PERIOD FORMULAS
FOR
CONCRETE SHEAR
W
ALL
BUILDINGS
By Rakesh K. Goel
l
and Anil K. Chopra
2
ABSTRACT
Most seismic codes specify empirical fonnulas to estimate the fundamental vibration period
of
buildings. Evaluated first
in
this paper are the fonnulas specified in present U.S. codes using the available data on the fundamental period
of
buildings measured from their motions recorded during eight California

quakes, starting with the
1971
San Fernando earthquake and ending
w ith
the 1994 Northridge
e rth~u~e
t
S
shown that current code fonnulas for estimating the fundamental penod
of
concrete shear wall
bUi1dmgs
are grossly inadequate. Subsequently, an improved fonnula is developed
by
calibr~ting
a the.oretical fonnula,
deriv~
using Dunkerley's method, against the measured period data through regress10n
analys1s
A]so recommended
IS
a factor to limit the period calculated
by
a rational analysis. such as Rayleigh's method.
INTRODUCTION
The fundamental vibration period
of
a building appears in the equation specified in building codes to calculate the design base shear and lateral forces. Because this building property cannot be computed for a structure that is yet to be designed, building codes provide empirical formulas that depend on the building material [steel, reinforced concrete
(Re),
etc.), building type (frame. shear wall. etc.). and overall dimensions. The empirical period formulas for concrete shearwall (SW) buildings
in
the 1997 UBC
( Uniform
1997) and the 1996 SEAOC blue book
( Recommended
1996) were derived by modifying the ATC306 formulas
( Tentative
1978) during development
of
the 1988 SEAOC blue book to more accurately reflect the configuration and material properties
of
these systems
( Recommended
1988, Appendix 1E2b(1)T). The period formulas in ATC306
( Tentative
]978) are based largely on motions
of
buildings recorded during the 1971 San Fernando earthquake. However, motions
of
many more buildings recorded during recent earthquakes. including the 1989 Loma Prieta and 1994 Northridge earthquakes, are now available. These recorded motions provide an opportunity to expand greatly the existing database on the fundamenta.1
vi~ra
tion periods
of
buildings. To this end. the natural
vlbr~tlon
periods
of
21
buildings have been measured by system Identification methods applied to the motions
of
buildings recorded during the 1994 Northridge earthquake (Goel and Chopra 1997a). These data have been combined with similar data from the motions
of
buildings recorded during the 1971 San Fernando, 1984 Morgan Hill, 1986 Mt. Lewis and Palm
prin~
1987 Whittier, 1989
Loma
Prieta, 1990 Upland, and
1991
S1
erra Madre earthquakes. The objective
of
this paper is to develop improved empirical formulas to estimate the fundamental vibration period
of
concrete SW buildings for use
in
equivalent lateral force analysis specified
in
building codes. Presented first is the expanded database for
measured
values
of
fundamental periods
of
SW buildings, against which the code formulas in present U.S. codes are evaluated; similar work on limited data sets has appeared previously (e.g., Arias and Husid 1962; Housner and Brady 1963; Cole et al. 1992;
Li
and Mau 1997).
t
is shown that current code formulas for estimating the fundamental period
of
concrete SW buildings are grossly inadequate. Subse
Asst
Prof.,
Dept
of
Civ. and Envir. Engrg., California Polytechnic Stale Univ., San Luis Obispo,
CA
93407. 2Johnson Prof., Dept.
of
Civ. Engrg., Univ.
of
California, Berkeley,
CA
94720.
quently. an improved formula is developed by calibrating a theoretical formula, derived using Dunkerley's method, against the measured period data through regression analysis. Finally. a factor to limit the period calculated by a
rational
analysis. such as Rayleigh's method,
is
recommended.
PERIOD D T B SE
The data that are most useful. but hard to
come
by. are from structures shaken strongly but not deformed into the inelastic range. Such data are slow to accumulate because relatively few structures are installed with permanent accelerographs, and earthquakes causing strong motions
of
these instrumented buildings are infrequent. Thus, it is very important to investigate comprehensively the recorded motions when they
do
become available, such as during the 1994 Northridge earthquake. Unfortunately, this obviously important goal
is
not always accomplished, as indicated by the fact that the vibration properties
of
only a few
of
the buildings whose motions were recorded during post1971 earthquakes have been determined. Available data on the fundamental vibration period
of
buildings measured from their motions recorded during several California earthquakes have been collected (Goel and Chopra 1997a). This database contains data for 106 buildings, including
21
buildings that experienced peak ground acceleration.
o
0 15g
during the 1994 Northridge earthquake. The remaining data come from motions
of
buildings recorded during the 1971 San Fernando earthquake and subsequent earthquakes (Hart et al. 1975; Hart and Vasudevan 1975; MacVerry 1979; Cole et al. 1992; Werner 1992; Gates et al. 1994; Marshall et al. 1994; Gael and Chopra 1996, 1997a). Shown
in
Table 1
is
the subset
of
this database pertaining to 16 concrete SW buildings (27 data points); buildings subjected to peak ground acceleration.
0 15g
are identified
o
with an asterisk
(*).
C and N denote buildings instrumented by the California Strong Motion Instrumentation Program (CSMIP) and National Oceanic and Atmospheric Administration (NOAA) and
ATC
denotes one
of
the buildings included in the ATC306 report
( Tentative
1978) for which the height and base dimensions were available from other sources, but these dimensions for other buildings could not be discerned from the plot presented in the ATC306 report. The number
of
data points exceeds the number
of
buildings, because the period
of
some buildings was determined from their motions recorded during more than one earthquake
or
was reported by more than one investigator for the same earthquake.
CODE FORMUL S
The empirical formula for the fundamental vibration
~er~od
of
concrete SW buildings specified in current U.S. bUlldmg codes NEHRP94
( NEHRP
1994) SEAOC96
( Recom
T BLE
1
Period Data for Concrete SW Buildings
Peak Ground Acceleration Period
T
Number
g)
sec) Building
ID
of Height LongiTransLongiTransWidth Length
num er
Location number stories ft) Earthquake tudinal verse tudinal verse ft) ft)
1
) (2)
3)
(4)
5)
(6)
7) 8) 9)
(10) (11 ) (12) 1 Belmont C58262 2 28.0 Lorna Prieta 0.10 0.11 0.13 0.20 NA NA 2* Burbank C24385
10
88.0 Northridge 0.26 0.30 0.60 0.56 75.0 215.0 3* Burbank C24385 10 88.0 Whittier 0.22 0.26 0.57 0.51 75.0 215.0 4 Hayward C58488 4 50.0 Lorna Prieta 0.05 0.04 0.15 0.22 NA NA 5 Long Beach C14311 5 71.0 Whittier 0.10 0.10 0.17 0.34 81.0 205.0 6 Los Angeles ATC 3 12 159.0 San Fernando NA NA 1.15
MRF
60.0 161.0 7* Los Angeles C24468 8 127.0 Northridge 0.16 0.11 1.54 1.62 63.0 154.0 8* Los Angeles C24601
17
149.7 Northridge 0.26 0.19 1.18 1.05 80.0 227.0 9 Los Angeles C24601
17
149.7 Sierra Madre 0.07 0.06 1.00 1.00 80.0 227.0 10* Los Angeles N2535
12
161.5 San Fernando 0.26 0.19 1.19 1.14 76.0 156.0 11* Los Angeles N2535
12
161.5 San Fernando 0.26 0.19 1.07 1.13 76.0 156.0 12 Palm Desert C12284 4 50.2 Palm Spring 0.07 0.12 0.50 0.60 60.0 180.0
13
Pasadena N2645 10 142.0 Lytle Creek 0.02 0.02 0.71 0.52 69.0 75.0 14* Pasadena N2645 10 142.0 San Fernando 0.18 0.22 0.98 0.62 69.0 75.0 15* Pasadena N2645 10 142.0 San Fernando 0.18 0.22 0.97 0.62 69.0 75.0
16
Piedmont C58334 3 36.0 Lorna Prieta 0.08 0.07 0.18 0.18 NA NA
17
Pleasant Hill C58348 3 40.6 Lorna Prieta 0.08 0.13 0.38 0.46 77.0 131.0
18
San Bruno C58394 9 104.0 Lorna Prieta 0.11 0.13 1.20 1.30 84.0 192.0
19
San Bruno C58394 9 104.0 Lorna Prieta 0.11 0.13 1.00 1.45 84.0 192.0 20 San Jose C57355 10 124.0 Lorna Prieta 0.09 0.11 MRF 0.75 82.0 190.0
21
San Jose C57355 10 124.0 Morgan Hill 0.06 0.06 MRF 0.61 82.0 190.0 22 San Jose C57355 10 124.0 Mount Lewis 0.03 0.03 MRF 0.61 82.0 190.0 23 San Jose C57356 10 96.0 Lorna Prieta 0.10 0.13 0.73 0.43 64.0 210.0 24 San Jose C57356 10 96.0 Lorna Prieta 0.10 0.13 0.70 0.42 64.0 210.0 25 San Jose C57356 10 96.0 Morgan Hill 0.06 0.06 0.65 0.43 64.0 210.0 26 San Jose C57356 10 96.0 Mount Lewis 0.04 0.04 0.63 0.41 64.0 210.0 27* Watsonville C47459 4 66.3 Lorna Prieta 0.39 0.28 0.24 0.35 71.0 75.0 Note: *Denotes building with
go
0.15g;
NA indicates data not available; MRF implies momentresisting frames form the lateral load resisting system; number followed by
C
or
N
indicates the station number and by
ATC
indicates the building number in ATC306 report
( Tentative
1978).
mended
1996), and UBC97
( Uniform
1997) is
of
the form
T=
C H
3/4
(1)
where
H
=
the height
of
the building in feet above the base; and the numerical coefficient
C,
=
0.02. UBC97 and SEAOC96 permit an alternative value for
C,
to be calculated from
C,
=
0.l/vA::
2)
where
A
c,
the combined effective area (in square feet)
of
the shear walls, is defined as
NW
Ac
=
2:
Ai
[0.2
+
D,/H)2];
Di/H:S
0.9 (3)
i I
in which
Ai
=
the horizontal crosssectional area (in square feet);
{
=
the dimension in the direction under consideration (in feet)
of
the
ith
SW
in the first story
of
the structure; and
NW
=
the total number
of
shear walls. The value
of
DJH
in (3) should not exceed 0.9. ATC306
( Tentative
1978) and earlier versions
of
other U.S. codes specify a different formula
T=
0.05H
(4)
Vi
where D
=
the dimension, in feet,
of
the building at its base in the direction under consideration. UBC97 and SEAOC96 codes specify that the design base shear should be calculated from
v cw
5)
in which
W
=
the total seismic dead load; and C
=
the seismic coefficient defined as
C
v
I 2.5C
a
C
=
;
O.llCal:S
C:S
I
T
R
0.8ZNJ
and for seismic zone 4
~ 
R
(6)
in which coefficients
C
v
and
C
a
depend on the nearsource factors,
N
v
and
N
a,
respectively, along with the soil profile and the seismic zone factor
Z;
I
=
the importance factor; and
R
=
the numerical coefficient representative
of
the inherent overstrength and global ductility capacity
of
the lateralload resisting system. The upper limit
of
2.5
Cal R
on C applies to very short period buildings, whereas the lower limit
of
O.llCal
(or
0.8ZNJ
R
for seismic zone 4) applies to very long period buildings. These limits imply that C becomes independent
of
the period for very short
or
very tall buildings. The upper limit existed, although in slightly different form, in previous versions
of
UBC and SEAOC blue book; however, the lower limit appeared only recently in UBC97 and SEAOC96. The fundamental period
T
calculated using the empirical formula
(1)
or (4), should be smaller than the
true
period to obtain a conservative estimate for the base shear. Therefore, code formulas are intentionally calibrated to underestimate the period by about
1020%
at first yield
of
the building ( Tentative 1978;
Recommended
1988). The codes permit calculation
of
the period by established methods
of
mechanics (referred to as rational analyses in this paper), such as Rayleigh's method
or
computerbased eigenvalue analysis, but specify that the resulting value should not
be longer than that estimated from the empirical formula
(I)
or
(4) by a certain factor. The factors specified in various U.S. codes are 1.2 in ATC306, 1.3 for high seismic region (Zone 4) and 1.4 for other regions (Zones 3, 2, and
I)
in UBC97 and SEAOC96, and a range
of
values with 1.2 for regions
of
high seismicity to 1.7 for regions
of
very low seismicity in NEHRP94. The restriction in SEAOC88 that the base shear calculated using the rational period shall not be less than 80%
of
the value obtained by using the empirical period corresponds to a factor
of
104
(Cole et al. 1992). These restrictions are imposed to safeguard against unreasonable assumptions in the rational analysis, which may lead to unreasonably long periods and hence unconservative values
of
base shear.
EVALUATION OF CODE FORMULAS
For buildings listed in Table
I,
the fundamental period identified from their motions recorded during earthquakes (subsequently denoted as
measured
period) is compared with the values given by the code empirical formulas [Figs. I(a), 2(a), and 3(a)]. Also compared are the two values
of
the seismic coefficient for each building calculated according to (6), with
=
1 for standard occupancy structures;
R
=
5.5 for concrete shear walls; and
C
v
=
0.64 and
Co
=
0.44 for seismic zone 4 with Z
=
004
soil profile type
Sv,
i.e., stiff soil profile with
Concrete SW Buildings
75 100 125 150 175 200 Height
H, ft
(a)
o_~
.150
oOgo
<
C
.159
I
.
,.
~.
.~
......
,.
.,.
.....
. .,
....
.
.......
:}..
1 2~
'
;.. 
'
....
.,..,.
,.,. .
....
; :::
31
...
~.
~T
~
\...
T=
O.02H~4
; ~
..
VI
0
'1
.25 0.5
2
lil1.25
1Il
r
:g
~0.75
1.75 1.5 25 50 0.3
~alues
I C Iro
~
>Cod
~
Perio(
UO.25
ã
MeE
sured F
rlod
0
10
~
0.1
9
'E
<Il
o<O.Hg
MeE
Isured F
rlod
0
'0
E
02
<3
.
1
.
·~0.15
.........
1Il
'Q)
l.
(/')
r
i
'
~
0.1
:1
ã
D
::l
0.05 25 50 75 100 125 150 175 200 Height
H,
ft
(b)
FIG. 1.
Comparison
of: (a) Measured and Code Periods; (b) UBC97
Seismic
Coefficients
from
Measured and
Code
Periods Concrete
SW
Buildings
2 1.75
1.5
lil1.25
1Il
r
:g
~0.75
0.5 0.25
ã
Q~
0.150
o
Ogo
(
0.159
ã
0 0
.
0
....
>.
.::
......
0
.
....
.....
....
....
ã<111
...
...
~
.............

ã 8
0
. .,
~
......
....
::::
J '
:
~T
=
0.1H
I4 A~/~
:\~.
.....
0
2
6
78
0.3 UO.25
'E
<Il
'0
~
0.2
U
·~0.15
1Il
jJ
G;
0.1
~
::l
0.05
Blues ( C Iron
I
COl
a Parlo arlod,
('
arlod, ( ã Me
o
Ma sured sured
go
~
0.1
go<
0.1
pg
9
ã
0
8
0
o
0
0
.
r .......
8
I
0
ã
0
2
6
7
8
FIG. 2.
Comparison
of: (a) Measured
and
Code Periods; (b) UBC·97 Seismic
Coefficients
from
Measured
and
Code
Periods: Code Periods
Are
Calculated
from
Alternate Formula
average shear wave velocity between 180 and 360
mis,
and
Nv
=
No
=
I [Figs. I(b), 2(b), 3(b)].
Code
Formula: (1)
with
C
l
=
0.02
For all buildings in Table I, the periods and seismic coefficients are plotted against the building height in Fig. I. The measured periods in two orthogonal directions are shown by circles (solid for
u
go
~
0.15g, open for
Ugo
<
0.15g) connected by a vertical line, whereas the code period is shown by a solid curve because the code formula gives the same period in the two directions
if
the lateralforce resisting systems are
of
the same type. Also included are the curves for
1 2T
and
IAT
representing the limits imposed by codes on a rational value
of
the period for use in high seismic regions like California. The seismic coefficients (6) corresponding to the measured periods in the two orthogonal directions are also shown by circles connected by a vertical line, whereas the value based on the code period is shown by a solid curve. Fig. 1 leads to the following observations. For a majority
of
buildings, the code formula gives a period longer than the measured value. In contrast, for concrete and steel momentresisting frame buildings, the code formula almost always gives a period shorter than measured value (Goel and Chopra 1996, 1997b). The longer period from the code formula leads to seismic coefficients smaller than the value based on the
Concrete SW Buildings
2
ãC
~0.15g
1.75
o
C
go
< 0.15g
ã ã
1.5
0
1.21/
0
~
1.25
0
ã
.
.
l
.. 
0,
· ·
8
'1::
75
.
0
.
o.
~
.,
ã
0.5
0
T=O
05H D112
o
.25
0
5 1015
20
25
H D112
(a)
0.3
Valu s of C from
0.25
)

.
Code Perlcd Measured erlod,
ago
~
0.15g
1::
0
Measured erlod,
ago
<
0 15g
0.2
ã
0
ã
~
0
0
0.15
0
'E
l
CD
n
0 1
0
v

~
::::>
~
0
ã
0.05
5
1015
20
25
H D112
(b)
FIG. 3. Comparison of: a) Measured and Code Periods; b) UBC97 Seismic Coefficients from Measured and Code Periods: Code Periods Are Calculated from ATC306 Formula
measured period
if
the period falls outside the flat portion
of
the seismic coefficient spectrum; otherwise, the two periods lead to the same seismic coefficient. For most
of
the remaining buildings, the code formula gives a period much shorter than the measured value and a seismic coefficient much larger than the value based on the measured period. Because the code period for many buildings is longer than the measured period, the limits
of
1 2T
or
I 4T
for the period calculated from a rational analysis are obviously inappropriate. The building height alone is not sufficient to estimate accurately the fundamental period
of
SW
buildings because measured periods
of
buildings with similar heights can
be
very different, whereas they can be similar for buildings with very different heights. For example, in Table 1 the measured longitudinal periods
of
Buildings 4 and 12
of
nearly equal heights differ by a factor
of
more than three; the heights
of
these buildings are
50
ft and 50.2 ft, whereas the periods are 0.15 sec and 0.50 sec, respectively.
On
the other hand, measured longitudinal periods
of
Buildings
13
and 23 are close even though Building
13
is 50% taller than Building 23; periods
of
these buildings are 0.71 sec and 0.73 sec, whereas the heights are 142 ft and
96
ft, respectively. The poor correlation between the building height and the measured period is also apparent from the significant scatter
of
the measured period data [Fig. l(a)].
Alternate Code Formula: 1) with
C
from 2) and 3)
Table 2 lists a subset
of
nine buildings (17
data
points) with their
c
values calculated from (3) using
SW
dimensions obtained from structural drawings; for details, see Appendix G
of
Goel and Chopra (1997a). These dimensions were not available for the remaining seven buildings in Table
1.
In Fig. 2 the alternate code formula for estimating the fundamental period is compared with the measured periods
of
the nine buildings. The code period is determined from
(1)(3)
using the calculated value
of
c
and plotted against
H
3I4
VIC.
This comparison shows that the alternate code formula almost always gives a value for the period that is much shorter than the measured periods and a value for the seismic coefficient that is much higher than from the measured periods. The measured periods
of
most buildings are longer than the code imposed limits
of
1 2T
and
1 4T
on the period computed from a rational analysis. Although the code period formula gives a conservative value for the seismic coefficient, the degree
of
conservatism seems excessive for most buildings considered in this investigation.
ATC3Q6 Formula
In Fig.
3,
the ATC306 formula for estimating the fundamental period is compared with the measured periods
of
all buildings listed in Table I. The code period
is
determined from (4) using the
Hand
D
dimensions
of
the building (Table 1) and plotted against
H
+
YD.
This comparison demonstrates that (4) significantly underestimates the period and considerably overestimates the seismic coefficient for many buildings and the ATC306 imposed limit
of
1 2T
is too restrictive. The ratio
H
+
Vl
is not sufficient to estimate accurately the fundamental period
of
concrete
SW
buildings because measured periods
of
buildings with similar values
of
this ratio can be very different, whereas they can be similar for buildings with very different values
of
H
+
YD.
For example, in Table 1 the measured transverse period
of
Building 18 and measured longitudinal period
of
building
27two
buildings with similar values
of
H
+
YDdiffer
by nearly a factor
of
five;
H
YD
=
7.51 and 7.87, and measured periods
=
1.30 sec and 0.24 sec, respectively. On the other hand, the measured longitudinal and transverse periods
of
Building 9 are the same, equal to 1 sec, even though the values
of
H
+
YD
in the two directions are 16.7 and 9.93. The poor correlation between the ratio
H
+
YD
and the measured periods is also apparent from a large scatter
of
the measured period data [Fig. 3(a)].
THEORETICAL FORMULAS
The observations in the preceding section clearly indicate that the current code formulas for estimating the fundamental period
of
concrete
SW
buildings are grossly inadequate.
For
this purpose, equations for the fundamental period are derived using established analytical procedures. Based on Dunkerley's method (Jacobsen and Ayre 1958, pages
119120
and
502
505; Veletsos and Yang 1977; Inman 1996, pages
442449),
the fundamental period
of
a cantilever, considering flexural and shear deformations, is
(7)
in which
T
F
and
T
s
are the fundamental periods
of
pureflexural and pureshear cantilevers, respectively. For uniform cantilevers
T
F
and
T
s
are given by (Jacobsen and Ayre 1958, pages
471496;
Timoshenko et al. 1974, pages
424431;
Chopra 1995, page 592)
T ~
F

3.516
YEi
H
2
(8)