str 1 dynamics

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Table of Contents

Section Page

1.0 Introduction........................................................................................................................31.1 Objective..................................................................................................................3

1.2 Abstract....................................................................................................................3

2.0 Theoretical Considerations ...............................................................................................3

2.1 Static and Dynamic Loads .......................................................................................32.2 Static Lateral Load Analysis....................................................................................4

2.2.1 Problem Statement....................................................................................4

2.2.2 Solution.....................................................................................................5

2.3 Dynamic Lateral Load Analysis ..............................................................................52.3.1 Problem Statement....................................................................................5

2.3.2 Purpose of Dynamic Analysis...................................................................7

2.4 System Identification ...............................................................................................82.4.1 Free-Vibration Test...................................................................................8

2.4.2 Sine-sweep Test ......................................................................................10

2.5 Dynamic Response Analysis..................................................................................142.5.1 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 ..............................................................................................193.2 Instructional Shake Table ......................................................................................19

3.3 Power Supply.........................................................................................................20

3.4 Data Acquisition and Control System....................................................................203.5 Software .................................................................................................................20

3.6 Measurement Sensors ............................................................................................21

3.7 Test Structure.........................................................................................................213.8 Seismic Isolation Systems......................................................................................22

4.0 Procedures for Conducting Experimental Tests ...........................................................26

4.1 System Identification: Free-Vibration Tests..........................................................264.2 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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SHAKING TABLE DEMONSTRATION OF DYNAMIC RESPONSE

OF BASE-ISOLATED BUILDINGS

1.0INTRODUCTION

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 howan 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 ina conventional configuration in which the building is rigidly attached to the shaking tableplatform (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 thestructure are explored through system identification tests (free vibration and forced vibration)

and earthquake tests.

2.0THEORETICAL CONSIDERATIONS

2.1 Static and Dynamic LoadsStructures 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 ofeffective 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 asequivalent 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 theconservatism), 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 equivalentstatic analysis. Instead, dynamic analysis must be performed. Thus, it is important for structural

engineers to understand the fundamentals of structural dynamics. The ensuing discussion onstructural 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 theroof 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, sf , which is proportional (assumingelastic 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 Newtons Second Law for translational motion of a rigid body (recall your course ondynamics):

= amF

(1)

which states that the net force acting on a rigid body is equal to the mass of the body multipliedby 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 Newtons Second Law to a free-

body-diagram of the roof (mass) of the structure results in (see Figure 2(b)):

0PfS =+ (2)

where the acceleration is taken as zero since the load is applied statically. The restoring force forlinear elastic behavior is given by


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kufS= (3)

Substituting Eq. (3) into Eq. (2) and rearranging gives

Pku= (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):

k

Pu= (5)

Thus, we have derived the simple result that application of a static load to the structure producesa displacement response equal to the magnitude of the force divided by the lateral stiffness of the

structure.

2.3DYNAMIC 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, ( )tP , is applied at the roof level in the north-southdirection, the resistance to lateral motion will be provided by three components: 1) the restoring

force, ( )tfS , associated with the displacement of the moment resisting frames, 2) the inertialforce, ( )tfI , associated with the acceleration of the mass, and 3) the damping forces, ( )tfD ,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 Newtons Second Law for translational

motion of a rigid body (recall your course in dynamics):

( ) ( ) = tamtF

(6)

Applying Newtons Second Law to a free-body-diagram of the roof (mass) of the structureresults in (see Figure 3(b)):

( ) ( ) ( ) ( )tumtPtftf DS =+ (7)

where the acceleration, ( )ta , is written as ( )tu where ( )tu is the second derivative of thedisplacement with respect to time. Note that the two overdots indicate second-order ordinarydifferentiation with respect to time. The term on the right-hand side of Equation (7) may be

regarded as the inertial force, ( )tfI . Thus, Eq. (7) may be rearranged into the following moreconvenient form

( ) ( ) ( ) ( )tPtftftf SDI =++ (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 equationwhile 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 isrepeated below except that now the force and displacement exhibit time-dependence.

( ) ( )tkutfS = (9)


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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 TestWhen 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 1and 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:

( ) ( ) ( ) 0tkutuctum =++ (13)

A more convenient form of Eq. (13) is obtained by dividing through by the mass. In this case,we have

( ) ( ) ( ) 0tum

ktu

m

ctu =++ (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, . The damping ratiois given by

nm2

c

= (15)

where

m

kn= (16)

The undamped circular natural frequency, n , characterizes the frequency of the free vibration ifno damping were present. The undamped circular natural frequency is related to the undamped

cyclic natural frequency, nf , and the undamped natural period, nT , by the following relations:

nn

n

2

f

1T

== (17)

Substituting Eq. (15) and (16) into (14) results in


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( ) ( ) ( ) 0tutu2tu 2nn =++ (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 constantcoefficients. From your differential equations course, you may recall that the solution to this

homogeneous equation consists of a damped harmonic function. Specifically, the solution isgiven by:

( ) ( ) ( ) ( ) ( )

( ) ( )texptsin0u0u

tcos0utu ndd

nd

++=

(19)

where ( )0u and ( )0u are the initial displacement and velocity that induce the free vibrationresponse and d is the damped natural frequency (i.e., the actual frequency of the free vibrationresponse) which is related to the natural frequency by

2nd 1 = (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

( %100


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( )( )

+=

djTtu

tuln

j2

1 (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 dT 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, nT , 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, , 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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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 resonantfrequency, 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

( ) ( ) ( ) ( )tPtkutuctum =++ (24)

In the case of harmonic loading, the forcing function is given by

( ) ( )tsinPtP o = (25)

where oP is the amplitude and is the circular frequency of the harmonic load. To perform asine 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 complementarysolution 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 completeresponse 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

( ) ( )+= tsinutu o (26)

where

( )[ ] ( )2n22

n

oo

21

1

k

Pu

+= (27)

( )

=

2

n

n1

1

2tan (28)

Note from Eq. (26) that ou is the amplitude of the steady-state harmonic response and is thephase angle between the response and the loading. Equation (27) may be simplified as follows

do

o Rk

Pu = (29)


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where dR is the displacement response factor given by

( )[ ] ( )2n22

n

d

21

1R

+= (30)

What is the significance of the displacement response factor? Let us examine this question by

first rearranging Eq. (29) to obtain

( )ost

o

o

od

u

u

kP

uR == (31)

where ( )ost

u is the amplitude of the response to harmonic loading when the harmonic loading is

applied very slowly (i.e., is very small). If is very small, the loading is considered to bequasi-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. 5with P replaced by oP . Thus, we have:

( )k

Pu o

ost = (32)

To answer our query regarding the significance of the displacement response factor, it is

apparent from Eq. (31) that the displacement response factor, dR , gives the ratio of the dynamic

response amplitude to the static response amplitude. In other words, dR 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 ofloading 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)

dR tends toward unity and tends toward zero. Thus, the dynamic response amplitude is equalto 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)

dR tends toward zero and 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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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 occurat a frequency equal to the natural frequency. Rather, the frequency at which the maximum

response occurs, max , is given by

n2

nmax 21


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( ) ( )

== =2

1RR

ndmaxd

(35)

Rearranging Eq. (35) gives the damping ratio of the structure

( )maxd

R2

1= (36)

Note from Eq. (35) that structures with low ability to dissipate energy (i.e., low ) will havelarge values of ( )

maxdR . Conversely, structures with a large ability to dissipate energy will have

low values of ( )maxd

R . 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 displacementresponse 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 anglecurves) 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. Estimate the mass, m, of the structure.2. Identify n from the location of the peak in the dR curve.3. From the natural frequency and mass, evaluate the stiffness using Eq. (16).4. Evaluate the damping ratio, , using Eq. (36).5. Having the mass, stiffness and damping ratio, determine the damping coefficient from

Eq. (15).

2.5 DYNAMIC RESPONSE ANALYSISThus 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 baseof 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, ( )tu g , thestructure 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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structure with respect to the ground, ( )tu , induces restoring forces, ( )tfS , and damping forces,( )tfD , 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, ( )tfI .

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 Newtons Second Law to the free-body-diagram shown in Figure 5(b) results in

( ) ( ) ( ) 0tftftf DSI = (37)

where

( ) ( )tumtf tI = (38)

The total acceleration of the mass, ( )tu t , may be written as the sum of the ground accelerationand the relative acceleration of the mass with respect to the ground:

( ) ( ) ( )tututu gt

+= (39)

Substituting Eq. (9), (10), (38) and (39) into Eq. (37) and rearranging gives

( ) ( ) ( ) ( )tumtkutuctum g =++ (40)

Recall Eq. (24) which gives the equation of motion for the case of applied loading ( )tP at theroof level

( ) ( ) ( ) ( )tPtkutuctum =++ (41)


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A comparison of Eq. (40) and (41) indicates that the analysis of the response of a structure to

lateral ground displacement, ( )tu g , is completely equivalent to the analysis of the structure

subjected to an effective applied load at the roof level, ( )tPeff , given by

( ) ( )tumtPgeff

= (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 forsinusoidal 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 twoderivatives 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 identificationmethod 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 devicecalled 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, dR (since the response of the structure and the shaking table motion are often

measured using accelerometers). The acceleration transfer function is given by:

go

to

u

uH

= (43)


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where tou is the maximum value of the total acceleration of the mass and gou is the maximum

ground acceleration. It can be shown that the relationship between the acceleration transferfunction and the displacement response factor is given by:

( )[ ] d2

n R21H += / (44)

from which it is apparent that the acceleration transfer function and the displacement responsefactor 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 motionsare 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 availableat discrete time steps. As a result, the dynamic analysis of a structure subjected to an earthquakeground 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 directsolution 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 ISOLATIONA 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 flexibleinterface consists of a number of isolation bearings. The bearings have high vertical stiffness [tosupport 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 systemresults 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 superstructureinterstory 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 morecomplex 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 bearingsmay 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 restoringforces. 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 energydissipation capacity.

Sliding Isolation Bearing Elastomeric Isolation Bearing

Figure 10 Sliding and Elastomeric Isolation Bearings.

3.0DESCRIPTION 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 anelectric 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 slidingplatform 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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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 shaketable servomotor (see Figure 11). In addition, it includes an independent 12 volt DC powersupply to provide power to sensors.

3.4 Data Acquisition and Control SystemThe 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 theboard 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 boardcontains an 8-channel digital-to-analog converter with an output range of 5 volts and 12 bitresolution. The computer system used to perform the tests described herein consisted of aPentium 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 byQuanser Consulting, Inc., Visual C++ is produced by Microsoft, Inc., and Matlab, Simulink, andReal-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 themotion of the shaking table while simultaneously recording various responses and performing

digital signal processing.

Shaking Table

Power ModuleComputer and MonitorData Acq. & Control Board

Servomotor


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3.6 Measurement Sensors

One accelerometer is used to measure the motion of the shaking table and two accelerometers areused 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 buildingframe 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 base-isolated 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 whenresponding 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. Thepurpose 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) I-shaped 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 areattached (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 aboutthe axis of bending is 1.713 x 10

-4in

4. 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 rollerbearings, 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 butexhibit 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 eachspring 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 (seeFigure 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 weightsupported 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 Description

1 12 x 12 x inch Plexiglas Plates

1a) Roof Attached to top of columns.

1b) Basemat For fixed-based configuration, basemat is not used. For base-isolatedconfiguration, basemat is attached to bottom of columns and top of isolation bearings.

1c) Foundation For fixed-base configuration, foundation is attached to bottom ofcolumns 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 onthe shaking table.

For base-isolated configuration, three accelerometers are used, one on the roof, one onthe basemat, and one on the shaking table.


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4.0PROCEDURES 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 shakingtable and allows for adjustment of the displacement amplitude and frequency of themotion. 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 ofthe excitation so as to induce appreciable motion of the structure while preventingdamage 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 shouldbe 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 theroof level of the structure.

5) Save the measured acceleration as a Matlab.mat file. Load the .matfile into Matlab andplot the free vibration acceleration response as a function of time.

6) From the free vibration plot, identify the system properties using the five-step procedureoutlined at the end of Section 2.4.1. Note that, for the base-isolated structures, you are toassume 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 thetwo base-isolated structures (see Table 2 for template).

Table 2 Template for summary of results from free-vibration system identification testing.

Configuration m (lb-s2/in) c (lb-s/in) k (lb/in) Tn(sec) (%)

Fixed

Sliding

Elastomeric

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 ofa 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 theexcitation so as to induce appreciable motion of the structure while preventing damage to

the structure. Be particularly careful about avoiding excessive response when theexcitation frequency passes through the resonant frequency.


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3) Throughout the duration of the test, the acceleration response should be measured at theroof and foundation level of the structure.

4) During the test, observe the structural response, watching for the development of modeshapes as the sine sweep excitation passes through the natural frequencies.

5) Save the measured accelerations in a Matlab.mat file. Load the .matfile into Matlab andplot the roof and foundation acceleration responses as a function of time. Also, plot thetransfer function relating the roof acceleration to the ground acceleration.

6) From the transfer function, identify the system properties using the five-step procedureoutlined 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 frequencypeak 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 m (lb-s2/in) c (lb-s/in) K (lb/in) Tn(sec) (%)

Fixed

Sliding NA NA

Elastomeric NA NANote: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 earthquakerecords 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 .matfile into Matlab.

3)

From the measured accelerations, plot the base shear coefficient (base shear normalizedby 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 asthe 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 base-isolated structures, extract the peak values from each of the shear force and drift ratiotime histories.

6) Provide a tabular summary of the peak response values for the fixed-base structure andthe two base-isolated structures (see Table 4 for template).

Table 4 Template for peak response of structures subjected to seismic testing.

Earthquake Configuration Drift Ratio (%) Story Shear /Roof Weight

Base Shear /

Total Weight

Fixed

SlidingEl Centro

Elastomeric

Fixed

SlidingHachinohe

ElastomericFixed

SlidingNorthridge

ElastomericNote: 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 madeabout the form of damping in the structure? If the decay is not exponential, offer an explanationas 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 structuralsystem? 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 sine-

sweep 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 theuse 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 naturalperiod. 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 isreasonable?

5. In a sine sweep system identification test, the circular frequency at which the maximumdisplacement 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 isat 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 sinusoidalloading, do you expect the displacement response to be approximately in phase, 180

oout of

phase, or 90

o

out of phase with respect to the applied load?

7. Numerical analysis of structural systems often requires the solution of second-orderdifferential equations. For example, the following equation of motion for earthquake loading

must be solved numerically: ( ) ( ) ( ) ( )tumtkutuctum g =++ For numerical analysis, this equation can be solved by rewriting it as a system of two first-

order differential equations via a state-space transformation. Perform this transformation.

Hint: Let the state-space variables be ( )tx1 and ( )tx2 where ( ) ( )tutx1 = and( ) ( )tutx2 = . The derivatives of ( )tx1 and ( )tx2 are ( ) ( )tutx1 = and ( ) ( )tutx2 = . Now

write the two first-order equations by solving the equation of motion for ( )tu andsubstituting in the state-space variables.

( ) ....tx1 = ( ) ....tx2 =

The above two equations will be first-order equations that are coupled in the variables ( )tx1 and ( )tx2 . The benefit of the transformation is that first-order equations can be solvedinstead 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 numericalpredictions of the response of the structure when subjected to the Northridge earthquake.

Compare the numerical predictions with the experimental test data.

str 1 dynamics

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