SOIL AND FOUNDATION DYNAMICS
SOIL-STRUCTURE INTERACTION
FOUNDATIONS VIBRATIONS
Jean-Georges Sieffert
Günther Schmid
Ruhr Universtity Bochum
Department of Civil Engineering
Lecture at
Computational Engineering Ruhr University Bochum
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).
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).
K C P(t) = P e i ω t
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:
i
(t)= e
P P (2.3) Consequently the displacement u(t) can be written as:
( ) i t
u t =ueω (2.4) Note, that the entire time-variance is expressed by the function ei tω 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 Mω iCω 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- ω ) iCω
= +
K (2.6) This complex impedance, which depends on frequency, can also be written in a more general way : R I ( )ω = ( )ω +i ( )ω 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 2 K KI M C ω ω = − =
Fig. 2.2. Impedance functions of a SDOF system with mass
We definethe circular eigenfrequency of the undamped SDOF system ωe, the critical damping Ccrit, the damping ratio ξ and the frequency ratio the relation between
excitation frequency ω and eigenfrequency ωe.
(2.9)
Using these definitions, the impedance may be written as the static stiffness multiplied by a dimension-less impedance function:
(
2)
( )η =K 1−η +i2ξη
K (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.
K
C
P(t) = P e i
ω
t
u(t) = u ei
ω
t
Fig. 2.3. Mass-less SDOF system; Voigt’s model The impedance is thus reduced to :
K iCω = + K (2.11) , 2 , crit e crit e e K C M M C C ω ω ω ξ η ω = = = =
or else R K = K I Cω = K (2.12) Writing equation (2.11) as 1 C K i K ω = + K
we may define the damping factor as:
tan
C K
ω
ϑ = = φ (2.13)
where the term tan C
K
ω
φ = is obtained from the complex representation of K (see Fig. 2.4) Whence
(
1)
(
1 tan)
K iϑ K i φ = + = + K (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 single-degree-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.
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.1 − =
F K (2.15)
For a single-degree-of-freedom system with mass, the compliance function is written as:
(
)
2(
)
2 2 2 1 1 1 2 ( ) ( ) 1 2 i K M i C K η ξη ω ω ω η ξη − − = = − + − + 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 :
( ) ( ) I( )
F FR F
i
ω = ω + 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 :
( ) ( )
( ) ( )
( ) ( )
( ) ( )
R R R R R I R I I I I I R I R I = = + + = = + + 2 2 2 2 2 2 2 2 K F F K K K F F - K - F F K K K F F (2.18) a)K
F
R = 0.02 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 = 0.05 = 0.1 = 0.5 = 0.7 b)K
F
I 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 = 0.02 = 0.05 = 0.1 = 0.5 = 0.7Fig. 2.6 a) Real and b) Imaginary part of compliance function of a SDOF system with mass
2.4 Solution of the equation of motion
From equation (2.1) we obtain:R I P u=u +iu = =FP K (2.19)
(
)
(
)
2 2 2 2 1 2 1 2 P i u K η ξη η ξη − − = − + (2.20)Usually we write the solution as modulus |u| and phase angle . Modulus:
(
2)
2(
)
2 1 2( )
1 , 1 2 P P u P D K η ξη K ξ η = = = − + 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: tan 2 2
1 I I R R u u ξη ψ η = = − − − K K (2.22)
phase lag: tan 2 2
1 I I R R u u ξη ϕ η = − = = − K K (2.23)
The phase lag ϕ 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 3 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
phase lag [deg]
= 0 = 0.707 = 2.0 = 4.0 = 0.15 = 0.25 = 0.35 = 0.50 = 1.0
2.4.1 |P| independent of
If |P| is independent on (the so-called constant or simple excitation) we define by
st
P u
K
= the static displacement of the system due to amplitude |P| of the time-varying force P t( )= P ei tω .
In this case is the dynamic magnification factor given as:
(
2)
2(
)
2 1 2 1 ( , ) 1 2 st u D u ξ η η ξη = = − + (2.24)One can show, that the maximal amplitude (resonance) does occur at the frequency ratio
2
1 2
res
η = − ξ . The dynamic magnification factor at this frequency ratio yields:
2 1 2 1 res D ξ ξ = − (2.25) 0 0.5 1 1.5 2 2.5 3 0 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
and linearly dependent on the unbalanced mass m and its distance r from the center of the rotation:
2
( ) i t i t
P t =mrω eω = P eω (2.26)
with the amplitude
2
P =mrω (2.27)
Usually m<<M and therefore m doesn’t ..contribute to the structural mass M.
Fig. 2.9. Rotating unbalanced mass – quadratic excitation
Displacement response of a SDOF system due to quadratic excitation yields:
. (2.28)
The dynamic magnification factor shown in Fig 2.10 is now given as
… … … ...(2.29)
For quadratic excitation any static displacement ust cannot be defined due to the fact, that the amplitude of the acting force at = 0 is zero. Therefore the factor
mr
M has been chosen, to provide a relevant reference value.
Resonance response occurs at
2 1 1 2 res η ξ = − (2.30)
(
)
(
)
2 2 2 1 2 2 2 2 2 ( , ) ( , ) 1 2 e m r m r m r u D D K M M ω ξ η ω ξ η η ω η ξη = = = − + (
)
(
)
2 1 2 2 2 2 ( , ) 1 2 u M D r m η ξ η η ξη = = − + The resonance dynamic magnification factor has the same form as in the case of constant excitation: 2 1 2 1 res D ξ ξ = − (2.31) 0 0.5 1 1.5 2 2.5 3 0 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 ξ, as already defined before, is given, by 2 2 c e C C C C M M K ξ ω = = = (2.32)
Equation (2.32) clearly states that this definition of ξ 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 :
( ) cos
P t = P ωt (2.33)
The displacement response function in term of its modulus |u| and phase lagϕis than as follows :
( ) cos( - )
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, ! "# $
the same cycle, W. One has:
2 2 2 W C u W K u π ω ∆ = = (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 ϑ:
1 W 2 W ϑ π ∆ = (2.36)
which gives for a single-degree-of-freedom system with viscous damping C
K
ω
ϑ = (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 ϑ we obtain for a system with
mass
(2.38)
We see that for viscous damping the factor ϑ is proportional to the frequency ratio
e
= ω η
ω
It also may be seen that ϑ =2ξ if the excitation frequency is the eigenfrequency (i.e., 1 e = → = ω ω η ). 2 2 2 e e M C K M ξ ω ω ω ϑ ξη ω = = =
The above definition of the damping coefficient defined in the equation (2.36) leads for Voigt’s model to
(
)
(
)
1 1 1 2 K K i C K i K i K ω = + = + ϑ = + ξη (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 ϑ is taken to be a constant value often having the form ϑ= 2β such that for a spring-damper element without mass (Voigt’s model) one gets
(
1 2)
K i = + β K (2.40) Or more general(
)
1 1 1 2 with 2 I I R I R R R R i i i = + = + = + β β = K K K 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)
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 ω
= − 2
K K M (3.2)
where K is the impedance matrix of the mass-less foundation (therefore index F) and F
Mis the mass matrix of the block related to the chosen point of reference. The solution to the equation of motion is
− =K 1 =F
u P P (3.3)
with amplitudes |ui| and phase angles i
( ) ( )
, tan i=1,2,..6 I R I i i i i i R i u u u u u ψ = 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 u u a u (3.4)
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 h u u h θ θ = + = − (3.5)
The transformation yields
0 0 0 0 0 0 0 0 0 0 T T G G G G G T T G G G G G = = = = K K K K a a P a P a a P a P (3.6)
3.2 Vertical Displacements
2 2 1 v v v v v F F v v v P P u P P M K M i C ω ω ω = = = = − − + v F K K (3.7)3.3 Torsional Rotation
2 2 1 t t t t t v F F t It Kt It i Ct θ ω ω ω = = = = − − + t M M M F M K K (3.8)3.4 Horizontal Displacement and Rotation (motion in x-z plane)
Reference point 0 0 0 0 0 0 1 0 1 hx hx G ry P u h θ − = = = Mry a 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 θ = K K K K M (3.10a) (3.10b) with 2 2 0 0 0 0 0 F F T G G G ω ω = − = − 0 K K M K a M a (3.11) , , 0 0 2 , , 0 0 0 0 F F hx hx hx ry F G F F ry ry hx ry ry G r M M hM I hM h M I − = = =− + K K K K K M M (3.12)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 P h P ω ω ω ω ω ω = − + = − + = − + + = = − + K K K M M (3.13) Reference point G 0 1 0 1 hx hx G G G ry G G P u h θ = = = a u P ry M (3.14) G G = G K u P or , , , , hx hx hx ry hx hx ry hx ry ry G ry G ry G u P θ = K K K K M (3.15a), (3.15b) with 2 0 T F G =aoG aoG−ω G K K M (3.16) , , , 2 , , , , , 0 2 0 F F F hx hx hx ry hx hx F G F F F F F G ry hx hx hx ry ry hx ry hx hx G r G h M h h h I + = = + + + K K K K K K K K K M (3.17)More explicitly equation (3.15):
2 , , , , , , 2 2 , , , , , , , , 2 F hx hx G hx hx F hx ry G hx ry F F F ry ry G ry ry hx ry hx hx r hx G hG ry G hG ry G M h h I P P P ω ω = − = = + + − = = + K K K K K K K K M M (3.18) where , ; , ; , ; , F i =Ki +i Cω i where i=hx hx hx ry ry hx ry ry K
Finally we get for the complex displacement for the point of reference:
( ) ( )
2 2 tan tan R I hx hx hx R I hx hx hx I I hx hx hx R hx R hx hx u u u u u u u u u u ψ ϕ = + = + = = − (3.19)and
( ) ( )
2 2 tan tan R I ry ry ry R I ry ry ry I I ry ry ry R ry R ry ry θ θ θ θ θ θ θ θ ψ ϕ θ θ = + = + = = − (3.20)The time harmonic motion is:
( ) ( ) ( ) i t i t i i t i hx hx hx hx u t =u eω = u e ω ψ+ = u e ω ϕ− (3.21) and ( ) ( ) ( ) i t i t i i t i ry t rye ry e ry e ω ψ ω ϕ ω θ =θ =θ + =θ − (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.
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 (a )= R +i I k k k (4.1) or the compliance : ( ) R I f a0 = f +i f (4.2)
In both cases, a0 is the dimension-less circular frequency defined by :
0 S a = c (4.3) in which :
- ω 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
Mode 2-D 3-D Translations G =K k G B = K k Rotations G B = K 2 k G B = K 3 k Coupled Transl.-Rot. G B = K k G B = K 2 k
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.
x x hG a H 2 B G, ν, ρ β G Phx e i ω t M = 6 000 kg/m Iry = 4 000 kgm2/m a = 1,5 m hG = 0,7 m G = 45 MPa ν = 0,3 ρ = 2 000 kg/m3 β = 5 % B = 1m H = 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 :
- ν = 0,3 - β = 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*π*16,71 * 1
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 fhx R - fry I - fhx I fry R a0 = 0,75 a0 = ω B / cS 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 : ωe = π
2 cS
H
or in dimensionless form and with our numerical values : a0 e = π
2 B
H = 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)
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 (or from Figure 4.3)
khxR (0,7) = 0,595 0,5952+0,292 =1,358 kry R(0, 7) 0,43 0,432+0, 042 =2,306 khxI (0,7)= 0,29 0,5952+0,292 = 0,662 kIry(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 khx R a0 = ω B / cS kry R khx I kry I
Fig. 4.3. Impedance functions Numerical calculation
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 R hx I hx R ry I ry = = = = = = = = (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 ω t x x G A x xG A Mry = Mry e i ω t Phx e i ω 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 7 11 7 7 12 7 7 22 (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 i k i i k i i i = + − = − + = + = + = + + + − = + (NC.6) uhxR = −3,99I PhxI10−8 uhxI = −2,78I PhxI 10−8 θryR =0,91I P hxI 10−8 θryI =2,01I PhxI 10−8 (NC.7) I uhx (16, 71)I=4,87IPhxI 10−8 Phase (uhx)=145,1° Iθry(16, 71)I=2,21IPhxI 10−8 Phase (θry)= −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 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 u
hxI / I P
hxI
I
θ
ryI / I P
hxI
14,3 Hz
22,7 Hz
Hz
10
-8Fig. 4.5. Amplitudes versus frequency
-200
-150
-100
-50
0
50
100
150
200
0
10
20
30
40
50
Phase of
θ
ryHz
Phase of uhx
14,3 Hz
degrees
90 °
- 90 °
Fig. 4.6. Phases versus frequency
4.4 Available results
- 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 0
K = K [k (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 β of the soil.
a0 = ω r
cs (4.7)
c= C cs
r Kst (4.8)
Mode Static stiffness Kst Vertical 4 G r 1− ν Horizontal 8 G r 2− ν Rocking 8 G r 3 3 (1− ν) Torsion 16 G r 3 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.
2 B 2 B Gazetas 2 B Gazetas Gazetas - Huh 2 B H Huh H 2 B D 2 r
Luco - Gazetas - Veletsos r 2 r Luco Kausel - Tassoulas H 2 r D d Kausel - Luco 2 r H Apsel 2 r D Dominguez 2 L 2 B
Dominguez - Wong - Rücker - Schmid 2 B
D
2 B
4.5 Simplified methods
4.5.1. Circular footingsIn 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 cS 2 Kst r2 Horizontal 1 0,58 M cS 2 Kst r2 Rocking 1 0, 30 1+B r Ir cS2 Kst r2 * Torsion 1 B t 1+32Bt/ 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 r 1− ν 1+1, 28 r H 1+ D 2 r 1+(0,85−0,28 D r) D H 1−D H Horizontal 8 G r 2− ν 1+ r 2H 1+ 2D 3 r 1+ 5D 4 H Rocking 3 8 G r r 2D D 1 1 1 0, 7 3(1 ) 6H r H Coupled Horizontal - Rocking 0,40 Ksth D Torsion 16 G r 3 3 1+2,67 D r
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 D / H
Torsion H / r
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 π 1 / 2 Rocking
around the x-axis
16 B3 L 3π 1 / 4 Rocking
around the x-axis
16 B L3 3π 1 / 4 Torsion 8 B L (B2+L2) 3π 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.
0,6 machine 2617 kg 4 cos ω t (kN) 1,5 m half space G = 54 MPa ν = 0,3 ρ = 1850 kg/m3 0,8 square footing ρ = 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 π 1 / 2 =0,846 m - the vertical static stiffness (see table 4.4) :
6 8 4 * 54 *10 * 0,846 0, 6 K 1 3,536 *10 / 1 0, 3 2* 0,846 st 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):
3 0 0,846 4,952 *10 54000 1,85 a = ω = − ω
We can also make explicit the formulation of the impedance function (see equation 4.5):
8 3 3, 536*10 (1 4,952 *10 *0,85 ) K = +i − ω = K+ i C ω 8 6 3.538 10 1.488 10 K = ⋅ +i ⋅ ω 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 v u ω ω = − +
and the amplitude of the vertical acceleration :
2 v u =ω 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/s2
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 methodWe 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)
( )t = ei tω
u u (5.2)
where u is determined from:
(
)
{
2}
i ω ω − + = K M C u P (5.3) where(
)
s i ω ω = − + K 2 C K 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 u (6,1) at the 0
point 0 (center of the base interface). The corresponding forces acting at the point 0 are
0 P . 1 2 3 1 2 3 u u u θ θ θ = 0 u 1 2 3 1 2 3 P P P M M M = 0 P (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 P
5.1.1.3 Coupling of Structure And Soil Step 1: Kinematic constraint of basement
Structural model rigid rigid interaction nodes 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 S S S SS SI S S I I IS II = u P K K u P K K (5.7)
The kinematical connection between u and I u is expressed by the transformation: 0
0
I =
u au (5.8)
By applying the principle of virtual works one obtains:
0 0 S S S S SS SI T S T S IS II = u P K K a u P a 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 m 0 0 0 S S S S SS SI T S T S IS II = + u P K K a u P a K a K a KF (5.10)
5.2 Flexible Structure On Flexible, Mass-less Foundation
5.2.1 SoilThe 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) FEM+ Thin Layer Method brick elements
Thin Layer Method/ Flexible Volume Method
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 K (index F for foundation) interface can be obtained as the relation F between displacement and forces related to the m degrees of freedom at the interface nodes (Fig. 5.3a, b).
F
K can be understood as hyper-element matrix. The direct stiffness matrix couples the two sub-structures. n-m m n-m m S S S S SS SI S S S F I IS II II I = + u K K P u K K K P (5.11)
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 P or i u i = 1, 2, 3...m has to be given. Their i corresponding unknown values are calculated.
2) Since K is regular for unbounded soil (no rigid body motion),F
( , )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 1 ( , ) ( , ) F m m m m − = K F
The elements Fij of the matrix F or obtained as the displacements ui 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 , of the foundation volume (to be excavated) discretized by FII 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 S S S SS SI S S E F I I IS II II II = − + u P K K u P K K K K (5.12)
5.3 Load Cases
a) Specified loads - Wind - Traffic - Explosion - MachinesWind
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, P would be zero. If I traffic loads are considered, P would be zero and S P would be specified at “ interaction I
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:
’’
( )
S S F
IS S + II I + II I − I =
K u K u K u u 0 (5.13)
Whence the equation of motion becomes
S S S SS SI S S F I I IS II II = + u 0 K K u P K K K (5.14) with PI =K uFII ′′I
In the case, when the free-field is known, the equation of motion is: S S S SS SI S S E F I I IS II II II = − + − ′ u 0 K K u u 0 K K K K or S S S SS SI S S E F I I IS II II II = − + u 0 K K u P K K K K (5.15)
where PI =K u and FII ’I KEII 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 i i i u u t = u ω ψt+ (5.17)( ) ( )
( ) ( )
2 2 2 2 tan tan i i i i i R I i i i i R I i i i i I R I i i i i P R i I R I i i i i u R i P P iP P e u u iu u e P P P P P u u u u u ψ ϕ ψ ψ = + = = + = = + = = + =Literature
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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 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 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
Vertical and Torsional Stiffness of Cylindrical Footings 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)
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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 = 0
u au (A.2)
where the vector u0 has dimension (m,1), m<n. Its corresponding force vector is P0.
They are related by
0 = 0 0
P K u (A.3)
The energy of the deformation may be expressed in term of both pairs of variables:
0 0 T T E=P u=P u (A.4) or using (A.2) 0 0 0 T T E=P au =P u (A.5) from there 0 T = P a P (A.6)
Connecting (A.4) and (A.2) yields
0 0 0
T = T
P au P u (A.7)
From (A.4), (A.1), (A.3) and then (A.2) one obtains
0 0 0 0 0 0 0 0 T T T T T = = u K u u K u u a Kau u K u (A.8) From there 0 T = K a K a (A.9)
Appendix B – Dimension-less equation of motion in frequency domain
The equation of motion with dimensions is
2 K K K K M hh hr hh hr h h rh rr rh rr r r M M u P M M ω ϕ − = (B.1)
Definitions of dimension-less quantities:
0 2 3 M M h h r h h r s u P B a u P c B GB GB ω ′ ϕ ϕ′ ′ ′ = = = = = (B.2) or reciprocal 2 3 0 s M M h h h h r r a c u Bu P GB P GB B ω= = ′ ϕ ϕ= ′ = ′ = ′ (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 = cs 2 ) – soil density (kg/m 3 ). ,
u′ ′ϕ – dimension-less displacement (-) and rotation (-), respectively ,M
h r
P′ ′ - dimension-less force (-) and moment (-), respectively
2 2 2 0 2 3 0 0 0 1 0 K K K K M hh hr s hh hr h h rh rr rh rr r r M M B u GB P c a M M B ϕ GB ′ ′ − = ′ ′ (B.4) 2 2 2 0 2 3 1 0 0 1 0 1 0 K K K K M hh hr s hh hr h h rh rr rh rr r r M M B u P c GB a M M B GB ϕ − ′ = ′ ′ ′ (B.5) 2 2 2 2 0 2 2 3 3 1 1 K K M K K hh hr hh hr h h s rh rr r r rh rr B M B M u P c GB GB a M B M B GB B B B GB GB ϕ ′ − ′ = (B.6) 3 4 2 2 0 2 3 4 5 K K M K K hh hr hh hr h h r r rh rr rh rr M M u P B B GB GB a M M GB GB B B ρ ρ ϕ ρ ρ ′ ′ − = ′ ′ (B.7)
Finally the equation of motion in dimension-less notation:
2 0 K K K K M hh hr hh hr h h rh rr rh rr r r M M u P a M M ϕ ′ ′ ′ ′ ′ ′ − = ′ ′ ′ ′ ′ ′ (B.8)
