International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
Non-Linear Static Pushover Analysis of Real Life Reinforced Concrete
Frame with ATENA 3-D Program
Beena Kumari 1
1
Research Scholar, Department of Civil Engineering,
Thapar University, Patiala, India
Abstract— The non-linear static analysis (push over
analysis) provides good understanding of seismic behavior
of structures. Pushover analysis produces pushover curve
which consists of capacity spectrum. In the current paper,
the non-linear response of four storeyed RC frame under
the pushover loading (monotonically incremental load)
has been obtained to study the response and load-carrying
capacity of RC frame using non-linear finite element
analysis. An analysis model for four storeyed RC frame
using software ATENA, a computer program using stress
analysis with finite element method, is presented. The
results obtained from FE analysis are compared with the
experimental data for four storeyed RC frame, at the same
location as used in experimental test. The correctness of
the finite element model is assessed by the comparison
with experimental results which are to be in good
agreement. The pushover curves (base shear versus
displacement curves) obtained from finite element analysis
agree with the experimental results in linear and nonlinear range.
Keywords: Finite Element modeling, Finite Element
Analysis, Reinforced concrete frame
I. INTRODUCTION
The concrete has been the most preferred construction
material for many decades due to its wonderful properties
There are a variety of factors which are responsible for the
degradation of RC structures such as increasing load,
deterioration of steel due to corrosion, seismic forces,
environmental degradations and accidental impacts on the
structure. The increase in height of modern structures is
intimately related with the extent and impact of disaster in
terms of human and economical loss in the occurrence of
structural failure. The existing structures can also become
seismically deficient since design code requirements are
constantly upgraded due to advancement in engineering
knowledge. In India most of the buildings constructed over
past two decades are seismically inadequate due to lack of
knowledge regarding seismic behavior of structures. Hence a
careful and systematic structural safety analysis becomes
compulsory. To evaluate RC structures against failure an
accurate estimation of the ultimate load is necessary and the
prediction of the load-deformation behavior of the structure
throughout the range of linear and non-linear response is
advantageous. To obtain a global mechanism for the structure
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with a ductile behavior the non-linear analysis is required to
be carried out.
Nowadays the statically nonlinear method called pushover
method, is becoming a popular tool for seismic performance
evaluation of new and existing structures. The pushover
analysis is a nonlinear static analysis for a reinforced concrete
framed structure subjected to lateral loading. The gravity
loads are applied and then lateral loading is applied first in Xdirection starting at the end of the gravity push and next in ydirection again starting at the end of gravity push[1]. The
pushover analysis provides ample information on seismic
demands imposed by the design ground motion on the
structural system and its components. The objective of
pushover analysis is to evaluate the expected performance of
structural systems by estimating performance of a structural
system by estimating its strength and deformation demands in
design earthquakes by means of static inelastic analysis, and
comparing these demands to available capacities at the
performance levels of interest. The inelastic static pushover
analysis can be viewed as a method for predicting seismic
force and deformation demands, which accounts in an
approximate manner for the redistribution of internal forces
that no longer can be resisted within the elastic range of
structural behavior [2].
Fig. 1 Diagram depicting the development of an equivalent SDOF system for
a pushover curve (FEMA 440, 2005).
While a concrete element undergoes large deformations in
the post-yield stage, it is assumed that all the deformation
takes place at a point called “plastic hinge”, a typical response
at a plastic hinge may be as shown in Fig.2. Here, Point A is
the origin; B is the point of yielding; BC represents the strainhardening area; C is the point corresponding to the maximum
force; and DE is the post-failure capacity region[3].
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
correct procedure to evaluate the characteristics can be judged
by comparing the analytical results with those of the
experiments.
A Finite Element Modeling in ATENA 3-D
The fast development in computer technology has led to
the development of computer programs that have
implemented finite element analysis where the stresses and
strains of the element are computed. Among them is ATENA
3-D.The program has a computing platform based on the
graphical user interface through which all the inputs are
entered. The computation is based on Lagrange formula. The
software is designed for nonlinear analysis for solids,
especially for reinforced concrete elements. The properties
and the characteristics of the non-linear materials can be
Fig. 2 Idealized load deformation curve [3]
defined manually or can be selected from the common ones,
Mathematically, nonlinear static analysis does not lead to a such as the standardized concrete and steel classes, the values
unique solution. Small changes in properties or sequence of are predefined in the program [5]. The program system
loading can lead to large variations in the nonlinear response. ATENA offers a variety of material models for different
The pushover analysis may be carried out using force control materials and purposes. The most important material models
or deformation control. In the first option, the structure is in ATENA for RCC structure are concrete and reinforcement.
subjected to an incremental distribution of lateral force, and These highly developed models take into account all the
incremental displacements are calculated. In the second option, important aspects of real material behavior in tension and
the structure is subjected to a deformation profile, and lateral compression. The program has three main functions:
forces needed to generate those displacements are computed.
1) Pre-processing: The geometry of the elements
Since the deformation profile is unknown, the first option is
including the spatial position of the reinforcement by
commonly used.
grid coordinates, the nonlinear characteristics for the
II. FINITE ELEMENT MODELING
To model the complex behavior of reinforced concrete
analytically in its non-linear zone is difficult. Due to this
engineers in the past had to rely on empirical formulas
derived from several experiments for the design of reinforced
concrete structures. The finite element method has thus
become a foremost computational tool, which allows complex
analyses of the nonlinear response of RC structures to be
carried out in a usual fashion. It
is able to model RC
structure easily and is able to calculate the non-linear behavior
of the structural members. The combination of the
introduction of digital computers and the finite element
method have led many efforts to develop analytical solutions
which would evade the need for experiments undertaken by
researchers.
FEM is useful for obtaining the load deflection behavior in
various loading conditions but the results of any analytical
model have to be verified by comparing them with
experiments in the lab or with full-scale structure as
experimentation is an excellent tool for the verification
purpose and to provide a firm basis for design equations,
which are invaluable for design but at the same time it is very
difficult and un-economical to perform experiment on full
scale structures [4] because conducting these tests are timeconsuming and expensive and often do not simulate exactly
the loading and support conditions of the actual structure. The
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materials, load assignment, bounding conditions for
the macro-elements, finite element meshing and
various other parameters like loading steps used for
the analysis are defined.
2) Analysis: In this part the results can be monitored
and accessed in real time, as the analysis enters from
one step to another.
3) Post-processing: This function provides access to the
results at each loading step expressed graphically and
numerically.
Modeling in ATENA 3-D is a little bit difficult unlike
other programs , but a comprehensive virtual model can be
achieved. The precise positioning of the reinforcement in the
concrete is possible which gives an idea about the problems
that may take place and permits to find solution in the design
stage. The areas with a plastic potential are better described
and analysed. The potential plastic areas can be characterized
directly through stress development until it reaches the yield
limit and through crack development for concrete. Based on
stresses and non-linear properties of the materials, results for
each material can be obtained.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
B Modeling of Concrete
In ATENA modeling of concrete is done using 3D solid
brick element with 8 up to 20 nodes having three degree of
freedom at each node: translations in the nodal x, y and z
directions. This is an iso-parametric which is capable of
plastic deformation, cracking in three orthogonal directions,
and crushing. The most significant feature of this element is
the treatment of non-linear material properties. Concrete
exhibits a large number of micro-cracks at the interface
between coarser aggregates and mortar, even before subjected
to any load. The presence of these micro-cracks has a great
effect on the behavior of concrete, since their propagation
during loading contributes to the nonlinear behavior. Many of
these micro-cracks are caused by segregation, shrinkage or
thermal expansion of the mortar. Since the aggregate-mortar
interface has a significantly lower tensile strength than mortar,
it constitutes the weakest linkage in the composite system and
is the main cause for the low tensile strength of concrete. The
stress-strain relation of the constituent materials and the
magnitude of stress mainly affects the response of a structure .
The stress-strain relation in compression is of main concern as
concrete is used normally in compression.
1) Equivalent uniaxial law: The complete equivalent uniaxial stress-strain diagram for concrete is shown in Fig. 3.The
numbers of the diagram parts (Material state numbers) are
used in the results of the analysis to indicate the state of
damage of concrete. Unloading is a linear function to the
origin. Thus, the relation between stress σcef and strain εeq is
not unique and depends on a load history. A change from
loading to unloading occurs, when the increment of the
effective strain changes the sign. If subsequent reloading
occurs the linear unloading path is followed until the last
loading point U is reached again. Then, the loading function is
resumed. The peak values of stress in compression fcef and in
tension ftef are calculated according to the biaxial stress
state. Thus, the equivalent uni-axial stress-strain law reflects
the biaxial stress state[6].
2) Behavior of cracked concrete: The progressive cracking
results in nonlinear response of concrete which results in
localized failure. The structural member is cracked at discrete
locations where the concrete tensile strength is exceeded. As
cracking is the major source of material nonlinearity, realistic
cracking models need to be developed in order to accurately
predict the load-deformation behavior of reinforced concrete
members. The selection of a cracking model depends on the
purpose of the finite element analysis. There are two crack
models in ATENA-3D, smear crack model and discrete crack
model. If overall load-deflection behavior is of primary
interest, without much concern for crack patterns and
estimation of local stresses, the "smeared" crack model is
probably the best choice. If detailed local behavior is of
interest, the adoption of a "discrete" crack model might be
necessary.
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Fig. 3 Uniaxial stress-strain law for concrete [6]
Fig. 4 Stages of crack openings [6]
C Modeling of reinforcement
Modeling of reinforcement in finite element is much
simpler than the modeling of concrete. Reinforcement
modeling could be discrete or smeared. In the present work, a
discrete modeling reinforcement has been modeled using bar
elements in ATENA. Reinforcement steel is a 3D bar element,
which has three degrees of freedom at each node; translations
in the nodal x, y and z direction. Bar element is a uniaxial
tension-compression element. The stress is assumed to be
uniform over the entire element. This element is capable of
plastic deformation, also creep, swelling and large deflection
capabilities are included in the element.
1) Stress-Strain laws for reinforcement: Reinforcement
can be modeled in two different forms: discrete and smeared.
Discrete reinforcement is in form of reinforcing bars. In
present study discrete approach has been used. The smeared
reinforcement is a component of composite material and can
be considered either as a single material in the element under
consideration or as one of the more such constituents. In both
cases the state of uni-axial stress is assumed and the same
formulation of stress-strain law is used in all types of
reinforcement. The bilinear law, elastic-perfectly plastic, is
assumed as shown in Fig. 5. The initial elastic part has the
elastic modulus of steel Es. The second line represents the
plasticity of the steel with hardening and its slope is the
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
hardening modulus Esh. In case of perfect plasticity Esh =0.
Limit strain εL represents limited ductility of steel.
.
Fig. 6 Floor plan , roof plan, and loading pattern [7]
Fig. 5 Bilinear stress-strain law for reinforcement [6]
III CASE STUDY
A full–scale four storeyed structure tested by Reactor
Safety Division, Bhabha Atomic Research Centre under
lateral monotonically increasing pushover loads at tower
testing facility at Central Power Research Institute, Bangalore,
is taken as case study. The test was conducted under steadily
increasing monotonic lateral load in an inverted triangular
pattern till failure. The main objectives of this work was to
test a real- life structure under pushover loads. In order to
keep the structure as close to reality as possible, no special
design for the structure was performed.
Fig. 7 (a) Experimental loading pattern and (b) loading, meshing and
monitoring points in ATENA
B. Summary of material Properties
The portion of a real life existing office building which
was un-symmetric in plan was intentionally selected so that it
had certain eccentricities. Also the column sizes and sections
were varied along the storey as in the case of original real life
structure [7]. A complete data base of the experiment carried
on a full scale real life structure mentioned above has been
used for the for the finite element analysis for the validation of
analytical results in the shape of Push-over curve,
displacements at various floor levels and damage patterns at
different loads. In the present study, non-linear response of
RCC control frame using FE modeling under the incremental
loading has been studied with the intention to investigate the
relative importance of several factors in the non-linear finite
element analysis of RCC frames using Atena3D. These
include the variation in base-shear v/s displacement graph,
crack patterns, propagation of the cracks and the crack width
on the analytical results.
Concrete, reinforcement steel and steel plates have been used to
model the RCC frame. The specification and the properties of these
materials are as under:
1) Concrete: In ATENA, concrete material is modeled as a 3D
nonlinear cementitious2. The physical properties of 3D nonlinear
cementitious2 material are given in Table 1. The values
are
calculated as per IS code 456:2000 and remaining are the default
values.
2) Reinforcement Bars: HYSD steel of grade Fe-415 of 28mm,
25mm, 20mm, 16mm and 12mm diameter are used as main steel
while 8mm and 10mm diameter bars are used as shear reinforcement.
The properties of these bars are shown in Table 1.
3) Steel Plate: The function of the steel plate in the ATENA is
for support and for loading. Here, the property of steel plate is same
as the reinforcement bar except its yield strength. The HYSD steel of
grade Fe-415 is used for steel plate.
A. Model Details
The structure analyzed is a four-storeyed moment-resisting
frame of reinforced concrete with properties as specified
above. The concrete floors are modeled as rigid. The plan of
the building is shown in the Fig. 6. The load on the structure
was applied in an inverted triangular fashion. The
reinforcement and ratio of force at “1st floor: 2nd floor: 3rd
floor: 4th floor” was kept as “1:2:3:4” as shown in Fig. 7.
Properties
Elastic
Modulus
Poisson Ratio
Specific
weight
Compressive
Strength
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Table 1
Material Properties of Concrete and Reinforcement
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Concrete
2,360 MPa
Properties
Elastic
modulus
0.2
24 KN/M3
Poisson Ratio
Specific
weight
0.3
785 KN/M3
20
Yield Strength
415 MPa
MPa
Reinforcement
200000 MPa
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
curve is same as per other floor levels. The maximum
deflection as base shear 950 KN has been found to be 156 mm
which increases up to 345mm, even after without any increase
of load. It can be observed from this figure at level 2 that the
structure showed the same behavior as that of third and fourth
floor level. The maximum deflection at base shear 950 KN
has reached to the value of 265mm, after this it started
increasing and has reached to the maximum value of
598mm.The displacements at various floor levels for different
base shear have been tabulated in Table 2:
TABLE 2
COMPARISON OF DISPLACEMENTS AT DIFFERENT BASE SHEAR AT
VARIOUS FLOOR LEVELS
Fig. 8 Modelling of reinforcement in ATENA
Sr.
No.
IV RESULTS AND DISCUSSIONS
All Finite element analysis of RCC frame under the static
incremental loads has been performed using ATENA software.
Subsequently these results are compared with experimental
results of four storey full Scale reinforced concrete structure
under monotonic Push-over Loads. This is followed by load
deflection curve and the cracking behavior obtained from the
analysis.
A. Push Over curves at various floor levels
The load on the structure has been gradually increased in
the steps till failure. The behavior of frame at every step has
been studied. Major observations made are discussed as given
below:
Floor level-4 It can be seen from Fig. 10 that the structure
behaved linearly elastic up to the value of base shear around
350 KN. At this point the minor cracks started to get
generated at top floor level. After this point there is a slight
curvature in the plot and deflection started increasing with the
load increments. When the base-shear reached to the value of
500 KN, the graph showed non-linearity in its behavior. It is
clear from the figure. that at base shear 750 KN, increase in
deflection is more with load increments. At 850 KN base
shear deflection has been observed to be 150mm and a rapid
increase in displacement is observed. After base shear value of
880 KN there is constant increase in deflections and plot has
become almost flat. After the value of base-shear 925 KN
deflection started increasing without any significant increment
in load; it has reached to the value of 340 mm with the baseshear value 950 KN.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Base
Shear
In
(KN)
100
150
200
250
300
350
402
452
497
547
601
652
702
752
802
850
882
952
962
947
Deflection
at
1th Floor
(mm)
0.837
1.29
1.8
2.5
3.06
4.06
5.34
6.66
9.2
13.2
15.2
20.6
23.8
26.4
43.4
74.7
87.4
158
230
346
Deflection
at
2th Floor
(mm)
1.32
2.02
2.8
3.83
4.7
6.25
8.1
9.99
14.1
19.1
22.5
31.6
37.1
41
69.2
119
139
267
391
598
Deflection
at
3th Floor
(mm)
1.74
2.67
3.68
4.94
6.05
7.91
10.1
12.6
17.7
23.1
27.3
38.8
45.5
50.2
86.4
142
165
331
488
766
Deflection
at
4th Floor
(mm)
2.08
3.18
4.5
5.81
7.0
9.15
11.6
14.1
20.2
25.9
30.6
43.1
50.6
55.8
93.4
151
177
345
502
787
B. Comparison between the FE model and the experimental
results of the RC frame
The pushover curves (Base shear v/s Floor Displacement)
as obtained from experimental results for extreme right side
column are plotted in Fig. 9. As can be seen from the figure,
the maximum displacement has been obtained as 765 mm.
The comparison of base shear versus deflections of
experimental & FE model results of RC frame are given in
Table3.
Floor level-3 has been depicted through Fig. 11. It can be
observed from this figure that the load-deformation behavior
is same as of fourth floor level. The maximum value of
deflection has been observed to be 328 mm at the base shear
of 950 KN.
Floor level-1 and 2 has been depicted through Fig.12-13. It
can be seen from the figure that floor level-1 has experienced
the minimum deflection even though the damage experienced
by this floor is maximum. The variation of load-deflection
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Fig. 9 Combined Pushover Curve from experimental data [7]
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
TABLE3.
COMPARISON OF EXPERIMENTAL & FE M ODEL RESULTS AT TOP FLOOR
LEVEL
Sr.
No.
Experimental Results
Base Shear Deflection
FE model Results
Base Shear Deflection
(KN)
(KN)
(mm)
(mm)
1.
196.2
7
196
4.32
2.
294.3
10
294
6.86
3.
392.4
20
392
10.6
5.
686.7
94
687
49.1
6.
784.8
160
787
84.3
7.
882.9
470
883
177
8.
882.9
765
922
208
9.
392.2
720
950
340
10.
--
--
947
787
The behavior of the frame has been observed to be linear
up to the value of base shear around 360 KN and the first
crack has been observed, whereas the structure has been found
to be linear up to the value of base shear 300 KN in case of
experiment. At this point the flexural tension cracks at the
base of the columns depicting reduction in stiffness have been
observed. At base shear value of approximately 500 KN, the
cracks at the base of the columns have been found to open
wider and failures at other location like beams and beam –
column joints started. Whereas in case of experiment at a base
shear value of approximately 500 KN, the cracks at the base
of the columns have been observed and failures at beam –
column joints start to show up. After reaching the base shear
values of 750 KN, displacements have found to be increasing
at fast rate whereas in case of experiment at the base shear
values of 700 KN, the joints of the frame have found to be
displaying rapid degradation and the inter storey drift
increasing rapidly. After this stage, at further increase in the
lateral load 880 KN, the FE model has been observed to
display soft behavior and displacement increased for the same
increase in the base shear but experimentally after reaching a
base shear of 800 KN, the frame has been found undergoing
increasing displacement at almost constant load. Maximum
deflection has been found to be more than 900mm at 950 KN.
Maximum deflection at fourth level at 880 KN is 770 mm
experimentally which is 173 mm in case of FE model.
Fig. 10 Pushover Curve at floor level-4 (Experimental & FE model)
Fig. 11 Pushover Curve at floor level-3 (Experimental & FE model)
Fig. 12 Pushover Curve at floor level-2 (Experimental & FE model)
Variety of failures like beam-column joint failure, flexural
failures and shear failures have been observed almost in the
same way as seen in the case of experiment. Prominent
failures shown by both models have been the joint failures.
Severe damages have been observed at joints of lower floors,
moderate damages at first and second floors and minor
damage at top floor has been observed.
Fig. 13 Pushover Curve at floor level-1 (Experimental & FE model)
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International Journal of Engineering Trends and Technology (IJETT) – Volume 10 Number 2 - Apr 2014
C. Crack Patterns
The first shear crack has been observed at value of base
shear 300 KN, when frame is in linear zone. This crack
appeared at left side beam-column joint of first storey. At
value of base shear 350 KN shear cracks have been observed
near beam column joints at1st and 2nd floor level. The beamcolumn joint failures at 1st floor level only along with few
flexural cracks in beams at value of base shear 450 KN is
observed. The beam- column joint failures can be observed in
sides of the frame at 1st, 2nd and at some points of 3rd floor
level along with some flexural cracks in beams at value of
base shear 625 KN . Lower level floor slab is also showing
cracks. At value of base shear 850 KN at 1st, 2nd ,3rd floor
level and few points of 4th floor level have experienced the
beam- column joint failures along with flexural cracks in
beams.
The elements of lower two storeys have experienced
moderate damage whereas upper storeys experienced minor
damage.At value of base shear 962 KN the beam- column
joint failures at 1st, 2nd ,3rd and at few points of 4th floor level
along with flexural cracks in almost all the flexural members
can be visualised from different side views All floor slabs
except top level floor slab are showing cracks. Major damage
has been noticed in elements of lower two storeys whereas 3rd
storey experienced moderate damage. The top storey
experienced minor damage.
IV CONCLUSIONS
The main observations and conclusions drawn are
summarized below:
1.
The frame behaved linearly elastic initially. Then it
depicted non-linearity in its behavior. After this,
increase in deflection has been observed to be more
with load increments. After reaching the maximum
value of base-shear it has kept increasing even at
decremented load steps.
2.
The joints of the structure have displayed rapid
degradation and the inter storey deflections have
increased rapidly in non-linear zone. Severe damage
has occurred at joints at lower floors whereas
moderate damage has been observed in the first and
second floors. Minor damage has been observed at
roof level.
3.
The frame has shown variety of failures like beamcolumn joint failure, flexural failures and shear
failures. Prominent failures shown by FE model are
joint failures. Flexural failures have been seen in
beams due to X-directional loading.
4.
The results of FE model of the control frame have
found to be higher by 8% of the
experimental
results. So it can be concluded that FE model push
over results holds fairly good with the experimental
results though they are on slightly higher side.
5.
After comparing experimental and FE Model baseshear v/s deflection curves of the frame at various
floor levels the FE model results can be said to be
reasonably in close agreement with the experimental
data from the full-scale frame test.
Fig. 14 Crack patterns in columns, beams and slab at various loads
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