×10-3

Exp. ANSYS Diff% Exp. ANSYS Diff.% Exp. ANSYS %Diff WBS 0.19 0.17 +10.53 0.73 0.70 +4.11 1.78 1.80 -1.12

WBP3-1 0.16 0.14 +12.50 0.58 0.55 +5.17 0.82 1.11 -35.36 WBP4 0.21 0.19 +9.52 0.51 0.49 +3.92 0.63 1.18 -87.30 WBP5 0.17 0.15 +11.74 0.52 0.50 +3.85 0.93 1.80 -93.55

Avg. of % Diff +11.07 Avg. of % Diff +4.26 Avg. of %Diff +8.14

Fig. 28. Variation of stress in shear steel plates of WBP3-1 at ultimate load.

Fig. 29. Variation of stress in shear steel plates of WBP4 at ultimate load.



Fig. 30. Variation of stress in shear steel plates of WBP5 at ultimate load.

5.6. The compression strain in concrete face Table 9 explains the strain in concrete of the experimental wide beams as reported by Ahmed [9]. The relationship with the strain in concrete from the models of finite element. This illustrates that the experimental strain is quite higher than the finite element models strain by 11.22% and 7.71% on average at crack and yield load respectively. It also shows that the results of experimental and finite element of concrete strain are compatible. The differences between experimental and theoretical results existed due the reason already mentioned in 5.2. Figures 31 to 34 show a variation of stress in the shear reinforcement of wide beams at the ultimate load.

Table 9. Experimental and FE results of strain in concrete.

Modelled Beams

Strain at crack load×10-3

Strain at yield load×10-3

Strain at ultimate load×10-3

Exp. ANSYS %Diff Exp. ANSYS %Diff Exp. ANSYS %Diff WBS -1.00 -0.85 +15.0 -1.27 -1.19 +6.21 -2.30 -2.55 -10.87

WBP3-1 -0.81 -0.74 +8.64 -1.74 -1.65 +5.17 -2.63 -2.82 -7.22 WBP4 -1.55 -1.35 +12.9 -2.58 -2.37 +8.14 -4.58 -4.75 -3.71 WBP5 -1.32 -1.21 +8.33 -1.68 -1.49 +11.31 -2.92 - 3.22 -10.27

Avg. of Diff % +11.22 Avg. of Diff% +7.71 Avg. of Diff % - 8.02

Fig. 31. Variation in strain concrete of WBS at ultimate load



Fig. 32. Variation in strain concrete of WBPL3-1 at ultimate load.

Fig. 33. Variation in strain concrete of WBP4 at ultimate load.

Fig. 34. Variation in strain concrete of WBP5 at ultimate load.



5.7. Cracks patterns Figures 35 to 38 demonstrate the advancement of the development of crack patterns for each beam at the last loading step. The ANSYS program show circles at locations of cracking or crushing in concrete elements. Cracking appears as a circle outline in the plane of the crack, and crushing appears as by an octahedron outline. The first crack at an integration point appears as a red circle outline, the second crack with a green outline, and the third crack with a blue outline [7]. These figures explain a compression between the crack patterns of experiment and finite element model. As a rule, flexural breaks occur right on time at mid-range. At the point when connected burdens increment, vertical flexural cracks broaden on a level plane from the mid-span to the support. At a higher applied load, diagonal tensile cracks appear. Increasing applied loads induces additional diagonal and flexural cracks. Compressive cracks show up at about the last applied load step. Failure modes of the finite element models demonstrate a decent concurrence with perceptions and information picked up from the exploratory full-scale beams.

Fig. 35. Cracks patterns in wide beam (WBS) and FE model of wide beam (WBS).

Fig. 36. Cracks patterns in wide beam (WBP3-1) and FE model of wide beam (WBP3-1).



Fig. 37. Cracks patterns in wide beam (WBP4) and FE of wide beam (WBP4).

Fig. 38. Cracks patterns in wide beam (WBP5) and FE model of wide beam (WBP5).

6. Conclusions Through the result obtained out of the analysis, the researchers have come up with the following conclusions:

• The models of finite element indicated by load -deflections curves at the central of span behave similarly to the experimental results. However, they are stiffer than the experimental test.

• According to the experimental results, the yield load results are less than the yield load ones as indicated by the analysis of finite element by 1.15 % on average and its related deflection are higher by 14.31% on average. Moreover, the ultimate load from experiment results are less than the ultimate load from the finite element by 1.05 % on average and its related deflection are higher by 10.65% on average. This is due to employing certain hypothetical of the material properties rather than neglect the effects of concrete toughening mechanisms.

• The strain in longitudinal reinforcement from experiment result are higher than that obtained from the finite element results by 7.65%, 10.36% and 8.14 % on average at crack, yield and ultimate load.



• The strain in shear reinforcement from experiment result are higher than that obtained from the finite element results by 11.07%, 4.26 % and 8.14 % on average at crack, yield and ultimate load

• The strains in concrete from experiment result are higher than that obtained from the finite element results by 11.07% and 4.26% on average at crack and yield load.

• At final loads, the cracks of the models of the finite elements show great correspond with they observed failure mode of the experimental beams.

Nomenclatures d Effective depth of concrete beam 𝐸𝐸𝑐𝑐 Modulus of elasticity of concrete 𝑓𝑓𝑐𝑐̀ Compressive strength of concrete 𝑓𝑓𝑟𝑟 Modules of rupture ft Uniaxial tensile strength of concrete 𝑓𝑓𝑢𝑢 Stress at ultimate load 𝑓𝑓𝑦𝑦 Yield stress 𝑃𝑃𝑢𝑢 Ultimate load of the beam 𝑃𝑃𝑦𝑦 Yield load of longitudinal reinforcement 𝑈𝑈𝑥𝑥 Horizontal displacement in x- direction Uy Deflection or vertical displacement Uz Displacement in z- direction x, y, z Global coordinates system Greek Symbols 𝛽𝛽0 Shear transfer coefficients for an open crack 𝛽𝛽𝑐𝑐 Shear transfer coefficients for a closed crack ∆u Deflection at ultimate load ∆y Deflection at yield load ɛ Strain ρ Reinforcement ratio σ Stress Abbreviations

ACI American Concrete Institute ANSYS Analysis System FE Finite element method SCC Self-compacted concrete

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swimmer bars as shear reinforcement in the reinforced concrete beams. International Journal of Civil and Structural Engineering, 3(2), 313-320.

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3. Adam, S.L.; Evan, C.B.; and Michael, P.C. (2009). Shear reinforcement spacing in wide members. ACI Structural Journal, 106(2), 205-214.

4. ACI Committee 318. (2011). Building code requirements for structural concrete (ACI 318-11) and commentary (318R-11) Farmington Hills: American Concrete Institute.

5. Mohamed, M.H.; Hatem, M.M.; and Nabil, A.B.Y. (2012). On the contribution of shear reinforcement in shear strength of shallow wide beams. Life Science Journal, 9(3), 484-498.

6. Amer, M.I.; Wissam, D.S.; and Qusay, W.A. (2015). Effect of steel plates on shear strength of wide reinforced concrete beams. Journal of Engineering and Development, 19(3), 1813-7822.

7. Al, S.J.; Ibrahim, A.M.; and Agarwal, V.C. (2015). Effect steel plate of shear reinforced wide beam concrete. International Journal of Science and Research (IJSR), National Institute of Science Communications and Information Resources-NISCAIR, 4(2), 596-601.

8. Amer, M.I.; Zeyad, S.M.K.; and Iman, M.A.A. (2017). Effect of using internal steel plates for shear reinforcement on flexural behaviour of self-compacting concrete beams. Al-Nahrain Journal for Engineering Sciences (NJES), 20(5), 1071-1082.

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reinforced concrete plate and shell structures using 20.-nodded isoperimetric brick elements. Computers and Structures, 25(6), 845-869.

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FINITE ELEMENT ANALYSIS OF REINFORCED CONCRETE WIDE …