Nonlinear Finite Element Flexural Analysis of RC Beams

Transcription

1 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp Nonlinear Finite Element Flexural Analysis of RC Beams Mazen Musmar Civil Engineering Department, The University of Jordan, Aan, Jordan Abstrat: The availability of nonlinear material models, and the existene of omputer software that an model the behavior of reinfored onrete (R) strutural members until failure, made finite element (FE) strutural modelling more attrative. FE omputer simulation for R strutural members is ost effetive. It also gives detailed results of the evolution of the omplex distribution of internal stresses and strains in onrete and steel reinforement in the R strutural member, in a matter that annot be attained upon arrying out the muh more expensive experimental testing. This paper presents a threedimensional nonlinear FE analysis to study the flexural behavior of RC beams. Partiular attention is paid to the nonlinear material modeling and to the failure surfaes of both onrete and steel reinforement and the heterogeni physial onrete properties and onrete raking. Obtained results from the FE model are ompared with analytial values alulated using ACI (18-14) provisions. The work involves investigating all the evolution of the internal ations assoiated with the time dependent applied inremental loading up to failure. In this study, the FE model is intended to simulate the behavior of under reinfored R beam that is designed to fail in flexure. The study showed that the performane of the FE model of the R beam, in terms of rak initiation, propagation and pattern, as well as load defletion relationship, flexural strength and mode of failure, realistially simulate the atual behavior when the proper material models are implemented in the FE model. This work aims to illustrate the vital role of FE strutural modeling in simulating the performane of R elements. The detailed obtained results that an be obtained from FE analysis would lead to a better understanding of the internal ations and stress resultants in steel reinforement and onrete that form the main omponents of R beams. FE omputer simulation would help in further development of design rules for R members. Keywords: Performane of Reinfored onrete beams, Nonlinear Finite element analysis, Flexural behavior of RC beams INTRODUCTION Safety and servieability assessment of omplex strutures, neessitate the development of aurate and reliable methods and models for their analyses, both analytial and experimental studies are to be arried out side by side. Experimental studies are expensive, and time onsuming but they do not give detailed information [1]. Aording to kwak [], it is very important to desribe the behavior of the strutures under overload onditions and estimate their ultimate strength aurately. The earliest publiation on the appliation of the finite element method to the analysis of R strutures was presented by Ngo and Sordelis []. In their study, simple beams were analyzed with a model in whih onrete and reinforing steel were represented by onstant strain triangular elements, and a speial bond link element was used to onnet the steel to the onrete and desribe the bond slip effet. A linear elasti analysis was performed on beams with predefined rak patterns to determine prinipal stresses in onrete, stresses in steel reinforement and bond stresses. Nilson [4] introdued nonlinear material properties for onrete and steel and a nonlinear bond-slip relationship into the analysis and used an inremental load method of nonlinear analysis. Four onstant strain triangular elements were ombined to form a quadrilateral element by ondensing out the entral node. Craking was aounted for by stopping the solution when an element reahed the tensile strength. For the analysis of RC beams with material and geometri nonlinearities, Rajagopal [5] developed a layered retangular plate element with axial and bending stiffness in whih onrete was treated as an orthotropi material. R beams have also been treated by many other investigators [6,7], using similar methods. Reza and Seyed [8] experimentally and theoretially investigated six under-reinfored onrete beams. Eah onrete beam was reinfored with two 16 diameter steel bars for tension. Mohaadhassani et al [9] studied the effet of different web reinforement on the stress strain distribution of R deep beams, they onluded that strain distribution aross the setion is nonlinear even before the yielding of steel reinforement. Nasr et al [10] arried out FE analysis to study the effet of transverse openings on the performane of R beams. Also Amir et al [11] studied the effet of fly ash on the flexural apaity of R beams. Metwally [1] suggested inluding the formulation of a onstitutive model for time dependent effets suh as onrete reep, shrinkage, and fire exposure, in order to realistially simulate their long term effet on the performane of R beams. 014

2 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp Tiejiong et al [1 ] undertook a FE analysis to examine the behavior of prestressed onrete beams. Kim et al [14] experimentally tested a full sale prestressed girder. In their study, numerial analysis using the FE method was onduted in parallel, in order to inrease the reliability of the experimental results and to understand the unpreditable additional behaviors due to the laboratory environment. Aording to Kwak [], onrete stress strain relation in ompression exhibits a nearly linear elasti response up to about 0% of the ompressive strength. This is followed by gradual softening, influened by rak spread and expansion. This nonlinear stage extends up to the onrete ompressive strength, when the material stiffness drops to zero. Beyond the ompressive strength, the onrete stress strain relation exhibits strain softening until failure takes plae by rushing. The onrete in the tension side behaves as linear elasti up to the onrete tensile strength, defined as the onrete modulus of rupture. On the other hand the steel reinforement stress strain relation in both ompression and tension zones is linear elasti up to steel yield strength. This is followed by the strain hardening stage up to steel ultimate strength. astly, the strain softening stage extends up to failure. Currently, the existene of omputer software that an model the nonlinear behavior of reinfored onrete strutural members up to failure, made omputer numerial modelling more attrative as it minimizes the ost and is muh faster than the experimental testing. It is ruial to know that implementing the material models that best desribe the mehanial properties of the materials would give the results that would more realistially simulate the experimental testing. The objetive of this paper is to perform a nonlinear FE analysis on a simply supported reinfored onrete beam. The work involves onsidering the load dependent variation of the material models of onrete and steel reinforement in tension and in ompression sides. The numerial model development involves the following: Construting a -D finite element model of the RC beam. The model involves the geometry of the beam and reinforement bars detailing, appropriate material models, meshing, applied boundary onditions and loading. Apply inremental loading aimed to apture the flexural response of R beam up to failure. Performing strutural stress analysis to obtain defletion, stress strain results in both onrete and steel reinforement. As well as traing and reording rak formation, development, and pattern. Validating the strutural model outomes by omparing them by analytial values alulated using ACI [15] provisions. Analytial alulations are shown in Appendix. Evaluating the load defletion relationship, investigating rak initiation, pattern and propagation up to failure, and ultimately identifying the mode of failure.. Finite Element Model. Geometry and oading. The RC beam has a 5.1 m span. The retangular setion width is 00, and depth is 00. The one layer bottom reinforement involves 1 orner rebars and 114 rebar at the middle. Stirrups are 18 / 150 /. Top reinforement is 10. Conrete over for rebars is 5. Rebars detailing is given in Figure (1) Figure 1. Detail of longitudinal and ross setion of RC beam Material Models It is not an easy task to implement in the FE model, the mutual behavior of the brittle onrete of load dependent material model, and the dutile steel reinforement load dependent material model. Conrete itself, exhibits a material model in tension that is different from that in ompression. In the ase of 015

3 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp onrete in the ompression side, onrete behaves as linear elasti in the first stage, followed by a nonlinear behavior in the seond stage that involves gradual stiffness softening. Nonlinear isotropi hardening ontrols onrete behavior in the seond stage. This is followed by strain softening ontrolled by non-metal plastiity and non-assoiated flow rule until failure takes plae as onrete rushing as shown in Figure (). Stress (MPa) Figure (): Conrete Material Model[16]. Conrete in the tension side, behaves as linear elasti up to the onrete tensile strength. Conrete properties are listed in Tables (1,). Table 1: Conrete properties prior to yield surfae. Material Conrete Strain (/) Material model inear Elasti Modulus of elastiity MPa.574x Poisson's ratio Table : Conrete parameters beyond yield surfae. Open shear transfer oeffiient, t 0. Closed shear transfer oeffiient, 0.9 Uniaxial raking stress Uniaxial rushing stress f'.40 Mpa 0 Mpa Stress (MPa) Strain () Figure. Steel Material Model Steel reinforement material is modelled as bilinear in the ompression and in tension sides, (Figure). The first stage is linear elasti up to steel yield strength. Followed by linear kinemati strain hardening stage, with a tangent modulus of elastiity of 10% of steel. Steel properties are listed in Table (). Table. Properties for the steel reinforement Material Model prior to initial yield surfae Poisson's ratio Yield stress, f y inear elasti = MPa Density 7800 Kg/m Elasti modulus, E s 00 GPa Finite Elements. Due to syetry about a vertial plane at b/, and about a vertial plane at / of the beam, in geometry, loading, boundary onditions and material, it was deided to build a quarter beam model in order to save the omputational time, (Figure 4). The support at the left support is pin. The applied load on width b/ beomes P/4. Figures (5,6) illustrate the steel reinforement distribution in the quarter span FE R model. Figure 4: FE Model for quarter beam illustrating oad and boundary onditions. 016

4 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp expressed in the form: f F S 0 (1) Where: F: a funtion of prinipal stress state S: failure surfae f: uniaxial onrete strength. Figure 5. Isometri view of the quarter beam FE model If equation 1 is satisfied, the material will rak or rush. Solid65 is apable of plasti deformation, raking in three orthogonal diretions, at eah integration point. The raking is modelled through an adjustment of the material properties that is done by hanging the element stiffness matries. Element ink180 was adopted to model the steel reinforement. It is a uniaxial element that has a single degree of freedom at eah nodal point. Bilinear plastiity model ontrols the failure of steel reinforement. It implements Von Mises yield surfae with assoiated plasti flow and kinemati hardening, available in Ansys [17]. Figure 6: Isometri view of Steel reinforement details The solid65 element [17] models the nonlinear response of the reinfored onrete. The behavior of the onrete material is based on a onstitutive model for the triaxial behavior of onrete after Williams and Warnke [18]. Solid65 is apable of plasti deformation, raking in three orthogonal diretions at eah integration point. RESUTS AND DISCUSSION. Model Validation. The proposed finite element analysis was validated by omparing the FE model solution with the analytial values alulated using ACI [15]. Table (4) presents a omparison of obtained results. It indiates that the results ompare well. Table 4. Results of FE simulation and analytial values. Case FE Model Analytial approah Initial oad 9.6 kn kn raking Defletion 1.6. Prior to failure oad 4.8kN 6.1 kn Defletion Figure 7: Solid65 element [18] The raking is modelled through an adjustment of the material properties that is done by hanging the element stiffness matries. If the onrete at an integration point fails in uniaxial, biaxial, or triaxial ompression, the onrete is assumed rushed at that point. Crushing is defined as the omplete deterioration of the strutural integrity of the onrete. The riterion for failure of onrete due to a multi axial stress is FE analysis results The performane of the FE omputer simulation of the RC beam is investigated. The R beam was subjeted to inremental transverse two point loading up to failure. Figures (8a,8b) depit the development of raks within the R beam. As the applied load exeeded the raking load of 9.6 kn, flexural raks started at the regions in whih onrete tensile stresses reahed modulus of rupture. The initial raking of the R beam. The first raks appeared in the onstant moment region. They are vertial flexural raks. It was notied that 017

5 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp flexural raks development was flat and sudden. This implies that the tensile stress relaxation is not funtioning properly, leading to a sudden stress drop. At larger loading, flexural shear raks appeared. Subsequent raking ourred as more load was applied to beam. Craking inreased and expanded in the onstant moment region. Then the beam started raking out towards the supports, and expanded towards the ompression zone. Yielding of steel reinforement took plae at a fore of.0 kn, and at 0. mid-span defletion. Beyond steel yielding, raking notieably spread and expanded towards the ompression zone as depited in Figure (8b). a. Craks at the raking load whih is assoiated with small inrements of steel stresses that are aompanied with large plasti strain values. Ultimately, the resulting large onrete strain values in the ompressive side aused onrete rushing in the ompression side, when onrete strain reahed ultimate ompressive strain oad (N) Mid span defletion () Figure 9. oad defletion relationship b. Craks just before failure Figure 8. Craks in the FE model Due to kinematis of deformation and ompatibility, onrete ompressive strain also inreased signifiantly. Ultimately, when the onrete strain in the ompression side reahed the onrete ultimate ompressive strain, the R beam failed by rushing of onrete at a load P= 4.8 kn, and mid-span defletion of 7.5. Figure (9) depits the reorded midspan defletion versus the inremental applied loading. The shape of the plot is trilinear with a kink that depits the first rak formation. Initially and up to a load of 9.60 kn, the load defletion urve is linear elasti. In this stage, the linear elasti material properties for both steel reinforement and onrete define the flexural rigidity. At the raking load P r, raks formed in tension onrete when onrete tensile stresses reahed the modulus of rupture. This is followed by nonlinear onrete behavior. At load =.0 kn, tension steel stresses reahed Fy, the load defletion relationship refleted the substantial loss of stiffness in tension steel. This stage extended until failure took plae. Figure (10) depits the rebars axial stresses just before failure. It shows that axial stresses in ompressive steel are very low as the R beam is designed as under-reinfored. On the other hand, maximum reorded axial stress in tension steel is 46 MPa, (Figure10). This indiates that the stresses surpassed the steel tensile strength Fy value of 460 Mpa, showing that at the verge of failure, Steel was in the stress hardening bilinear stage, Figure 10. Isometri view of the Rebars and CONCUSIONS Stirrups for quarter beam FE model A nonlinear material FE model has been used to simulate the behavior of R beams. The FE model has been verified by omparison between FE model and anlytial results aording to ACI [15]. The following onlusions were obtained upon arrying out inremental applied loading up to failure: The initial raking in the FE model formed at the maximum positive moment region. As the load inreased with time upon the appliation of inremental loading, the flexural 018

6 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp raks inreased and extended towards the top. Subsequently diagonal shear raks appeared in regions loser to the supports. Finally, a plasti hinge formed at the positive moment region, ultimately the beam failed in flexure. The load defletion response is linear elasti up to the raking moment strength, then the urve inlines more towards the horizontal exhibiting degradation of flexural rigidity. After yielding of tension steel reinforement, the urve inlines appreiably towards the horizontal. The FE simulation of the under reinfored onrete beam gave the same performane as that antiipated by ACI ode [15] for under reinfored beams that are designed to fail in flexure, in terms of dutility, onrete progressive raking, steel reinforement yielding, showing lues for ample warning before it ultimately fails by rushing of onrete in ompression zone. Finally it an be onluded that with the availability of the proper nonlinear material models, and the existene of omputer software that an model the nonlinear behavior of reinfored onrete strutural elements until failure, FE strutural modeling would give results that would more realistially simulate the experimental testing. Appendix Analytial evaluation based on ACI ode [15]. Calulations are arried by means of an exel sheet. Beam selfweight is not onsidered neither in the FE model nor in the analytial analysis. 10 f ' 0MPa, E MPa, fy 460MPa, E s.010 MPa, 0. beamwidth( b) 0.0m, depth( h) 0. m As 80, moment arm( a). 5m onover 0.05m, d 0. 75m ' fr 0.6 f. 4MPa E n E s h ( n 1) As( d ) y b. h ( n 1) As I ut 1 1 ( ). b. h b. h. y0 ( n 1). As( d y0 s h ) I ut At first rak initiation As b. d kd (. n (. n). n). d I t 1 b( kd) n. As( d kd) fr. Iut M r kNm h ( y0) P. a M r P kN.5 48E I At failure M kNm P. a M 1 (M (l 4a ) l ut * 40.6 P 6. 1kN.5 I e M M r ) 4. M r 8 4. Iut (1 ). It (M (l 4a ) l ) E I 4. 7 REFERENCES e M 1 [1] Park R, Pauley, "Reinfored Conrete Struture", New York, John Wiley and Sons,

7 International Journal of Applied Engineering Researh ISSN Volume 1, Number 4 (018) pp [] Hyo-Gyoung Kwak and Sun-Pil Kim, "Nonlinear Analysis of RC beams based on moment-urvature relation", Computers and Strutures, 00 [] Ngo D, Sordelis AC, "Finite Element Analysis of Reinfored Conrete Beams", ACI J 1967,64():156. [4] Nilson A," Internal Measurement of Bond Slip", Journal of ACI, 69-7(197) [5] Rajagopal K.R. "Nonlinear Analysis of Reinfored Conrete Beams, Beam-Columns and Slabs by Finite Elements. Ph.D Dissertation, Iowa State University [6] Rots, J.G., Nauta, P., Kusters, G.M.A. and Blaauw endraad, J. (1985), Smeared Crak approah and Frature oalization in Conrete, HERON, Vol.0,no.1. [7] Bergmann, R. and Pantazopoulou, V.A., Finite element for R Shearwalls under Cyli oads, Department of Civil Engineering, Report UCB/ SEMM-88/09, University of California, Berkely, [8] Reza Mahjoub, and Seyed Hamid, Finite element Analysis of R beams strengthened with FRP sheets under bending, Australian Journal of Basi and Applied Sienes, Vol.4, no.7, 010 [9] Mohaadhassani Mohaad, Mohd Zamin Bin Jumaat, Mohamed Chemrouk, Ali Ghasemi, SJS Hakim, Rafieipour Najmeh, "An experimental investigation of the stress strain distribution in high strength onrete deep beams", SieneDiret, Proedia Engineering 14(011) [10] Nasr Z, Alaa G, Amal H,"Finite Element analysis of Reinfored Conrete Beams with opening Strengthened Using FRP", Ain Shams Engineering Journal, 017 [11] Amir M. A, Amin O, Shima N, Peyman B, Naseri M.H," The Effet of Fly Ash On Flexural Capaity Conrete Beams", Advanes in Siene and Tehnology Researh, Vol10, no0, 016. [1] Metwally I.M," Three dimensional nonlinear finite element analysis of onrete deep beam reinfored with GFRP bars", Conrete Strutures Researh Inst, HBRC Journal(017) 1. [1] Tiejiong, Sergio M, Adelino V," Flexural Response Of Continuous Conrete Beams Prestressed with External tendons", Journal of Bridge Engineering, ASCE, 01. [14] Kim T, Do-Hak K, Moon K, Yung M" Flexural behavior of a half-deked bulb tee pre-stressed Conrete Girder: full Sale Test and Nonlinear Finite Element Analysis", Struture and Infrastruture Engineering, 017. [15] Building Code Requirements for Strutural Conrete ACI (18M-14) and Coentary, Amerian Conrete Institute. [16] Knaak M. Adam, Yahya C. Kurma, David J, Kirkner, " Compressive Stress Strain Relationships for North Amerian Conrete under Elevated Temperatures", ACI Materials Journal, Vol. 108, No., 011. [17] Ansys (Release 15), Theory Referene for the Mehanial APD and Mehanial appliations, 009. [18] William, K.J. and Warnke, E.P., Constitutive Model for the Triaxial Behavior of Conrete. Proeedings of the International Assoiation for Bridge and Strutural Engineering, Vol. 19, ISMES, Bergamo, Itali,