SHAKING TABLE DEMONSTRATION OF DYNAMIC
RESPONSE OF BASE-ISOLATED BUILDINGS
***** Student Manual *****
A PROJECT DEVELOPED FOR THE
UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES
http://wusceel.cive.wustl.edu/ucist/
Required Equipment:
• Instructional Shake Table
• Power Supply (Universal Power Module)
• Data Acquisition and Control System
(MultiQ Board and Computer)
• Software
• Measurement Sensors (Three
Accelerometers)
• Test Structure (One-Story Building
Model)
• Seismic Isolation Systems
Developed at Washington State University by:
Undergraduate Students:
Graduate Student:
Professor:
Stuart Bennion and Jason Collins
Nat Wongprasert
Michael D. Symans
This project was supported by the
National Science Foundation (Grant Nos. DUE-9950340 and CMS-9624227), the
Pacific Earthquake Engineering Research Center, and Washington State University.
Student Manual
Table of Contents
Section
Page
1.0 Introduction........................................................................................................................3
1.1 Objective ..................................................................................................................3
1.2 Abstract ....................................................................................................................3
2.0 Theoretical Considerations ...............................................................................................3
2.1 Static and Dynamic Loads .......................................................................................3
2.2 Static Lateral Load Analysis....................................................................................4
2.2.1 Problem Statement ....................................................................................4
2.2.2 Solution .....................................................................................................5
2.3 Dynamic Lateral Load Analysis ..............................................................................5
2.3.1 Problem Statement ....................................................................................5
2.3.2 Purpose of Dynamic Analysis...................................................................7
2.4 System Identification ...............................................................................................8
2.4.1 Free-Vibration Test...................................................................................8
2.4.2 Sine-sweep Test ......................................................................................10
2.5 Dynamic Response Analysis..................................................................................14
2.5.1 Ground Displacement Analysis ..............................................................14
2.5.2 Sinusoidal Ground Displacement ...........................................................16
2.5.3 Earthquake Ground Displacement ..........................................................17
2.6 Effects of Base Isolation ........................................................................................18
3.0 Description of Lab Equipment .......................................................................................19
3.1 Required Equipment ..............................................................................................19
3.2 Instructional Shake Table ......................................................................................19
3.3 Power Supply .........................................................................................................20
3.4 Data Acquisition and Control System....................................................................20
3.5 Software .................................................................................................................20
3.6 Measurement Sensors ............................................................................................21
3.7 Test Structure .........................................................................................................21
3.8 Seismic Isolation Systems......................................................................................22
4.0 Procedures for Conducting Experimental Tests...........................................................26
4.1 System Identification: Free-Vibration Tests..........................................................26
4.2 System Identification: Sine-Sweep Tests...............................................................26
4.3 Seismic Response Evaluation ................................................................................27
5.0 Questions...........................................................................................................................28
5.1 Questions Related to System Identification and Seismic Response Tests.............28
5.2 General Dynamic Analysis Questions ...................................................................28
6.0 References.........................................................................................................................30
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Student Manual
SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE
OF BASE-ISOLATED BUILDINGS
1.0 INTRODUCTION
1.1 Objective
The objective of this lab is to expose undergraduate civil engineering students to the topics of
structural dynamics and earthquake engineering; topics which are rarely introduced at the
undergraduate level. The primary vehicle for introducing these topics is a small-scale bench-top
shaking table. Furthermore, the lab will introduce the concept of base isolation and explain how
an isolation system affects the dynamic behavior of a structure.
1.2 Abstract
This lab concentrates on the behavior of base-isolated buildings subjected to dynamic loading. A
three-dimensional, one-story building frame is used in the experiments. The structure is tested in
a conventional configuration in which the building is rigidly attached to the shaking table
platform (foundation) and in an alternate configuration in which the building is supported on a
base isolation system. Two different isolation systems are tested; one that incorporates sliding
bearings and a second which incorporates elastomeric bearings. The effects of isolating the
structure are explored through system identification tests (free vibration and forced vibration)
and earthquake tests.
2.0 THEORETICAL CONSIDERATIONS
2.1 Static and Dynamic Loads
Structures must be designed to resist a variety of loads. Some of the loads are static (e.g., gravity
and snow loads) while others are dynamic (e.g., wind and earthquake loads). For simplified
analysis and design, the effects of dynamic loads are often accounted for by the application of
effective static loads. For example, in the analysis of structures subjected to earthquake ground
motion, the time-dependent inertial forces associated with the acceleration of the mass of the
structure can be accounted for in an approximate way by applying the inertial forces as
equivalent static loads. Such methods are commonly employed in building and bridge design
codes. Of course, the approximate nature of such an analysis/design approach requires an
appreciable level of conservatism. To increase the efficiency of the design (i.e., to reduce the
conservatism), the time-dependency of the inertial forces should be accounted for in the analysis.
Furthermore, for complex structures, design codes do not permit the use of simplified equivalent
static analysis. Instead, dynamic analysis must be performed. Thus, it is important for structural
engineers to understand the fundamentals of structural dynamics. The ensuing discussion on
structural dynamics is largely based on material from the textbook by Chopra (2001). Other
excellent resources on the topic of structural dynamics include the textbooks by Clough and
Penzien (1993) and Tedesco et al. (1999).
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2.2 Static Lateral Load Analysis
2.2.1 Problem Statement
Consider a simple one-story industrial building whose plan and elevation views are shown in
Figure 1. The structure resists lateral loads via moment resisting frames in the north-south
direction and braced frames in the east-west direction. Rigid diaphragm action is provided at the
roof level via a horizontal bracing system located at the bottom chord of the roof trusses. The
weight of the structure may be regarded as concentrated at the roof level.
Figure 1 (a) Plan view; (b) east and west elevations; and (c) north and south elevations of a
single-story industrial building (from Chopra, 2001).
If a static lateral load, P, is applied at the roof level in the north-south direction, the resistance to
lateral motion will be provided by the columns of the moment resisting frames. Specifically, the
stiffness of the columns, k, results in a restoring force, f s , which is proportional (assuming
elastic behavior) to the lateral displacement, u, of the columns (see Figure 2). The equation
relating the applied static force and the induced restoring forces may be obtained by application
of Newton’s Second Law for translational motion of a rigid body (recall your course on
dynamics):
K
K
∑ F = ma
(1)
which states that the net force acting on a rigid body is equal to the mass of the body multiplied
by the acceleration of the body. Note that Eq. (1) is a vector equation and thus the resulting
acceleration is in the same direction as the net force. Applying Newton’s Second Law to a freebody-diagram of the roof (mass) of the structure results in (see Figure 2(b)):
− fS + P = 0
(2)
where the acceleration is taken as zero since the load is applied statically. The restoring force for
linear elastic behavior is given by
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fS = ku
(3)
Substituting Eq. (3) into Eq. (2) and rearranging gives
ku = P
(4)
and thus we see that, in the case of static application of a force, the total force that develops in
the columns is equal to the applied static force.
Figure 2 (a) Deformed shape of frame under static lateral load; and (b) free body diagram
of mass of structure (adapted from Chopra, 2001).
2.2.2 Solution
The lateral displacement of the structure at the roof level is obtained by rearranging Eq. (4):
u=
P
k
(5)
Thus, we have derived the simple result that application of a static load to the structure produces
a displacement response equal to the magnitude of the force divided by the lateral stiffness of the
structure.
2.3 DYNAMIC LATERAL LOAD ANALYSIS
2.3.1 Problem Statement
Consider the same simple one-story industrial building whose plan and elevation views are
shown in Figure 1. If a dynamic lateral load, P(t ) , is applied at the roof level in the north-south
direction, the resistance to lateral motion will be provided by three components: 1) the restoring
force, fS (t ) , associated with the displacement of the moment resisting frames, 2) the inertial
force, f I (t ) , associated with the acceleration of the mass, and 3) the damping forces, f D (t ) ,
associated with energy dissipation in the structure. Note that the variable t represents time. The
equation relating the applied dynamic force and the induced restoring forces, inertial forces and
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damping forces may be obtained by application of Newton’s Second Law for translational
motion of a rigid body (recall your course in dynamics):
K
K
∑ F(t ) = ma (t )
(6)
Applying Newton’s Second Law to a free-body-diagram of the roof (mass) of the structure
results in (see Figure 3(b)):
− fS (t ) − f D (t ) + P(t ) = mu(t )
(7)
where the acceleration, a (t ) , is written as u(t ) where u(t ) is the second derivative of the
displacement with respect to time. Note that the two overdots indicate second-order ordinary
differentiation with respect to time. The term on the right-hand side of Equation (7) may be
regarded as the inertial force, f I (t ) . Thus, Eq. (7) may be rearranged into the following more
convenient form
f I ( t ) + f D (t ) + f S ( t ) = P ( t )
(8)
which states that, at each time instant, the sum of the inertia force, damping force and restoring
force is equal to the applied force. Comparing Equations (2) and (8), we see two differences
between the static and dynamic problem: 1) the static problem involves an algebraic equation
while the dynamic problem involves a time-dependent equation and 2) the dynamic problem
involves inertia and damping forces which are not present for static problems. The displacement
response for the case of static loading was easily obtained by rearranging the equilibrium
equation (see Eq. (5)). Is the displacement response for dynamic loading as easy to determine?
Let us answer this question by further examining Eq. (8).
Figure 3 (a) Deformed shape of frame under dynamic lateral load; and (b) free body diagram
of mass of structure (from Chopra, 2001).
The restoring force for linear elastic behavior has been given previously in Eq. (3) and is
repeated below except that now the force and displacement exhibit time-dependence.
f S (t ) = ku (t )
(9)
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The damping (energy loss) in the structure may be modeled by a “rate-dependent” damping force
that is proportional to the rate of change of the displacement.
f D (t ) = cu (t )
(10)
The proportionality constant c is known as the viscous damping coefficient and the damping
model given by Eq. (10) is known as a linear viscous damping model. Damping in real
structures comes from a variety of sources (e.g., friction at steel connections, opening/closing of
cracks in concrete, friction between the structural frame and nonstructural elements such as
partition walls). The complexity of damping in real structures is such that quantifying the
damping contribution from each source is an essentially impossible task. Thus, we often utilize
the simple linear viscous damping model to account for the energy dissipation from multiple
sources. Note that in Figure 3 the linear viscous damper model is represented by a dashpot that
is located at the center of the frame. The dashpot is characterized by its damping coefficient, c,
just as the columns are characterized by the stiffness, k.
The inertial force develops due to inertia of the mass. Inertia may be defined as the resistance to
acceleration. Structures with large mass have large inertia. Interestingly, the inertial force is not
a force that is applied to the mass. Rather, it is a force that develops due to the resistance of the
mass to its own motion. As noted above, the term on the right-hand side of Equation (7) may be
regarded as the inertial force, f I (t ) . Thus, we have
f I (t ) = mu(t )
(11)
Substituting Eq. (9), (10) and (11) into Eq. (8) results in
mu(t ) + cu (t ) + ku (t ) = P(t )
(12)
which is the standard form of a second-order, linear, ordinary differential equation with constant
coefficients. Although you may not remember, you have solved this type of equation previously
in your course on Differential Equations. In Eq. (12), t is the independent variable and u is the
dependent variable. Of course, the solution to Eq. (12) depends on the form of the timedependent loading function, P(t ) .
2.3.2 Purpose of Dynamic Analysis
Dynamic analysis can be used for two general interrelated purposes. First, dynamic analysis can
be utilized to identify the dynamic properties of a structure (i.e., m, c, and k in Eq. (12)). This
type of dynamic analysis is called system identification. Second, dynamic analysis can be
utilized to evaluate the response of structure to general dynamic loading, P(t ) . In order to
evaluate the response of structure to general dynamic loading, one must first have completed the
system identification so that the constant coefficients of Eq. (12) (i.e., m, c, and k) are available.
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2.4 SYSTEM IDENTIFICATION
There are a variety of methods for identifying the dynamic properties of a structure. We will
discuss two such methods.
2.4.1 Free Vibration Test
When a structures vibrates on its own (i.e., with no applied forces), the structure is said to be in
free vibration. For example, if we were to pull on the mass of the structure shown in Figure 1
and then release the mass, the structure would be in free vibration. What is the purpose of such a
test? The free vibration displacement response can be used to identify the dynamic properties of
the structure.
When a structure is in free vibration, the applied loading is zero. In this case, Eq. (12) becomes:
mu(t ) + cu (t ) + ku (t ) = 0
(13)
A more convenient form of Eq. (13) is obtained by dividing through by the mass. In this case,
we have
u(t ) +
c
k
u (t ) + u (t ) = 0
m
m
(14)
The ability of a structure to dissipate energy (i.e., its damping capacity) is typically defined, not
in terms of the damping coefficient, c, but in terms of the damping ratio, ξ . The damping ratio
is given by
ξ=
c
2mω n
(15)
k
m
(16)
where
ωn =
The undamped circular natural frequency, ω n , characterizes the frequency of the free vibration if
no damping were present. The undamped circular natural frequency is related to the undamped
cyclic natural frequency, f n , and the undamped natural period, Tn , by the following relations:
Tn =
1
2π
=
f n ωn
(17)
Substituting Eq. (15) and (16) into (14) results in
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u(t ) + 2ξω n u (t ) + ω 2n u (t ) = 0
(18)
which is the standard form of the free vibration problem. Equation (18) may be mathematically
described as a homogeneous, second-order, ordinary differential equation with constant
coefficients. From your differential equations course, you may recall that the solution to this
homogeneous equation consists of a damped harmonic function. Specifically, the solution is
given by:
⎡
⎤
⎛ u (0 ) + ξω n u (0 ) ⎞
⎟⎟ sin (ω d t )⎥ exp(− ξω n t )
u (t ) = ⎢u (0 ) cos(ω d t ) + ⎜⎜
ωd
⎢⎣
⎥⎦
⎠
⎝
(19)
where u (0 ) and u (0 ) are the initial displacement and velocity that induce the free vibration
response and ω d is the damped natural frequency (i.e., the actual frequency of the free vibration
response) which is related to the natural frequency by
ωd = ωn 1 − ξ 2
(20)
The free vibration response of three structural systems is shown in Figure 4. Realistic structures
exhibit damping ratios less than about 10%. Such systems are said to be underdamped
( ξ < 100% ) and thus, according to Figure 4, exhibit a damped harmonic free vibration response
(as indicated by Eq. (19)). The damped natural period of the response is given by
Td =
Tn
2π
=
ωd
1− ξ 2
(21)
where Eq. (17) and (20) were utilized. For typical structural systems having ξ < 10% , the
damped natural period is approximately equal to the undamped natural period (i.e., Td ≅ Tn ).
Thus, the undamped natural period, Tn , can be estimated from the free vibration response by
measuring the time required for one cycle of damped motion. It should be noted that the freevibration response is often measured in terms of acceleration rather than displacement. Taking
two derivatives of Equation (19) indicates that the natural period can be obtained from the
acceleration response in the same way that it is obtained from the displacement response.
The damping ratio may also be estimated from the free vibration response. Taking the ratio of
the free vibration displacement response at time t [i.e., u (t ) ] and j cycles later at time t + j Td
[i.e., u (t + jTd ) ], we find that the damping ratio may be written as
ξ=
1 ω d ⎛ u (t ) ⎞
⎟
ln⎜
2 jπ ω n ⎜⎝ u (t + jTd ) ⎟⎠
(22)
For realistic structures ( ξ < 10% ), ω d ≅ ω n . Therefore, Eq. (22) can be simplified to
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ξ=
1 ⎛ u (t ) ⎞
⎟
ln⎜
2 jπ ⎜⎝ u (t + jTd ) ⎟⎠
(23)
For practical application of Eq. (23), the time t is selected as the time at which a peak occurs
early in the response and the time t + j Td occurs at a peak j cycles later. Furthermore, it can be
shown that the damping ratio can be calculated from Equation (23) when the displacements are
replaced by accelerations. This is important since accelerometers (sensors for measuring
acceleration) are frequently used in dynamic testing of structures.
Now we have sufficient information from the free vibration test to evaluate the dynamic
properties of the structure. The procedure is as follows:
Figure 4 Free vibration response of structures (from Chopra, 2001).
Procedure for System Identification From Free Vibration Test Data
1. Estimate the mass, m, of the structure.
2. Measure the undamped natural period, Tn , from the free vibration response.
3. From the undamped natural period and mass, evaluate the stiffness using Eq. (16) and
(17).
4. Evaluate the damping ratio, ξ , using Eq. (23).
5. Having the mass, stiffness and damping ratio, determine the damping coefficient from
Eq. (15).
2.4.2 Sine Sweep Test
Instead of evaluating the dynamic properties of a structure from free vibration test data, a forced
vibration test can be performed using sinusoidal (harmonic) loading over a range of frequencies.
The idea is to excite the structure with harmonic loading over a range of frequencies such that at
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Student Manual
some frequency the structure experiences resonance. As you may recall, resonance occurs when
the frequency of the excitation is equal to the natural frequency of the structure. At the resonant
frequency, the structure experiences its largest response (as compared to any other frequency of
loading).
Recall Eq. (12) which gives the equation of motion for forced vibration
mu(t ) + cu (t ) + ku (t ) = P(t )
(24)
In the case of harmonic loading, the forcing function is given by
P(t ) = Po sin (ωt )
(25)
where Po is the amplitude and ω is the circular frequency of the harmonic load. To perform a
sine sweep test, the frequency of the loading is varied over a range that includes the natural
frequency of the structure.
As you may recall from your course in differential equations, the complete solution to Eq. (24),
with the forcing function given by the harmonic loading of Eq. (25), consists of a complementary
solution combined with a particular solution. The complementary solution is obtained by solving
the homogenous form of Eq. (24). The particular solution may be determined by various
methods (e.g., the method of undetermined coefficients). You may also recall that the complete
response contains both a transient component and a steady-state component. The transient
component dies out quickly with time while the steady-state component persists. For system
identification, the harmonic loading at each frequency is applied long enough that the transient
component of the response may be neglected. The steady-state solution may be written as
u (t ) = u o sin (ωt + θ )
(26)
where
uo =
Po
k
1
[1 − (ω ω ) ]
2 2
n
+ (2ξ ω ω n )
(27)
2
⎡ 2ξ ω ω n ⎤
θ = tan −1 ⎢
⎥
2
⎣⎢1 − (ω ω n ) ⎦⎥
(28)
Note from Eq. (26) that u o is the amplitude of the steady-state harmonic response and θ is the
phase angle between the response and the loading. Equation (27) may be simplified as follows
uo =
Po
Rd
k
(29)
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where R d is the displacement response factor given by
Rd =
1
[1 − (ω ω ) ]
2 2
n
+ (2ξ ω ω n )
(30)
2
What is the significance of the displacement response factor? Let us examine this question by
first rearranging Eq. (29) to obtain
Rd =
uo
uo
=
Po k (u st )o
(31)
where (u st )o is the amplitude of the response to harmonic loading when the harmonic loading is
applied very slowly (i.e., ω is very small). If ω is very small, the loading is considered to be
quasi-static. The loading is not static but it is applied very slowly. In this case, the inertia forces
and damping forces are very small and the displacement response is therefore given by Eq. 5
with P replaced by Po . Thus, we have:
(u st )o = Po
(32)
k
To answer our query regarding the significance of the displacement response factor, it is
apparent from Eq. (31) that the displacement response factor, R d , gives the ratio of the dynamic
response amplitude to the static response amplitude. In other words, R d tells us how much the
amplitude of the response is affected by applying the harmonic load dynamically rather than
statically. A plot of the displacement response factor and phase angle as the frequency of
loading varies is shown in Figure 5. Note that each curve shown in Figure 5 is for a specific
structural damping ratio.
Let us examine the displacement response factor [Eq. (30)] and phase angle [Eq. (28)] for three
different cases of harmonic loading frequency.
Case 1: Very small frequencies of loading (i.e., quasi-static loading)
R d tends toward unity and θ tends toward zero. Thus, the dynamic response amplitude is equal
to the static response amplitude and the phase angle between the dynamic response and the
loading is zero. In other words, the dynamic response follows the form of the quasi-static
loading.
Case 2: Very large frequencies of loading (i.e., very fast loading)
R d tends toward zero and θ tends toward 180o. Thus, the dynamic response amplitude is zero
and the phase angle between the dynamic response and the loading is 180o. In other words, the
structure is barely moving and the motion is in the opposite direction to the harmonic loading.
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Student Manual
Figure 5 Displacement response factor and phase angle for steady-state harmonic response
(from Chopra, 2001).
Case 3: Frequency of loading equal to natural frequency
When the frequency of loading is equal to the natural frequency, the structure experiences
resonance. Note from Figure 5 that the maximum displacement response does not exactly occur
at a frequency equal to the natural frequency. Rather, the frequency at which the maximum
response occurs, ω max , is given by
ω max = ω n 1 − 2ξ 2 < ω n
(33)
For realistic structures ( ξ < 10% ), ω max is essentially equal to ω n and thus we have
(R d )max = (R d )ω=ω
(34)
n
Substituting ω = ω n into Eq. (30) gives
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(R d )max = (R d )ω=ω
n
=
1
2ξ
(35)
Rearranging Eq. (35) gives the damping ratio of the structure
ξ=
1
(36)
2(R d )max
Note from Eq. (35) that structures with low ability to dissipate energy (i.e., low ξ ) will have
large values of (R d )max . Conversely, structures with a large ability to dissipate energy will have
low values of (R d )max . This behavior is clearly evident in Figure 5. Interestingly, at resonance
the phase angle is equal 90o for any value of the damping ratio. Thus, for all structures subjected
to resonant conditions, the dynamic response is 90o out of phase with respect to the loading.
Physically speaking, this means that the when the loading is maximum, the displacement
response is zero and when the loading is zero, the displacement response is maximum. This is in
strong contrast to static loading where the response is completely in phase with the loading (i.e.,
if the load is zero, the response is zero and if the load is maximum, the response is maximum).
In summary, the sine sweep test results (i.e., the displacement response factor and phase angle
curves) may be utilized to identify the dynamic properties of the structure. The procedure is as
follows:
Procedure for System Identification From Sine Sweep Test Data
1.
2.
3.
4.
5.
Estimate the mass, m, of the structure.
Identify ω n from the location of the peak in the R d curve.
From the natural frequency and mass, evaluate the stiffness using Eq. (16).
Evaluate the damping ratio, ξ , using Eq. (36).
Having the mass, stiffness and damping ratio, determine the damping coefficient from
Eq. (15).
2.5 DYNAMIC RESPONSE ANALYSIS
Thus far, we have discussed the case of dynamic response induced by a lateral force applied to
the mass of a structure. Alternatively, dynamic response may be induced by motion at the base
of the structure. For example, earthquakes induce motion at the base of structures.
2.5.1 Ground Displacement Analysis
When a structure is subjected to a time-dependent lateral ground displacement, u g (t ) , the
structure moves with the ground and, in addition, the flexibility of the structure results in
additional displacements with respect to the ground (see Figure 6). The displacement of the
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Student Manual
structure with respect to the ground, u (t ) , induces restoring forces, fS (t ) , and damping forces,
f D (t ) , in the lateral resisting elements (e.g., columns, shear walls, etc.) of the structure. In
addition, the acceleration of the mass results in the development of inertia forces, f I (t ) .
Figure 6 (a) Displaced shape of structure and (b) free-body-diagram of roof mass for
structure subjected to lateral ground displacement (from Chopra, 2001).
Application of Newton’s Second Law to the free-body-diagram shown in Figure 5(b) results in
− f I (t ) − f S ( t ) − f D ( t ) = 0
(37)
f I (t ) = mu t (t )
(38)
where
The total acceleration of the mass, u t (t ) , may be written as the sum of the ground acceleration
and the relative acceleration of the mass with respect to the ground:
u t (t ) = u g (t ) + u(t )
(39)
Substituting Eq. (9), (10), (38) and (39) into Eq. (37) and rearranging gives
mu(t ) + cu (t ) + ku (t ) = −mu g (t )
(40)
Recall Eq. (24) which gives the equation of motion for the case of applied loading P(t ) at the
roof level
mu(t ) + cu (t ) + ku (t ) = P(t )
(41)
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Student Manual
A comparison of Eq. (40) and (41) indicates that the analysis of the response of a structure to
lateral ground displacement, u g (t ) , is completely equivalent to the analysis of the structure
subjected to an effective applied load at the roof level, Peff (t ) , given by
Peff (t ) = −mu g (t )
(42)
The equivalence between Eq. (40) and (41) is illustrated in Figure 7.
Figure 7 Effective lateral loading for ground displacement (from Chopra, 2001).
2.5.2 Sinusoidal Ground Displacement
One implication of Eq. (42) is that the results from the analysis presented previously for
sinusoidal loading applied at the roof of a structure can be applied to the case of imposed ground
displacement if the resulting ground acceleration is sinusoidal. What type of ground
displacement is associated with a sinusoidal ground acceleration? Consider this: If we take two
derivatives of a sinusoidal ground displacement, we will get a sinusoidal ground acceleration.
Therefore, a sinusoidal ground displacement applied to a structure is completely equivalent to a
sinusoidal loading applied to the roof level of the structure. Thus, the sine sweep identification
method can be performed by applying a sinusoidal ground displacement to a structure rather than
applying sinusoidal loading at the roof level. For full scale structures, sine sweep system
identification tests are performed by applying sinusoidal loading at the roof level using a device
called an eccentric mass vibrator. In contrast, for model-scale structures sine sweep system
identification tests are often performed by subjecting the base of the structure to sinusoidal
motion using a shaking table.
For a sine sweep test with sinusoidal motion applied by a shaking table, it is often convenient to
characterize the response by the acceleration transfer function, H, rather than the displacement
response factor, R d (since the response of the structure and the shaking table motion are often
measured using accelerometers). The acceleration transfer function is given by:
H=
uot
ugo
(43)
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where uot is the maximum value of the total acceleration of the mass and ugo is the maximum
ground acceleration. It can be shown that the relationship between the acceleration transfer
function and the displacement response factor is given by:
H = 1 + [2ξ(ω / ωn )] R d
2
(44)
from which it is apparent that the acceleration transfer function and the displacement response
factor are nearly equal at resonance. For example, the error introduced by using the acceleration
transfer function for a system with a 10% damping ratio is only about 2%.
2.5.3 Earthquake Ground Displacement
In the case where the ground displacement can not be described analytically, the equation of
motion given by Eq. (40) must be solved numerically. For example, for earthquakes the ground
displacement cannot be described analytically (see Figure 8). In fact, earthquake ground motions
are measured using instruments that obtain the data in discretized (not continuous) form. In
other words, the ground motion data is not available at every time instant but rather is available
at discrete time steps. As a result, the dynamic analysis of a structure subjected to an earthquake
ground motion must be performed numerically (i.e., the equation of motion is solved numerically
rather than analytically). Commercial computer programs are available for solving differential
equations numerically. A procedure that is often used by such programs avoids the direct
solution of second-order differential equations by transforming the equations [e.g., Eq. (40)] to a
system of coupled first-order differential equations through a so-called state-space
transformation.
1940 Imperial Valley Earthquake, El Centro record, Component S00E
Figure 8 Typical earthquake ground motion record with acceleration, velocity and displacement
shown from top to bottom (from Chopra, 2001).
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2.6 EFFECTS OF BASE ISOLATION
A very brief description of the effects of base isolation is provided below. Additional
information on this subject can be found in Naeim and Kelly (1999) and Skinner et al. (1993).
In a base-isolated structure, the structure is “isolated” from the earthquake ground motion by
introducing a laterally flexible interface at the base of the building (see Figure 9). The flexible
interface consists of a number of isolation bearings. The bearings have high vertical stiffness [to
support the weight of the structure above (i.e., the weight of the superstructure)] but very low
horizontal (lateral) stiffness. In addition, to ensure that the bearings act in unison, a rigid
basemat is installed between the superstructure and the top of the isolation bearings. The bottom
of the isolation bearings is attached to the foundation. The introduction of the isolation system
results in a significant increase in the fundamental period of the structure, resulting in a reduction
in shear force demands due to the relatively low seismic input energy at large periods. In
addition to force reductions, an isolated structure typically experiences lower superstructure
interstory drifts since, if the bearings are flexible enough, the superstructure essentially behaves
as a rigid body that translates on top of the bearings (see Figure 9). For the one-story building
frame shown in Figure 1, the introduction of a basemat and isolation bearings results in a more
complex dynamic system. Specifically, an additional degree-of-freedom is introduced. Thus, a
base-isolated one-story building may be regarded as having two degrees-of-freedom and thus
two natural modes of vibration.
Fixed-Base Structure
Base-Isolated Structure
Figure 9 Qualitative illustration of seismic response of fixed-base and base-isolated structures.
Two common isolation bearings are sliding bearings and elastomeric bearings. Sliding bearings
may be flat or curved. For flat sliding bearings, a restoring force mechanism (e.g., a spring) must
be included to prevent large and possibly permanent displacements at the isolation level. For
curved sliding bearings, the curved shape of the bearings results in the development of restoring
forces. A photograph and schematic of a curved sliding bearing, known as the Friction
Pendulum System bearing, is shown in Figure 10. Elastomeric bearings consist of a rubber
(elastomeric) pad that is interspersed with relatively thin horizontal steel plates. The steel plates
ensure that the bearing is rigid in the vertical direction and the rubber provides lateral flexibility.
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A photograph and schematic of a elastomeric bearing is shown in Figure 10. Note that a
cylindrical lead plug is often included in the center of the bearing to provide additional energy
dissipation capacity.
Sliding Isolation Bearing
Elastomeric Isolation Bearing
Figure 10 Sliding and Elastomeric Isolation Bearings.
3.0 DESCRIPTION OF LAB EQUIPMENT
3.1 Required Equipment
• Instructional Shake Table
• Power Supply (Universal Power Module)
• Data Acquisition and Control System (MultiQ Board and Computer)
• Software
• Measurement Sensors (Three Accelerometers)
• Test Structure (One-Story Building Model)
• Seismic Isolation Systems
3.2 Instructional Shake Table
The bench-top shake table consists of an 18” x 18” sliding platform which is driven by an
electric servomotor (see Figure 11). The servomotor drives a lead screw which, in turn, drives a
circulating ball nut which is coupled to the sliding platform. The servomotor is driven by an
amplifier which is embedded within the Universal Power Module (power supply). The sliding
platform slides on low friction linear ball bearings which are mounted on two ground-hardened
shafts. The platform surface has 36 tie-down points located on a 3” x 3” grid.
The operational frequencies of the shake table range from 0 to 20 Hz. The maximum
displacement, velocity, and acceleration of the table are 6 in. (±3in.), 33 in/sec, and 2.5 g,
respectively. The maximum payload is 33 lbf. The computer-controlled table is capable of
reproducing both simple waveforms (e.g., sine waves and sine sweeps) and complex waveforms
(e.g., white noise and historical earthquake records).
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Data Acq. & Control Board
Computer and Monitor
Power Module
Shaking Table
Servomotor
Figure 11 Photograph showing components of instructional shaking table.
3.3 Power Supply
The Universal Power Module (power supply) includes a power amplifier for driving the shake
table servomotor (see Figure 11). In addition, it includes an independent ± 12 volt DC power
supply to provide power to sensors.
3.4 Data Acquisition and Control System
The data acquisition and control board used to collect data and drive the power amplifier (see
Figure 11) is a MultiQ I/O board manufactured by Quanser Consulting, Inc. Features of the
board that are of interest for this lab include an 8-channel analog-to-digital converter with an
input range of ± 5 volts, 13 bit resolution, and single-ended input. In addition, the board
contains an 8-channel digital-to-analog converter with an output range of ± 5 volts and 12 bit
resolution. The computer system used to perform the tests described herein consisted of a
Pentium II 300 MHz processor with 32 MB RAM and running Windows 98.
3.5 Software
Operation of the shaking table involves the use of the following five different software programs:
WinCon, Visual C++, Matlab, Simulink, and Real-Time Workshop. WinCon is produced by
Quanser Consulting, Inc., Visual C++ is produced by Microsoft, Inc., and Matlab, Simulink, and
Real-Time Workshop are produced by The MathWorks. WinCon is used for real-time feedback
control and digital signal processing. Specifically, WinCon converts a Simulink block diagram
to controller code using Real-Time Workshop, compiles and links the controller code using
Visual C++, and runs the controller code in real-time. The controller code is used to control the
motion of the shaking table while simultaneously recording various responses and performing
digital signal processing.
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3.6 Measurement Sensors
One accelerometer is used to measure the motion of the shaking table and two accelerometers are
used to record the response of the one-story base-isolated building. The accelerometers are
manufactured by Quanser Consulting, Inc. and produce an output of ± 5 volts with a range of
± 5 g. A photograph of one of the accelerometers is shown in Figure 12.
Figure 12 Photograph of accelerometer.
3.7 Test Structure
The test structure developed for this experiment is a one-story building frame. The building
frame can be configured as either a conventional fixed-base structure or a base-isolated structure
(see Figure 13). In the fixed-base configuration, the columns are attached to the roof above and
to the foundation below which is, in turn, rigidly attached to the shaking table. For the baseisolated configuration, the columns are attached to the roof above and to the basemat below. As
mentioned previously, in a base-isolated structure, a rigid diaphragm known as a basemat is
typically installed above the bearings so as to ensure that the bearings act in unison when
responding to the earthquake ground motion. In the case of the structure tested herein, the
basemat may also be considered to represent the first floor. The bottom of the bearings is
attached to the foundation below which is, in turn, rigidly attached to the shaking table. The
purpose of the foundation plate is to provide a convenient interface between the shaking table
and either the columns (fixed-base configuration) or isolation bearings (base-isolated
configuration). The roof, basemat and foundation plates have an area of 144 in2 (12” x 12”) and
a thickness of 0.5 in. The column height is 6 in. The four columns consist of styrene (plastic) Ishaped sections while the roof, basemat, and foundation levels are constructed from Plexiglas.
The top and bottom of each column is attached (glued) to styrene base plates which in turn are
attached (screwed) to the Plexiglas roof and basemat/foundation levels. The columns are
orientated such that bending occurs about the weak axis. An accelerometer is attached to the
roof and basemat levels to record the lateral accelerations.
The weight at each level of the structure (which includes one-half the column weight, column
connectors, and accelerometers) is 3.41 lb. The second moment of area of each column about
the axis of bending is 1.713 x 10-4 in4. Based on experimental testing, the modulus of elasticity
of the columns was determined to be approximately 230 ksi.
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Fixed-Base Structure
Base-Isolated Structure
Figure 13 Fixed-base and base-isolated test structures.
3.8 Seismic Isolation Systems
Two different seismic isolation systems are examined in this lab. The first isolation system is a
sliding system in which the structure was mounted on four roller bearings, one bearing being
located approximately under each column (see Figure 14). Due to the low pressure on the roller
bearings, the bearings behaved more like flat sliding bearings than roller bearings (i.e., the rollers
tended to slide rather than roll). The sliding bearings dissipate energy through friction but
exhibit no restoring force. To control the sliding displacements, a restoring force was provided
via two springs that were attached between the basemat and foundation (see close-up view of
Figure 14). The total weight supported by the isolation system is 6.82 lb. The stiffness of each
spring was experimentally determined to be 4.8 lb/ft.
Elevation View
Close-Up View of Isolation System
Figure 14 Test structure with sliding isolation system.
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The second isolation system was an elastomeric system in which the structure was mounted on
four elastomeric (rubber) bearings, each bearing being located under one of the columns (see
Figure 15). The rubber bearings provide both energy dissipation and restoring forces. In
addition, due to the relatively slender shape of the bearings, the building tended to exhibit some
rocking response. The rubber bearings were attached (glued) to styrene plates which were in
turn attached (screwed) to the basemat above and the foundation below. The total weight
supported by the isolation system is 6.82 lb.
Elevation View
Close-Up View of Isolation System
Figure 15 Test structure with elastomeric isolation system.
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A drawing of the test structure and a list that identifies each component and the materials used to
construct each component is provided in Figure 16 and Table 1.
(a)
(b)
(c)
Figure 16 (a) Elevation view of test structure with isolation system; (b) Sliding bearing isolation
system; and (c) Elastomeric bearing isolation system.
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Table 1 List of materials for construction of test structure and isolation systems
(numbers within description refer to Figure 16)
Item
1
Description
12 x 12 x ½ inch Plexiglas Plates
1a) Roof – Attached to top of columns.
1b) Basemat – For fixed-based configuration, basemat is not used. For base-isolated
configuration, basemat is attached to bottom of columns and top of isolation bearings.
1c) Foundation – For fixed-base configuration, foundation is attached to bottom of
columns and to shaking table. For base-isolated configuration, foundation is attached
to bottom of isolation bearings and to shaking table.
2
Isolation Systems
2a) Sliding Bearings
- Four roller bearings (5/8” diameter rollers)
- Two steel springs (5/16” dia. x 1-3/4” length x 016 thick; Century Spring Corp.,
Part No. C-660)
2b) Elastomeric Bearings
Rubber pad glued between two 1/8” thick styrene (plastic) base plates.
3
Columns
Styrene (plastic) I-beams (9/16” depth; Plastruct Part No. 90521)
Styrene (plastic) base plates (1/8” thick; Plastruct Part No. 91108)
4
Accelerometers
For fixed-base configuration, two accelerometers are used, one on the roof and one on
the shaking table.
For base-isolated configuration, three accelerometers are used, one on the roof, one on
the basemat, and one on the shaking table.
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4.0 PROCEDURES FOR CONDUCTING EXPERIMENTAL TESTS
4.1 System Identification: Free-Vibration Tests
The free vibration tests can be performed as follows:
1) Within Wincon, set up a control panel that generates sinusoidal motion of the shaking
table and allows for adjustment of the displacement amplitude and frequency of the
motion. Also, configure two scopes to monitor the roof acceleration and shaking table
acceleration.
2) Generate sinusoidal motion of the shaking table. Adjust the frequency and amplitude of
the excitation so as to induce appreciable motion of the structure while preventing
damage to the structure. Be particularly careful about exciting the structure at a
frequency near the natural frequency (i.e., near resonance).
3) After the structure reaches steady-state response, the frequency of the sine wave should
be reduced to zero very quickly such that structure goes into free vibration.
4) Throughout the duration of the test, the acceleration response should be measured at the
roof level of the structure.
5) Save the measured acceleration as a Matlab .mat file. Load the .mat file into Matlab and
plot the free vibration acceleration response as a function of time.
6) From the free vibration plot, identify the system properties using the five-step procedure
outlined at the end of Section 2.4.1. Note that, for the base-isolated structures, you are to
assume that the structure is primarily vibrating in its fundamental mode and thus the
estimated properties are associated with the fundamental mode.
7) Provide a tabular summary of the system properties for the fixed-base structure and the
two base-isolated structures (see Table 2 for template).
Table 2 Template for summary of results from free-vibration system identification testing.
Configuration
Fixed
Sliding
Elastomeric
m (lb-s2/in)
c (lb-s/in)
k (lb/in)
Tn (sec)
ξ (%)
4.2 System Identification: Sine-Sweep Tests
The sine-sweep tests can be performed as follows:
1) Within Wincon, set up a control panel that generates shaking table motion in the form of
a sine sweep from 0 to 20 Hz and with an adjustable displacement amplitude. Also,
configure two scopes to monitor the roof acceleration and shaking table acceleration.
2) Generate the sine-sweep motion of the shaking table. Adjust the amplitude of the
excitation so as to induce appreciable motion of the structure while preventing damage to
the structure. Be particularly careful about avoiding excessive response when the
excitation frequency passes through the resonant frequency.
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3) Throughout the duration of the test, the acceleration response should be measured at the
roof and foundation level of the structure.
4) During the test, observe the structural response, watching for the development of mode
shapes as the sine sweep excitation passes through the natural frequencies.
5) Save the measured accelerations in a Matlab .mat file. Load the .mat file into Matlab and
plot the roof and foundation acceleration responses as a function of time. Also, plot the
transfer function relating the roof acceleration to the ground acceleration.
6) From the transfer function, identify the system properties using the five-step procedure
outlined in Section 2.4.2. Note that the transfer function for the base-isolated structure
will exhibit two peaks, each peak being associated with a mode of vibration. The
fundamental natural period can be estimated from the location of the lowest frequency
peak and the damping ratios are related to the height of the peaks. However,
determination of the fundamental mode damping ratio from the sine sweep test is beyond
the scope of this project. Thus, for the base-isolated structures, do NOT attempt to obtain
system properties associated with damping.
7) Provide a tabular summary of the system properties for the fixed-base structure and the
two base-isolated structures (see Table 3 for template).
Table 3 Template for summary of results from sine-sweep system identification testing.
Configuration
Fixed
Sliding
Elastomeric
m (lb-s2/in)
c (lb-s/in)
K (lb/in)
NA
NA
Tn (sec)
ξ (%)
NA
NA
Note: NA = Not applicable since beyond scope of project.
4.3 Seismic Response Evaluation
Seismic testing can be performed as follows:
1) Generate shaking table motion corresponding to each of the following three earthquake
records that are pre-defined within the UCIST shaking table system: 1) El Centro, 2)
Hachinohe, and 3) Northridge. Configure four scopes to monitor: 1) roof acceleration, 2)
foundation acceleration, 3) roof displacement, and 4) foundation (for fixed-base) or
basemat (for base-isolated) displacement.
2) Save the measured accelerations and displacements in a Matlab .mat file. Load the .mat
file into Matlab.
3) From the measured accelerations, plot the base shear coefficient (base shear normalized
by total weight of structure) and story shear coefficient (story shear normalized by roof
weight) as a function of time. Note that the story shear and base shear may be taken as
the inertia force at the roof and the sum of the inertia forces at the roof and basemat,
respectively.
4) From the measured displacements, plot the interstory drift ratio as a function of time.
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5) For the purposes of evaluating the seismic performance of the fixed-base and baseisolated structures, extract the peak values from each of the shear force and drift ratio
time histories.
6) Provide a tabular summary of the peak response values for the fixed-base structure and
the two base-isolated structures (see Table 4 for template).
Table 4 Template for peak response of structures subjected to seismic testing.
Earthquake
El Centro
Hachinohe
Northridge
Configuration
Drift Ratio (%)
Story Shear /
Roof Weight
Base Shear /
Total Weight
Fixed
Sliding
Elastomeric
Fixed
Sliding
Elastomeric
Fixed
Sliding
Elastomeric
Note: Numbers in parenthesis indicate percentage change with respect to fixed-base configuration.
5.0 Questions
5.1 Questions Related to System Identification and Seismic Response Tests
1. For the free-vibration tests, is the free-vibration decay exponential for all three structural
systems? In the cases where the decay is exponential, what assumption can reasonably be made
about the form of damping in the structure? If the decay is not exponential, offer an explanation
as to why this is the case. Comment on the system properties obtained from the free-vibration
tests.
2. For the sine-sweep test, how many peaks appear in the transfer functions for each structural
system? What is the significance of the number of peaks? What is the significance of the
location and the height of the peaks? Comment on the system properties obtained from the sinesweep tests. Compare the results obtained from the free-vibration and sine-sweep tests.
3. For the seismic tests, discuss the effectiveness of the isolation systems in controlling the
response. Based on your discussion, what advantages and disadvantages are associated with the
use of an isolation system?
5.2 General Dynamic Analysis Questions
1. What are the main differences between static and dynamic analysis?
2. Why do structural engineers need to understand structural dynamics?
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3. The primary purpose of dynamic analysis is either system identification or response analysis.
Are these two purposes interrelated? How?
4. The damped natural period of a structure is often approximated by the undamped natural
period. What error is incurred by this assumption if the damping ratio of the structure is at
the approximate upper bound of 10%. Based on your result, do you think the assumption is
reasonable?
5. In a sine sweep system identification test, the circular frequency at which the maximum
displacement response factor occurs is often approximated as the undamped natural circular
frequency. What error is incurred by this assumption if the damping ratio of the structure is
at the approximate upper bound of 10%. Based on your result, do you think the assumption
is reasonable?
6. If you are evaluating the response of a very flexible structure subjected to very fast sinusoidal
loading, do you expect the displacement response to be approximately in phase, 180o out of
phase, or 90o out of phase with respect to the applied load?
7. Numerical analysis of structural systems often requires the solution of second-order
differential equations. For example, the following equation of motion for earthquake loading
must be solved numerically: mu(t ) + cu (t ) + ku (t ) = −mu g (t )
For numerical analysis, this equation can be solved by rewriting it as a system of two firstorder differential equations via a state-space transformation. Perform this transformation.
Hint: Let the state-space variables be x1 (t ) and x 2 (t ) where x1 (t ) = u (t ) and
x 2 (t ) = u (t ) . The derivatives of x1 (t ) and x 2 (t ) are x 1 (t ) = u (t ) and x 2 (t ) = u(t ) . Now
write the two first-order equations by solving the equation of motion for u(t ) and
substituting in the state-space variables.
x 1 (t ) = ....
x 2 (t ) = ....
The above two equations will be first-order equations that are coupled in the variables x1 (t )
and x 2 (t ) . The benefit of the transformation is that first-order equations can be solved
instead of second-order equations. The cost of the transformation is that it results in twice
as many equations to solve.
8. Bonus: Using the identified system properties for the fixed base structure, develop numerical
predictions of the response of the structure when subjected to the Northridge earthquake.
Compare the numerical predictions with the experimental test data.
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6.0 REFERENCES
Chopra, A.K. (2001). Dynamics of Structures: Theory and Applications to Earthquake
Engineering, Second Edition, Prentice Hall.
Clough, R.W. and Penzien, J. (1993). Dynamics of Structures, Second Edition, McGraw Hill.
Naeim and Kelly (1999). Design of Seismic Isolated Structures, J. Wiley & Sons.
Skinner, R.I., Robinson, W.H. and McVerry, G.H. (1993). An Introduction to Seismic Isolation,
J. Wiley & Sons.
Tedesco, J.W., McDougal, W.G. and Ross, C.A. (1999). Structural Dynamics: Theory and
Applications, Addison-Wesley.
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