SOIL STRUCTURE INTERACTION: DIFFERENT MODELS OF …
Transcript of SOIL STRUCTURE INTERACTION: DIFFERENT MODELS OF …
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SOIL STRUCTURE INTERACTION:
DIFFERENT MODELS OF ANALYSIS
Prof. Valério S. Almeida
April/2013
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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
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Desacopled Projects!
Structural Engineer
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Soil Structure Interaction (SSI)
Classical procedure
Geotechnical Engineer
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• FINITE ELEMENT METHOD (FEM)
• THOUSANDS OF 3D FINITE
ELEMENTS
• HIGH COMPUTATIONAL TIME• CUMBERSOME PROCESS
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MODELS OF ANALYSIS
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•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
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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
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3. BEM - ALGEBRAIC SYSTEM AND SOLUTIONS
Geometry discretization of boundary
Interpolate functions in boundary element 10
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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:
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SOLIDS MUST BE DISCRETIZED IN SURFACES ELEMENTS
g
48 m
12 m
Radier
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KELVIN SOLUTION:
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BEM
FEM
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KELVIN SOLUTION:
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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
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•WINKLER´S MODELHorizontal coefficient of
subgrade reaction (Kx)
Vertical coefficient of subgrade
reaction (Ky)
•Empirical and semiempirical values 18
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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
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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
13
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
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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
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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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