2010/2 PAGES 1 – 8 RECEIVED 21. 9. 2009 ACCEPTED 20. 1. 2010
Y. KOLEKOVÁ, M. PETRONIJEVIĆ,
G. SCHMID
Yvona KOLEKOVÁ
SPECIAL DYNAMIC SOILSTRUCTURE ANALYSIS
PROCEDURES DEMONSTATED
FOR TWO TOWER-LIKE
STRUCTURES
email: [email protected]
Research field: Soil-structure Interaction, Stability
problems, Seismicity
Address: Department of Structural Mechanics ,
Faculty of Civil Engineering,
Slovak University of Technology,
Radlinského 13, 813 68, Bratislava, Slovakia
Mira PETRONIJEVIĆ
email: [email protected]
Research field: Structural Analysis, Dynamic of
Structures, Trafic induced vibrations
Address: Građevinski fakultet, Bul. Kralja Aleksandra 73,
11000 Beograd
Günther SCHMID
email: [email protected]
Research field: Dynamics of Structures, Earthquake
structural response, Dynamics of Sub-soil
Address: Građevinski fakultet, Bul. Kralja Aleksandra 73,
11000 Beograd
ABSTRACT
KEY WORDS
Many problems in Earthquake Engineering require the modeling of the structure as a dynamic
system including the sub-grade. A structural engineer is usually familiar with the Finite
Element Method but has a problem modeling the sub-grade when its infinite extension and
wave propagation are the essential features. If the dynamic equation of a soil-structure system
is written in a frequency domain and the variables of the system are total displacements, then
the governing equations are given as in statics. The dynamic stiffness matrix of the system is
obtained as the sum of the stiffnesses of the structure and sub-grade sub-structures. To
illustrate the influence of the sub-grade on the dynamic behavior of the structure, the frequency
response of two tower-like structures excited by a seismic harmonic wave field is shown. The
sub-grade is modeled as an elastic homogeneous half-space. The structure is modeled as
a finite beam element with lumped masses.
• Soil-structure interaction,
• frequency domain,
• spectral elements
1. INTRODUCTION
Structures are founded on a sub-grade. In a dynamic analysis the
total system may therefore be understood as consisting of two substructures: the structure and the sub-grade. Depending upon the
relations of the mass and stiffness between the sub-structures, their
interaction effects may be neglected. If the sub-grade is much stiffer
than the structure, the sub-grade can be replaced by the kinematic
boundary conditions at the structure - sub-grade interface. If the
structure is much stiffer than the sub-grade, the structure can be
modeled as a rigid body resting on a deformable sub-grade. In other
cases the kinematic and inertial interaction between the two substructures should be considered in a dynamic analysis of the system.
Such a situation will be discussed in this presentation.
We will assume that the external forces are zero at the structure
and along the structure - sub-grade interface and that an excitation
of the system will arise from a seismic ground motion, which we
can assume as known at the location of the (future) structure - subgrade interface. These displacements are part of the seismic free
field (a wave field not disturbed by the structure). If the mass of the
structure is big, the inertial interaction will contribute considerably
to the structure - sub-grade interaction.
In this presentation we will further assume that the system behaves
linearly with respect to loading. One can observe in figure 1, where
the system is indicated, that the structure has a finite extension,
whereas the sub-grade may extend to infinity. Therefore, the
vibration energy introduced into the system is trapped by the finite
boundaries of the structure but can radiate away from the system
2010 SLOVAK UNIVERSITY OF TECHNOLOGY
Kolekova.indd 1
1
21. 6. 2010 9:43:34
2010/21 PAGES 1 — 8
in the sub-grade. This dissipation of energy results in the damping
of the motion of the structure even if no material damping is
assumed.
In the following the system is represented in the frequency
domain through its dynamic stiffness matrix (see section 2). It is
obtained according to the direct stiffness method as the sum of the
dynamic stiffness matrices
(upper index S for the structure) and
(upper index F for the foundation). The system displacements
are the discrete total displacements, including the inertial and the
kinematic parts of the interaction as well as the rigid body motion of
the structure. The tilde indicates a complex valued function.
The terms in equation (2) have the same meaning as in statics. But
in a frequency domain they are complex, representing harmonic
motions defined by teirs amplitudes and phases, such as, for
example
(3a)
which represents the harmonic motion
with
and
(3b)
2. EQUATION OF THE MOTION IN A FREQUENCY
DOMAIN
For the derivation of the equation of motion, we further assume
that at the source of the earthquake we know the soil-motion uG (t)
from which we obtain its frequency content
through the
Fourier Transform. The displacements
due to the source
excitation in the system can be represented by the discrete values
.
A corresponding nodal force value belongs to each term . These
values are related through the relations
Alternatively, the response in the time domain can be obtained by an
inverse Fourier Transform from equation (3a).
3. STRUCTURE-SOIL SYSTEM
Since the dynamic behaviour is different for the structure and the
sub-grade, one should model each “sub-structure” of the system by
the “appropriate” method.
Structure
or
.
(1)
The most well-established procedure for the structure is the Finite
Element Method. The linear equation of motion for a structure in
a time domain results in
In equation (1)
and
are the dynamic flexibility and stiffness
terms, respecctively of the equation of motion in the frequency
domain, which can be written as a partitioned matrix equation:
or
.
(2)
(4)
where
are the mass, damping and stiffness matrices of the
structure, respectively.
and
are the time histories of the
nodal displacements and nodal forces. The Fourier transformation
of equation (4) leads to the equation of motion in the frequency
domain:
(5a)
structure
or with the partitioning shown in equation (2)
sub-grade
(5b)
or
base rock
(wave source)
Fig. 1 Structure-soil model, left; system partitioning, right
2
Kolekova.indd 2
with
.
(5c)
is the dynamic stiffness of the structure. Note that equation (5)
can alternatively be obtained by assuming the load and response
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
21. 6. 2010 9:43:36
2010/2 PAGES 1 — 8
as the time harmonic functions
and inserting them
in equation (4). In equation (5) the dynamic stiffness matrix is
constructed from element matrices which are based on shape
functions that approximate the static deformation within the element
due to the unit deformations of its degrees of freedom (DOF). For
beam elements the shape functions are the solutions of the static
equilibrium equation of the beam, which satisfies the kinematic
beam end conditions. For static analyses the nodal values of the
beam structure are therefore exact. For dynamic analyses of beam
structures the shape functions of one structural beam element cannot
represent the vibration forms excited at high frequencies with its
shape functions. For higher frequencies the beam therefore has to
be subdivided into many elements to improve the approximation. If
the solution of the wave equation of the beam are chosen as shape
functions, then these shape functions are frequency dependent sin,
cos, sinh and cosh functions defined by the beam’s kinematic end
conditions. They are exact for any frequency. The corresponding
, are called
elements, with dynamic element stiffness
spectral elements. With spectral elements the dynamic stiffness
matrix of the structure is obtained through the usual FE-procedure
, with E as the number of spectral elements. For
as
.
(8)
Usually the motion at the source is not available, instead, the free
field motion
at the site, which is defined as the earthquake
motion of the sub-grade without the structure, is available. In this
case we can use equation (7) without the structure to express the
effective earthquake forces
, which were transferred from
the sub-grade to the soil-structure interface, through the free field
motion. The corresponding equation of motion is:
.
(9)
From the first equation in (9) we have at the soil-structure interface
the relation
.
(10)
Equations (8) and (10) give the dynamic system equation
further detail see [1, 2].
with
Foundation
We assume at the moment that we have numerical methods available,
which enable us to obtain the dynamic stiffness of the foundation,
including the sub-grade defined for the degrees of freedom at the
source G and the soil-structure interface I (see figure 1, right). The
corresponding dynamic stiffness matrix is
.
.
Equation (11) allows us to calculate the structural vibration at the
soil-structure interface and at the remaining nodal degrees of freedom
of the structure. These deformation are the total displacements (rigid
body motion plus elastic deformations) created by the free field
motion at the site. The system displacements contain the kinematic
interaction and the inertial interaction.
(6)
System
The kinematic interface conditions
, together with the
principle of virtual displacements (expressing the fact that the sum of
and
results in zero external forces), give the system equation
(7)
In equation (7) it is assumed that seismic motion
is
prescribed at the source.
are the reactions due to the prescribed
motion. The first two equations in (6) allow us to calculate the
system’s structural nodal displacements :
4. DYNAMIC STIFFNESS MATRIX OF THE
FOUNDATION
For general cases the dynamic stiffness matrix of the soil-structure
interface
(and
if the source is included in the discrete
model), has to be obtained numerically. There are powerful methods
available to model the dynamic stiffness matrix. We mention here
the Boundary Element Method (BEM) [3], the Thin Layer Method
[4] and an effective approach for inhomogeneous media [5]. Each
method has its advantages and disadvantages, which will not be
addressed here. In many practical applications, from the vibration of
a block foundation through the analysis of soil-structure interaction
in nuclear power plants, the simplification of a rigid soil-structure
interface may be introduced. Dynamic stiffness functions (also
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
Kolekova.indd 3
(11)
3
21. 6. 2010 9:43:38
2010/21 PAGES 1 — 8
called impedance functions) of rigid interfaces, where the sub-grade
is either a homogenous half-space or a homogenous layer, are found
in the literature [6]. In the following example the dynamic stiffness
of the foundation was obtained using the boundary element method.
Therefore, a short summary of the boundary element method within
the context of the derivation of the dynamic stiffness matrix of
a rigid foundation is given.
5. EXAMPLE
5.1 Beam on a rigid foundation block
The derivation of the dynamic stiffness matrix of a tower-like
structure based on a homogenous elastic half-space is illustrated.
For illustrative purposes, the structure is modeled through one axial
beam element with constant properties along the beam, which is
erected on a rigid foundation block. The mass of the beam is lumped
to its end cross sections.
Vertical excitation
For vertical loading (see figure 2, left), the dynamic stiffness matrix
of the structure results in
.
(12)
5.2 Sub-grade model
The boundary element method is applied here to model the wave
propagation in the sub-grade. The procedure is based on the solution
of the wave equation of the infinite space due to a unit load at
position xi (source point) in space. In the frequency domain this
unit load is a time harmonic function with a frequency ω and unit
amplitude. This unit load creates body waves, which propagate from
the source into the infinite space (the fundamental solution). At any
other position, these waves have xj (generic point) and response
Fij. If point xj represents the boundary point of our problem, we
can specify either the displacement or the traction at this position.
A numerical solution can be obtained by subdividing the boundary
of the problem – in our case the surface of the half-space - in the E
boundary elements, where (most simply) the displacement and the
traction are constant. These values could be allocated to the node
in the centre of the element. Either the displacement or the traction
in element j is known through the boundary conditions. In the case
of a half-space its boundary is the soil’s free surface discretized
(corresponding
by the boundary elements where the tractions
to the displacements
) are zero. By applying a ‘virtual’ unit
load in each element node at each degree of freedom, we obtain
N=3E equations (N=2E in 2-D problems) in order to calculate the
remaining unknowns, which in this case are the displacements
at
the surface of the half-space. One can demonstrate that the Principle
of Virtual Work leads to a linear equation system which relates the
displacement and traction variables at the boundary
i, j = 1, 2, ...N
For horizontal loading we correspondingly obtain (see figure 2,
right):
or in the matrix notation
.
(13)
(14a)
(14b)
Equation (14) is a generalization of the unit load principle that
a structural engineer uses to calculate the influence coefficients
of the flexibility matrix in statics. From equation (14) a relation
between the nodal displacements
and nodal forces
can be
established. This relation defines the dynamic stiffness of the subgrade:
(15)
with
(16)
Fig. 2 Models for vertical (left) and horizontal (right) excitation
4
Kolekova.indd 4
where A is a diagonal matrix which, element by element, transforms
the nodal traction in the nodal forces.
In the frequency domain one can obtain, without any loss of accuracy
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
21. 6. 2010 9:43:40
2010/2 PAGES 1 — 8
discretized
soil surface
kinematic constraint:
rigid interface
“static“ condensation:
flexible interface
soil-stucture
interface
Vi, v
Vi, v
Vi, h
Vo, h
Vi, h
i
half-space
Vo, v
0
i
ϕ0
Fig. 3 Models with different levels of condensation and with kinematic constraints
through condensation, a reduced model, which only contains the
nodes on the soil-structure interface I. The original model is
therefore condensed according to equation (17) to the dynamic
of the flexible foundation-soil interface. It can be
stiffness
considered to be the stiffness of a “finite element” representing the
sub-grade. If the sub-grade is given as in equation (6) we obtain
.
(17)
If the lower part of the structure which will come in contact with
the soil is very stiff, this part moves as a rigid body, which is
represented as degrees of freedom through its rigid body: 3 DFOs
in 2D problems and 6 DOFs in 3D problems. The same rigid body
motion is imposed onto the soil-structure interface. The point of
reference is usually chosen as the centre O of the soil-structure
interface. The condensation and kinematic constraints are illustrated
in Figure 3.
The constraint equation between the flexible soil-structure interface
and its rigid body motion is expressed in 2D problems as
(18)
Tab. 1 System properties
Structure
height
[m]
E
[N/m2]
A
[m2]
I
[m4]
ρ
[kg/ m3]
Chimney
180
28x109
22
517
3500
Antenna
62
21x109
0.075
2.19
7850
Foundation
Half edge-length
a [m]
Thickness
[m]
ρ [kg/ m3]
Foundation
block
7.5
1.5
2400
Sub-grade
Soil
.
ν
ρ [kg/ m3]
24x106
0.4
1900
sections of the beam’s ends. The beam is based on a rigid square
foundation block and founded on a homogenous half space. The
system’s properties are given in Table 1. A harmonic Rayleigh wave
is assumed as excitation. Since the vertical and horizontal motions
of the structure are decoupled, one can analyze the vertical and
horizontal motions separately. We chose a free field with amplitude
and
in the vertical and horizontal directions, respectively
at interface I.
For an interpretation of the results, it is helpful to calculate the
eigenfrequency
The dynamic flexibility matrix of the rigid foundation is then
G [N/m2]
,
of the fixed
structures (no structure – sub-grade-interaction) with
(19)
also compare the static stiffnesses of the structure,
,
and
,
, with the static stiffness of the sub-grade,
5.3 Numerical results
As an example of the computation we will consider a chimney and
an antenna, both of which were modeled through a 1 beam element.
The constant mass distribution is modelled as lumped at the cross
, at the
,
,
structure - sub-grade interface. The corresponding numerical values
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
Kolekova.indd 5
5
21. 6. 2010 9:43:42
2010/21 PAGES 1 — 8
Tab. 2 Eigenfrequencies (no soil-structure interaction) and static
stiffnesses of the system
ωe, v
ωe, h
[rad/s] [rad/s]
Kv [N/m]
Kh [N/m] [Kr [Nm]
Chimney
22.22
1.04
3.42x109
7.45x106
241x109
Antenna
117.97
17.81
2.50 x107
5.75 x105
2.23x109
-
-
Sub-grade
0.98.2x109 1.31 x109 57.41x109
are given in Table 2. With BEM we obtain the static stiffness terms
of the sub-grade from the stiffness functions for a0=0 (see figure7).
Chimney
Table 2 shows that in the case of the chimney, the static stiffnesses of
the structure for any vertical and rotational motions are about 3 and
4 times larger than the corresponding stiffnesses of the foundation,
whereas in a horizontal motion, the stiffness of the foundation is
about two orders of magnitude larger than that of the structure.
Without SSI (soil-structure interaction) the vertically base-excited
1DOF system approaches an infinite amplitude when the excitation
frequency approaches its eigenfrequency. With SSI the system
, where the mass of the structure
shows a first resonance at
essentially vibrates with only a little in-phase elastic elongation.
corresponds to a vibration where the
The second resonance at
top and base of the chimney vibrate in an opposite phase and equal
amplitude (figure 4, left). The damping is due to the large wave
radiation in the half-space.
For horizontal free field excitation the structure with a rigid subgrade (no SSI) again shows the singularity for the 1DOF system
at its eigenfrequency. With the SSI on the considered frequency
range, only one of the resonances of the 3DOF-system shows up
(see figure 4, right). The motion is essentially a rigid body rotation
of the chimney (see figures 6a, b). The radiation damping is small
Fig. 4 Frequency response of the vertical motion at the top of the
chimney, a), Frequency response of the horizontal motion at the top
of the chimney, b)
6
Kolekova.indd 6
due to the small foundation motion in a horizontal direction and due
to the small radiation-damping effect in rocking at low frequencies.
Figures 6a and 6b show snapshots of the deformation when the
harmonic free field motion has its maximal value for the excitation
frequencies β1=0.2 and β2=1.1 .
Antenna
In the case of the antenna the static stiffness of the foundation
is larger than the stiffness of the structure in all three directions;
in a horizontal direction the foundation´s stiffness can even be
considered as rigid. The result is a very small resonance amplitude
in the vertical vibration with the total mass of the antenna
and
a strong resonance at about the same frequency as the fixed
structure. For horizontal free field excitation SSI somewhat reduces
the resonance frequency with respect to the case with no SSI and
introduces a small amount of damping, since the radiation energy is
small for the relatively stiff half-space. Figures 6c and 6d show snap
shots of the deformation when the harmonic free field motion has its
maximal value for the excitation frequencies β1 and β2.
Fig. 5 Frequency response of the vertical motion at the top of the
antenna, a), Frequency response of the horizontal motion at the top
of the antenna, b)
Fig. 6 Maximal horizontal deformation of the harmonic motion:
a) chimney, β=0.2, b) chimney, β=1.1; c) antenna, β=0.2, d) antenna,
β=1.1
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
21. 6. 2010 9:43:44
2010/2 PAGES 1 — 8
Fig. 7 Non-dimensional dynamic stiffness functions for horizontal, vertical and rocking motions of a rectangular rigid soil-structure
interface. The foundation is not embedded; sub-grade: homogenous halfspace with ν=0.4. The coupling term between the swaying and
rocking is not shown.
SUMMARY
If the dynamic interaction between a structure and sub-grade is
formulated in total displacements (they include the kinematic and
inertial interaction effects), then the system response in the frequency
domain is obtained using the usual Finite Element procedure. The
given formulation can be easily extended and applied to general
structures and layered soil. In the two examples the soil-structure
interaction effects are shown for vertical and horizontal excitations.
It should be noted that all the material damping in the given
examples is set to zero, so the damping effects solely arise from the
outgoing waves. The results present the importance of the relation
of the dynamic stiffness of a structure to the dynamic stiffness of
a foundation and sub-grade. They also show that different vibration
forms create different damping effects through the outgoing waves
produced.
Aknowledgement
The authors acknowledge support by the Slovak Scientific Grant
Agency under Contract No. 1/0652/09.
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
Kolekova.indd 7
7
21. 6. 2010 9:43:48
2010/21 PAGES 1 — 8
REFERENCES
[1] Petronijevic, M., G. Schmid, Y. Kolekova: “Dynamic soilstructure interaction of frame structures with spectral elements
– Part I”, GNP2008, Žabljak 3-7 Mar, 2008
[2] Penava, D., N. Bajrami,G. Schmid, M. Petronijevic, G.
Aleksovski: “Dynamic soil-structure interaction of frame
structures with spectral elements – Part II”, GNP2008, Žabljak
3-7 Mar., 2008
[3] Dominguez, J. “Boundary elements in dynamics”, Computational
Mechanics Publications, Southampton, Boston, 1993
8
Kolekova.indd 8
[4] Kausel, E, “Thin-layer method”, International Journal for
Numerical Methods in Engineering, Vol. 37, 1994, pp. 927-941
[5] Kalinchuk, V.V., T.I. Belyankova, A. Tosecky: “The effective
approach to the inhomogeneous media dynamics modeling“, 5th
Structural Dynamics Conference EURODYN 2005, Vol. 2, pp.
1309-1314, 2005
[6] Sieffert, J.-G., F. and Cevaer, “Handbook of Impedance
Functions”, Ouest Edition, Nantes, 1992
SPECIAL DYNAMIC SOIL-STRUCTURE ANALYSIS PROCEDURES DEMONSTATED ...
21. 6. 2010 9:43:49