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Dynamic Behavior of Simple Soil-Structure Systems Department of Civil and Environmental Engineering University of California at Davis

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

Dynamic Behavior of Simple Soil-Structure Systems A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES

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

Developed by: Stefano Berton ([email protected]) Tara C. Hutchinson ([email protected]) Dr. John E. Bolander ([email protected]) University of California at Davis

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

Dynamic Behavior of Simple Soil-Structure Systems Department of Civil and Environmental Engineering University of California at Davis

Objective: This experiment illustrates the influence local geology and soil conditions can have on the intensity of earthquake induced ground shaking and structural vibration. A simple soil model will be constructed and connected in series with a 1-D structure model. Both the soil-structure system and the structure model alone are subjected to identical base excitations via a bench-scale shaking table. The experiment serves as an introduction to the modeling of soil-structure systems and demonstrates 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 ground surface has been observed and studied since the beginning of the nineteenth-century. However, only recently engineers have included local site effects in earthquakeresistance design procedures. Nowadays, the development of site-specific ground motions and response spectra represents one of the most challenging aspects of earthquake engineering. Important characteristics of the ground motion such as peak acceleration, frequency content, etc., are affected by local site conditions. During an earthquake, energy is released local to the fault and seismic waves are produced. These waves propagate from the hypocenter in all directions and during their travel may be reflected and refracted by the layered earth composition. The local soil conditions then affect the ground motion by increasing or reducing the amplitude of frequency components present in the bedrock motion. An important aspect of the dynamic behavior of a single or multi-degree of freedom system subjected to external excitation is the resonance phenomenon. Resonance occurs when a frequency component of the excitation is close to the fundamental frequency of a SDOF system or one of the natural frequencies of a MDOF system. Resonance is associated with an amplification of deformations that can lead to unforeseen consequences, including structural collapse. For a soil-structure system, resonance will occur when the site period is close to the fundamental period of the structure. A classical example of local soil effect on structural response is the 1985 Mexico City earthquake. This magnitude 8.1 earthquake caused only moderate damage to structures near the epicenter but caused extensive damage 350 km away from the epicenter in Mexico City. Furthermore, damage to structures in Mexico City was concentrated 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 reinforced concrete buildings) suffered the most damage.

2. Basic Theory: Dynamics of Soil-Structure Systems This laboratory makes use of analytical methods and several idealizations to characterize the soil and soil-structure models. These include the shear column representation of the soil mass as a generalized SDOF system, a method to determine the 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 system during earthquake-type excitations. 2.1 Shear Beam Approach for Ground Response Analysis A rigorous study of the response of a particular site involves sophisticated analyses using wave propagation theory. In general, the soil may be approximated as a multilayered medium, where each layer is given appropriate characteristics estimated from site exploration. Dynamic soil properties, such as damping ratios and shear wave velocities, may be estimated for each layer. Consider two sites, each composed of uniform layers of soil resting on rigid rock. The two sites have similar geometry and characteristics, except that one site is significantly stiffer than the other. Using wave propagation 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 shear beam approach. In this case, an isolated column of soil is modeled as an equivalent beam that can deform only in the shear mode (i.e. without contributions from bending), as shown in Fig. 1. If the soil is uniform, the natural frequencies of the system can be determined using π ωn = (2n − 1) 2



GA mh2

(1)

where n is the mode shape number; G is the shear modulus; A is the column cross section 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 soil column 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 be experimentally determined using the bench-scale shake table and Discrete Fourier Transform analysis. A periodic function f (t) can be expressed in terms of a Fourier series, i.e., as an infinite sum of sine and cosine terms. The same concept can be extended for non-periodic functions since, in the limit, the function can be regarded as periodic with infinite period. In this process a new function F (ω), known as the 2

soil column ground surface

h

uniform soil

k* m*

bedrock

a)

b)

c)

d)

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 Fig. 1

Fourier amplitude

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peak corresponding to the first natural frequency

10

5

f1

0 0

f2 5

f3 10

f4 15

20

25

frequency (Hz) Fig. 2

Transfer function

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

1 ∞ f (t)eiωt dt 2π −∞

(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) is applicable. The most efficient and fast algorithm to evaluate the DFT of a vector is the Fast Fourier Transform (FFT). This algorithm is very efficient and is implemented in many signal processing routines, including some of the routines available in MATLAB. Using the FFT allows us to process a finite number of data points and plot the Fourier amplitude spectra in the frequency domain. For example, the Fourier amplitude spectra of an acceleration time history, obtained via FFT, represents the values of the coefficient of the Fourier series versus frequency. Frequencies corresponding to peaks in the Fourier amplitude spectra are the dominant natural frequencies of the model. When the DFT of the initial data is normalized with the DFT of the input acceleration, the result is called the transfer function. Figure 2 shows an example of a transfer function indicating four natural frequencies within the frequency range plotted.

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1

amplitude 0.8

m

c

0.6

p

k

0.4

p

0.2 0 -0.2

t*+T

-0.4

t* -0.6

time -0.8

Fig. 3

0

2

4

6

8

10

12

14

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 obtained by solving the governing differential equation of equilibrium. For free vibrations, the governing differential equation is: m¨ u(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 linear system, the solution u(t) can be written as: u(t) = e−ξωn t [A cos(ωd t) + B sin(ωd t)]

(4)

where ξ = damping ratio; ωn = natural frequency; and ωd = damped natural frequency; 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 out a free vibration test and using the so-called logarithmic decrement method. The logarithmic decrement δ is defined as the logarithm of the ratio between two consecutive peaks of the free vibration solution (eq. 4), as shown in Fig. 3. Note that either displacement or acceleration solutions can be used to calculate δ. The theoretical value of δ can be determined using eq. 4: ∗

p1 e−ξωn t δ = ln = ln −ξωn (t∗ +T ) = ξωn T = 2πξ p2 e

(5)

where t∗ = the time corresponding to peak p1 and T = the period. Solving for ξ: ξ= 4

δ 2π

(6)

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

1 p1 ln (j − 1)2π pj

(7)

3. Required Equipment and Model Components The instructional shake table was developed for the University Consortium on Instructional Shake Table (UCIST) project. It supports uniaxial shaking and has a maximum design payload capacity of 33 lbs. The shake table itself is just one component of the package developed as part of the UCIST project, which includes a computer-based data acquisition system, a power unit, accelerometers and relevant cables. Hardware and software necessary to carry out this experiment include the following: - 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 explanation on 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 of local soil conditions on the dynamic response of a structure during earthquake type excitations. Two different model configurations will be tested for the same input motions and the results compared. In the first configuration, the test structure model will be rigidly connected to the shake table. In the second configuration a model of the soil will be introduced between the structure model and the shake table.

5

’

h=2

hinges

widt

thickness = 8"

3/8" Plexiglas

height = 1’

foam matting (thickness = 5’’)

Fig. 4

Soil column model

3.1 Structure Model The test structure provided with the original package is a simple model of a two-story building. For this experiment, the top story is removed and only the remaining one-story model is used. The base and roof of the model are a 1/2 Plexiglas plates, while the columns are thin steel plates. An accelerometer is attached to the top Plexiglas plate so that accelerations can be recorded at that level. Since the Plexiglas plates are essentially rigid relative to the column plates, and most of the mass of the system is concentrated at the floor levels, the structure can be modeled as 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 dimensions. The soil model is constructed using widely available, inexpensive materials, such as Plexiglas plates and foam rubber matting. Four Plexiglas plates are connected with small door hinges to form a rectangular box (with the front and rear sides left open). The rectangular section of foam should be slightly larger than the overall height and length of the Plexiglas frame, such that a small amount of prestressing is produced when it is inserted into the frame. The prestressing will provide bearing stresses against the Plexiglas to prevent slippage along the interfaces. Without such frictional resistance, slippage will occur and the foam element will not deform uniformly in pure shear. The fundamental period of the model can be tuned by adjusting the foam matting thickness or by attaching more or less mass to the top Plexiglas plate.

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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 different components of the educational shake table, please refer to the Bench-Top Shake Table Users Guide and the WINCON software manual that can be downloaded from the UCIST web site (http://ucist.cive.wustl.edu/). Is important that you follow all the safety instructions in order to prevent possible injuries and damage of the equipment. In particular remember the following: • The safety override button on the power supply unit should always remain in the off position. • The deadman switch must be depressed to excite the shake table. Press this button and hold it before hitting the Start button on the Wincon server and for all the duration of the experiment. Remember that the deadman switch is not 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 for the experiment. In order to excite and control the shake table you need Wincon and MATLAB software installed in your computer. Start by turning on the power supply and wait to see that the right and left indicator lights blink. Then turn on 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 you can open the Wincon software by opening a Wincon server window. Another important step that you have to do before starting any test is the center calibration of the table. You can do it by running the calibrate.wcp project located in C:/UCIST/Pc folder. 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 dynamic response of the structure, two different configurations are tested and the results compared. 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 soil model is introduced between the test structure and the shake table. As shown in the figure, accelerometers are attached to the table mounting plate and roof of the structure, 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 the accelerometer attached to the shake table platform. Having the sign convention consistent is particularly important for this experiment, since accelerations of the roof 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 the manual.

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1-D Structure Model accel.#3 1-D Structure Model

accel.#2 accel.#2

Shake table

Soil Column Model

accel.# 1 accel.# 1

direction of shaking

a)

b)

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

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 characteristics of the 1-D structure model, such as the fundamental frequency and the damping ratio. The fundamental frequency may be determined by running the sweep function and then determining the transfer function of the recorded data, as described in section 2.2. Begin by installing the 1-D test structure on the mounting plate of the shake table as shown in Fig. 5a. Make sure the shake table is centered and then run the sweep function test. Save the acceleration recorded for the top of the model (roof level) as an M-file. Now open MATLAB and import the data by typing the name of the corresponding M-file and then run the function “freqmax1 15”. Remember to set the appropriate path in MATLAB so that the file can be located. A window 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 fundamental 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 vibration test. Using the same test setup, center the shake table and then run the free

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vibration function fexp in the WINCON environment. Record the acceleration on the roof of the model by saving it as an M-file. Open MATLAB command window and run the file that you have just saved. This will create a vector in MATLAB workspace named PLOT DATA and a plot of this vector versus time. Find the values of peaks p1 and pj (separated by j −1 cycles) and use eq. 7 to determine the value 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. After centering the shake table, run the WINCON El Centro file. Save the acceleration data measured at the mounting plate and at the roof level. In order to evaluate the relative acceleration of the roof with respect to the ground surface, the absolute value 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 compared with 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 the same way you have done for configuration A, as described in section 4.3.1. Attach the soil column model on the mounting plate of the shake table. Run the sweep function on the shake table and save the acceleration recorded on the shake table platform and on the top of the soil column model as M-files. Now determine and record 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 approximate damping 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 influenced by the presence of a structure. When a structure exists at a site, the motion of the structure influences the motion of the ground and, of course, the motion of the ground influences the motion of the structure. This process of mutual influence is called soil-structure- interaction. After centering the shake table, run the WINCON El Centro file again. Save the accelerations at the shake table and at the top of the soil model as M-files. In the MATLAB environment, run the M-file for acceleration of the shake table and then calculate the DFT of this data by typing “freqmax1 15” in the MATLAB work window. Repeat this using the free-field acceleration.

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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 and centering the shake table). For this test you will use all three accelerometers; in particular you will measure the acceleration on the mounting plate of the shake table, on the top of the soil model (ground surface) and on the top of the test structure (roof level). Save all these acceleration histories as M-files. Now go back to MATLAB and run the files that you just saved. To compare the test results from configuration A and B, you will need the relative acceleration of the roof with respect to the ground surface. Therefore you need to subtract the absolute acceleration of the 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 two configurations. In particular, look at the peak accelerations and the frequency content of the two cases. Compare the free-field acceleration of section 4.4.3 with the absolute acceleration of the ground surface recorded during the test in section 4.4.4. The difference in peak acceleration and dominant frequencies provides an indication of 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 the soil model (used for the above work) was tuned to that of the 1-D structure by adjusting the soil model mass and/or stiffness. Here you are asked to adjust the amount of mass so that the soil model period, determined in section 4.4.1, is twice as large. By constraining the foam matting to deform uniformly in shear, as indicated in Fig. 1c, the soil model is a SDOF system with generalized mass m∗ and generalized stiffness k ∗ . For convenience, the single degree of freedom is defined by the horizontal displacement at the ground level. Equation 1 no longer applies toward calculating the system frequency. Rather, use the fundamental relation 

ωn =

k∗ m∗

(8)

to determine the amount of additional mass needed to double the period. Note that m∗ differs from the total mass of the model, since the mass components (i.e., the masses of the frame elements, the foam, and any additional metal plates) are not lumped at the degree of freedom. For the structure shown in Fig. 6, the generalized mass can be determined using m∗ =

 h

ρA(y)ψ 2 dy +

0

 i

10

mi ψi2

(9)

ψ(y)

mi h

y yi

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 summation term accounts for the possibility of having lumped mass, mi , at position yi above the base. The ideas expressed by Eq. 9 can be applied to determine m∗ for the soil system model, using ψ(y) = y/h and lumping the mass where appropriate. The quantity ρA(y) is simply the mass per unit length in the y-direction, which can be determined from the weight of an individual component of the model. After adjusting the mass, according to the preceding calculations, retest and analyze the soil and soil-structure models as described in sections 4.4 and 4.5. The main 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 site period 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 systems and demonstrates the potential effects of site period on the structural response. Before concluding, it must be emphasized that these models are only crude approximations of the actual soil and structural systems. The following questions and exercises are not only to reinforce the main points covered above, but also to help you 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 condition, where the site and structural periods are close to one another, one might expect a much greater amplification of the structural response. What mitigating factor is introduced when inserting the soil model between the structure and the table platform? The influence of this factor is evident when comparing the 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 freefield ground surface motion and the ground surface motion with the structure attached? 11

c. If the framing system shown in Fig. 4 is ideally hinged, and therefore provides no lateral resistance by itself, the lateral resistance is due only to the foam matting deforming uniformly in pure shear. Calculate the shear modulus of the 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 soil column height (assuming a proportional increase in m∗ , as well)? Estimate the natural period for the same setup, but with twice the soil column height and twice the generalized mass. e. How is the soil column model employed here different from an actual soil system? What factors are present during dynamic excitation of actual soilstructure 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, Prentice Hall, N.J., 1988.

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