soil structure interaction: different models of …

SOIL STRUCTURE INTERACTION:

DIFFERENT MODELS OF ANALYSIS

Prof. Valério S. Almeida

valerio.almeida@usp.br

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

3

Soil Structure Interaction (SSI)

Classical procedure

Geotechnical Engineer

• FINITE ELEMENT METHOD (FEM)

• THOUSANDS OF 3D FINITE

ELEMENTS

• HIGH COMPUTATIONAL TIME• CUMBERSOME PROCESS

4

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 =+

5

6

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,,

7

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

8

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

12

KELVIN SOLUTION:

SOLIDS MUST BE DISCRETIZED IN SURFACES ELEMENTS

g

48 m

12 m

Radier

13

KELVIN SOLUTION:

BEM

FEM

14

KELVIN SOLUTION:

15MINDLIN SOLUTION:

16

MINDLIN SOLUTION:

DISCRETIZE ONLY WHERE

THERE ARE TRACTION CONTACTS

17

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

19

•WINKLER´S MODEL

•Empirical and semiempirical values 20

21

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

13

13

13

13

13

13

22

Comparing BEM and Winkler´s models – Two Radiers

23

24

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)

25

Winkler´s Model – 1 column/footing

GEOMETRICALLY NON-LINEAR ANALYSIS OF

MULTI-STOREY BUILDINGS SUPPORTED ON THE

DEFORMABLE MASS

26

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

27

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)

28

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

29

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)

30

MEMBRANE EFFECT: FREE FORMULATION

a) Basic Order Stiffness: Linear Shape functions

b) High Order Stiffness: Quadratic Shape functions

31

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

32

)()(),()()(),()()( ** 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:

33

Mindlin´s Solution (1936) for a point load within

a semi-infinite elastic solid

mecmec PXK =

SEMI-CONTINUUM MEDIA

PGUH = ][][

34

+

+

=

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

35

−

+=

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

36

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

38

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

39

2) NUMERICAL EXAMPLE

40

2) NUMERICAL EXAMPLE

41

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.

42

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.

43

REFERENCES

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