Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
1/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
Organization (S):
EDF/AMA, CS IF
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
V7.11.001 document
HSLS01 - Thin square plate subjected to one
heat gradient in the thickness
Summary
The purpose of this test is to validate thermal dilation in the elements of plate, where the temperature is variable
in the thickness.
Two modelings make it possible to test modelings DKT, DST, Q4G on meshs TRIA3 and QUAD4 and
COQUE_3D on meshs TRIA7 and QUAD9.
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
2/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
D
C
With
B
B
has
= 53.1301°
y
X
Z
= 1.2 m have
B = 1.3 m
E = 0.01 m
E is the thickness of the plate
1.2
Properties of materials
Young modulus: E = 2.10
11
AP
Poisson's ratio:
= 0.3
Expansion factor:
= 1.10
5
°C
1
1.3
Boundary conditions and loadings
Sides AB, BC, CD and DA are embedded. The temperature is constant on the higher face and
is equal to
T
S
= 100°C.
The temperature is constant on the lower face and is equal to
T
I
= 0°C; the variation in temperature
is supposed to be linear in the thickness.
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
3/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The solution is analytical.
D
C
B
With
M
The thermal loading is equivalent to a loading defined by a uniform distribution of
moments on the edges such as it appears on the figure.
The value of these moments per unit of length is equal to:
(
)
(
)
M
T
T
E
E E
S
I
=
-
× ×
× +
-
3
2
1
12 1
.
That is to say:
(
)
(
)
M
T
T
E E
S
I
=
-
×
×
-
2
12 1
. This led to a uniform distribution of
M
in the plate
.
2.2
Results of reference
One thus has
M
= 2380.95238 NR; the plate being turned of an angle
= 53°.1301, one has
components whose absolute value is:
M
X cos
= 1428.5715 NR and
M
X sin
= 1904.76184 NR.
The reactions are defined by a distribution of moments equal to the preceding one in absolute value
and of contrary sign.
The meshs are squares of which the length is equal to 0.05 m, therefore the moments in each node
must be equal to
M
1
=
M
X cos
X 0.05 = 71.42857 N.m
and
M
2
=
M
X sin
X 0.05 = 95.2381 N.m
that is to say
M
M
M
=
+
=
12
22
119.0476 N.m
2.3
Uncertainty on the solution
Uncertainty is null.
2.4 References
bibliographical
[1]
TIMOSHENKO: Theory off punts and shells chapter 2, article 14.
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
4/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
The model consists of:
·
936 elements,
·
675 nodes,
of which:
-
114 elements Q4G,
-
84 elements DSQ,
-
84 elements DKQ,
-
312 DST elements,
-
312 elements DKT.
The elements are squares of which the length is equal to 0.05 Mr.
Edges AB, BC, CD and DA are embedded.
The plate is subjected to a variation in temperature of 100°C in the thickness. This gradient is
uniform on the plate.
3.2 Functionalities
tested
Order
CREA_CHAMP
NOM_CMP
TEMP
TEMP_INF
TEMP_SUP
AFFE_CHAR_MECA
TEMP_CALCULEE
EPSI_INIT
MECA_STATIQUE
OPTION
SIEF_ELGA_DEPL
CALC_NO
FORC_NODA
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
5/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
X-ray
reaction according to
R
O X
Ry
reaction according to
R
O y
Identification Reference
Aster %
difference
N104 (on edge AD in the part with a grid
in DKT)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
N260 (on edge AD in the part with a grid
in DSQ)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
N270 (on edge AD in the part with a grid
in Q4G)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
N8 (on edge AB in the part with a grid in
DKT)
DRx = 95.2381
DRy = 71.4286
DRx = 95.2381
DRy = 71.4286
0
0
N21 (on edge AB in the part with a grid
in DST)
DRx = 95.2381
DRy = 71.4286
DRx = 95.2381
DRy = 71.4286
0
0
N102 (on edge BC in the part with a grid
in DST)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
NR 466 (on edge BC in the part with a grid
in DKQ)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
NR 544 (on edge BC in the part with a grid
in Q4G)
DRx = 71.4286
DRy = 95.2381
DRx = 71.4286
DRy = 95.2381
0
0
4.2 Remarks
The nodes tested are about placed as follows:
Q4G
DSQ
DKQ
DKT
DST
N270
N260
N104
N8
N21
N102
N466
N544
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
6/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
5 Modeling
B
5.1
Characteristics of modeling
Modeling is COQUE_3D.
The model consists of:
·
662 elements,
·
2267 nodes,
of which:
-
462 triangles with 7 nodes,
-
200 quadrilaterals with 9 nodes.
Edges AB, BC, CD and DA are embedded.
The plate is subjected to a variation in temperature of 100°C in the thickness. This gradient is
uniform on the plate.
5.2 Functionalities
tested
Order
CREA_CHAMP
NOM_CMP
TEMP
TEMP_INF
TEMP_SUP
AFFE_CHAR_MECA
TEMP_CALCULEE
EPSI_INIT
MECA_STATIQUE
OPTION
SIEF_ELGA_DEPL
CALC_NO
FORC_NODA
STAT_NON_LINE
CALC_ELEM
OPTION EFGE_ELNO_DEPL
OPTION SIEF_ELNO_ELGA
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
7/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
6
Results of modeling B
6.1 Values
tested
One tests moments MXX and MYY. These values are given in the local reference mark to the plate, chosen
parallel at the sides.
One thus has: MXX = MYY = M = 2,38095 10
3
NR as the moment is uniform in the plate, it is enough
to test the values maximum and minimum of the moments and to check that they are both
equal to M:
Identification
Reference
Aster %
diff
Efforts obtained by
EFGE_ELNO_DEPL
:
MXX Maximum
2,38095 10
3
2,38095
10
3
0
MXX Minimum
2,38095 10
3
2,38095
10
3
0
Maximum MYY
2,38095 10
3
2,38095
10
3
0
Minimum MYY
2,38095 10
3
2,38095
10
3
0
Efforts obtained by
SIEF_ELNO_ELGA
:
MXX Maximum
2,38095 10
3
2,38095
10
3
0
MXX Minimum
2,38095 10
3
2,38095
10
3
0
Maximum MYY
2,38095 10
3
2,38095
10
3
0
Minimum MYY
2,38095 10
3
2,38095
10
3
0
Code_Aster
®
Version
6.3
Titrate:
HSLS01 - Thin square plate subjected to a heat gradient
Date:
20/08/02
Author (S):
J.M. PROIX, G. BERTRAND
Key
:
V7.11.001-A
Page:
8/8
Manual of Validation
V7.11 booklet: Thermomechanical linear statics of the plates and hulls
HT-66/02/001/A
7
Summary of the results
The perfect adequacy of the results with the analytical reference shows the good taking into account of
variation in the temperature.