of 18
Simulating Building Motions Using Ratios
of the Building
s Natural Frequencies
and a Timoshenko Beam Model
Ming Hei Cheng
a)
M.EERI
, and Thomas H. Heaton
a)
M.EERI
A simple prismatic Timoshenko beam model with soil-structure interaction
(SSI) is developed to approximate the dynamic linear elastic behavior of build-
ings. A closed-form solution with complete vibration modes is derived. It is
demonstrated that building properties, including mode shapes, can be derived
from knowledge of the natural frequencies of the first two translational modes
in a particular direction and from the building dimensions. In many cases, the
natural frequencies of a building
s first two vibrational modes can be determined
from data recorded by a single seismometer. The total building
s vibration
response can then be simulated by the appropriate modal summation. Preliminary
analysis is performed on the Caltech Millikan Library, which has significant
bending deformation because it is much stiffer in shear. [DOI: 10.1193/
011613EQS003M]
INTRODUCTION
The use of seismic data in buildings to study the deformations of a structure is a well-
established research area in earthquake engineering (e.g., Skolnik et al. 2006,
Kohler et al.
2007
,
Krishnan and Muto 2013
). Ideally, several stations are established in locations that can
best describe the spatial/temporal pattern of building motions. However, a new type of seis-
mic network is under development
crowd-sourced seismic networks where volunteers
install the seismometers. Examples include the Quake Catchers Network (
Cochran et al.
2011
) and the Community Seismic Network (
Clayton et al. 2011
). In a crowd-sourced net-
work, station locations are typically chosen for reasons that are unrelated to optimal seismic
network design. In particular, there may be instances in which a single seismometer is the
only record that is available from a building. Furthermore, it may not be practical to construct
a detailed finite-element model of the building; even if there are sufficient resources to con-
struct a dynamic model, the building
s structural design may not be available. However, if the
geometry of the building is known (e.g., obtained from Google Earth), it may be feasible to
infer its approximate deformations using only a single seismic station. Seismologists have
inferred the radial structure of the Earth using models that predict the frequencies of its nor-
mal modes (e.g.,
He and Tromp 1996
,
Rosat et al. 2007
, and
Romanowicz et al. 2008
). These
frequencies are the same at all stations; that is, knowledge of the modal frequencies recorded
at a single station is sufficient to infer the radial properties of a spheroid. This paper presents a
simple methodology for obtaining the approximate motions of a building based on at least
one seismogram and knowledge of the building
s geometry (especially the height).
Earthquake Spectra
, Volume 31, No. 1, pages 403
420, February 2015; © 2015, Earthquake Engineering Research Institute
a)
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA
403
Although buildings primarily consist of structural members (columns, walls, floors, etc.)
separated by void spaces (rooms, doors, windows, etc.), large-scale deformation of a struc-
ture can often be approximately described by an equivalent homogeneous elastic continuum.
In particular, many aspects of building motions can be understood in the context of a simple
cantilevered elastic beam on an elastic half-space (i.e., the Earth). The strains that develop at
the interface between the beam and the half-space are quite complex. Fortunately, it is pos-
sible to develop methodologies to approximately model the mechanics of a cantilevered
beam. The simplest approximation is to consider a beam that responds only in shear
(e.g.,
Westergaard 1933
,
Jennings and Newmark 1960
, and
Iwan 1997
), often referred to
as a shear beam. A shear beam is basically the same problem as a uniform layer on a
half-space that is subject only to horizontal shear tractions. The effective density of the build-
ing can be calculated from an estimate of the building mass and total volume. The effective
shear wave velocity in the building is
4
Lf
1
, where
L
is the building height and
f
1
, where
L
is
the frequency of the first (fundamental) mode.
As a beam becomes very narrow compared to its length, deformation primarily occurs as
bending
that is, contraction on one side of the beam and extension on the opposite side.
Much is known about the technical theory of bending, which relates the bending moment on a
beam to the bending displacements of the beam. If shear deformations are negligible, a beam
can be simulated using the Bernoulli-Euler equation, which relates fourth-order spatial deri-
vatives of horizontal deflections to horizontal forces and inertial accelerations (e.g.,
Foutch
and Jennings 1978
). Unlike shear waves, which have frequency-independent wave velocities,
bending waves are inherently dispersive; higher frequencies have higher wave speeds.
Although interstory shearing (Figure
1a
) is the primary deformation mode of buildings
that are wide in relation to their height, tall and thin buildings may experience significant
bending (Figure
1b
). This is especially true if the building has high shear stiffness, which is
the case for Caltech
s Millikan Library, which has extensive shear walls. Models ignoring the
effect of flexural deformation may cause noticeable errors. However, pure flexural models
are rarely appropriate, even for buildings with high effective rigidity [e.g., shear walls as the
(a)
(b)
(c)
Figure 1.
Building deformation (a) shear-beam behavior (vertically propagating SH-waves) is
expected when shear stiffness is less than flexural stiffness; (b) bending-beam behavior
(Bernoulli-Euler beam equation) is expected for tall and narrow buildings (small flexural stiff-
ness); (c) combined shearing and bending known as a Timoshenko beam.
404
M. H. CHENG AND T. H. HEATON
main lateral load
resisting systems (
Miranda 1999
)]. In reality, a building deforms in a mix-
ture of flexural and shearing deflections (Figure
1c
).
Because of the deficiencies of simple pure-shear or pure-flexural models, researchers
began using more complex models that incorporated more general building responses.
In particular, a Timoshenko beam (
Timoshenko 1937
,
Timoshenko and Goodier 1951
,
Heidebrecht and Smith 1973
, and
Rahgozar et al. 2004
) is a computational model with dif-
ferential equations coupling the effects of a shear beam and a bending beam, with the con-
straint that total deflections are caused by the sum of the shear deformations and the flexural
deformations.
Miranda (1999)
used a continuum structural model consisting of a flexural
cantilever beam and a flexural shear beam to predict deformations in buildings. He assumed
the particular case of a Timoshenko for which the roof deformations from flexure and shear
were equal.
Boutin et al. (2005)
and
Michel et al. (2006)
suggested that mode shapes could be
retrieved from the ratio of the first two natural frequencies of a building using a fixed-base
Timoshenko beam model, but they ignored section rotation inertia and soil-structure inter-
action. Soil-structure interaction can significantly affect the natural frequencies of a building.
A simple consideration of the first two recorded natural frequencies of the system may lead to
misinterpretation of the properties of a fixed-base Timoshenko beam. Dynamic Timoshenko
beam models are used in other engineering applications. For example, they are widely
applied in the simulation of micro- or nanobeams (e.g.,
Hsu et al. 2007
,
Wang et al. 2007
).
In this paper, a simple prismatic Timoshenko beam model with soil-structure interaction
(SSI) is developed to approximate the dynamic linear elastic behavior of buildings. A closed-
form response solution with complete vibration modes is derived. Once the dimensions of the
building are specified, the building properties, including mode shapes, can be derived by
knowing the ratios of the frequencies of the first two normal modes in a particular direction.
In many instances, the natural frequencies of the first two vibrational modes of a building can
be identified by spectral analysis of data from a single seismometer. If the modes and mode
shapes are known, the excitation of each mode can be determined from analysis of a single
building record. The entire spatial/temporal building vibration response can then be approxi-
mated by the appropriate modal summation. Preliminary analysis is performed on Caltech
s
Millikan library, which has high shear stiffness and so bending makes up a significant part of
the overall deformation.
TIMOSHENKO BEAM MODEL WITH SOIL-STRUCTURE INTERACTION
A multistory building can be modeled as an equivalent prismatic homogeneous
Timoshenko beam with SSI, as shown in Figure
2
. The building has a flexural rigidity
of
E

I
, where
E

is an effective Young
s modulus and
I
is the second areal moment
about the neutral axis of bending (horizontal in this case).
E

is typically much lower
than the intrinsic
E
of the structural materials because building volume comprises mostly
air. Because bending obeys a fourth-order differential equation, the beam
s overall flexural
stiffness decreases rapidly with its increasing length; that is; flexural stiffness is defined as
E

I
L
3
, where
L
is the building height.
The effective shear modulus of the building is
G

and it is useful to compare overall
building stiffness with soil stiffness to assess the importance of the soil-structure interaction.
In this case,
G

can be much lower than the
G
of the building materials. In particular,
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
405
moment-resisting frame buildings are often designed to be flexible in shear, in which case
G

can be very small. The overall shear stiffness of the building is defined as
kG

A
L
, where
A
is
the cross-sectional area and
k
is the shear factor (to adjust for different cross-sectional shapes:
k
¼ð
2
3
Þ
for a rectangular cross section). The building considered here is assumed to have
uniform stiffness and mass along its height and remains linearly elastic. The effect of the
soil is to cause horizontal motions of the building
s base and rocking about a horizontal
axis. To simulate the effect of the soil-structure interaction, a translational spring with stiff-
ness
K
T
and a rotational spring with stiffness
K
R
are incorporated at the base of the building.
Balendra et al. (1982)
studied the responses of a linear
N
-story shear building sitting on an
elastic homogeneous half-space and showed that
K
T
and
K
R
can be estimated from the soil
properties; that is,
EQ-TARGET;temp:intralink-;sec2;41;308
K
T
¼
32
ð
1

υ
Þ
7

8
υ
G
soil
r
o
and
K
R
¼
8
3
ð
1

υ
Þ
G
soil
r
3
o
where
G
soil
is the shear modulus of the soil,
υ
is Poisson
s ratio of the soil, and
r
o
is the
equivalent radius of the foundation.
Given the flexural stiffness and shear stiffness of a building, together with the soil spring
stiffness, a closed-form free vibrational response is expressed as follows (the derivation is
given in the Appendix):
EQ-TARGET;temp:intralink-;e1;41;195
w
¼
C
1
cosh

α
x
L

þ
C
2
sinh

α
x
L

þ
C
3
cos

β
x
L

þ
C
4
sin

β
x
L

(1)
where
w
is the horizontal deflection and
x
is the distance along the vertical axis. The values
α
and
β
depend on the building
s natural frequencies
ω
.
Using the characteristic equation derived in the online Appendix, the natural frequency
ratios of the one-dimensional Timoshenko beam with SSI can be determined with different
shear stiffness-to-flexural stiffness ratios
r
:
w
x
d
L
K
R
K
T
Building:
Flexural stiffness
Shear stiffness
z
kG A
*
L
E I
L
*
3
Figure 2.
Timoshenko beam with soil-structure interaction. Horizontal spring
K
T
simulates the
difference in horizontal ground motion of the building base caused by the inertial forces of the
building on the ground. This effect is typically minor because the average density of the building
is small compared to the ground. The rotational spring
K
R
simulates the rocking rotation of the
base of the building.
406
M. H. CHENG AND T. H. HEATON
EQ-TARGET;temp:intralink-;e2;62;640
r
¼
shear stiffness
flexural stiffness
¼
kG

A
L
E

I
L
3
¼
kG

AL
2
E

I
¼
8
G

L
2
E

d
2
for rectangular cross- section with
k
¼
2
3
(2)
To investigate the dynamic behavior of buildings on different soil conditions, three soil
types with shear wave velocities
V
S
of
60 m
s
(very soft),
300 m
s
(stiff), and
1,000 m
s
(fixed base) are chosen. Soil with shear wave velocity
V
s
¼
60 m
s
is considered the
very soft end of soft soil. In practice, buildings typically will not be constructed on such
soil. This study considers this very soft soil to investigate the effect of the soil-structure
interaction. Other parameters are arbitrarily chosen based on the values estimated for the
Millikan Library building (described in a later section). The study investigates how
model parameters, including building height
L
, stiffness ratio
r
, and building density
ρ
building
, are related to natural frequency ratios. The sensitivity analysis assumes the effective
shear modulus of the building to be
EQ-TARGET;temp:intralink-;sec2;62;449
G

¼
ρ
building
ð
4
Lf
1
Þ
2
¼
ρ
building

4
L
30
L

2
¼
14
;
400
ρ
building
where a story height of 3 m is used and a fundamental period of 1 s is assumed for every
10 stories (i.e.,
f
1
¼
30
L
). It is assumed that building densities,
ρ
building
, are on the order of
200 kg
m
3
for light and flexible frame buildings and
400 kg
m
3
for heavy, stiff shear wall
buildings. The aspect ratio of a building is defined as
L
d
. The values used in this analysis are
listed in Table
1
.
The natural frequency ratio of the
i
th mode is defined as
f
i
f
1
. In this study, the proposed
method focuses on the frequency ratios estimated using the Timoshenko beam. A pure shear
Table 1.
Parameter study for Timoshenko beam model
s-wave velocity
V
s
60
m
s
(very soft soil)
300
m
s
(stiff soil)
1,000
m
s
(fixed base)
Soil density
ρ
soil
1,850
kg
m
3
Poisson
s ratio
υ
0.33
Building depth
d
Varies
Building height
L
Varies
Building density
ρ
building
200
kg
m
3
or 400
kg
m
3
Equivalent foundation radius
r
o
12.4 m
Soil shear modulus
G
soil
6
.
67

10
6
Pa
1
.
67

10
8
Pa
1
.
85

10
9
Pa
Soil translational stiffness
K
T
4
.
06

10
8
N
m
1
.
02

10
10
N
m
1
.
13

10
11
N
m
Soil rotational stiffness
K
R
5
.
05

10
10
Nm
1
.
26

10
12
Nm
1
.
40

10
13
Nm
Effective shear modulus
G

14,400
ρ
building
Pa
Stiffness ratio
r
Varies
Effective Young
s modulus
E

Varies
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
407
beam with a rigid base exhibits its first three natural frequency ratios as 1, 3, and 5, whereas a
pure bending beam with a rigid base exhibits its first three natural frequency ratios as 1, 6.27,
and 17.55. With the effect of the soil-structure interaction, the frequency ratio
f
2
f
1
is
not necessarily bounded between 3 (pure shear building on a fixed base) and 6.27 (pure
bending building on a fixed base) (see Figure
3
, which is explained in the next paragraph).
Timoshenko beams that have the same ratios between flexural stiffness, shear stiffness, and
soil stiffness exhibit the same frequency ratios; that is, the frequency ratios are only functions
of the stiffness ratios and the dimensions of the building
given the height and the aspect
ratio of the building, the stiffness ratio between shearing, bending, and soil can be determined
from the ratios of the normal mode frequencies.
Figure
3
describes the effect of building height
L
on the frequency ratio
f
2
f
1
for a light-
weight frame building with a building density
ρ
building
¼
200 kg
m
3
, a fixed effective shear
modulus
G

¼
2.88 MPa
, and a fixed stiffness ratio
r
¼
20
. The result shows that building
height has negligible effect on frequency ratio for flexible buildings with a fixed base. A taller
building induces a larger frequency ratio
f
2
f
1
for buildings on very soft soil because of the
soil-structure interaction effect, and this effect is more significant on a small aspect ratio
L
d
.
For example, the frequency ratio
f
2
f
1
can increase from about 4.0 to 10.3 for a building with
L
¼
150 m
and
L
d
¼
2
sitting on soft soil.
60
90
120
150
4
5
6
7
8
9
10
Frequency ratio
f
2
/
f
1
Building Height (m)
60
90
120
150
Building Height (m)
60
90
120
150
Building Height (m)
Aspect ratio = 2
Aspect ratio = 5
Aspect ratio = 10
(a)
(b)
(c)
Figure 3.
Effect of building height on frequency ratio
f
2
f
1
: (a) very soft soil (
V
s
¼
60
m
s
);
(b) stiff soil (
V
S
¼
300
m
s
); (c) fixed base (
V
S
¼
1,000
m
s
). Effective shear modulus of the
building
G

¼
2
.
88
MPa
and stiffness ratio
r
¼
20 are fixed. The aspect ratio is the building
height divided by its width.
408
M. H. CHENG AND T. H. HEATON
Figure
4
(light, flexible frame buildings,
ρ
building
¼
200 kg
m
3
) and Figure
5
(heavy, stiff
shear wall buildings,
ρ
building
¼
400 kg
m
3
) describe the effect of stiffness ratio
r
on the
frequency ratio. In this case, a building height of 60 m and a building effective shear modulus
G

¼
14,400
ρ
building
are used. A Timoshenko beam with a small
r
ð
r
¼
0.1
Þ
behaves like a
pure shear beam, whereas a fixed-base Timoshenko beam with a large
r
ð
r
¼
1,000
Þ
does not
necessarily behave like a pure bending beam. The results show that the frequency ratio of the
Timoshenko beam matches that of a pure bending beam only when it has a large aspect ratio
(
L
d
¼
10
), which confirms that a building performs similarly to a pure bending beam only
when it is relatively tall and narrow. A point to note is that
Miranda and Taghavi (2005)
define a similar stiffness ratio
r
in their coupled shear wall frame structure, but their
model behaves like a shear beam for large
r
and a bending beam for small
r
.
Dym and
Williams (2007)
explain such a difference as being due to the fact that the Timoshenko
beam couples the shear and flexural stiffnesses in series but Miranda
s model couples
them in parallel.
Comparing the frequency ratio of a building sitting on very soft soil to one on a fixed
base suggests that the soil-structure interaction has a significant effect on shear-type buildings
but has a minimal effect on pure bending structures. In Figures
4
and
5
, it is seen that fre-
quency ratio
f
2
f
1
is amplified only for regions with stiffness ratio
r
<
50
, which represents
buildings with considerable shear-type behavior. Also, the amplification is more significant
0.1
1
10
100
1000
2.5
3
3.5
4
4.5
5
5.5
6
6.5
Frequency ratio
f
2
/
f
1
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
Aspect ratio = 2
Aspect ratio = 5
Aspect ratio = 10
(a)
(b)
(c)
Figure 4.
Effect of stiffness ratio
r
(shear stiffness/flexural stiffness) on the frequency ratio of
buildings with
ρ
building
¼
200
kg
m
3
(light and flexible frame): (a) very soft soil (
V
S
¼
60
m
s
);
(b) stiff soil (
V
S
¼
300
m
s
); (c) fixed base (
V
S
¼
1,000
m
s
). Effective shear modulus of the
building
G

¼
2
.
88
MPa
and building height
L
¼
60
m
are fixed. The aspect ratio is the building
height divided by its width.
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
409
on buildings with higher densities
ρ
building
and smaller aspect ratios. For example, the
frequency ratio
f
2
f
1
of a Timoshenko beam with an aspect
ratio
¼
2
increases from 3
to 5 for a change from fixed base to very soft soil for the light, flexible building with
ρ
building
¼
200 kg
m
3
; it increases to 5.5 for the heavy, stiff building
ρ
building
¼
400 kg
m
3
. This is similar to the phenomenon reported by
Gajan et al. (2010)
that foundation
displacement and rotation are relatively small compared to the structural deformation in
highly flexible structural systems (e.g., bending-type buildings) and so the effect can be
neglected; however, the soil-structure interaction has a significant effect on stiff structural
systems (e.g., shear-type buildings).
Stewart and Fenves (1998)
suggest that the soil-structure interaction predominantly
affects the fundamental mode frequency, and this observation agrees with the present
numerical study. Figure
6a
shows that the frequencies of a Timoshenko beam with
L
¼
60 m
,
L
d
¼
5
,
ρ
building
¼
200 kg
m
3
, and
r
¼
0.1
(shear-type building) are reduced
by 35.7
%
, 9.7
%
, and 5.4
%
, respectively for the first, second, and third modes with a change
from fixed base to very soft soil. The percentage of frequency reduction becomes relatively
the same at about 6
%
for
r
¼
50
, and such an effect is negligible for
r
¼
1,000
(bending-type
building). This study also investigates the effect of soil rotational spring stiffness
K
R
and soil
translational spring stiffness
K
T
on frequency change. As shown in Figure
6b
,
K
R
has a
significant effect on the first mode frequency but a minimal effect on the higher modes.
0.1
1
10
100
1000
2.5
3
3.5
4
4.5
5
5.5
6
6.5
Frequency ratio
f
2
/
f
1
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
Aspect ratio = 2
Aspect ratio = 5
Aspect ratio = 10
(a)
(b)
(c)
Figure 5.
Effect of stiffness ratio
r
(shear stiffness/flexural stiffness) on the frequency ratio of
buildings with
ρ
building
¼
400
kg
m
3
(heavy, stiff shear wall). (a) very soft soil (
V
S
¼
60
m
s
);
(b) stiff soil (
V
S
¼
300
m
s
); (c) fixed base (
V
S
¼
1,000
m
s
). Effective shear modulus of the
building
G

¼
5
.
76
MPa
and building height
L
¼
60
m
are fixed. The aspect ratio is the building
height divided by its width.
410
M. H. CHENG AND T. H. HEATON
In contrast,
K
T
in general has less than a 3
%
effect. In the combined situation, the frequency
change is mostly contributed by
K
R
. It is important to understand that the soil
s effect on the
horizontal motion of the base of the building is rather minor. That is, the horizontal motion of
the ground at the base of the building does not change much compared to the horizontal
motion of the ground without a building. However, the rotation of the base of the building
during rocking causes the forces in the building to differ significantly from those for a build-
ing without rocking.
As mentioned, the soil-structure interaction has a major effect on the fundamental mode.
Table
2
describes the effect of soil type and building type on the percentage change in funda-
mental frequency
f
1
. Some existing buildings are constructed on soft soil with
V
S
¼
150 m
s
(e.g.,
Kohler et al. 2005
), and maximum change in fundamental frequency corresponds to
9.5
%
for shear-type buildings. In general, buildings behave like a Timoshenko beam with
properties between a pure shear beam and a pure bending beam, so the soil-structure inter-
action should not be neglected in determining the dynamic behavior of buildings during
earthquakes.
0.1
1
10
100
1000
0
5
10
15
20
25
30
35
Percentage change in frequency
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
0.1
1
10
100
1000
Stiffness ratio
r
f
1
f
2
f
3
(b)
(c)
(a)
Figure 6.
Effect of stiffness ratio
r
on the percentage change in frequency of the first three
modes: (a) very soft soil (
V
S
¼
60
m
s
) with soil translational and rotational springs;
(b) very soft soil with soil rotational spring only; (c) very soft soil with soil translational spring
only. Effective shear modulus of the building
G

¼
2
.
88
MPa
, building height
L
¼
60
m
, and
aspect ratio
L
d
¼
5 are fixed.
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
411
BUILDING SYSTEM IDENTIFICATION
The following steps are taken to simulate the spatial/temporal dynamic behavior of an
existing building:
1. Estimate the soil
s s-wave velocity
V
S
, density
ρ
soil
, and Poisson
s ratio
υ
.
2. Estimate the building
s dimensions (width
d
and height
L
), which can usually be
approximated using Google Earth; underground substructures usually contain thick
and stiff shear walls, so they have minimal effect on the building
s deflection;
L
can
be roughly approximated as the height above ground.
3. Estimate the building
s density
ρ
building
, which can be approximated from the build-
ing
s structural type; assume that building densities are on the order of
200 kg
m
3
for light, flexible frame buildings and
400 kg
m
3
for heavy, stiff shear wall
buildings.
4. Determine the building
s first two natural frequencies (
f
1
and
f
2
) in a particular
direction from spectral analysis of motions (either ambient noise or driven by
minor earthquakes) recorded by an installed seismometer.
5. Calculate the soil spring properties. For the equivalent foundation radius,
EQ-TARGET;temp:intralink-;sec3;41;169
r
o
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cross-sectional area of building
π
r
For the soil shear modulus,
G
soil
¼
V
2
s
ρ
soil
. For the soil translational spring stiffness,
EQ-TARGET;temp:intralink-;sec3;41;118
K
T
¼
32
ð
1

υ
Þ
7

8
υ
G
soil
r
o
Table 2.
Effects of soil type (shear wave velocity
V
S
) and building type (stiffness ratio
r
i.e., shear stiffness/flexural stiffness) on percentage change in fundamental frequency
f
1
r
0.1 (shear-type
building)
5
50
250
1,000 (bending
type building)
V
S
60
m
s
(very soft soil)
35.74
%
24.18
%
5.86
%
1.34
%
0.38
%
90
m
s
21.61
%
13.18
%
2.77
%
0.62
%
0.17
%
120
m
s
13.93
%
8.05
%
1.55
%
0.35
%
0.10
%
150
m
s
(soft soil)
9.52
%
5.33
%
1.06
%
0.23
%
0.07
%
180
m
s
6.84
%
3.75
%
0.73
%
0.16
%
0.03
%
210
m
s
5.11
%
2.79
%
0.57
%
0.12
%
0.03
%
240
m
s
3.92
%
2.10
%
0.41
%
0.09
%
0.00
%
270
m
s
3.10
%
1.65
%
0.33
%
0.07
%
0.00
%
300
m
s
(stiff soil)
2.48
%
1.31
%
0.24
%
0.05
%
0.00
%
412
M. H. CHENG AND T. H. HEATON
For the soil rotational spring stiffness,
EQ-TARGET;temp:intralink-;sec3;62;627
K
R
¼
8
G
soil
r
3
o
3
ð
1

υ
Þ
6. Calculate the building properties. For the effective shear modulus,
G

¼
ρ
building
ð
4
Lf
1
Þ
2
.
The stiffness ratio
r
and the effective Young
s modulus of the building
E

(through Equation
2
) can be approximated from the defined parameters and the mea-
sured natural frequency ratio
f
2
f
1
using the characteristic equation (Equation
A26
in the online Appendix) or precalculated frequency ratio plots (e.g., Figures
4
5
).
7. The constructed Timoshenko beam model provides natural frequency ratios
f
i
f
1
as
well as the corresponding mode shapes, where
f
i
is the ith mode frequency and
f
1
is
the fundamental mode frequency. Higher mode frequencies can be estimated from
the measured fundamental natural frequency. From the numerical method for
solving the differential equations, the maximum estimated natural frequency is
bounded by the following condition:
EQ-TARGET;temp:intralink-;sec3;62;438
f
2
max
<
kGA
ð
2
π
Þ
2
ρ
building
I
(see the online Appendix).
EXISTING STRUCTURE ANALYSIS: MILLIKAN LIBRARY
Caltech
s Millikan Library (Figure
7
) is a nine-story reinforced-concrete building with
one level of basement embedded in stiff soil whose shear wave velocity is about
300 m
s
.
The building has moment-resisting frames and a very stiff core wall in both the NS and EW
directions. It also has continuous NS shear walls that extend from the foundation to the roof
on the east and west sides. The library is an unusually stiff building considering its height.
4.9 m
4.3 m
4.3 m
4.3 m
4.3 m
4.3 m
4.3 m
4.3 m
4.3 m
4.3 m
4.9 m
ROOF
9TH
8TH
7TH
6TH
5TH
4TH
3RD
2ND
BASE
1ST
Figure 7.
Millikan library profile and shaker on roof (from
Bradford 2006
).
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
413
It was instrumented when first constructed in 1966, so it is possible to compare the apparent
modal frequencies that have been recorded for many earthquakes and shaking experiments.
Furthermore, a 36-channel accelerometer network recorded by a Kinemetrics Mt. Whitney
system was installed in 1998; this is a triggered system that measures horizontal motions on
every building level including the basement and the roof. An almost continuous recording of
building motions has been available since 2000, when the Southern California Seismic
Network installed the station MIK on the ninth floor. In 2008, a three-axis rotational sensor
was added to MIK and a second identical station, MIKB, was installed in the basement.
A Kinemetrics harmonic shaker was installed on the roof of the library in the early 1970s,
and it has been used for many class projects and research studies over the past decades
(Figure
7
); its maximum frequency is 9 Hz. From forced-vibration experiments, the first
two measured NS modal frequencies are 1.75 and 7.21 Hz (
Bradford 2006
), corresponding
to natural frequency ratios of 1 and 4.1. The third NS mode has not yet been identified, and its
frequency is almost certainly higher than the 9-Hz limit of the roof shaker. The first two
measured EW modal frequencies are 1.22 Hz and 4.76 Hz (
Bradford 2006
), corresponding
to natural frequency ratios of 1 and 3.9. An apparent resonance at 7.83 Hz was suggested by
Bradford (2006)
to be the third EW mode. However, because that mode shape is not ortho-
gonal to the other two mode shapes, only the first two identified EW modes are considered in
this paper.
Table
3
shows the parameters used to construct the Timoshenko beam model for the
library. Values for soil properties and building properties are extracted from
Favela
(2004)
and
Todorovska (2009)
. The calculated stiffness ratio
r
is 27 and 17 for the NS
and EW directions, respectively. The frequency ratios
f
2
f
1
are similar for both directions,
which means that the shear stiffness relates in the same way to the flexural stiffness for both
directions. Using the Timoshenko beam model, the estimated third vibration mode occurs at
15.3 Hz (
f
3
f
1
¼
8.75
) and 10.05 Hz (
f
3
f
1
¼
8.24
) for the NS and EW directions, respec-
tively. These values are definitely higher than the 9-Hz safety limit of the roof shaker, so they
have not been identified.
Figures
8
and
9
show the estimated mode shapes. Those produced from the Timoshenko
beam are good estimations of the measured shapes. The frequency ratios
f
2
f
1
of 3.9 and 4.1
occur between that of a pure shear beam and a pure bending beam. In such a case, the mode
shapes of the Timoshenko beam provide a much better result compared to those of a pure
shear beam. The two estimated third mode shapes from the Timoshenko beam are also shown
in the figures.
DISCUSSION
Simple models of buildings represented as shear beams have been surprisingly success-
ful. They can be used to explain many features of building modes of vibration, and they can
aid understanding of traveling waves in a building (e.g.,
Kohler et al. 2007
,
Rahmani and
Todorovska 2013)
. However, there are cases where buildings exhibit global bending about
their base (tall and slender buildings that are stiff), and there are cases in which a better
understanding of the effect of foundation rocking is desired. All of these effects are incor-
porated in the present solution for a cantilevered Timoshenko beam on a translational as well
as a rotational spring. The good news is that the solution is expressible in a closed-form way
414
M. H. CHENG AND T. H. HEATON
(see Equations
A18
and
A19
in the online Appendix). Unfortunately, the expressions are far
more complex than for a simple shear beam.
The formal methodology presented here produces the mode shapes of a building assum-
ing that (1) records are available from at least one seismometer in the building from which the
first two modal frequencies in a given direction can be determined; (2) the exterior dimen-
sions of the building can estimated; and (3) the building can be approximately modeled as a
Timoshenko beam on a translational and rotational base. Of course the last assumption is both
critical and dubious. Few buildings are really prismatic; it is common to design tall buildings
with larger structural elements in the lower stories (they also have larger effective densities).
Miranda and Taghavi (2005)
investigated the effect of linear and parabolic variations of lat-
eral stiffness along the building height using finite-element models. They suggested that the
effect of nonuniform stiffness on mode shapes and frequency ratios is very small and can be
neglected for bending-type structures. The individual effect on shear-type structures is
greater, but the overall influence is relatively small and can be neglected. Nevertheless,
we have shown that this simple methodology does predict the type of mode shapes of a
building that has significant bending (e.g., Millikan Library).
Knowing the earthquake response from one sensor on a particular floor of a building can
allow estimating responses for the other floors using the calculated mode shapes through
modal decomposition (
Kohler et al. 2013
). In addition, the Timoshenko beam model can
Table 3.
Timoshenko beam model parameters for Millikan Library
NS direction
EW direction
Input
s-wave velocity
V
S
316
m
s
Soil density
ρ
soil
1,850
kg
m
3
Poisson
s ratio
υ
0.306
Building depth
d
21 m
23 m
Building height
L
44 m
Building density
ρ
building
400
kg
m
3
Fundamental frequency
f
1
1.75
1.22
Frequency ratio
f
2
f
1
4.1
3.9
Calculate
Equivalent foundation radius
r
o
12.4 m
Soil shear modulus
G
soil
1
.
67

10
8
Pa
Soil translational stiffness
K
T
1
.
01

10
10
N
m
Soil rotational stiffness
K
R
1
.
22

10
12
N
m
Effective shear modulus
G

3
.
79

10
7
Pa
1
.
84

10
7
Pa
Output
Stiffness ratio
r
27
17
Effective Young
s modulus
E

4
.
11

10
7
Pa
3
.
81

10
7
Pa
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
415
be used to estimate building responses for predicted ground motions. The time behavior of
each mode can be determined by solving the problem of a linear single-degree-of-freedom
oscillator with the appropriate frequency and damping (damping can be approximately
obtained from spectral analysis of the data). With an appropriate participation factor for
each modal response, a vector sum of the modes can be performed to find the entire build-
ing
s approximate deformations and floor accelerations for the the ground motion. The result
allows approximation of the performance of nonstructural elements (e.g., elevator, mechan-
ical piping, etc.) due to large floor accelerations while the structure of the building remains
elastic.
In practice, it may be best to estimate the approximate effect of foundation rocking by
estimating the rocking stiffness from the soil
s elasticity parameters and the building
s foot-
print. For example,
Bycroft (1956)
presented the solution for a rigid circular disk that forced
to rock on an elastic half-space. This approach is not pursued here, but if an independent
estimate of rocking stiffness were available, the present solutions could be used to estimate
the approximate building motions from estimates of the first two modal frequencies. This
study focuses on buildings without significant plan irregularities and so torsional effects
are ignored. Possible future studies can include the effect of three-dimensional seismic exci-
tation and an extension of the existing Timoshenko beam model to capture the torsional
−0.5
0
0.5
1
1.5
Building Height
L
Timoshenko beam with SSI
Measurement
Shear beam
−1
0
1
−1
0
1
(a)
(b)
(c)
Figure 8.
Mode shapes comparison for the Millikan library in the NS direction: (a) 1st mode,
(b) 2nd mode, (c) 3rd mode.
416
M. H. CHENG AND T. H. HEATON
responses of a tall building. Such analysis may require comparisons with three-dimensional
finite-element simulations, but the procedure described should provide a good estimate of
building motion and is relatively simple to apply. It is important, however, for any potential
user to recognize that this type of analysis cannot produce detailed results.
CONCLUSION
In this paper, a closed-form solution for the prismatic Timoshenko beam model with soil-
structure interaction is derived. Timoshenko beams with a small stiffness ratio (
r
¼
0.1
)
behave similarly to pure shear beams whereas those with a large stiffness ratio
(
r
¼
1,000
) and a high aspect ratio (
L
d
¼
10
) behave similarly to pure bending beams.
The soil-structure interaction affects shear-type buildings, but the corresponding effect is
small for bending-type buildings. In general, buildings behave like a Timoshenko beam
with properties between a pure shear beam and a pure bending beam, so the soil-structure
interaction should not be neglected in determining the dynamic behaviors of buildings during
earthquakes.
A formal methodology is presented that predicts the mode shapes of a building, assuming
that (1) records from at least one seismometer in the building are available from which the
−0.5
0
0.5
1
1.5
Building Height
L
Timoshenko beam with SSI
Measurement
Shear beam
−1
0
1
−1
0
1
(a)
(b)
(c)
Figure 9.
Mode shapes comparison for the Millikan library in the EW direction: (a) 1st mode,
(b) 2nd mode, (c) 3rd mode.
SIMULATING BUILDING MOTIONS USING A TIMOSHENKO MODEL
417
first two modal frequencies in a given direction can be determined, (2) the exterior dimen-
sions of the building can be estimated, and (3) the building can be approximately modeled as
a Timoshenko beam on a translational and rotational base. The proposed method is success-
fully applied to estimate the mode shapes of Millikan Library at Caltech.
ACKNOWLEDGMENTS
The authors thank Dr. Monica Kohler for her review of the manuscript and her valuable
suggestions. This work is partially supported by the National Science Foundation
(EAR-1027790), the George Housner Earthquake Engineering Research Endowment
(EAS-41212), the Fred L. Hartley Family Foundation, and the Croucher Foundation. The
support of these institutions is gratefully acknowledged.
APPENDIX
Please refer to the online version of this paper to access
Appendix: Derivation of the
one-dimensional Timoshenko beam model with soil-structure interaction.
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