INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN Parametric study on Nonlinear Finite Element Analysis on flexural behaviour of RC beams using Vasudevan. G 1, Kothandaraman.S 2 1- Assistant Professor, Perunthalaivar Kamarajar Institute of Engineering and Technology Karaikal , Puducherry UT, India 2- Professor and Head of Civil Engineering, Pondicherry Engineering College, Puducherry , India vasug1967@gmail.com doi: /ijcser ABSTRACT Nonlinear behaviour of RC beams is complex due to involvement of various parameters. Many attempts have been made by the past researchers to predict the behaviour using. The accuracy and convergence of the solution depends on factors such as mesh density, constitutive properties of concrete, convergence criteria and tolerance values etc., Past researchers have used various values of the above factors without providing much generalized guidelines. Hence, in order to lay a wider base for the behaviour prediction of RC beams using, a large number of trial analysis were carried out by changing various parameters. In this paper, results of the four point bending analysis conducted with respect to concrete constitutive properties, mesh density, use of steel cushion for the supports and loading points, effect of shear reinforcement on flexural behaviour, convergence criteria, and impact of percentage of reinforcement are analysed and discussed. The outcome of this work will provide a wider platform for further usage of in the analysis of RC beams. Keywords: Material nonlinearity, Convergence, Steel cushion, Shear reinforcement,. 1. Introduction Experimental study on flexural behaviour of Reinforced Concrete (RC) beam involves cost of materials, testing devices, labour and time. Usually, finite element (FE) analysis is also carried out to counter check the test values. This helps in refining the analytical tools, so that even without experimental proof or check the complex nonlinear behaviour of RC beams can be confidently predicted. Hence, wider attempts were made by various researchers to accurately predict the behaviour of RC beams till complete failure using various FE software. It has been found that due to quasi-brittle material behaviour of concrete, many parameters are to be properly taken into consideration in order to obtain an accurate solution. Hence, numbers of trial analyses are carried out using 12.0 by changing various parameters which influences the accuracy and convergence. Idealization of reinforcement in concrete, constitutive properties of concrete, mesh density, incorporation of boundary conditions for supports and symmetric planes, modeling of loading and support regions, effect of shear reinforcement on flexural behaviour, effect of convergence criteria, impact of percentage of reinforcement and Received on August 2011 published on September

2 other parameters which governs the analysis are considered for the present study. The results and discussion of the present study are compared with the findings available in the literature. 2. Problem considered for the study For the proposed study, beam model used by Wolanski, 2004 is considered by making suitable conversion to SI units. The length of the beam is mm with supports located at 76.2 mm from each end of the beam allowing a simply supported span of 4572 mm. The cross-section of the beam is 254 mm x mm with main reinforcement of 3 bars of each area 200 mm 2 and shear reinforcement of 25 nos. with area of each vertical link as 71 mm 2. The detail of the RC beam model is as shown in Figure 1. Figure 1: Beam considered for the study Figure 2: Idealization of rebar in concrete 3. Idealization of steel reinforcement in concrete elements The steel reinforcement is incorporated in concrete using either discrete model, embedded model or smeared model depending on the geometry of the system. In the discrete model, spar or beam elements with geometrical properties similar to the original reinforcing elements are connected to concrete mesh nodes and hence the concrete and the reinforcement mesh share the same nodes. Concrete mesh is restricted by the location of the reinforcement. Also, the concrete occupies the same regions occupied by the reinforcement and the volume of the steel reinforcement is not deducted from the concrete volume [Wolanski, 2004 and Kachlakev et al., 2001]. The embedded model overcomes the concrete mesh restriction because the stiffness of the reinforcing steel is evaluated separately from the concrete elements. The model is built in a way that keeps reinforcing steel displacements compatible with the surrounding concrete elements. For 99

3 complex reinforcement details, this model is advantageous. However, this model increases the number of nodes and degrees of freedom which increases the run time and computational cost. The smeared model assumes that the reinforcement is uniformly spread throughout the concrete elements in a defined region of the FE mesh. The effect of reinforcing is averaged within the pertaining concrete element [Dahmani et al., 2010]. This approach is used for large-scale models where the reinforcement does not significantly contribute to the overall response of the structure. The features of the above techniques are schematically shown in Figure 2. Hence, for the modeling of RC beams with well defined geometry and reinforcement details, the discrete modeling approach provides an accurate and true representation of the field reality. Earlier researchers [Wolanski, 2004, Kachlakev et al., 2001 and Dahmani et al., 2010] also suggested the discrete modeling strategy due to the facts stated above. Hence, the discrete modeling is followed for all the analysis presented in this report. 4. Elements used for modeling For modeling RC beam, eight noded Solid65 element with three degrees of freedom at each node (translations in the nodal x, y, and z directions), which handles nonlinear behaviour, cracking in three orthogonal directions due to tension, crushing in compression and plastic deformation is used. For modeling reinforcement, two noded Link8 spar element with three degrees of freedom at each node (translations in the nodal x, y, and z directions), which handles plasticity, creep, swelling, stress stiffening and large deflection is used. In order to avoid stress concentration problem, the supports and loading points are modeled with eight noded Solid45 element with three degrees of freedom at each node (translations in the nodal x, y, and z directions), which handles plasticity, creep, swelling, stress stiffening, large deflection and strain. 5. Effect of mesh density on accuracy and convergence of analysis For the analysis only a quarter of the beam is considered by using the symmetry of the geometry and loading so as to effectively utilize the computational time and available disk space [Figure 3]. In order to depict the behaviour of full size beam, nodes defining a vertical plane through centroid of the beam cross-section are given a degree of freedom constraint UX = 0 and all nodes selected at Z = 0 are given the constraint UZ = 0. The support nodes were constraint along UY and UZ directions in order to create roller condition. The accuracy and the convergence of the results mainly depend on the mesh density. An optimum mesh density is arrived by conducting few numbers of trial analyses by varying the mesh density. For the study on mesh density, four trial analyses are carried out using 2790, 4185, 5580 and 8370 Solid65 concrete elements [Figure 4]. A plot of load versus midspan deflection [Figure 5] shows that the behaviour remains almost same up to steel yielding stage. After the yielding of steel, there is a small variation in the load versus deflection behaviour. It is also observed that for model with 2790 elements, the analysis terminated at kn due to non-convergence problems. Figure 6 shows the plot of number of elements versus midspan deflection at ultimate load, which shows little variation of midspan deflection with respect to number of elements from 5580 to Hence as a preliminary step a few numbers of trial analyses are carried out to decide the optimum mesh density. 100

4 6. Properties of concrete Concrete is a quasi-brittle material and has different behaviour in compression and tension. In the present study, analysis is carried out by using three stress-strain models proposed by Hognestad [Park and Paulay, 1975], simple stress-strain model [Wolanski, 2004 and Kachlakev et al., 2001] and IS 456:2000 stress-strain model as shown in Figure 7. Stress-strain curves for concrete in compression arrived using the above models are shown in Figure 8. The load-deflection curves indicated that the behaviour of beam remains almost the same for all the above models. Modulus of elasticity of concrete determined by any reliable experimental and analytical method may be used. In this report, modulus of elasticity E 5000 f as per IS 456:2000 codal provision is c adopted. Shear coefficient of zero represents a smooth crack (i.e., complete loss of shear transfer due to no aggregate interlock) and one represents a rough crack (i.e., no loss of shear transfer due to full aggregate interlock). Uniaxial tensile cracking stress obtained using f 0. 7 f as per IS 456: 2000 is used in the analysis. For cracked tensile t ck condition, the effect of tension stiffening is incorporated using stiffness multiplier constant (T c ). After cracking, the uniaxial tensile strength of the concrete (f t ) drops abruptly to a fraction of it (T c f t ) and approaches to zero at a strain 6 times the cracking strain as shown in Figure 9. A parametric study has been carried out by the authors by varying the value of T c and found that the results remain unchanged. Hence, a default value of 0.6 incorporated in is used for all the analysis. From the literatures and the recommendations of the manual, the various values used in the analysis are listed in the Table 1. Parameters which are not stated in the report are taken as program default [, 2005]. ck Figure 3: Quarter beam FE model Figure 4: Model with varying mesh density 101

5 Figure 5: Effect of varying mesh density Figure 6: Mesh density on midspan deflection Figure 7: Stress-strain model for concrete in compression 102

6 Figure 8: Concrete stress-strain models Figure 9: Tensile strength of cracked Concrete 7. Material properties of reinforcing steel and steel plate cushion The steel reinforcement used for the finite element models is assumed to be an elasticperfectly plastic material, identical in tension and compression as shown in Figure 10. The bi-linear elastic-plastic stress-strain for steel reinforcement to be used with Link8 element is furnished in two sets of data. Modulus of elasticity of N/mm 2 and Poisson s ratio of 0.3 is used to setup a linear isotropic model, which is for the elastic range. For bilinear isotropic hardening model of Link8 element, the specified yield stress, the stress-strain curve of reinforcement continues along the second slope defined by the tangent modulus. It is also experienced that for tangent modulus a small value of 10 to 20 N/mm 2 shall be used to avoid loss of stability upon yielding. In the present study, yield stress (f y ) of 414 N/mm 2 and tangent modulus of 20 N/mm 2 is used for reinforcement [Wolanski, 2004]. The modulus of elasticity and the Poisson s ratio for Solid45 element for modeling steel cushion is considered as same as that of the steel reinforcement. Table 1: Material property for concrete Material property Value Characteristics strength of concrete at 28 days N/mm 2 Modulus of elasticity of concrete N/mm 2 Poisson s ratio 0.3 Shear coefficient for open crack 0.3 Shear coefficient for closed crack 1.0 Uniaxial crushing stress -1.0 Uniaxial tensile cracking stress N/mm 2 Stiffness multiplier for cracked tensile condition

7 Figure 10: Stress-strain curve for reinforcement steel 8. Effect of convergence criteria on accuracy and convergence of solution For nonlinear analysis of RC beams, use of default convergence criteria experiences nonconvergence problems after concrete starts to cracks. Various convergence criteria were followed by earlier researchers after the formation of first crack in concrete. Wolanski, 2004 used default convergence criteria up to the formation of initial crack. Thereafter, the force convergence criteria were dropped and a tolerance limit of 0.05 was used for displacement convergence criteria. Kachlakev et al., 2001 and Dahmani et al., 2010 used convergence tolerance limits as and 0.05 for force and displacement. Revathi et al., 2005 adopted a tolerance limit for convergence as at lower load levels and 0.04 at higher load levels for both force and displacement. Wu, 2006 followed a tolerance limit of 0.05 for force and displacement convergence. A detailed study has been conducted by the authors, for wide range of tolerance limits by keeping other values to program default and the salient features of the trials are presented in Table 2. Plot of load versus midspan deflection at ultimate load level for various convergence trials are shown in Figure 11. It is noted that the analysis with lower convergence limits (CON1) requires more number of trials and ultimately increase in computational time and disk space requirement. However, the maximum midspan deflection obtained by this trial ( mm) is in very close agreement with experimental value (92.71 mm) [Wolanski, 2004]. Also noted that, irrespective of the convergence limits used, the behaviour of beam remains same up to steel yielding stage. The variation of number of iterations, ultimate load and corresponding midspan deflection due to various convergence criteria are plotted in Figure 12. From the above plot, it shows that the number of iterations is not increased significantly due to higher convergence limits. Table 2: Convergence study Force Displacemen Number of Ultimate tolerance t tolerance iterations load (kn) CON CON CON CON CON Convergence scheme Midspan deflection (mm) 104

8 Figure 11: Convergence criteria on behaviour Figure 12: Convergence study 9. Importance of load step and load increment In order to predict the nonlinear behaviour, the total load is to be divided into series of load increments (or) load steps as required by Newton-Raphson method. The automatic time stepping in the program predicts and controls load step sizes for which the maximum and minimum load step sizes are furnished. The number of load steps, minimum and maximum step sizes is determined after attempting many trial analyses. During the initiation of concrete crack, the steel yielding stage and at the ultimate stage where large numbers of cracks occurs, the loads are applied gradually with smaller load increments. For the present analysis load step pattern followed by Wolanski, 2004 is used. Failure of the model is identified where the solution fails to converge even with very low load increment. 105

9 10. Need for steel cushion for supports and loading point In order to overcome the stress concentration problems at the support and loading points, Wolanski, 2004, Elavenil et al., 2007 and Ibrahim et al., 2009 had included steel cushion at the supports and at the loading points using Solid45 element bonded with the Solid65 elements at the nodal points. The restraint and loading was applied to the nodes of the Solid45 elements. Wu, 2006 had studied the effect of steel cushion on the behaviour of RC beams and stated that the response of the beam remains practically the same. Figure 13 shows the FE model with and without steel cushion. The load versus deflection at midspan shown in Figure 14 indicated that the responses of the analysis in both the cases are practically the same up to the yielding of steel, which is almost 95% of the ultimate load. It is also noted that the load at first crack also varies marginally as indicated in Figure 15. Hence, for the evaluation of flexural response of RC beams, the inclusion of steel cushion may not be necessary. However, by comparing the stress contour diagrams as indicated in Figure 15, for the detailed study on stress variation at the loading and support location, the steel cushion has to be included in the modeling. Figure 13: Model with and without steel cushion 106

10 Figure 14: Effect of steel cushion 11. Effect of shear reinforcement (stirrups) on flexural behaviour While modeling RC beams for flexural analysis, the beams are to be modeled including the shear reinforcements using Link8 elements so as to reflect the field reality. However, some of the researchers excluded the shear reinforcement for simplicity [Dahmani et al., 2010]. In order to study the effect of excluding shear reinforcement on the flexural behaviour, a comparative study is undertaken and the results are discussed. Figure 16 shows the FE model with and without shear reinforcement. Load versus deflection diagram shown in Figure 17 indicated that at ultimate load level, there is a small variation in the load versus deflection behaviour due to building up of more shear force. The crack pattern and stress distribution shown in Figure 18 and 19 indicated that the load at first crack for beams with shear reinforcement has marginally increased from kn to kn. Also, noted that at kn and at kn load more diagonal tension cracks are appeared for beams without shear reinforcement. Hence, for the more accurate prediction of nonlinear behaviour RC beams, the shear reinforcements are to be included in the modeling. Figure 15: Stress contours at first crack load with and without steel cushion 107

11 12. Impact of percentage of tension steel on flexural behaviour Flexural behaviour RC beams due to variation in percentage of reinforcement (p t ) is studied by using 0.33, 0.58, 0.91 (under reinforced), 1.53 (balanced), 2.05 and 2.34 (over reinforced) percentages. The plots of load versus deflection at midspan are displayed in Figure 20. It is noted that the behaviour in uncracked elastic range is almost the same for various percentage of steel, which is mainly dependent on the grade of concrete. It is also noted that for higher values of percentage of reinforcement the transition is smooth due to contribution of moment of inertia by the steel in lieu of loss of moment of inertia due to cracking. The effect of tension reinforcement on first cracking load, ultimate load and corresponding midspan deflections are shown in Figure 21. It is observed that the initial cracking behaviour is not much influenced by the percentage of reinforcement. However, it has more impact in the post-cracking behaviour. Also the ultimate capacity of the beam can be varied by varying the percentage of tension reinforcement. 13. Conclusions Based on the parametric study conducted on the four point bending nonlinear FE analysis of RC beams using software the following conclusions are drawn: Figure 16: With and without stirrups Figure 17: Behaviour with and without stirrups 108

12 Figure 18: Effect of shear reinforcement on crack pattern Figure 19: Effect of stirrups on longitudinal stress (X) distribution 109

13 Figure 20: Reinforcement % on behaviour Figure 21: Effect % of reinforcement 1. An optimum mesh density should be arrived by performing a few preliminary trial analysis. 2. Stress-strain model recommended by IS 456: 2000 can be used for concrete as the results are in close agreement with models used by past researchers. 3. The stiffness multiplier for cracked tensile condition has no effect on the behaviour of beams and hence default value can be used. 4. Near the first cracking stage, steel yielding stage and at the ultimate stage lower convergence limits are to be used for accurate prediction of behaviour. 5. The total load is to be divided into a number of suitable load steps (load increments) by conducting a few trial analyses until a smooth load versus deflection curve is obtained. 6. For prediction of general flexural behaviour, the use of steel cushion may not be required. However, for the detailed study on stress concentration at the loading and support location, the steel cushion is to be included. 7. The initial cracking behaviour is not varying much with varying percentage of reinforcement. However, in the steel yielding level the variation is much and the ultimate strength can be varied by varying the percentage of reinforcement. 8. The tension and shear reinforcements are to be precisely incorporated using discrete modeling technique in order to get more accurate behaviour. Acknowledgement The authors thankfully acknowledge Dr. M. C. Sundarraja, Assistant Professor, Department of Civil Engineering, Thiagarajar College of Engineering, Madurai , Tamilnadu, India, for having graciously permitted us to use the software for this work. 110

14 14. References 1. Commands Reference, (2005),, Inc. Southpointe, 275 Technology Drive, Canonsbury, PA 15317, 2. Dahmani, L., Khennane, A., Kaci, S. (2010), Crack identification in reinforced concrete beams using software, Strength of materials, 42 (2) pp Elavenil, S., Chandrasekar, V. (2007), Analysis of reinforced concrete beams strengthened with ferrocement, International Journal of Applied Engineering Research, Research India Publication, 2(3), pp , 4. Ibrahim, A.M., Sh.Mahmood, M. (2009), Finite element modeling of reinforced concrete beams strengthened with FRP laminates, European Journal of Sci. Research, Euro Journals Publishing, Inc., 30(4), pp IS 456:2000, Indian Standard: Plain and reinforced concrete code of practice, Bureau of Indian Standards, New Delhi. 6. Kachlakev, D., Miller, T., Yim, S., Chansawat, K., Potisuk, T. (2001), Finite element modeling of reinforced concrete structures strengthened with FRP laminates, SPR 316, Oregon Department of transportation Research Group, Salem, OR and Federal Highway Administration, Washington, DC Park, R, Paulay, T. (1975), Reinforced concrete structures, John Wiley & Sons, Inc., New York. 8. Revathi, P., Devdas Menon. (2005), Nonlinear finite element analysis of reinforced concrete beams, Journal of Structural Engineering, Structural Engineering Research Centre, Chennai, 32(2), pp , 9. Wolanski, A.J. (2004), Flexural behaviour of reinforced and pre-stressed concrete beams using finite element analysis, M.S.Thesis, Marquette University, Wisconsin. 10. Wu, Z. (2006), Behaviour of high strength concrete members under pure flexure and axial-flexural loadings, Ph.D.Thesis, North Carolina State University, Raleigh, North Carolina. 111