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NONLINEAR ANALYSIS OF SHEAR WALL STRUCTURES BY AN
EFFICIENT REINFORCED CONCRETE MEMBRANE ELEMENT
Leopoldo TESSER1, Michele PALERMO
2 and Filip FILIPPOU
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ABSTRACT
This paper discusses the parameter calibration of a recently proposed, efficient numerical model for
the inelastic response analysis of reinforced concrete panels under seismic excitations. The concrete
material model is based on damage plasticity with two damage variables. The reinforcing bars are
modelled as smeared steel layers with a uniaxial stress-strain relation in the bar direction following the
Menegotto-Pinto constitutive model with isotropic hardening. For in-plane analysis, a plane stress 2d
membrane model is used with an initial assumption of perfect bond between reinforcing steel and
concrete. After the calibration the model is used to simulate the cyclic response of planar reinforced
concrete shear walls and coupled walls. The specimens are representative of modern mid-rise wall
construction. The evaluation of the model response makes use of global and local response
measurements and compares damage parameter contours of the model against the observed state of the
specimen so as to draw conclusions for the seismic performance assessment of reinforced concrete
shear walls.
INTRODUCTION
The seismic assessment of reinforced concrete shear walls by international standard provisions suffers
from two main sources of uncertainty. The first is the evaluation of the shear demand by linear
analysis methods and the second concerns the determination of the shear wall capacity. In fact, current
design provisions, such as Eurocode 8 (CEN 2004) in Europe and ASCE/SEI 7-10 (2010) in the
United States, might lead to the underestimation of the seismic shear demand in the case of linear
analysis because of the following issues: the flexural over-strength, the higher mode effects for slender
walls in the nonlinear range, and the determination of the force reduction factor for low and mid-rise
walls (Rutenberg and Nsieri 2006, Priestely et al. 2007, Rejec et al. 2012, Pugh 2012). Furthermore,
the formulas for the determination of the shear strength provided by Eurocode 8 (CEN 2004), ACI 318
(2005), ASCE/SEI 43 (2005) do not seem to be effective for all resisting mechanisms and for all wall
configurations (Salonikios 2002, Gulec 2008, Turgeon 2011, Krolicki 2011). General formulas for the
shear strength of shear walls do not account for the presence of barbells, openings, coupling beams,
and special reinforcement detailing.
Consequently, there is a growing consensus that reinforced concrete shear walls should be
designed and assessed with nonlinear analysis methods. However, the use of these methods in
professional practice is still limited because of the lack of physical understanding and numerical
robustness for existing models (Lee and Kuchma 2007, Lee et al. 2008, Gulec and Wittaker 2009,
Hagen 2012). For this reason several studies propose the use of beam-column models with or without
1 Ph.D., Géodynamique & Structure, Bagneux – France, [email protected]
2 Ph.D. candidate, Dept. of Civil Engineering, University of Bologna –Italy.
3 Professor, Dept. of Civil Engineering, University of California, Berkeley – United States of America.
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shear effect consideration for reinforced concrete shear walls of high to moderate slenderness (Mazars
2006, Saritas 2006, Wallace 2007, Martinelli and Filippou 2009). These models are, however, not
suitable for squat shear walls or for the representation of the multi-axial response in the inelastic zone
of slender walls with the formation of inclined concrete struts and high shear deformations.
Furthermore, the computational complexity of several enhanced beam elements undermines their
advantage relative to membrane and plate finite elements.
In the interest of striking a balance between model accuracy and computational speed while
being able to capture important aspects of the nonlinear inelastic response of reinforced concrete shear
walls an efficient nonlinear reinforced concrete membrane model was recently proposed (Tesser et al.
2011). This paper simply aims at showing the practicality of nonlinear analysis and the reliability of
the developed model for the seismic performance assessment of reinforced concrete structures.
The calibration of the model is presented with the generalization of the relevant steps needed for
every concrete model with softening local response. The procedure to assure mesh objectivity and to
account for transverse confinement and tension stiffening is also clarified.
The model performance is assessed by correlation with the experimental results from the
MUST-SIM project (Lowes et al. 2011) on planar and coupled shear walls. The assessment is done by
comparison of global and local response measures from the nonlinear analysis with test measurements.
MODELLING CONSIDERATIONS
Reinforced concrete shear walls are subjected to a multi-dimensional stress state. Regardless of the
selected finite element, a multi-dimensional concrete material model is critical for capturing the local
material behaviour. Most of the existing macroscopic concrete models fall into two categories. The
models of the first category are directly developed in multi-dimensions starting with a surface limiting
the elastic range and a flow rule governing the nonlinear range (e.g. Darwin and Peknold 1982, Ju
1989). The models of the second category have multiple uniaxial stress-strain constitutive laws acting
in different directions and interacting with each other through empirical formulae (e.g. Vecchio and
Collins 1986, Hsu 1997).
A macroscopic concrete model has to take into account at least the failure modes of cracking in
tension and crushing in compression. The softening response following the peak strength is
fundamental to represent the local failure and the stress redistribution. It is also important to
approximate the shear transfer across cracks in case of non-proportional loadings such as seismic
actions. The model has also to account for the progressive stiffness reduction and for the inelastic
strains to determine the dissipated energy and the residual displacements in cyclic nonlinear analyses.
It is essential to select a method for ensuring the independence of the solution from different mesh
sizes (e.g. crack band theory, nonlocal models).
The reinforcement model should be nonlinear in order to represent the elastic-plastic steel
behaviour. The discrete or the smeared approach for modelling steel bars has to seek an adequate
representation of the complex steel-concrete interaction in the linear and nonlinear range. Other
advanced features are the steel-concrete bond nonlinear behaviour, the steel bar buckling in
compression, the steel fracturing in tension and other secondary shear mechanisms (e.g. dowel action
of steel bars).
Considering the multi-dimensional concrete model, the softening response always slows the
convergence when searching for global equilibrium. This issue can be tackled using an appropriate
nonlinear solution algorithm (e.g. Newton Raphson, quasi-Newton, accelerated Newton, arc-length)
together with the iterative stiffness matrix (i.e. initial tangent, secant stiffness and computational
tangent). The coupling of classical models for stiffness reduction (e.g. damage mechanics) and
residual displacements (e.g. theory of plasticity) strongly affects the convergence rate of the local
material state determination. The most effective algorithms are the explicit ones without internal
iterations. All additional sources of nonlinearity decrease the convergence rate of the model.
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CONCRETE MODEL
The concrete damage model in this paper belongs to the class of energy-based isotropic continuum
damage models (Tesser et al. 2011). It is a general three-dimensional law for use with any finite
element model. Both tensile and compressive damage modes are taken into account by means of two
scalar damage variables. The two variables represent the material damage by controlling the strain
energy density for tensile stress states (the variable is called tensile damage) and a modified Drucker-
Prager criterion for compressive states (the variable is called compressive damage). Tensile and
compressive stress states are identified by the spectral decomposition of the effective (hyperelastic)
stress tensor. The variables vary from 0 to 1 where 0 means virgin material and 1 meaning that
stiffness and strength are completely lost.
The constitutive model assumes that the damage criteria describe also the plastic surface so that the
development of material damage is simultaneous to the accumulation of irreversible strains for all
stress states. It is to be noted that this assumption does not allow to properly account for dilatancy
effects. A simplified plasticity evolution law represents the residual strains for all stress states. This
feature assures the efficiency of the concrete model since a straightforward predictor-corrector
algorithm is used for the material state determination without internal iterations. The best convergence
rate for the global equilibrium is obtained with the computational tangent for small problems and with
the secant stiffness for large-scale problems. The solution strategy for the global equilibrium is the
Newton method accelerated with the Krylov subspace (Scott and Fenves 2009).
The elastic loading-unloading response is governed by the Young modulus E and the Poisson
ratio . The other model parameters are: the uniaxial compressive strength fcc, the uniaxial tensile
strength fct, the plastic strain ratio , the tensile fracture energy Gft and its equivalent in compression
Gfc. In a uniaxial test, the plastic strain ratio is the ratio between the total strain and its plastic strain
part. It is a constant that replace the consistency parameter of the classical plasticity theory.
For the sake of clarity, the parameters of the model are summarized in Table 1.
Table 1. Parameters for three-dimensional concrete material model
Parameter E fcc fct Gft Gfc
Description elastic
modulus
Poisson’s
ratio
uniaxial
compressive
strength
uniaxial
tensile
strength
plastic
strain
ratio
tensile
fracture
energy
compressive
fracture
energy
REINFORCED CONCRETE MEMBRANE MODEL
For in-plane analysis, a plane stress 2d membrane model has been developed that accounts for the
concrete and reinforcing steel interaction. The general three-dimensional concrete model has been
developed for plane stress states in a plane stress state in agreement with the membrane element
mechanics. The reinforcements are modelled in a discrete or in a smeared way. In the present paper
the embedded approach is preferred and all the parallel rebars are simulated by an additional layer that
comply with the element kinematic and it is characterized by the uniaxial stress and stiffness response
in the direction of the bars. As a result the equivalence of the total strains of the different material
layers is imposed. Perfect bond between concrete and reinforcement is assumed. The smeared model
can be further extended to take into account the bond slip between the two materials at the cost of an
additional computational effort. An important benefit of the membrane model is the ease in handling
any reinforcement configurations.
The Menegotto-Pinto constitutive law with the isotropic hardening is selected for representing
the uniaxial reinforcement response in the direction of a set of steel bars (Filippou et al. 1983).
The main limitations which arise from the current formulation of the model for reinforced
concrete panels are: the concrete dilatancy may be not properly simulated due to the coupling between
damage and plasticity; the perfect bond between concrete and reinforcement bars is assumed; the
failure of reinforcement bars is not simulated.
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CALIBRATION OF THE CONCRETE CONSTITUTIVE LAW IN COMPRESSION
The model response under cyclic uniaxial compression is represented in Figure 1 and compared with
the experimental results from Sinha et al. 1964. The model parameters for this test are: E=35GPa;
fcc=32.5MPa; Gfc=14.7N/m; =0.55, where fcc is the experimental peak compressive strength. Once the
initial elastic limit, which is assumed to be half of the compressive strength in the model, is exceeded,
the plastic strain and the damage variable increase. The hysteresis of the reloading loop cannot be
simulated by the model because the unloading process is assumed to be elastic.
The three-dimensional concrete constitutive model takes into account the increase of strength
and ductility under biaxial and triaxial compression. In the case of membrane finite elements, the
concrete law needs to be projected in two dimensions and the effect of the transverse stress cannot be
accounted for. Hence the values of fcc and Gfc have to be properly calibrated to represent the
confinement exerted by the transverse reinforcement in the direction normal to the membrane. To this
end, numerical analysis as well as empirical or semi-empirical correlations can be used. In this study
the values of these parameters are selected with the helped of the well-known formulas by Park and
Kent 1971.
For instance, Figure 2a shows the concrete damage constitutive law in compression for the
unconfined case and for two transverse reinforcement ratios T equal to 0.5% and 1.0%. The evolution
of the damage variable dn for the three cases is compared in Figure 2b. The curves displayed in Figure
2 are obtained with a Lagrange four-node quadrilateral with 100 mm side length.
(a) (b)
Figure 1. Cyclic stress-strain response and damage evolution under uniaxial compression.
(a) (b)
Figure 2. (a) Constitutive law in compression for unconfined concrete and confined concrete
(calibrated according to Park and Kent 1971 ); (b) evolution of damage parameter dn.
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CALIBRATION OF THE CONCRETE CONSTITUTIVE LAW IN TENSION
The numerical response under cyclic uniaxial tension is represented in Figure 3 and compared with the
experimental results from Gopalaratnam and Shah 1985. The model parameters used in this test are:
E=29GPa; fct=3.45GPa; =0.55, Gft=170N/m. The finite element model consists of a Lagrange four-
node quadrilateral with 100 mm side length. For given fracture energy, the local stress-strain response
depends on the finite element size and integration method and differs from the average experimental
response. Nonetheless the example shows the evolution of residual strains and the progressive stiffness
degradation of the unloading-reloading cycles. The evolution of the damage variable, governing the
residual stiffness, and the plastic strain, related to the residual strain upon unloading, provides an
accurate representation of the physical response. Classical plastic flow rules coupled to any damage
model cannot achieve similar accuracy since the consistency parameter is strain dependent. In fact, the
extension of classical plasticity models for steel or soil to the simulation of the cyclic behaviour of
concrete does not seem reasonable.
(a) (b)
Figure 3. Cyclic stress-strain response and damage evolution under uniaxial compression.
If not provided by experiments, the value of the fracture energy can be assumed according to the
expression given by the Model Code 2010:
0.1873F cmG f (1)
where fcm is the mean compressive strength in MPa.
Figure 4a displays the constitutive law in tension for a fracture energy value of 136 N/m
(corresponding to a plain concrete of fcm = 30 MPa according to Eq. 1) and also for two other values of
fracture energy (270 N/m and 540 N/m). Figure 4b displays the evolution damage variable dp for the
three values of fracture energy.
For modelling reinforced concrete specimens, the tensile fracture energy Gft can be modified to
account for the tension stiffening effect. To this end, the model parameters are calibrated against
measurements from uniaxial tension experiments.
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(a) (b)
Figure 4. (a) Constitutive law in tension for three different values of fracture energy GF; (b) evolution
of damage parameter dp for three different values of fracture energy GF.
SIMULATION OF EXPERIMENTS
In this section the model is used to simulate the cyclic behavior of planar shear walls and coupled
walls which were tested at the NEES facility MUST-SIM (Multi-Axial Full-Scale Sub-Structured
Testing and Simulation) at the University of Illinois in Champaign-Urbana. The selected tests are part
of the research project entitled "Behavior, Analysis, and Design of Complex Wall Systems" which
were aimed at assessing the seismic behavior of modern reinforced concrete wall systems. A total of
four planar shear walls, one coupled wall and three C-shaped (or U-shaped) walls were tested under
lateral reversal loads. A complete description of the test program and test results related to the planar
shear walls and coupled wall is available online at the NEES Project Warehouse website. In the next
section, for the sake of conciseness, only selected experimental data will be briefly presented and
compared with the results of the numerical simulations.
The planar wall specimens are designed in order to be representative of a ten-story prototype
wall of 36.6 m height, 9.1 m length and 46 cm thick. Due to limitation of the facility the specimens are
one-third scale of the first three stories of the prototype. The test parameters are: (i) the lateral load
distribution; (ii) reinforcement layout; (iii) the presence of splice at the base. Two different lateral load
distributions are adopted: the equivalent lateral force distribution according to the ASCE 7-05 (which
is essentially an inverted triangular distribution whose resultant acts at an effective height, Heff equal to
0.71 H, where H is the total height of the prototype building.) and a uniform distribution whose
resultant is applied at 0.50 H. Two reinforcement layouts are adopted, one with longitudinal
reinforcement concentrated in the confined boundary region (referred to as Boundary Element (BE)
layout) and the other with a uniform distribution of longitudinal reinforcement (referred to as uniform
layout). The longitudinal reinforcement ratio in the boundary elements is equal to 3.41%, while it is
equal to 1.57% for the uniform layout. The horizontal reinforcement ratio is equal to 0.27% for both
layouts. Three specimens present spliced longitudinal reinforcement at the base, while one specimen
present continuous bars from the basement to the top of the wall.
The coupled walls test specimen was designed in order to be representative of a typical concrete
core in a 10-story building. The design process was conducted with the following objectives: (i) satisfy
the minimum requirements imposed by the ASCE 318-08 building code; (ii) obtain a prescribed
ductile behaviour under seismic excitation (i.e. performance-based design framework and capacity
design requirements); (iii) be consistent with the characteristics of the planar wall specimens tested
within the Planar Shear Wall Test program (see chapter 2).
During each test reversed lateral loads and a constant axial load were applied. Lateral loads
were applied in displacement control; the imposed drift was increased up to 1.5% for the
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planar walls and up to 2.0% for the coupled wall. The geometry of the specimen has been modelled using four-node square membrane elements.
The mesh of the planar wall model is composed of 12x14 quadrilateral elements, while the mesh of the
coupled wall is composed of 226 quadrilateral elements. The foundation and the wall cap are included
in the model as linear elastic elements with the same elastic modulus adopted for the non-linear
concrete model. Horizontal loads are applied at the mid height of the wall caps. In order to reproduce
the actual loading protocol and the boundary conditions at the top wall, the effect of the applied top
moment My is simulated by imposing appropriate distribution of the vertical displacements at the node
of the wall cap. A single load path is sufficient to reproduce the applied load protocol, due to the
proportionality between the lateral force Fx and the top moment My.
For the sake of conciseness only selected results of the numerical simulations are here
presented. Firstly, the comparisons between experimental and numerical results in terms of global
response (base shear vs top drift) are provided in Figures 5. In order to better appreciate the model
capabilities selected loading cycles are shown in Figure 6. It can be noted that the model results appear
reasonably accurate at both low and large imposed drifts with the accuracy increasing with the load
amplitude.
Also the capabilities of reproducing local response have been investigated. Figure 7a,b
compares the experimental and numerical strain history of the longitudinal reinforcement at the base
of the wall and at the base of the pier of the coupled wall, respectively. Although perfect bond between
concrete and reinforcement is assumed (i.e. relative slips cannot be explicitly reproduced) the local
strains seem also quite accurate at large drifts.
Figures 8 and 9 show the evolution of the compressive damage variable (dn) for Wall 1 and the
coupled wall, respectively, using contour maps.
The damage variables can be used for the performance assessment of the structure. First, the
local damage variables contours immediately show the cumulative damage distribution in the model
and can be correlated to the physical damage. For instance, a tensile damage greater than 0
corresponds to cracked wall regions and its value can be correlated to the micro-crack width. On the
other hand, a compressive damage of about 0.4-0.5 corresponds to regions with initiation of concrete
crushing following the attainment of the concrete compressive strength.
Secondly, the local damage variable distribution assists with the understanding of the structural
resisting mechanisms. For instance, Figure 7a shows that the compressive damage initially affects the
right side of the specimen and it develops mainly in the vertical direction. This means that the initial
wall resisting mechanism is mainly flexural. Figure 7b shows that the concrete compressive strength is
exceeded in the compression zone corresponding to the state of maximum wall resistance. Figure 7c
and 7d shows that the maximum compressive damage spreads horizontally with increasing lateral
drift, as the wall takes advantage of its shear resistance. The attainment of the unity value of the
compressive damage variable in the element at the bottom left precedes a sudden decrease of the wall
resisting strength. Furthermore, the damage evolution depicted by the contour maps of Figures 8 and 9
agrees very well with the physical damage distribution observed during the experiments (the details
regarding the mechanisms of failure observed during the experimental tests are available in the test
report, Lowes et al. 2011).
Third, the local damage variables can be used to define appropriate global damage indexes that
could be readily correlated with structural limit states so as to define safety margins. Such global
damage indexes were already defined for beam elements (Scotta et al. 2009) showing their suitability
in the seismic performance assessment of reinforced concrete moment resisting frames. The extension
of these global damage indexes to two-dimensional elements is under development.
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(a) (b)
Figure 5. Comparison between experimental and the numerical results in terms of base shear vs drift:
(a) Wall 1; (b) Coupled Wall.
(a) (b)
(a) (b)
Figure 6. Comparison between the experimental and the numerical results for a single load cycle for
Wall 1: (a) 0.1% drift; (b) 0.5% drift; (c) 0.75% drift; (d) 1.5% drift
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(a) (b)
Figure 7. Comparison between the experimental and the numerical strain history response for the
longitudinal reinforcement at the base: (a) Wall 1; (b) Coupled Wall.
(a) (b)
(c) (d)
Figure 8. Color contour maps of damage parameter dn for Wall 1: (a) at 0.1% drift; (b) at 0.5% drift;
(c) at 1% drift; (d) at 1.5% drift.
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(a) (b)
(c) (d)
Figure 9. Color contour maps of damage parameter dn for the coupled wall: (a) at 0.5% drift; (b) at
0.75% drift; (c) at .51% drift; (d) at 2.5% drift.
CONCLUSIONS
The paper highlights the capabilities and limitations of a recently proposed reinforced concrete
membrane model for large-scale seismic analysis. The concrete constitutive law belongs to the class of
energy-based isotropic continuum damage models.
The application of a concrete constitutive law with softening response requires the calibration of
the model parameters for mesh size independence. The effects of confinement and tension stiffening
can be taken into account with parameter calibration as well. After the model calibration with
available material response results, it is used to simulate the hysteretic response of single and coupled
reinforced concrete shear walls with great success.
The damage parameters of the concrete constitutive law permit the identification of the key
resisting mechanisms, the attainment of limit states, and the determination of cumulative physical
damage. These results show that the proposed reinforced concrete membrane model holds significant
promise for the seismic performance assessment of large-scale reinforced concrete structures.
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