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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

NUMERICAL ANALYSIS OF RC

SHEAR CRITICAL BEAMS

Sankar Jegadesh J S1* and S Jayalekshmi1

1 Resaerch Scholar, Department of Civil Engineering, National Institute of Technology, Tiruchirrappalli-15.

*Corresponding author:Sankar Jegadesh J S � jssj14@gmail.com

ISSN 2319 – 6009 www.ijscer.com

Vol. 3, No. 1, February 2014

© 2014 IJSCER. All Rights Reserved

Int. J. Struct. & Civil Engg. Res. 2014

Research Paper

INTRODUCTION

The analysis and understanding on the shearbehavior of RCC members is quite complexdue to the mechanism of shear transfer. Finiteelement (FE) analysis is a numerical analysisapplied to the concrete structures based onthe use of the nonlinear behavior of materials.FEA provides a tool that can simulate andpredict the responses of reinforced concretemembers. Nonlinear material models havebeen integrated in many of general purposefinite element codes, like ANSYS. MacGregorJ G (1992) proposed a model to determine

the shear capacity of the RCC beams. Thissimple model develops the multi-linearisotropic stress-strain curve for the concrete.A wide range of work had been done and manyexperimental data had been verified by usingthis model. This was the simplest model topredict the behavior of concrete.Vecchio F J,Collins M P (1986) proposed that thecompressive strength of the strut is a functionof both the compressive stress along the strutand the tensile stress perpendicular to the strut.They proposed Modified Compression Fieldtheory.

In the present study, a non-linear finite element analysis procedure is adopted with the SoftenedMembrane Model (SMM) and MacGregor Model that incorporates the material properties andbehavioural characteristics of RCC member which can be applied to obtain the shear behaviour.The Finite Element package ANSYS with its 3D Modelling is used as a tool for the analysis of theRCC beams. The 3D concrete Solid 65 finite element is employed in this research to model thenonlinear behaviour of reinforced concrete beams. The analysis is based on the 5 parameterplasticity failure surface due to Willam-Warnke. The load deflection curve analysis is done inANSYS and the results are compared with the available experimental results taken from theliterature. The prediction of the ultimate load by the two different models throws light on the factthat SMM predicts conservative results when compared with the experimental and analyticaldata.

Keywords: RC beams, SMM, Macgregor Model, ANSYS 3D Modelling, Load-Deflection Response

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

Hsu T T C (1988) proposed Softening TrussModel Theory for Shear applied to structureswhere Shear Behavior predominates. Thestructural applications are low-rise Shearwalls, framed wall panels, Deep Beams etc.The theoretical predictions shear and torsionstrengths, the deformation of the structuresthroughout their post cracking loading were ingood agreement with the test results. ThomasHsu T C and Ronnie R H Zhu (2002) concludedthat three rational analytical models had beenproposed to predict the nonlinear shearbehavior of reinforced concrete membraneelements. These models were the compressionfield theory (Vecchio, 1989) the rotating-anglesoftened truss models (Hsu, 1995) and thefixed-angle softened truss model (Hsu, 1997).These models could predict the pre-peakascending branches of the response curves,but not the post peak descending branches. Anew softened membrane model (Hsu, Ronnieand Zhu, 2002) was developed thatincorporated the new Hsu/Zhu ratios thatincorporates Poisson effect of concrete. Theconstitutive relationship of cracked concretein shear was based on a complicatedempirical formula and it was overcome by thetheoretical derivation of a rational shearmodulus (Zhu, 2001). In this study, the SMM isgreatly simplified and its accuracy is improvedby using the rational shear modulus. Thisimproved version of SMM can now be used topredict the nonlinear shear behavior ofreinforced concrete membrane elements.

PREDICTION MODELS

Based on the Literature survey, that has been

made two models that had been proposedhave been considered for the present study.This paper assesses the accuracy ofprediction of results by the two models.

1. MacGregor Model

2. Softened Membrane Model

REVIEW OF THE MODELS

MacGregor proposed a model to determinethe shear capacity of the RCC beams. Thecompressive uniaxial stress-strain relationshipfor the concrete model was obtained and themultilinear isotropic stress-strain curve for theconcrete is shown in Figure 1 (MacGregor1992). The multilinear curve is used to helpwith convergence of the nonlinear solutionalgorithm.

Figure 1: Simplified CompressiveUniaxial Stress-Strain Curve

for Concrete (MacGregor)

Thomas T C Hsu and Ronnie R H Zhuproposed Softened Membrane Model forReinforced Concrete Elements in Shear. Thismodel satisfies Navier’s three principles ofmechanics of materials, stress equilibrium,strain compatibility, and material constitutivelaws. The softened-membrane model, canconfidently predict the entire behavior of RCC

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

beams including both the pre-peak and thepost-peak behavior. In this general theory, thepostpeak behavior is rationally predicted bytaking into account the stresses and strainscaused by Poisson effect. The Poisson effectis characterized by two Hsu/Zhu ratios, definedas “the Poisson’s ratios of cracked reinforcedconcrete based on the smeared-crackconcept.” In the smeared-crack concept,cracked reinforced concrete is treated as acontinuous material, and the constitutive lawsof concrete and steel bars are expressed interms of smeared (average) stresses andsmeared (average) strains as shown inFigures 2 and 3 respectively.

EXPERIMENTAL DATA

In this paper, the models are comparedstatistically with the experimental resultsreviewed in the literature. Table 1 gives thebeam Cross-Section properties and Table 2gives the beam material properties.

FINITE ELEMENT

MODELLING BY ANSYS

The experimental results of the beams fromthe literature paper are verified analytically,using the finite element software ‘ANSYS’. Toobtain good results from the Solid65 element,the use of a mapped mesh is recommended.The solid element has eight nodes with threedegrees of freedom at each node translationsin the nodal x, y and z directions. Beams are

Figure 2: Constitutive Lawfor Concrete (SMM)

Figure 3: Constitutive Lawfor Steel Bars

Table 1: Beam Cross-Section Properties

Beam No. Lmm Dmm Bmm a/d

1. Sonia,(2000) 1800 250 150 3

2. Anthony, (2004)* 4572 457.2 254 10

3. Anthony, (2004)** 4572 457.2 254 10

4. Nasir-Uz-Zaman(2009) 700 150 75 4

Note: * – Beam with Stirrups; ** – Beam without Stirrups.

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

modeled using 8 nodded solid 65 elements.Link8 elements were used to create the flexuraland shear reinforcement. Shear stirrups aremodeled throughout the beam. The overallmesh of the concrete, plate, and supportvolumes is shown in Figure 4. The William andWarnke model is capable of predicting failurefor concrete materials both cracking andcrushing failure modes are accounted for. Thetwo input strength parameters i.e., ultimate uni-axial tensile and compressive strengths areneeded to define a failure surface for theconcrete. Consequently, a criterion for failureof the concrete due to a multi axial stress statecan be calculated (William and Warnke, 1975).Displacement boundary conditions are

needed to constrain the model to get a uniquesolution. To ensure that the model acts thesame way as the experimental beam, boundaryconditions need to be applied at where thesupports and loadings exist. Loading wasapplied at loading point. The Solution controlprocess is done by using Newton Raphsonmethod. The analytical load – deflectionscurves are determined for MacGregor andSMM Model. The load – deflection curvespredicted by finite element analysis iscompared with experimental results.

The nonlinear analysis of RCC shear criticalbeams have been carried out using ANSYSfor MacGregor and SMM Model individuallyand the load-deflection response of thepresent studies is compared with the availableexperimental results.

FINITE ELEMENT ANALYSIS OF

RC SHEAR CRITICAL BEAMS

The nonlinear analysis is based on theMacGregor Model and SMM for a reinforcedconcrete beam. The cracking pattern(s) in thebeam can be obtained using the Crack/Crushing plot option in ANSYS. Vector Modeplots must be turned on to view the cracking inthe model. The first cracking load for beam-1based on SMM is 10.26kN and that of

Table 2: Beam Material Properties

No. fc (N/mm2) fy (N/mm2) Reinforcements

1 26.7 396 2 No. 16 mm dia bars

2 33.095 415 2 No. 20 mm dia bars; 8mm dia stirrups at 150mm spacing

3 33.095 415 2 No. 20 mm dia bars

4 35 415 2-16 mm diameter steel bar as the main longitudinal reinforcement

and 6 mm diameter steel bars, placed at 100 mm c/c.

Figure 4: 3D Beam Modellingin ANSYS (BEAM1)

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

MacGregor Model is 10kN. The yielding ofsteel reinforcement occurs when a force of36.72kN (MacGregor) and 34.85kN (SMM) isapplied. At this point in the response, thedisplacements of the beam increase at ahigher rate as more load is applied and theability of the beam to distribute load throughoutthe cross-section is reduced and greaterdeflections occur at the beam centerline. Thebeam no longer can support additional loadas indicated by an insurmountableconvergence failure at a load of 46.44kN(MacGregor) and 44.45 kN(SMM). The crackpattern at the failure load for beam-1 is shownin Figures 5 and 6.

LOAD-DEFORMATION RESPONSE

The load-deformation response for beam-1predicted by different models is shown inFigure 7 .The nonlinear responses calculatedusing FEA based on MacGregor and SMM isplotted and compared with the experimentalresponse of the beam. The entire load-deformation response of the model producedcompares well with the experimental response.This gave confidence in the use of ANSYS andthe model developed. MacGregor model issiffer than the SMM and SMM predicts theultimate load of the beam conservatively whencompared with the experimental results.

Figure 5: Failure of the ConcreteBeam-1 (MacGregor)

Figure 6: Failure of the ConcreteBeam-1 (SMM)

Figure 7: Analytical Load-DeflectionResponse (BEAM-1)

COMPARISON OF ANALYTICAL

MODELS WITH THE EXPERI-

MENTAL RESULTS

The Table 3 tabulates the experimental andanalytical ultimate load of the beams. Theultimate load predicted by MacGregor is stifferSMM than model when compared to theexperimental result. The Table 3 elucidates thatthe ultimate load predicted by SMM isconservative than MacGregor model whencompared with the experimental results.

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

CONCLUSION

From the results of this study, the followingconclusions can be drawn:

1. The proposed SMM, which incorporates theHsu/Zhu ratios, can predict not only the pre-peak behavior, but also the post-peakbehavior of cracked reinforced concretemembrane beams.

2. The SMM is a rational theory that satisfiesNavier’s three principles of mechanics ofmaterials. The constitutive models ofconcrete include a rational shear modulus,which is derived from the smeared-crackconcept that greatly simplifies the SMMsolution algorithm and improves itsaccuracy.

3. The predictions of the SMM agree very wellwith the experimental results and predict theultimate load of the beams conservativelythan the MacGregor model, including boththe ascending and the descendingbranches of the load-deformation responsecurves.

ACKNOWLEDGMENT

The authors thank Principal, Department ofCivil Engineering, National Institute ofTechnology for constant cooperation and

extending the facilities to carry out thisresearch work.

REFERENCES

1. Anthony J and Wolanski B S (2004),“Flexure Behavior of Reinforced andPrestressed Concrete Beams UsingFinite Element Analysis”, Faculty ofGraduate School, Marquette University,Milwaukee, Wisconsin, May.

2. Hsu T T C (1988), “Softening Truss ModelTheory for Shear and Torsion”, StructuralJournal of the American ConcreteInstitute, Vol. 85, No. 6.

3. Hsu T T C, Zhang L X and Gomez T(1995), “Nonlinear Analysis of MembraneElements by Rotating-Angle Softened-Truss Model”, Journal of Testing andEvaluations ASTM, Vol. 23, No. 6,pp. 424-430.

4. Hsu T T C and Zhang L X (1997),“Nonlinear Analysis of MembraneElements by Fixed-Angle Softened-TrussModel”, ACI Structural Journal, Vol. 94,No. 5, Sept.-Oct., pp. 483-492

5. Mac Gregor J G (1992), “ReinforcedConcrete Mechanics and Design”,Prentice-Hall, Inc., Englewood Cliffs, NJ.

Table 3: Comparison of Calculated Analytical Ultimate Load with Experimental Values

S.No Ultimate Load (kN) P(exp)/P(MG) P(exp)/P(SMM)

Experimental MacGregor Model SMM

1 45Sonia,(2000) 46.44 44.45 0.969 1.0123

2 71.95Anthony, (2004) 78.65 70.65 0.915 1.018

3 274.5Anthony, (2004) 279.85 271.5 0.98 1.011

4 32.5Nasir-Uz-Zaman(2009) 36.85 31.65 0.882 1.027

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Int. J. Struct. & Civil Engg. Res. 2014 Sankar Jegadesh J S and S Jayalekshmi, 2014

6. Nasir-Uz-Zaman M and Sohel Rana M(2009), “Experimental and AnalyticalInvestigation of Flexural Behavior ofReinforced Concrete Beam”, Under-graduate Thesis Report, Department ofCivil Engineering, Khulna University ofEngineering and Technology, Khulna,April.

7. Sonia L Parvanova (2000), “Modellingthe Nonlinear Behavior of RCC Beamswith Moderate Shear Span and withoutStirrups”, Asian Journal of CivilEngineering (Building and Housing),Vol. 8, No. 5, pp. 479-489.

8. Thomas T C Hsu and Ronnie R H Zhu(2002), “Softened Membrane Model forReinforced Concrete Elements in Shear”,ACI Structural Journal, Vol. 99, No. 4.

9. Vecchio F J and Collins M P (1986), “The

Modified Compression Field Theory forReinforced Concrete ElementsSubjected to Shear,” ACI Journal,Proceedings, Vol. 83, No. 2, pp. 219-231.

10. Vecchio F J (1989), “Nonlinear FiniteElement Analysis of Reinforced ConcreteMembranes”, ACI Structural Journal,Vol. 86, No. 1, pp. 26-35.

11. William K J William and Warnke E D(1975), “Constitutive Model for theTriaxial Behavior of Concrete”, in Proc.of the Int. Association for Bridge andStructural Engineering, ISMES,Bergamo, Vol. 19, p. 174.

12. Zhu R R H, Hsu T T C and Lee J-Y (2001),“Rational Shear Modulus for SmearedCrack Analysis of Reinforced Concrete”,ACI Structural Journal, Vol. 98, No. 4,pp. 443-450.