Finite element analysis of RC rectangular shear walls

under bi-directional loading

2016 NZSEE Conference

A. Niroomandi, S. Pampanin & R. P. Dhakal

University of Canterbury, Christchurch, New Zealand.

M. Soleymani Ashtiani

Ian Connor Consulting Ltd.

ABSTRACT: Most of the experimental and numerical studies available in literature on

the seismic assessment of rectangular reinforced concrete (RC) shear walls have

concentrated on the two-dimensional response, using uni-directional cyclic loading

testing protocol. While investigating RC members under bi-directional loading started

several years ago, the effects of this type of loading regime on RC shear walls have not

yet been fully understood and possibly underestimated. Such lack of data is due to the

complexity of the test setup and the perception that rectangular shear walls would tend to

work as 2D elements, without being significantly affected by bi-directional loading (and

thus 3D response). However, recent observations in Chile, New Zealand and Japan

earthquakes have highlighted more complex failure mechanisms confirming the need for

further investigations on the bidirectional response of shear walls. Not only there are a

limited number of experimental studies on the effects of bi-directional loading on

rectangular RC shear walls, but also there is a lack of a numerical model capable of

simulating RC shear walls under bi-directional loading.

In this paper a finite element (FE) model, based on available elements within the library

of FE software, DIANA, has been adopted to simulate the seismic response of rectangular

doubly reinforced concrete shear walls subject to bi-directional loading. Curved shell

with embedded bar elements are used to simulate RC shear walls. This type of model

does not require plane sections to remain plane along the wall. The numerical results are

compared with a set of experimental tests available in the literature. In order to validate

the model, force-displacement curves and failure mechanisms of each shear wall under

uni- and bi-directional loadings are compared to the experimental results. This model will

be used at a later stage of the research program to identify the key parameters influencing

the seismic performance of rectangular RC shear walls under bi-directional loading.

1 INTRODUCTION

In recent earthquakes in Chile (2010) and New Zealand (2011), peculiar failure mechanisms were

observed in reinforced concrete (RC) shear walls which differed from the traditional 2D-based

response (Fig. 1). Of interest was the effect of bi-directional loading on the performance of RC shear

walls, typically ignored in code guidelines for both new design and assessment of existing buildings.

In the past decades the effects of bi-directional loading have been recognized and studied, through

experimental and numerical investigations for both columns (Umehara and Jirsa 1984, Low and

Moehle 1987, Zeris and Mahin 1991, Qiu et al. 2002, Boys et al. 2008) and beam-column joints (Jirsa

and Leon 1986 and Akguzel and Pampanin 2010). However, due to the inaccurate and simplistic

assumption that walls would mostly response as in-plane structural elements, the effect of bi-

directional loading was overlooked for shear walls. In fact, as long as a structural ‘column’ has a

section aspect ratio (length to width) longer than a certain value (e.g. four), it is considered to act as a

wall, and bi-directional loading effects are not considered.

2

(a) (b)

Figure 1 - Shear wall failure modes observed in recent earthquakes in (a) Chile (NIST 2014) and (b) New Zealand (Elwood et al. 2011)

A large number of studies experimental and numerical studies were conducted on the seismic

behaviour of RC walls; most of which focused on the uni-directional cyclic lateral load, simulating the

response of a proto-type structure during an earthquake. Only a small number of these studies

scrutinized the bi-directional loading effects.

Tatsuya (1996) tested five one-third scaled shear wall specimens; one under uni-directional and four

under bi-directional loadings. The tests showed a smaller deformation capacity for the specimens

tested under bi-directional loading compared to the benchmark specimen subjected to uni-directional

loading. However the same failure mode (web crushing) was observed in both cases. In terms of shear

strength and flexural stiffness, instead, not much difference was observed between the specimens

tested under uni-and bi-directional loading regimes.

In a more recent experimental study by Kabeyasawa et al. (2014), eight one-third scaled specimens

representing the first story of a medium-rise multi-story shear wall structure were tested (four under

uni-directional and four under bi-directional loading). The flexural strengths were not much different

between the two loading regimes. However, relatively smaller ultimate inelastic deformations under

bi-directional loading were observed in those tested under uni-directional loading by 1.2 to 1.33 times.

In this paper a FE model capable of simulating RC shear walls under bi-directional loading is utilized

and validated with experimental results conducted by other researchers. This paper is part of a larger

research project investigating the seismic response of rectangular RC shear walls under bi-directional

loading. In the next phases, the key parameters affecting the performance of rectangular RC shear

walls under bi-directional loading will be numerically investigated and an experimental campaign on

shear wall specimens subjected to cyclic bi-directional loading will be carried out in the structural

laboratory.

2 PROPOSED FINITE ELEMENT MODEL

In this study, DIANA software (DIANA 2015) is used to simulate RC walls under cyclic loading. A

description of the adopted material and elements is provided below.

3

2.1 Material Models

2.1.1 Concrete: Total Strain Rotating Crack Model

Total strain rotating crack model which is developed based on the Modified Compression Field

Theory (MCFT), originally developed by Vecchio et al. (1986) is used to model concrete. This model

follows a smeared crack approach for the fracture energy (Litton 1974).

Various methods are reported to model the behaviour of unconfined and confined concrete under

compression such as Popovics (1973), Thorenfeldt et al. (1987), Mander et al. (1988) and Saatcioglu

and Razvi (1992). Here the model proposed by Mander et al (1988) is used for both confined and

unconfined concrete. The modulus of elasticity of concrete, Ec, is estimated using Eq. 1 (Priestley et

al. 2007).

cc fE '5000 (1)

Where, f’c is the compressive strength of concrete.

The ultimate compression strain in concrete, εcu, is calculated using Eq. (2) which is proposed by

Priestley et al. (1996) and is a conservative estimate of Mander et al. (1988) equation.

'4.1004.0 ccsuyhscu ff (2)

Where ρs = volumetric ratio of confining steel (ρs=ρx+ρy), ρx=Avx/(dc×s), ρy=Avy/(bc×s), ρx and ρy =

transverse reinforcement ratios in the x and y directions, respectively, Avx and Avy = the total area of

transverse bars running in the x and y directions, respectively, bc and dc = core dimensions to

centrelines of perimeter hoop in the x and y directions, respectively, s = vertical spacing between

hoops, fyh = yield stress of transverse reinforcement, εsu = steel strain at maximum tensile stress and fcc'

= maximum confined concrete stress (Mander et al. 1988).

According to Priestley et al. (1996), when the member is subjected to bending or combined bending

and axial compression, ultimate compression strain resulting from Eq. (2) tends to be conservative by

at least 50%. Therefore, the ultimate compression strain of concrete is considered 1.5 times of that

resulting from Eq. (1).

One of the other inputs required for the total strain rotating crack model is the concrete response under

uniaxial tensile loading. The uniaxial concrete response under tension is defined following the model

proposed by Hordijk (1991). The fracture energy ‘Gf’ may be determined from Eq. (3) proposed by

Remmel (1994).

10/'1ln065.0 cf fG (3)

2.1.2 Steel Reinforcement: Menegotto-Pinto Model

Cyclic performance of reinforced concrete elements is highly dependent on nonlinear response of

reinforcing bars under cyclic loading. Reinforcing bars undergo yielding during loading reversal,

which is an important nonlinear behaviour; more commonly known as the Bauschinger effect. Many

researchers have successfully investigated the Bauschinger effect in the behaviour of reinforcing steel

(Menegotto et al. 1973, Dodd et al. 1995, Balan et al. 1998 and Hoehler et al. 2006). Some have also

accounted for buckling effect in their models (Rodriguez et al. 1999, Dhakal and Maekawa 2002a,

2002b and Rodriguez et al. 2013).

Out of the above mentioned steel hysteresis response models, currently the Menegotto et al. (1973)

model coupled with a bilinear backbone curve is implemented in DIANA which can be incorporated

to model reinforcements in RC elements. Although the Menegotto et al. model does not consider

buckling of longitudinal reinforcements, this is controlled manually using the method proposed by

Rodrigues et al. (2013).

4

2.2 Solid vs. Curved Shell Elements

DIANA provides a wide selection of different elements including but not limited to the truss, beam,

flat shell, curved shell, solid, and interface elements. Solid or shell elements are the two possible

options to model shear walls. Solid elements tend to produce large system of equations. These are

usually utilized only when other elements are unsuitable or produce inaccurate results. If selection of

solid elements increases the computational effort and time required for the analysis without adding

significant value to the final outcome, two-dimensional elements (shell) may be used. Thus, curved

shell elements, (Fig. 2) which proved to be suitable for modelling shear walls (Dashti et al. 2014) are

used. By employing the curved shell elements, location of steel reinforcements (longitudinal and

transverse) along the thickness of shear wall can still be defined and different diameter bars can be

modelled (Fig. 7 and 8).

The curved shell elements (in this case Q20SH) in DIANA (Fig. 2) are defined based on isoparametric

degenerated-solid approach by introducing two shell hypotheses (DIANA 2015) as follows.

Straight-normals: assumes that normals remain straight, but not necessarily normal to the reference

surface. Transverse shear deformation is included according to the Mindlin–Reissner theory (Mindlin

1951, Reissner 1945 and Reissner 1947).

Zero-normal-stress: assumes that the normal stress component in the normal direction of a lamina

basis is forced to zero (i.e. σzzl(ξ,η,z) = 0). The element tangent plane is spanned by a lamina basis which

corresponds to a local Cartesian coordinate system (xl, yl) defined at each point of the shell with xl and

yl tangent to the ξ,η plane and zl perpendicular to it.

(a) (b)

Figure 2 - Curved shell elements (a) Characteristics and (b) Q20SH (DIANA 2015)

Five degrees of freedom are defined in every element node: three translations and two rotations (Fig.

3). The basic variables in the nodes of the curved shell elements are the translations uX, uY and uZ in the

global XYZ directions (Fig. 3a) and the rotations ϕx and ϕy around the local +x and +y axes in the

tangent plane (Fig. 3b).

(a) (b)

Figure 3 - Displacements (a) Translations, (b) Rotations (DIANA 2015)

3 VALIDATION OF THE PROPOSED FINITE ELEMENT MODEL

In order to validate the numerical model proposed in the previous section, four out of the eight shear

walls tested by Kabeyasawa et al. (2014) were chosen; namely WC-1D, WC-2D, WD-1D and WD-

2D. Table 1 and Fig 4 summarize the basic characteristics of the shear wall specimens WC and WD

(the extensions 1D and 2D indicate the uni- and bi-directional loading regimes, respectively). The

5

other four shear walls are still under investigation and the results will be presented in future

publications. A comparison between the numerical vs experimental results is carried out with

reference to the force-displacement curves and cracking patterns.

Table 1. Shear wall specimens characteristics (Kabeyasawa et al. 2014)

Specimen WC WD

Boundary

Elements

Cross section (mm) 150×300 100×450

Longitudinal reinforcements 8 D13 12 D10

Transverse reinforcements D4@33 D4@33

Web

Length (mm) 1150 850

Thickness (mm) 150 100

Longitudinal and transverse reinforcements 2 D4@50 2 D4@75

Axial force (kN) 675 675

Wall height / In-plane shear span ratio, M/V (mm) 1925 1925

Figure 4 - Details of tested wall specimens (WC and WD) (Kabeyasawa et al. 2014)

Table 2 and 3 show the material properties of each specimen. In the absence of information on the

ultimate strain of steel reinforcements, an ultimate limit state value of 0.06 was adopted for as

recommended by Kowalsky (2000) and NZSEE (2006).

Table 2. Concrete properties (Kabeyasawa et al. 2014)

Specimen WC1D WC2D WD1D WD2D

Compressive Strength (N/mm2) 29.1 30.6 31.2 32.1

Tensile Strength (N/mm2) 1.4 2.1 2.1 1.78

Table 3. Reinforcing bars properties (Kabeyasawa et al. 2014)

Specimen WC, WD

Nominal Diameter D13 D10 D4

Yield Strength (N/mm2) 345 365 380

Young Modulus (N/mm2) 186000 185000 167000

Ultimate Strength (N/mm2) 480 505 494

The test set up and the bi-directional loading protocol adopted by Kabeyasawa et al. (2014) are shown

in Fig. 5 and 6, respectively. The out-of-plane displacement was attained first, then at each peak the

in-plane displacement was applied maintaining the out-of-plane peak displacement (Fig. 6).

The target peak deformation angle (drift) for the in-plane loading reversals were 1/400, 1/300, 1/200,

1/150, 1/100, 1/75, 1/50, 1/37.5, while those for out-of-plane loading were 1/300, 1/200, 1/150, 1/100,

1/75, 1/50, 1/37.5, 1/25. The corresponding peak drifts in the out-of-plane direction were 33% or 50%

larger than those in the in-plane direction, simulating the earthquake response in buildings with

moment resisting frames in the one direction and walls in the other direction. In such cases, the walls

are expected to be subjected to larger drift demands in the out-of-plane direction due to the higher

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flexibility of the frame systems. Furthermore, for the out-of-plane direction of loading, the lateral

jacks were set at the mid-height of the inner column height and a lever arm was adopted to apply shear

and moment on the top of the wall, so that asymmetric bending moment could be applied in the

orthogonal direction (Fig. 5).

Figure 5 - Test setup for uni- and bi-axially loaded walls (Kabeyasawa et al. 2014)

Figure 6 - Bi-directional loading pattern (Kabeyasawa et al. 2014)

Fig. 7 and 8 show the FE models of WD-1D and WD-2D, respectively. As explained before, the shear

walls tested by Kabeyasawa et al. (2014) are cantilever in the in-plane and double curvature in the out-

of-plane directions; thus, there were two challenges involved in simulating the test setup shown in Fig.

5. First, to make sure that the out-of-plane loading system (the rigid liver arm in Fig. 8a) does not

affect the in-plane behaviour and that the shear wall acts as a cantilever in the in-plane direction (Fig.

9a). Second, the rigid liver arm should have enough rigidity to give a double curvature displacement

for the out-of-plane direction (Fig. 9b).

7

(a) (b)

Figure 9 - Finite Element Model of WD-1D (a) Materials Assignment (b) boundary conditions and loadings

(a) (b)

Figure 8 - Finite Element Model of WD-2D (a) boundary conditions and loadings (b) steel reinforcements

Linear Concrete

Concrete Cover

(Unconfined)

Boundary Zone

(Confined)

Web (Unconfined)

Gravity loads

Lateral load

Rigid zone

Out-of-Plane

Loading

Boundary

Condition

In-Plane

Loading

BZ Longitudinal

and Transverse

Reinforcements

Web Flexural

and Shear

Reinforcements

8

(a) (b)

Figure 9 - Shear wall deformation in each direction for the ones under bi-directional loading (a) in-plane

(b) out-of-plane

3.1 Force-Displacement curve

Figs. 10 and 11 show the force-displacement curves of specimens WC and WD, respectively. As it is

seen, numerical results predict the experimental ones satisfactorily. According to Fig. 10 and 11, not

much difference is observed in the flexural strengths when the walls were under bi-directional loading.

Flexural strength was decreased by 3.6% and 2.1% for WC and WD, respectively. However,

displacement capacity was decreased by 47.2% and 32.0% for WC and WD, respectively due to bi-

directional loading. It should be noted that the ultimate displacement is considered when 20%

reduction in strength is reached. The energy dissipation was also decreased in cases where shear walls

underwent bi-directional loading.

(a) (b)

Figure 10 - Force-Displacement curves of WC (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

-800

-600

-400

-200

0

200

400

600

800

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Ho

riz

on

tal

Fo

rce

(kN

)

In-plane Drift Angle (rad.)

WC-1D

WC-2D

Cantilever

deformation

(in-plane)

Double

curvature

deformation

(out-of-plane)

9

(a) (b)

Figure 11 - Force-Displacement curves of WD (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

3.2 Crack Patterns

Figs. 12-15 show the numerical and experimental comparison in terms of crack patterns at collapse.

Kabeyasawa et al. (2014) reported that the brittle shear failure modes under uni-axial loading were

diagonal shear tension in the wall panel while those under bi-directional loading tended to be

horizontal sliding shear in the wall panel above the wall base. FE analysis results also show similar

failure modes to those of experimental ones.

(a) (b)

Figure 12 - Cracking pattern of WC-1D (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

(a) (b)

Figure 13 - Cracking pattern of WC-2D (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

-800

-600

-400

-200

0

200

400

600

800

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Ho

riz

on

tal

Fo

rce

(kN

)

In-plane Drift Angle (rad.)

WD-1D

WD-2D

10

(a) (b)

Figure 14 - Cracking pattern of WD-1D (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

(a) (b)

Figure 15 - Cracking pattern of WD-2D (a) Experiments (Kabeyasawa et al. 2014) (b) DIANA

4 CONCLUSION

This paper is one of the first steps of a comprehensive study dealing with some concerns arisen after

recent earthquakes in Chile (2010) and New Zealand (2011), and specifically regarding the effects of

bi-directional loading on the performance of RC shear walls. The main goal of the whole study is to

identify the key parameters affecting the performance of rectangular RC shear walls under bi-

directional loading and possible recommendations (based on the experimental and analytical

investigations) to improve current practice (taking into account bidirectional loading effect) for both

the design of new walls as well as the assessment of existing ones. Thus, in this paper, a finite element

model capable of simulating cyclic response of RC walls under uni- and bi-directional loading is

proposed and validated with the experimental results available in the literature. For the next step the

proposed finite element model will be used for a numerical parametric study on rectangular RC shear

walls to identify the key parameters influencing the seismic performance of these shear walls under bi-

directional loading.

5 AKNOWLEDGEMNTS

Financial support of this research was provided by the SAFER Concrete Research Project funded by

the Natural Hazards Research Platform (NHRP) and the MBIE Wall project.

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