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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:   nunziante.valoroso@uniparthenope.it (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.

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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.

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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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−0.5

0

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