Transcript Presentation

Different models of soil-structure
interaction and
consequent reliability of foundation
structure
Radim ČAJKA
Technical University Ostrava,
Faculty of Civil Engineering
http://www.vsb.cz
Czech Republic
June 13 - June 15, 2012
REC 2012, Brno, Czech Republic
1
Analyses of structures
• Reliability of the reinforced and pre-stressed concrete
structure and foundation depends on a lot of factors, for
example
– strength and safety factors of materials (fck, gc, fyk, gs)
– static and dynamic force loads (g, q, Fg, Fq,….)
– deformation loads (e, g,…), for example creep,
shrinkage, temperature, subsoil deformation due to
undermining or flooding
• Suitable soil – structure interaction model
June 13 - June 15, 2012
REC 2012, Brno, Czech Republic
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Application on real structure
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FEM analyses of reinforced space
structure
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Subsoil models used in FEM
Principle condition for all subsoil models
Settlement of subsoil (s) is equal to deformation of foundation (w)
s( x, y)  wx, y )
Space 3D FEM model
Surface model
• Boussinesque halfspace
• Winkler
• Pasternak
• Modification and combination of these models
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Settlement calculation model and
active zone determination
si, j  f  p z , Aef , Edef , g , m)
coefficient of structural strength
- Czech Standard ČSN 73 1001
m = 0,1 to 0,5
- Eurocode EC 7
m = 0,2
(recommended value)
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Analyses of subsoil settlement
- Condition for active zone determination
 z ( x, y, zz )   ol ( x, y, zz )  m( x, y)  or ( x, y, zz )  0
- Condition for subsoil settlement in each nodal point
zz
zz
m( x, y, z )   or ( x, y, z )
s( x, y)  
 dz  
 dz  sol  sor
Eoed ( x, y, z )
Eoed ( x, y, z )
0
0
 z ( x, y, z )
- Then contact stress and contact function in all nodal points
 ci, j  pzi , j
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C zi , j 
 ci, j
si , j
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Active zone determination with
respect to ČSN or EC Standard
• The depth of compressible subsoil layers (active zone)
depends on size and shape of foundation, changes of subsoil
compressibility and distribution of foundation elements
• The coefficient of structural strength varies in Czech Standard
ČSN 73 1001 from m = 0,1 to m = 0,5
• In accordance ČSN EN 1997-1 Design of geotechnical
structures, the effective vertical stress from foundation contact
pressure is equal to 20 % of effective geotechnical stress, i.s.
practically for m=0,2
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Explicit expression of stress under
rectangular area
pz 

z 
 arctg
2 
z

 arctg
z
 arctg
 arctg
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z
z
x  a )   y  b)
 x  a )2   y  b )2  z 2

x  a )   y  b)

 x  a )2   y  b )2  z 2
x  a )  y  b)

2
2
2
x  a )   y  b)  z
x  a )  y  b)
 x  a )2   y  b )2  z 2
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
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

x  a )
2


2

 z 2   y  b)  z 2 
2

2
x  a )
2




  y  b)  z 2
2
z  x  a )   y  b)  x  a )   y  b)  2  z 2
x  a )


z  x  a )   y  b )  x  a )   y  b )  2  z 2

2

 z 2   y  b)  z 2 
2
2

2
 x  a )2   y  b )2  z 2
z  x  a )   y  b )  x  a )   y  b )  2  z 2
x  a )
2

2

 z 2   y  b)  z 2 
2

2
x  a )
2

  y  b)  z 2
2
z  x  a )   y  b)  x  a )   y  b)  2  z 2
x  a )
2
z
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2
  y  b)
2
z
2
2

2
 x  a )2   y  b )2
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



2
 z 
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Analysis of stress in subsoil
pz 3  z 3
z  
 5  dA
2  r
A
r 2  x2  y 2  z 2
pz ( ,) 3  z 3
z   
 5  detJ  d  d
2 
r
1 1
1 1
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Numerical integration
Gauss quadrature formulae
1 1
n
n
  f ( ,)  d  d  
p 1 q 1
1 1
where
p
 q  f ( p ,q )
 p , q are weighting factor for interval <-1,1>,
integration points of function f
 p ,q
number of integration points
n
Numerical integration of vertical stress
pz ( p ,q ) 3  z 3
 z   p  q
 5  det J ( p ,q )
2 
r
p 1 q 1
n
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
n
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
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Shape functions of a 4-node
element
N1 ( ,)  0,25 (1       )
N2 ( ,)  0,25 (1      )
N3 ( ,)  0,25 (1       )
N4 ( ,)  0,25 (1       )
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Shape functions of a 8-node
element
- corner nodes
N1 ( ,)  0,25 (1     2   2   2   2 )
N2 ( , )  0,25 (1     2  2   2   2 )
N3 ( ,)  0,25 (1     2  2   2   2 )
N4 ( ,)  0,25 (1     2   2   2   2 )
- intermediate nodes
N5 ( , )  0,5  (1    2   2 )
N6 ( ,)  0,5  (1    2   2)
N7 ( ,)  0,5  (1    2   2)
N8 ( ,)  0,5  (1    2   2 )
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Jacobian of transformation
 f   x
    
 f    x
  
    
y   f 
 f 
 x 
   x 
   f   J    f 
y 
 
 
 y 
   y 
detJ   ( x, y) / ( ,)
 r N i

 f 
 f 
 fi 




 x 

1   
1  i 1 
 f   J    f   J    r N

 
 
 i  f i 
 y 
  
 i 1 

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Contact stress course in element
r
p z ( x, y )   N i ( x, y )  p zi
i 1
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Convergence to exact solution
• Division of loaded area into finite elements and degree of
polynomial approximation (convergence of shape and size
of loaded area),
• Accuracy of approximation of stress course in subsoil, i.e.
number of Gauss integration points (convergence of
subsoil).
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Solved test examples
• Square area
• Triangle area
• Circle area
• The results are practically same for exact solution and
numerical integration to 6 integration points
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Application of designed solution
• Analysis of stress components under arbitrary area
constructed from 4-noded and 8-noded isoparametric
elements
• Analysis of settlement of non-linear elastic half-space
modified with soil structural strength coefficient following
EC and Czech standard
• Solution of soil – foundation and soil – structure
interaction problems by means of FEM
June 13 - June 15, 2012
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Rigidity matrix of isoparametric
plate element
1 1
Ke     G  D G detJ  d  d
T
1 1
Ke    p  q  G p ,q )T  D G p ,q ) detJ  p ,q )
n
n
p 1 q 1
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Subsoil rigidity matrix
K   
n
ep
n
p 1 q 1
p

 

 
 q  N ( p ,q )  C ( p ,q )  N ( p ,q )  det J ( p ,q )
Matrix of contact
function
T
C1z ( , )

0
0


C ( p ,q )   0
C2 x ( , )
0

 0
0
C2 y ( , )



r
Contact function
C1z   N i ( , ).C1zi
i 1
Nodal contact parameter
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C1zi 
 ci
si
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
Four - noded isoparametric
semispace element
X
Y
Z
p4
p3=0,5
p2
4
3
Z
p1=2
p2
ol
p2
1
3'
Z3
ol
2
Z
Z4
4'
Z
Z2
ZZ1
m
ori
2'

1'
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Test example of circular plate on
elastic halfspace
• Shallow plate foundation
-
modulus of elasticity of concrete
Poisson’s ratio
plate thickness
radius of circular plate
Ec
mc
hc
r
= 22,95.103 MPa
= 0,2
= 0,1 m
= 1,0 m
• Subsoil
-
modulus of deformation, F5
Poisson’s ratio
coefficient of structural strenght
density of subsoil
Edef = 5,0 MPa
mp = 0,4
m = 0,2
g = 19 kN.m-3
• Load
- uniformly distributed load
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pz = 100 kPa
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Settlement s(x,y) of Halfspace
under Circular Plate after 0th and
10th Iteration
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Contact stress of Circular plate
under Centre and Edge
Contact Stress under Centre of Circle Plate
75
IntPoints=2
65
IntPoints=4
60
Contact Stress under Edge of Circle Plate
IntPoints=6
55
50
IntPoints=8
450
45
40
35
1
2
3
4
5
6
Iteration
7
8
9
Contact Stress [kPa]
Contact Stress [kPa]
70
IntPoints=10
400
IntPoints=2
350
10
IntPoints=4
300
IntPoints=6
250
IntPoints=8
200
IntPoints=10
150
1
2
3
4
5
6
7
8
9
10
Iteration
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Test comparison with TRIMAS
published example
• Shallow plate foundation
-
modulus of elasticity of concrete
Poisson’s ratio
plate thickness
Dimensions of rectangular plate
Ec = 30.103 MPa
mc = 0,154
hc = 0,3 m
lx = 8,0 m, ly = 12,0 m
• Subsoil
-
modulus of deformation, F5
Poisson’s ratio
coefficient of structural strength
density of subsoil
Edef = 4361,5 kPa
mp = 0,38
m = 0,001 0,01 0,1…
g = 18,5 kN.m-3
• Load
- uniformly distributed line load
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pz = 68,6 kN.m-1
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Example of Foundation Plate
TRIMAS (RIB software)
• Foundation plate with
rigid walls in cross
section
• Solution of the 2D plate
on 3D space soil elements
(TRIMAS software)
• Solution with proposed
surface subsoil model
• Active zone in subsoil in
accordance ČSN 73 1001
and EC7
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Bending moments and Contact
Stresses
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Dependence of bending moment on subsoil stiffness, IntBody=8
Bending moment [kNm/m']
-20
-15
-10
-5
m = 0,5
m = 0,4
0
m = 0,3
m = 0,2
m = 0,1
m = 0,01
m = 0,001
5
10
15
20
1
2
3 4
5 6
7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Number of element
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Design of Testing Equipment
Structure
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Soil – foundation interaction test
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Conclusion
• Original and general method for stress solution in elastic
halfspace was presented
• Presented subsoil model is suitable for soil – structure
interaction task and can save 3D subsoil elements
• The mentioned solution eliminates difficulties encountered up
until now, when trying to apply a soil CSN EN standard
model in FEM interaction tasks.
• Comparison of various subsoil model shows great scattering
results, sometimes more than 100 %
• Field soil – foundation interaction experimental tests are now
performed
• Designed models will be verified and uncertainty eliminated
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Thank you for your attention
This paper was supported by the research project No. FR-TI2/746,
program TIP, Ministry of Industry and Trade ,
Czech Republic
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REC 2012, Brno, Czech Republic
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