1. No category

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Impedance matrix for four adjacent rigid surface foundations
1
M. Radisic1, G. Mueller2, M. Petronijevic1
Department for Engineering Mechanics and Theory of Structures, Faculty of Civil Engineering, University of Belgrade
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia
2
Tehnical University Munich, Arcisstrasse 21, D-80333 Munich, Germany
email: mradisic@grf.bg.ac.rs, gerhard.mueller@bv.tu-muenchen.de, pmira@grf.bg.ac.rs
ABSTRACT: In this paper, a frequency dependent impedance matrix for four adjacent rigid foundations resting on the
viscoelastic halfspace is presented. The frequency dependent impedance matrix is developed using the integral transform
method. Non-relaxed boundary conditions, without separation or sliding, between the soil and foundation are assumed. First, the
compliances for one rigid foundation on the half space are calculated and compared with the results from the literature. Than the
impedance matrix of a single foundation as well as the impedance matrix of four rigid foundations are obtained. The influence
between adjacent rigid surface foundations resting on the half space is presented.
KEY WORDS: Soil-structure interaction; Integral Transform Method; Impedance; Halfspace; Frequency domain; Wave-number
domain; Foundation.
1
INTRODUCTION
Soil-structure interaction (SSI) is an important part of the
dynamic analysis of structures. Although it is often neglected
by assuming that the structure is mounted on a rigid bedrock,
it is well known that the SSI can significantly change the
structural response [1]. SSI can be taken into account by
considering displacements of the soil in the soil-structure
interaction nodes with the help of impedance functions at the
foundations [2]. If we consider a system subjected to a force P
with the resulting displacement u, the impedance of the
system here is defined as the quotient of the load P to the
response u. Impedance function is often presented as a nondimensional dynamic stiffness of the foundation against a
non-dimensional frequency [3]. While the literature is focused
on impedance functions of a single foundation, this paper
describes the process of the calculation of impedance
functions of a system of four adjacent foundations.
The basis of the calculation of the impedance functions is
the solution of the halfspace subjected to a dynamic unit force
load. There are several methods established for solving this
type of problem: Finite Element Method (FEM) [4], [5], Thin
Layer Method (TLM) [6], Boundary Element Method (BEM)
[7] and Integral Transform Method (ITM) [8]. In this paper
ITM is chosen as the most appropriate method that provides a
semi-analytical solution which fulfills the radiation condition
and can easily be expanded to a layered medium [9].
The main goal of this work is to develop a computer
program for calculation of impedance functions of a single
foundation and four coupled foundations on the halfspace
using Matlab [10].
The schematic usage of ITM with the theoretical
background is presented in the second chapter. The third
chapter describes the procedure of obtaining the impedance
functions for a single foundation and for the system of
foundations. Chapter four deals with the comparison of the
results of the numerical analysis between two different
models: a single foundation and a system of four foundations.
2
INTEGRAL TRANSFORM METHOD
The Integral Transform Method is based on the analytical
solution of Lamé’s differential equations of motion of the
continuum:
2u  (   )·u  u ,
(1)
where  is mass density of the material, u is displacement
vector and  and  are Lamé’s material constants through
which the damping model is introduced, Eq. (2).
E
1  2i
1  2i
.
,   2 E
2(1  )
1  2
(2)
In Eq. (2) E is elasticity modulus,  is Poisson’s coefficient
and  is damping ratio.
Lamé’s equations of motion (1) can be brought into the
form of wave equations:
2 
1
,
c 2p
2 ψ 
1
ψ,
cs2
(3)
where the displacement vector is expressed by the scalar field
 and the vector field ψ that have to satisfy the relation (4),
according to the Helmholtz’s principle.
u      ψ .
(4)
In Eqs. (3) c p and cs are the velocities of the dilatational and
shear waves, respectively.
c 2p 
  2
,

cs2 

.

(5)
In order to find the solution of the system of equations (3)
they are transferred from the space/time domain (x,y,z,t) into
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
the wave number/frequency domain (kx,ky,z,ω) by using
threefold Fourier transform, Eq (6).
ˆf ( k ,k , ) 
x
y
  

f ( x, y,t )e
 i( k x x  k y y t )
dxdydt .
harmonic forces in x, y, z direction at the surface of the
halfspace, presented in Figure 2, using ITM.
(6)
  
By assuming that ψz=0, the system of partial differential
equations (3) becomes a system of three decoupled ordinary
differential equations with six unknown coefficients of
integration, C. Taking into account Sommerfeld’s radiation
condition and the boundary conditions at the surface of the
halfspace, the coefficients of integration are obtained. The
connection between the displacement vector and the
coefficients of integration is derived from Eq. (4) in the
following form:
ˆ  Au · C ,
u
Figure 2. Load cases
These load cases can be considered as a fundamental solution
and can be established in the same manner for a layered halfspace.
(7)
where
 ik x
uˆ x 

 
u
ˆ  uˆ y  , A   ik y
u
uˆ 
 1
 z

2 
0
2
ik y
 A2 
 
C   B2 x  ,
B 
 2y 

0 ,
ik x 
(8)
and
12  k x2  k y2  k p2 ,
kp 
22  k x2  k y2  ks2 ,
ks 

,
cp

cs
(9)
.
Once the displacements, u , in the transformed domain are
obtained, the displacements, u , in the original domain are
calculated by using threefold inverse Fourier transform [11]:
f ( x, y,t ) 
  
·

1
( 2 )3
·
(10)
ˆf ( k ,k , )ei( kx x  k y y t ) dk dk d  .
x
y
x
y
  
The ITM is presented on Figure 1 [12].
Figure 3. Shifting process - position 3
Figure 3 and Figure 4 explain the process of obtaining the
dynamic flexibility matrix Ff. Red grid defines the nodes of
the soil, while the blue surface represents the displacements of
the surface of the soil. On these figures, only the vertical
displacements due to the vertical force are presented. If n
nodes are assumed, the size of the flexibility matrix of the soil
is (3n,3n). As the vertical displacement is the third component
of the displacement vector, Eq. (8), Figure 3 shows the values
of the third column of the dynamic flexibility matrix Ff. By
shifting the blue surface to the arbitrarily chosen position j,
the values of the 3jth column of the flexibility matrix are
obtained, (Figure 4). Successively changing the position and
direction of the unit force, leads to the dynamic flexibility
matrix Ff .
Figure 1. ITM Scheme.
3
3.1
IMPEDANCE FUNCTIONS
Single Foundation
The impedance functions are calculated in the Cartesian, three
dimensional coordinates system. First, the dynamic flexibility
matrix Ff of the soil is determined. The displacements of the
surface of the halfspace are calculated for single unit
654
Figure 4. Shifting process - position j
The size of the area for which the halfspace displacements
are calculated has to cope with the size of the problem. The
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
discretization in the wave number domain depends on the
calculation frequency range and the soil characteristics,
resulting in characteristic wave-lengths. Undesired effects of
the Fourier Transform, such as aliasing [12], [12], [13] have to
be considered.
The dynamic stiffness matrix of the soil Kf is obtained by
inverting the flexibility matrix Ff:
K f  Ff1 .
(11)
This step is very sensitive in a way that can easily cause
important numerical errors, due to the flexibility matrix
condition.
In order to achieve the convergence of the results, a
distributed load over a significantly small area is used instead
of a unit force load. That could cause the flexibility matrix to
be close to singular, as the load cases are overlapped.
Therefore the displacements of the neighboring nodes, which
correspond to adjacent matrix columns, are similar. Therefore,
a new, coarser grid is introduced for the purpose of calculating
the flexibility matrix. Besides that, the elements of the
flexibility matrix are calculated as mean value of
displacement over the chosen area around the nodes, instead
of the single point value at the node location, Equation (12).
uˆ 
1 0 0
ai  0 1 0
0 0 1
F
(12)
F
The dynamic stiffness matrix of the rigid foundation is
obtained by using the energy principle that equates the
deformation energy of the flexible and the rigid foundation,
[14].
ˆ Tu
ˆ Tu
ˆ f P
ˆr.
P
f
r
(13)
It is adopted that the rigid foundation has six degrees of
freedom, three translations and three rotations of the centroid.
ˆ tr  uˆ x
u
uˆ y
uˆ z
ˆ x ˆ y ˆ z  ,
ˆt  P
ˆ
P
r
x
ˆ
P
y
ˆ
P
z
ˆ
M
x
ˆ
M
y

ˆ .
M
z
ai
The submatrices ai are obtained by the kinematic
consideration, as presented in Figure 5 and Equation (17). In
Eq. (17), xi and yi are coordinates of the node A=i and O is the
centroid of the foundation. The size of the matrix a is (3n,6),
where n is the number of the foundation nodes.
ˆ =K
ˆ u
ˆf,
P
f
f
(18)
ˆ K
ˆ u
ˆr .
P
r
r
(19)
(20)
f
r
The stiffness matrix of the rigid foundation Eq. (21) has the
form of a diagonal matrix with non-diagonal elements
regarding additional rotational stiffnesses to the horizontal
translational stiffnesses, and vice versa.
 K xx
 0

 0
Kr  
 0
 K my , x

 0
(14)
(16)
(17)
Regarding previously adopted assumptions, the sizes of the
ˆ are (3n,3n) and (6,6), respectively.
ˆ and K
matrices K
(15)
anxn  .
 yi 
xi  .
0 
ˆ  aT K
ˆ a.
K
r
f
where
at  a1
0
0
 xi
Taking into account equations (13) and (15) the following
relation is obtained:
The vectors of the nodal displacements ûf and ûr are related
with kinematics matrix a:
ˆ f =a  u
ˆr,
u
0
0
yi
The relation between the nodal displacements and the
corresponding force vectors for the flexible and rigid
foundation is given by equations:
 uˆ( x, y)dF
 dF

Figure 5. Kinematic transformation
3.2
0
0
0
K x , my
K yy
0
K y , mx
0
0
K zz
0
0
K mx , y
0
K mx
0
0
0
0
K my
0
0
0
0
0 
0 
0 
.
0 
0 

K mz 
(21)
System of the Foundations
The procedure for calculating the impedance for the system of
foundations is similar to the calculation of the impedance
function of a single foundation.
Once the displacements of the soil are calculated, the
flexibility matrix of the system of the flexible foundations is
obtained by using the shifting method described in the
previous section. For the system of four foundations, where
each one is divided into n nodes, the size оf the global
flexibility matrix is (4×(3n,3n))=(12n,12n). The global
stiffness matrix of the flexible foundations is obtained by
inverting the flexibility matrix. The global dynamic stiffness
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
matrix of rigid foundations is calculated using the same
kinematics principle, as for one foundation:
ˆ  aT K
ˆ a.
K
r
f
(22)
In Eq. (22) a is a diagonal block matrix, consisting of
kinematics matrices a I , a II , a III , a IV for each foundation,
given by Eqs. (16) and (17):
a I
0
a
0

0
0
a II
0
0
0
0
a III
0
0 
0 
.
0 

a IV 
(23)
Figure 6. Single Foundation Model
Therefore, the size of the matrix a is (4×(3n,6))=(12n,24).
Considering Equation (22) it is evident that the size of the
stiffness of the system of rigid foundation is
(4×(6,6))=(24,24).
3.3
Dynamic Stiffness
The stiffness matrix of the group of four foundations is
calculated for every frequency inside the chosen interval. A
dimensionless frequency is defined as:
a0 
B
cs
(24)
Figure 7. Four Foundations Model
where  is the radial frequency, B is the half size of the
foundation dimension and cs is the shear wave velocity in the
soil.
The dimensionless stiffness is defined as:
K0 
K
coeff
(25)
where K is the observed dynamic stiffness and coeff is the
corresponding coefficient of division. In the case of the
translational stiffness, coeff is defined as:
coeff  GB
(26)
where G=μ is the shear modulus of the soil, Eq. (2).
In the case of rotational stiffness, coeff is defined as:
coeff  GB3
4
(27)
NUMERICAL ANALYSIS
The results of two numerical models are analyzed and
compared. One model is a single foundation system, shown in
Figure 6, while the other model is a system of four
foundations, presented in Figure 7.
656
In section 4.1 the dimensionless flexibilities Fj, j=h,v,r
(compliance functions) are presented instead of impedance
functions in order to be able to compare the results with the
results from the literature [3]. However, impedance functions
are more convenient for the analysis of the soil-structure
interaction problems. Therefore, section 4.2 shows the
comparison between a single foundation and a system of
foundations, regarding impedance functions.
4.1
Single Foundation Model
Horizontal, vertical and rotational compliance functions of a
single, square foundation are calculated using ITM. The
results are compared with the results from the literature [3].
The damping ration ξ (treated as zero in [3]), is chosen to be
2% in order to avoid numerical errors from spatial aliasing.
Poisson’s coefficient ν is equal to 1/3. The foundation is
square, B/L = 1. Real and imaginary values of the compliance
functions are presented in the terms of dimensionless
frequency a0 in Figures 8, 9 and 10, respectively. The dashed
line presents the results from the literature [3], the solid line
the results obtained by the method described in this paper
(ITM).
While the vertical impedances are equal compared to the
results from the literature, Figure 9, the horizontal and
rotational impedances show some discrepancies, Figure 8 and
Figure 10. That might be due to the fact that horizontal and
rotational impedances, obtained by using the procedure
described in this paper, are coupled (Equation (21)), unlike
those from the literature.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
4.2
Four Foundations Model
The influence of coupling of four foundations is analyzed
using the model presented in Figure 7. All foundations are
square (BxL=1x1m). The results are calculated for X=2m and
X=5m. Soil properties are the same as in section 4.1. The
symbols Fijk, used in Figures 11 and 12 indicate the vertical
compliance of the foundation i, in a group of k foundations,
when the foundation j is loaded with a vertical force. The
symbol Kijk is used for the impedances, likewise.
Figure 11 shows the vertical compliance of the foundations
2 and 3, when the foundation 1 is loaded and the distance
between adjacent foundations is 2m. F214 has greater peak
amplitudes than F414 since foundation 2 is closer to foundation
1. F314 is equal to F214 while F114 is equal to the vertical
compliance of the single foundation system (Figure 9).
Figure 8. Real (Re) and Imaginary (Im) part of horizontal
compliance for single foundation
Figure 11. Real (Re) and Imaginary (Im) part of vertical
compliances for system of foundations, X = 2m
Figure 9. Real (Re) and Imaginary (Im) part of vertical
compliance for single foundation
Figure 12. Real (Re) and Imaginary (Im) part of vertical
compliance F214 for system of foundations
Figure 10. Real (Re) and Imaginary (Im) part of rocking
compliance for single foundation
In Figure 12 the influence of the distance between
foundations is examined. The vertical compliances of the
foundation 2 due to the unit load at the foundation 1, F21,
when distances between two adjacent foundations are X=2m
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
and X=5m is presented. It shows that the values of the
compliance depend on the distance X and the frequency.
The sign of the compliance function of the adjacent
foundation depends of the wavelength of the shear waves of
the halfspace. Therefore, compliance functions Fij4, i≠j,
oscillate around zero value (Figures 11 and 12).
To validate obtained results, the ratio F114/F214 is calculated
for the static loading case (a0=0), when the distance X
between the foundations is 2m and 5m. The solutions
proposed by Steinbrenner and Kany (SB-K) [15] are used.
F114 for rigid foundation is calculated applying the Kany’s
method of equivalent displacement of the flexible foundation.
For F214 only the Steinbrenner’s method for flexible
foundation can be applied. The comparison of the results is
given in Table 1. The results can be accepted as satisfactory,
since the applied method for the static solution, SB-K, is
approximate in comparison with the solution obtained by
ITM.
parts of K214 and K414 oscillate with a frequency taking
negative values, opposite to the imaginary part of K114 which
is always positive and ascendant with frequency.
Table 1. F114/F214 ratio
X [m]
2
5
SB-K
4.99
13.19
ITM
5.61
10.92
Figure 14. Real (Re) and Imaginary (Im) part of vertical
impedance Ki1, X = 2m
Figure 13 shows the vertical impedances of the foundation 1
for a single and two coupled cases. There are just
slightdifferences between K111 and K114 for X=5m. However,
the differences became visible for smaller distance between
foundations, X=2m. The mutual influence of adjacent
foundations is more prominent for lower values of X so it
affects the values of impedances.
Figure 13. Real (Re) and Imaginary (Im) part of vertical
impedance K11
4
4
Figure 14 shows the impedances K21 and K41 of the
adjacent foundations 2 and 4, for X=2m, when the unit load is
applied on the foundation 1. K314 is not shown since it is equal
to K214. The absolute values of the real parts are more than 8
times less than the real part of K114 , Fig. 13. The imaginary
658
CONCLUSIONS
Impedance functions are important input parameter for the
soil-structure interaction analysis that can significantly change
the structural response. They can be calculated using different
numerical or analytical methods. This paper presents the
procedure for determining the impedance functions of single,
rigid square foundation on the halfspace, using the Integral
Transform Method. The proposed procedure is extended to a
system of four coupled foundations and the frequency
dependent impedances of the soil-foundations system are
obtained.
Numerical results presented in this paper correspond to two
different numerical models: a single foundation and a system
of four foundations. In order to verify the proposed
methodology, impedance functions of the single foundation
system are compared with the results from the literature. The
results of the comparison are satisfactory.
The response of the system of the foundations shows that
the mutual impact of the adjacent foundations decreases with
increasing the distance. Still, the coupling of the foundations
is weak, since they are not coupled with the structure, but with
the soil only.
The continuation of this research will include the influence
of adjacent rigid foundations resting on the layered halfspace
with parametric analysis regarding the influence of the effect
of the soil layer depth, foundations mass, distance ratios and
damping levels.
ACKNOWLEDGMENTS
We are grateful that this research is partly financially
supported through the Project TR 36046 by the Ministry of
Education, Science and Technology, Republic of Serbia and
partly by the DAAD (The German Academic Exchange
Service) in the frame of SEEFORM Project (The South East
European Graduate School for Master and PhD Formation).
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
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