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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
vasug1967@gmail.com
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
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 100
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.
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 101
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
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 102
Figure 5: Effect of varying mesh density Figure 6: Mesh density on
midspan deflection
Figure 7: Stress-strain model for concrete in compression
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 103
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
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 104
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
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 105
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.
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 106
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
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 107
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
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 108
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
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 109
Figure 18: Effect of shear reinforcement on crack pattern
Figure 19: Effect of stirrups on longitudinal stress (X) distribution
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 110
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.
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 111
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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