Code_Aster ®
Version
8.1
Titrate:
SSNL130 ­ indeformable Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
1/8

Organization (S): EDF-R & D/AMA
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
Document: V6.02.130
Indeformable SSNL130 ­ Plaque on a carpet of
springs

Summary:

The objective is to test and validate one of the possibilities of the command AFFE_CARA_ELEM, option
RIGI_PARASOL, coupled with behavior DIS_CHOC. This case test models a plate, considered as
indeformable, posed on a carpet of springs.

· The springs are modelled by DIS_T (K_T_D_L), that makes it possible to impose conditions on
limits at the ends of the springs which are not related to the solid.
· Behavior DIS_CHOC allows a unilateral behavior of the springs, which leaves one
possibility of separation of the plate with respect to the carpet of spring.

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
2/8

1
Problem of reference

1.1 Geometry

A rectangular plate of width “is” and length “B”, pressed on a carpet of springs.

Z
With
ep B y
B

Appear 1.1-a: Schéma of the plate and springs in plan (y, Z)

Dimensions:
= 1m has
B = 2 m
ep = 0.30m

1.2
Properties of material

Young modulus: 2.0E+11 Pa
Poisson's ratio: 0.3
Total stiffness of the carpet of springs: K = 10000.0 NR/m

1.3
Boundary conditions and loadings

The loading is a loading of pressure of the form P = p. (y-b)2, with p = 5N/m2

Displacements imposed at the ends of the springs off-line to the plate:

· in the interval of time [0,1] displacement is imposed on 0.0 following DX, DY and DZ,
· in the interval of time [1,2] displacement is imposed on 0.0 following DX and DY. According to DZ
it is imposed by the Dz function = (t-1.0) * 0.5E-02.

1.4 Conditions
initial

Without object for a static analysis.

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
3/8

2
Reference solution

2.1
Method of calculation of the continuous problem

Z
B
With
y
y 0

Appear 2.1-a: Schéma of the plate and springs after loading

The resolution of the problem consists in calculating vertical displacements of the corners of the plate and
position of the point of separation with respect to the carpet of springs.

The equilibrium equations are as follows:

Effort resulting due to the loading:
3
B
FP =
.
P ds =.
a.
p


éq
2.1-1
3
S

Moment resulting at point “A” due to the loading:
4
B
Mp =
.
P.
y ds =.
a.
p
With


éq
2.1-2
12
S


y
The plate is regarded as rigid, its displacement is form Z (y) = Ua .1-

. With

y0
U has the vertical displacement of point “A” and y0 the position of separation.

Effort of reaction of the springs:
K

y
y
Fr =
Ua
. 1
ds = K Ua
. 0


éq
2.1-3
B has
.


y
B
.
2
S

0
Moment of reaction of the springs at point “A”:

K

y
y2
Mr. =
U
.
. 1
.y ds
.
= K U
.
. 0
With


éq
2.1-4
B has
has
.


y
has


B
.
6
S

0
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
4/8

The resolution of the equations [éq 2.1-1], [éq 2.1-2], [éq 2.1-3], [éq 2.1-4] (balance of the efforts and of
moments) gives the following result:

3.b
.
8 p has
. B
. 3
U
y =
U = -
one deduces some
has
U = -

0
4
has
K
.
9
B
3

2.2
Method of calculation of the discretized problem

In this analysis the carpet of springs is not regarded any more as continuous. The springs are
regularly distributed. As previously vertical displacements of the corners of the plate and
position of the line of separation with respect to the carpet of springs will be calculated.

Z
With
B
y
y 0
Appear 2.2-a: Schéma of the plate and springs after loading
k4
C
D
be
Ag
k1
k2
p
U
éco
D
X
N
With
B
k3
ny cuttings


Appear 2.2-b: Discrétisation of the plate in the plan (X, y)

The figure above locates the springs according to their stiffness. This stiffness is calculated by
option RIGI_PARA_SOL of command AFFE_CARA_ELEM. The assignment of the values is done in
function of the surface of the zone which they affect. If K is the total stiffness of the carpet of spring, one has
thus:

K
k4
K
k4
K
k4 =
k2 = k3 =
=
k1 =
=

nx ny
.
2
nx
.
2
ny
.
4
nx
.
4
ny
.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
5/8

The equilibrium equations are as follows:

Effort of reaction of the springs:

N
'

B
(
Fr
U. K
K.
1 J

éq
2.2-1
J) =
has
X +
X -



j=1
ny.y0

Moment of reaction of the springs along line “AB”:

N

B
B
(
Mr.
The U.K.
.
.
1 J
. J

éq
2.2-2
J) =
has
X -


j=1
ny.y0 ny

'
K

K
with
K = (K
.
2 1+ K.(
4 nx -))
1 =
K = (K
.
2 3 + K.(
4 nx
X
-))
1 =
ny
X
.
2
ny
B

N.
y0 (+) B
N 1
ny
ny

The resolution of the equations [éq 2.1-1], [éq 2.1-2], [éq 2.2-1], [éq 2.2-2] (balance of the efforts and of
moments) the solution of balance gives:

3
.
p.
B .ny has (.3.ny - 8.n - 4)
.
B N (.1+ N) (.3.ny - 8.n - 4)
U =
y
has
O =

6.K (
2
.1 + N + N)
3.ny (.ny + 2.n (.ny - 2)
2
- 4.n)

or N and must there observe the following conditions:
O

B
N.
y +

0
(
) B
N 1
ny
ny
0 y

0 B
N whole

2.3
Sizes and results of reference

The sizes tested will be vertical displacements with the 4 corners of the plate.

2.4
Uncertainties on the solution

Aucunes, the solution is analytical.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
6/8

3 Modeling
With

3.1
Characteristics of modeling

The plate is modelled by elements DKT. The springs are modelled by affected SEG2
of a modeling DIS_T whose characteristics are K_T_D_L. They are the discrete ones in
translation having a diagonal matrix, cf the documentation of AFFE_CARA_ELEM.

3.2
Characteristics of the grid

The plate is cut out with ny = 16 and nx = 4. Dimensions of the plate are has = 1m and B = 2m.

3.3 Functionalities
tested

Commands



AFFE_CARA_ELEM HULL


RIGI_PARA_SOL
K_T_D_L
“GLOBAL”
RIGI_PARA_SOL
GROUP_MA_SEG2

ORIENTATION
CARA
“ANGL_NAUT”
AFFE_CHAR_MECA_F FORCE_COQUE
GROUP_MA

DEFI_MATERIAU ELAS


DIS_CONTACT
DIST_1

STAT_NON_LINE COMP_INCR
RELATION
“DIS_CHOC”

3.4
Sizes tested and results

For the step of time n°1, displacements of the ends of the springs, off-line to the plate,
are imposed on zero. The results of Code_Aster are compared with the discrete solution, which
corresponds to the solution of the modelled problem. This solution is obtained for N = 12.

Nature of the results
UA=UB
UC=UD
Continuous solution
4
-
-
4
555555555
.
3
E - 03
185185185
.
1
E - 03
1125
3375
Discrete solution
208
-
-
176
532908705
.
3
E - 03
149763188
.
1
E - 03
58875
153075
Code_Aster results - 3.5334390124E-03
1.142045709828E-03
Relative error
1.50E-04 6.71E-03
Discrete Code_Aster/Solution

For the step of time n°2, displacements of the ends of the springs, off-line to the plate,
are moved of +5.0E-03m. Les results of Code_Aster are compared with the discrete solution, which
corresponds to the solution of the modelled problem.

Nature of the results
UA=UB
UC=UD
Continuous solution
1.444444444E-03
6.185185185E-03
Discrete solution (n=12)
1.467091295E-03
6.149763188E-03
Results Code_Aster 1.466560457E-03
6.142046322E-03
Relative error
3.62E-04
1.25E-03
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
7/8

4
Summary of the results

The use of the discrete, affected elements on nodes or segments, with a material of the type
DIS_CONTACT and used with STAT_NON_LINE (behavior COMP_INCR and relation DIS_CHOC)
allows to model a unilateral behavior of the springs.
The use of key word RIGI_PARASOL of command AFFE_CARA_ELEM makes it possible to affect to
springs of the stiffnesses proportional to the surface of the elements to which they are connected.

The behavior being unilateral, it is necessary that Code_Aster makes several iterations for
to find the position of balance. It is also possible to encounter problems of convergence
dependant on a loss of precision, had with a bad conditioning of the matrix of stiffness during
iterations. Stiffness of the springs being able to cancel itself from one iteration to another.

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

Code_Aster ®
Version
8.1
Titrate:
Indeformable SSNL130 ­ Plaque on a carpet of springs
Date
:

01/09/05
Author (S):
J.L. Key FLEJOU
:
V6.02.130-A Page:
8/8

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Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A

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