Finite Element Analysis of Reinforced and Pre-Tensioned Concrete Beams

Finite Element Analysis of Reinforced and Pre-Tensioned Concrete Beams Nimiya Rose Joshuva 1, S. Saibabu 2, P. Eapen Sakaria 3, K. N. Lakshmikandhan 4, P. Sivakumar 5 1 M.Tech Student, 3 Head of Department, Department of Civil Engineering, SAINTGITS College of Engineering, Kottayam, Kerala, India 2 Sr. Principal Scientist, 4 Scientist, 5 Chief Scientist, CSIR-SERC, Chennai, India Abstract Concrete is strong in compression but weak in tension. Reinforced concrete, in which steel rods are provided to resist tensile stresses, however, does not meet the satisfactory structural demands. The concept of prestressing was introduced to generate compressive stresses in concrete prior to loading, by means of prestressing tendons inserted in the member. These compressive stresses resist the tensile forces, thereby effectively increasing the tensile strength of the concrete member. In this study, reinforced and pre-tensioned concrete beams are analysed for their nonlinear behaviour under external loading using the finite element method of analysis. ANSYS 12.0, an efficient finite element software package, is used for the analysis of the concrete members. Load-deflection responses, variations of stresses in concrete and steel and the crack patterns at critical stages of loading are studied. The numerical predictions are compared to the data obtained using the theories of structural analysis. In comparison to the theoretically predicted data, the numerical method of analysis using ANSYS was seen to satisfactorily predict the behavioural responses of the beams up to ultimate, but was not as effective in predicting the strain variation in the prestressing tendons. Keywords finite element analysis, prestressed concrete beams, reinforced concrete beams. I. INTRODUCTION The efficient application of concrete structures requires an understanding of their response to a variety of loadings. There are a number of approaches for the study of the behaviour of concrete structures, viz., experimental, numerical, theoretical, etc. Finite Element Analysis (FEA) is a numerical one which provides a tool that can accurately simulate the behaviour of concrete structures. Finite Element Analysis, as used in structural engineering, determines the overall behaviour of the structure by dividing it into a number of simple elements, each of which has well-defined mechanical and physical properties. A number of commercial finite element analysis codes are available (ABAQUS, ATENA, ANSYS, NASTRAN, Hypermesh, etc.) for the analytical study of structures. 449 The use of computer software to model the elements has been proved to be convenient, faster and extremely costeffective compared to experimental analyses. This study presents an analytical investigation of the nonlinear behaviour of reinforced and pre-tensioned concrete beams in the finite element software package ANSYS 12.0. II. LITERATURE REVIEW Literature survey was carried out to comprehend the nonlinear behaviour of reinforced and prestressed concrete beams and the applicability of the finite element software packages in simulating the nonlinear behaviour of the beams. Barbosa and Rebeiro, (1998) considered the practical application of nonlinear models in the analysis of reinforced concrete structures and the consequences of small changes in modelling. The best results were obtained from the elastoplastic-perfectly plastic, work-hardening models that reached ultimate loads, very close to the predicted values. It was also concluded that the highest analysis loads could be considered as the ultimate loads of the models and the actual beams. Revathi and Menon, (2005) conducted finite element and experimental studies on under-reinforced, over-reinforced and shear test beams in ANSYS, to validate the potential of numerical simulation in predicting the nonlinear response of the elements. The numerical and test results were seen to compare well. The ductile behaviour of under-reinforced beams and the brittle mode of failure in the over-reinforced and shear beams were produced well by the numerical model. The crack patterns of the specimens were also seen to be in good correlation with the patterns obtained from the numerical analysis. The study recommended the use of convergence criteria in terms of force to get more reliable and accurate results. Dahmani, Khennane, et.al., (2010) conducted an investigation into the applicability of ANSYS software for analysis and prediction of crack patterns in RC beams and the advantage of performing numerical simulation instead of experimental tests. For this purpose, different phases of the behaviour of the FE model of an RC beam was studied from initial cracking to failure of the beam.

The entire load-deformation response produced correlated well with the hand-calculated results and it was inferred that failure model of concrete adopted by ANSYS was adequate to determine the nonlinear response of reinforced concrete structures. Fanning, (2001) studied the experimental load-deflection response of ordinary reinforced concrete beams and post-tensioned concrete T- beams and used it to assess the suitability of numerical modelling implemented in the FE software ANSYS, in predicting the ultimate response of RC beams. The correlation of test and numerical data was found to depend on the values of linear and nonlinear material properties assigned to the materials, most importantly the Young s modulus of elasticity of concrete and the yield strengths of the reinforcing bars and the post-tensioning tendons. Anthony J. Wolanski, (2004) in his thesis work, studied reinforced and prestressed concrete beams using Finite Element Analysis (FEA) to understand their loaddeformation response. The results were compared to experimental data. Characteristic points on the loaddeformation curve predicted using FEA were then compared to theoretical results. The nonlinear analysis of the model yielded results that compared well to the calculated values. Bursting was also seen to occur in concrete at the area of prestressing where maximum stress and localised cracking were observed. It was concluded from the work that the failure mechanism of the beams could be modelled well using the finite element package. III. ANALYSIS OF REINFORCED CONCRETE BEAM The RC beam of concrete grade M25 is 150mm x 300mm in cross-section and simply supported over a span of 3m. The beam is loaded symmetrically at third points along its span. Internal longitudinal reinforcement consists of two numbers of 20mm diameter Fe415 bars placed at an effective cover of 25mm. Two-legged 8mm diameter Fe415 stirrups are provided at a spacing of 50mm c/c as shear reinforcement. The hangar bars have a diameter of 10mm and are placed at a cover of 25mm. Fig. 1 shows the dimensional details of the RC beam. In the finite element modelling of the beam, concrete, steel support and loading plates and the steel reinforcing bars were represented using SOLID65, SOLID45 and LINK8 elements respectively. Shear reinforcements were neglected. The SOLID65 element required linear isotropic and multilinear isotropic material properties to be defined. Table I summarises the material properties assumed for reinforced concrete. Table I Material Properties Of Concrete Density of concrete 2.5485 x 10-6 kg/mm 3 Modulus of Elasticity 25000 N/mm 2 Uniaxial cracking stress 3.5 N/mm 2 Poisson s Ratio 0.2 Open shear transfer 0.3 coefficient Closed shear transfer 1 coefficient Uniaxial crushing stress -1 Biaxial crushing stress 0 Hydrostatic pressure 0 Hydrostatic biaxial 0 crushing stress Hydrostatic uniaxial 0 crushing stress Tensile crack factor 0.6 The compressive uniaxial stress-strain relationship for concrete was obtained using the equations developed by MacGregor (1992). Fig. 2 represents the stress-strain relationship of reinforced concrete. The steel plates were modelled as linear isotropic materials with modulus of elasticity of steel (2 x 10 5 N/mm 2 ) and Poisson s ratio (0.3). The reinforcing steel was assumed to have bilinear isotropic properties with yield stress of 415 N/mm 2 and hardening modulus 20 N/mm 2. Linear isotropic properties of the steel rebars were the same as that of the steel plates. Fig. 1 Dimensional Details Of The Reinforced Concrete Beam 450

Fig. 2 Stress-Strain Relationship Of Concrete The beam and steel plates were modelled using separate volumes. The beam was meshed such that it consisted of square or rectangular elements of size 25mm. The necessary mesh attributes were set before the volumes were meshed. Fig. 3 shows the meshed beam in ANSYS. Merging of nodes and key points were carried out to avoid errors due to multiple nodes at the same location. The longitudinal reinforcement bars were then modelled by creating individual link elements through the nodes of the concrete volume mesh at the desired depth after setting the rebar element attributes. The supports were modelled such that roller and hinged supports were created at either ends of the beam. Self weight of the concrete beam was taken into account by providing the value of acceleration due to gravity (9.81m/s 2 ). The external loads were applied as concentrated forces distributed equally among the nodes forming the centre line of the two loading plates located at third points along the span. Fig. 3 Meshed Beam For the analysis of the model, the static analysis type was utilized. Prestress effects were not considered and the analysis was carried out for Small Displacement Static condition. The rest of the commands were set to ANSYS defaults. The beam was analysed with its self-weight and the loads in one load step. The analysis was seen to terminate at the sub step corresponding to 119.58kN due to non-convergence of solution. 451 Therefore, it was concluded that failure of the beam had taken place and 119.58kN was taken as the failure load for the model. The first crack in the FE model was a flexural crack (vertical) in the constant moment region of the beam, formed at 18.8025kN. The deflections and stresses in the beam were seen to increase with loading. Cracking also progressed consequentially and was observed to increase in the constant moment region before it spread out towards the supports in the form of diagonal cracks. The beam was seen to fail due to excessive cracking of concrete in the tension side. Therefore, failure of the beam was attributed to flexural failure and yielding of the steel reinforcement. The ultimate mid-span deflection of the model was 20mm at 119.58kN. The numerical crack and stress distributions at ultimate load are illustrated in Fig. 4. Fig. 4 Crack Distribution In The Beam At Failure Using the data obtained from the numerical analysis of the beam, mid-span deflection, compressive stress in concrete and stress in the reinforcing steel were plotted against the applied load and compared with theoretical computations of the same. From Fig. 5, it is seen that the ANSYS model has captured the entire load-deflection response of the beam from zero loads up to failure. The curve exhibits three approximately straight segments reflecting three different stages during the loading process. The initial linear portion of the curve represents the elastic uncracked region where the numerical and theoretical results are seen to compare well. The sudden change in linearity represents commencement of cracking in the beam. The initial cracking load, obtained as 20.65kN from the finite element analysis and 20.16kN from the structural analysis of the beam, also show good agreement in values. The nonlinear region that follows represents the behaviour of the beam in the cracked stage, where cracking propagates throughout the constant moment region, reducing the stiffness of the section. The effective concrete area thus decreases and the steel reinforcement bears the tensile stresses developed. This leads to a faster rate of increase of deflections with applied load. The theoretical curve in this region is observed to deviate from the ANSYS curve. The transition from the second stage to the third stage marks the start of yielding of reinforcement.

Classical reinforced concrete theory predicted the yielding to commence at 115.3kN which is observed to be consistent with the change in slope of the numerical curve at 116.77kN. The finite element model was assumed to fail in flexure shortly after yielding of steel due to excessive cracking in the beam at a load of 119.58kN producing an ultimate deflection of 20.3mm. Fig. 6 Load Vs Compressive Stress In Concrete Fig. 5 Load Vs Mid-Span Deflection Fig. 6 shows a plot between the applied load and stress in the extreme concrete fibre under compression. The theoretical computation of compressive stress in the top concrete fibre was carried out only up to initial cracking in the beam owing to lack of formulations that accounted for the reduction in effectiveness of the section due to cracking. In the uncracked stage, the theoretical and ANSYS curves are seen to compare well. The numerical curve beyond that stage represents nonlinear variation of stress with load with an ultimate compressive stress value of 25.08N/mm 2 in concrete at failure. The variation of steel stress with applied load is represented in Fig. 7. The plot shows good agreement between the analytical values and the results from theoretical computations. Stress in the tensile reinforcement increases linearly with load, at a faster rate beyond initial cracking of the beam compared to that in the uncracked stage. The theoretical computation of steel stress was terminated at yield stress of steel (415N/mm 2 ) due to the lack of means to compute the stress beyond yielding of steel. The ultimate value of steel stress at failure of the beam was observed to be 419.76N/mm 2 from the finite element analysis. Fig. 7 Load Vs Stress In Reinforcing Steel IV. ANALYSIS OF PRE-TENSIONED CONCRETE BEAM The pre-tensioned concrete beam of grade M40 is 150mm x 300mm in cross-section with a simply supported span of 3m and loaded symmetrically at third points along the span. Internal prestressing is provided through two numbers of straight, 7 ply 12.7mm diameter prestressing strands placed at an eccentricity of 100mm and tensioned to 80kN each. Internal longitudinal reinforcement consists of two numbers of 8mm diameter Fe415 bars placed at 25mm from the soffit of the beam. Two-legged 8mm diameter Fe415 stirrups are provided at a spacing of 150mm c/c as shear reinforcement. The hangar bars have a diameter of 8mm and are placed at an effective cover of 25mm. Fig. 8 shows the dimensional details of the beam. 452

The stress-strain curve for prestressing steel was developed using the following equations (Anthony J. Wolanski, 2004). ε ps 0.008 : f ps = 28000ε ps (ksi) ε ps > 0.008 : f ps = 268 - <0.98f pu (ksi) Fig. 8 Dimensional Details Of The Pre-Tensioned Concrete Beam Finite element model for the analysis of the pretensioned concrete beam is very similar to the RC beam model. Numerical modelling was carried out neglecting the tensile and shear reinforcements. The initial effective strain of 0.0035328 was entered in the Real Constant set for the prestressing strands in addition to the cross-sectional area. Table II lists the material properties of the prestressed concrete. Where f pu is the ultimate strength of the prestressing tendons, taken as 1771.58N/mm 2 or 256.974ksi. The values obtained in units of ksi were then converted to SI units and used for the analysis. Fig. 10 shows the stress-strain relationship for the prestressing strands. Table II Material Properties Of Prestressed Concrete Density of concrete 2.3955 x 10-6 kg/mm 3 Modulus of elasticity 36049.965 N/mm 2 Uniaxial cracking stress 4.427 N/mm 2 Poisson s Ratio 0.2 Open shear transfer coefficient Closed shear transfer coefficient 0.3 Uniaxial crushing stress -1 1 Fig. 9 Stress-Strain Curve For Prestressed Concrete Biaxial crushing stress 0 Hydrostatic pressure 0 Hydrostatic biaxial crushing stress Hydrostatic uniaxial crushing stress Tensile crack factor 0.6 The stress-strain relationship of concrete was obtained using the equations developed by MacGregor (1992) and is shown in Fig. 9. Modulus of elasticity for the type of strands used in the analysis was taken as 195000MPa as specified in IS 1343:1980 and Poisson s Ratio as 0.3. 0 0 Fig. 10 Stress-Strain Curve For Prestressing Steel For the analysis of the model, prestress effects were included and the analysis was carried out in a number of load steps as listed in Table III. The analysis was seen to terminate at the sub step corresponding to 143.6kN due to non-convergence of solutions. Hence, it was concluded that failure of the beam had taken place at 143.6kN. 453

Table III Load Steps For The Analysis Of Pre-Tensioned Beam Beginning Time (N) Time at the End of Load Step (N) Load Step Number of Sub steps Load Increment (N) 0 1 1 1 Prestress 1 2 2 1 Selfweight 2 36600 3 2 18300 36600 56000 4 2 9700 56000 58000 5 50 40 58000 80000 6 100 220 80000 100000 7 100 200 100000 120000 8 100 200 Failure of the beam at 143.6kN produced an ultimate mid-span deflection of 39.63mm. The crack distribution in the beam at failure is illustrated in Fig. 11. Fig. 11 Crack Distribution In The Beam At Failure From the numerical study carried out, the response of the beam to loading in terms of deflection, compressive stress in the extreme concrete fibre and steel strains was evaluated. 120000 140000 9 100 200 140000 143600 10 60 60 Camber in the beam at the mid-span section due to application of prestress was observed to be 1.042mm. As the load was increased, tensile stresses were induced in the bottom concrete fibres due to development of bending stresses. A stage was reached when the compressive stress in concrete at the soffit was balanced by the flexural tensile stresses so that the net stress was zero. The load at this stage is called decompression load and was observed to be 36.6kN. With further increase in load, the tensile stresses at the soffit of the beam increased. At a load of 57.72kN, the tensile stress approximately equalled the flexural strength of concrete and flexural (vertical) cracking was observed to commence in the constant moment region of the numerical model. As the loading on the beam progressed, vertical cracking was seen to propagate throughout the flexure zone and subsequently towards the supports in the form of diagonal cracks. Later, multiple cracking was also observed at the same location. 454 Fig. 12 Load Vs Mid-Span Deflection Fig. 12 shows the load-deflection response of the beam. The FE analysis is seen to predict the entire behaviour of the beam up to failure, the results closely related to the theoretically predicted values. Initially, the beam deflection increases linearly with the applied load. On appearance of flexural cracks at approximately 25% of the ultimate load, the beam stiffness is reduced after which the deflections again increase linearly, but at a faster rate. This continues till yielding of the internal reinforcement at approximately 119.7kN as predicted by the theoretical analysis. After the section becomes sufficiently plasticized, the deflection increases substantially with very small increase in load. The beam shows considerable ductility at ultimate after which it was observed to fail in flexure due to excessive cracking of concrete and yielding of tension steel producing an ultimate deflection of 38.54mm. The variation of compressive stress at the top concrete fibre at mid-span section with load is shown in Fig. 13. The theoretical computations were carried out only in the uncracked stage where the linear curve was seen to compare well with the numerical results.

The numerical curve, after commencement of cracking in the beam, was seen to follow a highly nonlinear profile till failure of the beam at which the compressive stress in concrete was noted to be 40.93N/mm 2. Fig. 13 Load Vs Compressive Stress In Concrete The strain in the prestressing strands exhibited approximately bilinear variation with load, the rate faster after the onset of cracking. The theoretical and numerical predictions compared almost well up to steel yielding load. The numerical prediction of steel strain extended up to failure load where the ultimate strain was obtained as 0.0089. However, it was noticed that the FE model produced an initial effective strain value of 0.00336, a value lower than the applied effective prestrain of 0.0035328, the reason for which is unclear. Fig. 14 shows the plot of load vs strain in the prestressing strands. V. COMPARISON OF REINFORCED AND PRE-TENSIONED CONCRETE BEAMS Fig. 15 shows the load-deflection responses of the reinforced and pre-tensioned concrete beams predicted using finite element analysis. The numerical models were seen to be capable of producing the entire load-deflection behaviour of the beams till failure. The reinforced concrete beam model produced a linear curve till a load of 18.8kN after which it cracked. The linear portion of the curve for the pre-tensioned concrete beam extended to 57.72kN, providing a much higher service load range. This is attributed to the pre-compression applied to concrete which in turn effected in increased service load capacity. Both the beam types exhibited sudden increase in deflections beyond initial cracking, up to 1.6mm approximately. This was followed by a nearly linear portion again where the pretensioned concrete beam was observed to register a higher rate of increase of deflections compared to the reinforced concrete beam. A sudden change of slope followed by a nonlinear curve in the response of the reinforced concrete beam indicated yielding of the steel reinforcement to commence at a load of 116.77kN whereas this was not as pronounced in the case of pre-tensioned concrete beam which showed more ductility. Both the beams failed in flexure due to yielding of steel and excessive cracking in the tension zone. The pre-tensioned concrete beam produced a higher ultimate load capacity of 143.54kN in comparison to 119.58kN of the RC beam. Fig. 14 Load Vs Strain In Prestressing Strands Fig. 15 Load Vs Mid-Span Deflection The comparison of the numerical prediction of the variation of compressive stress in the extreme concrete fibres at the mid-span section, in the reinforced and pretensioned concrete beams is shown in Fig. 16. The curves represent linear variation of stresses up to commencement of cracking. 455

The curves, beyond the stage of initial cracking, are seen to show uneven increase in stress which reaches an ultimate of 25.08N/mm 2 for the RC beam and 40.5N/mm 2 for the pre-tensioned concrete beam. The correctness of the curves cannot be assured as the beams have been modelled under a number of approximations. Fine tuning the model can result in better solutions. Fig. 16 Load Vs. Compressive Stress In Concrete Fig. 17 depicts the variation in steel strain in the bonded reinforcement with the applied load. The prestressing tendons show an initial effective stain of 0.00335. From the figure, the curves for both the beam types are observed to have similar bilinear profiles. The curves show a small rate of strain increase in the uncracked, elastic region. The rate of strain increase, however, rapidly develops in the cracked, inelastic region, with a major portion occurring just after cracking. Ultimate strains in the reinforcement bar and the prestressing strands in the reinforced and pretensioned concrete beams were found to be 0.002 and 0.0089 respectively. Fig. 17 Load Vs Strain In Steel 456 VI. CONCLUSIONS In the present study, the response of reinforced and pretensioned concrete beams to vertical loading was investigated using the finite element software package ANSYS 12.0. The load-deflection response, variations of stresses in concrete and strains in the steel reinforcements and prestressing tendons with increasing loads were evaluated and compared to theoretical data obtained using the theories of structural analysis. In comparison to the theoretically predicted data, the numerical method of analysis using ANSYS was seen to satisfactorily predict the behavioural responses of the beams up to failure. However, a discrepancy was observed in the initial value of effective prestrain in the tendons predicted by the numerical analysis, the reason for which is unclear. The variation of compressive stress in concrete beyond the stage of initial cracking could not be estimated using the theories of structural analysis owing to the absence of formulations that took into account the decreasing effectiveness of the section in the cracked stage. On comparing the behaviour of the RC beam with that of the prestressed concrete beam, the advantage of prestressing was verified as the prestressed concrete beam was seen to show a higher service load range and higher ultimate load capacity. Acknowledgement The paper is published with the permission of the Director, CSIR-SERC, Chennai, India. The help rendered by Shri. K. Saravana Kumar, Scientist is acknowledged. REFERENCES [1] Antonio F. Barbosa and Gabriel O. Ribeiro, Analysis Of Reinforced Concrete Structures Using Ansys Nonlinear Concrete Model, Computational Mechanics, New Trends And Applications, Barcelona, Spain 1998. [2] P. Revathy and Devdas Menon, Nonlinear finite element analysis of reinforced concrete beams, Journal of Structural Engineering, Vol. 32, No.2, June-July 2005, pp. 135-137. [3] L. Dahmani, A. Khennane and S. Kaci, Crack Identification in Reinforced Concrete Beams using Ansys Software, Strength of Materials, Vol. 42, No. 2, 2010, pp. 232-244. [4] R. Srinivasan and K. Sathiya, Flexural Behaviour of Reinforced Concrete Beams Using Finite Element Analysis (Elastic Analysis), Bulletin of the Polytechnic Institute of Jassy, Vol. LVI (LX), No. 4, 2010, pp. 31-42. [5] I. Saifullah, M. Nasir-uz-zaman, S.M.K. Uddin, M.A. Hossain and M.H. Rashid, Experimental and Analytical Investigation of Flexural Behavior of Reinforced Concrete Beam, International Journal of Engineering & Technology IJET-IJENS, Vol: 11 No: 01, February 2011, pp. 146-153. [6] Vasudevan. G and Kothandaraman.S, Parametric study on Nonlinear Finite Element Analysis on flexural behaviour of RC beams using ANSYS, International Journal of Civil and Structural Engineering, Vol. 2, Issue 1, 2011, pp.98-111.

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