final article chai and kunnath

8/11/2019 Final Article Chai and Kunnath

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Y.H. Chai, S.K. Kunnath / Engineering Structures 27 (2005) 10521063 1053

minimum wall thickness requirement is reproduced here; for

walls with two layers of vertical reinforcement, it needs to

satisfy the following inequality:

twkm ( +2)(Ar+2)lw

1700

(1)

wherekm= 1.0, unless it can be shown thatkm=

ln

(0.25+0.055Ar)lw0.1 (3)

where is the displacement ductility factor, Ar is the

aspect ratio of the wall, i.e. total wall height divided by wall

length,ln is the clear vertical distance between floors, lw is

the length of the wall, l is the vertical reinforcement ratio,

fy is the yield strength of the reinforcement, and fc is the

concrete compressive strength. Although not immediatelyapparent, the minimum wall thickness required by [4] is

quite different from that required by [10]. For example,

for a 10-story wall with a displacement ductility factor of = 4, aspect ratio of Ar = 6, vertical reinforcementratio ofl = 0.02, concrete compressive strength of fc=34.5 MPa, and steel yield strength of fy = 414 MPa,the [4] minimum wall thickness is 9.7% of the floor-

to-floor (story) height, where the unsupported height has

been assumed to be 90% of the story height, i.e. ln =0.9hs . This minimum wall thickness is significantly larger

than that required by [10], which is 5.6% of the story

height. In view of the significant difference between the

two codes, an assessment of the minimum wall thicknesses

and their comparison under different design parameters is

warranted.

2. Stability of structural walls

The phenomenon of inelastic buckling of structural

walls under reversed cyclic loading is admittedly complex;

for tractable solutions, appropriate simplifications must be

made. A basic understanding of the inelastic buckling of

walls may be provided by a cyclic axially loaded reinforced

concrete column, the response of which is assumed to berepresentative of the high tension/compression region of

the wall. Although the effect of strain gradient across the

wall is not included, the idealization is nonetheless useful

in identifying critical parameters governing the buckling

mechanism.

The reversed cyclic response of an axially loaded

reinforced concrete column is demonstrated using the

following test data from [2]. The photographs in Fig. 1(a)

and (b) show the condition of the test column after

being loaded to different tensile strain amplitudes. The test

column, which had a rectangular cross-section of 100 mm 200 mm, was reinforced with six No. 3 bars (db

=9.5 mm)

providing a longitudinal reinforcement ratio of 2.1%. The

length of the test column wasL o=1500 mm, correspondingto a length-to-width ratio of 14.75. Transverse ties were

provided at close spacing of six times the longitudinal bar

diameter simulating a well-confined condition in the end

region of a ductile wall. The close spacing of the transverse

reinforcement was also intended to prevent local buckling ofthe longitudinal reinforcement.

Fig. 1(c) shows the hysteretic response of the test column

under axial tension/compression cycles. The nominal axial

strain a , plotted on the y-axis of both plots, corresponds

to the average strain measured over several cracks in the

middle portion of the column, while the normalized out-of-

plane displacement, plotted on thex -axis of the left hand plot

ofFig. 1(c), corresponds to the out-of-plane displacement

measured at mid-height of the column divided by the

width of the column tw. The x -axis on the right hand

plot ofFig. 1(c) corresponds to the applied axial force

on the column. The loading history consisted of a tensile

strain cycle followed by a compression cycle with a targetcompressive strain amplitude that was about 1/7 of the

tensile strain amplitude. The axial tensile strains imposed on

the test column, namelya= 0.0078, 0.0108, 0.0133and0.0161, are shown in both plots ofFig. 1(c). It canbe seen that, for axial tensile strain less than or equal to

0.0133, the test column remained stable with relativelysmall out-of-plane displacement of /tw 0.05. For anaxial tensile strain to0.0161, however, the out-of-planedisplacement of the column upon subsequent compression

increased significantly, causing the column to buckle. The

stable condition following a tensile strain ofa= 0.0133is shown inFig. 1(a), in contrast to the buckled column aftera tensile strain of a = 0.0161 in Fig. 1(b). Althoughonly one test specimen is provided, Fig. 1 nonetheless

demonstrates the importance of the tensile strain cycle for

the cyclic stability of structural walls. Further test results are

given in [2].

3. Methodology and basic equations

Since the current use of structural walls in buildings

often involves different height and length, amount of

reinforcement, wall-to-floor area ratio etc., an assessment

of the minimum wall thickness with these parameters isinstructive. The methodology and basic equations governing

the minimum wall thickness are first discussed, and this is

followed by a presentation of results for various parameters

including ground motions that are representative of current

design earthquakes.

3.1. Elastic response acceleration spectrum

The required thickness of structural walls depends on

the ductility demand, which is a function of the ratio of

wall lateral strength to elastic force demand. In this study,

the elastic force demand uses a NewmarkHall type elastic


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1054 Y.H. Chai, S.K. Kunnath / Engineering Structures 27 (2005) 10521063

(a) Stable column up to tensile strain of 1.33%. (b) Buckled column after tensile strain

of 1.61%.

(c) Hysteretic response of test column.

Fig. 1. Stability of a reinforced concrete column when subjected to large amplitude reversed cyclic tensile/compression loading.

response acceleration spectrum, which was used earlier byChai et al. [3], and is shown inFig. 2. The elastic response

acceleration, as characterized in terms of an amplification

factor a , is specified in three period regions:

a=

1.0+2.5(ca1)T/ Tcca2cv ( xg)max/[( xg)maxT]

for

0< T 0.4Tc0.4Tc < T TcTc < T

(4)

where the coefficient ca is the ratio of spectral elastic

response acceleration to peak ground acceleration in the

acceleration-controlled region, and the coefficient cv is the

ratio of spectral elastic response velocity to peak groundvelocity in the velocity-controlledregion. On the basis of the

average values obtained by Vidic et al. [11], coefficients caandcv are taken as 2.5 and 2.0 respectively. The term Tc in

Eq.(4) corresponds to the characteristic period of the ground

motion, which is given by Vidic et al. [11]:

Tc=2cv

ca

( xg)max( xg)max

(5)

where the ratio( xg)max/( xg)max corresponds to the inverseof the peak ground acceleration to peak ground velocity

(referred to asa/vratio herein), which has been shown to be


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Fig. 2. Amplification factor for elastic response acceleration.

a useful parameter for characterizing the frequency contentas well as the duration of the earthquake ground motion [9].

For rock or firm soil sites located near the source of the

earthquake, the ground motion tends to be characterized by

a high a/v ratio with a relatively short duration, whereas

for soft sites at moderate distance from the source, the

ground motion tends to be characterized by a low a /v ratio

with an increased duration. The elastic response acceleration

amplification factor inFig. 2corresponds to foura/vratios,

i.e. ultrahigh, high, normal and low a/v ratios. Note that

the shape of the elastic response acceleration spectrum

changes with thea/vratio as the period range for maximum

response shifts to the longer period region for decreasinga/vratio.

3.2. Fundamental period

The use of elastic response acceleration spectrum for

elastic force demand requires an estimate of the period of the

wall. To this end, the expression proposed by Wallace [13]

for prismatic reinforced concrete cantilever walls is used:

T= 8.8 hwlw

n

whs

Ec gp(6)

where hw = total height of wall, lw = length of wall,n= number of stories, w= tributary floor weight per unitarea, hs = floor-to-floor height (assumed uniform), g=acceleration due to gravity (= 9.81 m/s2), Ec= elasticmodulus of concrete, which is taken as 4730

fc where f

c

is the concrete cylinder strength in MPa, and p= wall-to-floor area ratio in the direction of lateral loading. It

should be noted that a cracked-section stiffness with Icr=0.5Ig has been used in Eq. (6). A study by Wallace and

Moehle [12] has shown that Eq. (6) correlates well with

the measured small amplitude vibration periods of mid-rise

US and Chilean structural wall buildings with a wall-to-floor

area ratio in the range of 1% to 4.7%.

Fig. 3. Kinematic and equilibrium conditions for the ultimate limit state of

rectangular wall sections.

3.3. Lateral strength

The level of ductility demand, and hence the magnitude

of tensile strain in the end region of the wall, depends on

the lateral strength of the wall. The flexural strength of a

concrete section can be estimated using the traditional [1]

equivalent stress block approach in conjunction with a linear

strain profile for the ultimate condition. For a rectangular

section with distributed longitudinal reinforcement plus

end reinforcement, such as that shown in Fig. 3, the

normalized neutral axis depth cu may be determined usingthe equilibrium equation for vertical forces:

cucu

lw

= 10{2k1+k2[2w+( )(t+w(21))]}171+40k2w

(7)

where cu= neutral axis depth measured from the extremecompression fiber, w = vertical reinforcement ratio inthe middle portion of the wall, t= overall (or average)reinforcement ratio, i.e. total longitudinal steel area divided

by gross wall area, k1 = axial force ratio Pu/fclwtwwhere Pu is the axial force acting on the wall, k2= ratioof the steel yield strength to concrete compressive strength

fy /fc , and1 is the equivalent stress block parameter asdefined in [1]. The axial force Pu may be estimated from


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the tributary floor weight w and the number of stories n,

in which case the result is Pu = nwlwtw/p, where p isthe wall-to-floor area ratio defined before. The parameter

in Eq.(7) is defined as the end length divided by the wall

length (see Fig. 3). It should be noted that in estimating

the vertical forces, the end reinforcement is assumed to

be stressed to fy and fy for tension and compressionreinforcement, respectively, where and >1. The higher

stress level in the end reinforcement is intended to reflect

the increased likelihood of the actual yield strength being

higher than the nominal value as well as the possible strain

hardening of reinforcement at the ultimate limit state. A

value of = = 1.25 is used in this paper. For thelongitudinal reinforcement in the middle region, however,

the reinforcement is assumed to be stressed to fy instead

of 1.25fy for both tension and compression since the strain

in that region is smaller. Note that the flexural strength of

the wall is not expected to be significantly affected by the

assumed stress level in the middle portion of the wall since

the reinforcement ratio in the middle region tends to besmaller and their corresponding forces operate at a smaller

lever arm with respect to the centerline of the wall.

Upon the determination of the neutral axis depth, the

flexural strength of a rectangular section may be determined

by taking moments about the centerline of the wall, which in

normalized form is given by

MuMu

fc twl2w

= 140

{cu(171+40k2w)(cu)2(1721+40k2w)

10k2(1)[(+ )(tw)+ 2(+2)w ]}. (8)Since current seismic design is to ensure a ductile response,

the lateral strength of structural walls may be established on

the basis of the flexural strength at the critical section. By

adopting the commonly assumed linearly distributed lateral

force on the wall, the lateral strength of the wall may be

estimated as

Vu=3Mu

2hw(9)

where an effective height of two-thirds of the overall height

has been assumed for the wall.

3.4. Ductility-based force reduction factor

The amount of yielding and hence ductility demand on

a structural wall may be determined in a manner similar to

the construction of inelastic response spectra using a force

reduction factor R, which is defined as the ratio of minimum

elastic force to actual lateral strength. Extensive numerical

studies have been conducted to determine the appropriate

level of force reduction for a given ductility capacity, and

various expressions have been proposed in the literature

(e.g. [6,11]). In this study, the RT relation proposed

by Vidic et al. [11] for a stiffness-degrading hysteretic model

is used:

R=

( 1)T/ To+1

for T ToT > To

(10)

where

To=0.65Tc0.3 (11)where To is the period of transition between the ascending

region and the constant region of R, and Tc is the charac-

teristic period as defined previously in Eq. (5). The force

reduction factor R increases linearly with period T in the

short period range (T To) and remains constant andequal to the displacement ductility factor in the long period

range(T > To). Note that the transition period To increases

slightly with increased displacement ductility factor.

3.5. Maximum tensile strain

The maximum tensile strain imposed in the end regionof the wall is related to the curvature demand on the wall.

By assuming that the inelastic deformation is facilitated by

a plastic hinge rotation at the base of the wall, the curvature

ductility factor may be written as [7]

=( 1)

3

Lphe

1 Lp

2he

+1 (12)where lp is the equivalent plastic hinge length, he is an

effective height, taken as 2/3 of the height of the wall. The

equivalent plastic hinge length depends on the geometry

of the wall, with generally longer plastic hinge length for

slender walls. In this paper, the equivalent plastic hinge

length proposed by Paulay and Priestley [7] is used:

0.3lwlp= (0.20+0.044Ar)lw0.8lw. (13)For walls with aspect ratios in the range of 2 Ar 10, theequivalent plastic hinge length lp varies between 0.29lwand

0.64lw.

Since non-zero strain gradient exists across the section,

the location for calculating the critical tensile strain must be

properly identified. To this end, the critical tensile strain is

calculated at the center of the end region, i.e. at a distance(1

0.5)lw from the extreme compression fiber. The

critical tensile strain, denoted as sm in Fig. 3, may bewritten in terms of the curvature ductility factor as

sm= 2y (10.5cu ) (14)where y is the yield strain of the wall reinforcement, and

other parameters have been previously defined. In deriving

Eq.(14), the equivalent elasto-plastic yield curvature y of

the wall has been taken as [8]

y=2y

lw. (15)

It should be noted that, even though Eq.(15) was originally

proposed for rectangular sections with w =

0.5%


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Fig. 4. Maximum tensile strain in end regions of cantilever structural walls.

distributed reinforcement [8], it is also used for walls with

w=0.25% distributed reinforcement in this paper.To illustrate the potentially large tensile strain in the

end region of the structural wall, Fig. 4 shows a plot of

the critical tensile strain for a 10-story wall subjected to a

peak ground acceleration of 0.4gand a peak ground velocity

of 66.7 cm/s. The floor-to-floor height is taken as 3.05 m

and the tributary floor weight is taken as 8.38 kN/m2. The

wall-to-floor area ratio is 1% and the end length is taken

as 5% of the wall length, i.e. = 0.05. It can be seenfrom Fig. 4 that the critical tensile strain increases with

increased aspect ratio, but decreases with increased overallwall reinforcement. For an aspect ratio of Ar = 10, themaximum tensile strain may reach as high as sm = 3.5%for the case of a low reinforcement ratio of t = 0.5%.Although Fig. 5 corresponds to only one particular set of

ground motion parameters, the magnitude of the critical

tensile strain is expected to increase with increased intensity

of the ground motion.

3.6. Stability criterion

Recognizing that the tensile strain in the end region of the

wall is a critical parameter, the stability of structural wallsmay be ensured by limiting the magnitude of the tensile

strain in that region. The following inequality was proposed

by Chai and Elayer [2] for the critical tensile strain:

sm 2

2

tw

lo

2c+3y (16)

where lo is the buckled length of the wall. Although

the buckled length is generally difficult to assess,

recommendations nonetheless exist in the literature. For

example, [7] recommends that the buckled length lo be

equal to the equivalent plastic hinge length lp . Such a

recommendation may result in an excessively long buckled

length for tall slender walls. In this paper, the buckled length

is taken to be equal to the equivalent plastic length, as

recommended by Paulay and Priestley [7] in Eq. (16), but

is limited to one-half of the story height in order to reflect

the restraint against buckling by the top and bottom floor

slabs. The parameter c in Eq.(16) was originally derived

by Paulay and Priestley [7], and is given by

c=0.5

1+2.35m

5.53m2 +4.70m

(17)

where m is the mechanical reinforcement ratio in the end

region of the wall, and is defined as

m= endfy

fc(18)

and end is the end region reinforcement ratio, which may

be written in terms of the overall reinforcement ratio t and

reinforcement ratio in the middle portion of the wall w:

end=

tw(12 )2

. (19)

The tensile strain limit of Eq.(16) and the stability criterion

of Eq. (17) may be used in conjunction with the seismic

demand and force reduction to calculate the minimum wall

thickness of structural walls.

4. Results

Minimum wall thickness is assessed for a range of

ground motion a /v ratios, wall reinforcement ratios, wall-

to-floor area ratios, floor tributary weights and numbers of

stories. Results are presented in terms of story-height-to-

wall-thickness ratio hs /tw . In this paper, a uniform storyheight ofhs= 3.05 m is assumed and the length over whichthe end reinforcement is concentrated is taken as 5% of the

wall length, i.e. = 0.05. A concrete compressive strengthof fc= 34.5 MPa and steel yield strength of fy=414 MPaare also assumed.

4.1. Ground motion parameters

Since the stability of structural walls is influenced by

the magnitude of the tensile strain, the minimum wall

thickness is expected to vary with the intensity of the ground

motion. A peak ground acceleration of 0.4g, assumed to berepresentative of the seismic intensity in zone IV of [10], is

used in this paper. The effect of site conditions on minimum

wall thickness is studied by varying the a/v ratio. In this

case, four a/v ratios, namely, 2.0g/m s1, 1.4g/m s1,1.0g/m s1 and 0.6g/m s1, are used, and these ratios arereferred to as ultrahigh, high, normal and low a/v ratios,

respectively. For a peak ground acceleration of 0.4gand for

these a/v ratios, the peak ground velocities are 20 cm/s,

28.6 cm/s, 40 cm/s and 66.7 cm/s, respectively.

Fig. 5(a)(d) show plots of the story-height-to-wall-

thickness ratio hs /tw versus wall aspect ratio Ar for a

range of reinforcement ratios from t =

0.5% to 2%.


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(a) Lowa/vratio=0.6g/m s1. (b) Normala/vratio=1.0g/m s1.

(c) Higha/vratio=1.4g/m s1. (d) Ultrahigha /vratio=2.0g/m s1.

Fig. 5. Story-height-to-wall-thickness ratio for varyinga /vratios. Assumed parameters are n=10,h s= 3.05 m,= 0.05, p=1% andw=8.38 kN/m2.

The solid line corresponds to results obtained in this

study, while the short dashed line corresponds to the

minimum wall thickness required by [4] and the long dashed

line corresponds to the minimum wall thickness required

by [10]. It should be noted that, for plotting the [10]

story-height-to-wall-thickness ratio, the unsupported heightis taken as 90% of the floor-to-floor height, i.e. ln =0.9hs . For the[4] minimum wall thickness in Eq. (1), the

displacement ductility factor is calculated using the

force-reduction factor of Eq. (10) instead of a constant

displacement ductility factor. For the plots inFig. 5(a)(d),

the number of stories is n= 10, the tributary floor weightis w = 8.38 kN/m2, and the wall-to-floor area ratio isp=1%.

It can be seen from Figs. 5(a)(d) that the hs /tw ratio

generally decreases with increasing aspect ratio of the wall,

i.e. thicker wall is required for slender walls. For the low

a/v ratio of 0.6g/m s1 in Fig. 5(a), which represents a

soft soil condition, the h s /tw ratio varies from about 14 at

an aspect ratio of Ar = 3 to about 10 at Ar = 10. Forthis a/v ratio, the hs /tw ratio is not sensitive to the wall

reinforcement ratio from Ar= 3 to 10. For normal and higha/v ratios in Figs. 5(b) and (c), however, the hs /tw ratio

becomes increasingly sensitive to wall reinforcement ratio,where a rapid increase in hs /tw ratio is noted for increasing

wall reinforcement. The increase in hs /tw ratio is due to

limited yielding of the wall, which occurs as a result of

increasing lateral strength with increased wall reinforcement

ratio. For the ultrahigh a/v ratio in Fig. 5(d), yielding is

reduced significantly, resulting in a very large h s /tw ratio.

The actual hs /tw ratio is out of range of the y-axis and

hence not plotted inFig. 5(d). Note that, although results for

the h s /tw ratio are sensitive to the wall reinforcement ratio

for the case of normal and high a/v ratios, the minimum

wall thickness required by [4] is not sensitive to the wall

reinforcement ratio.


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It can also be seen from Figs. 5(a)(d) that the hs /twratio increases with increasing a /v ratio, with the smallest

hs /tw ratio being calculated for the low a/v ratio of

0.6g/m s1. The small hs /tw ratio for low a/v ratiois due to the changing shape of the elastic response

acceleration spectrum. Since the fundamental period of 10-

story walls generally lies in the velocity-controlled regionof the response spectrum, and since the predominant period

of the ground motion tends to shift towards a longer period

for low a/v ratio, a decreasing a/v ratio tends to result

in an increased spectral response, thereby increasing the

ductility demand, and hence a larger tensile strain and wall

thickness.

Results in Figs. 5(a)(d) also show that the hs /tw ratio

may be smaller than that required by [10] and[4], depending

on the ground motion parameters, wall reinforcement ratios

and wall aspect ratios. For the case of low a/v ratio, the

hs /tw ratio is smaller than that of [10] for an aspect ratio

of Ar

3, which means that the required wall thickness

is larger than that of [10]. On the other hand, the hs /twratio compares fairly well with that of [4], even though the

ratio tends to be slightly smaller than that of [4] in the large

aspect ratio range, e.g. Ar 8. In the case of the normala/v ratio, the hs /tw ratio compares fairly well with that

required by both [10] and [4] for t = 0.5% and 1% inthe large aspect ratio range. For higher reinforcement ratios

oft= 1.5% and 2%, however, the hs /tw ratio is largerthan that required by [10] and [4]. In the case of high or

ultrahigh a/v ratios, the h s /tw ratio obtained in this study

is significantly larger than that required by [10] and [4],

indicating that the current code provision is conservative

with respect to these ground motion parameters. It is alsoof interest to note that, while results from this study indicate

a monotonically decreasinghs /tw ratio, the trend forh s /twratio required by [4] indicates a slight increase in hs /twratio

for aspect ratio Ar 7 for alla/vratios.

4.2. Wall-to-floor area ratio

The fundamental period of structural walls depends on

the percentage of wall area, as evident in Eq. (6), and

consequently, the hs /tw ratio is expected to be influenced

by the wall-to-floor area ratio.Fig. 6(a)(d) show the hs /tw

ratio for wall-to-floor area ratios of p = 1%, 2%, 3%and 4%. In this case, the hs /tw ratio is assessed for a low

a/v ratio of 0.6g/m s1 since a low a/v ratio generallyresults in greater spectral response. The same peak ground

acceleration of 0.4g is used and the tributary floor weight

is taken as w= 8.38 kN/m2, and the number of stories istaken asn=10.

Trends observed for differenta/vratios inFigs. 5(a)(d)

are also observed inFigs. 6(a)(d) for different wall-to-floor

area ratios. For the wall-to-floor area ratio of p = 1%in Fig. 6(a), the hs /tw ratio is not sensitive to the wall

reinforcement ratio. For increased wall-to-floor area ratios,

however, the h s /tw ratio becomes increasingly sensitive to

the wall reinforcement ratio. For example, for the wall-to-

floor area ratio of p= 4% inFig. 6(d) and for an aspectratio of Ar = 10, the hs /tw ratio increases from 9 fort= 0.5% to h s /tw= 14 for t= 2%. Also for the caseof a low reinforcement ratio oft= 0.5%, the h s /tw ratioremains relatively constant with respect to the wall-to-floor

area ratio, as can be seen by comparing the solid lines fort= 0.5% for all wall-to-floor area ratios in Fig. 6(a)(d).As an example, for a reinforcement ratio of t = 0.5%and for an aspect ratio of Ar = 10, the hs /tw ratio is10.4 for p= 1%, compared to the hs /tw ratio of 8.7 forp=4%.

Results inFig. 6(a)(d) also indicate that theh s /tw ratio

for low a/v ratio is generally smaller than that required

by [10] or [4]. The hs /tw ratio required by [4] is also

smaller than that required by the [10] for large aspect ratios.

However, comparison between the two codes depends on the

value of the a/v ratio since the [4] requirement is taken

to be dependent on the displacement ductility factor. As

such, the [4] hs /tw , which is expected to increase withincreasing a/v ratio, may increase above that of [10] in

some cases.

4.3. Tributary floor weight

Tributary floor weight, which affects the period of the

wall, is expected to influence the hs /tw ratio of the wall.

Figs. 7(a)(d) show the story-height-to-wall-thickness ratio

for tributary floor weight from w = 6.49 kN/m2 to10.39 kN/m2. The range of floor weight is considered to

represent reasonable limits in current building design. The

same lowa/vratio of 0.6g/m s1 and the same peak groundacceleration of 0.4g are used. The number of stories is

kept at n= 10 and the wall-to-floor area ratio is taken asp=1%.

It can be seen from Figs. 7(a)(d) that the hs /tw ratio

is relatively insensitive to the tributary floor weight, even

though an increased tributary floor weight results in a

slightly larger hs /tw ratio. For example, for the tributary

floor weight of w = 6.49 kN/m2 in Fig. 7(a) and for anaspect ratio of Ar = 8, the hs /tw ratio is 9.9 for thelow reinforcement ratio oft= 0.5%. With an increasedtributary floor weight of w = 10.39 kN/m2 in Fig. 7(d)but for the same low reinforcement ratio oft= 0.5% andwall aspect ratio of Ar= 8, the hs /tw ratio is 11.2. Theincrease inhs /twratio is about 11% and is mainly due to the

longer period of the wall as a result of increased tributary

floor weight, which tends to reduce the spectral response,

and hence reduced ductility demand in the wall. The relative

insensitivity of the h s /tw ratio to the tributary floor weight

is also observed for higher longitudinal reinforcement ratio.

For the same aspect ratio of Ar = 8 but for a higherlongitudinal reinforcement ratio oft= 2%, thehs /twratiois 10.4 for w = 6.49 kN/m2 inFig. 7(a). In comparison,the hs /tw ratio increases only slightly to 10.6 for w =10.39 kN/m2 inFig. 7(d). Comparison of results with code


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(a) Floor weightw=6.49 kN/m2. (b) Floor weightw=7.79 kN/m2.

(c) Floor weightw=9.09 kN/m2. (d) Floor weightw=10.39 kN/m2.

Fig. 7. Story-height-to-wall-thickness ratio for varying tributary floor weight. Assumed parameters are ( xg )max = 0.4g, ( xg )max= 66.7 cm/s, n = 10,hs= 3.05 m,= 0.05 and p=1%.

contrast to the monotonically increasing trend of 10-, 15-

and 20-story walls.

5. Summary and conclusions

This paper examines the out-of-plane stability of planar

structural walls under seismic loading and its implications

in design. Factors governing out-of-plane buckling as well

as the methodology for determining the required thickness

of planar walls are discussed; this is followed by an

assessment of the minimum wall thickness for a peak ground

acceleration of 0.4g, assumed to be representative of the

ground motion for Zone IV of 1997 Uniform Building Code.

The effect of site conditions is studied by varying the ratio

of the peak ground acceleration to the peak ground velocity

(a/v) ratio. Although results are presented primarily in

terms of story-height-to-wall-thickness ratio, concluding

remarks will be made with reference to the minimum wall

thickness. Also because of the large number of parameters

involved and only results from a limited set of parameters

being presented, concluding remarks are only intended for

those parameters.

Results for 10-story walls with a wall-to-floor area ratio

of 1% and tributary floor weight of 8.38 kN/m2 indicated

that the minimum wall thickness increases with decreasing

a/v ratio, i.e. thicker walls are required for the lower a/v

ratios. The increase in wall thickness is due to the changing

shape of the elastic response acceleration spectrum, which

generally indicates a softening of the site for decreasing

a/v ratio. Since the period of mid-height walls generally

falls in the velocity-controlled region of the spectrum and

since a decreasing a/v ratio tends to result in a longer

predominant period, an increase in spectral response and

hence larger thickness is expected for the low a/v ratio.


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(a) Number of stories=5. (b) Number of stories=10.

(c) Number of stories=15. (d) Number of stories=20.

Fig. 8. Story-height-to-wall-thickness ratio for different number of stories. Assumed parameters are( xg )max=0.4g,( xg )max=66.7 cm/s,w=8.38 kN/m2,hs= 3.05 m,= 0.05 and p=1%.

Results for the same set of 10-story walls also indicated

that the minimum wall thickness is not conservative with

respect to the 1997 Uniform Building Code and 1995 New

Zealand Concrete Code (NZS 3101) for the low a/v ratio

of 0.6g/m s

1. For the normal a/v ratio of 1.0g/m s

1,

however, results for minimum wall thickness generally agree

well with those for the codes, whereas for high and ultrahigh

a/v ratios of 1.4g/m s1 and 2.0g/m s1, both codesare conservative. Assessment based on the low a/v ratio

of 0.6g/m s1 indicated that the minimum wall thicknessgenerally decreases with increase in the wall-to-floor area

ratio, even though the amount of decrease depends on the

wall reinforcement ratio. For the low a/v ratio and wall-

to-floor area ratios between 1% and 4%, the minimum

wall thickness is larger than that required by both building

codes, even though results agree better with [4]. Study in

this paper also indicated that the minimum wall thickness

is not sensitive to the tributary floor weight. In contrast,

however, the minimum wall thickness is rather sensitive to

the wall height, or equivalently, the number of stories. For

the low a/v ratio of 0.6g/m s1, wall-to-floor area ratioof 1%, and tributary weight of 8.38 kN/m2, large wall

thickness is predicted for short walls. The large thickness

is primarily due to shortening of the structural period,

which tends to increase the spectral response of the wall

and hence increased ductility demand. For the case of the

lowa /v ratio, the minimum wall thickness required by [4]

also shows reasonable agreement with results for 10- and

15-story walls. Although results presented in this paper

are instructive for prescribing the minimum thickness of

ductile planar structural walls, future code development

or revision should nonetheless be verified by experimental

testing of actual structural walls. Analyses in this paper

were also carried out on the assumption of isolated planar


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walls whereas many real buildings may exhibit significant

torsional response when subjected to seismic load. Further

investigation of structural wall stability under the influence

of torsional response of buildings is warranted.

References

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commentary. Detroit: American Concrete Institute; 1999.

[2] Chai YH, Elayer TD. Lateral stability of reinforced concrete columns

under axial reversed cyclic tension and compression. ACI Structural

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[3] Chai YH, Fajfar P, Romstad KM. Formulation of duration-dependent

inelastic seismic design spectrum. Journal of Structural Engineering,

ASCE 1998;124(8):91321.

[4] NZS 3101. Concrete structures standard part 1 and 2. Private bag

2439. Wellington 6020, New Zealand, 1995.

[5] Paulay T. The design of ductile reinforced concrete structural walls

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[6] Paulay T, Priestley MJN. Seismic design of reinforced concrete and

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[7] Paulay T, Priestley MJN. Stability of ductile structural walls. ACI

Structural Journal 1993;90(4):38592.

[8] Priestley MJN, Kowalsky MJ.Aspects of drift and ductility capacity of

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Society for Earthquake Engineering 1998;31(6):7385.

[9] Tso WK, Zhu TJ, Heidebrecht AC. Seismic energy demands on re-

inforced concrete moment-resisting frames. Earthquake Engineering

and Structural Dynamics 1993;22:53345.

[10] UBC. Uniform building code, vol. 2. In: International conference of

building officials. 1997.

[11] Vidic T, Fajfar P, Fischinger M. Consistent inelastic design spectra:

strength and displacement. Earthquake Engineering and Structural

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[12] Wallace JW, Moehle JP. Ductility and detailing requirements of

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[13] Wallace JW. New methodology for seismic design of RC shear walls.

Journal of Structural Engineering, ASCE 1994;120(3):86384.

final article chai and kunnath

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