Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
1/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
Document: V6.02.130
SSNL130 indeformable Plaque on a carpet of
springs
Summary:
The objective is to test and validate one of the possibilities of the control
AFFE_CARA_ELEM
, option
RIGI_PARASOL
, coupled with the behavior
DIS_CHOC
. This case test modelizes a plate, considered as
indeformable, posed on a carpet of springs.
·
The springs are modelized 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.
·
The behavior
DIS_CHOC
a unilateral behavior of the springs allows, which leaves one
possibility of separation of the plate with respect to the carpet of spring.
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
2/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
1
Problem of reference
1.1 Geometry
A rectangular plate of width “is” and length “B”, pressed on a carpet of springs.
y
Z
B
ep
With
B
Appear 1.1-a: Diagram 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 AP
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. (there-b)
2
, with p = 5N/m
2
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.
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
3/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
2
Reference solution
2.1
Method of calculation of the continuous problem
With
B
y
Z
y 0
Appear 2.1-a: Diagram 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
.
.
.
3
B
p
has
ds
P
FP
S
=
=
éq
2.1-1
Moment resulting at point “A” due to the loading:
12
.
.
.
.
4
B
p
has
ds
y
P
Mp
S
With
=
=
éq
2.1-2
The plate is regarded as rigid, its displacement is form
-
=
0
1
.
)
(
y
y
Ua
y
Z
. With
has
U
the vertical displacement of point “A” and
0
y
the position of separation.
Effort of reaction of the springs:
B
y
Ua
K
ds
y
y
Ua
B
has
K
Fr
S
.
2
.
.
1
.
.
.
0
0
=
-
=
éq
2.1-3
Moment of reaction of the springs at point “A”:
B
y
U
K
ds
y
y
y
U
B
has
K
Mr.
has
S
has
With
.
6
.
.
.
.
1
.
.
.
2
0
0
=
-
=
éq
2.1-4
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
4/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
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:
4
.
3
0
B
y
=
K
B
has
p
U
has
.
9
.
.
.
8
3
-
=
one deduces some
3
has
B
U
U
-
=
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.
With
B
y
Z
y 0
k2
k1
k3
k4
With
B
C
D
ny cuttings
N
X
D
éco
U
p
Ag
be
Appear 2.2-b: Discretization of the plate in the plan (X, y)
The figure above identifies the springs according to their stiffness. This stiffness is calculated by
the option
RIGI_PARA_SOL
control
AFFE_CARA_ELEM
. The assignment of the values is done in
function of the surface of the area which they affect. If K is the total stiffness of the carpet of spring, one has
thus:
ny
nx
K
K
K
ny
nx
K
K
K
K
ny
nx
K
K
.
.
4
4
4
1
.
.
2
2
4
3
2
.
4
=
=
=
=
=
=
Appear 2.2-a: Diagram of the plate and springs after loading
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
5/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
The equilibrium equations are as follows:
Effort of reaction of the springs:
()
-
+
=
=
N
J
X
X
has
J
y
ny
B
J
K
K
U
Fr
1
0
“
'
.
1
.
.
éq
2.2-1
Moment of reaction of the springs along line “AB”:
()
=
-
=
N
J
X
has
J
ny
B
J
y
ny
B
J
K
U
Mr.
1
0
“
.
.
1
.
.
éq
2.2-2
(
)
ny
B
N
y
ny
B
N
ny
K
nx
K
K
K
ny
K
nx
K
K
K
X
X
1
.
))
1
.(
4
3
.
2
(
.
2
))
1
.(
4
1
.
2
(
0
“
'
+
=
-
+
=
=
-
+
=
with
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:
(
)
(
)
(
) (
)
(
)
(
)
2
2
3
.
4
2
.
.
2
.
.
3
4
.
8
.
3
.
1
.
.
1
.
.
6
4
.
8
.
3
.
.
.
.
N
ny
N
ny
ny
N
ny
N
N
B
y
N
N
K
N
ny
ny
B
has
p
U
O
has
-
-
+
-
-
+
=
+
+
-
-
=
or
N
and
O
y
must observe the following conditions:
(
)
ny
B
N
y
ny
B
N
1
.
0
+
B
y
0
0
N
entirety
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.
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
6/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
3 Modeling
With
3.1
Characteristics of modeling
The plate is modelized by elements
DKT
. The springs are modelized by
SEG2
affected
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 mesh
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
Controls
AFFE_CARA_ELEM HULL
RIGI_PARA_SOL
K_T_D_L
“TOTAL”
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 pitch 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 modelized problem. This solution is obtained for N = 12.
Nature of the results
U
With
=U
B
U
C
=U
D
Continuous solution
03
555555555
.
3
1125
4
-
-
-
E
03
185185185
.
1
3375
4
-
E
Discrete solution
03
532908705
.
3
58875
208
-
-
-
E
03
149763188
.
1
153075
176
-
E
Code_Aster results - 3.5334390124E-03
1.142045709828E-03
Relative error
Code_Aster/discrete Solution
1.50E-04 6.71E-03
For the pitch of time n°2, displacements of the ends of the springs, off-line to the plate,
of +5.0E-03m. the results of Code_Aster are moved are compared with the discrete solution, which
corresponds to the solution of the modelized problem.
Nature of the results
U
With
=U
B
U
C
=U
D
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
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
7/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
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 modelize a unilateral behavior of the springs.
The use of the key word
RIGI_PARASOL
control
AFFE_CARA_ELEM
allows 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.
Code_Aster
®
Version
8.1
Titrate:
SSNL130 indeformable Plaque on a carpet of springs
Date
:
01/09/05
Author (S):
J.L. FLEJOU
Key
:
V6.02.130-A
Page:
8/8
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/05/005/A
Intentionally white left page.