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Performance limits for structural walls: An analytical perspective

_Ilker Kazaz a, Polat Glkan b,, Ahmet Yakut c

a Department of Civil Engineering, Atatrk University, 25240 Erzurum, Turkeyb Department of Civil Engineering, ankaya University, 06810 Ankara, Turkeyc Department of Civil Engineering, Middle East Technical University, 06800 Ankara, Turkey

a r t i c l e i n f o

Article history:

Received 4 July 2011

Revised 13 May 2012

Accepted 14 May 2012

Available online 18 June 2012

Keywords:

Damage limits

Seismic codes

Plastic rotation

Shear walls

ASCE/SEI 41

Eurocode 8

Turkish Seismic Code

a b s t r a c t

Recently proposed changes to modeling and acceptance criteria in seismic regulations for both flexure

and shear dominated reinforced concrete structural walls suggest that a comprehensive examination is

required for improved limit state definitions and their corresponding values. This study utilizes nonlinear

finite element analysis to investigate the deformation measures defined in terms of plastic rotations and

local concrete and steel strains at the extreme fiber of rectangular structural walls. Response of finite

elements models were calculated by pushover analysis. We compare requirements in ASCE/SEI 41, Euro-

code 8 (EC8-3) and the Turkish Seismic Code (TSC-07). It is concluded that the performance limits must

be refined by introducing additional parameters. ASCE/SEI 41 limits are observed to be the most accurate

yielding conservative results at all levels except low axial load levels. It is shown that neither EC8-3 nor

TSC-07 specifies consistent deformation limits. TSC-07 suggests unconservative limits at all performance

levels, and it appears to fall short of capturing the variation reflected in the calculated values. Likewise

EC8-3 seems to fail to represent the variation in plastic rotation in contrast to several parameters

employed in the calculation. More accurate plastic rotation limits are proposed.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

One of the most important steps of performance based assess-

ment of RC buildings relies on comparison of deformations

obtained from nonlinear structural analyses (static or dynamic)

with the performance based limits. These deformation limits

significantly affect the assessment result so their accuracy plays

a critical role. Provisions for performance assessment of reinforced

concrete structures, such as FEMA356 [1], Eurocode 8 [2] and

ASCE/SEI 41 [3] include deformation limits for both flexure and

shear controlled wall members at specific limit states to estimate

the performance of components and structures. The criteria are de-

fined in terms of plastic hinge rotations and total drift ratios for the

governing behavior modes of flexure (ductile members) and shear(brittle members), respectively. Recently, strain limits are defined

for concrete in compression and steel in tension at serviceability

and damage-control limit states as a vital component of direct

displacement-based design procedures [4]. The recently revised

Turkish Seismic Code (TSC-07)[5] specifies limiting strain values

associated with different performance levels of reinforced concrete

members. While deformations are specified in relation to global

parameters, local damage indicators in terms of strain limits are

used inconsistently to determine the expected performance. For

results of nonlinear pushover analyses to be evaluated according

to either of the acceptance criteria, i.e. whether local or global

response will imply similar performance states is a matter that

must be established because full calibration of the requirements

is lacking.

Another criticism raised against the rotations associated with

different limit states is that they may turn out to be lower than

the actual rotations expected to develop in reinforced concrete sec-

tions[6]. So, it is postulated that the given limits may be unduly

conservative. In a way this is a direct consequence of adaptations

performed for the plastic hinge analysis method employed in the

guidelines. In most applications the momentcurvature relation

of a section is calculated using the plane section assumption, the

limiting plastic rotations in codes were adjusted to conform tothe resulting plastic rotations calculated by multiplying the as-

sumed plastic hinge length and plastic curvature rather than the

actual rotations. These issues will be examined in this study to test

adequacy of the limits specified by codes and guidelines. In previ-

ous studies [7,8]analytical modeling techniques for shear wall ele-

ments both in micro- and macro-levels were investigated and

closed-form equations for estimation of building collapse capacity

of moment resisting frame and shear wall structural systems

subjected to seismic excitations were developed. We employ non-

linear finite element analysis for reinforced concrete structural

components that has been thoroughly verified by benchmark prob-

lems as can be found in Kazaz [9].

0141-0296/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.05.011

Corresponding author. Tel.: +90 312 233 1403.

E-mail address: [email protected](P. Glkan).

Engineering Structures 43 (2012) 105119

Contents lists available atSciVerse 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
http://dx.doi.org/10.1016/j.engstruct.2012.05.011mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2012.05.011http://www.sciencedirect.com/science/journal/01410296http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://www.sciencedirect.com/science/journal/01410296http://dx.doi.org/10.1016/j.engstruct.2012.05.011mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2012.05.011

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Structural performance assessment, done either by the family of

methods grouped under nonlinear static procedures (NSPs) or by

using the more reliable nonlinear dynamic analysis needs to pro-

vide insight into how severe individual members are deformed.

The bridge that gaps the global deformations and member end

deformations must be reliable and stable. In this article we develop

a tool that has been derived for walls that will provide improved

estimates for member level deformations. Current modeling and

acceptance criteria available in the provisions and codes will be

summarized in the next section.

2. Code performance limits

A performance level describes a limiting damage condition that

may be considered likely to be brought into existence for a given

building under seismically induced deformations. Building seismic

performance level is determined on the basis of structural member

damage states. Basically all deformation-based provisions employ

similar damage state definitions for reinforced concrete members.

Structural members are classified as ductile and brittle with re-

spect to their mode of failure in determining the damage limits.

Fig. 1 shows the conceptualized force versus deformation curveused in ASCE/SEI 41[3], TSC-07[5]and EC8-3[2]to specify mem-

ber modeling and acceptance criteria for deformation-controlled

actions. Three discrete Component Performance Levels and two

intermediate Component Performance Ranges are defined in

Fig. 1 to identify the performance level of a member. The terminol-

ogy used in reference to damage states of a member differs among

the documents. In ASCE/SEI 41[3], the discrete Performance Levels

are Immediate Occupancy (IO), Life Safety (LS), and Collapse Pre-

vention (CP). The intermediate Structural Performance Ranges are

designated as Damage Control Range and the Limited Safety Range.

In EC8-3 [2] the discrete Limit States are named as Damage Limita-

tion (DL), Significant Damage (SD), and Near Collapse (NC). In TSC-

07 [5] Damage Limits are Minimum Damage Limit (MD), Safety

Limit (SL) and Collapse Limit (CL) (Fig. 1).

As indicated in Fig. 1, at the Collapse Prevention level (CP)

member deformation capacities are taken at ultimate strength or

at lateral displacement demand at which capacity begins to rapidly

degrade for primary components. At the Life Safety level (LS),

member deformation capacities are reduced by a (safety) factor

of 4/3 over those applying at Collapse Prevention. For the Immedi-

ate Occupancy (IO) two definitions arise in reference to Fig. 1.

While ASCE/SEI 41 and TSC-07 anticipate some degree of nonlinear

deformation beyond the global yield for the immediate occupancy

level and minimum damage, respectively, EC8-3 adopts the global

yield point as the limit state for the damage limitation on the

member.

Seismic assessment provisions investigated in this study estab-

lish the damage states on semantically similar definitions as stated

above, but the differences between the deformation measures

(drift, rotation, curvature and strain) and limiting values used as

modeling and acceptance criteria for structural members can lead

to significant differences in the estimations of global structural

performance. For instance, while ASCE/SEI 41 and EC8-3 uses

plastic rotation as the primary deformation parameter, TSC-07 uses

section strains to assess the performance level of a member.

Considering that these measures are interchangeable, the need

for consistent limit values in different deformation measures is

apparent.

2.1. ASCE/SEI 41 performance limits

ASCE/SEI 41[3] basically adopts the same performance limits

proposed in the wall provisions of FEMA 356 [1]for the seismic

assessment and rehabilitation of existing buildings. According to

ASCE/SEI 41 shear walls shall be considered slender if their aspect

ratio isHw/Lw> 3.0, and shall be considered short or squat if their

aspect ratio is Hw/Lw< 1.5. Slender shear walls are normally con-

trolled by flexural behavior; short walls are normally controlled

by shear. The response of walls with intermediate aspect ratios is

influenced by both flexure and shear. For walls deforming inelasti-

cally under lateral loading governed by flexure, the rotation (h)

over the plastic hinging region at the base of member will be used.

For shear walls whose inelastic response is controlled by shear, the

deformation limits are expressed in terms of the lateral drift ratios.

For multi-story shear walls the drift shall be the story drift.

Table 1gives the ASCE/SEI 41 plastic rotation limits for mem-

bers controlled by flexure where P/Po is the axial load ratio and v

is the maximum average shear stress in the member normalized

with respect to concrete compressive strengthffiffiffiffifc

p calculated as

v Vmax

twLwffiffiffiffifc

p 1HereVmaxis the maximum shear force carried by the member. The

knowledge inherited in normalized shear stress expression given inEq.(1)covers the parameters that affect the wall response signifi-

cantly, so normalized shear is a useful parameter that discriminates

the distinct behavior modes of wall response. ASCE/SEI 41 adopts

the ACI 318-02 [10]requirements for the definition of a confined

boundary.

Elwood et al.[11]proposes further changes to acceptance and

modeling criteria for walls controlled by both flexure and shear,

in order to make them more consistent with experimental results.

For flexural walls the limiting average shear stress in Table 1was

increased from 0:25ffiffiffiffifc

p to 0:33

ffiffiffiffifc

p (MPa) to obtain a better match

with experimental results. Linear interpolation between tabulated

values is to be used if the member under analysis has conditions

that are between the limits given in the tables.

2.2. Eurocode 8

The deformation capacity of beam-columns and walls is defined

as the chord rotation h, i.e., the angle between the tangent to the

axis at the yielding end and the chord connecting that end with

the end of the shear span (Lv= M/V= moment/shear), i.e., the point

of contra-flexure. The chord rotation is also equal to the element

drift ratio, i.e., the deflection at the end of the shear span divided

by the length. The state of damage in a member is defined in

EC8-3[2]by three Limit States:

2.2.1. Limit state of Near Collapse (NC)

The value of the total chord rotation capacity (elastic plus

inelastic part) at ultimate hum of concrete members under cyclicloading may be calculated from the following expression:Fig. 1. Component performance levels.

106 _I. Kazaz et al. / Engineering Structures 43 (2012) 105119

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hum 1

cel0:0160:3v

max0:01;x0max0:01;x

fc

0:225Lvh

0:3525

aqsxfywfc

1:25

100qd 2

where cel = 1.5 for primary elements and 1.0 for secondary ele-ments, h= depth of cross-section, m= N/bhfc (b width of compres-sion zone, N axial force positive for compression), x andx0 = reinforcement ratio of the longitudinal tension (including theweb reinforcement) and compression reinforcement, respectively,

fcis the concrete compressive strength (MPa), qsx=Asx/shbw= ratioof transverse steel parallel to the direction x of loading (sh= stirrup

spacing), qd= steel ratio of diagonal reinforcement (if any), in eachdiagonal direction,a = confinement effectiveness factor. In walls thevalue given by Eq.(2)is multiplied by 0.625.

The value of the plastic part of the chord rotation capacity of

concrete members under cyclic loading may be calculated from

the following expression

hpl

um

1

cel 0:01450:25v

max0:01;x0

max0:01;x 0:3

f0:2

c

Lvh

0:3525

aqsxfyw

fc

1:275

100qd 3

cel= equal to 1.8 for primary elements and 1.0 for secondary ele-ments in Eq.(3).

2.2.2. Limit state of Significant Damage (SD)

The chord rotation capacity corresponding to significant dam-

agehSD may be assumed to be 75% of the ultimate chord rotation

hum given by Eq. (2).

2.2.3. Limit state of Damage Limitation (DL)

The capacity for this limit state used in the verifications is the

yielding bending moment under the design value of the axial load.

In case the verification is carried out in terms of deformations the

corresponding capacity is given by the chord rotation at yielding hy,

evaluated for walls using the following equation

hy /yLv avz

3 0:002 1 0:125

h

Lv

0:13/y

dbfyffiffiffiffifc

p 4where /y is the yield curvature andaVzis the tension shift of thebending moment diagram, db is the (mean) diameter of the tension

reinforcement. zis the internal lever arm, taken equal to 0.8Lw in

walls with rectangular section. avshould be set equal to 1 if shearcracking is expected to precede flexural yielding, otherwise

av= 0.0. The first term in the above expressions accounts for flexure,

the second term for shear deformation and the third for bond slip ofbars.

2.3. Turkish Seismic Code (TSC-07) limit states

In a perplexing divergence from either of these two approaches

the Turkish Seismic Code[5]specifies strain limits to evaluate the

performance of reinforced concrete members. Depending on the

analysis tool, concrete compression and steel tension strain

demands at the member section extreme fibers can be obtained

directly (if the fiber section modeling technique is used) or must

be transformed from section rotations, which are obtained from

pushover analyses or time history analyses (if generalized load

deformation models were used in the member modeling). When

the latter case holds the plastic rotations obtained at the member

plastic hinge locations are used for calculating the plastic curvature

demands at these critical sections followed by the calculation of

total curvature,/t, by adding the yield curvature, /yusing the fol-

lowing equation:

/p hp

lp; /t /p /y 5

Concrete compressive strains and steel tensile strain demands

corresponding to the calculated total curvature demand at the

plastic regions are calculated from the momentcurvature dia-

grams obtained by conventional sectional analyses of the critical

section. Momentcurvature diagrams of the critical sections are

obtained by using appropriate stressstrain rules for concrete

and steel. Finally, the calculated strain demands are compared

with the damage limits given below to determine the member

damage states.

Concrete and steel strain limits at the fibers of a cross section

for Minimum Damage Limit (MD) are given as

ecuMD 0:0035; esMD 0:010 6

Concrete and steel strain limits at the fibers of a cross section for

Safety Limit (SL) are

ecgSL 0:0035 0:01qs=qsm 0:0135; esSL 0:040 7

and for Collapse Limit (CL) they are specified as

ecgCL 0:004 0:014qs=qsm 0:018; esCL 0:060 8

In Eqs.(6)(8),ecu is the concrete strain at the outer fiber, ecgis theconcrete strain at the outer fiber of the confined core, esis the steelstrain and (qs/qsm) is the ratio of existing confinement reinforce-ment at the section to the confinement required by the Code. The

ASCE/SEI 41 [3] standard recommends that the maximum compres-

sive strain in longitudinal reinforcement shall not exceed 0.02, and

maximum tensile strain in longitudinal reinforcement shall not

exceed 0.05 in confined concrete. The Turkish requirements seem

to transcend both limits, but no justification is provided as to theprovenance of these generous limits.

Table 1

Plastic rotation limits for shear wall members controlled by flexure in ASCE/SEI 41.

Shear walls and wall segments Acceptable plastic hinge rotation (rad)

AsA0sfyP

twlwf0

c

Shear force

twlwffiffiffiffif0c

pConfined boundary Performance level

IO LS CP

60.10 60.25 (3)a Yes 0.005 0.010 0.015

60.10 P0.50 (6)a Yes 0.004 0.008 0.010

P0.25 60.25 (3)a Yes 0.003 0.006 0.009P0.25 P0.50 (6)a Yes 0.0015 0.003 0.005

60.10 60.25 (3)a No 0.002 0.004 0.008

60.10 P0.50 (6)a No 0.002 0.004 0.006

P0.25 60.25 (3)a No 0.001 0.002 0.003

P0.25 P0.50 (6)a No 0.001 0.001 0.002

a The values in parentheses are in psi.

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The limits given in TSC-07 [5]are mostly based on the studies

and proposals of Priestley et al.[4], where strain limits for tension

and compression in relation to serviceability and damage-control

limit states to be used in momentcurvature analysis were listed.

Damage-control limit state corresponds to the collapse limit in

TSC-07. The substitution of strain limits for performance accep-

tance criteria is not appropriate for walls, because Priestley

et al.s work is based on momentcurvature analysis that relies

on the plane section hypothesis.

3. Analytical framework for the investigation of performance

limits

3.1. Parameters

A parametric study calculating the static response of idealized

cantilever structural wall models under monotonically increasing

lateral point load applied at the top was conducted. Instead of an

inverted triangular load distribution mimicking the first mode

response, a point load applied at the effective height (2Hw/3)

was used as shown in Fig. 2. The effective heightin such case is also

referred as shear span (Lv). Elastic solutions obtained from bothcases result in only minor difference in lateral deflection calculated

at the effective height. Effect of different design and wall parame-

ters on the deformation measures of structural walls was investi-

gated. The variables of the parametric study are summarized

below. The schematic description of these variables is given in

Fig. 2. The parameters are

Wall length (Lw): 3 m, 5 m and 8 m.

Effective shear span (Lv): 5 m, 6 m, 9 m, 15 m, 24m.

Wall boundary element longitudinal reinforcement ratio (qb):0.5, 1, 2, 4%.

Wall axial load ratio at the base (P/fc/Aw): 0.02, 0.05, 0.1, 0.15,

0.5.

3.2. Design requirements

The walls were designed according to TSC-07 specifications.

Concrete strength was taken as 25 MPa for all cases. Wall boundary

elements were assumed to extend over a region of 0.2Lw at the

edges according to TSC-07. A constant value of tw= 250 mm was

assigned for the wall thickness. A sketch of a typical wall section

is displayed inFig 3. For any given combination of above parame-

ters, such as wall length (Lw), ratio of boundary element longitudi-

nal reinforcement area to the boundary region cross section area

(qb) and axial load ratio (P/Po), the wall yield moment (My) wascalculated. For a given qb the steel area (As= 0.2Lwtwqb) in theboundary element was calculated and converted to discrete longi-

tudinal bars of specific diameter. In the following step, using the

specified shear span length (Lv

) the design shear force was calcu-

lated (Vd= My/Lv). The ratio of the horizontal and vertical web rein-

forcement was assumed to be nominally 0.0025. If the factored

shear force (Ve= kVd) exceeded the shear safety limit calculated

with Vn Aw0:65fctd qtfyd according to TSC-07, the requiredamount of web horizontal reinforcement was recalculated employ-

ing the same equation. Since codes specify that the amount of ver-

tical reinforcement should not be less than the horizontal

reinforcement in the web, the same steel ratio of web reinforce-

ment was used in the vertical direction. The design shear force

was factored only for flexural over-strength. The amplification in

the base shear due to higher mode effects was disregarded.

The deformation capacity of structural walls is controlled by the

level of confinement in the boundary elements. TSC-07 and ACI

318-02 calculate the amount of transverse reinforcement that is

required at the wall boundaries with similar expressions. The

expression in TSC is given as Ash 0:05sbcfck=fytk. This is 2/3 of

the amount of transverse reinforcement used to confine the col-

umn elements. The same equation with a multiplier of 0.09 is given

in ACI 318. Since the thickness of walls was taken as constant,

8 mm bars at 100 mm spacing (/8/100 mm) was used as trans-

verse reinforcement at the boundaries. The yield strength of trans-

verse bars was assumed to be 420 MPa. If the ACI 318-02[10]had

governed the design,/8 hoops at 85 mm spacing would have been

required as confinement steel at the boundary elements. In conclu-

sion wall boundaries can be considered as well confined for TSC-07

and adequately confined for ACI 318-02. Obviously confinement

should be considered among the variables of the parametric study,

but since this would have increased the analysis permutations sig-

nificantly, the study was limited to confined members. Since theconfined concrete model employed here requires only area and

spacing of the transverse reinforcement to achieve the intended

confined section, no specific detailing of the section has been

shown herein.

3.3. Finite element model

Response of the designed walls was calculated using nonlinear

finite element analysis program ANSYS [12]. A new wall model

was developed for analysis purposes. Trial analysis of cantilever

walls under monotonically increasing uniform and inverted trian-

gular load patterns demonstrated that even when cracking may ex-

tend up to mid-height of the wall, significant steel yielding extends

over only lower one or two stories. The upper stories can be effec-tively treated as a cracked beam. Using this analogy the finite

element model displayed in Fig. 4 was developed to reduce the

computation time. The regions with different colors at the edges

designate the boundary elements. Confined concrete material

model was used in this region. Web concrete was taken as uncon-

fined concrete. The first two stories of the cantilever wall were

discreticized with solid continuum elements whereas the upper

stories were modeled with Timoshenko beam elements. The non-

conformance between the nodal degrees of freedom of beam and

solid elements was overcome by providing the transition with

constraint elements. To define the behavior of beam elements

generalized nonlinear section properties were used. The load

deformation behavior of beam elements was assigned in the form

of bilinear forcedistortion angle (Fc) and momentcurvature(M/) relation. The initial flexural rigidity was taken as 0.5EIwFig. 2. Illustration of variables of the parametric study.


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[1]. This model proved to be adequate because all the response

parameters under investigation were focused at the lower stories.

Nonlinear loaddeformation response of the wall models under

monotonically increasing force was determined using static analy-

sis with a point load applied at the top as displayed in Fig. 4.

3.4. Material models

Several physical effects causing nonlinearity in the system wereincluded in the material models adopted for the concrete and steel

materials. For concrete material the five-parameter WillamWarn-

ke[13]criterion was used with the solid element. The model as-

sumes linear elastic stressstrain relationship until crushing.

When used without a plasticity law it underestimates the deforma-

tion capacity of concrete because it neglects the nonlinearity in the

ascending branch and the post-crushing strength of concrete in

compression. ANSYS offers a number of rate independent kine-

matic and isotropic hardening plasticity options that can be used

with the concrete element to model the compression behavior.

Multi-linear isotropic work hardening plasticity (MISO) was com-

bined with the tensile failure criteria of WillamWarnke material

model (CONC). When plasticity based models are combined with

the WillamWarnke concrete material option (CONC), the plastic-ity check is done before the cracking and crushing checks. MISO is

similar to von Mises yield criterion except that a multilinear curve

is used instead of a bilinear curve. Yielding or cracking of any mate-

rial point within the element is evaluated on the basis of principal

stresses. If the wall response is simplified to a plane stress condi-

tion, it is obvious fromFig. 5a that in the quadrants for tension

tension and tensioncompression the WillamWarnke model will

prevail until the cracking of concrete. Upon cracking a plane of

weakness will form orthogonal to the crack direction which re-

duces the principal stress in this direction to zero as the solution

converges. Following the stress relaxation due to cracking in the

quadrant tensioncompression both models will interact. In the

quadrant for compression-compression pure plastic behavior will

apply. The combined multi-surface failure model is displayed in

Fig. 5a. The failure (yield) stress value in each surface is input

through the multilinear stressstrain curve displayed in Fig. 5b.

This curve was obtained using available concrete models.

The confined concrete compressive stressstrain curve at the

boundary elements was calculated according to the Saatcioglu

and Razvi [14] model. Then the smooth curve was brought to a

multilinear formin five segments. Typical confined concrete curves

are plotted in Fig. 5b. For the unconfined concrete classical Hognes-

tad model was used.

Uniaxial behavior of longitudinal and transverse steel bars was

modeled with a bilinear isotropic hardening using the von Mises

yield criterion. Modulus of elasticity of the steel material was takenas 200,000 MPa. The yield stress and tangent modulus at the strain

hardening was taken as 420 MPa and 1500 MPa, respectively. A

strain hardening stiffness helps in achieving a quicker

convergence.

Buckling of longitudinal reinforcement in the compression

boundary element was modeled according to Dhakal and Maekawa

[15].In this model the average compressive stressstrain relation-

ship including the softening in the post-buckling range can be

completely described in terms of the product offfiffiffiffify

p and the slen-

derness ratio,s/db, of the reinforcing bar, wheres is the unconfined

length of the longitudinal reinforcement between the two trans-

verse reinforcement and db is the diameter of the longitudinal

bar. Stressstrain curves modified to take into account bar buckling

were implemented for the longitudinal bars at the compressionboundaries. Typical curves are plotted inFig. 5c.

A sample simulation study is described next. The finite element

model and analysis results of the wall specimen with rectangular

cross section (RW2) tested by Thomsen and Wallace [20] is

displayed in Fig.6. The wall was 3.66 m tall and 102 mm thick. Wall

length was 1.22 m. Reinforcement configuration being similar to

Fig. 3, well-detailed boundary elements of 153 mm length

(0.125Lw) were provided at the edges of the wall over the bottom

1.22 m of the wall. The volumetric ratio of the longitudinal rein-

forcement in the boundary element was approximately

qb = 0.033. Vertical web reinforcement amounts to qsh= 0.00325.The ratio transverse reinforcement for the confinement of bound-

aries wasqs= 0.01. The average compressive strength of concrete

at the time of testing was measured to be 42.8 MPa. The longitudi-nal bars were Grade 60 (fy= 414 MPa) and the ultimate steel strain

Fig. 3. Sketch of a typical wall section and reinforcement detailing of analyzed models.

Fig. 4. Finite element model.


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at rupture from material tests was obtained as 0.08 for the #3 rebar

that was used as longitudinal reinforcement at the boundary ele-

ments. The specimen was loaded cyclically by hydraulic actuators

at the top. An axial stress of approximately 0.075Agfc (P/Po= 0.075) was maintained throughout the duration of the test.

Analysis of the test specimen was performed under both cyclic

and monotonic static loads. The comparison of calculated and

experimental loaddeformation curves are given in Fig. 6f. The

FEM model successfully captured the experimental globalresponse. The wall shear stress normalized with respect to square

Fig. 5. Description of the material models for concrete and steel: (a) Biaxial stress state representation of the combined material model for concrete, (b) Multilinear isotropic

hardening plasticity used to model the behavior of confined concrete under compression, (c) Bilinear steel uniaxial base stressstrain curve and curves modified for buckling.

Fig. 6. (a) Dimensions of the wall tested by Thomsen and Wallace [20], (b and c) Finite element models, (d) Vertical strain plot, (e) Variation of vertical strain profile over

229 mm gage length along the web in the plastic hinge region, (f) Force displacement response.


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root of concrete strength was m= 0.19.Fig. 6d displays the verticaldistribution of strains along the edges at the ultimate limit state.

The maximum concrete compression and steel tensile strains were

calculated as 0.0113 and 0.0335, respectively. The distribution of

vertical strains over the 221 mm high gage length along the web

of the wall at the base is plotted in Fig. 6e. These strain distribu-

tions give the curvature profiles for intermediate and severe dam-

age levels at the base section of the wall. Although the response of

the wall was governed by flexure (m= 0.19 < 0.33), the curvatureprofile is not linear. Using the edge strains, the base curvature at

2% drift was calculated as (0.0335 + 0.0113)/1.22 = 0.0367 rad/m.

Using the given base section properties, the momentcurvature

relation (M/) at the base of the wall was calculated by section

analysis. At the calculated curvature the concrete and steel strains

were obtained as 0.007 and 0.0344, respectively, from section

analysis. Wall edges would be considered as well confined in both

TSC-07 and ACI 318-02 (qs= 0.01). Under such conditions Eq. (8)calculates the ultimate strains for concrete and steel as 0.018 and

0.06, respectively. The same strain limit states were proposed for

the damage-control by Priestley et al.[4]. These limits are greater

than the actual strains. This situation jeopardizes the reliability of

the performance evaluation procedures based on strain limits for

reinforced concrete members.

Further details of the modeling approach and verification

studies on 23 shear wall specimens that were tested under

monotonic, cyclic and dynamic conditions can be found in Kazaz

[9]and Kazaz et al. [16]. It is believed that further discussion on

the finite element modeling aspects, which can be found in the

given references, will contribute much to the scope of this

article.

4. Relation between drift, curvature, rotation and section

strains

In the interest of consistency with the conventional analyses

procedures and deformation measures of modeling and acceptance

criteria specified in codes and standards, curvatures and rotations

on the model should be evaluated in a way that will not contravene

the plastic hinge analysis method [17]. The method is especially

appealing for structural wall buildings because it is simple and it

is possible to idealize a wall member inside the building as isolated

cantilevers as displayed inFig. 7. In the plastic hinge analyses the

tip displacement of a cantilever is obtained as the sum of its flex-

ural yield displacement, Dy, and plastic displacement component,

Dp. While the yield displacement is calculated by double integrat-

ing the curvature distribution along the cantilever, plastic displace-

ment component is calculated by multiplying the height of the

cantilever (measured from the center of plastic hinge region) by

the plastic rotation, hp, at the base as expressed in the following

equation:

D Dy Dp /yH

2

3 / /yLpH 0:5Lp 9

The term, (//y)Lp, in Eq. (9)is the plastic rotation hp and is

based on the governing assumption that the plastic curvature is

lumped at the center of the equivalent plastic hinge length, Lp.

The actual physical length over which the plasticity spreads may

be larger and referred to as plastic hinge region, Lpz. The plastic

zone length that yields accurate plastic rotation can best be deter-mined from experiments. Eq.(9)may then be used to calculate the

equivalent plastic hinge length.

Relating the local deformation demands (strains) of the wall to

the curvatures so that flexural deformations can be interpreted

correctly requires correct determination of plastic hinge length or

spread of plasticity along the member. The method that was

employed to evaluate the length of the plastic zone,Lpz, can be de-

scribed as follows. Curvature profile computed from element

strains calculated at the same height at the two wall ends were

used to determine the spread of plasticity along the wall. The lim-

iting yield curvature to determine the spread of plasticity along the

wall can be calculated with the expression proposed by Priestley

et al.[4]

/y 2eyLw

10

whereey is the yield strain of the reinforcement and Lw is the walllength.

As a second step, the base curvatures and rotations were calcu-

lated over the identified plastic zone lengths. The rotations (hb),

which were assumed to represent the rotation of the base section,

were calculated just above the plastic zone length by using the ver-

tical displacements calculated at tensile and compressive edges in

the same row. The sketch inFig. 8illustrates the calculation of the

base section curvature and rotation in a way that is consistent with

ASCE/SEI 41. The base curvature was calculated in two different

ways to ensure the accuracy in the calculation of this parameter,

because all the performance criteria and assessment procedure de-

pends on it. /b1was obtained by using the moment-area theorem.

Since the rotation above the plastic zone is known (hb) now by inte-

grating the curvature profile, whichwas assumed to be linear along

the plastic zone lengthLpz,/b1was obtained from 2hb/Lpz. Then/b2was obtained by fitting a best line to the curvature profile along the

plastic zone length. The intercept of the best fit line equation at the

base level was adopted as /b2. The two methods of curvature cal-

culation yielded very similar results. The base curvatures used in

this study are those calculated by best line fit method, i.e./b= /b2.

Violation of the plane section remains plane after deformation

hypothesis due to concentration of compression strains at the

section of maximum moments at the base of the wall precludes a

direct comparisonof the curvatures calculated fromexperimentally

Fig. 7. Definition of plastic hinge length and analysis [17].


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measured strains and those obtained from finite element analyses

such as in this article with the curvatures obtained from mo-

mentcurvature based section analysis. Base curvature calculated

in the plastic hinge length may provide a more appropriate means

of linking local deformations measured experimentally (or results

of FE analysis) at the base of the wall to the results from mo-

mentcurvature analysis. The plastic rotations calculated above

the plastic hinge region should be compatible with the modeling

and acceptance criteria values tabulated in ASCE/SEI 41[3].

In the current study the strains and curvatures were evaluatedaccording to section analyses, because the current state-of-the-art

analysis methods for shear walls depend primarily on plane sec-

tion deformations (plastic rotation-plastic hinge, fiber elements,

multiple vertical spring line elements). So, the presented results

and recommendations must be consistent with section based anal-

ysis methods. Matching the momentcurvature relations of finite

element analysis and section analysis on the basis of curvature,

the strain limits were determined.

The section based momentcurvature analyses were conducted

with a modified version of CUMBIA [18]. Comparison of typical

momentcurvature relations obtained from section analyses and

finite element analyses is shown in Fig. 9. The curves agree in

the initial segment and in terms of moment capacities, but the ulti-

mate curvature capacities obtained from the two different analysesdiffer significantly. The ratio of ultimate curvatures obtained from

section and finite element analyses are plotted as a function of nor-

malized shear stress inFig. 10. It is seen from the figure that the

section analysis results deviate from the finite element analysis

results in terms of curvature limits as the wall length and unit

shear on the member increase. In most cases the sectional analysis

results overestimated the deformation capacity of walls, which

may lead to un-conservative assessment of the structural walls.

Even when section analyses indicate very large deformation

capacities, the limiting value for the curvatures was adopted as

the capacities obtained from the finite element analysis in this

study.

The second floor drift ratio was used to display the deformation

capacity of the walls. Roof drift of cantilever walls is not a mean-

ingful measure to investigate the deformation capacity because if

the drift value is used as modeling criterion for structural walls it

would be unconservative. Additionally, the base stories are the

most critical regions when the deformability of the walls is consid-

ered. In this way the deformation components can be directly com-

pared with code specified values. The second floor displacements

were preferred over first floor values because the total drift com-

posed of shear and flexural deformation components can be calcu-

lated more representatively over two story height due to excessive

diagonal cracking and localized damage at the base story.

5. Calculated damage states

The response quantities of wall models are presented at the

three damage levels given above namely, global yield, ultimate

and an intermediate damage level, i.e. life safety, defined as the

percentage of ultimate. In the analysis the ultimate point was

determined on the basis of one of the criteria defined as the pointon the loaddeformation curve where strength drops abruptly or

degrades to 85% of the ultimate strength (Vmax), or the steel strain

at the tension side exceeds es= 0.1, or the reinforcing bars at thecompression side buckles (accompanied by significant crushing

of concrete). The degrading effect of cyclic loading regimes on

the stiffness and strength of reinforced concrete was not consid-

ered in the analyses carried out in this study. Vallenas et al. [19]

proposed that as a general rule the overall deformation capacity

under a realistic ground motion could be expected to be over

75% of the deformation capacity under monotonic loading condi-

tions. This is due to deterioration of concrete, and the development

of cyclic failure mechanisms associated with the load history and

characteristics of the specimens. Typical loaddeformation curves

validating this assumption was obtained from analyses of theRW2 wall specimen tested by Thomsen and Wallace [20]as dis-

played in Fig. 6f. 75% of the ultimate displacement capacity of

the analytical model analyzed under monotonically increasing

loading agrees well with the ultimate displacement capacity of

the specimen tested under cyclic loading regime. Further valida-

tion can be found in Kazaz [9]. In conclusion, it is assumed that col-

lapse prevention performance level is taken at the 75% of the

ultimate point of the finite element analyses in this study.

For damage limitation or minimum damage for immediate

occupancy two definitions arise in reference to Fig. 1,global yield

Fig. 8. Schematic descriptions of base curvature and rotation calculation.

Fig. 9. Comparison of typical momentcurvature relationships obtained from section and finite element analysis.


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and a point slightly greater than the yield point including slight

nonlinear action. Global yielding of wall models was determined

on the basis of momentcurvature relation at the wall base section

employing Eq.(10). As discussed previously ASCE/SEI 41 and TSC-

07 anticipate that slight nonlinear action can be accommodated by

the member. Explicitly the damage limitation point was adopted as

the point where concrete compressive strain of 0.0035 and steel

tensile strain of 0.01 is first reached in the section level.

6. Results of analyses

6.1. Evaluation of global and local deformations

Fig. 11presents the various deformation measures at particular

damage states, corresponding to global yield, ultimate deformation

and 75% of the ultimate deformation. These general terms are used

to define damage states of the wall models, because although seis-

mic design documents indicate similar damage condition, they usedifferent terminology for these damage states. Another objective is

to present the limiting values for global yielding of the member as

a benchmark point, because ASCE/SEI 41 and TSC-07 anticipate the

point of immediate occupancy and minimum damage beyond the

global yield. Drift limits at different damage states are plotted as

a function of normalized shear stress in the wall. The shear stress

limits used to determine the plastic rotations in ASCE/SEI 41 are

superposed on the artwork. ACI 318-02 states that the normalized

shear carried by a wall member should not exceed 0:83Awffiffiffiffifc

p in

MPa. The deformation limits are categorized with respect to axial

load ratio as well. As seen inFig. 11, increased shear stress and ax-

ial load ratio significantly reduce the deformation capacity of

structural walls. The data presented here agrees with the normal-

ized shear stress limits used to differentiate between the flexuraland shear type of behavior in ASCE/SEI 41. For walls with

m< 0.33 flexural behavior governs.According toFig. 11a, the drift at yield varies between 0.1% and

0.4%. The drift capacities of walls that respond in the flexural mode

range from 1.5% to 4.5% depending on the level of axial load as

shown inFig. 11c. For flexural walls (m< 0:33ffiffiffiffifc

p ) and walls under

combined flexure and shear action (0:33ffiffiffiffifc

p < m< 0:50

ffiffiffiffifc

p ) the

lower bound of drift can be taken as 11.5% for life safety and col-

lapse prevention performance levels, respectively, even under very

high axial load conditions. Under moderate loading conditions

(m< 0:50ffiffiffiffifc

p andP/Po< 0.10) the limiting values can be extended

to 1.52.5% for life safety and collapse prevention performance

levels, respectively. These limits should be interpreted as the inter-

story drift ratio. The roof drift at these performance levels may takea slightly higher value.

The second row plots inFig. 11display the dimensionless base

curvature obtained by multiplying the calculated base curvature

with the wall length. It is seen inFig. 11d that yield base curvature

can be effectively calculated using Eq. (10). Dimensionless curva-

ture yields the total strain excursion (/Lw= etec) on the section.Restrepo-Posada[21]states that the total strain excursion (etec)is better suited for characterizing the onset of buckling and frac-

ture of reinforcing bars than the tensile strainet

alone. Priestley

et al. [4], investigating the effect of several parameters on the

ultimate curvature set the damage control curvature limit as /

Lw= 0.072, and assert that the variation in limit curvatures is

small, and average values provide an adequate estimate, within

10% of the data for all except high axial load combined with high

reinforcement. From measurements on six flexural wall specimens

Dazio et al. [22] proposed that fracture of boundary reinforcing

bars occurs at an average value of/Lw= 0.7esu, where esu is thestrain at rupture of the reinforcing bar, yet the scatter was rela-

tively high. Dazio et al. [22] proposed that the ultimate damage

state limit should be set to 0.50.6esu for plastic hinge analysis(which yields/Lw= 0.050.06 foresu= 0.1). WhenFig. 11f is eval-uated in the light of this discussion, it is seen that the proposed

limits can only be assumed to form the upper bound of the data

at the low-to-medium shear stress range (m< 0:50ffiffiffiffifc

p ).

The total rotations calculated above the plastic zone as illus-

trated in Fig. 8 are plotted in Fig. 11gi. The smooth trend observed

in Fig. 11i confirms the correctness of including shear stress and

axial load in specifying structural wall rotation limits at ultimate

limit state. However, no pronounced effect of shear stress and axial

load is observed on the yield rotation of walls. When three distinct

data clouds are investigated inFig. 11g, it is noticed that wall yield

rotation correlates well with wall length. This is the direct conse-

quence of the procedure used to calculate the rotations described

above. Yield rotation is calculated over the region where the yield

curvature specified in Eq.(10)is exceeded for the first time along

the wall. Since yielding starts at the base, where the moment is

maximum and then moves up along the wall as the top drift is

increased, for all models the first yield occurs on the first row ofelements on the finite element model. This length over which the

yield rotation is calculated varies between 0.25 and 0.30 m. Then

rotation becomes the product of the yield curvature and that

length. It must be noticed that if a greater length, such as the plas-

tic hinge length that will fully develop in the inelastic response

phase, is used in deriving the yield rotation the order of rotations

will be much higher. The order of yield rotations calculated at

the very base of the walls agrees with reported experimental val-

ues [23]. Although guidelines enforce the use of plastic rotations

as assessment criteria, the section curvature may be a more appro-

priate means of drawing performance limits since curvature is the

direct product of simple section analyses. This way, the uncertainty

due to plastic hinge length assumption is avoided.

The last two rows inFig. 11present the compressive (ec)SECandtensile strains (et)SECobtained by matching section curvatures frommoment curvature analysis with the base curvature (/b2) at limit

states determined from finite element analyses. There is a signifi-

cant difference between the finite element analyses strains and

section analyses strains calculated at the same curvature that arise

from plane sections assumption.Fig. 12 displays the comparison of

compressive strains from two types of analyses at the global yield-

ing and ultimate limit states. The compressive strains calculated

from finite element analysis are especially larger than the com-

pressive strains from section analysis at the ultimate limit state

by an order of 2.1 in average due to localized compression damage

at the boundary element. The tensile strains from the two separate

analyses methods vary but agree in the mean. The strains

presented in this study are from section analysis because finite ele-ment analysis strains are meaningless unless the tools employed in

Fig. 10. Ratio of ultimate curvature capacities obtained from section (/SEC) and

finite element analysis (/SIM).


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Fig. 11. Variation of deformation parameters with normalized wall shear stress and axial load ratio.


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seismic assessment of reinforced concrete members and structures

incorporate such effects in the analysis.

The limiting concrete compressive strain at wall yield can be ta-

ken as 0.002 as shown inFig. 11j. For walls responding in the flex-

ural mode (m< 0:33ffiffiffiffifc

p ) under low axial load ratios (P/Po< 0.1)

compressive strains are much lower than 0.002. Conversely, thesteel strains are larger at the flexure controlled region and decrease

as the shear carried by wall increases. The total strain excursion

etec, which is given indirectly in Fig. 11d, is nearly constant sum-ming to approximately 0.004. However, etec= 2ey, which stemsfrom the position of the neutral axis located at the center of the

section assumption, is valid only for medium range shear stresses

whenFig. 11j and m are examined together.

Fig. 11l and o display the compressive and tensile strains calcu-

lated at the ultimate limit state, respectively. The strain limits

proposed by Priestley et al.[4] and TSC-07[5]are superposed on

the same figure for comparison. The available limits lead to uncon-

servative estimates of the deformation capacity at the ultimate

limit state. The limiting strains are sensitive to the axial load ratio

on the section.

6.2. Comparisons with codified limits

InFig. 11five different damage parameters were addressed at

the described limit states. Among these parameters rotations are

of primary importance since structural elements are modeled as

nonlinear frame elements with lumped plasticity by defining plas-

tic hinges at both ends of the members in structural analysis pro-

grams. While ASCE/SEI 41 and EC8-3 directly specify the plastic

rotation limits for the modeling and acceptance criteria, TSC-07

dictates strain limits for momentcurvature analysis that are re-

quired to be converted to plastic rotations. Figs. 1315 display

the comparison of the calculated rotation limits with the limitsspecified in these documents. Data in each plot is also classified

with respect to axial load ratio. The description of the damage lim-

its used in the analysis was given previously.

Fig. 13ac compare ASCE/SEI 41 limits with the calculated plas-

tic rotation limits. At the immediate occupancy performance level

the ASCE/SEI 41 limits yield conservative estimates for medium

and high axial load ratios, but for low axial load ratios the limitsare on the unsafe side. This situation contradicts expectations,

yet it is the consequence of the procedure used in the calculation

of plastic rotations. Analysis results indicate that while yielding

initiates at a limited region near the base of the wall under low

axial load ratios, it has a distributed pattern in walls subjected to

high axial load ratios. Consequently the plastic region length is

larger in high axial load cases yielding larger plastic rotations. At

the collapse prevention performance level, ASCE/SEI 41 limits are

below 0.02 rad, yielding conservative estimates. However, it

appears that a cap has been applied for greater values. The capped

data falls into the region characterized by low shear stress and

flexural response as displayed inFig. 11i. Since Eq.(10)seems to

yield good estimation of yield curvature in reference to Fig. 11d,

the yield rotation according to ASCE/SEI 41 can be obtained usingEq.(10)as hy= (2ey/Lw)0.5Lw= 0.0021 rad.

The rotation limits presented in EC8-3 that are defined in terms

of chord rotation differ from the rotations that will be advocated in

this study where it is intended to obtain rotation limits that ac-

count for the member deformation characteristics within a story.

The chord rotation cannot be representative of base rotation of

shear walls under higher mode effects and walls interacting with

frames, especially with the strong ones. In order to be consistent

with EC8-3 definitions, chord rotations were calculated and com-

pared with EC8-3 limits inFig. 14. The chord rotation is calculated

as the tip drift ratio using the flexural displacement component.

EC8-3 adopts the yield point as the damage limitation point.

Fig. 14a displays the correlation of the yield rotation calculated

using Eq. (4) with the element drift ratio calculated at the tip (roof)of cantilever finite element model. The inference ofFig. 14a is that

Fig. 12. Comparison of extreme fiber compression strains at global yield and ultimate limit states for section and finite element analyses.


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the yield rotation given by Eq. (4)significantly overestimates the

analysis results, especially as the shear-span-to-wall-length ratio

increases. No pronounced effect of axial load ratio is observed on

the yield rotation. The order of yield rotations calculated according

to EC8-3 ranging from 0.003 to 0.009 rad is high for stiff shear wall

elements. The yield rotation limits proposed by EC8-3 are uncon-

servative for shear walls especially when the shear-span-to-wall-

length ratio is high.

For shear wall elements, Eq.(3)proposed in EC8-3 used to cal-

culate the ultimate limit state plastic rotation yields conservative,

yet unrealistic limits as shown in Fig. 14c. The equations seem to

be insensitive to the most of the design parameters, except the

shear-span-to-wall-length ratio and axial load ratio. The resultingplastic rotation limits vary between 0.006 rad and 0.019 rad on

average. The plastic rotation limits calculated according to EC8-3

are observed to be smaller than the ones given in ASCE/SEI 41,

which is contrary to the expectations. EC8-3 defines chord rotation

with respect to shear-span (M/V) as opposed to the plastic rotation

in ASCE41 defined over the plastic hinge region. The life safety lim-

its are calculated as 3/4 of collapse prevention limits.

Fig. 15displays the correlation of plastic rotation limits calcu-

lated according to strain limits given in TSC-07 using section based

momentcurvature analysis with the plastic rotations calculated

from the analytical model. The plastic hinge length was taken as

0.5Lw as proposed by TSC-07 in the calculation of plastic rotations

from the curvatures from section analyses. For all damage states

(or performance levels) TSC-07 overestimates the rotationsignificantly without showing much variation. The horizontally

Fig. 13. Comparison of calculated plastic rotation limits at specific performance levels with the limits available in ASCE/SEI 41.

Fig. 14. Comparison of calculated plastic rotation limits at specific performance levels with the plastic chord rotation limits in EC8-3.

Fig. 15. Comparison of calculated plastic rotation limits at specific performance levels with the limits available in TSC-07.


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extending trend inFig. 15ac reflects that the given TSC-07 limits

fall short of predicting the variation in the rotation due to varying

design parameters. As discussed previously the same is also valid

for EC8-3. The average limit curvatures in dimensionless form for

serviceability (/sLw) and damagecontrol (/dcLw) states were pro-

posed as 0.0175 and 0.072, respectively, by Priestly et al.[4] as dis-

cussed previously. The procedure implemented in TSC-07 for the

seismic assessment results in rotations as hMD

= 0.5/sLw

=

0.00875 rad for minimum damage and hCL= 0.5/dcLw= 0.036 rad

for collapse limit. These limits are consistent with what is pre-

sented inFig. 15, yet they are insufficient to calculate the actual

rotations for a range of walls. Another reason that causes differ-

ences between the analytical results and TSC-07 predictions is

the basic crudeness of plastic hinge length.

7. Discussion

The analysis results suggest that the deformation capacity of

structural walls with confined boundary elements is larger than

the limits given in ASCE/SEI 41 provisions. It is seen that ASCE/

SEI 41 yields conservative estimations of the structural perfor-

mance. On the other hand, if the strain based performance criteriadefined in TSC-07 or as suggested by Priestley et al. [4]is used in

the determination of structural performance, unconservative

estimates of performance are obtained for reinforced concrete

rectangular walls. With reference to trends in Fig. 11i and l it can

be concluded that plastic rotation is a much more stable parameter

than the strains to establish the limit states of reinforced concrete

members. The dispersion in the strain data in Fig. 11k and l indi-

cates that this measure of deformation is excessively sensitive to

member dimensions, material properties, reinforcement and the

level of axial load than the plastic rotations. So, the results of this

study favor the use of plastic rotations as performance limits in

the assessment of reinforced concrete wall members. Equations gi-

ven in EC8-3 for calculating performance based rotation limits lead

to unconservative values at yield. The plastic rotation limits show

insignificant variation and appear to be inadequate in estimating

the finite element results. It is worth noting that similar equations

are used for beams, columns and walls despite significant differ-

ences in their behavior within a structure. We address one part

of this oversimplification by the refinement for walls.

Fig. 16 displays the lower bound plastic rotation limits of the

data obtained from analyses at different performance levels. In this

figure, ASCE/SEI 41 limits are also displayed for comparison. As

seen here, the major differences are for limits in the immediate

occupancy level and at low and high shear stress ranges for the

other two performance levels. ASCE/SEI 41 limits are specified at

0:33ffiffiffiffifc

p and 0:50

ffiffiffiffifc

p , where intermediate values can be obtained

by linear interpolation. However, outside these limits we have lit-

tle basis about what the extrapolated trend may be. The proposed

Fig. 16. Lower bound plastic rotation limits at different performance levels.


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limits for the plastic rotations of walls controlled by flexure are

tabulated inTable 2. These limits are given as alternative values

to Table 6.18 in ASCE/SEI 41 for conforming members. The limits

are derived as a function of normalized shear stress (m) and axialload level (P/Po) for different ranges of these variables in order to

obtain more accurate representation of plastic rotation limits at

the specified performance levels. Limits in relation to mid-range

axial load levels (P/Po= 0.15) are also introduced to increase the

accuracy of the assessment procedure. In case the ASCE/SEI 41

format, i.e. numerical limit values at specific m andP/Po, the under-lined number corresponds to the existing ASCE/SEI 41 limit and the

left side number is the value proposed by this study.

Alternative to limits proposed inTable 2, the lower bound plas-

tic rotation limits can also be expressed with a more general expo-

nential expression as given in the following equation:

hp 0:031 0:053P=Po e

1:75m if P=Po 0:10

0:031 0:053P=Po e1:901:47P=Pom if P=Po > 0:10

(

11

As seen inFig. 16the proposed values correspond to the lower

bound limits of the collective results. A general expression is

derived through regression analysis to calculate the actual rotation

limits at collapse prevention performance level. Using the param-

eters that govern the response of shear walls an equation is derived

to estimate the rotation capacity of shear walls with moderately

confined boundary elements. The equation from a regression anal-

ysis reads as

hp AqbB

eCmDLw rhp 12

whereA,B,CandD are coefficients defined inTable 3as a function

of axial load ratio, the boundary element reinforcement ratio (qb),

the normalized shear stress (m and the wall length (Lw). Whilenormalized shear and axial load have already been used to express

the plastic rotations in the seismic documents, the existence ofLwin

the expression can be legitimized by the fact that the rotations in

our model were calculated above the plastic zone where the spread

of plasticity depends primarily on the wall length. Strain hardening

and tension shift affecting the spread of plasticity are related to the

amount of boundary element longitudinal reinforcement (qb).r(hp)is the standard deviation of the calculated plastic rotation limits de-

fined as a function of plastic rotation. The following set of equations

can be employed in the calculation ofr(hp).

rhp

0:1429hp 0:00 05 if hp 0:014 rad

0:0025 if 0:014 rad< hp 0:03 rad

0:8125hp 0:0219 if 0:03 rad< hp 0:038 rad

0 if 0:038 rad< hp

8>>>>>:

13

The limit obtained through Eq.(12) is greater than the limits gi-

ven inTable 2. The predictions omitting the standard deviation are

compared with the analytical results in Fig. 17. The predictedvalues agree quite well with the computational results. If the pre-

dicted values are reduced by 0.75 the limits for life safety perfor-

mance level is obtained.

8. Conclusions

Nonlinear static or response history analysis procedures are the

tools with which the deformation response of structural compo-

nents are estimated in displacement based procedures. Regardless

of the method of analysis employed, local and global quantities in

terms of internal forces and deformations form the basis of judg-

ment. These are then used to assess performance of structural

assemblies. The most challenging part of the displacement based

assessment procedures is the determination of the deformationlimits that strongly influence the results. Therefore, the primary

objective of the study here was to evaluate the limits recom-

mended by the codes and guidelines we have chosen to examine

in detail. The results of this study that were obtained from compre-

hensive parametric analyses of the walls provide data to ade-

quately test performance based deformation limits specified in

different documents. Among the documents evaluated, ASCE/SEI

41 limits were observed to be the most accurate ones yielding con-

servative results at all levels except the low axial load levels. It has

been shown that neither EC8-3 nor TSC-07 specifies adequately

consistent deformation limits. TSC-07 suggests unconservative

limits at all performance levels, and it appears to fall short of

capturing the variation reflected in the calculated values. Likewise

EC8-3 seems to fall short of representing the variation in plasticrotation in contrast to several parameters employed in the

Table 2

The proposed plastic rotation limits for shear wall members controlled by flexure

(underlined values corresponds to ASCE/SEI 41 limits).

IO LS CP

P/Po 6 0.10

m< 0.33 0.0050.0106m 0.0180.0242m 0.0250.0303mm= 0.33 0.0015/0.005a 0.01/0.01 0.015/0.0150.33 < m< 0.50 0.0015 0.01390.0118m 0.02470.0294m

m> 0.50 0.0015 0.0120.008m 0.0150.01mm= 0.50 0.0015/0.004 0.008/0.008 0.01/0.01

P/Po= 0.15

m< 0.33 0.0050.0045m 0.0180.0273m 0.0250.0394mm= 0.33 0.0025/0.0043 0.009/0.0087 0.012/0.0130.33 < m< 0.50 0.00350.0029m 0.01480.0176m 0.0160.012mm> 0.50 0.00250.001m 0.00850.005m 0.0160.012mm= 0.50 0.002/0.0032 (0.006/0.0063 0.01/0.0083

P/Po P 0.25

m< 0.33 0.0040.0045m 0.0150.0242m 0.020.0333mm= 0.33 0.0025/0.003 0.007/0.006 0.009/0.0090.33 < m< 0.50 0.00350.0029m 0.01180.0109m 0.0120.009mm> 0.50 0.00250.001m 0.0070.004m 0.0120.009mm= 0.50 0.002/0.0015 0.005/0.003 0.0075/0.005

a This study/ASCE/SEI 41.

Table 3

Coefficients of Eq. (12) to calculate the ultimate plastic rotation limit of structural

walls with conforming boundary elements.

P/Po A B C D

60.10 0.138 0.220 1.814 0.071

=0.15 0.087 0.148 1.779 0.066

=0.25 0.034 0.037 1.485 0.037

Fig. 17. Correlation of predicted plastic rotations with analysis results.


7/26/2019 1-s2.0-S0141029612002477-main

15/15

calculation. The estimations are unconservative at damage limita-

tion. Although conservative estimations are obtained at life safety

and collapse prevention levels, the values are not logical.

For displacement-based design and assessment structural per-

formance limits are best defined in terms of slowly varying

structural deformation measures. Member end rotations and drifts

are such measures because they represent integration along de-

formed sections where the effect of local singularities is evened.

Strain is a poor indicator not only because it is a local index, but

also because its recovery from other measures is vulnerable to sig-

nificant modeling errors. The ASCE/SEI 41 and EC8-3 requirements

are reasonable because they refer to plastic deformations in terms

of chord rotations. The TSC-07 requirements are inconsistent, and

seem not to be based on any careful prior evaluation, victimized

by the fundamental false premise of arriving at global indices of re-

sponse from very local quantities of deformation such as strain.

The computational modeling approach for a range of parame-

ters applied to a structural wall governed by flexural deformation

described in this article permits derivation of expressions for plas-

tic rotations that expand and improve ASCE/SEI 41 limit states.

These are summarized in Fig. 16 and Table 2. These expressions

are adjusted to err on the safe side, and represent a significant

improvement over existing provisions. The provisions of the Turk-

ish Code seem to lack basis in fact, and must be revised.

Nonlinear static procedures or response history analyses must

ultimately be interpreted in terms of member deformations formu-

lated as acceptability criteria. The superiority of one set of proce-

dures against another becomes moot when acceptability criteria

have fallen prey to internal inconsistencies. By incorporating

explicitly the effects of axial load and unit shear the expressions

that have been derived in this study would permit an improved

judgment basis for performance.

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