Soil and Foundation Dynamics

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  • SOIL AND FOUNDATION DYNAMICS

    SOIL-STRUCTURE INTERACTION

    FOUNDATIONS VIBRATIONS

    Gnther Schmid Andrej Tosecky

    Ruhr Universtity Bochum

    Department of Civil Engineering

    [email protected]

    Lecture for the Master Course Earthquake Engineering at IZIIS

    University SS. Cyril and Methodius Skopje

    Supported by Stability Pact for South Eastern Europe and German Academic Exchange Service (DAAD)

    May 2003

    Acknowledgement: The authors are grateful to Prof. J.J. Sieffert, member of INRIA Strasbourg who has contributed in a large extent to this script.

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    1. Introduction Initially intended for the calculation of the vibrations of the massive foundations of heavy machines, the analyses of dynamic soil-structure interaction have also been long used for seismic calculations. Whereas in the first case the machine (or the rail or road traffic) is in general the source of the vibrations, in the second case the soil directly provides the loads (Fig.1.1). In both cases however, the objectives are identical, i.e. to evaluate the movements of the foundation under the action of external loads, and consequently anticipate the displacements of the machine or of the structure keeping in mind both the characteristics of the foundation and the properties of the soil.

    Fig. 1.1. General applications of dynamic soil-structure interaction The purpose of this work is to give an introduction to the use of impedance functions for the analysis of dynamic soil-structure systems. Impedance functions may be obtained through numerical methods as the Thin Layer Method or the Boundary Element Method. For special cases they can be found in the literature and provide the user with an adequate aid in many cases (Sieffert et al., 1992).

    Fig. 1.2. Equivalence of soil and spring-dashpot system

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    Of course, the dynamic stiffness is a function of : - the soil characteristics: - shear modulus, - Poissons ratio, - density, - internal damping, - boundary conditions, - - the foundation characteristics: - shape, - embedment,

    - stiffness - - the frequency of vibrations. It is very important to know that the impedance functions of the soil are always given for the for mass-less foundations or in other words for the interface of the structure and the soil. Our purpose is not to develop here the theoretical aspects in order to establish the equations or the values of the impedance functions, but only to present the basic uses of these functions. We consider first as the most simple case a rigid foundation block resting on the elastic or viscous-elastic soil. 2. GENERAL DESCRIPTION OF IMPEDANCE FUNCTIONS 2.1. General definition of the impedance functions (dynamic stiffness) Using complex notation, we consider a general visco-elastic system subjected to a harmonic force (or to a moment) P(t), with the resulting harmonic response (displacement or rotation) u(t). By definition, the impedance K of the system is the relation between the load P and the response u. Generally, the load, the impedance and the response are complex quantities. The relation among impedance, displacement response and applied load is:

    =K u P (2.1) The value K is also called dynamic stiffness and is generally frequency dependent. The impedance can be easily illustrated considering a single-degree-of-freedom system. 2.1.1. Single-degree-of-freedom system with mass A single-degree-of-freedom system comprises a mass M, a spring with stiffness K, and a dashpot with viscous damping C (Fig.2.1).

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    K C

    P(t) = P e i t

    u(t) = u e i t

    Fig. 2.1. SDOF system The equation of motion of the mass is :

    ( )M Cu Ku P t+ + =& (2.2) We assume that the external load is represented by a complex force with amplitude P and circular frequency :

    it(t)= eP P (2.3) Consequently the displacement u(t) can be written as:

    ( ) i tu t ue = (2.4) Note, that the entire time-variance is expressed by the function i te and that P and u generally are complex and depend on . After substitution of the equations (2.3) and (2.4) in the equation (2.2), we obtain

    ( )2-K M iC u P + = (2.5) In applying the definition (2.1), the impedance function of this single-degree-of-freedom system is obtained as :

    2( - )K M iC = +K (2.6)

    This complex impedance, which depends on frequency, can also be written in a more general way : R I( ) ( ) ( )i = +K K K (2.7) with:

    (2.8)

    Figure 2.2 shows the evolution of the real and imaginary term of the impedance as a functions of the frequency.

    M

    R 2KK I

    MC

    = =K

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    Fig. 2.2. Impedance functions of a SDOF system with mass We define the circular eigenfrequency of the undamped SDOF system e, the critical damping Ccrit, the damping ratio and the frequency ratio the relation between excitation frequency and eigenfrequency e.

    (2.9) Using these definitions, the impedance may be written as the static stiffness multiplied by a dimension-less impedance function:

    ( )2( ) 1 2K i = + K (2.10)

    2.2. Mass-less single-degree-of-freedom system (Voigts model) In case of a mass-less single-degree-of-freedom system, the notions of resonance frequency and critical damping are rendered meaningless.

    K C

    P(t) = P e i t

    u(t) = u ei t

    Fig. 2.3. Mass-less SDOF system; Voigts model

    The impedance is thus reduced to : K iC= +K (2.11)

    , 2

    ,

    crit e

    crit e

    eK C MM

    CC

    = =

    = =

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    or else R K=K I C=K (2.12) Writing equation (2.11) as

    1 CK iK = + K

    we may define the damping factor as:

    tanCK = = (2.13)

    where the term tan CK

    = is obtained from the complex representation of K (see Fig. 2.4) Whence ( ) ( )1 1 tanK i K i = + = +K (2.14)

    Fig. 2.4. Impedance representation in complex plane is called the damping factor (usually given in %), is the loss angle. Figure 2.5 represents the evolution of the impedance functions of this particular single-degree-of-freedom system, i.e. of the Voigts model. In both cases (SDOF with or without mass), the imaginary part of the impedance which is related to damping is proportional to the frequency. Concerning the real part, it is the inertial effect of the mass which renders this term a function of the frequency.

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    Fig. 2.5. Impedance functions of a mass-less SDOF system, Voigts model 2.3. General definition of compliance function (dynamic flexibility) By definition, the compliance function is the inverse of the impedance function. 1=F K (2.15)

    For a single-degree-of-freedom system with mass, the compliance function is written as:

    ( ) ( )22 221 1 1 2( )

    ( ) 1 2

    iK M i C K

    = = + +

    F (2.16)

    The compliance is also, of course, complex and a function of the frequency. The compliance function is also called transfer function (it transfer the input (load) to the output (displacement)). As has been done for the impedance, the compliance function can be written in a more general way : ( ) ( ) ( )IF F FR i = + (2.17) Fig. 2.6. a) and b) presents the real and the imaginary part of the compliance function multiplied by K, i.e. the real and the imaginary part of the compliance function in dimension-less form. As the modulus of the compliance function is related to the amplification factor of the displacement (see chapter 2.4) the compliance function is also called displacement function. As a consequence of the fact, that the compliance is the inverse of the impedance, there exist certain equations linking impedance functions K R and K I to the compliance functions F R and F I. In the case of single-degree-of-freedom system, the relations are easily obtained as follows :

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    ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

    R RR R

    R I R I

    I II I

    R I R I

    = =+ += =+ +

    2 2 2 2

    2 2 2 2

    K FF K

    K K F F

    - K - FF K

    K K F F

    (2.18)

    a)KF R

    = 0.02

    0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3765432101234567

    = 0.05 = 0.1

    = 0.5 = 0.7

    b)

    KF I

    0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 31312111098765432101

    = 0.02

    = 0.05

    = 0.1

    = 0.5

    = 0.7

    Fig. 2.6 a) Real and b) Imaginary part of compliance function of a SDOF system with mass

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    2.4 Solution of the equation of motion From equation (2.1) we obtain:

    R I Pu u iu P= + = = FK

    (2.19)

    ( ) ( )2

    2 22

    1 2

    1 2

    P iuK

    =

    + (2.20)

    Usually we write the solution as modulus |u| and phase angle . Modulus:

    ( ) ( ) ( )1 22 221 ,

    1 2

    P Pu P D

    K K

    = = = +

    F (2.21)

    where |P| is the amplitude of the loading given in equation (2.3). By ust we denote the static displacement and D(,) is the dynamic magnification factor. From the equation (2.20) and (2.10) we have:

    phase angle: 22tan

    1

    I I

    R Ruu

    = = KK

    (2.22)

    phase lag: 22tan

    1

    I I

    R Ruu

    = = = KK

    (2.23)

    The phase lag is shown in Fig. 2.7 for various values of damping ratio.

    0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 30

    10203040506070

    8090

    100110120130140150160170180

    phase lag [deg]

    = 0

    = 0.707 = 2.0

    = 4.0

    = 0.15

    = 0.25 = 0.35

    = 0.50

    = 1.0

    Fig. 2.7 Phase lag (with respect to the load) vs. frequency ratio

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    2.4.1 |P| independent of If |P| is independent on (the so-called constant or simple excitation) we define by

    st

    Pu

    K= the static displacement of the system due to amplitude |P| of the time-varying

    force ( ) i tP t P e = . In this case is the dynamic magnification factor given as:

    ( ) ( ) 1 22 221( , )

    1 2st

    uD

    u

    = = +

    (2.24)

    One can show, that the maximal amplitude (resonance) does occur at the frequency ratio

    21 2res = . If 1/ 2 < the dynamic magnification factor at this frequency ratio yields:

    2

    12 1

    resD = (2.25)

    0 0.5 1 1.5 2 2.5 30

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    = 0.15

    = 0.25

    = 0.35

    = 0.5

    = 0.707 = 1 = 2

    = 4

    = 0

    D(,)

    Fig. 2.8 Dynamic magnification factor constant harmonic excitation

    2.4.2 |P| dependent on Dynamic excitation caused by rotation of an unbalanced mass m at circular frequency is called quadratic excitation. For this kind of dynamic excitation the amplitude of the resulting force becomes quadraticaly dependent on the excitation circular frequency

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    and linearly dependent on the unbalanced mass m and its distance r from the center of the rotation: 2( ) i t i tP t mr e P e = = (2.26) with the amplitude 2P mr= (2.27)

    Usually m

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    The resonance dynamic magnification factor has the same form as in the case of constant excitation:

    2

    12 1

    resD = (2.31)

    0 0.5 1 1.5 2 2.5 30

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    = 0.15

    = 0.25

    = 0.35

    = 0.5

    = 0.707 = 1

    = 2 = 4

    = 0

    D(,)

    Fig. 2.10 Dynamic magnification factor quadratic harmonic excitation

    2.5 Damping For a SDOF system with mass, the damping ratio , as already defined before, is given, by

    2 2c e

    C C CC M M K

    = = = (2.32) Equation (2.32) clearly states that this definition of is not possible for a mass-less SDOF system for which the natural frequency does not exist. We assume without loss of generality that the external load is given in form :

    ( ) cosP t P t= (2.33)

    The displacement response function in term of its modulus |u| and phase lag is than as follows : ( ) cos( - )u t u t= (2.34)

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    It is known that the area of the loop (see Figure 2.11) on the force-displacement diagram (usually recorded by cyclic loading material testing) is the dissipated energy per cycle, W, and the area of the triangle the maximum elastic energy stored during the same cycle, W. One has:

    2

    2 2

    W C u

    W K u

    == (2.35)

    u

    P

    u (t)

    P(t)

    W

    W

    Fig. 2.11 Load - displacement loop Consequently we can use W to define a new damping measure. We introduce the dimension-less damping factor :

    1 W

    2 W

    = (2.36)

    which gives for a single-degree-of-freedom system with viscous damping

    CK = (2.37)

    Since this factor does not depend on the mass of the system, it can be used as a damping measure for a mass-less system. Comparing the damping ratio and the damping factor we obtain for a system with mass

    (2.38) We see that for viscous damping the factor is proportional to the frequency ratio

    e

    = . It also may be seen that 2 = if the excitation frequency is the eigenfrequency (i.e.,

    1e= = ).

    2

    2 2ee

    MCK M

    = = =

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    The above definition of the damping coefficient defined in the equation (2.36) leads for Voigts model to

    ( ) ( )1 1 1 2K CK i K i K iK

    = + = + = + (2.39)

    A great deal of cyclic material tests has proved, that the energy dissipated per cycle is essentially independent of frequency. Therefore it is desirable to remove the frequency dependency as it is presented in the formulation (2.39) and formulate the damping being frequency-independent instead, leading to the so-called hysteretic damping. In this formulation is taken to be a constant value often having the form = 2 such that for a spring-damper element without mass (Voigts model) one gets

    ( )1 2K i= + K (2.40) Or more general

    ( ) 11 1 2 with2

    I IR I R R

    R Ri i i = + = + = + =

    K KK K K K K

    K K (2.41)

    Note: a) The name hysteretic damping is not adequately chosen because every form of

    damping produces a hysteretic loops on the load-displacement diagram recorded by cyclic loading.

    b) Making the damping independent of frequency violates the laws of mechanics because for = 0 the imaginary part of the impedance does not vanish. This results in a non-causal behavior of the response. But nevertheless, this model of damping is widely used in practice.

    3. SOIL-FOUNDATION INTERACTION; MULTI-DEGREE-OF-FREEDOM- SYSTEM 3.1 General Remarks We consider a rigid block resting on the soil under harmonic excitation. The 6 DOFs of the block, 3 translations and 3 rotations, may be referred to its center of gravity G or to the center of the soil-structure interface 0. The mass of the block is M. Mass moments of inertia with respect to the x, y and z axis are Ix, Iy and Iz, respectively. The equation of motion for the reference point is

    K u = P (3.1) Note: K is a symmetric matrix.

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    Fig. 4.1. Rigid block on the soil

    u is the displacement vector (6,1) and P is loading vector corresponding to the 6 rigid body DOFs (u and P are complex quantities). K is the impedance matrix defined as

    F = 2K K M (3.2)

    where FK is the impedance matrix of the mass-less foundation (therefore index F) and M is the mass matrix of the block related to the chosen point of reference. The solution to the equation of motion is

    = =1K Fu P P (3.3) with amplitudes |ui| and phase angles i

    ( ) ( ) , tan i=1,2,..6IR I ii i i i Ri

    uu u uu

    = + =2 2 (3.3) If the block-soil system has two planes of symmetry the vertical and torsion motion are decoupled. Only swaying and rocking are coupled in each plane of symmetry. The impedance matrix K is usually given with respect to the center of the lower soil-foundation interface 0 whereas the mass matrix M is given as a diagonal matrix with respect to the center of gravity of the rigid block G. For surface foundations the coupling between swaying and rocking may be neglected. It is obvious that the equation of motion has to be established with respect to the one selected reference point. The transformations of the equation of motion from G to 0 and from 0 to G is determined by the kinematic constrained equations

    0

    0 0

    oG G

    G G

    ==

    u a uu a u

    (3.4)

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    where a is constructed with the kinematic relations, that the vertical translation and the 3 rotations are the same for G and 0 and

    ,0 ,

    , ,0

    h h G r

    h G h r

    u u hu u h

    = += (3.5)

    The transformation yields

    0 0 0 0 0

    0 0 0 0 0

    T TG G G G GT T

    G G G G G

    = == =

    K K

    K K

    a a P a Pa a P a P

    (3.6)

    3.2 Vertical Displacements

    2 2

    1 v vv v v F F

    v v v

    P Pu P PM K M i C = = = = +vFK K (3.7)

    3.3 Torsional Rotation

    2 2

    1 t tt t t v F F

    t t t t tI K I i C = = = = +t

    M MM F M

    K K (3.8)

    3.4 Horizontal Displacement and Rotation (motion in x-z plane) Reference point 0

    0 0 00 0

    10 1

    hxhxG

    ry

    Puh

    = = = ryMa u P

    (3.9)

    0 0 0=K u P or , ,, , 0 0 0

    hx hx hx ry h hx

    ry hx ry ry ry ry

    u P

    = K KK K M

    (3.10a) (3.10b)

    with

    2 20 0 0 0 0F F T

    G G G = = 0K K KM a M a (3.11)

    , ,0 0 2

    , , 00

    00

    F Fhx hx hx ryF

    GF Fryry hx ry ry rG

    M M hMI hM h M I

    = = = + K

    K KK K

    M M (3.12)

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    Explicitly we write for the equation (3.10)

    ( )

    2, ,0 , ,

    2, ,0 , ,

    2 2, ,0 , ,

    ,0

    ,0 ,

    hx hx hx hx hx hx

    hx ry hx ry hx ry

    ry ry ry ry r ry ry

    hx hG

    ry hG ry G

    K M i C

    K hM i C

    K h M I i C

    P Ph P

    = += += + +

    == +

    K

    K

    K

    M M

    (3.13)

    Reference point G

    0

    10 1

    hxhxG G G

    ry G G

    Puh

    = = = ryMa u P (3.14)

    G G G=K u P or , ,, ,

    hx hx hx ry hx hx

    ry hx ry ry ry ryG G G

    u P

    = K KK K M

    (3.15a), (3.15b)

    with

    20

    T FG oG oG Ga a = K K M (3.16)

    , , ,2

    , , , , ,

    02 0

    F F Fhx hx hx ry hx hxF

    G GF F F F Fry hx hx hx ry ry hx ry hx hx r GG

    h Mh h h I

    + = = + + + K

    K K KK K K K K

    M (3.17)

    More explicitly equation (3.15):

    2, , ,

    , , ,

    2 2, , , , ,

    ,

    , ,

    2

    Fhx hx G hx hx

    Fhx ry G hx ry

    F F Fry ry G ry ry hx ry hx hx r

    hx G hG

    ry G hG ry G

    M

    h h IP P

    P

    = == + +

    == +

    K K

    K K

    K K K K

    M M

    (3.18)

    where

    , ; , ; , ; ,Fi i iK i C where i hx hx hx ry ry hx ry ry= + =K Finally we get for the complex displacement for the point of reference:

    ( ) ( )2 2tan tan

    R Ihx hx hx

    R Ihx hx hx

    I Ihx hx

    hx hxR Rhx hx

    u u u

    u u u

    u uu u

    = += +

    = =

    (3.19)

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    and

    ( ) ( )2 2tan tan

    R Iry ry ry

    R Iry ry ry

    I Iry ry

    ry ryR Rry ry

    = += +

    = =

    (3.20)

    The time harmonic motion is:

    ( ) ( )( ) i ii t i ti thx hx hx hxu t u e u e u e + = = = (3.21) and ( ) ( )( ) i ii t i ti try ry ry ryt e e e + = = = (3.22)

    4 USE OF IMPEDANCE FUNCTIONS 4.1 History of impedance functions We can appreciate the interest of scientists and designers through three relatively recent publications on the State of the Art : WHITMAN et al (1967), McNEIL (1969) and GAZETAS (1983). Concurrently, five works on Soil Dynamics have been published respectively by : RICHART et al (1970), DAS (1983), PECKER (1984), HAUPT (1986) and SIEFFERT et al. (1992). Without going into an exhaustive and detailed account of the diverse approaches and methods developed since the beginning of the century, several points of reference can be cited which focus on surface footings : In 1904, LAMB studies the vibrations of a linear elastic half-space due to a harmonic load acting on a point. This, in fact, dealt with the generalization of Boussinesq's problem in dynamics. In 1936, REISSNER analyses the response to a vertical harmonic excitation of a plate placed at the surface of a homogeneous elastic half-space. Credit must be given to him as having been the first to cast light on an aspect which today seems obvious, namely the existence of energy dissipated by radiation. The footing vibrations give rise to volume waves and surface waves whose energy contents is noteworthy. In a half-space, these waves propagate indefinitely, and so do not, in any way, return the energy they contain. There is consequently dissipation of energy and everything occurs as if the medium induced damping although it is supposed to be elastic, linear and non-dissipative. From 1953 to 1956, SUNG, QUILAN, ARNOLD et al and BYCROFT referred to, clarified and generalized the work of REISSNER on movements corresponding to the six degrees of freedom of the footing.

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    Between 1962 and 1967, whereas AWOJOBI et al and ELORDUY et al. were perfecting the proceeding methods, HSIEH and especially LYSMER were introducing for the first time the idea that soil - footing behavior in vertical displacement can be represented by a single-degree-of-freedom system with stiffness and damping as constants independent of frequency (lumped parameters). This simplified approach commonly designated as Lysmer's analogy, has been extended to all movements by RICHART and WHITMAN. Fictitious masses are used to allow an easier adjustment of the resonance frequencies. The end of the 60's and the beginning of the 70's brought about the perfecting of resolution methods of soil-structure interaction due to an improvement in the means of assessment. One can say that these results are almost systematically presented in the form of two frequency dependant functions, the first being the real, the second the imaginary part of the complex dynamic stiffness. These functions are also referred to as impedance functions. The use of a possible additional fictitious mass whose role was to correct errors based on a consideration of constant functions therefore becomes needless. This method implicitly comes down to replacing the mass-less foundation - soil system by a spring and a dashpot in which the characteristics depend on the frequency. 4.2 Use of complete impedance functions The more recent publications in the literature give the impedance functions in dimensionless forms : 0( )

    R Ia i= +k k k (4.1) or the compliance :

    ( )IRf a i= +0 f f (4.2)

    In both cases, a0 is the dimension-less circular frequency defined by :

    0

    S

    Ba =c (4.3)

    in which : - is the circular frequency, - B a characteristic dimension of the footing (generally the one half of the shorter

    fundament edge for a rectangular footing, the radius for a circular footing), - and cS the shear wave velocity in the soil. The relations between the values without and with dimensions are given in table 4.1. For derivation of the dimension-less equation of motion see Appendix B. Note: for surface foundations without embedment the off-diagonal terms in the impedance matrix of the foundation-soil interface K0 are much more smaller then the diagonal terms and therefore may be neglected.

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    Mode 2-D 3-D Translations

    G= Kk

    G B= Kk

    Rotations

    G B= 2Kk G B= 3

    Kk

    Coupled Transl.-Rot.

    G B= Kk

    G B= 2Kk

    Table 4.1. Dimensionless impedance functions

    4.3 Example 1 This first example illustrates the use of the impedance functions. We wish to know the movements of the two-dimensional structure (see figure 4.1) loaded by a linear horizontal harmonic force.

    xx

    hGa

    H2 B

    G, ,

    G

    Phx e i t M = 6 000 kg/mIry = 4 000 kgm2/ma = 1,5 mhG = 0,7 m

    G = 45 MPa = 0,3 = 2 000 kg/m3 = 5 %

    B = 1mH = 2 m

    A

    bedrock Fig. 4.1. Example 1 In order to have an easier understanding, we will give the details of the numerical calculation for one frequency : N = 16,71 Hz. Step 1 HUH (1986) has published the displacements (compliance) functions in form of dimensionless charts concerning a strip foundation without embedment resting on a horizontal layer for : - = 0,3 - = 5 % - H/B = 2 These values correspond exactly to our case, so that we can use directly these published results (see figure 4.2). The dimensionless circular frequency is defined by relation (4.3). Numerical calculation (see fig. 4.2)

    a0 = 2* *16,71 * 145000 2 = 0,7 (NC.1)

    x z

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    0,0

    0,1

    0,2

    0,3

    0,4

    0,5

    0,6

    0,7

    0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

    fhxR

    - fryI- fhxI

    fryR

    a0 = 0,75

    a0 = B / c S

    I fhx I

    Fig. 4.2. Displacement functions (after Huh)

    Comments A two-dimensional structure has only three degrees of freedom : - a vertical displacement along the z-axis, - a horizontal displacement along the x-axis, - a rotation around the horizontal y-axis. In our case, the horizontal load does not induce vertical oscillation of the center of gravity : for this reason, we do not present the vertical displacement functions on figure 4.2. Since the foundation is not embedded : the coupled term is close to zero. A layer without load has a circular eigenfrequency in horizontal direction given by :

    e = 2

    cSH

    or in dimensionless form and with our numerical values :

    a0 e = 2

    BH

    = 0,785

    (4.4) This value is close to the value (~ 0,75) corresponding to the maximum of IfhxI. Numerical calculation

    fhxR (0,7) = 0,595 fryR (0,7) = 0,430

    fhxI (0,7) = 0, 290 fryI (0,7) = 0, 040

    (NC.2)

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    In order to use the equations of the motion presented earlier (chapter 3), we need the values of the impedance functions. To transform the displacement functions to impedance functions, we use the relations (2.18). The results are presented on figure 4.3. Comments In comparison with the results obtained for a classical mass-less SDOF (chapter 2.2) the figure 4.3 shows clearly that kR are not constant and kI are not proportional to the frequency : that means that the equivalent stiffness and damping are functions of the frequency contrary to the classical mass-less SDOF. Numerical calculation (from equ. 2.18; or from Figure 4.3)

    khxR (0,7) = 0,595

    0,5952 +0,292 = 1,358 kryR (0, 7) 0,43

    0,432 + 0, 042 = 2,306

    khxI (0,7) = 0,29

    0,5952 +0,292 = 0,662 kryI (0,7) 0,04

    0, 432 +0,042 = 0,214

    (NC.3) Step 2 We need the dimensional values of the components of the impedance functions : these are obtained by using the relations in table 4.1.

    0

    1

    2

    3

    4

    5

    6

    0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

    khxR

    a0 = B / c S

    kryR khxI

    kryI

    Fig. 4.3. Impedance functions Numerical calculation Non-diagonal terms are neglected for surface foundations.

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    6 7

    6 7

    6 7

    6 7

    (16,71) 1,358 * 45*10 6,11*10(16,71) 0,662 * 45*10 2,98*10(16,71) 2,306* 45*10 *1 10,38*10(16,71) 0,214* 45*10 *1 0,96*10

    RhxI

    hxR

    ryI

    ry

    = == == == =

    KKKK (NC.4)

    Step 3 The load is not applied to the center of gravity : it is necessary to translate this load to the center of gravity with addition of a rocking moment (see fig. 4.4).

    Phx e i t xxG

    A xxG

    A

    Mry = Mry e i t

    Phx e i t

    Fig. 4.4. Load at the center of gravity The amplitude of the rocking moment is as follows (see Figure 4.1) :

    Mry = Phx (hG a)

    (4.5) Numerical calculation

    Mry = Phx (0,7 1,5) = 0,8 Phx (NC.5)

    Step 4 We have yet all the values necessary to calculate the horizontal displacement of the center of gravity and the rotation about the center of gravity by using relations derived in the chapter 3. Numerical calculation

    7 711

    7 712

    7 722

    (6,11 2,98 1,10*6) *10 ( 0,50 2,98)*10(6,11 2,98) *0,7*10 (4,28 2,09)*10(10,38 0,96 (6,11 2,98) *0,49 1,10*4) *10 (8,96 2,43) *10

    k i ik i ik i i i

    = + = + = + = + = + + + = + (NC.6)

    uhxR = 3,99 I PhxI108 uhxI = 2,78 I PhxI 108

    ryR = 0,91 I PhxI 108 ryI = 2,01 I PhxI 108

    (NC.7)

    I uhx (16, 71) I = 4,87 IPhx I 108 Phase (uhx ) = 145,1I ry (16, 71) I = 2,21 IPhx I 108 Phase (ry ) = 65,6

    (NC.8)

    The complete curves are given on figure 4.5 and 4.6. The curves of the amplitudes (Fig. 4.5) show that the maximum of the amplitudes of displacement and rotation are obtained for 14,3 Hz (first mode). Of course, this value is not the same as the

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    eigenfrequency of the layer without footing (18,7 Hz) : the dynamic effect of the soil-structure interaction appears clearly. The second mode (22,7 Hz) is only obvious for the rotation. The phase curves (Fig. 4.6) show that the phases are close to 90 degrees at the first maximum of amplitudes.

    0

    2

    4

    6

    8

    10

    12

    0 10 20 30 40 50

    I uhx I / I Phx I

    I ry I / I Phx I

    14,3 Hz

    22,7 Hz

    Hz

    10-8

    Fig. 4.5. Amplitudes versus frequency

    -200

    -150

    -100

    -50

    0

    50

    100

    150

    200

    0 10 20 30 40 50

    Phase of ry Hz

    Phase of uhx

    14,3 Hz

    degrees

    90

    - 90

    Fig. 4.6. Phases versus frequency

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    4.4 Available results Figure 4.7 presents the available results for : - circular footings, - strip footings, - and rectangular footings. Circular footings Due to its symmetry, the theoretical solution of circular foundation is easier to obtain than that of rectangular foundation which needs three-dimensional calculations. The first results were presented by Reissner in 1936, before the development of computers. We dispose on very complete results for : - footing on a half-space medium (with or without embedment depth), - footing on a layer resting on an horizontal substratum (with or without embedment depth), - footing on an horizontal layer resting on an half-space medium (without embedment depth). Rectangular footings It is the more classical geometry for a foundation. But significant results concern only foundations on a half-space medium, with or without embedment depth. Strip footings Available results concern : - footing on a half-space medium (without embedment depth), - footing on a layer resting on an horizontal substratum (with or without embedment depth), - footing on an horizontal layer resting on an half-space medium (without embedment depth). The first formulation of the impedance function was obtained for circular footings on a half-space, and expressed as follows : st 0 0 0K = K [k (a , ) + i a c (a , )] (4.6) in which Kst is the static stiffness of a circular foundation , k a dimensionless coefficient in order to introduce the dynamic effect on the stiffness and c a dimensionless coefficient in order to describe the loss of energy. Both coefficients k and c are depending on the frequency and on the internal damping of the soil.

    scra =0 , c = C csr Kst for circular foundation, (4.7)

    scBa =0 ,

    st

    s

    KBcC

    c = for rectangular and strip foundation; B

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    Mode Static stiffness Kst

    Vertical 4 G r1

    Horizontal 8 G r2

    Rocking 8 G r3

    3 (1 )

    Torsion 16 G r3

    3

    Table 4.2. Static stiffness. Circular footing without embedment on a half-space Many other results are given in Sieffert et al. 1992. Some more impedance functions for rectangular foundation are also available in Appendix C.

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    - 26 -

    2 B

    2 B

    Gazetas

    2 B

    Gazetas

    Gazetas - Huh

    2 BH

    Huh

    H2 B

    D

    2 r

    Luco - Gazetas - Veletsos

    r

    2 r

    Luco

    Kausel - Tassoulas

    H2 rD d

    Kausel - Luco

    2 rH

    Apsel

    2 rD

    Dominguez

    2 L

    2 B

    Dominguez - Wong - Rcker - Schmid

    2 B

    D2 B

    Fig. 4.7. Available results

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    4.5 Simplified methods 4.5.1. Circular footings In order to simplify the calculations, it is often assume that k and c can be considered as independent of the frequency and of the internal damping of the soil.

    Mode k c B

    Vertical 1 0,85 M cS2

    Kst r2

    Horizontal 1 0,58 M cS2

    Kst r2

    Rocking 1

    0,301 + B r

    Ir cS2

    Kst r2 *

    Torsion 1

    B t1 + 32 B t/ 3

    It cS2

    Kst r2

    Table 4.3. Coefficients. Circular footing on half-space *) Ir refers to the center of the soil-foundation interface. Every engineer in geotechnical engineering knows that : - the more classical situation of a footing soil is a layer on a rigid substratum, - in practice, a footing has ever an embedment depth corresponding to soil

    freezing depth (between 0.6 and 1 meter depending on geographical situation and climatic conditions).

    The static stiffness of a layer resting on a rigid substratum obviously depends on its thickness H and are greater than the static stiffness of the half-space given in table 4.2. These equivalent stiffness are also functions of the embedment D. GAZETAS recommends the values recorded in table 4.4.

    Mode Static stiffness Kst

    Vertical 4 G r1 1+ 1, 28

    rH

    1+

    D2 r

    1+ (0,85 0,28

    Dr

    )D H

    1 D H

    Horizontal 8 G r2 1 +

    r2H

    1 +

    2D3 r

    1 +

    5D4 H

    Rocking 38 G r r 2D D1 1 1 0,7

    3(1 ) 6H r H + + +

    Coupled Horizontal - Rocking

    0,40 Ksth D

    Torsion 16 G r3

    31 + 2,67 D

    r

    Table 4.4. Static stiffness. Circular footing on a layer

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    These values are sufficiently accurate provided that the various parameters remain within the limits given in table 4.5.

    Mode Range of validity Vertical H / r > 2 D / r < 2 Horizontal H / r > 1 Rocking 4 H / r > 1 D / H 0,5 Torsion H / r 1,25

    Table 4.5. Range of validity Needless to say, the static stiffness in table 4.2 (surface footing on a half space) can be obtained again taking D = 0 and extending H to infinity. The static stiffness of torsion is independent of layer thickness. 4.5.2. Rectangular footings If the impedance functions do not exist in the literature, one possible way is to replace the rectangular footing by an equivalent circular footing which radius is obtained by requiring that it has the same area for the translation movements and the same moment of inertia for rotation movements (see table 4.6) respectively. In table 4.6, 2L is the length of the footing, and 2 B the width, where L>B.

    Mode Radii

    Translation 4 B L

    1/ 2

    Rocking

    around the x-axis 16 B3 L

    3

    1/4

    Rocking

    around the x-axis 16 B L3

    3

    1/ 4

    Torsion 8 B L (B2 + L2)

    3

    1/ 4

    Table 4.6. Equivalent radii 4.5.3. Example 2 One wishes to verify that the vertical acceleration of the machine-footing system (Fig. 4.8) does not exceed the limit of 0,15 ms-2.

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    - 29 -

    0,6

    machine2617 kg

    4 cos t (kN)

    1,5 m

    half spaceG = 54 MPa = 0,3 = 1850 kg/m3

    0,8

    square footing = 2400 kg/m3

    Fig. 4.8. Machine fixed on a footing partially embedded on a half-space Step 1 We assume that the machine is fixed rigidly at the footing, so that the machine and the footing can be considered as one rigid block. We need : - the mass of the system : M = 2617 + 1,5*1,5*0,8*2400 = 6937 kg - the vertical equivalent radius (see table 4.6) :

    r = 4 * 0, 75* 0,75

    1/ 2

    = 0,846 m - the vertical static stiffness (see table 4.4) :

    6

    84 *54 *10 *0,846 0,6K 1 3,536*10 /1 0,3 2*0,846st

    N m = + =

    D / r = 0,6 / 0,846 = 0,709 < 2 (in range of validity) - the dimensionless dynamic coefficients (see table 4.3) : k = 1 c= 0,85 - the dimensionless circular frequency (equation 4.6):

    300,846 4,952*10

    54000 1,85a = =

    We can also make explicit the formulation of the impedance function (see equation 4.5): 8 33,536*10 (1 4,952*10 *0,85 )K i = + = K+ i C

    8 63.538 10 1.488 10K i = +

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    Step 2 We know yet the stiffness K and the damping C corresponding to the dynamic soil-structure interaction, and the system machine-footing-soil can be calculated as a SDOF system. The amplitude of the vertical movement is following equation (2.21) :

    8 2 2 12 2

    4000(3,536*10 6937 ) 2,214*10

    vu = + and the amplitude of the vertical acceleration : 2 vu u=&&

    0

    0,1

    0,2

    0,3

    0,4

    0,5

    0,6

    0,7

    0 20 40 60 80 100

    17,3 Hz

    Hz

    m/s 2

    Fig. 4.9. Amplitude of acceleration versus frequency Figure 4.9 shows the amplitude of the acceleration as a function of the frequency. In order to respect the limit value 0,15 ms-2, the frequency must be in the frequency range (0 - 17,3 Hz). For frequencies out of this range, it is necessary to modify the dimensions of the footing. 5 Soil Structure Interaction 5.1 Flexible Structures; Rigid Foundation 5.1.1 Sub-Structure method We assume the system consist of sub-structures. And it is also further assumed that all sub-structures and then also the complete system behave linearly. 5.1.1.1 Structure We assume that the structure is discretized by finite elements leading to the equation of motion:

    ( )t=&& &Mu + Cu + Ku P (5.1)

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    - 31 -

    For harmonic excitation i te P with frequency we obtain the steady-state response: ( ) i tt e =u u (5.2)

    where u is determined from:

    ( ){ }2 i + =K M C u P (5.3) where ( )s i = +K C2K M (5.4) is the impedance of the structure. 5.1.1.2 Soil The soil is represented by the impedance matrix of the mass-less rigid foundation described with respect to the center of the lower soil-structure interface. In general case we have an embedded 3-D rectangular foundation.

    rigid

    0

    Foundation

    Soil

    X1

    X3

    X2

    Fig. 5.1. Rigid foundation, degrees of freedom The motion of the rigid basement may be described by the displacement 0u (6,1) at the point 0 (center of the base interface). The corresponding forces acting at the point 0 are

    0P .

    1

    2

    3

    1

    2

    3

    uuu

    =

    0u

    1

    2

    3

    1

    2

    3

    PPPMMM

    =

    0P (5.5)

    Forces and displacements are related through the dynamic matrix of the foundation

    0 0 0(6,6) (6,1) (6,1)

    F =u PK (5.6)

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    5.1.1.3 Coupling of Structure And Soil Step 1: Kinematic constraint of basement

    Structuralmodel

    rigid

    rigid

    interactionnodes

    a) b)

    Fig. 5.2 Kinematics constraint through rigid basement: a) soil discretized by BEM b) soil discretized by TLM/FVM

    We define all nodes of the structure, the motion of which is restricted through the rigid foundation at interaction node (index I), the remaining nodes are the structure nodes (index S) and partition the equation of motion of the structure correspondingly:

    S SS SSS SI

    S SI IIS II

    = u PK Ku PK K

    (5.7)

    The kinematical connection between Iu and 0u is expressed by the transformation:

    0I =u au (5.8) By applying the principle of virtual works one obtains:

    0 0

    S SS SSS SI

    T S T SIS II

    = u PK K au Pa K a K a

    (5.9)

    Step 2: Coupling The soil-structure system is coupled by considering the dynamic stiffness of the rigid foundation as a hyper-element stiffness matrix. The direct stiffness method yielding immediately:

    n-6 6

    n-6

    m0 00

    S SS SSS SI

    T S T SIS II

    = + u PK K au Pa K a K a KF

    (5.10)

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    5.2 Flexible Structure On Flexible, Mass-less Foundation

    5.2.1 Soil The impedance matrix (dynamic stiffness matrix) of the soil is defined by the dynamic stiffness matrix of the nodes on the interface between structure and soil. Boundary element method; Thin Layer Method/Finite Element Method We assume theory of elasticity and assume a discretization of the (excavated) soil with elements and (if necessary) condense the number of degrees of freedom to those of the interface.

    Boundary elementdiscretization(Integral equationmethod)

    FEM+Thin LayerMethod

    brick elements

    Thin Layer Method/Flexible VolumeMethod

    a)

    b)

    c)

    Fig. 5.3 Various methods for solving soil-structure interaction problems Through semi-analytical or numerical methods the impedance matrix of the soil-structure interface FK (index F for foundation) interface can be obtained as the relation between displacement and forces related to the m degrees of freedom at the interface nodes (Fig. 5.3a, b).

    FK can be understood as hyper-element matrix. The direct stiffness matrix couples the two sub-structures.

    n-m m

    n-m

    m

    S S SSSS SI S

    S S FIIS II II I

    = + uK K PuK K K P

    (5.11)

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    - 34 -

    All matrices involved are in general complex and frequency dependent. For a selected frequency for a specified harmonic loads and/or displacements the solution can be obtained from the complex linear system of equations. Note:

    1) For each degree of freedom either iP or iu i = 1, 2, 3...m has to be given. Their corresponding unknown values are calculated.

    2) Since FK is regular for unbounded soil (no rigid body motion), ( , )n nK is also

    regular. Thin Layer Method/Flexible Volume Method In the TLM/FVM, the m interaction nodes are defined as the nodes of the intersection of the horizontal layers and the volume elements representing the volume to be excavated (Fig.5.3c). The impedance matrix of the interaction node is calculated as the inverse of the dynamic flexibility matrix of the m degrees of freedom of the interaction nodes. where 1

    ( , ) ( , )

    F

    m m m m

    =K F The elements Fij of the matrix F or obtained as the displacements u i due to a harmonic unit load 1jP = . ,i jF corresponds to the numerical Greens function of the interaction region. To couple the soil with the structure, the foundation volume has to be excavated. This is done by subtracting from the dynamic stiffness matrix of the interaction node the dynamic stiffness matrix FIIK , of the foundation volume (to be excavated) discretized by volume brick elements. In practical calculations, this matrix will be subtracted from the structural matrix (therefore the basement nodes of the structure have to be identical with the finite volume model). Soil-structure coupling results finally in:

    S SS SSS SI

    S S E FI IIS II II II

    = + u PK Ku PK K K K

    (5.12)

    5.3 Load Cases a) Specified loads

    - Wind - Traffic - Explosion - Machines

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    - 35 -

    Wind

    Traffic

    Machine

    Traffic

    Fig. 5.4 Possible sources of dynamic loading

    Loads may be specified at any nodal point of the structure, above the soil surface or below the soil surface. If, for example, wind loads are specified, IP would be zero. If traffic loads are considered, SP would be zero and IP would be specified at interaction nodes on the soil surface or below it. (Note that we define here as interaction nodes these points, where loads are specified).

    b) Seismic loads

    Most simple case

    Fig. 5.5 Seismic loading; left: soil layer on rigid bed-rock; right: soil as infinite half-

    space Definitions:

    Free-field 'u : wave field at the site without structure. Scattered field u : wave field at the side with excavation but without structure.

    At the interaction nodes the forces stemming from the structure and the forces stemming from the soil have to add up to zero. In case the scattered field is known the forces stemming from the soil are proportional to the difference of the total displacement field minus the scattered field:

    =>Free field problem

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    - 36 -

    ' '( )S S FIS S II I II I I+ + =K u K u K u u 0 (5.13) Whence the equation of motion becomes

    S SSSS SI

    S S FI IIS II II

    = + u 0K Ku PK K K

    (5.14)

    with FI II I=P K u In the case, when the free-field is known, the equation of motion is:

    S SSSS SI

    S S E FI IIS II II II

    = + u 0K K

    u u 0K K K K or

    S SSSS SI

    S S E FI IIS II II II

    = + u 0K Ku PK K K K

    (5.15)

    where 'FI II I=P K u and EIIK is the stiffness of the excavated soil. 5.4 Solution Of The Equation Of Motion From equation (5.10), (5.11), (5.12), (5.14) and (5.15) the displacements for the DOFs can be obtained for the given loads. For each degree of freedom, i, we have:

    (5.16) The time-harmonic displacements are: ( )( ) cos ii i uu t u t = + (5.17)

    ( ) ( )( ) ( )

    2 2

    2 2

    tan

    tan

    i

    i

    i

    i

    iR Ii i i

    iR Ii i i i

    IR I i

    i i i P Ri

    IR I i

    i i i u Ri

    P P iP P e

    u u iu u e

    PP P PPuu u uu

    = + == + == + =

    = + =

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    Literature LAMB, H. (1904) On the Propagation of Tremors over the Surface of an Elastic Solid Phil. Trans. of the Royal Soc., Lond., Vol. 203, pp 1-42 REISSNER, E. (1936) Stationre, Axialsymmetrische, durch eine Schuttelnde Masse erregte Schwingungen eines Homogenen Elastischen Halbraumes Ing. Arch., Vol. 7, Part 6, Dec., pp 381-396 SUNG, T.Y. (1953) Vibration in Semi-infinite Solids Due to Periodic Surface Loading Harvard University, Sc.D. Thesis Symp. on Dyn. Testing of Soils, ASTM-STP No 156, pp 35-64 QUILAN, P.M. (1953) The Elastic Theory of Soil Dynamics Symp. on Dyn. Test. of Soils, ASTM STP, No 156, pp 3-34 ARNOLD, R.N., BYCROFT, G.N. and WARBURTON, G.B. (1955) Forced Vibrations of a Body on an Infinite Elastic Solid ASME, J. Appl. Mech.,Vol. 77, pp 391-401 BYCROFT, G.N. (1956) Forced Vibration of a Rigid Circular Plate on a Semi-infinite Elastic Space and an

    Elastic Stratum Phil. Trans. Royal Soc., Lond., Vol. 248, pp 327-368 AWOJOBI, A.D. and GROOTENHUIS, P. (1965) Vibration of Rigid Bodies on Elastic Media Proc. Royal Soc. Lond., Vol. 287, pp 27 LYSMER, J. (1965) Vertical Motions of Rigid Footings Univ. of Michigan, Ann Arbor, Ph.D. Thesis, Aug. WHITMAN, R.V. and RICHART, F.E. (1967) Design Procedures for Dynamically Loaded Foundations ASCE, J. Soil Mech. and Found. Div., Vol. 93, No SM6, Nov., pp 169-193 RICHART, F.E. and WHITMAN, R.V. (1967) Comparison of Footing Vibration Tests with Theory ASCE, J. Soil Mech. and Found. Engrg. Div., Vol. 93, No SM6, Nov., pp 143-168 ELORDUY, J., NIETO, J.A. and SZEKELY, E.M. (1967) Dynamic Response of Bases of Arbitrary Shape Subject to Periodic Vertical Loading Proc. Int. Symp. Wave Prop. & Dyn. Prop. Earth Mat., Univ. of New Mexico,

    Albuquerque, Aug., pp 105-121

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    DELEUZE G. (1967) Rponse un mouvement sismique d'un difice pos sur un sol lastique Annales ITBTP, No 234, pp 884-902 McNEIL, R.L. (1969) Machine Foundations : The State-of-the Art Proc. Soil Dyn. Spec. Sess., 7th ICSMFE, pp 67-100 RICHARD, F.E., WOODS, R.D. and HALL, E.R. (1970) Vibrations of Soils and Foundations Prentice-Hall, Inc., Englewood Cliffs, New Jersey VELETSOS, A.S. and WEI, Y.T. (1971) Lateral and Rocking Vibrations of Footings ASCE, J. Soil Mech. Found. Div., Vol. 97, SM 9, pp 1227-1248 LUCO, J.E. and WESTMANN, R.A. (1971) Dynamic Response of Circular Footing ASCE, J. Engng. Mechanics Div., Vol. 97, No EM 5, pp 1381 WAAS, G. (1972) Analysis Method for Footing Vibration through Layered Media Ph. D. thesis, Univ. of California, Berkeley KAUSEL, E. (1974) Forced Vibrations of Circular Foundations on Layered Media MIT, Research Rep. R 74-11 LUCO, J.E. (1974) Impedance Functions for a Rigid Foundation on a Layered Medium Nuclear Engineering and Design, Vol. 31, pp 204-217 WONG, H.L. and LUCO, J.E. (1976) Dynamic Response of Rigid Foundations of Arbitrary Shape Earthquake Engng and Structural Dynamics, Vol. 4, pp 579-587 GAZETAS, G. and ROESSET, J.M. (1976) Forced Vibrations of Strip Footings on Layered Soils Meth. Strct. Anal., ASCE, Vol. 1, No. 115 ELSABEE, F. and MORRAY, J.P. (1977) Dynamic Behavior of Embedded Foundations MIT, Research Rep. R 77-3 DOMINGUEZ, J. and ROESSET, J.M. (1978) Dynamic Stiffness of Rectangular Foundations MIT, Research Rep. R 78-20 KAUSEL, E. and USHIJIMA, R.A. (1979) Vertical and Torsional Stiffness of Cylindrical Footings

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

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    MIT, Resaerch Rep. R 79-6 TASSOULAS, J.L. (1981) Elements for the Numerical Analysis of Wave Motion in Layered Media MIT, Research Rep. R 81-2 RCKER, W. (1982) Dynamic Behaviour of Rigid Foundations of Arbitrary Shape on a Halfspace Earthquake Engng and Structural Dynamics, Vol. 10, pp 675-690 GAZETAS, G. (1983) Analysis of Machine Foundation Vibrations : State of the Art Soil Dynamics and Earthquake Engng., Vol. 2, No 1, pp 2-42 DAS, B.M. (1983) Fundamentals of Soil Dynamics Elsevier Science Publishing Co., Inc., New York PECKER, A. (1984) Dynamique des Sols Presses de l'Ecole Nat. des Ponts et Chausses, Paris HAUPT, W. (1986) Bodendynamik - Grundlagen und Anwendung Friedr. Vieweg & Sohn, Braunschweig HUH, Y. (1986) Die Anwendung der Randelementmethode zur Untersuchung der dynamischen

    Wechselwirkung zwischen Bauwerk und geschichtetem Baugrund RUB, SFB 151 - Mitteilung Nr. 86-13, Dezember APSEL, R.J. and LUCO, J.E. (1987) Impedance Functions for Foundations embedded in a layered Medium : an Integral

    Equation Approach Earthquake Engrg. and Structural Dynamics, Vol. 15, pp 213-231 DOMINGUEZ, J. and ABASCAL, R. (1987) Dynamics of Foundations Springer - V., Topics in Boundary Element Research, Vol. 4, Applications in

    Geomechanics, pp 27-75 SCHMID, G., WILLMS, G., HUH, Y. und GIBHARDT, M. (1988) Ein Programmsystem zur Berechnung von Bauwerk-Boden-Wechsel- wirkungs-

    problemen mit der Randelementmethode RUB, SFB 151 - Berichte Nr. 12 , Dezember SIEFFERT, J.G. and CEVAERT,F. (1992) Handbook of Impedance Functions. Surface foundations Ouest Editions, 174 p.

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

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    Appendix A Energy considerations Given a displacement vector u (n,1) and its corresponding force vector P (n,1) of an elastic system. They are related by stiffness K

    =P K u (A.1) Assume a kinematic constraint equation

    = 0u au (A.2) where the vector u0 has dimension (m,1), m

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 41 -

    Appendix B Dimension-less equation of motion in frequency domain The equation of motion with dimensions is

    2K KK K M

    hh hr hh hr h h

    rh rr rh rr r r

    M M u PM M

    =

    (B.1)

    Definitions of dimension-less quantities:

    0 2 3

    MMh h rh h r

    s

    u PBa u Pc B GB GB

    = = = = = (B.2) or reciprocal

    2 30 M Ms h h h h r ra c u Bu P GB P GBB

    = = = = = (B.3) where: a0 dimension-less frequency (-) circular frequency (rad/s) B one half of the shorter fundament sides (m) cs shear wave velocity (m/s) G shear modulus (N/m2) ( G = cs2 )

    soil density (kg/m3). ,u dimension-less displacement (-) and rotation (-), respectively ,Mh rP - dimension-less force (-) and moment (-), respectively

    22

    20 2 3

    0 00 1 0

    K KK K M

    hh hr hh hr h hs

    rh rr rh rr r r

    M M u PB GBcaM MB GB

    = (B.4)

    22

    20 2

    3

    1 0 01 0 10

    K KK K M

    hh hr hh hr h hs

    rh rr rh rr r r

    M M u PBcGB aM MB

    GB

    = (B.5)

    22 2

    20 2 2

    3 3

    11

    K K

    MK K

    hh hrhh hr

    h hsrh rr

    r rrh rr

    B M B M u PcGB GB a M B MB GB BB BGB GB

    = (B.6)

    3 42

    20

    2 3 4 5

    K K

    MK K

    hh hrhh hr

    h h

    r rrh rhrr rr

    M Mu PB BGB GB a

    M MGB GB B B

    = (B.7)

    Finally the equation of motion in dimension-less notation:

    20K KK K M

    hh hr hh hr h h

    rh rr rh rr r r

    M M u Pa

    M M =

    (B.8)

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 42 -

    Appendix C Impedance functions of square foundation Impedance function in this appendix are given for the four ensuing cases of embedment depth t:

    - t : 2a = 0 - t : 2a = 1/3 - t : 2a = 2/3 - t : 2a = 1

    x

    y z

    mx

    mzmy

    0

    2a

    2b

    t = 0

    rigid, mass-less

    s

    3

    x

    y z

    mxmzmy

    0

    2a

    2b

    t = 2

    arigid, mass-less

    3

    s

    2b :2a = 1, t :2a = 0 2b :2a = 1, t :2a = 1/3

    2b :2a = 1, t :2a = 2/3 2b :2a = 1, t :2a = 1

    x

    y z

    mxmzmy

    0

    2a

    2b

    t = 2

    /3 2

    a

    rigid, mass-less

    s

    3

    rigid, mass-less

    Impedance functions have been calculated using Boundary Element Method in frequency domain. Rigid, mass-less square foundation lays on elastic half-space with Poissons ratio = 0.4. Presented are in dimension-less notation separately as

    - real part KR and - imaginary part KI

    - damping coefficient C (I

    I a C Ca

    = =00

    KK )

    Partitioning of the impedance matrix:

    ,

    ,

    ,

    ,

    xx x my

    yy y mx

    zz

    mx y mx

    my x my

    mz

    =

    K 0 0 0 K 00 K 0 K 0 00 0 K 0 0 0

    K0 K 0 K 0 0

    K 0 0 0 K 00 0 0 0 0 K

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 43 -

    Kxx Real

    02468

    1012141618

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'xx

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -10

    0

    10

    20

    30

    40

    50

    60

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'xx

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    0

    5

    10

    15

    20

    25

    30

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'x

    x

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 44 -

    K yy

    Real

    02468

    1012141618

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'yy

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -100

    10203040

    5060

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'yy

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    0

    5

    10

    15

    20

    25

    30

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'y

    y

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 45 -

    K zz Real

    0246

    8101214

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'zz

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -100

    10203040

    5060

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'zz

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    0

    5

    10

    15

    20

    25

    30

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'z

    z

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 46 -

    K mx Real

    0

    1020

    3040

    5060

    70

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    x

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -200

    20406080

    100120

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    x

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    -100

    1020

    3040

    5060

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'm

    x

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 47 -

    K my Real

    010

    2030

    4050

    6070

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    y

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -200

    2040

    6080

    100120

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    y

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    -100

    1020

    3040

    5060

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'm

    y

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 48 -

    K mz Real

    0

    10

    20

    30

    40

    50

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    z

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Imaginary

    -10

    0

    10

    20

    30

    40

    50

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    z

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    -10-5

    05

    1015

    2025

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'm

    z

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 49 -

    K x,my Real

    -20

    -15

    -10

    -5

    0

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'x,

    my

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Imaginary

    -70-60-50-40-30-20-10

    010

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'x,

    my

    t :2a = 0

    t :2a = 1/3

    t :2a = 2/3

    t :2a = 1

    Damping coefficient

    -35-30-25-20-15-10

    -505

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'x

    ,my

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 50 -

    K my,x Real

    -22-20-18-16-14-12-10

    -8-6-4-20

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    y,x

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Imaginary

    -70-60-50-40-30-20-10

    010

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    y,x

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Damping coefficient

    -35-30-25-20-15-10

    -505

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'm

    y,x

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 51 -

    Ky,mx Real

    -5

    0

    5

    10

    15

    20

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'y,

    mx

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Imaginary

    -100

    10203040506070

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'y,

    mx

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Damping coefficient

    -505

    101520253035

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'y

    ,mx

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

  • Soil Structure Interaction & Foundations Vibrations __________________________________________________________________________________________________________

    - 52 -

    Kmx,y Real

    -5

    0

    5

    10

    15

    20

    25

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    x,y

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Imaginary

    -100

    10203040506070

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    k'm

    x,y

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1

    Damping coefficient

    -505

    101520253035

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

    a0

    C'm

    x,y

    t :2a = 0t :2a = 1/3t :2a = 2/3t :2a = 1