Soil and Foundation Dynamics

SOIL AND FOUNDATION DYNAMICS

SOIL-STRUCTURE INTERACTION FOUNDATIONS VIBRATIONS Günther 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.

Soil – Structure Interaction & Foundations Vibrations

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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, - Poisson’s 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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M

P(t) = P e i ω t

K

C

u(t) = u e i ω t

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

Mü + Cu& + Ku = P(t )

(2.2)

We assume that the external load is represented by a complex force with amplitude P and circular frequency ω: P(t)=Peiωt

(2.3)

Consequently the displacement u(t) can be written as: u (t ) = ueiωt

(2.4)

Note, that the entire time-variance is expressed by the function eiωt and that P and u generally are complex and depend on ω. After substitution of the equations (2.3) and (2.4) in the equation (2.2), we obtain ⎡( K - M ω 2 ) + iCω ⎤ u = P ⎣ ⎦

(2.5)

In applying the definition (2.1), the impedance function of this single-degree-offreedom system is obtained as : K = ( K - M ω 2 ) + iCω

(2.6)

This complex impedance, which depends on frequency, can also be written in a more general way : K (ω ) = K R (ω ) + i K I (ω ) (2.7) with: K R = K − M ω2 K I = Cω

(2.8)

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

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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 ωe, the critical damping Ccrit, the damping ratio ξ and the frequency ratio η the relation between excitation frequency ω and eigenfrequency ωe.

ωe =

K , M

C ξ= , Ccrit

Ccrit = 2 M ω e

(2.9)

ω η= ωe

Using these definitions, the impedance may be written as the static stiffness multiplied by a dimension-less impedance function: K (η ) = K ⎡⎣(1 − η 2 ) + i 2ξη ⎤⎦

(2.10)

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

P(t) = P e i ω t

K

C

u(t) = u e i ω t

Fig. 2.3. Mass-less SDOF system; Voigt’s model The impedance is thus reduced to : K = K + iCω

(2.11)

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or else KR =K K I = Cω

(2.12)

Writing equation (2.11) as Cω ⎞ ⎛ K = K ⎜1 + i ⎟ K ⎠ ⎝ we may define the damping factor as:

ϑ=

Cω = tan φ K

where the term tan φ =

(2.13)

ωC K

is obtained from the complex representation of K

(see Fig. 2.4) Whence K = K (1 + iϑ ) = K (1 + i tan φ )

(2.14)

Fig. 2.4. Impedance representation in complex plane

ϑ is called the damping factor (usually given in %), φ is the loss angle. Figure 2.5 represents the evolution of the impedance functions of this particular singledegree-of-freedom system, i.e. of the Voigt’s 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, Voigt’s model

2.3. General definition of compliance function (dynamic flexibility) By definition, the compliance function is the inverse of the impedance function. F = K −1

(2.15)

For a single-degree-of-freedom system with mass, the compliance function is written as: F (ω ) =

1 (K − M ω ) + i C ω 2

=

1 1 − η − i 2ξη K (1 − η 2 )2 + ( 2ξη )2

(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 : F (ω ) = F R (ω )+ i F I (ω )

(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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FR = FI =

a)

KF

KR

(K ) + (K ) R 2

I 2

-K I

(K ) + (K ) R 2

I 2

KR = KI =

FR

(F ) + (F ) R 2

I 2

(2.18)

-FI

(F ) + (F ) R 2

I 2

R

7 6

ξ = 0.02

5 4

ξ = 0.05

3

ξ = 0.1

2 1 0

ξ = 0.5

1

ξ = 0.7

2 3 4 5 6 7

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

1.5

1.75

2

2.25

2.5

2.75

3

η

b)

KF

I

1

ξ = 0.7

0 1

ξ = 0.5

2 3 4

ξ = 0.1

5 6 7 8

ξ = 0.05

9 10

ξ = 0.02

11 12 13

0

0.25

0.5

0.75

1

1.25

η

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: u = u R + iu I = u=

P =FP K

(2.19)

P 1 − η 2 − i 2ξη K (1 − η 2 )2 + ( 2ξη )2

(2.20)

Usually we write the solution as modulus |u| and phase angle ψ. Modulus: u =F P =

P P 1 = D (ξ ,η ) 1 2 2⎤ K ⎡ K 2 2 ⎢⎣(1 − η ) + ( 2ξη ) ⎥⎦

(2.21)

where |P| is the amplitude of the loading given in equation (2.3). By ust we denote the static displacement and D(ξ,η) is the dynamic magnification factor. From the equation (2.20) and (2.10) we have: uI KI 2ξη phase angle: tanψ = R = − R − u K 1 −η 2

phase lag: tan ϕ = −

(2.22)

uI K I 2ξη = R = R u K 1 −η 2

(2.23)

The phase lag ϕ is shown in Fig. 2.7 for various values of damping ratio. phase lag ϕ [deg] 180 170 160 150 140 130 120 110 100

ξ = 4.0

90

ξ = 1.0

80

ξ = 2.0

ξ = 0.707

70 60

ξ = 0.15

50

ξ=0

40 30

ξ = 0.50

20

ξ = 0.25

10 0

ξ = 0.35

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

Fig. 2.7 Phase lag ϕ (with respect to the load) vs. frequency ratio η -8-

3

η

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2.4.1 |P| independent of ω

If |P| is independent on ω (the so-called constant or simple excitation) we define by P the static displacement of the system due to amplitude |P| of the time-varying K force P(t ) = P eiω t . ust =

In this case is the dynamic magnification factor given as: D(ξ ,η ) =

u 1 = 12 2 ust ⎡ (⎣⎢ 1 −η 2 ) + ( 2ξη )2 ⎤⎦⎥

(2.24)

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

ηres = 1 − 2ξ 2 . If ξ < 1/ 2 the dynamic magnification factor at this frequency ratio yields: Dres =

1

(2.25)

2ξ 1 − ξ 2

D(ξ,η) 3.5

ξ=0

3 ξ = 0.15

2.5 ξ = 0.25

2 ξ = 0.35

1.5 ξ = 0.5

1

ξ = 0.707 ξ=1

ξ=2

0.5

0

ξ=4

0

0.5

1

1.5

2

2.5

3

η

Fig. 2.8 Dynamic magnification factor – constant harmonic excitation

2.4.2 |P| dependent on ω

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

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and linearly dependent on the unbalanced mass m and its distance r from the center of the rotation: P (t ) = mrω 2 eiω t = P eiω t

(2.26)

with the amplitude P = mrω 2

(2.27)

Usually m1

Torsion

H / r ≥ 1,25

D/r B.

Mode

Radii

Translation

1/ 2 ⎡4 B L ⎤ ⎣ π ⎦

Rocking around the x-axis Rocking around the x-axis

⎡16 B3 L ⎤ ⎢ 3π ⎥ ⎦ ⎣

1/4

1/ 4

⎡16 B L3 ⎤ ⎢ 3π ⎥ ⎦ ⎣

1/ 4

Torsion

⎡ 8 B L (B2 + L2 ) ⎤ ⎢⎣ ⎥⎦ 3π

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. - 28 -

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4 cos ω t (kN) square footing ρ = 2400 kg/m3

machine 2617 kg

half space G = 54 MPa ν = 0,3 ρ = 1850 kg/m3

0,8

0,6 1,5 m

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) : 1/ 2 ⎡ 4 * 0, 75* 0,75 ⎤ = 0,846 m r= ⎣ ⎦ π the vertical static stiffness (see table 4.4) : K st =

-

4 *54 *106 * 0,846 ⎡ 0,6 ⎤ 1+ = 3,536 *108 N / m ⎢ ⎥ 1 − 0,3 ⎣ 2* 0,846 ⎦ 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): ω 0,846 a0 = = 4, 952 *10−3 ω 54000 1,85 We can also make explicit the formulation of the impedance function (see equation 4.5): K = 3,536*108 (1 + i 4,952*10−3 *0,85ω ) = K+ i C ω

K = 3.538 ⋅108 + i1.488 ⋅106 ω

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Step 2 We know yet the stiffness K and the damping C corresponding to the dynamic soilstructure 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) : uv =

4000 (3,536 *108 − 6937 ω 2 ) 2 + 2, 214 *1012 ω 2

and the amplitude of the vertical acceleration : u&& = ω 2 uv 0,7

m/s 2

0,6 0,5 0,4 0,3 0,2

17,3 Hz

0,1 Hz

0 0

20

40

60

80

100

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:

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

(5.1)

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For harmonic excitation Peiωt with frequency ω we obtain the steady-state response: u(t ) = ueiωt (5.2) where u is determined from:

{( K − ω M ) + iωC} u = P

(5.3)

K s = ( K − ω 2 M ) + iωC

(5.4)

2

where

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. Foundation

rigid 0 X1 X2

Soil

X3

Fig. 5.1. Rigid foundation, degrees of freedom The motion of the rigid basement may be described by the displacement u0 (6,1) at the point 0 (center of the base interface). The corresponding forces acting at the point 0 are P0 . ⎡ u1 ⎤ ⎢u ⎥ ⎢ 2⎥ ⎢u ⎥ u0 = ⎢ 3 ⎥ ⎢θ1 ⎥ ⎢θ 2 ⎥ ⎢ ⎥ ⎣⎢θ 3 ⎦⎥

⎡ P1 ⎤ ⎢P ⎥ ⎢ 2⎥ ⎢P ⎥ P0 = ⎢ 3 ⎥ ⎢ M1 ⎥ ⎢M 2 ⎥ ⎢ ⎥ ⎣⎢ M 3 ⎦⎥

(5.5)

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

K 0F u0 = P0 (6,6) (6,1)

(5.6)

(6,1)

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5.1.1.3 Coupling of Structure And Soil

Step 1: Kinematic constraint of basement Structural model

interaction nodes

rigid rigid

b)

a)

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: ⎡ K SSS ⎢ S ⎣ K IS

K SIS ⎤ ⎡u S ⎤ ⎡ PS ⎤ ⎥⎢ ⎥ = ⎢ ⎥ K SII ⎦ ⎣ u I ⎦ ⎣ PI ⎦

(5.7)

The kinematical connection between u I and u 0 is expressed by the transformation: u I = au 0 (5.8) By applying the principle of virtual works one obtains: ⎡ K SSS ⎢ T S ⎣a K IS

K SIS a ⎤ ⎡u S ⎤ ⎡ PS ⎤ ⎥⎢ ⎥ = ⎢ ⎥ aT K SII a ⎦ ⎣ u 0 ⎦ ⎣ P0 ⎦

(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:

⎡ K SSS ⎢ T S m ⎣a K IS

n-6

n-6

⎤ ⎡u S ⎤ ⎡ PS ⎤ K SSI a ⎥ F ⎢ ⎥=⎢ ⎥ aT K SII a + K 0 ⎦ ⎣ u 0 ⎦ ⎣ P0 ⎦ 6

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(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 element discretization (Integral equation method)

a)

b)

FEM+ Thin Layer Method

c)

brick elements Thin Layer Method/ Flexible Volume Method

Fig. 5.3 Various methods for solving soil-structure interaction problems Through semi-analytical or numerical methods the impedance matrix of the soilstructure interface K F (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). K F can be understood as hyper-element matrix. The direct stiffness matrix couples the two sub-structures. n-m m

⎡K SSS ⎢ S ⎣ K IS n-m

⎤ ⎡u S ⎤ ⎡ PSS ⎤ ⎥⎢ ⎥ = ⎢ ⎥ K SII + K IIF ⎦ ⎣ u I ⎦ ⎣ PI ⎦ K SIS m

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(5.11)

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All matrices involved are in general complex and frequency dependent. For a selected frequency ω 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 Pi or ui i = 1, 2, 3...m has to be given. Their corresponding unknown values are calculated. 2) Since K F is regular for unbounded soil (no rigid body motion), K is also ( n,n )

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 K F = F −1 ( m ,m )

( m,m )

The elements Fij of the matrix

F or obtained as the displacements u i

due to a

harmonic unit load Pj = 1 . Fi , j corresponds to the numerical Green’s 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 K FII , 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: ⎡ K SSS ⎢ S ⎣ K IS

⎤ ⎡u S ⎤ ⎡ PS ⎤ ⎥⎢ ⎥ = ⎢ ⎥ K − K + K ⎦ ⎣ u I ⎦ ⎣ PI ⎦ K SSI

S II

E II

F II

5.3 Load Cases a) Specified loads - Wind - Traffic - Explosion - Machines

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(5.12)

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Wind

Machine

Traffic

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, PI would be zero. If traffic loads are considered, PS would be zero and PI 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 =>Free field problem

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

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K SIS u S + K SII u I + K FII (u I − u 'I' ) = 0

(5.13)

Whence the equation of motion becomes ⎡ K SSS ⎢ S ⎣ K IS

⎤ ⎡u S ⎤ ⎡ 0 ⎤ ⎥⎢ ⎥ = ⎢ ⎥ K + K ⎦ ⎣ u I ⎦ ⎣ PI ⎦ K SIS

S II

(5.14)

F II

with PI = K IIF u′′I In the case, when the free-field is known, the equation of motion is: ⎡ K SSS ⎢ S ⎣ K IS

⎤ ⎡ u S ⎤ ⎡0⎤ ⎡ K SSS K SIS or = ⎢ S ⎥⎢ ⎥ ⎢ ⎥ K SII − K EII + K IIF ⎦ ⎣u I − u′I ⎦ ⎣ 0 ⎦ ⎣ K IS

⎤ ⎡u S ⎤ ⎡ 0 ⎤ K SIS ⎥ = ⎢ ⎥ (5.15) S E F ⎥⎢ K II − K II + K II ⎦ ⎣ u I ⎦ ⎣ PI ⎦

where PI = K FII u 'I and K EII 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: Pi = Pi R + iPi I = P eiψ i ui = uiR + iuiI = ui eiϕi Pi =

(P ) + (P )

ui =

(u ) + (u )

R 2

i

R 2 i

I

i

I i

2

2

(5.16) tanψ Pi =

I

Pi Pi R

uiI tanψ ui = R ui

The time-harmonic displacements are:

(

ui (t ) = ui cos ωt + ψ ui

)

(5.17)

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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) Stationäre, Axialsymmetrische, durch eine Schuttelnde Schwingungen eines Homogenen Elastischen Halbraumes Ing. Arch., Vol. 7, Part 6, Dec., pp 381-396

Masse

erregte

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 143168 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) Réponse à un mouvement sismique d'un édifice posé sur un sol élastique 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

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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 RÜCKER, 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 Chaussées, 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- wirkungsproblemen 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.

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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 = Ku

(A.1)

Assume a kinematic constraint equation

u = au0

(A.2)

where the vector u0 has dimension (m,1), m