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Limit state analysis of reinforced shear walls
Nunziante Valoroso a,⇑, Francesco Marmo b, Salvatore Sessa c
a Dipartimento di Ingegneria, Università di Napoli Parthenope, Centro Direzionale Isola C4, 80143 Napoli, Italyb Dipartimento di Strutture per l’Ingegneria e l’Architettura, Università di Napoli Federico II, via Claudio 21, 80125 Napoli, Italyc Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 12, 20133 Milano, Italy
a r t i c l e i n f o
Article history:
Received 13 August 2013
Revised 20 December 2013
Accepted 28 December 2013
Available online 6 February 2014
Keywords:
Reinforced concrete
Nonlinear static analysis
Shear wall
Limit state analysis
Wall–frame interaction
Finite elements
a b s t r a c t
The nonlinear analysis of reinforced concrete structures is revisited. In particular, reference is made to the
nonlinear static analysis of structures containing shear walls, for which we present a dedicated shell ele-
ment. The developed element formulation is flexible enough to allow for nonlinear material constitutions
for concrete and re-bars and arbitrarily distributed steel reinforcement. Moreover, for shear walls coex-
isting with frame elements within the same structural system, it allows to capture the effects of localized
actions such as concentrated forces or moments transmitted by the nearby framed part the structure.
Numerical results are presented that demonstrate the effectiveness of the proposed approach in finite
element computations.
2014 Elsevier Ltd. All rights reserved.
1. Introduction
Nonlinear static analysis is emerging as a paradigm for the eval-
uation of the seismic performances of structures. In the civil engi-
neering literature it is probably better known as the pushover
analysis method, see e.g. [1], introduced in the 1980s aiming to
construct a viable alternative to nonlinear transient dynamic anal-
ysis, whose cost is generally prohibitive for large-scale engineering
structures. In this context, the success of nonlinear static analysis
procedures is probably due to the directness and flexibility of the
method that, for a moderately wide class of structural systems,
can predict the seismic force and deformation demands due to
the redistribution of internal forces in the nonlinear regime at an
affordable computational cost [2,3].
Nonlinear static computations produce results that are certainly
less accurate compared to a fully nonlinear dynamic analysisowing to the implicit assumptions made about the dominant
deformation modes when selecting the load patterns to be used
in the analysis. However, a nonlinear static analysis can provide
valuable information on the structural response provided that
the inelastic behavior of each element in the structural system is
consistently described. This is particularly true for those systems
containing shear walls that, if not properly modeled, may lead to
unsafe estimations of limit loads and of failure mechanisms.
Accurate computations such as those presented e.g. in [4,5] for
the CAMUS I wall require to specify a moderately large set of mate-rial parameters in order to describe the behavior of the bulk con-
crete material and concrete-steel interfaces. Actually, highly
refined 2D and 3D material models for reinforced concrete (RC)
walls usually require parameters such as the softening modulus,
the critical fracture energy, the strength of concrete under pure
tension and/or in biaxial compression and so forth. Such parame-
ters are seldom at hand to professional users who, in most cases,
need to analyze either new structures or existing constructions
based only on the prescriptions of building codes and a few exper-
imental data, whenever available. These considerations motivate
the development of alternative computational models for the limit
state analysis of RC walls, either stand-alone or included in more
complex structural systems, able to capture the failure mecha-
nisms based upon a reduced set of material data.One of such alternatives is provided by macroelement-based
models [6–8], in which entire portions of a RC wall can be
described via 1D nonlinear springs connecting rigid beams. Other-
wise, RC walls have been represented using one-dimensional
beam–column finite elements [9–11], mostly relying on Timo-
shenko kinematics to account for shear deformation. Finite ele-
ment models based upon 1D or macro-elements allow for
successful descriptions of shear walls, see e.g. [12–14], provided
that they are stand-alone, i.e. not connected to other structural
elements. Actually, this approach may dramatically fail whenever
RC walls coexist with frame elements within the same structural
system, since the 1D kinematics is intrinsically unable to capture
0141-0296/$ - see front matter 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.12.032
⇑ Corresponding author. Tel.: +39 081 5476720; fax: +39 081 5476777.
E-mail address: [email protected] (N. Valoroso).
URL: http://www.uniparthenope.it (N. Valoroso).
Engineering Structures 61 (2014) 127–139
Contents lists available at ScienceDirect
Engineering Structures
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t
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such interactions, particularly in the nonlinear range. In such cases
RC walls are typically subject to localized actions that can be visu-
alized as concentrated forces or moments transmitted by the near-
by framed part of a building structure. These actions produce in
turn effects that are almost undetectable if the wall follows the
Euler–Bernoulli or Timoshenko kinematics, whereby cross sections
remain plane during deformation.
An example of such a counter-intuitive behavior is illustrated in
Fig. 1. Here are shown the vertical displacements at the top cross
section of a RC wall where acts a concentrated force or moment.
The different deformed shapes refer either to a beam or to a shell
formulation and are computed using the same material constitu-
tion for concrete and reinforcement. In short, Fig. 1 shows that a
beam model is not adequate to describe high strain gradients in
the vicinity of concentrated loads and that beam kinematics is un-
able to capture non-symmetric behavior of the wall under opposite
stress couples M x. In this respect, it is worth noting that the dif-
ference of deformations due to M x and M x is not limited to the
global loss of symmetry because the two strain distributions are
completely different due to the nonlinear and non-symmetric con-
stitutive behavior.
These aspects are particularly relevant for the analysis of exist-
ing RC structures, which is among the leading objectives of the
present work. Actually, in such cases the differences between ten-
sile and compressive response are even more enhanced by the ef-
fects of damage in concrete, which usually makes existing
structures particularly sensitive to localized actions.
The nonlinear behavior of RC shear walls can also be conve-
niently modeled based on the so-called lattice models [15,16]. In
this case nonlinear beam and rod elements are suitably assembled
to reproduce the most common load-carrying mechanisms of rein-
forced concrete, typically flexural, shear and arch behaviors. This
approach can be in many cases successful since it is potentially
able to capture the complex structural behavior typical of RC walls.
Nonetheless, the definition of mechanical properties for the com-
ponents of the lattice is often arguable and strongly reliant on engi-
neering judgment and end-user experience.A completely different methodology for the analysis of RC struc-
tures relies upon use of layered shell elements [17–19]. In such
case the presence of steel reinforcement within a concrete matrix
is described by stacking up a sequence of homogeneous layers that
may alternatively represent concrete or uniform reinforcement.
The mechanical properties of the composite are then allowed to
vary along the thickness of the slab or wall, see e.g. Fig. 2. The
layered approach allows in many cases to effectively describe RC
plates and shells in bending. Among its advantages are the fact that
it enables one to better simulate the behavior of walls or slabs in
which cross sections are not supposed to remain plane during
deformation in the sense specified above. However, the assump-
tion that reinforcement is uniformly distributed within layers
may represent a true limitation of this procedure. For instance,
concrete walls typically need more dense reinforcement at the
edges and in highly stressed regions such as corners and hole-bor-
ders. Moreover, even when reinforcement bars are fairly distrib-
uted in the plane of the wall, they still need to be considered in
their actual position rather than being spread along the width of
the wall to capture the correct response.
In the present study a model is proposed for computing the re-
sponse of RC structural walls suitable to nonlinear static analysis.In particular, a newly developed shell element is presented in
which arbitrary distributions of steel re-bars and nonlinear mate-
rial constitutions for both concrete and reinforcement are allowed.
Nonlinearity due to the axial–flexural behavior of the wall is dealt
with based on the closed-form stress integration presented in
[20,21], thereby bypassing the inaccurate and computationally
expensive subdivisions of the cross section into fibers, see e.g.
Fig. 1. RC walls subject to concentrated actions. Schematic comparison of vertical displacement distributions at the top cross section obtained using either a beam or a shell-like model. Note the non-symmetric response computed with the shell model for opposite values of the stress couple M x.
σ
Fig. 2. Schematic of a layered shell element with reinforcement. Stress distribution
is as results from usual through-the-thickness integration.
128 N. Valoroso et al. / Engineering Structures 61 (2014) 127–139
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[22,23] and references therein. Alternative sectional integration
methods, either relying on numerical quadrature or based on a
closed-form analytical can be found e.g. in Refs. [24–28],
respectively.
The driving rationale of our approach to the limit state analysis
of reinforced structural walls is to develop a formulation meeting
two basic requirements. The first one is to end up with a reinforced
shell element whose kinematics is sufficiently rich to overcome
known limitations of beam and layered elements. The second is a
robustness requirement, whereby the present approach can be
confidently used to analyze full-scale structures using a minimal
set of material parameters. As shown in the paper, both objectives
can be attained based on minimal kinematic assumptions that al-
low to carry out the stress integration in closed form and result
in a fairly good computational stability of the solution procedure.
The outline of the paper is as follows. In Section 2 we describe
the cross sectional analysis, which is one of the primary ingredi-
ents of the newly developed RC shell element presented in Sec-
tion 3. The performances of the proposed formulation are then
discussed with the aid of numerical examples in Section 4, with
special emphasis on the effects of localized actions engendered
by the interaction of RC walls with the framed parts of the struc-
ture. Closure and future research directions are finally outlined in
Section 5.
2. Analysis of a reinforced cross section
A RC cross section can be regarded as a domain X R2 whose
geometry is described via a Cartesian coordinate system, see
Fig. 3. Points on the cross section are identified by the position vec-
tor r ¼ ½ x; yT .
In classical beam analysis axis z is directed along the element
axis and oriented orthogonal to the xy plane in a way to obtain a
right-handed coordinate system. Moreover, cross sections remain
plane during deformation and the longitudinal strain along z -direc-
tion is the linear function:
e z ðr Þ ¼ e þ g r ð1Þ
e being the strain at origin O and g the vector collecting the bending
curvatures, i.e. g ¼ ½v y;v xT
.
The heterogeneous nature of concrete and the complexity of
many interacting degradation mechanisms has motivated the
development of a number of highly refined and regularized consti-
tutive models, see e.g. [29,30] and reference therein. Such models
do often require a large set of material parameters which may lack
a clear physical meaning and, by far more important, be unavail-
able in professional practice, see e.g. the seventeen-parameter con-
crete model of Sittipunt and Wood [31]. Since our ultimate goal is
the computational limit state analysis of full-scale reinforced con-
crete structures, assumptions adopted in this work will be based
on a simplified modeling and require basic material data only.
Namely, we assume perfect bond between steel bars and concrete,
whereby the longitudinal strain will be the same function of the
form (1) for both materials, and prescribe the normal stress r z as
a nonlinear scalar function of the longitudinal strain e z at each
placement r . RC sections are obtained as the superposition of a
concrete polygon X with n vertices of position r i; i ¼ 1 n, and
a set of nr steel reinforcement of area Arj and position r rj,
j ¼ 1 nr . Therefore, two distinct constitutive functions rs and
rc for concrete and steel are needed.
Stress resultants are evaluated by considering the contribution
of the steel and concrete parts of the section as:
N ¼Xnr j¼1
rs½e z ðr rjÞ Arj þ
Z X
rc ½e z ðr Þ dX
M y
M x ¼ X
nr
j¼1
rs½e z ðr rjÞr rj Arj þ Z X
rc ½e z ðr Þr dX
ð2Þ
The tangent stiffness of the section, is then computed as the
gradient of the stress resultants with respect to the generalized
strain parameters as:
K ¼
@ N @ e
@ N @ v x
@ N @ v y
@ M x@ e
@ M x@ v x
@ M x@ v y
@ M y@ e
@ M y@ v x
@ M y@ v y
26664
37775 ð3Þ
Since the domain of integration X does not depend upon the
strain parameters, the derivatives of the stress resultants are com-
puted as:
@ N
@ e ¼Xnr j¼1
E r ½e z ðr jÞ Arj þZ XE c ½e z ðr Þ dX ð4Þ
@ N @ v y
@ N @ v x
" # ¼
@ M y@ e
@ M x@ e
" # ¼
Xnr j¼1
E r ½e z ðr jÞr j Arj þ
Z X
E c ½e z ðr Þr dX ð5Þ
@ M y@ v y
@ M y@ v x
@ M x@ v y
@ M x@ v x
24
35 ¼
Xnr j¼1
E r ½e z ðr jÞr j r j Arj þ
Z X
E c ½e z ðr Þr r dX ð6Þ
where use has been made of the chain rule to evaluate the
derivatives:
@ rðÞ
@ e ¼
@ rðÞ
@ e z
@ e z @ e
¼ E ðÞ;@ rðÞ
@ g ¼
@ rðÞ
@ e z
@ e z @ g
¼ E ðÞr ð7Þ
the functions E ðÞ ¼ @ rðÞ=@ e being the tangent moduli of re-bars (E r )
and concrete(E c ).
2.1. Integrals over polygonal domains
Computation of the sums reported on the right-hand sides of
Eqs. (2), (4)–(6) is straightforward. This is not so for the evaluation
of the integrals appearing in the same formulas, that require some
explanation instead. In fiber-based elements [22] these integrals
are evaluated by subdividing the domain X into a number of
sub-elements (fibers) so to compute all integrals as the sum of
the (constant) contribution of each fiber. This approach is widely
used in research and professional implementations; however, its
computational efficiency is generally quite poor due to the highnumber of fibers required to get accurate results. An alternative,
Fig. 3. A general cross section of a reinforced concrete element. Any polygonalshape is allowed with arbitrarily distributed reinforcement.
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more effective approach is proposed in [20,21], where the stress
integrals are computed in closed-form as functions of the position
of the vertices of X and of the values that the primitives of the
integrand functions take at such points.
To this end, the integrals appearing in formulas Eqs. (2), (4)–(6)
are first expressed in the form:
A f ¼
Z X
f ½e z ð
r Þ
dX ð
8Þ
s f ¼
Z X
f ½e z ðr Þr dX ð9Þ
J f ¼
Z X
f ½e z ðr Þr r dX ð10Þ
where f is a piecewise continuous scalar function that coincides
either withrc or E c , while its argument e z is function of r via Eq. (1).
Theintegrationformulas presented in [20,21] have beeninitially
conceivedfor nonlinear cyclic analyses; assuch, they canaccount for
plastic deformations. Since in the present context we consider only
monotonic loading, the integration procedure can be simplified in
that plastic strains do not need to be explicitly computed. In partic-
ular, integrals (8)–(10) are evaluated as follows:
A f ¼Xni¼1
U0i ½ f
ðþ1Þðg niÞ l i ð11Þ
s f ¼Xni¼1
U0i ½ f
ðþ1Þr i þ U1i ½ f
ðþ1ÞDr i U0i ½ f
ðþ2Þg n o
ðg niÞ l i ð12Þ
J f ¼Xni¼1
fU0i ½ f
ðþ1Þðr i r iÞ þ 2U1i ½ f
ðþ1Þsymðr i Dr iÞ
þ U2i ½ f
ðþ1ÞðDr i Dr iÞ 2U0i ½ f
ðþ2Þsymðr i g Þ
2U1i ½ f
ðþ2ÞsymðDr i g Þ þ 2U0i ½ f
ðþ3Þðg g Þgðg niÞli ð13Þ
In the previous relationships symðÞ denotes the symmetric part
of the argument ðÞ; f ðþ jÞ is the jth primitive of f , i.e. @ j
f ðþ jÞ=@ e j z ¼ f ; li
is the length of the ith side of the boundary of X, that is
li ¼ jr iþ1 r ij ¼ jDr ij and n i is the outward unit vector orthogonal
to the ith side of X defined as n i ¼ e z Dr i=li, see also Fig. 4. In
the above relationships the vector g is defined as g ¼ g =ðg g Þ,
whereby formulas (11)–(13) are not well defined for vanishing g .
Accordingly, in this case the function f is replaced with its Taylor
expansion truncated at the third order derivatives, so that the inte-
grals (8)–(10) are also of simpler evaluation.
Functions Uki ½ f ðþ jÞ; k ¼ 0; 1; 2, are evaluated as a function of the
values that f ðþ jÞ and its primitives take at the vertices of X by
means of
U0i ½ f
ðþ jÞ ¼ f ðþ jþ1Þðeiþ1Þ f ðþ jþ1ÞðeiÞ
eiþ1 eið14Þ
U1i ½ f
ðþ jÞ
¼ f ðþ jþ1Þðeiþ1Þ
eiþ1 ei f ðþ jþ2Þðeiþ1Þ f ðþ jþ2ÞðeiÞ
ðeiþ1 eiÞ2 ð15Þ
U2i ½ f
ðþ jÞ ¼ f ðþ jþ1Þðeiþ1Þ
eiþ1 ei 2
f ðþ jþ2Þðeiþ1Þ
ðeiþ1 eiÞ2
þ 2 f ðþ jþ3Þðeiþ1Þ f ðþ jþ3ÞðeiÞ
ðeiþ1 eiÞ3
ð16Þ
where ei ¼ e z ðr iÞ and eiþ1 ¼ e z ðr iþ1Þ. These functions are not defined
when eiþ1 ei ! 0; also in this case the function f is replaced with
its truncated Taylor expansion. Additional details on the evaluation
of the integrals (8)–(10) are given in [20,21], which the interested
reader may refer to.
3. Reinforced walls and finite element computations
Main purpose of this work is to present an original procedure
for describing the behavior of shell-like reinforced concrete ele-
ments in which we account for nonlinear material behavior and
heterogeneities due to the presence of arbitrarily distributed flex-
ural reinforcement. In particular, aiming to keep the computational
effort as low as possible though preserving a good accuracy, in the
present approach use is made of a multi-scale integration tech-
nique able of dealing with material heterogeneities and nonlinear
constitution at a reduced computational cost. In particular, in our
modeling we assume that the response of structural walls is flex-
ure-dominated and make reference to a representation of concrete
and steel reinforcing bars as in Section 2, where it is analyzed the
cross section of a RC beam with reinforcement along its longitudi-nal axis.
It is worth emphasizing that in the present context the plane
sections assumption (1) introduced in Section 2 is relevant exclu-
sively to the quadrature subdomains and for the only purpose of
evaluating the material response. This is indeed one of the main
novelties of the present scheme, i.e. that based on (1) the compu-
tation of stress resultants for the non-homogeneous material is
carried out in analytical closed-form. This reduces significantly
the cost of integration and also alleviates problems related to pos-
sible non-convergence at the fine scale, which would require
restarting the analysis or resort to sub-incrementation [32].
3.1. Planar thin shell formulation
The analysis of reinforced concrete structural walls is carried
out based on a planar thin shell model. The motivations beyond
this choice are spread over different areas. First is the fact that
we are mainly interested to the analysis of structural walls, for
which curved geometries are seldom encountered. Second is the
possibility of effectively describing curved geometries using facets,
thus removing any potential membrane locking that curved shell
finite elements may suffer from, see e.g. [33]. Last, but not least,
is that flat shell elements are more economical, particularly if the
desired results for verification purposes are stress resultants
(forces and moments). Moreover, the numerical implementation
of flat elements is simpler and more flexible since a robust low or-
der shell element with the desired characteristics can be obtained
from the mere superposition of the most suitable membrane andplate bending elements [34].
Fig. 4. RC cross section. Geometry and deformation characteristic vectors. Axialstrain diagrams are for illustrative purposes only.
130 N. Valoroso et al. / Engineering Structures 61 (2014) 127–139
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The three-dimensional shell body under consideration is
embedded in a 3D Euclidean space and the shell domain V is de-
fined as:
V ¼ f xj x3 2 ½l3=2; þl3=2; ð x1; x2Þ 2 Ag ð17Þ
where x ¼ ½ x1; x2; x3T
is the vector of local rectangular coordinates,
l3 is the shell thickness and A the midsurface area. The displacement
vector for linearized kinematics is given as:
U ð x1; x2; x3Þ ¼ uð x1; x2Þ þ x3bð x1; x2Þ ð18Þ
where uið x1; x2Þ are the displacements of the midsurface, bð x1; x2Þ is
a rotation vector defined as b ¼ h e3; h being the infinitesimal con-
tinuum rotation vector and e3 the unit vector in the thickness direc-
tion. As for the components of the rotation that are associated to
bending, the Kirchhoff hypothesis is assumed, i.e.
h1ð x1; x2Þ ¼ b2ð x1; x2Þ ¼ @ U 2@ x3
¼ þ@ U 3@ x2
¼ þ@ u3
@ x2
h2ð x1; x2Þ ¼ þb1ð x1; x2Þ ¼ þ@ U 1@ x3
¼ @ U 3@ x1
¼ @ u3
@ x1
ð19Þ
The generalized strain components in the local rectangular
coordinate system are obtained as the symmetric in-plane gradient
of in-plane displacements:
eð x1; x2Þ ¼ sym ruð x1; x2Þ ¼
e1
e2
c12
264
375 ¼
@ u1
@ x1
@ u2
@ x2
@ u1
@ x2þ @ u2
@ x1
2664
3775 ð20Þ
As for bending curvatures, they can be expressed as:
jð x1; x2Þ ¼ sym rbð x1; x2Þ ¼
j1
j2
j12
264
375 ¼
@ b1
@ x1
@ b2
@ x2
@ b1
@ x2þ @ b2
@ x1
2664
3775 ð21Þ
The vectors of stress resultants work-conjugate to the mem-
brane and bending strains are respectively the normal forces and
stress couples per unit length:
N ¼
Z l3
T d x3 ð22Þ
M ¼
Z l3
T x3 d x3 ð23Þ
In the previous relationships T denotes the stress vector, which
has only three non-zero components due to the plane stress condi-
tion of thin shells, while the thickness integral is indicated with the
shorthand notation:Z þl3=2
l3=2
ðÞ d x3 ¼
Z l3
ðÞ d x3 ð24Þ
3.2. Stress resultants computation
Due to the heterogeneous nature of reinforced concrete, the
deformation-driven process that ends up with the integration of
the constitutive behavior is carried out here at a lower level with re-
spectto the finiteelements of the mesh, i.e. a level wherethenonlin-
ear response of the constituent materials is more easily interpreted.
In usual multi-scale procedures this lower level corresponds to the
scale of a suitable Representative Volume Element (RVE) that is
attached to each quadrature point of the structural model.
For the case at hand of RC structures we assume that the mate-
rial has no heterogeneities except for an arbitrary distribution of
inclusions represented by the reinforcement bars. Therefore, for
computing the stress state we attach to each Gauss point a 3Dquadrature cell within which a cross section in the sense specified
in Section 2 is defined and the kinematic hypothesis (1) is postu-
lated. The quadrature subdomains and the local axes associated
to a 2 2 grid of Gauss points in a typical finite element is sche-
matically depicted in Fig. 5.
Based on the orientation of reinforcement, here denoted with
subscript b, the deformation state on the quadrature subdomains
is obtained by identifying the strain components acting on the
quadrature subcell with the mean dilatational strain and the bend-
ing curvatures as follows:
e ¼ e z ð^ x0Þ ¼ ebð^ x0Þ for b 2 f1; 2g ð25Þ
v x ¼ @ e z @ y
^ x0
¼@ eb
@ xs
^ x0
¼ for b ¼ 1; s ¼ 2
þ@ eb@ xs
^ x0
¼ for b ¼ 2; s ¼ 1
8><>: ð26Þ
v y ¼ @ e z @ x
^ x0
¼ @ eb@ x3
^ x0
¼ jbð^ x0Þ for b 2 f1; 2g ð27Þ
having denoted by x 0 the position vector of the quadrature point on
the midsurface. Axes xs; xb; xt in Fig. 5 are introduced for the only
purpose of making fully compatible the coordinate system of the
shell element with that of the cross section. Namely, axis xt does al-ways coincide with x3, axis xb is always directed parallel to the rein-
forcement and xs is such that the reference xs; xb; xt is right-
handed. Once the strain components affecting the nonlinear re-
sponse are determined in the form (1) using (25)–(27), the stress
resultants and the relevant linearization can be computed using
the closed-form integration of Section 2 with minor modifications.
In particular, the explicit expressions of the nonlinear shell resul-
tants that are later being transformed into equivalent nodal forces
for finite element computations read:
N b ¼ 1
ls
Z ls
N b d xs ¼ N
l yð28Þ
M bs ¼ 1ls
Z ls
N b xs d xs ¼ 1
l yR X
r z y dX ¼ M x
l yfor b ¼ 1; s ¼ 2
1l y
R Xr z y dX ¼ M x
l yfor b ¼ 2; s ¼ 1
(ð29Þ
M b ¼ 1
ls
Z ls
M b d xs ¼ M yl y
ð30Þ
As for the tangent stiffness terms, i.e. the linearizations of the
stress resultants N ; M x and M y, they can be obtained similarly to
the averaged stress resultants, i.e. scaling by the appropriate length
the gradients of the stress resultants defined in (3) that are
computed on the quadrature subdomains.
4. Numerical examples
The numerical examples developed in this section focus on
illustrating the main features of the present RC shell formulation.
The first example is intended to validate the proposed element
by comparing numerical simulations against experimental results.
Two further examples are then presented aiming to demonstrate
the capabilities of the RC shell in capturing the effects of localized
actions that arise in structures where shear walls interact with
frames. In particular, in such two examples the shear walls have
been modeled either with 1D beam–column elements either using
the shell elements presented in the paper to highlight their
differences.
The beam–column elements we used in computations are
almost standard since they are displacement-based and rely on
Timoshenko kinematics. The relevant basis functions have beencomputed starting from the exact elasticity solution and no locking
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behavior has ever been experienced. Moreover, in our examples
beams and columns have always been subdivided at least into
three elements to obtain a sufficiently flexible and converged
solution.
The shell element we implemented is a four-node element with
6 degrees of freedom per node, which makes the shell element
fully compatible with beam elements. This is an essential require-ment in order to correctly capture the effects of stress concentra-
tions that may arise in the vicinity of beam-to-wall connections.
As outlined in Section 3.1, our derivation departs from planar thin
shell kinematics, thus allowing the implementation of the ap-
proach presented in the paper into any finite element code starting
from a standard shell element. We emphasize that in order to ob-
tain a non-zero in-plane curvature the interpolation of the mem-
brane displacement field should be at least quadratic, which is
standard practice for flat shells. This leads naturally to 6-degrees
of freedom per node formulations and avoids membrane error
terms to dominate the behavior of the shell problem solutions,
see e.g. the discussion and relevant references in [35], pp. 435-440.
As for the material constitution we adopted in computations,
the point of departure is represented by Eurocode 2 prescriptions
[36] for the design of reinforced concrete cross sections, see e.g.
Fig. 6. Namely, reinforcement bars are assumed to behave accord-
ing to a bilinear stress–strain relationship both in traction and
compression:
rr ðeÞ ¼
r y if e < e
y
E e if e y 6 e 6 eþ
y
rþ y if eþ
y < e
8><>:
where E is the elastic modulus, e y and r y ¼ E e y are the yield strain
and stress and the superscripts þ and stand for tension and
compression, respectively. Concrete behavior is described by the
parabolic-rectangular stress block, i.e.
rc ðeÞ ¼
0 if 0 < erco
eco2e e2
eco
if eco 6 e 6 0
rco if e < eco
8><>:
where the tensile strength is altogether neglected, rco is the peak
compressive stress and eco is the corresponding strain.
Conventional elastic limit states have been defined for steel and
concrete materials. The elastic limits for steel are obvious and are
set equal to e y and eþ y while for concrete the elastic limit for the
compressive strain has been conventionally set equal to 0:1eco.
For the purpose of designing reinforced concrete cross sections
the conventional ultimate limit state is assumed to be attained
when any of the two materials attains a limit strain which is set
to ecu for concrete in compression, typically 0.35%, and to eþru and
eru for steel reinforcement, usually 1%. Beyond the ultimate lim-
its the stress–strain relationships employed for section design are
not defined at all and no softening behavior is admitted in the ver-ification stage [36].
Unlike cross sectional design, for our purposes of nonlinear lim-
it state structural analysis we shall assume in numerical computa-
tions that concrete and steel possess unlimited deformation
capacity, i.e. that material behavior is indefinitely ductile. This
assumption is consistent with the fundamental hypothesis of clas-
sical limit analysis, which aims at finding the collapse mechanism
of structures with ideally plastic behavior. In the same spirit, in our
limit state analysis we allow strains and curvatures to indefinitely
increase, the final goals being the determination of the failure
mechanism of the structure.
Fig. 5. A typical finite element and its partition into quadrature subcells.
(a) (b)
Fig. 6. Stress–strain laws adopted for limit state analysis of reinforced concrete structures. Concrete (a) and steel (b).
132 N. Valoroso et al. / Engineering Structures 61 (2014) 127–139
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All numerical computations presented in this section have been
carried out with the above assumptions and using a customized
version of the finite element code FEAP [37], in which we imple-
mented both the beam–column and RC shell elements.
4.1. T-shaped wall – numerics vs experimental
The first presented example concerns a T-shaped wall extractedfrom the NEES shear wall database [38] that has been tested by
Thomsen and Wallace [39]. The wall specimen is a one-quarter
scale representation of a wall pertaining to a prototype multi-sto-
rey office building located in a seismic area. Specimen height
equals 1:657 m; the geometry of the cross section and the rein-
forcement setup are given in Fig. 7.
Concrete strength and peak strain as retrieved from [39] equal
rco ¼ 41:7 MPa and ec 0 ¼ 0:002 while the steel Young modulus
and yielding strain are E ¼ 206 GPa and e y ¼ 0:00201. Prior to the
application of the lateral load an axial compression of 365 kN has
been applied to the wall and maintained constant throughout the
analysis. The horizontal load has then been applied at the top of
the wall parallel to the wall web and directed in a way to compress
the flange of the wall cross section. The lateral load has been pro-gressively increased via force control up to a final value of about
200 kN.
The wall has been modeled via a regular mesh consisting of
5 6 of shell elements for the wall’s web and 6 6 elements for
the flange. For nodes located on the top of the wall a master–slave
relationship corresponding to a rigid diaphragm has been pre-
scribed to simulate the load transfer assembly used during exper-
imental testing, conceived to prevent twisting of the wall. Nodes at
the base of the wall have been fully restrained.
The computed top displacement is plotted versus the lateral
load in Fig. 8 along with the experimental curve provided in [39].
In the original experimental test carried out by Thomsen and Wal-
lace the lateral load was of cyclic type with lateral drifts increasing
up to 2.5% of the wall height. However, the quasi-static solution
under monotonic loading (thick line) computed based on the shell
formulation presented in the paper compares pretty well to the
envelope of the hysteresis loops measured during the test.
The limit states of reinforced concrete are defined in terms of
magnitude of the axial strain at each reinforcing bar and over the
concrete cross section associated to the quadrature points of finite
elements. Therefore, a synthetic graphical representation of the
limit states for a RC structure is naturally obtained by plotting
the level sets of the axial strain and using the numerical values cor-
responding to the elastic (ELS) and ultimate (ULS) strain limits in
concrete and re-bars as delimiters. The deformed shape of the
TW2 wall and the contour plot of the limit states are depicted on
the right-hand side of Fig. 8. Here the failure mechanism of the
structure is easily recognized to involve the ultimate limit state
only for reinforcement in tension at the base of the wall’s web.
4.2. Planar wall–frame structure
The second example concerns a planar symmetric reinforced
concrete structure consisting of a shear wall connected to two
frames, see Fig. 9. Such wall–frame structures are almost ubiqui-
tous in buildings located in seismic regions owingto the high resis-
tance offered by structural walls to lateral loads induced by
earthquake ground motions.
Concrete behavior has been described usingrco ¼ 40:0 MPa andec 0 ¼ 0:002 as material parameters for the parabola-rectangle
stress block while for steel constitution E ¼ 206 GPa and
e y ¼ 0:00208 have been adopted, see also Fig. 6. Loading consists
of a vertical uniform load distribution that is applied over horizon-
tal beams and progressively increased using force control up to the
final value of 175 kN/m. For comparison purposes the structure has
been modeled using either the present reinforced shell element or
one-dimensional beam–column elements by adding rigid end-off-
sets to account for the width of the shear wall. In both cases beams
and columns have been modeled by adopting a mesh of 10 ele-
ments per member; for the shell model of the wall we used a reg-
ular mesh of 10 20 elements, while 20 elements have been used
when modeling the wall via beam elements. Nodes at the base of
the wall and of columns are fully restrained.
Fig. 10 shows the deformed shape of the structure along with
the contour plot of the limit states obtained for the two kinematic
models considered in the analysis. Here is noted that when using a
1.2192
3x0.0508
0.019050.01905
0.0635
0.019050.01905
0.0635
3x0.1016
0.1016
1.2192
φ6.4 / 0.1905
φ6.4 / 0.1397
0.0508 φ6.4
φ4.75 / 0.1016
φ4.75 / 0.0381
φ4.75 / 0.03175
8φ9.5
3x0.0508 0.01905
0.0635
φ9.5 0.01905φ4.75 / 0.0762
DETAIL A
DETAIL B
DETAIL C
DETAIL A
DETAIL B
DETAIL C
Typ.
Fig. 7. Cross section and reinforcement of the TW2 wall tested by Thomsen and Wallace [39]. All dimensions given in meters.
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beam element to represent the shear wall the latter is subject to
normal force only and no stress concentration arise in the vicinity
of the connections with the frame. Basically, this occurs due to the
symmetry of the geometrical model and because beam cross sec-
tions are constrained to stay plane. The resulting effect is that hor-
izontal beams are perfectly built-in at the connection with the
wall, whereby bending moments in the midspan region are signif-
icantly reduced and so do the size and rotations of plastic hinges.
On the contrary, use of the developed RC shell elements for
modeling the shear wall allows to capture the stress concentra-tions occurring near wall-beam connections. The plastic hinges
that arise consequent to such stress concentrations produce a sig-
nificant modification in the end boundary conditions of the hori-
zontal beams that change from built-in to simply supported.
Hence, for increasing vertical load spread of inelasticity in the mid-
span region of the horizontal beams is much more pronounced,
which explains the differences in magnitude of the deformed
shapes as well as of the failure mechanisms depicted in Fig. 10.
The differences in the response obtained based on the 1D
beam–column and RC shell model are further appreciated by com-
paring the distribution of bending curvatures computed in the two
cases for beams labeled A and B in Fig. 9. The comparison is shown
in Fig. 11; here the position of plastic hinges correspond to the
peaks of the curvature plots at beam ends and at the midspan. Itcan be noted that, except for the right end of beam A, curvatures
computed based on use of the RC shell model are always higher
compared to those obtained when using a beam-like modeling
for the shear wall, with differences that can be as high as 500%.
4.3. Shear walled building
Unlike the previous numerical example, that could be under-
stood as rather academic, we consider hereafter a full-scale struc-
ture previously described in [40] that is representative of shear-
walled buildings, see Fig. 12. The structure is a four-storey buildingconsisting of six shear walls interconnected by RC frames symmet-
rically arranged in the x direction. In the y direction symmetry is
lost because of the presence of additional beams due to a stair
intermediate landing placed at mid-height of the central span of
the frame located at y ¼ 13:60 m.
Concrete and steel behavior are described using rco ¼ 40:0 MPa
and ec 0 ¼ 0:002 as material parameters for the parabola-rectangle
stress block and E ¼ 206 GPa and e y ¼ 0:00208 for steel constitu-
tion, see also Fig. 6. As in the previous example, shear walls are
modeled using either the present shell element or one-dimensional
beam–column elements by adding rigid end-offsets to account for
the width of the walls. In both cases rigid diaphragms have also
been introduced at each story and all the base nodes of the struc-
ture have been fully restrained. The framed part of the structurehas been discretized with the same number of elements for both
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
20
40
60
80
100
120
140
160
180
200
L a t e r a l l o a d [ k N ]
Top displacement [m]
Experimental curve
Computed curve
Limit states
ULS (bars)
ELS (bars)
ELS (conc)
ULS (conc)
Fig. 8. Load–deflection curve and limit states plot for the TW2 wall [39].
Fig. 9. Wall–frame reinforced concrete structure. Geometry, loading and details of reinforcement.
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models, namely 6 elements for each column and beam. Walls have
been discretized either with a regular mesh of 6 4 shell elements
per story or using 6 beam–column elements per story.
Vertical load is prescribed as a line distributed load
qv ¼ 30:0 kN/m for horizontal beams and as a force per unit vol-
ume c ¼ 2:5 kN/m3 for shear walls. Both have been increased pro-
gressively up to their final value, reached at time T ¼ 1:00, and
kept constant until the end of the analysis. Uniformly distributed
horizontal forces along the height of the structure have been ap-
plied in x direction at the centroid of each floor. Forces start actingat time T ¼ 1:00 and are progressively amplified via load control.
The global response obtained for the two examined cases is
summarized in Fig. 13, where has been plotted the base shear
against the top displacement of the building. The curves show that
the structural model in which shear walls are modeled via beam–
column elements overestimates both the global stiffness and the
capacity of the structure. Although the difference in global strength
is relatively small (less than 15%), a more detailed analysis of the
computed results reveals higher discrepancies. To this end we re-
port in Figs. 14 and 15 the deformed shapes and the limit states
corresponding to the points highlighted in Fig. 13. These plotsshow that the global collapse mechanism of the structure
Limit states
ULS (bars)
ELS (bars)
ELS (conc)
ULS (conc)
(a)
(b)
Fig. 10. Wall–frame reinforced concrete structure. Deformed shape and the limit states computed using the present RC shell formulation (a) and 1D beam–column elements
(b).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
−0.1
−0.05
0
0.05
0.1
0.15
x [m]
χ [ m
− 1 ]
Beam A
Shell model
Beam model
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
−0.1
−0.05
0
0.05
0.1
0.15
x [m]
χ [ m
− 1 ]
Beam B
Shell model
Beam model
Fig. 11. Curvatures distribution along beams A and B of Fig. 9 obtained when modeling the shear wall using the shell formulation and beam–column elements.
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correspondsto theattainmentof high bending curvatures atthe base
of the wall and at the beam and column ends. This mechanism is
equally predicted by the two models; however, differences appear
in the local strain distributionat different locations in the structure.
In particular, Fig. 14 shows that fora top displacement of 0.20 m the
global mechanisminvolves crushing of concrete at both the sides of
thewallsas a consequence of theinteraction betweenthewall anthe
nearby frames. This behavior is captured only when using shell
elements; on the contrary, it is completely undetectable if beam
elements are used, in which case only a plastic zone at the bottom
of the walls can be recognized at this load level.
At the last step of the analysis the collapse mechanismis almost
fully developed. The limit states registered at this stage are con-
tour-plotted in Fig. 15, where the previously described local effects
are even more pronounced. Actually, one can remark that the
inelastic regions that start developing from the external regions
of the walls have now spread within the walls that, at this load le-
vel, are almost completely damaged. On the opposite side, beam
elements do recognize only the attainment of limit strain values
only at some isolated cross sections of the wall.
A more precise estimation of the approximation introduced by
use of one-dimensional elements for modeling shear walls can be
gained considering the vertical strain distribution computed at
the three cross sections A, B and C highlighted in Fig. 12. Results
are compared in Fig. 16, where the effect of localized actions
engendered on the wall by the interaction with the frame are
clearly visible. Namely, section A exhibits a high concentration of
vertical strain at both extremities due to the presence of beams
Fig. 12. Shear walled building: structural plans.
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of the first floor. Similarly, section B undergoes high strains only at
the left-hand side because of the presence of the beam of the inter-
mediate landing level, while it remains approximatively plane
nearby the right-hand side. These details of such a complex re-
sponse are out of reach for one-dimensional beam elements that,
owing to their intrinsic kinematic limitations, in this case dramat-
ically fail to estimate the strain state.
On the contrary, when analyzing the strain distribution along
section C located at the base of the wall, the response computed
using the RC shell and the beam–column element are found to
be identical. This is not in contradiction with the previous results
since, unlike sections A and B, section C is not subject to localized
actions due to wall–frame interactions and thereby it stays plane
during deformation.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
0
2000
4000
6000
8000
10000
12000
14000
16000
B a s
e s h e a r [ k N ]
Top displacement [m]
T=2.00
T=2.09
T=2.30
T=2.20
Fig. 13. Shear walled building. Base shear vs top displacement.
Fig. 14. Shear walled building. Limit states for a top displacement equal to 0.20 m.
Fig. 15. Shear walled building. Limit states for a top displacement equal to 0.75 m.
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5. Summary and conclusions
A shell model has been presented for the nonlinear analysis of
reinforced concrete structural walls in which nonlinear material
behavior and arbitrarily distributed steel reinforcement are
allowed.
The capabilities of the present model have been demonstrated
via three numerical examples concerning typical engineering
structures for which the limit state analysis is carried out based
upon a reduced set of material data. Besides the reduced computa-
tional cost of numerical simulations, the presented results also
show that one of the most attracting features of the proposed mod-el resides in its effectiveness in capturing the effects of localized
actions on shear walls due to the interaction with nearby framed
structures. Actually, such actions do produce effects that are out
of reach if walls are modeled via beam–column elements owing
to the intrinsic limitations of classical beam kinematics that con-
strains element cross sections to stay plane during deformation.
In order to recognize the details of global and localized failure
mechanisms, the limit states attained in the examined structures
are plotted by associating a conventional numerical value to the
nominal elastic and ultimate limit states evaluated at the element
quadrature points. Specifically, when analyzing a multi-storey 3D
building the limit states plots show that a difference of a few per-
cent units in terms of the global pushover curve may correspond to
distinct locations of plastic hinges that also behave very differentin terms of deformation. It is therefore expectable that a more
refined analysis based upon material constitutions including dam-
age in the bulk material and/or at concrete-steel interfaces would
greatly enhance the effects of localized actions and modify the
failure mechanisms as well.
In view of their relevance for applications, these last topics de-
serve further research work and will be addressed in forthcoming
papers.
Acknowledgements
This work has been entirely carried out at Department of Engi-
neering of University of Napoli Parthenope. The support of the
hosting institution is gratefully acknowledged.
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