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Titrate:
Processing of offsetting for the elements of plate
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
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Organization (S):
EDF-R & D/AMA
Manual of Reference
R3.07 booklet: Machine elements on average surface
Document: R3.07.06
Processing of offsetting for the elements of
plate DKT, DST, DKQ, DSQ and Q4G
Summary:
The elements of plate [R3.07.03] are intended for the three-dimensional mean structural analyzes.
average layer of these structures always does not coincide with the plan of diagram or plan of mesh. One introduces
thus concept of offsetting of the average layer compared to the plan of diagram. It is usable for
elements with taking into account of transverse shearing, or without this assumption.
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Processing of offsetting for the elements of plate
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Count
matters
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Appendix 1
Factors of transverse correction of shearing for orthotropic plates or
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1 Introduction
With an aim of being able to analyze the behavior of slim structures of type plates, or surfaces
curves approached by breakages, whose average layer is excentré compared to the plan
of load application, one introduces the concept of offsetting of the average layer compared to
surface mesh. Fields of displacement varying linearly in the thickness of the plate
originate in the surface of mesh, i.e. on the level of the surface of mesh, only
degrees of freedom of translation are necessary to the description of displacement.
The introduction of kinematics into the expression of the work of deformation makes it possible to obtain them
rigidities of membrane, bending and transverse shearing of the excentré element from those
element are equivalent nonexcentré and of the distance from offsetting. The whole of calculations (out
specific postprocessing) is thus made in a reference mark of diagram attached to the plan of the mesh. By
defect the results are thus obtained in the reference mark of the mesh. For certain postprocessings, it is
possible to have automatically these results in other reference marks insofar as the user
indicate the position of the plan of postprocessing compared to the plan of the mesh.
The distance from offsetting between the plan of the mesh and the average layer of the plate is given in
AFFE_CARA_ELEM
on the same level as the thickness. A offsetting D positive means that surface
average of the plate is actually at a distance DNN of the element of plate with a grid, direction N
being given by the normal to the element (see [§4.1] reference material [R3.07.03] of
elements of plate for the construction of this normal).
The adopted notations are those of the note [R3.07.03] on the elements of plates
DKT
,
DST
,
DKQ
,
DSQ and
Q4G
.
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Processing of offsetting for the elements of plate
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2 Formulation
2.1 Geometry
For the offset elements of plate, the surface of reference is given by the plan of diagram or plan
mesh (plane X y for example). The average layer of the element is positioned compared to this
surface reference. The thickness H (X, y) must be small compared to other dimensions (extensions,
radii of curvature) of the structure to be modelized. The figure [Figure 2.1-a] below illustrates our
matter. Concerning the value of offsetting D, and the conditions of linearization of the bending
adopted in the theory, D will be taken so that an element thickness d+h remains in the theory
plates.
Thickness H < L, B, R
1
, R
2
Solid 3D
X
Y
Z
H
L
B
Mesh
L
B
H
X
y
Z
N
R
1
R
2
Plate
offsetting D > 0
Z
X
y
+
-
+
=
2
H
D
;
2
H
D
Z
D
Z
Z
Appear 2.1-a
One attaches to the plan of diagram (the plan of the mesh) a local reference mark orthonormé 0xyz associated the plan of
mesh different from total reference mark OXYZ. The position of the points of the plate is given by
Cartesian co-ordinates (X, y) in the plan of diagram (plane of the mesh) and front elevation Z compared to it
plan.
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2.2 Kinematics
The cross-sections which are the sections perpendicular to the average layer of the plate remain
straight lines. The material points located on a normal at not deformed average surface remain on
a line in the deformed configuration. It results from this approach that the fields of
displacement vary linearly in the thickness of the plate. If one indicates by U, v, W them
displacements of a point of the plan of diagram Q (X, y, Z) according to X, y and Z, the kinematics of Hencky-Mindlin
us gives:
+
=
-
+
=
0
)
y
,
X
(
)
y
,
X
(
Z
)
y
,
X
(
W
)
y
,
X
(
v
)
y
,
X
(
U
0
)
y
,
X
(
)
y
,
X
(
Z
)
y
,
X
(
W
)
y
,
X
(
v
)
y
,
X
(
U
)
Z
,
y
,
X
(
U
)
Z
,
y
,
X
(
U
)
Z
,
y
,
X
(
U
y
X
X
y
Z
y
X
where: U, v, W are displacements of the plan of diagram;
X
and
y
are respectively rotations of this plan compared to respectively axis X and
axis Y.
One prefers to introduce two rotations
)
y
,
X
(
)
y
,
X
(
,
)
y
,
X
(
)
y
,
X
(
X
y
y
X
-
=
=
. Deformations
three-dimensional in any point, with kinematics introduced previously, are thus given
by:
y
yz
X
xz
xy
xy
xy
xy
yy
yy
yy
xx
xx
xx
2
2
Z
2
E
2
2
Z
E
Z
E
=
=
+
=
=
+
=
+
=
where: E
xx
, E
yy
and E
xy
are the membrane deformations of average surface;
X
and
y
deformations associated with transverse shearings;
xx
,
yy
,
xy
the deformations of bending of average surface, which are written:
y
W
X
W
X
y
2
y
X
y
U
X
v
E
2
y
v
E
X
U
E
y
y
X
X
y
X
xy
y
yy
X
xx
xy
yy
xx
+
=
+
=
+
=
=
=
+
=
=
=
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
Date:
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
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R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
Note:
·
in the theories of plate, the introduction of
X
and
y
allows to symmetrize them
formulations of the deformations and the equilibrium equations [R3.07.03]. In the theories
of hull, one uses rather
X
and
y
and associated couples
X
M
and
y
M
compared to X
and y,
·
the degrees of freedom which one chose are displacements and rotations of the plan of diagram
and not those of the average layer. Indeed if one considers the superposition of several
plates offset to carry out a material sandwich it cannot correspond to
nodes of the mesh that only one field of displacement and not the various fields
displacements of the layers composing material.
2.3
Law of behavior
The behavior of the plates is a behavior 3D in “plane stresses”. The stress
transversal
zz
is taken null because negligible compared to the other components of the tensor of
stresses (assumption of the plane stresses). The most general law of behavior is written then
as follows:
+
+
=
=
C
C
It
C
Z
)
,
(
y
X
xy
yy
xx
yz
xz
xy
yy
xx
with
=
=
y
X
xy
yy
xx
xy
yy
xx
0
0
0
=
and
0
0
2
,
0
0
E
2
E
E
E
where: C (
,
) is the matrix of local tangent rigidity in plane stresses;
represent the whole of the internal variables when the behavior is nonlinear.
For behaviors (for example of multi-layer) for which the distortions are coupled
with the deformations of membrane and bending, C (
,
) puts itself in the form:
=
H
H
H
H
C
T
C
C
where:
(
,
) is a symmetrical matrix 3x3;
(
,
) a symmetrical matrix 2x2;
C
(
,
) a matrix 3x2 of coupling between the effects of membrane or bending and of
transverse shearing.
If it is uncoupled, one has
C
(
,
) =0. Determination of
(
,
) within the framework of the theory
of Reissner ([§2.2.3.2] of [R3.07.03]) is given in appendix. It is shown that it is equivalent to that
not offset plates.
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3
Principle of virtual work
3.1
Work of deformation
The general expression of the work of deformation 3D for the element of excentré plate of the distance D
compared to the datum-line is worth:
FD
)
(
W
S
yz
y
xz
X
xy
xy
2
/
H
D
2
/
H
D
yy
yy
xx
xx
def
+
+
+
+
=
+
-
where S is average surface, dV=dxdydz and where the position in the thickness of the plate varies between
d-h/2 and d+h/2.
3.1.1 Expression of the resulting efforts
By adopting the kinematics of [R3.07.03], one identifies the work of the interior efforts:
dS
)
T
T
M
2
M
M
NR
E
2
NR
E
NR
E
(
W
y
y
X
X
xy
xy
yy
yy
xx
xx
xy
xy
yy
yy
xx
S
xx
def
+
+
+
+
+
+
+
=
where:
dz
NR
NR
NR
2
/
H
D
2
/
H
D
xy
yy
xx
xy
yy
xx
+
-
=
=
NR
dz
Z
M
M
M
2
/
H
D
2
/
H
D
xy
yy
xx
xy
yy
xx
+
-
=
=
M
dz
T
T
2
/
H
D
2
/
H
D
yz
xz
y
X
+
-
=
=
T
where: NR
xx
, NR
yy
, NR
xy
are the efforts resulting from membrane (in NR/m);
M
xx
, M
yy
, M
xy
are the efforts resulting from bending or moments compared to the plan of diagram (in
NR);
T
X
, T
y
are the efforts resulting from shearing or sharp efforts (in NR/m).
3.1.2 Relation resulting efforts generalized deformations
The expression of the work of deformation is also written:
dSdz
)]
Z
(
Z
Z
Z
Z
[
FD
]
)
,
(
[
W
S
2
/
H
D
2
/
H
D
2
S
2
/
H
D
2
/
H
D
def
+
+
+
+
+
+
+
+
=
=
+
-
+
-
E
C
C
C
It
EC.
EC.
eCe
C
where: C (
,
) is the matrix of local tangent rigidity (symmetrical matrix).
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This is still written:
+
+
+
+
+
+
+
+
+
+
+
+
+
=
-
dSd
)]
)
D
(
(
)
D
(
)
D
(
)
D
(
)
D
(
[
W
S
2
/
H
2
/
H
2
def
E
C
C
C
It
EC.
EC.
eCe
By using the expression obtained for W
def
in the preceding paragraph, one finds the relation following
between the resulting efforts and the généraliées deformations:
+
+
+
=
+
+
+
+
+
+
=
+
+
+
=
ct
m
F
m
m
F
m
MF
F
m
MF
m
m
MF
m
H
H
H
E
H
T
H
H
H
H
H
E
H
H
M
H
H
H
E
H
NR
)
D
(
)
D
(
)
D
D
2
(
)
D
(
)
D
(
T
T
T
2
with:
+
-
+
-
+
-
=
=
=
2
/
H
2
/
H
2
/
H
2
/
H
2
2
/
H
2
/
H
D
D
D
H
H
H
H
H
H
F
MF
m
+
-
+
-
+
-
=
=
=
2
/
H
2
/
H
C
2
/
H
2
/
H
C
2
/
H
2
/
H
D
D
D
H
H
H
H
H
H
F
m
ct
and:
=
=
=
y
X
xy
yy
xx
xy
yy
xx
,
2
,
E
2
E
E
E
The matrices H
m
, H
F
and H
ct
are the matrices of rigidity out of membrane, bending and shearing
transverse, respectively, for the element of nonexcentré plate. The matrix H
MF
is a matrix of
rigidity of coupling between the membrane and the bending for the element of nonexcentré plate. It is
null if the element of plate is symmetrical compared to its average layer. The matrix H
m
is one
stamp rigidity of coupling between the membrane and the transverse distortion. The matrix H
F
is one
stamp rigidity of coupling between the bending and the transverse distortion. These matrices are null
for a null offsetting, except in the case of the multi-layer ones where they remain nonnull.
For an isotropic homogeneous elastic behavior, these matrices have as an expression:
+
=
-
-
=
-
-
=
1
0
0
1
)
v
1
(
2
kEh
,
2
v
1
0
0
0
1
v
0
v
1
)
v
1
(
12
Eh
,
2
v
1
0
0
0
1
v
0
v
1
v
1
Eh
2
3
2
ct
F
m
H
H
H
and H
MF
= H
m
= H
F
= 0 bus there is a material symmetry compared to the plan
=0.
For the determination of the coefficient of shearing K one returns to [§2.2.3] of [R3.07.03].
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The system of relation between the resulting efforts and the generalized deformations can be also written:
+
+
=
+
+
=
+
+
=
ct
F
m
F
F
MF
m
MF
m
H
H
E
H
T
H
H
E
H
M
H
H
E
H
NR
T
T
with:
=
+
=
+
+
H
H
H
H
H
H
H
MF
MF
m
F
F
MF
m
D
D
D
2
2
=
+
H
H
H
F
F
m
D
Thus, in the case of a plate having material symmetry compared to the plan
=0, one has H
MF
= 0
but
m
MF
H
H
D
=
. The offsetting of the plate involves a coupling between the terms of membrane
and of bending.
Note:
Relations binding H
m
, H
F
, H
MF
with H and H
ct
with H
are valid whatever the law of
elastic behavior tangent, with anelastic deformations (thermoelasticity,
plasticity,…).
For a plate made up of NR orthotropic layers in elasticity, the matrices H
m
, H
F
, H
MF
and H
ct
are written:
=
=
=
=
=
+
+
=
+
=
=
NR
1
I
NR
1
I
I
I
2
I
I
I
I
NR
1
I
I
I
I
NR
1
I
I
,
)
2
(
,
)
(
,
ct
ct
m
MF
F
F
m
MF
MF
m
m
H
H
H
H
H
H
H
H
H
H
H
where:
(
)
I
1
I
I
Z
Z
2
1
+
=
+
H
semi
, H
fi
, H
mfi
, H
I
the matrices of membrane, bending, coupling membrane bending represent and
of transverse shearing for layer I. One notices the analogy between these expressions with
form established above:
m
MF
F
F
m
MF
MF
H
H
H
H
H
H
H
2
D
D
2
D
+
+
=
+
=
One from of deduces whereas offsetting for such a plate is obtained in substituent
D
I
+
with
I
.
3.1.3 Energy interns elastic of plate
Taking into account the preceding remarks, energy interns elastic plate is expressed more
usually for this kind of geometry in the following way:
dS
)]
(
)
(
)
(
[
2
1
T
T
S
int
+
+
+
+
+
+
+
+
=
ct
F
m
F
F
MF
m
MF
m
H
H
E
H
H
H
E
H
H
H
E
H
E
.
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
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3.1.4 Notice
One can choose to express the efforts resulting from bending or moments compared to the average layer
element and either compared to the datum-line. In this case one obtains:
dz
NR
NR
NR
2
/
H
D
2
/
H
D
xy
yy
xx
xy
yy
xx
+
-
=
=
NR
,
dz
)
D
Z
(
M
M
M
2
/
H
D
2
/
H
D
xy
yy
xx
xy
yy
xx
+
-
-
=
=
M
,
dz
T
T
2
/
H
D
2
/
H
D
yz
xz
y
X
+
-
=
=
T
and the expression of the work of the interior efforts becomes:
dS
)
T
T
)
dN
M
(
2
)
dN
M
(
)
dN
M
(
NR
E
2
NR
E
NR
E
(
W
y
y
X
X
xy
xy
xy
yy
yy
yy
xx
xx
xx
xy
xy
yy
yy
xx
S
xx
def
+
+
+
+
+
+
+
+
+
+
=
One then deduces from it by using the expression 3D from work from deformation that:
+
+
+
=
+
+
+
+
+
+
=
+
+
+
+
=
ct
m
F
m
m
F
m
MF
F
m
MF
m
m
MF
m
H
H
H
E
H
T
H
H
H
H
H
E
H
H
NR
M
H
H
H
E
H
NR
)
D
(
)
D
(
)
D
D
2
(
)
D
(
D
)
D
(
T
T
T
2
That is to say still:
+
+
+
=
+
+
+
=
+
+
+
=
ct
m
F
m
F
MF
F
MF
m
m
MF
m
H
H
H
E
H
T
H
H
H
E
H
M
H
H
H
E
H
NR
)
D
(
)
D
(
)
D
(
T
T
T
.
The expression of the internal energy of the plate remains unchanged of course as for it. In the case of
elasticity, it is always written:
dS
)]
(
)
(
)
(
[
2
1
T
T
S
int
+
+
+
+
+
+
+
+
=
ct
F
m
F
F
MF
m
MF
m
H
H
E
H
H
H
E
H
H
H
E
H
E
The question of the choice of the plan interesting to use for the expression of the moments can vary from one
situation with another.
M
M
M
M
In the case of the figure of straight line, the approach developed above is preferable because the expression of
loadings is defined compared to the average layer of each plate. In the case of the figure of
left, if one wishes to replace the multi-layer hull by two offset hulls, the axis of
reference is the average layer of the multi-layer hull. One thus may find it beneficial with all to define compared to
plan of diagram. It is this approach which is adopted in the code. All the loadings applied are
regarded as being defined by defect in the reference mark of diagram or plan of the mesh. If ever
certain loadings are defined compared to other plans (average layer, higher layer or
inferior) is with the user to make the adapted changes of reference mark, with the hand or by the means
command file by specifying the plan of load application when that is possible (see
[§5]), to bring back itself to a loading defined in the plan of the mesh.
Code_Aster
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Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
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:
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R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
3.2
Work of the forces and couples external
The work of the forces and couples being exerted on the plate is expressed in the following way:
+
-
+
-
+
+
=
C
2
/
H
D
2
/
H
D
S
S
2
/
H
D
2
/
H
D
ext.
ds
dz
.
dS
.
FD
.
W
U
F
U
F
U
F
C
S
v
where: F
v
, F
S
, F
C
are the voluminal, surface efforts and of contour being exerted on the plate,
respectively.
C is the part of the contour of the plate on which efforts of contour F
C
are applied.
With the kinematics of [§2.2], one determines as follows:
-
+
+
+
+
-
+
+
+
=
+
+
+
+
+
+
+
+
+
=
C
y
X
X
y
Z
y
X
S
y
X
X
y
Z
y
X
C
y
y
X
X
Z
y
X
S
y
y
X
X
Z
y
X
ext.
ds
)
W
v
U
(
dS
)
C
C
W
F
v
F
U
F
(
ds
)
W
v
U
(
dS
)
C
C
W
F
v
F
U
F
(
W
where are present on the plate:
·
Z
y
X
F
,
F
,
F
surface forces acting according to X, y and Z;
·
I
S
I
v
E
F
E
F
.
dz
.
F
2
/
H
2
/
H
I
+
=
+
-
where E
X
and E
y
are the basic vectors of the tangent plan and E
Z
their
normal vector;
·
y
X
C
,
C
: surface couples acting around axes X and y;
·
C
Z D
dz
D H
I
H
H
=
+
+
±
-
+
[(
)
].
[(
)
].
/
/
E F E
E F E
Z
v
I
Z
S
I
2
2
2
where E
X
, E
y
, E
Z
are the basic vectors
previously definite.
and where are present on the contour of the plate:
·
X
y
Z
,
,
linear forces acting according to X, y and Z;
·
+
-
=
2
/
H
2
/
H
I
dz
.
I
C
E
F
where E
X
, E
y
, E
Z
are the basic vectors previously definite;
·
X
y
,
linear couples around axes X and y;
·
+
-
+
=
2
/
H
2
/
H
I
dz
].
)
D
Z
[(
I
C
Z
E
F
E
where E
X
, E
y
, E
Z
are the basic vectors previously definite.
Note:
The moments compared to Z are null. The efforts and the couples are expressed in
identify mesh. All calculations are made by defect in the reference mark of diagram. If
efforts or of the couples are expressed in another reference mark (that of the average layer of
plate for example) the user will have to make conversions with the hand if it uses the options
by defect or to specify the plan of load application (see the paragraph [§ 5]).
3.3
Principle of virtual work and equilibrium equations
This paragraph is unchanged compared to the paragraph [§3.3] of [R3.07.03].
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
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R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
4
Numerical discretization of the variational formulation
exit of the principle of virtual work
4.1 Introduction
The variational formulation for energy interns enables us to write:
dS
)]
(
)
(
)
(
[
W
T
T
S
int
+
+
+
+
+
+
+
+
=
ct
F
m
F
F
MF
m
MF
m
H
H
E
H
H
H
E
H
H
H
E
H
E
with:
+
+
=
+
=
+
=
y
y
,
X
X
,
X
,
y
y
,
X
y
,
y
X
,
X
X
,
y
,
y
,
X
,
W
W
,
,
v
U
v
U
E
The five degrees of freedom are displacements in the plan of the mesh U and v, except plan W and them
two rotations
X
and
y
.
Elements
DKT
and
DST
are triangular isoparametric elements. Elements
DKQ
,
DSQ
and
Q4
are quadrilateral isoparametric elements. They are represented below:
y
X
1
2
3
4
2
1
3
Appear 4.1-a: real Elements
Code_Aster
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Processing of offsetting for the elements of plate
Date:
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Key
:
R3.07.06-A
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:
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R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
The elements of reference are presented below:
(0,0)
(1,0)
(0,1)
(- 1,1)
(1,1)
(1, - 1)
(- 1, - 1)
1
2
3
4
1
2
3
Appear 4.1-b: Elements of reference triangle and quadrangle
One defines the reduced reference mark of the element as the reference mark (
,
) of the element of reference. The reference mark
room of the element, in the plan of diagram (X, y) is defined by the user, by the key word
ANGLE_REP
.
X1 direction of this local reference mark is the projection of a direction of reference D in the field of the element.
This direction of reference D is chosen by the user who defines it by two nautical angles in
the total reference mark. The normal NR in the plan of the element (12
13 for a triangle numbered 123 and 12
14
for a quadrangle numbered 1234) the second direction fixes. The vector product of the two vectors
previously definite Y1=N
X1 makes it possible to define the local trihedron in which they will be expressed
generalized efforts representing the state of stresses. The user will have to take care that the axis of
reference selected is not found parallel with the normal of certain elements of plate. By defect,
the direction of reference D is axis X of the total reference mark of definition of the mesh.
Note:
For the elements of plate
QUAD4
, the use of a noncoplanar element can lead to
irregularities ([bib1]). In this case, the user is alerted.
4.2
Discretization of the field of displacement
The matrix jacobienne
J (,)
is:
=
=
=
=
=
=
=
22
21
12
11
NR
1
I
I
,
I
NR
1
I
I
,
I
NR
1
I
I
,
I
NR
1
I
I
,
I
,
,
,
,
J
J
J
J
y
NR
X
NR
y
NR
X
NR
y
X
y
X
J
Moreover:
21
12
22
11
11
21
12
22
1
22
21
12
11
J
J
-
J
J
det
J
where
J
J
J
J
J
1
J
J
J
J
with
y
X
=
=
-
-
=
=
=
=
-
J
J
J
J
Code_Aster
®
Version
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Titrate:
Processing of offsetting for the elements of plate
Date:
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
The field of displacement is discretized by:
()
=
=
NR
1
K
K
K
K
v
U
,
NR
v
U
W
NR
W
P
P
X
y
K
K
xk
yk
K
NR
xk
yk
K NR
NR
K
=
+
=
= +
(,)
[
(,)
(,)
]
1
1
2
0
In this last expression, the term between hooks is present for the elements of the type
DKT
,
DST
,
DKQ
or
DSQ
, but not for the elements
Q4
.
4.3
Taking into account of the transverse distortion
It is reminded the meeting that the essential difference between the elements
DKT
,
DKQ
on the one hand and
DST
,
DSQ
,
Q4
in addition comes owing to the fact that for the first the transverse distortion is null is still
= 0.
difference enters
Q4
and elements
DST
and
DSQ
comes from a choice different of interpolation for
representation of transverse shearing. The introduction of offsetting leads to a processing
private individual of transverse shearing.
One replaces in the expression of the internal energy established with [§4.1]
by
where them
are
deformations of substitution checking
=
in a weak way (integral on the sides of the element), and
such as:
+
+
=
+
+
=
+
+
=
ct
F
m
F
F
MF
m
MF
m
H
H
E
H
T
H
H
E
H
M
H
H
E
H
NR
T
T
One checks thus that on the sides ij of the element, one a:
=
-
J
I
S
S
0
ds
)
(
with
S
S
,
S
W
+
=
.
4.3.1 For the elements
Q4
The field linearly is discretized
constant by side so that:
-
+
-
+
+
-
=
=
41
23
34
12
2
1
2
1
2
1
2
1
By using the relations then:
(
(
))
;
(
(
))
,
,
-
+
=
-
+
=
-
+
-
+
W
D
W
D
0
0
1
1
1
1
Code_Aster
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Version
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Titrate:
Processing of offsetting for the elements of plate
Date:
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
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:
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HT-66/03/005/A
it is established that:
)
W
W
(
2
1
)
W
W
(
2
1
K
p
K
p
kp
J
I
I
J
ij
+
+
-
=
+
+
-
=
for (ij) = (12,34) and (kp) = (23,41).
By deferring the two results above in the expression of
, it is established that:
= =
B U
where:
U
=
W
W
NR
NR
NR
1
1
1
M
and
=
B
B
B
(
,
)
1
L
NR
with
=
B
K
K
K
K
K
K
K
NR
NR
NR
NR
,
,
,
,
0
0
Moreover, like:
I
I
xi
yi
J
J
J
J
=
11
12
21
22
one deduces from it that
=
B U
F
where: U
F
X
y
NR
xN
yN
W
W
=
1
1
1
M
and B
B
B
=
(
,
)
1
L
NR
with: B
K
K
K
K
K
K
K
K
K
K
K
NR
NR
J
NR
J
NR
NR
J
NR
J
=
,
,
,
,
,
,
11
12
21
22
Finally:
=
=
=
X
y
C
F
J
J
J
J
11
12
21
22
B U with B
jB
C
NR
[
]
2 3
×
=
Note:
This processing is equivalent to that of the elements
Q4
not offset of [§4.3.2.1] of
[R3.07.03].
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
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P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
4.3.2 For the elements of the type
DKT
,
DST
,
DKQ
,
DSQ
With regard to the transverse distortions, one knows that:
X
,
xy
y
,
yy
y
y
,
xy
X
,
xx
X
M
M
T
and
M
M
T
+
=
+
=
with
+
+
=
F
F
MF
H
H
E
H
M
One deduces from it that:
xx
,
C
xx
,
C
+
=
F
m
H
U
H
T
Calculation of:
C
C
F
m
H
H
where:
(
)
xy
,
y
yy
,
y
xx
,
y
xy
,
X
yy
,
X
xx
,
X
T
xx
,
=
(
)
xy
,
yy
,
xx
,
xy
,
yy
,
xx
,
T
xx
,
v
v
v
U
U
U
=
U
with:
+
+
=
MF
23
MF
22
MF
33
MF
33
MF
12
MF
23
MF
13
MF
33
MF
12
MF
23
MF
13
MF
13
MF
33
MF
11
C
H
2
H
H
H
H
H
H
H
H
H
H
H
2
H
H
m
H
+
+
=
F
23
F
22
F
33
F
33
F
12
F
23
F
13
F
33
F
12
F
23
F
13
F
13
F
33
F
11
C
H
2
H
H
H
H
H
H
H
H
H
H
H
2
H
H
F
H
where them
MF
ij
H
are the terms (I, J) of
MF
H
and where them
F
ij
H
are the terms (I, J) of
F
H
.
Like:
K
NR
2
1
NR
K
,
yk
21
11
,
yk
21
12
22
11
,
yk
21
11
NR
1
K
yk
xy
,
K
NR
1
K
NR
2
1
NR
K
K
xy
,
yk
yk
xy
,
K
xy
,
y
K
NR
1
K
NR
2
1
NR
K
,
yk
2
22
,
yk
22
21
,
yk
2
21
NR
1
K
yk
yy
,
K
NR
2
1
NR
K
K
yy
,
yk
yk
yy
,
K
yy
,
y
K
NR
1
K
NR
2
1
NR
K
,
yk
2
12
,
yk
12
11
,
yk
2
11
NR
1
K
yk
xx
,
K
NR
2
1
NR
K
K
xx
,
yk
yk
xx
,
K
xx
,
y
K
NR
2
1
NR
K
,
xk
21
11
,
xk
21
12
22
11
,
xk
21
11
NR
1
K
xk
xy
,
K
NR
1
K
NR
2
1
NR
K
K
xy
,
xk
xk
xy
,
K
xy
,
X
K
NR
2
1
NR
K
,
xk
2
22
,
xk
22
21
,
xk
2
21
NR
1
K
NR
1
K
xk
yy
,
K
NR
2
1
NR
K
K
yy
,
xk
xk
yy
,
K
yy
,
X
K
NR
2
1
NR
K
,
xk
2
12
,
xk
12
11
,
xk
2
11
NR
1
K
xk
xx
,
K
NR
1
K
NR
2
1
NR
K
K
xx
,
xk
xk
xx
,
K
xx
,
X
)
P
J
J
P
]
J
J
J
J
[
P
J
J
(
xx
1
)
,
(
NR
)
,
(
P
)
,
(
NR
)
P
J
P
J
J
2
P
J
(
)
,
(
NR
)
,
(
P
)
,
(
NR
)
P
J
P
J
J
2
P
J
(
)
,
(
NR
)
,
(
P
)
,
(
NR
)
P
J
J
P
]
J
J
J
J
[
P
J
J
(
)
,
(
NR
)
,
(
P
)
,
(
NR
)
P
J
P
J
J
2
P
J
(
)
,
(
NR
)
,
(
P
)
,
(
NR
)
P
J
P
J
J
2
P
J
(
)
,
(
NR
)
,
(
P
)
,
(
NR
+
+
+
+
=
+
=
+
+
+
=
+
=
+
+
+
=
+
=
+
+
+
+
=
+
=
+
+
+
=
+
=
+
+
+
=
+
=
+
=
=
=
+
=
=
+
=
=
+
=
=
+
=
=
+
=
+
=
=
=
+
=
+
=
=
=
+
=
+
=
=
=
+
=
4
4
4 3
4
4
4 2
1
with:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
=
yk
xk
K
NR
1
K
,
K
22
12
,
K
21
12
22
11
,
K
21
11
,
K
2
22
,
K
22
21
,
K
2
21
,
K
2
12
,
K
12
11
,
K
2
11
,
K
22
12
,
K
21
12
22
11
,
K
21
11
,
K
2
22
,
K
22
21
,
K
2
21
,
K
2
12
,
K
12
11
,
K
2
11
xx
,
1
W
NR
J
J
NR
]
J
J
J
J
[
NR
J
J
0
0
NR
J
NR
J
J
2
NR
J
0
0
NR
J
NR
J
J
2
NR
J
0
0
0
NR
J
J
NR
]
J
J
J
J
[
NR
J
J
0
0
NR
J
NR
J
J
2
NR
J
0
0
NR
J
NR
J
J
2
NR
J
0
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
18/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
the first contribution to
xx
,
in the expression above and:
=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
N
1
K
K
K
,
K
22
12
,
K
21
12
22
11
,
K
21
11
,
K
2
22
,
K
22
21
,
K
2
21
,
K
2
12
,
K
12
11
,
K
2
11
,
K
22
12
,
K
21
12
22
11
,
K
21
11
,
K
2
22
,
K
22
21
,
K
2
21
,
K
2
12
,
K
12
11
,
K
2
11
xx
,
v
U
NR
J
J
NR
]
J
J
J
J
[
NR
J
J
0
NR
J
NR
J
J
2
NR
J
0
NR
J
NR
J
J
2
NR
J
0
0
NR
J
J
NR
]
J
J
J
J
[
NR
J
J
0
NR
J
NR
J
J
2
NR
J
0
NR
J
NR
J
J
2
NR
J
U
that is to say still in matric form that:
+
+
=
+
+
=
+
+
=
+
+
=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
+
=
+
=
=
=
+
=
=
=
+
=
=
=
C
C
C
C
C
C
C
C
NR
2
1
NR
K
K
C
NR
1
K
yk
xk
K
K
C
C
NR
1
K
K
K
K
C
C
K
NR
2
1
NR
K
,
K
K
,
K
K
,
K
K
,
K
K
,
K
K
,
K
K
C
NR
1
K
NR
1
K
yk
xk
K
K
C
C
K
K
K
C
C
,
K
21
11
,
K
21
12
22
11
,
K
21
11
K
,
K
2
22
,
K
22
21
,
K
2
21
K
,
K
2
12
,
K
12
11
,
K
2
11
K
,
K
21
11
,
K
21
12
22
11
,
K
21
11
K
,
K
2
22
,
K
22
21
,
K
2
21
K
,
K
2
12
,
K
12
11
,
K
2
11
K
NR
2
1
NR
K
K
C
NR
1
K
NR
1
K
yk
xk
K
K
C
C
K
K
K
C
C
xy
,
y
yy
,
y
xx
,
y
xy
,
X
yy
,
X
xx
,
X
C
xy
,
yy
,
xx
,
xy
,
yy
,
xx
,
C
W
v
U
P
S
P
S
P
S
P
C
P
C
P
C
W
v
U
)
P
J
J
P
]
J
J
J
J
[
P
J
J
(
S
)
P
J
P
J
J
2
P
J
(
S
)
P
J
P
J
J
2
P
J
(
S
)
P
J
J
P
]
J
J
J
J
[
P
J
J
(
C
)
P
J
P
J
J
2
P
J
(
C
)
P
J
P
J
J
2
P
J
(
C
W
v
U
v
v
v
U
U
U
B
U
B
U
B
T
T
H
U
P
H
U
P
H
T
T
H
P
H
P
H
T
H
P
H
P
H
H
P
H
P
H
H
H
T
F
m
m
2
F
F
F
m
m
m
ck
2
F
F
m
m
2
F
F
m
m
F
F
m
m
F
m
Where:
U
m
NR
NR
U
v
U
v
=
1
1
M
T
T
T
=
+
(
)
(
)
C NR
C NR
1
2
L
T
T
T
2
2
2
0
0
=
with T
2
11
2
12
2
11 12
21
2
22
2
21 22
11 21
12 22
11 22
12 21
2
2
=
+
J
J
J J
J
J
J J
J J
J J
J J
J J
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
19/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
=
yN
xN
NR
1
y
1
X
1
F
W
W
M
U
One can also write:
+
+
=
+
=
C
C
C
xx
,
C
xx
,
C
B
U
B
U
B
H
U
H
T
F
m
m
F
m
By using the relation
=
-
J
I
S
S
0
ds
)
(
with
S
S
,
S
W
+
=
for each side ij of the element, one
can obtain them
K
since this relation is still written:
sk
K
K
K
yj
K
xj
K
yi
K
xi
K
K
I
J
L
L
3
2
)
S
C
S
C
(
2
L
W
W
=
+
+
+
+
+
-
where:
]
)
(
)
(
)
[(
)
S
C
(
]
[
)
S
C
(
)
S
C
(
C
C
C
1
-
ct
K
K
1
-
ct
K
K
K
K
sk
-
+
-
+
-
=
-
-
=
=
F
T
F
F
F
T
F
m
m
T
m
m
T
F
T
m
B
H
B
U
B
H
B
U
B
H
B
H
H
E
H
T
H
The relation above is still written in matric form:
m
m
F
U
With
U
With
With
With
+
+
=
)
(
W
with:
)
(
S
L
C
L
S
L
C
L
L
0
0
0
0
0
0
L
3
2
T
C
1
ct
NR
2
NR
2
NR
2
NR
2
1
NR
1
NR
1
NR
1
NR
NR
2
1
NR
-
+
+
+
+
+
-
-
=
F
F
B
H
B
H
With
M
M
O
-
-
-
-
-
=
-
-
-
-
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
NR
2
NR
2
NR
2
NR
2
NR
2
NR
2
NR
2
NR
2
1
NR
2
1
NR
2
1
NR
2
1
NR
2
1
NR
2
1
NR
2
1
NR
2
1
NR
2
1
K
1
K
1
K
1
K
1
K
1
K
1
K
1
K
1
NR
1
NR
1
NR
1
NR
1
NR
1
NR
1
NR
1
NR
W
S
L
C
L
2
0
0
0
0
S
L
C
L
2
S
L
C
L
2
S
L
C
L
2
0
0
0
0
0
0
0
0
0
S
L
C
L
2
S
L
C
L
2
0
0
0
0
0
0
0
0
0
S
L
C
L
2
S
L
C
L
2
2
1
L
L
With
)
(
S
L
C
L
S
L
C
L
T
C
1
ct
NR
2
NR
2
NR
2
NR
2
1
NR
1
NR
1
NR
1
NR
-
+
+
+
+
-
=
F
F
B
H
B
H
With
M
M
)
(
S
L
C
L
S
L
C
L
T
C
1
ct
NR
2
NR
2
NR
2
NR
2
1
NR
1
NR
1
NR
1
NR
m
m
m
m
B
H
B
H
With
-
+
+
+
+
-
=
M
M
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
20/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
As follows:
m
m
U
P
U
P
+
F
=
with:
)
(
W
1
-
+
=
With
With
With
P
m
1
m
With
With
P
-
=
what implies:
+
+
+
=
F
C
C
C
C
)
(
)
(
U
P
B
B
U
P
B
B
T
m
m
m
Note:
For the elements of the type
DKT
and
DST
, one has
0
C
C
=
=
B
B
m
. It results from it from the expressions
simplified preceding equations.
4.4
Stamp elementary rigidity
4.4.1 Stamp elementary rigidity for the elements
Q4
One takes again the forms of the matrices of rigidity given to [§4.4.1] of the documentation of
reference [R3.07.03] and one replace
MF
H
by
MF
H
,
F
H
by
F
H
and
F
H
by
F
H
. It will be noted that
in [R3.07.03] the results were presented without term of coupling membrane shearing
transverse or transverse bending shearing. They here are added.
4.4.2 Stamp elementary rigidity for the elements
DKT
,
DKQ
One takes again the forms of the matrices of rigidity given to [§4.4.1] of the documentation of
reference [R3.07.03] and one replace
MF
H
by
MF
H
,
F
H
by
F
H
. Since the relation
0
=
is
satisfied the couplings transverse membrane shearing or transverse bending shearing are
non-existent.
4.4.3 Stamp elementary rigidity for the elements
DST
,
DSQ
One a:
dS
)
]
[
]
([
)
]
[
]
([
dS
)
(
)
(
W
1
ct
T
F
1
ct
F
F
T
m
1
ct
F
MF
T
F
1
ct
m
MF
T
m
1
ct
m
m
E
1
ct
1
ct
F
F
F
MF
1
ct
m
m
MF
m
E
E
int
T
TH
H
H
H
H
E
H
H
H
H
H
H
H
H
E
H
H
H
H
E
T
TH
T
H
H
H
H
E
H
T
H
H
H
H
E
H
E
-
-
-
-
-
-
-
-
+
-
+
-
+
-
+
-
=
+
-
+
+
+
-
+
+
=
That is to say still:
dS
)
(
)
(
W
1
ct
F
T
MF
MF
m
E
E
int
T
TH
H
E
H
H
E
H
E
-
+
+
+
+
=
where:
T
1
ct
T
1
ct
T
1
ct
-
-
-
-
=
-
=
-
=
F
F
F
F
F
m
MF
MF
m
m
m
m
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
21/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
From where:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
+
+
=
F
T
F
m
m
T
F
F
m
T
m
F
T
T
T
F
m
T
m
T
m
T
m
T
m
Fm
T
F
F
MF
T
m
F
F
T
F
m
m
T
m
m
E
cm
1
ct
T
C
T
F
F
E
C
1
ct
T
cm
T
m
F
E
C
1
ct
T
C
T
F
E
C
1
ct
T
C
T
F
F
E
C
1
ct
T
C
T
m
E
cm
1
ct
T
cm
T
m
E
C
1
ct
T
cm
T
m
m
E
cm
1
ct
T
C
T
E
C
1
ct
T
C
T
m
E
m
T
MF
T
F
T
F
F
E
F
MF
T
m
T
m
F
E
F
F
T
F
T
F
m
E
m
m
T
m
T
m
m
cm
1
ct
T
C
T
F
F
C
1
ct
T
cm
T
m
F
C
1
ct
T
C
T
F
C
1
ct
T
C
T
F
F
C
1
ct
T
C
T
m
cm
1
ct
T
cm
T
m
C
1
ct
T
cm
T
m
m
cm
1
ct
T
C
T
C
1
ct
T
C
T
F
F
F
T
F
T
F
m
m
T
MF
T
F
T
F
F
F
MF
T
m
T
m
m
m
m
T
m
E
T
m
E
int
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
)
dS
(
dS
)
(
W
U
K
U
U
K
U
U
K
U
U
K
K
U
U
K
K
U
K
U
K
U
U
K
U
U
K
U
U
K
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
B
H
B
U
U
B
H
B
U
B
H
B
U
B
H
B
U
U
B
H
B
B
H
B
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
B
H
B
U
U
B
H
B
U
B
H
B
U
B
H
B
U
U
B
H
B
B
H
B
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
U
B
H
B
U
with:
-
+
=
S
cm
1
ct
T
cm
m
m
T
m
m
dS
]
[
B
H
B
B
H
B
K
It is also known that
)
,
(
F
F
=
U
U
from where it results that:
=
22
T
12
F
12
F
11
F
F
K
K
K
K
K
with:
=
=
=
S
F
F
T
F
22
F
S
F
F
T
F
12
F
S
F
F
T
F
11
F
dS
dS
dS
B
H
B
K
B
H
B
K
B
H
B
K
(
)
12
MF
11
MF
MF
K
K
K
=
with:
=
=
S
F
MF
T
m
12
MF
S
F
MF
T
m
11
MF
dS
dS
B
H
B
K
B
H
B
K
T
MF
Fm
K
K
=
Using the fact that
m
m
F
U
P
U
P
+
=
one deduces from it that:
m
Fm
T
F
F
MF
T
m
F
F
T
F
m
m
T
m
int
W
U
K
U
U
K
U
U
K
U
U
K
U
+
+
+
=
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
22/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
where:
T
MF
Fm
22
F
T
m
T
T
12
F
T
m
m
12
MF
m
11
MF
MF
T
T
12
F
T
12
F
22
F
T
11
F
F
T
m
T
12
MF
T
m
m
m
12
MF
m
22
F
T
m
m
m
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
K
K
P
K
K
P
K
K
P
P
K
K
K
K
K
K
K
P
P
K
K
P
K
K
P
K
K
K
K
K
P
P
K
K
P
K
K
P
K
K
=
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
=
+
+
+
+
+
+
=
This is still written:
(
)
=
F
m
F
m
E
int
,
W
U
U
K
U
U
where:
=
×
×
×
×
×
]
NR
3
NR
3
[
F
T
]
NR
2
NR
3
[
MF
]
NR
3
NR
2
[
MF
]
NR
2
NR
2
[
m
]
NR
5
NR
5
[
K
K
K
K
K
is the elementary matrix of rigidity for an element of
excentré plate DST.
4.5
Stamp of elementary mass
The terms of the matrix of mass are obtained after discretization of the variational formulation
following:
dS
)
) (
D
D
2
(
)
v
U
v
U
) (
D
(
)
W
W
v
v
U
U
(
dzdS
W
y
y
X
X
m
2
MF
F
y
X
y
S
X
m
MF
m
2
/
H
D
2
/
H
D
S
ac
farmhouse
+
+
+
+
+
+
+
+
+
+
+
=
=
+
-
&&
&&
&&
&&
&&
&&
&&
&&
&&
& & U
U
with
+
-
+
-
+
-
=
=
=
2
/
H
2
/
H
2
F
2
/
H
2
/
H
2
/
H
2
/
H
MF
m
dz
Z
and
,
zdz
,
dz
.
Note:
If the plate is homogeneous or symmetrical compared to its average layer then
MF
=0.
4.5.1 Stamp of conventional elementary mass
4.5.1.1 Element
Q4
The discretization of displacement for this isoparametric element is:
NR
,…,
1
K
W
v
U
NR
NR
1
K
yk
xk
K
K
K
K
=
=
=
U
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
23/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
The matrix of mass, in the base where the degrees of freedom are gathered according to the directions of
translation and of rotation, has then as an expression:
=
F
T
MF
F
T
MF
m
MF
m
MF
m
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
M
M
M
M
M
M
M
M
M
with:
dS
S
T
m
m
=
NR
NR
M
dS
)
D
(
S
T
m
MF
MF
+
=
NR
NR
M
dS
)
D
D
2
(
S
T
m
2
MF
F
F
+
+
=
NR
NR
M
where:
(
)
NR
=
NR
NR
K
1
L
.
For the continuation, one poses
m
MF
MF
D
+
=
and
m
2
MF
F
F
D
D
2
+
+
=
.
4.5.1.2 Elements of the type
DKT
,
DST
Like:
K
NR
2
1
NR
K
yk
xk
NR
1
K
yk
xk
K
K
y
X
)
,
(
P
)
,
(
P
0
W
)
,
(
NR
W
+
=
+
=
=
where:
=
+
P U
P U
m
m
F
one deduces from it that:
=
=
NR
1
K
yk
xk
K
K
K
kyy
kyx
kyw
kyv
kyu
kxy
kxx
kxw
kxv
kxu
K
y
X
W
v
U
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
0
0
)
,
(
NR
0
0
W
.
The matrix of mass has then as an expression:
=
F
Fm
MF
m
M
M
M
M
M
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
24/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
The membrane part
m
M
elementary matrix of mass is composed of the blocks kp (kth line
and pième column) following:
+
+
+
+
+
+
+
+
+
+
pyv
kyv
pxv
kxv
kyv
pyu
kxv
pxu
pyv
kyu
pxv
kxu
pyu
kyu
pxu
kxu
F
p
kyv
pyv
K
p
kxv
pyu
K
p
kyu
pxv
K
p
kxu
pxu
K
MF
p
K
p
K
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
0
0
NR
NR
The bending part
M
F
is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
+
+
+
pyy
kyy
pxy
kxy
pyx
kyy
pxx
kxy
pyw
kyy
pxw
kxy
pyy
kyx
pxy
kxx
pyx
kyx
pxx
kxx
pyw
kyx
pxw
kxx
pyy
kyw
pxy
kxw
pyx
kyw
pxx
kxw
pyw
kyw
pxw
kxw
F
p
K
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
0
0
0
0
0
0
0
0
NR
NR
The coupling part between the membrane and the bending
MF
M
of the blocks kp (kth line is composed and
pième column) following:
+
+
+
+
+
+
+
pyy
kyv
pxy
kxv
pyx
kyv
pxx
kxv
pyw
kyv
pxw
kxv
pyy
kyu
pxy
kxu
pyx
kyu
pxx
kxu
pyw
kyu
pxw
kxu
F
pyy
K
pyx
K
pyw
K
pxy
K
pxx
K
pxw
K
MF
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The coupling part between the bending and the membrane
M
Fm
of the blocks kp (kth line is composed and
pième column) following:
+
+
+
+
+
+
+
pyv
kyy
pxv
kxy
pyu
kyy
pxu
kxy
pyv
kyx
pxv
kxx
pyu
kyx
pxu
kxx
pyv
kyw
pxv
kxw
pyu
kyw
pxu
kxw
F
p
kyy
p
kxy
p
kyx
p
kxx
p
kyw
p
kxw
MF
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
4.5.2 Stamp of improved elementary mass
As the arrow of a flexbeam only can be represented with difficulty by one
linear approximation, one can enrich the functions by form for the terms of bending. This
approach is used in Code_Aster for the elements of the type
DKT
,
DST
and
Q4G
where functions
of form used in the calculation of the matrix of mass of bending are of command 3. The interpolation for
W is written as follows:
=
+
-
+
-
+
-
+
+
=
NR
1
K
K
,
3
)
1
K
(
3
K
,
2
)
1
K
(
3
K
1
)
1
K
(
3
W
)
,
(
NR
W
)
,
(
NR
W
)
,
(
NR
W
4.5.2.1 Elements of the type
DKT
It is known that in the approximation of one Coils-Kirchhoff has
X
,
X
W
-
=
and
y
,
y
W
-
=
in any point of
the element.
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
25/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
Because of discretization stated above one a:
=
+
-
+
-
+
-
+
-
+
-
+
+
+
+
=
NR
1
K
yk
,
3
)
1
K
(
3
22
2
)
1
K
(
3
12
xk
,
3
)
1
K
(
3
21
2
)
1
K
(
3
11
K
1
)
1
K
(
3
W
))
,
(
NR
J
)
,
(
NR
J
(
W
))
,
(
NR
J
)
,
(
NR
J
(
W
)
,
(
NR
W
since:
=
yk
,
xk
,
22
21
12
11
K
,
K
,
W
W
J
J
J
J
W
W
This is still written:
=
=
+
-
+
-
+
-
+
+
=
+
+
=
NR
1
K
yk
kwy
xk
kwx
K
kww
NR
1
K
yk
3
)
1
K
(
3
xk
2
)
1
K
(
3
K
1
)
1
K
(
3
)
,
(
NR
)
,
(
NR
W
)
,
(
NR
)
,
(
NR
)
,
(
NR
W
)
,
(
NR
W
where:
)
,
(
NR
J
)
,
(
NR
J
)
,
(
NR
)
,
(
NR
J
)
,
(
NR
J
)
,
(
NR
)
,
(
NR
)
,
(
NR
3
)
1
K
(
3
22
2
)
1
K
(
3
12
3
)
1
K
(
3
3
)
1
K
(
3
21
2
)
1
K
(
3
11
2
)
1
K
(
3
1
)
1
K
(
3
1
)
1
K
(
3
-
-
=
-
-
=
=
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
.
As follows:
=
=
NR
1
K
yk
xk
K
K
K
kyy
kyx
kyw
kyv
kyu
kxy
kxx
kxw
kxv
kxu
kwy
kwx
kww
y
X
W
v
U
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
0
0
W
By not taking account of the effects of inertia, the matrix of mass has the following form thus:
=
F
Fm
MF
m
M
M
M
M
M
The membrane part
m
M
elementary matrix of mass is composed of the blocks kp (kth line
and pième column) following:
+
+
+
+
+
+
+
+
+
+
pyv
kyv
pxv
kxv
kyv
pyu
kxv
pxu
pyv
kyu
pxv
kxu
pyu
kyu
pxu
kxu
F
p
kyv
pyv
K
p
kxv
pyu
K
p
kyu
pxv
K
p
kxu
pxu
K
MF
p
K
p
K
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
0
0
NR
NR
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
26/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
The membrane-bending part
MF
M
is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
pyy
kyv
pxy
kxv
pyx
kyv
pxx
kxv
pyw
kyv
pxw
kxv
pyy
kyu
pxy
kxu
pyx
kyu
pxx
kxu
pyw
kyu
pxw
kxu
F
pyy
K
pyx
K
pyw
K
pxy
K
pxx
K
pxw
K
MF
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The bending-membrane part
Fm
M
is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
pyv
kyy
pxv
kxy
pyu
kyy
pxu
kxy
pyv
kyx
pxv
kxx
pyu
kyx
pxu
kxx
pyv
kyw
pxv
kxw
pyu
kyw
pxu
kxw
F
p
kyy
p
kxy
p
kyx
p
kxx
p
kyw
p
kxw
MF
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The term
F
M
of bending is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
+
+
+
pyy
kyy
pxy
kxy
pyx
kyy
pxx
kxy
pyw
kyy
pxw
kxy
pyy
kyx
pxy
kxx
pyx
kyx
pxx
kxx
pyw
kyx
pxw
kxx
pyy
kyw
pxy
kxw
pyx
kyw
pxx
kxw
pyw
kyw
pxw
kxw
F
pwy
kwy
pwx
kwy
pww
kwy
pwy
kwx
pwx
kwx
pww
kwx
pwy
kww
pwx
kww
pww
kww
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
4.5.2.2 Elements of the type
DST
It is known that for these elements one has
X
,
X
X
W
-
=
and
y
,
y
y
W
-
=
where the distortion
is constant
on the element.
Like:
=
+
-
+
-
+
-
+
-
+
-
+
+
+
+
=
NR
1
K
yk
,
3
)
1
K
(
3
22
2
)
1
K
(
3
12
xk
,
3
)
1
K
(
3
21
2
)
1
K
(
3
11
K
1
)
1
K
(
3
W
))
,
(
NR
J
)
,
(
NR
J
(
W
))
,
(
NR
J
)
,
(
NR
J
(
W
)
,
(
NR
W
one can also write:
)
,
(
NR
)
J
J
(
)
,
(
NR
)
J
J
(
)
,
(
NR
)
,
(
NR
W
)
,
(
NR
W
3
)
1
K
(
3
y
22
X
21
2
)
1
K
(
3
y
12
X
11
NR
1
K
yk
3
)
1
K
(
3
xk
2
)
1
K
(
3
K
1
)
1
K
(
3
+
+
+
+
+
+
=
+
-
+
-
=
+
-
+
-
+
-
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
27/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
where:
)
,
(
NR
J
)
,
(
NR
J
)
,
(
NR
)
,
(
NR
J
)
,
(
NR
J
)
,
(
NR
)
,
(
NR
)
,
(
NR
3
)
1
K
(
3
22
2
)
1
K
(
3
12
3
)
1
K
(
3
3
)
1
K
(
3
21
2
)
1
K
(
3
11
2
)
1
K
(
3
1
)
1
K
(
3
1
)
1
K
(
3
-
-
=
-
-
=
=
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
=
+
-
+
-
=
+
-
+
-
=
+
-
+
-
=
=
=
NR
1
K
3
)
1
K
(
3
3
)
1
K
(
3
NR
1
K
2
)
1
K
(
3
2
)
1
K
(
3
NR
1
K
1
)
1
K
(
3
1
)
1
K
(
3
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
+
=
+
+
+
=
-
yN
xN
NR
1
y
1
X
1
W
NR
NR
1
1
U
yN
xN
NR
1
y
1
X
1
C
C
NR
NR
1
1
C
C
1
ct
y
X
W
W
v
U
v
U
]
W
W
)
(
v
U
v
U
)
[(
M
M
M
M
T
T
P
B
B
P
B
B
H
m
m
One obtains the interpolation for W then:
=
+
-
+
-
+
-
=
+
-
+
-
+
+
+
+
=
NR
1
K
yk
5
)
1
K
(
5
xk
4
)
1
K
(
5
K
3
)
1
K
(
5
NR
1
K
K
2
)
1
K
(
5
K
1
)
1
K
(
5
)
,
(
NR
)
,
(
NR
W
)
,
(
NR
v
)
,
(
NR
U
)
,
(
NR
W
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
28/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
where:
)
,
(
NR
))
3
)
1
K
(
3
,
2
(
T
J
)
3
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
))
3
)
1
K
(
3
,
2
(
T
J
)
3
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
)
,
(
NR
)
,
(
NR
))
2
)
1
K
(
3
,
2
(
T
J
)
2
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
))
2
)
1
K
(
3
,
2
(
T
J
)
2
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
)
,
(
NR
)
,
(
NR
))
1
)
1
K
(
3
,
2
(
T
J
)
1
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
))
1
)
1
K
(
3
,
2
(
T
J
)
1
)
1
K
(
3
,
1
(
T
J
(
)
,
(
NR
)
,
(
NR
)
,
(
NR
))
2
)
1
K
(
2
,
2
(
T
J
)
2
)
1
K
(
2
,
1
(
T
J
(
)
,
(
NR
))
2
)
1
K
(
2
,
2
(
T
J
)
2
)
1
K
(
2
,
1
(
T
J
(
)
,
(
NR
)
,
(
NR
))
1
)
1
K
(
2
,
2
(
T
J
)
1
)
1
K
(
2
,
1
(
T
J
(
)
,
(
NR
))
1
)
1
K
(
2
,
2
(
T
J
)
1
)
1
K
(
2
,
1
(
T
J
(
)
,
(
NR
3
)
1
J
(
3
W
22
W
21
2
)
1
J
(
3
W
12
W
11
3
)
1
K
(
3
5
)
1
K
(
5
3
)
1
J
(
3
W
22
W
21
2
)
1
J
(
3
W
12
W
11
2
)
1
K
(
3
4
)
1
K
(
5
3
)
1
J
(
3
W
22
W
21
2
)
1
J
(
3
W
12
W
11
1
)
1
K
(
3
3
)
1
K
(
5
3
)
1
J
(
3
U
22
U
21
2
)
1
J
(
3
U
12
U
11
2
)
1
K
(
5
3
)
1
J
(
3
U
22
U
21
2
)
1
J
(
3
U
12
U
11
1
)
1
K
(
5
+
-
+
+
-
+
+
-
+
+
-
+
=
+
-
+
+
-
+
+
-
+
+
-
+
=
+
-
+
+
-
+
+
-
+
+
-
+
=
+
-
+
+
-
+
+
-
+
+
-
=
+
-
+
+
-
+
+
-
+
+
-
=
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
This can be still written in the following way:
=
=
NR
1
K
yk
xk
K
K
K
kyy
kyx
kyw
kyv
kyu
kxy
kxx
kxw
kxv
kxu
kwy
kwx
kww
kwv
kwu
y
X
W
v
U
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
)
,
(
NR
W
The matrix of mass has the following form thus:
=
F
Fm
MF
m
M
M
M
M
M
Code_Aster
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Processing of offsetting for the elements of plate
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The membrane part
m
M
elementary matrix of mass is composed of the blocks kp (kth line
and pième column) following:
+
+
+
+
+
+
+
+
+
+
+
+
pyv
kyv
pxv
kxv
kyv
pyu
kxv
pxu
pyv
kyu
pxv
kxu
pyu
kyu
pxu
kxu
F
p
kyv
pyv
K
p
kxv
pyu
K
p
kyu
pxv
K
p
kxu
pxu
K
MF
pwv
kwv
p
K
pwu
kwv
pwv
kwu
pwu
kwu
p
K
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The membrane-bending part
MF
M
is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
+
pyy
kyv
pxy
kxv
pyx
kyv
pxx
kxv
pyw
kyv
pxw
kxv
pyy
kyu
pxy
kxu
pyx
kyu
pxx
kxu
pyw
kyu
pxw
kxu
F
pyy
K
pyx
K
pyw
K
pxy
K
pxx
K
pxw
K
MF
pwy
kwv
pwx
kwv
pww
kwv
pwy
kwu
pwx
kwu
pww
kwu
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The bending-membrane part
Fm
M
is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
+
pyv
kyy
pxv
kxy
pyu
kyy
pxu
kxy
pyv
kyx
pxv
kxx
pyu
kyx
pxu
kxx
pyv
kyw
pxv
kxw
pyu
kyw
pxu
kxw
F
p
kyy
p
kxy
p
kyx
p
kxx
p
kyw
p
kxw
MF
pwv
kwy
pwu
kwy
pwv
kwx
pwu
kwx
pwv
kww
pwu
kww
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
The term
F
M
of bending is composed of the blocks kp (kth line and pième column) following:
+
+
+
+
+
+
+
+
+
+
pyy
kyy
pxy
kxy
pyx
kyy
pxx
kxy
pyw
kyy
pxw
kxy
pyy
kyx
pxy
kxx
pyx
kyx
pxx
kxx
pyw
kyx
pxw
kxx
pyy
kyw
pxy
kxw
pyx
kyw
pxx
kxw
pyw
kyw
pxw
kxw
F
pwy
kwy
pwx
kwy
pww
kwy
pwy
kwx
pwx
kwx
pww
kwx
pwy
kww
pwx
kww
pww
kww
m
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
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Processing of offsetting for the elements of plate
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4.5.2.3 Elements of the type
Q4
One proceeds in the same way that for the elements of the type
DST
but with:
=
yN
xN
NR
1
y
1
X
1
C
y
X
W
W
M
B
where: B
C
is the matrix established with [§4.3.1].
One deduces from it that:
=
=
NR
1
K
yk
xk
K
K
K
K
K
kwy
kwx
kww
y
X
W
v
U
)
,
(
NR
0
0
0
0
0
)
,
(
NR
0
0
0
)
,
(
NR
)
,
(
NR
)
,
(
NR
0
0
W
The matrix of mass has the following form thus:
=
F
m
0
0
M
M
M
The membrane part
m
M
elementary matrix of mass is composed of the blocks kp (kth line
and pième column) following:
p
K
p
K
m
NR
NR
0
0
NR
NR
The term
F
M
of bending is composed of the blocks kp (kth line and pième column) following:
+
p
K
p
K
F
pwy
kwy
pwx
kwy
pww
kwy
pwy
kwx
pwx
kwx
pww
kwx
pwy
kww
pwx
kww
pww
kww
m
NR
NR
0
0
0
NR
NR
0
0
0
0
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
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P. MASSIN, J.M. PROIX, A. ASSIRE
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4.5.2.4 Notice
One neglects in the form of the elementary matrix of mass without offsetting the terms
of inertia of rotation
dS
)
(
S
y
y
X
X
F
+
&&
&&
because the latter are negligible compared to
others. Indeed a multiplicative factor of H
2
/12 the dregs with the other terms and they become negligible
for a thickness report/ratio over characteristic length lower than 1/10. When offsetting is
introduced, these terms of the form
dS
)
) (
D
D
2
(
S
y
y
X
X
m
2
MF
F
+
+
+
&&
&&
are not any more
negligible and are introduced into the form of the matrix of mass.
5
Implementation and postprocessings
Offsetting is introduced by the optional key word
OFFSETTING
on the level of
AFFE_CARA_ELEM
of
same manner as the thickness according to methods' defined in introduction. When this key word is not
not present offsetting is worth zero per defect.
5.1
Load application and couples
All calculations are made in the reference mark of diagram (plane of the mesh). If one defines forces or
couples compared to another reference mark, the user will have to make for
FORCE_ARETE
and
FORCE_NODALE
transformations necessary to be reduced to the reference mark mesh. For
FORCE_COQUE
the user will be able to specify the plan of load application and conversion towards the reference mark of calculation
will be automatic.
One introduces thus into
AFFE_CHAR_MECA
concept of plan of load application by the key word
PLAN
under
FORCE_COQUE.
This plan of application is different from the datum-line or plan from diagram
on which the mesh rests. For this key word one will define the four following possibilities
of application of the forces:
“INF” “MOY” “SUP” “MALL”. “INF”
“MOY”
and
“SUP”
mean that one
respectively apply the efforts in lower, average and higher skin of plate.
“MALL”
mean that one applies the efforts to the level of the datum-line or plan of the mesh. By defect them
efforts will be applied to the plan of the mesh of the plate. The efforts of the type are concerned
FORCE_COQUE
TE0032
.
In local reference mark with the element, when the forces and the couples are brackets on
“MOY”
one uses
simple relation of passage:
X
y
y
y
X
X
df
C
C
df
C
C
+
=
-
=
to bring back the efforts and the couples in the reference mark of the mesh where calculations are made.
In local reference mark with the element, when the forces and the couples are applied to
“SUP”
one uses
simple relation of passage:
X
y
y
y
X
X
F
)
2
/
H
D
(
C
C
F
)
2
/
H
D
(
C
C
+
+
=
+
-
=
In local reference mark with the element, when the forces and the couples are applied to
“INF”
one uses
simple relation of passage:
X
y
y
y
X
X
F
)
2
/
H
D
(
C
C
F
)
2
/
H
D
(
C
C
-
+
=
-
-
=
Code_Aster
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Processing of offsetting for the elements of plate
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If the efforts are given in the total reference mark of the element, one uses relations of passage of
type:
= +
+
C
C
N F
(
/)
D
H
2
where
C
is defined compared to the reference mark
“INF” “MOY” “SUP”
with
equal
with - 1, 0 and 1, respectively. When there is no offsetting, the preceding formula is reduced to
F
N
C
C
+
=
2
/
H
.
Note:
For the loadings of the type
FORCE_ARETE
or
FORCE_NODALE
efforts and couples
can be expressed that compared to the reference mark of the mesh. If the user does not know them
that compared to the average layer of the plate, it will have to carry out the change of reference mark with
the hand to have the expression of the efforts and the couples compared to the surface of mesh.
The relation to be used is
F
N
C
C
+
=
D
where D is the distance between the plan of calculation and the plan
of loading directed by the normal with the hull. It is obvious that the user has interest with it
that the loading plan is the plan of the mesh, but it is not always possible to make
to coincide these two plans as one can see it on the left part of the figure of page 6.
5.2
Application of the boundary conditions in displacement
For the boundary conditions of the displacement type the user will have to pay attention to the fact that they
can apply that to the reference mark of mesh. Relations of passage compared to
conditions given on the average layer are as follows:
N
U
U
D
moy
moy
ref.
moy
ref.
-
=
=
5.3 Postprocessings
For postprocessings, the results owing to lack of generalized efforts type are given in
identify corresponding to the plan of diagram. To have them in the other reference marks, it will be necessary that the user
indicate the plan of postprocessing and the changes of reference mark will be automatic.
For the postprocessing of the efforts generalized in
TE0033
, the four possibilities will be defined
following of postprocessing of the efforts by the key word
PLAN: “INF” “MOY” “SUP” “MALL”
controls
CALC_ELEM
and
CALC_CHAM_ELEM
with the same direction as previously. The defect is
put at
“MALL”.
All calculations are made in the plan
“MALL”
mesh (in particular calculation
nodal forces). When there is no offsetting it is the average layer of the plate: one
thus find postprocessing by defect. To pass from the efforts results generalized of
“MALL”
with
“MOY”
the simple relation of passage is used:
T
T
NR
M
M
NR
NR
=
-
=
=
D
To pass from the efforts results generalized of
“MALL”
with
“SUP”
one uses the simple relation of
passage:
T
T
NR
M
M
NR
NR
=
+
-
=
=
)
2
/
H
D
(
To pass from the efforts results generalized of
“MALL”
with
“INF”
one uses the simple relation of
passage:
T
T
NR
M
M
NR
NR
=
-
-
=
=
)
2
/
H
D
(
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
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6
Static and modal validation
6.1 Validation
initial
The first part of the validation consists in testing a simple plate subjected to forces and
couples and whose plan of mesh does not coincide with the plan of the average layer on which are
applied efforts. For the plate subjected to forces and couples, results with and without
offsetting must take account of the change of reference mark for the couples as indicated Ci
below.
M+dn^F
F
F
M
M
F
F
M+dn^F
N
D
Displacements are in the following way dependant for a point located at a height Z compared to
average layer:
N
U
N
U
U
)
D
Z
(
Z
ref.
ref.
moy
moy
+
+
=
+
=
what is still written:
N
U
U
D
ref.
ref.
moy
ref.
moy
+
=
=
what enables us to establish the relations of passage between displacements compared to the layer
means and those compared to the datum-line.
For the generalized efforts, in the two preceding cases of figure, there are the same results on
layers means, inferior and superior of plate.
6.2
Case-test SSLS111: offsetting for simple plates
It is about a calculation in bending of double-layered made up of two different isotropic materials. One studies
the coupling membrane-bending. The calculation of reference is that of double-layered defined by
DEFI_COQU_MULT
composed of two different isotropic materials (not symmetry according to Z). The other
modeling is made up of two plates offset compared to average fiber of the plate
used with
DEFI_COQU_MULT
. The results, identical of one modeling to the other, are given in
term of displacements and generalized efforts. Moreover one carries out on the geometry of this test one
analyze modal for two modelings: the found Eigen frequencies are identical.
6.3
Case-test SSLS112: offsetting for composite plates
It is about a calculation in bending of a quadricouche having a material not-symmetry compared to
its average plan. The calculation of reference uses a definite quadricouches by
DEFI_COQU_MULT.
The other
modeling uses two double-layered definite by
DEFI_COQU_MULT
but offset compared to fiber
average of the quadricouche. The results, identical of one modeling to the other, are given in
term of displacements.
Code_Aster
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Processing of offsetting for the elements of plate
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7 Conclusion
The finite elements of plate which we describe here are used in the mean structural analyzes
hurled whose thickness report/ratio over characteristic length is lower than 1/10. The average layer
of these structures does not coincide with the plan of the mesh (plane of diagram). Offsetting corresponds
thus with the distance from the average layer compared to the layer of diagram. A offsetting D positive means
that the average surface of the plate is at a distance DNN of the element of plate with a grid, the direction
N being given by the normal to the element.
The values of displacements and generalized efforts obtained are given by defect in the reference mark
mesh. For the generalized efforts, one can however define a reference mark of postprocessing -
identify associated with the average layer - different from the reference mark of diagram. Same manner, efforts
applied are regarded as being given by defect in the reference mark of diagram. In the case of
FORCE_COQUE
, one can however specify a reference mark of load application and couples - reference mark
associated the average layer - different from the reference mark of diagram.
Equivalent elements are not available in thermics; thermomechanical chainings
are thus not available for the offset elements of plates.
8 References
bibliographical
[1]
J.L. BATOZ, G.DHATT: “Modeling of the structures by finite elements: beams and plates
“, Hermès, Paris, 1992.
[2]
D. BUI: “Shearing in the plates and the hulls: modeling and calculation”, Note
HI-71/7784, 1992.
[3]
J.G. REN: “A new theory off laminated punt”, Composite Science and Technology, Vol.26,
p.225-239,1986.
[4]
T.A. ROCK'N'ROLL, E. HINTON: “A finite element method for the free vibration off punts Al for
transverse shear deformation “, Computers and Structures, Vol.6, p.37-44,1976.
[5]
T.J.R. HUGHES: “The finite element method”, Prentice Hall, 1987.
[6]
E. HINTON, T. ROCK'N'ROLL and O.C. ZIENKIEWICZ: “A notes one Farmhouse Lumping and Related
Processes in the Finite Element Method “, Earthquake Engineering and Structural Dynamics,
Vol4, p. 245-249, 1976.
[7]
F. VOLDOIRE: “Modeling by thermal and thermo homogenization elastic of
thin mechanical components ", CR MN/97/091.
[8]
P. MASSIN, F. VOLDOIRE, S. ANDRIEUX: “Model of thermics for the thin hulls”,
Manual of Reference of Code_Aster [R3.11.01].
[9]
F. VOLDOIRE: “Thermoelastic Hollow roll”, Manual of Validation of Code_Aster
[V7.01.100].
[10]
A.K. NOOR, W.S. BURTON: “Assessment off shear deformation theories for multilayered
composite punts “, ASME, Applied Mechanics Review, Vol.42, N°1, p.1-13,1989.
[11]
A.K. NOOR, W.S. BURTON, J.M. PETERS: “Assessment off computational models for
multilayered composite cylinders “in Analytical and Computational Models off Shells, Noor and
Al Eds, ASME, CED - Vol.3, p.419-442,1989.
Code_Aster
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Titrate:
Processing of offsetting for the elements of plate
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Key
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R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
Appendix 1 Factors of transverse correction of shearing
for orthotropic or laminated plates
offset
The matrix
H
ct
is defined so that the surface density of transverse energy of shearing obtained
in the case of the three-dimensional distribution of the stresses resulting from the resolution of balance is equal to
that of the model of plate based on the assumptions of Reissner, for a behavior in pure bending. One
must thus find
H
ct
such as:
1
2
1
2
1
2
2
2
H
TH T
H
1
ct1
ct
-
-
+
-
=
=
H
H
/
/
with
=
xz
yz
and
T
H
ct
=
=
-
+
dz
H
H
/
/
2
2
.
To obtain
H
ct
one uses the distribution of
according to Z obtained starting from the resolution of the equations
of balance 3D without external couples:
xz
xx X
xy y
H
Z
yz
xy X
yy y
H
Z
D
D
= -
+
= -
+
-
-
(
)
;
(
)
,
,
/
,
,
/
2
2
with
xz
yz
=
=
0
for z=±h/2.
If there is no coupling membrane bending (symmetry compared to z=0), stresses in
plan of the element
xx
yy
xy
,
,
in the case of have as an expression a behavior of pure bending:
=
Z
Z
With
M
()
with
With
H
H
F 1
()
()
Z
Z
=
-
.
If
()
H Z
and
H
F
do not depend on X and y one can determine
H
ct
. Indeed:
()
()
()
Z
Z
Z
=
+
D
T D
1
2
where
T
=
=
+
+
T
T
M
M
M
M
X
y
xx X
xy y
xy X
yy y
,
,
,
,
and
=
-
-
M
M
M
M
M
M
xx X
xy y
xy X
yy y
yy X
xx y
,
,
,
,
,
,
like:
D
1
= -
+
+
+
+
-
2
11
33
13
32
31
23
22
33
2
With
With
With
With
With
With
With
With D
H
Z
/
,
D
2
= -
-
-
-
-
-
2
2
2
2
2
11
33
13
32
12
31
31
23
33
22
32
21
2
With
With
With
With
With
With
With
With
With
With
With
With D
H
Z
/
.
It results from it that
1
2
1
2
2
2
-
+
-
=
H
H
/
/
H
T C
C
C
C
T
1
11
12
12
T
22
with:
C
D H D
C
D H D
C
D H D
11
1T
1 1
12
1
T
1 2
22
2
T
1 2
=
=
=
-
+
-
-
-
+
-
+
-
H
H
H
H
H
H
dz
dz
dz
/
/
/
/
/
/
;
;
2
2
2
2
2
2
Code_Aster
®
Version
6.3
Titrate:
Processing of offsetting for the elements of plate
Date:
15/07/03
Author (S):
P. MASSIN, J.M. PROIX, A. ASSIRE
Key
:
R3.07.06-A
Page
:
36/36
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
As in addition
1
2
1
2
2
2
-
+
-
-
=
H
H
/
/
H
TH T
1
ct1
one proposes to take
H
C
ct
111
=
-
to satisfy them as well as possible
two equations whatever T and
.
While comparing
H
ct
thus calculated with
H
H
ct
=
-
+
dz
H
H
/
/
2
2
one reveals the coefficients of correction of
following transverse shearing:
K
H
H
K
H
H
K
H
H
ct
ct
ct
ct
ct
ct
1
11
11 12
12
12
2
22
22
=
=
=
/
;
/
;
/
.
For a homogeneous, isotropic or anisotropic plate, one finds as follows:
H
ct
=kh
H
with k=5/6.
Note:
This method is valid only when the composite plate is symmetrical compared to z=0.
·
For a multi-layer material, one establishes that:
C
With
WITH H
With
With
WITH H
With
With
With
WITH H A
11
p
T
I
T
1
p
I
I
T
1
p
I
p
T
I
T
1
I
=
-
-
+
-
-
+
-
+
-
=
-
-
=
=
-
+
-
=
-
=
-
-
+
H
H
Z
H
Z
Z
Z
H
Z
H
Z
Z
Z
I
p p
p
I
I
I
NR
p p
p
I
I
I
I
p p
p
I
I
p p
p
I
I
I
I
4
1
2
1
2
1
24
1
2
1
2
1
80
1
1
2
1
1
1
2
1
3
3
1
1
2
1
1
2
1
5
5
(
)
(
)
(
) [
(
) (
)
]
(
)
WITH H A
I
T
1
I
-
where:
(
)
H
Z
Z
Z
Z
I
I
I
I
I
I
=
-
=
+
+
+
1
1
1
2
,
and
With
I
represent the matrix
With
With
With
With
With
With
With
With
11
33
13
32
31
23
22
33
+
+
+
+
for
layer I.
·
Validity of the choice
H
C
ct
111
=
-
can be examined a posteriori when one has an estimate of
the solution (fields of displacements and plane stresses, in particular). One can then estimate
the variation enters the two estimates on energy. A step of calculation in two stages for
multi-layer plates and hulls (with
H
ct
diagonal and two coefficients K
1
and K
2
) was besides
developed by Noor and Burton [bib10] [bib11].
·
In the case of an isotropic or anisotropic homogeneous plate the equality between two energies is
satisfied in a strict sense since D
2
= 0. The choice makes above is then valid and no examination has
posteriori is not necessary.