Increasing requirements ofindustries and research institutes to analytically results ofinteractionsoil-foundation related systems, reveals the importance of the ...
International Journal ofGeotechnical Earthquake Engineering , 6(1), 1-14, January-June 2015 1
Dynamic Response of Different Types of Shallow Foundation over Sandy Soils to Horizontal Harmonic Loading Taha Ashoori, University of Texas at Arlington, Tehran, Iran Keivan Pakiman, Kayson .Inc., Tehran, Iran
ABSTRACT Increasing requirements ofindustries and research institutes to analytically results ofinteractionsoil-foundation related systems, reveals the importance of the dynamic impedance functions than ever before. The dynamic impedance function relations are presentedfor mass less rigidfoundations which is possible to obtain dynamic response offoundations/or different frequencies and masses accordingly. Jn this study, the dynamic impedance fimctions were investigated using physical modeling tests on sandy soil with.finite thickness soil stratum over bedrock. The tests were carried out inside a steel container of dimensions I x f x0.8m in length, width and height respectively which was filled into container with Babolsar sand by using air - pluviation technique after calibration test with relative density of55.1 percent. The selected foundations were square and circular with same surface area and rectangular with length to width ratio of 2 that were investigated to determine effects ofshape, inertia, embedment ratio, dynamic force amplitude and bedrock on horizontal impedance. Keywords:
Bedrock, Dynamic Stiffness and Damping Coefficient, Horizontal Impedance Function, Physical Modeling, Shallow Foundation
1. INTRODUCTION One of the fundamental problems in dynamic soil-structure interaction is the characterization of the dynamic response of surface foundations resting on a soil medium under time-dependent loads. Despite its importance to various soil dynamics and earthquake applications, a clear understanding of the problem has yet to be established owing to the complexity of real soil behavior and its constitutive modeling, the in-situ and stress-induced heterogeneity in the soil's modulus, and the three-dimensional nature of the underlying wave propagation phenomenon. To provide a basis for improvement, a comprehensive experimental data base that can permit
DOI: 10.4018/IJGEE.20 15010101
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2 International Journal of Geotechnical Earthquake Engineering, 6(1 ), 1-14, January-June 2015
y
a direct identification of key physical parameters for the problem would clearly be valuable. For such purpose, one should note the dynamic field tests by Dobry & Gazetas (1986), who studied vertical, torsional, horizontal and rocking vibrations of circular and rectangular surface foundations. Some information on the effect of foundation embedment and shape can be found in researches of Stokoe & Richart ( 1974). In addition to providing important insights into the complex nature of the subject, these field data have been helpful as a direct check of the applicability of relevant theories to practice. As to the use of the field studies for basic research and scientific generalization, however, such an approach is inherently limited due to the significant expenses and technical difficulties in performing systematic parametric variations and characterization of aspects such as soil properties and site formations in the prototype environment. Scaled modeling on the other hand doesn't present such problems. Owing to the small scale of models, numerous tests with well-defined can be performed at a fraction of the cost offull-scale tests. One of the key steps in the current methods of dynamic analysis of foundation soil system under seismic or machine type loading is to estimate the dynamic impedance functions (spring and dashpot coefficients) associated with rigid but massless foundations. However literature presents experimental results from studying variation of the dynamic impedance function is scanty. Jn scaled modeling simulation of dynamic problems, however, it is well known that one of the major difficulties is the boundary/box effect caused by desirable wave reflections from the finite boundary of the soil model. Without proper treatment, the latter can contaminated the response to the extent that it may become meaningless. In what follows, a series of physical model tests on the vertical dynamic characterization of surface and embedded foundations in circular, rectangular and square shapes on sand will be presented!. The paper includes an analytical synthesis of the results of an extensive parametric study on the effect of foundation contact pressure, footing shape, and footing embedment on the dynamic foundation stiffness as a function of frequency. The subject of vibratory response of foundations bas attracted the attention of many researchers since the classical work. The significant advances during the last three decades in developing the theoretical solution to problems of foundation vibrations resting on a thick homogeneous soil deposit has been achieved (e.g. Lysmer & Ri.chart 1966; Gazetas 1983, 1991).Today2D and JD solutions of finite element and boundary element methods are available for embedded footings in layered soil deposits. Approximate solutions provide relief from the detailed input and execution of analytical and numerical methods and also provide clarity of the problem that rigorous methods do not allow. The approximate methods include the half-space analogs of Richart, Hall, and Woods (1970) and Cone models are likely the most advanced of the approximate methods for application in engineering practice. Wolfand Deeks (2004) provide a detailed description of the implementation of cone models to foundation vibration problems. Wong and Luco ( 1978) have presented a detailed tabulation of numerical values for the impedance functions for a massless, rigid, rectangular foundation perfectly bonded to the surface of a viscoelastic, homogeneous half-space. To explore the influence of foundation embedment, Mita and Luco ( 1989) have presented a detailed tabulation of numerical values for the impedance functions of a massless, rigid, square foundation embedded in and perfectly welded to a viscoelastic, homogeneous half-space. Numerical models using the finite and boundary element method (BEM) are under development for the analysis of wave propagation problems in solids with emphasis on dynamic soil-structure interaction (SSI) due to dynamic excitations. Celebi et al (2006) studied the numerical analysis by using the substructure approach in the frequency domain which was formulated on basis of the BEM based on the fundamental solution for a homogeneous, isotropic, linear-elastic continuum. To further demonstrate in practical applications and to show the solutions of this type of problems to civil engineers, a comprehensive
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International Journal of Geotechnical Earthquake Engineering, 6(1 ), 1-14, January-June 2015 3
parametric analysis and systematic calculations was performed with various controlling parameters to evaluate the dynamic response of the vibrating soil-foundation system.
2. TEST SETUP AND PROGRAM In this investigation, the experiments were conducted in a container with rectangular geometry which was chosen to avoid the usual severe wave-focusing effects caused by cylindrical boundaries. The size of the container at the bottom was fixed as 1.0 m x 1.0 m where it was 0.8 m deep. To realize the non-wave-reflecting characteristics of a semi-infinite medium more closely for container, a thin layer of sawdust with 5 cm thickness was placed and compacted carefully between container and sand layer. To ensure the workability of sawdust and also avoid t.he mixing with sand, a wooden frame covered with nylon was placed in the container. The simulation of bedrock was using 4 rectangular concrete slabs with dimension of 0.4 x0.4 at base and 0.15m in height. In addition, to model the graduate layer change from rock to the sandy soil, a 3 centimeter grout layer was spread out over the stiff concrete slab. The non-existing coarse materials in the transition layer will best simulate the soil layer change in the reality. To construct a homogeneous sample, the method of pluviation through air was employed with satisfactory results. Existing raining devices were calibrated to yield relative density of 55% by controlling the raining height and the flux of the sand. To permit adjustment of the foundation contact pressure (foundation weight), which is a parameter of critical interest, four steel disks with total of I. I kg weight was also fabricated that can be attached to the vibrating assembly. The calibration test conducted with the aid of 9 little boxes to investigate the uniformity of the sand fall in the container. Dynamic excitation system consists of a permanent magnetic shaker, a power amplifier and a signal generator. A BrUel and Kjrer 4814 vibration exciter was used in this study. The unit can provide acceleration, velocity, and displacement over the frequency range of 1-10,000 Hz. The exciter is supported by a stee.1 adjustable frame. The excitation signal was provided by a BrUel and Kjrer portable pulse type 3560 signal analyzer. To monitor the loading and the response, all model footings were instrumented with 2 BrUel and KjrerAS-065m piezoelectric accelerometers in both vertical and horizontal axes, along with a Brilel and Kjrer 8200 load cell. Two other Delta Shear 4384 and Brilel and Kjrer4507piezoelectric accelerometers also placed horizontally in the soil in different soil depths to monitor the wave response beneath the foundation. The instrumentation configuration is shown in Figure 1. The shaker, foundation- frame connections are also shown in Figure 2. The soil tested was a uniformly graded, air dried (moisture less than 1%), medium-fine Babolsar sand. The properties for sand for this loading condition were shear modulus 4700 kN/ m2, unit (dry) weight 17.80kN/m3, Relative density 55%, and Poisson's ratio 0.33 (assumed). Table I shows soil properties of the test. Shear modulus of the sand was obtained based on static stiffness of foundation and equation K, = SGR
,
2-v
[1 + !B..] where R and v are equiva2H
lent radius of model and Poisson's ratio of soil respectively. Magnitude of the average shear strain during the tests of the homogeneous material was on the order of 10-4 • The shear modulus of the soil becomes strain dependent when the assoc.iated shear strain magnitude exceeded I 0-4 • Figure l(b) set up configuration in two layer bed rock, Figure l(a) set up configuration in one layer bed rock and Figure I ( c) system's set up configuration.
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4 International Journal of Geotechnical Earthquake Engineering, 6(1 ), 1-14, January-June 2015
Figure 1. Test geometry and instrumentation 0
0
.
r.
0
Sand
n
0
d
\0
3. MEASURMENT AND DATA PROCCEDING In this study, the dynamic characteristics of surface foundations undergoing small amplitude motion are of primary interest. For such problems, the soil-foundation system can be conveniently characterized in the frequency domain in terms of its frequency response function (FRF). Mathematically, the FRF is defined by:
FRF(w)
=
Y(w)IX(w)
(1)
where X(oo) and Y(oo) are Fourier transforms of the analog input [x(t)] and output [y(t)] signals; respectively. In general, the FRF is a complex-valued function that can be represented in terms of its real and imaginary parts. Usually, the horizontal dynamic force applied to the footing is taken as the input to the system while the horizontal displacement oftlhe foundation is taken as the output. ln this investigation, the total interfacial load at the soil-foundation interface experimentally is being measured and used . So the dynamic foundation characteristic is characterized by this
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International Journal of Geotechnical Earthquake Engineering, 6(1 ), 1-14, January-June 2015 5
Figure 2. Foundation - shaker connection
Table 1. Soil properties
Unified Gradation
SP
G, 2.753
Minimum density (gr/cm 3 )
Maximum density (gr/cm 3)
1.551
1.777
Mass density p
(gr/cm 3 ) 1664
complex-valued dynamic interfacial impedance function commonly referred to as the "massless impedance" function which is defined as the ratio of total interfacial load at the soil-foundation interface to the displacement response as a function frequency. Inverse of the impedance function is called compliance function. Results of experiments in this investigation are shown in the form of impedance function. Subsequently minimums in results represent the maximum values in pertinent compliance function. The use of the term "interfacial compliance" (or impedance) is strongly preferred for the foundation problems (Pak &Guzina 1995). To examine the linearity of the dynamic soil-foundation system in small-amplitude vibration , the compliance function was determined for different forcing levels, which yielded displacement amplitudes ranging from 10-6 xa to 10-5 xa, where a being the foundation radius.
4. ANALYTICAL FRAMEWORK For each particular harmonic excitation with frequency w, the dynamic impedance function is defined as the ratio between the steady-state force and the resulting displacement at the base of the massless foundation :
(2)
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6 International Journal of Geotechnical Earthquake Engineering, 6(1 ), 1-14, January-June 2015
t)=Rh (u) exp {iwt} is the harmonic horizontal force applied at the base of
In which R,. (
t) =
uexp {iwt } is the uniform harmonic displacement of the soil-foundation
the disk, and u (
interface. It is evident that is the total soil reaction against the foundation, it's made up of the Rh normal stresses against the basement plus, in the case of embedded foundations, the shear stresses along the vertical side walls. As dynamic force and dis1Placement are generally out of phase, impedance may also be written in the form:
K" (w) where
=
K" (0) K1i.
1
(
ro) + iK,, ro)
(3)
2 (
K111 ( m) and Kh 2 ( ro) are dimensi.onless frequency functions associated with the uniform
half-space solution and:
Kh = 8GR (1+!~) 2-v 2H
(4)
m = 2n Bf I ~GI p For some appropriate shear modulus
G, Poisson 's Ratio v,
and soil density p.
Kh 1 ( m)
reflects the stiffness and inertia of the supporting soil; its dependence on frequency is attributed solely to the influence which frequency has on inertia, since so.ii properties are essentially frequency independent. The imaginary component reflects the radiation and material damping of the system. The former, being the result of energy dissipation by waves propagating away from the foundation, is frequency dependent; the latter, arising chiefly from the hysteretic cyclic behavior of soil, is practically frequency independent. For horizontal or moment loading contained within a single plane, the horizontal sliding and rocking modes are often coupled, and a two-degree-of-freedom dynamic model of rigid body motion is required. In general, these equations involve four impedances, namely, horizontal , rocking, the coupling ofrocking with horizontal sliding, and the coupling of horizontal sliding with rocking, to describe the horizontal displacement and rotation response of the foundation. Jnitially, this strategy is problematic since only two .independent responses can be measured (e.g., horizontal displacement and rotation), yet four unknown impedances are to be determined. As a remedy, it is possible to assume that the cross-coupling impedances are negligible, thus allowing the horizontal sliding and rocking impedances to be computied from the two measurements (similar to vertical mode discussed above) as presented in Equation 8 and Equation 9:
S h
F
=-s
u
(5)
h
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International Journal of Geotechnical Earthquake Engineering , 6(1 ), 1-14, January-June 2Q.15 7
S =Ms I'
(6)
