Dynamic Behavior of Simple Soil-Structure Systems

Department of Civil and Environmental EngineeringUniversity of California at Davis

Please wait while the document loads.

Dynamic Behavior of Simple

Soil-Structure SystemsA PROJECT DEVELOPED FOR THE

UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES

http://ucist.cive.wustl.edu/

Developed by:

Dr. John E. Bolander (jebolander@ucdavis.edu)

Tara C. Hutchinson (thutchin@uci.edu)Stefano Berton (ssberton@ucdavis.edu)

University of California at Davis

This project is supported in part by theNational Science Foundation Grant No. DUE-9950340

**** STUDENT'S GUIDE ****

Dynamic Behavior of Simple Soil-Structure Systems

Department of Civil and Environmental EngineeringUniversity of California at Davis

Objective: This experiment illustrates the influence local geology and soil condi-tions can have on the intensity of earthquake induced ground shaking and structuralvibration. A simple soil model will be constructed and connected in series with a1-D structure model. Both the soil-structure system and the structure model aloneare subjected to identical base excitations via a bench-scale shaking table. Theexperiment serves as an introduction to the modeling of soil-structure systems anddemonstrates some potential effects of the site period on structural response.

1. Introduction

The influence local soil conditions have on the intensity of shaking at the groundsurface has been observed and studied since the beginning of the nineteenth-century.However, only recently engineers have included local site effects in earthquake-resistance design procedures. Nowadays, the development of site-specific groundmotions and response spectra represents one of the most challenging aspects ofearthquake engineering. Important characteristics of the ground motion such aspeak acceleration, frequency content, etc., are affected by local site conditions. Dur-ing an earthquake, energy is released local to the fault and seismic waves are pro-duced. These waves propagate from the hypocenter in all directions and duringtheir travel may be reflected and refracted by the layered earth composition. Thelocal soil conditions then affect the ground motion by increasing or reducing theamplitude of frequency components present in the bedrock motion.

An important aspect of the dynamic behavior of a single or multi-degree of free-dom system subjected to external excitation is the resonance phenomenon. Reso-nance occurs when a frequency component of the excitation is close to the funda-mental frequency of a SDOF system or one of the natural frequencies of a MDOFsystem. Resonance is associated with an amplification of deformations that canlead to unforeseen consequences, including structural collapse. For a soil-structuresystem, resonance will occur when the site period is close to the fundamental periodof the structure.

A classical example of local soil effect on structural response is the 1985 MexicoCity earthquake. This magnitude 8.1 earthquake caused only moderate damage tostructures near the epicenter but caused extensive damage 350 km away from theepicenter in Mexico City. Furthermore, damage to structures in Mexico City wasconcentrated in areas where the site period was in the range of 1.9 to 2.8 seconds.

1

Buildings with fundamental periods near this range (e.g. some 5 to 20-story rein-forced concrete buildings) suffered the most damage.

2. Basic Theory: Dynamics of Soil-Structure Systems

This laboratory makes use of analytical methods and several idealizations tocharacterize the soil and soil-structure models. These include the shear column rep-resentation of the soil mass as a generalized SDOF system, a method to determinethe natural frequency of the models, and a determination of their damping ratios.The results are used to interpret the behavior of the composite soil-structure systemduring earthquake-type excitations.

2.1 Shear Beam Approach for Ground Response Analysis

A rigorous study of the response of a particular site involves sophisticated analysesusing wave propagation theory. In general, the soil may be approximated as a multi-layered medium, where each layer is given appropriate characteristics estimated fromsite exploration. Dynamic soil properties, such as damping ratios and shear wavevelocities, may be estimated for each layer. Consider two sites, each composed ofuniform layers of soil resting on rigid rock. The two sites have similar geometry andcharacteristics, except that one site is significantly stiffer than the other. Using wavepropagation theory, it can be shown that the softer site will amplify low-frequency(longer period) bedrock motions more than the stiff site.

The simplest approach to model the dynamic behavior of the soil is the shearbeam approach. In this case, an isolated column of soil is modeled as an equivalentbeam that can deform only in the shear mode (i.e. without contributions frombending), as shown in Fig. 1. If the soil is uniform, the natural frequencies of thesystem can be determined using

ωn = (2n− 1)π

2

√GA

mh2(1)

where n is the mode shape number; G is the shear modulus; A is the column crosssection area; m is the mass per unit depth; and h is the depth of the soil layer.

For the most basic case of uniform shearing of a soil layer on rigid rock, the soilcolumn can be modeled as a generalized single degree of freedom (SDOF) system(as shown in Fig. 1c or, equivalently, in Fig. 1d).

2.2 Discrete Fourier Transform

The fundamental frequencies of the soil system and structural models can beexperimentally determined using the bench-scale shake table and Discrete FourierTransform analysis. A periodic function f(t) can be expressed in terms of a Fourierseries, i.e., as an infinite sum of sine and cosine terms. The same concept can beextended for non-periodic functions since, in the limit, the function can be regardedas periodic with infinite period. In this process a new function F (ω), known as the

2

uniform soil

ground surface

bedrock

soil column

a) b) c) d)

h

m*k*

Fig. 1 a) Uniform soil on solid bedrock, b) Shear deformable soil column model,c) Uniform shearing of soil column, and d) Single degree of freedom (SDOF) model

5

10

15

00 5 10 15 20 25

frequency (Hz)

Four

ier

ampl

itude

peak corresponding to thefirst natural frequency

f1 f2 f3 f4

Fig. 2 Transfer function

Fourier Transform of f(t), can be defined as:

f(ω) =1

2π

∫ ∞−∞

f(t)eiωtdt (2)

where ω = circular frequency. If the data in the space or time domain is a vector(i.e. a finite number of data points), then the Discrete Fourier Transform (DFT) isapplicable. The most efficient and fast algorithm to evaluate the DFT of a vectoris the Fast Fourier Transform (FFT). This algorithm is very efficient and is imple-mented in many signal processing routines, including some of the routines availablein MATLAB. Using the FFT allows us to process a finite number of data pointsand plot the Fourier amplitude spectra in the frequency domain. For example, theFourier amplitude spectra of an acceleration time history, obtained via FFT, repre-sents the values of the coefficient of the Fourier series versus frequency. Frequenciescorresponding to peaks in the Fourier amplitude spectra are the dominant naturalfrequencies of the model. When the DFT of the initial data is normalized with theDFT of the input acceleration, the result is called the transfer function. Figure 2shows an example of a transfer function indicating four natural frequencies withinthe frequency range plotted.

3

0 2 4 6 8 10 12 14-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1amplitude

time

pp

t*

t*+T

k

mc

Fig. 3 Exponential decay of the free vibration solution for a SDOF system

2.3 Experimental Determination of the Damping Ratio

For a damped SDOF system, the displacement history can be directly obtainedby solving the governing differential equation of equilibrium. For free vibrations,the governing differential equation is:

mu(t) + cu(t) + ku(t) = 0 (3)

where u(t) = displacement, m = system mass, c = damping coefficient, and k= system stiffness. The superior dot(s) indicate time derivatives of u. For a linearsystem, the solution u(t) can be written as:

u(t) = e−ξωnt[A cos(ωdt) +B sin(ωdt)] (4)

where ξ = damping ratio; ωn = natural frequency; and ωd = damped natural fre-quency; A and B are constants that depend on the initial displacement and velocity,respectively. An estimate of the damping ratio ξ can be obtained by carrying outa free vibration test and using the so-called logarithmic decrement method. Thelogarithmic decrement δ is defined as the logarithm of the ratio between two con-secutive peaks of the free vibration solution (eq. 4), as shown in Fig. 3. Note thateither displacement or acceleration solutions can be used to calculate δ.

The theoretical value of δ can be determined using eq. 4:

δ = lnp1

p2

= lne−ξωnt

∗

e−ξωn(t∗+T )= ξωnT = 2πξ (5)

where t∗ = the time corresponding to peak p1 and T = the period. Solving for ξ:

ξ =δ

2π(6)

4

Using two consecutive peak values of the solution p1 and p2, the damping ratio ξof the SDOF system model can be easily estimated using eq. 6. To reduce errorsassociated with experimental measurement, however, the calculation of ξ shouldmake use of peak measurements made j−1 cycles apart.

ξ =1

(j − 1)2πlnp1

pj(7)

3. Required Equipment and Model Components

The instructional shake table was developed for the University Consortium onInstructional Shake Table (UCIST) project. It supports uniaxial shaking and hasa maximum design payload capacity of 33 lbs. The shake table itself is just onecomponent of the package developed as part of the UCIST project, which includes acomputer-based data acquisition system, a power unit, accelerometers and relevantcables. Hardware and software necessary to carry out this experiment include thefollowing:

- Data acquisition system (Multi board and computer)

- Instructional shake table

- Three accelerometers

- Power unit and cables

- Software: WINCON and MATLAB

- 1-D structure model

- Soil column model

Please refer to the Bench-Top Shake Table Users Guide for a detailed explanationon how to connect the different components and how to operate the shake table.This guide is available on the UCIST web site (http://ucist.cive.wustl.edu/).

The primary goal of this experiment is to demonstrate the potential effects oflocal soil conditions on the dynamic response of a structure during earthquake typeexcitations. Two different model configurations will be tested for the same inputmotions and the results compared. In the first configuration, the test structuremodel will be rigidly connected to the shake table. In the second configuration amodel of the soil will be introduced between the structure model and the shake table.

5

width = 2’

height = 1’

thickness = 8"

foam matting (thickness = 5’’)

3/8" Plexiglas

hinges

Fig. 4 Soil column model

3.1 Structure Model

The test structure provided with the original package is a simple model of atwo-story building. For this experiment, the top story is removed and only theremaining one-story model is used. The base and roof of the model are a 1/2′′ Plex-iglas plates, while the columns are thin steel plates. An accelerometer is attached tothe top Plexiglas plate so that accelerations can be recorded at that level. Since thePlexiglas plates are essentially rigid relative to the column plates, and most of themass of the system is concentrated at the floor levels, the structure can be modeledas a single DOF system defined in terms of the horizontal movement of the roof plate.

3.2 Soil Column Model

Figure 4 shows a schematic representation of the soil system model and its dimen-sions. The soil model is constructed using widely available, inexpensive materials,such as Plexiglas plates and foam rubber matting. Four Plexiglas plates are con-nected with small door hinges to form a rectangular box (with the front and rearsides left open).

The rectangular section of foam should be slightly larger than the overall heightand length of the Plexiglas frame, such that a small amount of prestressing is pro-duced when it is inserted into the frame. The prestressing will provide bearingstresses against the Plexiglas to prevent slippage along the interfaces. Withoutsuch frictional resistance, slippage will occur and the foam element will not deformuniformly in pure shear. The fundamental period of the model can be tuned byadjusting the foam matting thickness or by attaching more or less mass to the topPlexiglas plate.

6

4. Experimental Procedure

4.1 Initialization of the Shake Table and Safety Operations

For a detailed explanation on how to operate and how to connect the differentcomponents of the educational shake table, please refer to the Bench-Top ShakeTable Users Guide and the WINCON software manual that can be downloadedfrom the UCIST web site (http://ucist.cive.wustl.edu/). Is important that youfollow all the safety instructions in order to prevent possible injuries and damage ofthe equipment. In particular remember the following:

• The safety override button on the power supply unit should always remain inthe off position.

• The deadman switch must be depressed to excite the shake table. Press thisbutton and hold it before hitting the Start button on the Wincon server andfor all the duration of the experiment. Remember that the deadman switch isnot an on/off button!

• Turn the power supply off if you turn off or reboot the computer.

After checking all the connections, you can start to prepare the shake table forthe experiment. In order to excite and control the shake table you need Winconand MATLAB software installed in your computer. Start by turning on the powersupply and wait to see that the right and left indicator lights blink. Then turnon your computer and run the boot.exe program located in the UCIST directory:C:/UCIST/boot. This operation will initialize the shake table and after that youcan open the Wincon software by opening a Wincon server window. Another impor-tant step that you have to do before starting any test is the center calibration of thetable. You can do it by running the calibrate.wcp project located in C:/UCIST/Pcfolder. Now your table is centered and you are ready to start the experiments.

4.2 Experiment Overview

To investigate the potential effects that local soil conditions have on the dynamicresponse of the structure, two different configurations are tested and the results com-pared. Figure 5 shows these two configurations. At first the 1-D structure model(configuration A) is attached directly to the shake table. In configuration B, the soilmodel is introduced between the test structure and the shake table. As shown inthe figure, accelerometers are attached to the table mounting plate and roof of thestructure, as well as at the base of the structure when the soil column is present.Each accelerometer should be mounted consistent with the positive direction of theaccelerometer attached to the shake table platform. Having the sign conventionconsistent is particularly important for this experiment, since accelerations of theroof relative to the base are needed for comparing the results for each configuration.Record all results and anticipate the questions given in section 6 at the end of themanual.

7

accel.#3

accel.#2accel.#2

accel.# 1

direction of shaking

accel.# 1Shaketable

1-D StructureModel

1-D StructureModel

Soil Column Model

a) b)

Fig. 5 a) 1-D structure model attached to the table platform and b) 1-D structuremodel fixed on the soil column model

4.3 Configuration A – Structure Model Only

4.3.1 Natural frequency of the structure model

The first step in this experiment consists of determining the dynamic characteris-tics of the 1-D structure model, such as the fundamental frequency and the dampingratio. The fundamental frequency may be determined by running the sweep func-tion and then determining the transfer function of the recorded data, as describedin section 2.2. Begin by installing the 1-D test structure on the mounting plate ofthe shake table as shown in Fig. 5a. Make sure the shake table is centered andthen run the sweep function test. Save the acceleration recorded for the top of themodel (roof level) as an M-file. Now open MATLAB and import the data by typingthe name of the corresponding M-file and then run the function “freqmax1 15”. Re-member to set the appropriate path in MATLAB so that the file can be located. Awindow will open showing a plot of the transfer function in the frequency domain.From the MATLAB command window, read and record the value of the fundamen-tal frequency corresponding to the peak value of the transfer function.

4.3.2 Damping ratio of the structure model

The damping ratio of configuration A may be determined by running a free vi-bration test. Using the same test setup, center the shake table and then run the free

8

vibration function fexp in the WINCON environment. Record the acceleration onthe roof of the model by saving it as an M-file. Open MATLAB command windowand run the file that you have just saved. This will create a vector in MATLABworkspace named PLOT DATA and a plot of this vector versus time. Find thevalues of peaks p1 and pj (separated by j−1 cycles) and use eq. 7 to determine thevalue of the damping ratio ξ.

4.3.3 Earthquake input motion

Now it is time to subject the test structure to a record of a real earthquake. Aftercentering the shake table, run the WINCON El Centro file. Save the accelerationdata measured at the mounting plate and at the roof level. In order to evaluatethe relative acceleration of the roof with respect to the ground surface, the absolutevalue recorded at the ground must be subtracted from the value at the roof level.This may be done using MATLAB. The relative acceleration data will be comparedwith the corresponding values for configuration B.

4.4 Configuration B – Structure on Soil Column Model

4.4.1 Natural frequency of the soil column model

The fundamental frequency of the soil column model may be determined in thesame way you have done for configuration A, as described in section 4.3.1. Attachthe soil column model on the mounting plate of the shake table. Run the sweepfunction on the shake table and save the acceleration recorded on the shake tableplatform and on the top of the soil column model as M-files. Now determine andrecord the fundamental frequency as outlined in section 4.3.1

4.4.2 Damping ratio of the soil column model

As already done for the test structure in section 4.3.2, determine an approximatedamping ratio of the soil column model.

4.4.3 Earthquake input motion – Free-field motion

The free-field motion is defined as the motion of the ground that is not influencedby the presence of a structure. When a structure exists at a site, the motion of thestructure influences the motion of the ground and, of course, the motion of theground influences the motion of the structure. This process of mutual influence iscalled soil-structure- interaction.

After centering the shake table, run the WINCON El Centro file again. Savethe accelerations at the shake table and at the top of the soil model as M-files. Inthe MATLAB environment, run the M-file for acceleration of the shake table andthen calculate the DFT of this data by typing “freqmax1 15” in the MATLAB workwindow. Repeat this using the free-field acceleration.

9

4.4.4 Earthquake input motion test – Structure on uniform soil layer

Install the test structure on the top of the soil model and subject the assembly(configuration B) to the El Centro motion (after checking all the connections andcentering the shake table). For this test you will use all three accelerometers; inparticular you will measure the acceleration on the mounting plate of the shaketable, on the top of the soil model (ground surface) and on the top of the test struc-ture (roof level). Save all these acceleration histories as M-files. Now go back toMATLAB and run the files that you just saved. To compare the test results fromconfiguration A and B, you will need the relative acceleration of the roof with respectto the ground surface. Therefore you need to subtract the absolute acceleration ofthe ground surface to the acceleration recorded at the roof level.

4.5 Comparison of Results

Compare the graphs of the relative accelerations at the roof level for the twoconfigurations. In particular, look at the peak accelerations and the frequency con-tent of the two cases. Compare the free-field acceleration of section 4.4.3 with theabsolute acceleration of the ground surface recorded during the test in section 4.4.4.The difference in peak acceleration and dominant frequencies provides an indicationof the influence the structure has on the soil layer response.

5. Site Period Modification (optional)

To demonstrate the consequences of soil-structure resonance, the period of thesoil model (used for the above work) was tuned to that of the 1-D structure byadjusting the soil model mass and/or stiffness. Here you are asked to adjust theamount of mass so that the soil model period, determined in section 4.4.1, is twiceas large.

By constraining the foam matting to deform uniformly in shear, as indicated inFig. 1c, the soil model is a SDOF system with generalized mass m∗ and generalizedstiffness k∗. For convenience, the single degree of freedom is defined by the horizontaldisplacement at the ground level. Equation 1 no longer applies toward calculatingthe system frequency. Rather, use the fundamental relation

ωn =

√k∗

m∗(8)

to determine the amount of additional mass needed to double the period. Note thatm∗ differs from the total mass of the model, since the mass components (i.e., themasses of the frame elements, the foam, and any additional metal plates) are notlumped at the degree of freedom.

For the structure shown in Fig. 6, the generalized mass can be determined using

m∗ =∫ h

0ρA(y)ψ2dy +

∑i

miψ2i (9)

10

mi

ψ(y)

yi

yh

Fig. 6 Example structure for the determination of m∗

where ρ is the mass density of the material, A(y) is the member cross-section area,ψ(y) is a function describing the mode shape of deformation, and the summationterm accounts for the possibility of having lumped mass, mi, at position yi abovethe base. The ideas expressed by Eq. 9 can be applied to determine m∗ for the soilsystem model, using ψ(y) = y/h and lumping the mass where appropriate. Thequantity ρA(y) is simply the mass per unit length in the y-direction, which can bedetermined from the weight of an individual component of the model.

After adjusting the mass, according to the preceding calculations, retest and an-alyze the soil and soil-structure models as described in sections 4.4 and 4.5. Themain points are: 1) whether the new soil model period is close to the target period(i.e., twice the previous soil model period); and 2) how the modification of the siteperiod affects the acceleration history experienced by the 1-D structure.

6. Questions and Exercises

This laboratory serves as an introduction to the modeling of soil-structure sys-tems and demonstrates the potential effects of site period on the structural response.Before concluding, it must be emphasized that these models are only crude approx-imations of the actual soil and structural systems. The following questions andexercises are not only to reinforce the main points covered above, but also to helpyou question the validity of the models and the significance of the comparisons made.

a. How does the soil model affect the structural response? For the resonant condi-tion, where the site and structural periods are close to one another, one mightexpect a much greater amplification of the structural response. What mitigat-ing factor is introduced when inserting the soil model between the structureand the table platform? The influence of this factor is evident when comparingthe acceleration histories of configurations A and B for the El Centro motion.

b. What is the significance of the comparison made in section 4.5 between the free-field ground surface motion and the ground surface motion with the structureattached?

11

c. If the framing system shown in Fig. 4 is ideally hinged, and therefore providesno lateral resistance by itself, the lateral resistance is due only to the foammatting deforming uniformly in pure shear. Calculate the shear modulus ofthe foam from the results obtained in section 4.4.1 and the derivation of m∗

given in section 5.

d. Continuing from the previous question, what is the effect of increasing the soilcolumn height (assuming a proportional increase in m∗, as well)? Estimatethe natural period for the same setup, but with twice the soil column heightand twice the generalized mass.

e. How is the soil column model employed here different from an actual soilsystem? What factors are present during dynamic excitation of actual soil-structure systems that are not accounted for here?

References

1. Clough, R.W. and Penzien, J., Dynamics of Structures, McGraw-Hill, Inc.,N.Y., 1993.

2. Chopra, A.K., Dynamics of Structures, Prentice Hall, N.J., 1995.

3. Kramer, S.L., Geotechnical Earthquake Engineering, Prentice Hall, N.J., 1996.

4. Scherbaum, F., Basic Concepts in Digital Signal Processing for Seismologists,Springer-Verlag, Berlin, 1994.

5. Brigham, E.O., The Fast Fourier Transform and Its Applications, PrenticeHall, N.J., 1988.

12