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INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 1, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on August 2011 published on September 2011 98

Parametric study on Nonlinear Finite Element Analysis on flexural

behaviour of RC beams using ANSYS Vasudevan. G

1, Kothandaraman.S

2

1- Assistant Professor, Perunthalaivar Kamarajar Institute of Engineering and

Technology Karaikal - 609 603, Puducherry UT, India

2- Professor and Head of Civil Engineering, Pondicherry Engineering College,

Puducherry - 605 014, India

[email protected]

doi:10.6088/ijcser.00202010096

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

ANSYS. 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 ANSYS, 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

ANSYS in the analysis of RC beams.

Keywords: Material nonlinearity, Convergence, Steel cushion, Shear reinforcement,

ANSYS.

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 ANSYS 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

Parametric study on Nonlinear Finite Element Analysis on flexural behaviour of RC beams using

ANSYS

Vasudevan. G, Kothandaraman.S

International Journal of Civil and Structural Engineering

Volume 2 Issue 1 2011 99

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 4724.4 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 457.2 mm with main reinforcement of 3

bars of each area 200 mm2 and shear reinforcement of 25 nos. with area of each vertical

link as 71 mm2. 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


ANSYS




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 71.728 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 8370. Hence as a preliminary step a few numbers of trial analyses are

carried out to decide the optimum mesh density.


ANSYS




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 ckc fE 5000 as per IS 456:2000 codal provision is

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 ckt ff 7.0 as per IS 456: 2000 is used in the analysis. For cracked tensile

condition, the effect of tension stiffening is incorporated using stiffness multiplier

constant (Tc ). After cracking, the uniaxial tensile strength of the concrete (ft) drops

abruptly to a fraction of it (Tcft) 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 Tc and found that the results remain unchanged. Hence, a default

value of 0.6 incorporated in ANSYS is used for all the analysis. From the literatures and

the recommendations of the ANSYS 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 [ANSYS, 2005].

Figure 3: Quarter beam FE model Figure 4: Model with varying mesh density


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Figure 5: Effect of varying mesh density Figure 6: Mesh density on

midspan deflection

Figure 7: Stress-strain model for concrete in compression


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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 elastic-

perfectly 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 200000 N/mm2

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/mm2 shall be used to avoid loss of stability upon yielding. In the present study, yield

stress (fy) of 414 N/mm2 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 33.095 N/mm2

Modulus of elasticity of concrete 27227.9 N/mm2

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 3.585 N/mm2

Stiffness multiplier for cracked tensile condition 0.6


ANSYS




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 non-

convergence 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 0.005 and 0.05 for force and displacement. Revathi et

al., 2005 adopted a tolerance limit for convergence as 0.001 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 (93.245

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 Convergence

scheme

Force

tolerance

Displacemen

t tolerance

Number of

iterations

Ultimate

load (kN)

Midspan

deflection (mm)

CON1 0.1 0.1 1038 69.926 93.245

CON2 0.3 0.3 789 72.06 112.89

CON3 0.4 0.4 784 72.87 96.647

CON4 0.7 0.7 729 72.87 80.922

CON5 0.8 0.8 708 72.87 96.809


ANSYS




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 ANSYS 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.


ANSYS




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


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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 23.17 kN to

23.35 kN. Also, noted that at 62.27 kN and at 68.06 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


ANSYS




12. Impact of percentage of tension steel on flexural behaviour

Flexural behaviour RC beams due to variation in percentage of reinforcement (pt) 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 ANSYS software the following conclusions are drawn:

Figure 16: With and without stirrups Figure 17: Behaviour with and without stirrups


ANSYS




Figure 18: Effect of shear reinforcement on crack pattern

Figure 19: Effect of stirrups on longitudinal stress (X) distribution


ANSYS




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 625 015,

Tamilnadu, India, for having graciously permitted us to use the ANSYS software for this

work.


ANSYS




14. References

1. ANSYS Commands Reference, (2005), ANSYS, Inc. Southpointe, 275

Technology Drive, Canonsbury, PA 15317, http:/www.ansys.com.

2. Dahmani, L., Khennane, A., Kaci, S. (2010), “Crack identification in reinforced

concrete beams using ANSYS software”, Strength of materials, 42 (2) pp 232-240.

3. 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 431-440,

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 526-541.

5. 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 97301-5192 and Federal Highway Administration, Washington, DC

20590.

7. 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 135-137,

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.

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