SOIL STRUCTURE INTERACTION: DIFFERENT MODELS OF …
Transcript of SOIL STRUCTURE INTERACTION: DIFFERENT MODELS OF …
SOIL STRUCTURE INTERACTION:
DIFFERENT MODELS OF ANALYSIS
Prof. Valério S. Almeida
April/2013
Soil Structure Interaction (SSI)
SSI is a vast field of interest in the area of civil engineering
The realistic representation of
its behavior must take into
account:
• superstructure
• infrastructure
• supporting soil
Complex numerical task2
Desacopled Projects!
Structural Engineer
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Soil Structure Interaction (SSI)
Classical procedure
Geotechnical Engineer
• FINITE ELEMENT METHOD (FEM)
• THOUSANDS OF 3D FINITE
ELEMENTS
• HIGH COMPUTATIONAL TIME• CUMBERSOME PROCESS
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MODELS OF ANALYSIS
•BOUNDARY ELEMENT METHOD (BEM)
LINEAR ELASTICITY
Equations of Equilibrimum
Weighting the equation by an arbitrary funtion u*,
it known as ‘fundamental solution’,
the integration of the product over the domain results
0bkj,kj =+Ω
Γ
x
1
x
2
n
( ) 0dub kkj,kj =+
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Integrating by parts the derivative term
Integrating by parts again the derivative term
Then, a Betti’s Reciprocal Theorem is obtained
( ) ( ) ( ) −=+−
dundubd kjkjkkkjkj
( ) ( ) ( ) ( )
==
+−=+
kjkjkjkj
jkjkkjkjkkj,kjk
pnandpn
with
dnudundubdu
( ) ( ) ( ) ( ) +−=+
dpudupdubdu kkkkkkj,kjk
FUNDAMENTAL SOLUTION
Considering
∆ℓ the Dirac’s Delta Distribuition
and eℓ unit tensor in direction ℓ
at s (load point),
The previous integration of the product over the domain results
in which uℓs is a component of displacement in direction ℓ at
point s and
are the responses in the domain at q (field point) in direction m
0ei
i
j,ij =+
( ) ( ) ss
kj,kj ueduedu −=−=
mmmm p,u,,
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Applying this definition,
the integral equation of displacement at the point s is obtained
The last equation is known as Somigliana’s Identity
To consider points at the boundary of the body an extension ofthe boundary is considered, hemispheric with centre in sand radius ξ
( ) ( ) ( )
( ) ( ) ( ) +=+
+−=+−
dubdupdpuu
or
dpudupdubu
kkkkkk
s
kkkkkk
s
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Taking the two integrals on the boundary Γ considering theextension one, the integrals can be written as
and taken to the limit ξ→0, the follow results can be proved
Therefore, the equation for points at the boundary results
( ) ( )
( ) ( ) +
+
−
−
dpudpu
and
dupdup
kkkk
kkkk
( )
( ) i
kk21
kk0
k0
udpulim
and
0dulim
−=
→
→
( ) ( ) ( ) +=+
dubdupdpuuc kkkkkk
i
k
i
k 9
3. BEM - ALGEBRAIC SYSTEM AND SOLUTIONS
Geometry discretization of boundary
Interpolate functions in boundary element 10
ALGEBRAIC SYSTEM
The integral equation can be written for every nodal points j,considering the kernals of the integrals being calculatednumerically over every boundary elements ℓ,
resulting in a linear system of algebraic equations as follow
Introducing the boundary conditions, this system results in finalsystem of equations
( ) ( ) ( )
jj
ii
PPUU
with
BduPduUdpUC
==
+=+
BGPHU +=
FAX = 11
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KELVIN SOLUTION:
SOLIDS MUST BE DISCRETIZED IN SURFACES ELEMENTS
g
48 m
12 m
Radier
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KELVIN SOLUTION:
BEM
FEM
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KELVIN SOLUTION:
15MINDLIN SOLUTION:
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MINDLIN SOLUTION:
DISCRETIZE ONLY WHERE
THERE ARE TRACTION CONTACTS
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FEM BEM
Consolidated numerical
method
Numerical method in development on several
analysis like dynamic, porous media, damage
and fracture, biological analysis
Real problem dimension Integral formulation in a dimension below
of real problem
Discretization of domain Discretization of boundary
Banded and symmetrical
matrices
Full and non-symmetrical Matrices
Integral over domain
elements (cells)
Integral over boundary elements
Numerical sensibility with
physical singularities
Numerical sensibility with physical
singularities and on fundamental solutions
singularities (1/r, 1/r2, 1/r3, ln(1/r) with r→0 )
Infinite or semi-infinite
problems – large cells
Infinite or semi-infinite problems
– fundamental solutions obtained on infinite
domain, discretization of semi-infinite border
•WINKLER´S MODELHorizontal coefficient of
subgrade reaction (Kx)
Vertical coefficient of subgrade
reaction (Ky)
•Empirical and semiempirical values 18
F = k . d
d
FF
k
P
P
d
P = k . dv
kv
a) b)
)(1 3−== FLdd
Pkv
( )
−
−−−
= BA
E
bpd
1
211 2
POULOS & DAVIS(1974)
−++
++++
−++
+++=
11
11
1
1
2
122
22
222
222
nm
nmnm
mnm
mnmnA
++=
2212 nmn
marctg
nB
bLm =b
zn =
•WINKLER´S MODEL
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•WINKLER´S MODEL
•Empirical and semiempirical values 20
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COMMERCIAL SOFTWARE FOR CONSIDER SSI
•USING WINKLER´S MODEL
g=2,8 tf/m2
E = 3921 tf/m2
s
= 0,2s
A B
C
h = 0,4m
E = 2,8E+6 tf/m
= 0,2sapata
2
sapata
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13
13
13
13
13
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Comparing BEM and Winkler´s models – Two Radiers
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E = 9,1 MPa
5m
10 m
h
10mE = 21000 MPa
5mC t = 0,26m
lâmina = 0,15lâmina
solo = 0,3solo
p A Bp=0,01 MPa
lâmina
h = 10m
a) b) c)
Comparing BEM and Winkler´s models
Footing supported by a finite layer
E = 9,1 MPa
5m
10 m
h
10mE = 21000 MPa
5mC t = 0,26m
lâmina = 0,15lâmina
solo = 0,3solo
p A Bp=0,01 MPa
lâmina
h = 10m
a) b) c)
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Winkler´s Model – 1 column/footing
GEOMETRICALLY NON-LINEAR ANALYSIS OF
MULTI-STOREY BUILDINGS SUPPORTED ON THE
DEFORMABLE MASS
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OBJECTIVE
Present a numerical model to simulate Soil Structure
Interaction (SSI), considering:
• 3D multi - storey buildings (3D frames)
• Semi-continuum media
• Flexible shallow foundation
• Geometrically non-linear analysis
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ANALYSIS TECHNIQUES
3D multi - storey buildings using Finite Element
Method (FEM) to simulates 3D frames
• Columns and Beams (slabs are not considered)
• Continuum joint
• Linear Stress-Strain relationships (Hooke´s Law)
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Flexible shallow foundation: FEM using laminar elements
Two independent formulations, one to represent the
membrane effect and the other the plate effect.
• Membrane Effect: Free Formulation
• Plate Bending Effect: DKT (Discrete Kirchhoff Theory)
ANALYSIS TECHNIQUES
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a) Rotations varies quadratically along the sides
b) Kirchhoff hypotesis are considered in the corners and
in the middle of the edge:
c) Variation of w along the sides is cubic
d) Displacements and rotations are compatible along the sides
(interelement continuous)
Plate Bending Effect: DKT (Discrete Kirchhoff Theory)
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MEMBRANE EFFECT: FREE FORMULATION
a) Basic Order Stiffness: Linear Shape functions
b) High Order Stiffness: Quadratic Shape functions
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SEMI - CONTINUUM MEDIA
ELASTOSTATICS BOUNDARY ELEMENT FORMULATION
)1(0)(
)(21
1)( ,, =+
−+
G
sbsusu i
jijjji
)2()(2)()( sGss ijkkijij +=
)21()1(
−+
=
E
• Essencial conditions:
• Natural conditions:
uii SuSu =)(
pijiji SpSSp == )()(
E,
(s)u (s)ij
i
p (S)iu (S)
i
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)()(),()()(),()()( ** SdSpSPuSdSuSPpPuPC ikiikikki =+
= =
=+
NE
k
kiij
NE
k
kiijjij PSSPuUSSPpPUPC
1
*
1
* )(),()(),()()(
i
e
ii
e
i
e
PSSpUSSu
functionsshapelinear
==
)()()()(
:)(
==
=NE
j
j
i
kiNE
j
j
i
ki PGUH11 PGUH = ][][Absence of body forces
SEMI - CONTINUUM MEDIASomigliana’s Identity:
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Mindlin´s Solution (1936) for a point load within
a semi-infinite elastic solid
mecmec PXK =
SEMI-CONTINUUM MEDIA
PGUH = ][][
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+
+
=
22
2
1
x
w
x
v
x
ux
( ) +−++=
m
dxvMwMwwvvuNW zyx
´́´́´´´´´int
m
T
extW rq =
GEOMETRICALLY NON-LINEAR ANALYSIS
Green-Lagrange Strain
Appling Green-Lagrange Strain with Navier-Bernoulii hypothesis
The virtual work equation can be expressed as
Piola-Kirchhoff stress
The integration of the undisturbed volume – Total Lagrangian formulation
The work of
Internal forces
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−
+=
m
dx
MwN
MvN
N
wywx
vzvx
ux
m
´́´´
´́´´
´
f
GEOMETRICALLY NON-LINEAR ANALYSIS
The vector of internal forces
−
+
+
+
=
m
dx
MN
Nw
MN
Nv
N
T
y
wwvux
T
xw
T
zvwvux
T
xv
T
xu
T
q00
q
q00
q
q
k
´́´´´
´́´´´
´
The tangent stiffness matrix
derivative of fm
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Shape functions:
Derivatives of Nx, My and Mz in relation to q:
GEOMETRICALLY NON-LINEAR ANALYSIS
Using a degenerated form of the Green strain, it was necessary, with respect to the
continuity requirements, use a quintic for u (with a cubic w), but it is extremely
cumbersome, thus causing for low-order function the “membrane locking”
But for this application no problem was encoutered, small deformation are envolved37
THE BUILDING-FOUNDATION-SOIL SYSTEM
BEM/FEM COUPLING
=
F
C
F
C
FFFC
CFCC
F
F
U
U
KK
KK
CCC FUK =
FC1
FFCFCCC KKK-KK =−
F1
FFCFCC FKK-FF−
=
Static condensation
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1) NUMERICAL EXAMPLE
2
1
3
4
5
6
7
19
18
17
16
15
14
13H
P P
8 9 10 11 12
240
in
240 in
P = 350 kipsH = 1 kip
A = 2in
I = 100 in
E = 30000 ksi
2
4
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2) NUMERICAL EXAMPLE
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2) NUMERICAL EXAMPLE
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REMARKS:
• Differential settlement is the main cause of changes of the
structure behavior;
• It is mandatory to compute geometrically non-linear effects
for the building analisys;
• In the 1st and 2nd floors occur the major changes of the
structure behavior.
• Material non linearity (plasticity) in the building and
dynamics effects must be included in the present model.
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Brebbia,C.A. (1978)."The boundary element method for engineers",
Pentech, London.
Fraser, R.A.; Wardle, L.J. (1974). Numerical analysis of rectangular
rafts on layered foundations. Géotechnique, v. 26, p. 613-630.
Poulos, H.G.; Davis, E.H. (1974). Elastic solutions for soil and rock
mass. New York, John Wiley & Sons 535p.
Sadecka, L. (2000). A finite/infinite element analysis of thick plate on
a layered foundation.Computers & Structures, v. 76, p. 603-610.
Burmister, D.M. Theory of stresses and displacements and
applications to the design of airport runways. 23rd proc. Highway
Research Board, pp.127-248, 1943.
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REFERENCES
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