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Organization (S): EDF-R & D/AMA
Handbook of Référence
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 grid. 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
1 Introduction ............................................................................................................................................ 4
2 Formulation ............................................................................................................................................ 5
2.1 Geometry ........................................................................................................................................ 5
2.2 Kinematics ..................................................................................................................................... 6
2.3 Law of behavior ....................................................................................................................... 7
3 Principle of virtual work .................................................................................................................. 8
3.1 Work of deformation ..................................................................................................................... 8
3.1.1 Expression of the resulting efforts ........................................................................................... 8
3.1.2 Relation resulting efforts generalized deformations ........................................................... 8
3.1.3 Energy interns elastic of plate ..................................................................................... 10
3.1.4 Notice ............................................................................................................................. 11
3.2 Work of the forces and couples external ........................................................................................ 12
3.3 Principle of virtual work and equilibrium equations ......................................................................... 12
4 numerical Discretization of the variational formulation resulting from the principle of virtual work ............ 13
4.1 Introduction .................................................................................................................................... 13
4.2 Discretization of the field of displacement ..................................................................................... 14
4.3 Taking into account of the transverse distortion ................................................................................. 15
4.3.1 For the elements Q4 .......................................................................................................... 15
4.3.2 For the elements of type DKT, DST, DKQ, DSQ ...................................................................... 17
4.4 Stamp elementary rigidity ...................................................................................................... 20
4.4.1 Stamp elementary rigidity for the elements Q4 ......................................................... 20
4.4.2 Stamp elementary rigidity for elements DKT, DKQ ............................................... 20
4.4.3 Stamp elementary rigidity for elements DST, DSQ ............................................... 20
4.5 Stamp of elementary mass ...................................................................................................... 22
4.5.1 Stamp of traditional elementary mass ............................................................................. 22
4.5.1.1 Element Q4 ............................................................................................................. 22
4.5.1.2 Elements of the type DKT, DST ..................................................................................... 23
4.5.2 Stamp of improved elementary mass ............................................................................ 24
4.5.2.1 Elements of type DKT .............................................................................................. 24
4.5.2.2 Elements of the type DST .............................................................................................. 26
4.5.2.3 Elements of the type Q4 ............................................................................................... 30
4.5.2.4 Notice ................................................................................................................ 31
5 Implementation and postprocessings ...................................................................................................... 31
5.1 Load application and couples .................................................................................................. 31
5.2 Application of the boundary conditions in displacement ................................................................ 32
5.3 Postprocessings ............................................................................................................................. 32
6 static and modal Validation ............................................................................................................... 33
6.1 Initial validation ............................................................................................................................ 33
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Processing of offsetting for the elements of plate
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6.2 Case-test SSLS111: offsetting for plates simple ...................................................... 33
6.3 Case-test SSLS112: offsetting for composite plates ................................................ 33
7 Conclusion ........................................................................................................................................... 34
8 bibliographical References ............................................................................................................... 34
Appendix 1
Factors of transverse correction of shearing for orthotropic plates or
laminated offset ...................................................................................................................... 35
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Processing of offsetting for the elements of plate
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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 facets, 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 grid. Fields of displacement varying linearly in the thickness of the plate
originate in the surface of grid, i.e. on the level of the surface of grid, 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, inflection 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 grid. By
defect the results are thus obtained in the reference mark of the grid. 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 grid.
The distance from offsetting between the plan of the grid 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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2 Formulation
2.1 Geometry
For the offset elements of plate, the surface of reference is given by the plan of diagram or plan
grid (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 modelled. The figure [Figure 2.1-a] below illustrates our
matter. Concerning the value of offsetting D, and the conditions of linearization of the inflection
adopted in the theory, D will be taken so that an element thickness d+h remains in the theory
plates.
Solid 3D
Z
H
Y
B
X
L
R1
R2
Thickness H < L, B, R1, R2
Z = Z + D
H
H
Z
Z D -;D +
y
2
2
H
N
Plate
X
B
L
Z
offsetting D > 0
y
X
Grid
Appear 2.1-a
One attaches to the plan of diagram (the plan of the grid) a local reference mark orthonormé 0xyz associated the plan of
grid 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 grid) and rise 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:
U (X,
y Z)
X
U (X, y)
(X, y)
y
U (X, y)
(X, y)
X
U (X,
y Z)
y
= v (X, y) + Z (X, y)
X
= v (X, y) + Z (X, y)
y
U (X,
y Z)
Z
W (X, y)
0
W (X, y)
0
where: U, v, W are displacements of the plan of diagram;
and
X
y are respectively rotations of this plan compared to respectively axis X and
axis Y.
One prefers to introduce two rotations (X, y) = (X, y,
) (X, y) = - (X, y). Deformations
X
y
y
X
three-dimensional in any point, with kinematics introduced previously, are thus given
by:
= E + Z
xx
xx
xx
= E + Z
yy
yy
yy
2 = = 2nd + 2z
xy
xy
xy
xy
2 =
xz
X
2 =
yz
y
where: exx, eyy and exy are the membrane deformations of average surface;
X and y deformations associated with transverse shearings;
xx, yy, xy the deformations of inflection of average surface, which are written:
U
E =
xx
X
v
E =
yy
y
v
U
2nd =
+
xy
X
y
X
=
xx
X
y
=
yy
y
2
y
X
=
+
xy
y
X
W
= +
X
X
X
W
= +
y
y
y
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Note:
· in the theories of plate, the introduction of and makes it possible to symmetrize them
X
y
formulations of the deformations and the equilibrium equations [R3.07.03]. In the theories
of hull, one uses rather
M
M
X and
and associated couples
and
compared to X
y
X
y
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 grid 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 constraints”. The constraint
transversal zz is taken null because negligible compared to the other components of the tensor of
constraints (assumption of the plane constraints). The most general law of behavior is written then
as follows:
E
0
xx
xx
xx
xx
E
0
yy
yy
yy
yy
E = 2nd
,
2
and
= 0
xy
= xy
xy
= C (,) xy = This + Z
C +
C with
0
0
X
xz
X
0
0
yz
y
y
where: C (,) is the matrix of local tangent rigidity in plane constraints;
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
at the deformations of membrane and inflection, C (,) is put in the form:
H
H C
C =
H T
H
C
where: (,) is a symmetrical matrix 3x3;
(,) a symmetrical matrix 2x2;
C (,) a matrix 3x2 of coupling between the effects of membrane or inflection and of
transverse shearing.
If it is uncoupled, there are 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:
d+h/2
W
= (+
+ + +
FD
)
def
xx
xx
yy
yy
xy
xy
X
xz
y
yz
S
d-h/2
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:
W
= (E NR + E NR + E
2
NR + M + M + 2 M + T + T) dS
def
xx
xx
yy
yy
xy
xy
xx
xx
yy
yy
xy
xy
X
X
y
y
S
where:
NR
xx
d+h/2
xx
NR = NR =
dz
yy
yy
NR d-h/2
xy
xy
M
xx
d+h/2
xx
M = M =
dz
Z
yy
yy
M d-h/2
xy
xy
T
d+h/2
X
xz
T =
=
dz
T
y d-h/2 yz
where: Nxx, Nyy, Nxy are the efforts resulting from membrane (in NR/m);
Mxx, Myy, Mxy are the efforts resulting from inflection or moments compared to the plan of diagram (in
NR);
Tx, Ty 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:
d+h/2
d+h/2
W
= [(
C,) FD
]
= [eCe + Z
EC. +
EC. + zCe + z2
C + Z
C + (
C E + Z + dSdz
)]
def
S d-h/2
S d-h/2
where: C (,) is the matrix of local tangent rigidity (symmetrical matrix).
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This is still written:
H/2
W
= [eCe + (+ D)
EC. +
EC. + (+ D) This + (+ D) 2 C
C
C E
def
+ (+ D) + (+ (+ D) + dSd
)]
S - H/2
By using the expression obtained for Wdef in the preceding paragraph, one finds the relation following
between the resulting efforts and the généraliées deformations:
NR = H E (H
dH)
H
m
+
MF +
m +
m
M = (H
dH
E
)
(H
2dH
D 2H)
(H
dH)
MF +
m
+
F +
MF +
m +
F +
m
T = HT E (HT dHT)
H
m
+
F +
m +
ct
with:
+h/2
+h/2
+h/2
H
m =
Hd Hmf = Hd Hf = H2 D
- H/2
- H/2
- H/2
+h/2
+h/2
+h/2
H
ct =
H D Hm = H D H
H
D
C
F
=
C
- H/2
- H/2
- H/2
and:
E
xx
xx
X
E = E
,
,
yy
= yy
=
y
2nd
2
xy
xy
The matrices Hm, Hf and Hct are the matrices of rigidity out of membrane, inflection and shearing
transverse, respectively, for the element of nonexcentré plate. The Hmf matrix is a matrix of
rigidity of coupling between the membrane and the inflection for the element of nonexcentré plate. It is
null if the element of plate is symmetrical compared to its average layer. The Hm matrix is one
stamp rigidity of coupling between the membrane and the transverse distortion. The matrix Hf is one
stamp rigidity of coupling between the inflection 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 v
0
1 v
0
Eh
Eh3
kEh 1 0
H
H
H
m =
v 1
0
,
F =
v 1
0
,
ct =
2
2
1 - v
1 - v
1
(
12 - v)
1 - v
1
(
2 + v) 0 1
0 0
0 0
2
2
and Hmf = Hm = Hf = 0 bus there is a material symmetry compared to the =0 plan.
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:
NR = H E H
H
m
+ MF + m
M = H E H
H
MF
+ F + F
T = HT E H T
H
m
+ F + ct
with:
Hmf = Hmf + H
D m
H
2
F = Hf + 2 H
D MF + D Hm
H = H + dH
F
F
m
Thus, in the case of a plate having material symmetry compared to the plan =0, one has Hmf = 0
but H = dH. The offsetting of the plate involves a coupling between the terms of membrane
MF
m
and of inflection.
Note:
The relations binding Hm, Hf, Hmf with H and Hct 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 Hm, Hf, Hmf and Hct
are written:
NR
NR
NR
NR
H
H
H
H
H
H
H
H
H
H
H
m =
,
(
,
)
(
2
,
)
semi
MF =
mfi + I
semi
F =
fi +
I mfi + semi 2i
ct =
I
ct
i=1
i=1
i=1
i=1
1
where: =
Z + + Z
I
(I 1 I)
2
Hmi, Hfi, Hmfi, Hi represent the matrices of membrane, inflection, coupling membrane inflection and
of transverse shearing for layer I. One notices the analogy between these expressions with
form established above:
H = H
+ H
D
MF
MF
m
H = H +
H
2
D
2
+ D H
F
F
MF
m
One from of deduces whereas offsetting for such a plate is obtained in substituent + D with.
I
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:
1
=
[E (H E H
H
H E H
H
H E H
H
.
m
+ MF +
m) + (
MF + F + F) + (T
T
m
+ F + ct dS
)]
int
2 S
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3.1.4 Notice
One can choose to express the efforts resulting from inflection or moments compared to the average layer
element and either compared to the datum-line. In this case one obtains:
NR
M
xx
d+h/2
xx
xx
d+h/2
xx
T
d+h/2
NR = NR =
dz
, M = M =
(Z - D dz
),
X
xz
T =
=
dz
yy
yy
yy
yy
T
NR d-h/2
-
y
yz
M
D H/2
d-h/2
xy
xy
xy
xy
and the expression of the work of the interior efforts becomes:
W
= (E NR + E NR + E
2
NR + (M + dN) + (M + dN) + 2 (M + dN) + T + T) dS
def
xx xx yy yy
xy
xy
xx
xx
xx
yy
yy
yy
xy
xy
xy
X
X
y
y
S
One then deduces from it by using the expression 3D from work from deformation that:
NR = H E (H
dH)
H
m
+
MF +
m +
m
M + dN = (H
dH E
)
(H
2dH
D 2H)
(H
dH)
MF +
m
+
F +
MF +
m +
F +
m
T = HT E (HT dHT)
H
m
+
F +
m +
ct
That is to say still:
NR = H E (H
dH)
H
m
+
MF +
m +
m
M = H E (H
dH)
H
.
MF
+
F +
MF +
F
T = HT E (HT
dHT)
H
m
+
F +
m +
ct
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:
1
=
[E (H E H
H
H E H
H
H E H
H
m
+ MF +
m) + (
MF + F + F) + (T
T
m
+ F + ct dS
)]
int
2 S
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 grid. 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 grid.
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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:
d+h/2
d+h/2
W
F. FD
U
F. dS
U
F .U ds
dz
ext. = v
+ S
+ C
S d-h/2
S
C d-h/2
where: Fv, Fs, FC are the voluminal, surface efforts and of contour being exerted on the plate,
respectively.
C is the part of the contour of the plate to which the efforts of contour FC are applied.
With the kinematics of [§2.2], one determines as follows:
W
(F U F v F W.C.
C
) dS
(U
v
W
) ds
ext. =
X
+ y + Z + xx + yy
+ X +y +z +xx +yy
S
C
= (F U F v F W.C.
C
) dS
(U
v
W
) ds
X
+ y + Z + yx - xy
+ X +y + Z +yx - xy
S
C
where are present on the plate:
· F, F, F surface forces acting according to X, y and Z;
X
y
Z
+h/2
·
F =
F E
. dz + F E
. where E
I
v I
S
I
X and ey are the basic vectors of the tangent plan and ez them
- H/2
normal vector;
· C, C: surface couples acting around axes X and y;
X
y
+h/2
H
·
C =
Z
[(+d) E
where E
Z F] .e dz
v
I
+ [(D ±) ezF] .e
I
S
I
2
X, ey, ez are the basic vectors
- H/2
previously definite.
and where are present on the contour of the plate:
· X, y, Z linear forces acting according to X, y and Z;
+h/2
·
F .e dz where E
I =
C I
X, ey, ez are the basic vectors previously definite;
- H/2
· X, there linear couples around axes X and y;
+h/2
·
[(Z D) E
F] .e dz where E
I =
+ Z C I
X, ey, ez are the basic vectors previously definite.
- H/2
Note:
The moments compared to Z are null. The efforts and the couples are expressed in
locate grid. 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].
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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:
W
= [E (H E H
H
H E H
H
H E H
H
m
+ MF +
m) + (
MF + F + F) + (T
T
m
+ F + ct dS
)]
int
S
with:
U
, X
X, X
W, X + X
E =
v
,
,
, y
=
y, y
=
W, y + y
U
v
, y +
, X
X, y + y, X
The five degrees of freedom are displacements in the plan of the grid U and v, except plan W and them
two rotations X and Y.
The elements DKT and DST are triangular isoparametric elements. Elements DKQ, DSQ and
Q4 are quadrilateral isoparametric elements. They are represented below:
4
3
3
1
y
2
1
2
X
Appear 4.1-a: Eléments realities
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The elements of reference are presented below:
(0,1)
(- 1,1)
(1,1)
3
4
3
1
2
(0,0)
(1,0)
1
2
(1, - 1)
(- 1, - 1)
Appear 4.1-b: Eléments 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 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 make 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 grid.
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:
NR
NR
X
y
NR X
NR y
I,
I
I,
I
J
J
,
,
11
12
i=1
i=1
J =
=
NR
NR
=
X
y
J
J
,
,
21
22
NR X
NR y
I,
I
I, I
i=1
i=1
Moreover:
J
J
X
-
1 J
- J
11
12
1
22
12
= J
with
J =
= J =
J
where
= det J = J J - J J
11 22
12 21
J
J
J - J
J
21
22
21
11
y
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The field of displacement is discretized by:
U
NR
K
=
U
NR,
K ()
K
v k=1
v
W
NR
wk
2 NR
0
= NR (,) + [
P (,)
X
K
xk
xk
K]
K =1
K = NR
+1 P
(,
y
yk
yk
)
In this last expression, the term between hooks is present for the elements of type DKT,
DST, DKQ or DSQ, but not for the Q4 elements.
4.3
Taking into account of the transverse distortion
It is pointed out that the essential difference between 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 between 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 are them
deformations of substitution checking = in a weak way (integral on the sides of the element), and
such as:
NR = H E H
H
m
+ MF + m
M = H E H
H
MF
+ F + F
T = HT E H T
H
m
+ F + ct
J
One checks thus that on the sides ij of the element, one a: (
) ds 0 with = W +.
S - S
=
S
, S
S
I
4.3.1 For the Q4 elements
One linearly discretizes the constant field by side so that:
1 -
1
12
+ 34
+
=
= 2
2
1 -
1
23
-
41
+
2
2
By using the relations then:
+1
(- (
W +))
D = 0;
,
- 1
+1
(- (
W +))
,
D = 0
- 1
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it is established that:
ij
1
= (W - W + +)
2
J
I
I
J
for (ij) = (12,34) and (kp) = (23,41).
kp
1
= (W - W + +)
2
p
K
p
K
By deferring the two results above in the expression of
, it is established that:
= = B U
w1
1
1
Nk, K Nk,
0
where: U = M and B = (B, B
B
1 L
)
NR with K = Nk,
0
K Nk,
wN
NR
NR
Moreover, like:
I J
J
11
12 xi
=
I J
J
21
22 yi
one deduces from it that
= Drunk F
w1
x1
y
1
where: U
M
F =
and B = (B, B)
1 L
NR
W
NR
xN
yN
Nk, K Nk J
,
11
K Nk J
,
12
with: B
K =
NR
K, K Nk J
,
21
K Nk J
,
22
Finally:
X J
J
11
12
=
= B U with B
= jB
=
C
F
C [2×3N]
y
J
J
21
22
Note:
This processing is equivalent to that of the Q4 elements not offset of [§4.3.2.1] of
[R3.07.03].
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4.3.2 For the elements of type DKT, DST, DKQ, DSQ
With regard to the transverse distortions, one knows that:
T = M
+ M
and T = M
+ M
with M = H E H
H
MF
+ F + F
X
xx, X
xy, y
y
yy, y
xy, X
One deduces from it that:
C
C
T = H U
H
m
+ F
, xx
, xx
Calculation of:
C
C
H H
m
F
where: T
=
, xx
(X, xx X, yy X, xy y, xx y, yy y, xy)
T
U
= U
U
U
v
v
v
, xx
(, xx, yy, xy, xx, yy, xy)
MF
MF
MF
MF
MF
MF
H
H
2H
H
H
H
H
C
11
33
13
13
23
12 +
MF
with: H
m =
33
MF
MF
MF
MF
MF
MF
MF
H
H
H
H
H
H
2H
13
23
12 +
33
33
22
23
F
F
F
F
F
F
H
H
2H
H
H
H
H
C
11
33
13
13
23
12 +
F
H
F =
33
F
F
F
F
F
F
F
H
H
H
H
H
H
2H
13
23
12 +
33
33
22
23
where them
MF
H are the terms (I, J) of H and where F
H are the terms (I, J) of H.
ij
MF
ij
F
Like:
NR
2N
NR
2N
2
2
= NR (,)
+ P
(,
)
= NR (,)
+ (J P + 2j J P + J P
)
X, xx
K, xx
xk
xk, xx
K
K, xx
xk
11 xk,
11 12
xk,
12
xk,
K
K 1
=
K =N 1
+
K 1
=
K =N 1
+
NR
2N
NR
2N
2
2
= NR (,)
+ P
(,
)
= NR (,)
+ (J P + 2j J P + J P
)
X, yy
K, yy
xk
xk, yy
K
K, yy
xk
21 xk,
21 22
xk,
22
xk,
K
K 1
=
k=N 1
+
K 1
=
k=N 1
+
NR
2N
NR
2N
= NR (,)
+ P
(,
)
= NR (,)
+ (J J P + [J J + J J P
]
+ J J P
)
X, xy
K, xy
xk
xk, xy
K
K, xy
xk
11 21 xk,
11 22
12 21
xk,
11 21 xk,
K
K 1
=
K =N 1
+
K 1
=
K =N 1
+
NR
2N
NR
2N
2
2
= NR (,)
+ P
(,
)
= NR (,)
+ (J P + 2j J P + J P
)
y, xx
K, xx
yk
yk, xx
K
K, xx
yk
11 yk,
11 12
yk,
12
yk,
K
K 1
=
K =N 1
+
K 1
=
K =N 1
+
NR
2N
NR
2N
2
2
= NR (,)
+ P
(,
)
= NR (,)
+ (J P + 2j J P + J P
)
y, yy
K, yy
yk
yk, yy
K
K, yy
yk
21 yk,
21 22
yk,
22
yk,
K
K 1
=
K =N 1
+
K 1
=
K =N 1
+
NR
2N
NR
2N
= NR (,)
+ P
(,
)
= NR (,)
+ (J J P + [J J + J J P
]
+ J J P
)
y, xy
K, xy
yk
yk, xy
K
K, xy
yk
11 21 yk,
11 22
12 21
yk,
11 21 yk,
K
K 1
=
K =N 1
+
K 1
=
K =N 1
+
1 4
4 2
4
4
4 3
1
xx
with:
2
2
0
J NR
2j J NR
J NR
0
11
K, +
11 12
K, +
12
K,
2
2
0
J NR
2j J NR
J NR
0
21
K, +
21 22
K, + 22
K,
W
K
NR
0 J J NR
[J J
J J] NR
J J NR
0
1
11 21
K, +
11 22 + 12 21
K, +
, xx =
12 22
K,
2
2
xk
0
0
J NR
2j J NR
J NR
k=1
11
K, +
11 12
K, + 12
K,
2
2
yk
0
0
J NR
2j J NR
J NR
21
K, +
21 22
K, + 22
K,
0
0
J J NR
[J J
J J] NR
J J NR
11 21
K, +
11 22 + 12 21
K, + 12 22
K,
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the first contribution to in the expression above and:
, xx
2
J NR
2j J NR
J NR
0
11
K,
+ 11 12 K, + 2
12
K,
2
J NR
2j J NR
J NR
0
21
K,
+ 21 22 K, + 222 K,
N
J J NR
[J J
J J] NR
J J NR
0
U
U
, xx =
11 21
K,
+ 11 22 + 12 21 K, +
12 22
K,
K
2
2
0
J NR
2j J NR
J NR
v
K 1
11
K,
+ 11 12 K, +
=
12
K,
K
2
2
0
J NR
2j J NR
J NR
21
K,
+ 21 22 K, +
22
K,
0
J J NR
[J J
J J] NR
J J NR
11 21
K,
+ 11 22 + 12 21 K, +
12 22
K,
that is to say still in matric form that:
U, xx
X, xx
U, yy
X, yy
U
C
T
H, xy
C
H X, xy
= m
+
v
F
, xx
y, xx
v, yy
y, yy
v, xy
y, xy
2
2
C (J P
2j J P
J P
)
K
11 K, +
11 12 K, +
12 K,
2
2
C (J P
2j J P
J P
)
K
21 K, +
21 22 K, + 22 K,
W
NR
NR
K
2N
U
C (J J P
[J J
J J P
]
J J P
)
C
K
K
C
K
C
K 11 21 K, + 11 22 + 12 21 K, + 11 21 K,
= Hm P
H
P
H
cm
+ F C xk
F
+
K
v
2
2
S (J P
2j J P
J P
)
k=1
K
k=1
k=N+1
K
11 K, +
11 12 K, +
12 K,
yk
2
2
S (J P
2j J P
J P
)
K
21 K, +
21 22 K, + 22 K,
S (J J P
[J J
J J P
]
J J P
)
K
11 21 K, +
11 22 + 12 21
K, + 11 21 K,
C P
K K,
C P
K K,
W
NR
NR
K
2N
U
C P
C
K
K
C
K
C
H
P
H
P
H T
= m cm
+ F C xk
K K,
+
F
2
v
S P
K 1
K
K 1
K NR 1
K K, K
=
=
= +
yk
S P
K K,
S P
K K,
W
NR
U
C
K
K
NR
K
2N
= Hm P
H
P
H T
T
cm
+ cf K
C
C
xk
F
2
+
ck
K
v
k=1
K
k=1
k=N+1
yk
= C
H P U
H P U
H T T
B U
B U
B
m cm
m +
C
C
F
C
F +
F
2 =
cm
m +
C
F +
C
Where:
u1
v
1
U =
m
M
one
v
NR
T
(T
T
=
C NR +
)
(
)
1 L C2 NR
2
2
J
J
2 J J
T
0
11
12
11 12
T
2
2
2
2 =
with T = J
J
2 J J
0 T
2
21
22
21 22
2
1j1 j21
12
J j22
11
J j22 + 12
J j21
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w1
x1
1y
U
F
= M
W NR
xN
yN
One can also write:
T = C
H U
H
B U
B U
B
m,
+ C
xx
F, xx =
cm
m +
C
F +
C
J
By using the relation (
) ds 0 with = W + for each side ij of the element, one
S - S
=
S
, S
S
I
can obtain K since this relation is still written:
L
2
K
W - W +
(C + S + C + S) + L = L
J
I
K
xi
K
yi
K
xj
K
yj
K
K
K
sk
2
3
where:
= (C
S) = (C
S)
1
-
H [T - T
T
H E H
m
- F]
sk
K
K
K
K
ct
= (C
S) 1
H [(B
H B U
B
H B U
B
H B
m -
T
)
T
T
m
m
m + (
-
)
F
F
F + (
-
)
F
F]
K
K
ct
C
C
C
The relation above is still written in matric form:
WITH = (A
W + A) U
+ A U
F
m
m
L
0
0 L
C
L
S
2
NR 1
+
NR 1
+
NR 1
+
NR 1
+
NR 1
+
with: To = 0
0
1
-
O
-
M
M
H (
T
B
- H B)
3
ct
C
F
F
0
0
L
L C
L S
2N
2N2 NR
2N
2N
- 2 L C
L S
2
L
C
L S
0
0
0
0
0
0
N+1 N+1
N+1 N+1
N+1 N+1
N+1 N+1
1 0
0
0
- 2 L C
L S
2
L C
L S
0
0
0
K +1 K +1
K +1 K +1
K +1 K +1
K +1 K +1
Aw = -
2 0
0
0
0
0
0
- 2 L
C
L
S
2
L
C
L
S
2N-1 2N-1
2N-1 2N-1
2N-1 2N-1
2N-1 2N-1
2
L C
L S
0
0
0
0
2
L C
L S
2N
2N
2N2 NR
L
L
-
2N
2N
2N2 NR
L C
L
S
NR 1
+
NR 1
+
NR 1
+
NR 1
+
1
With =
M
M
H (
T
B - H B)
ct
C
F
F
L C
L S
2N2 NR
2N
2N
L C
L
S
NR 1
+
NR 1
+
NR 1
+
NR 1
+
1
With =
M
M
H (
T
B
- H B)
m
ct
cm
m
m
L C
L S
2N2 NR
2N
2N
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As follows:
= P U + P U
F
m
m
with:
- 1
P = A (A + A)
W
- 1
P = A A
m
m
what implies:
T = (B
B P) U
(B
B P) U
cm +
C
m
m +
C
+
C
F
Note:
For the elements of type DKT and DST, there is B
. It results from it from the expressions
m = B = 0
C
C
simplified preceding equations.
4.4
Stamp elementary rigidity
4.4.1 Stamp elementary rigidity for the Q4 elements
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 H
by H, H by H and H per H. One will note that
MF
MF
F
F
F
F
in [R3.07.03] the results were presented without term of coupling membrane shearing
transverse or transverse inflection shearing. They here are added.
4.4.2 Stamp elementary rigidity for 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 H
by H, H per H. Puisque the relation = 0 is
MF
MF
F
F
satisfied the couplings transverse membrane shearing or transverse inflection shearing are
non-existent.
4.4.3 Stamp elementary rigidity for elements DST, DSQ
One a:
W E
=
E
(
1
H E + H + H - H H - T) + (
1
H E + H + H - H H - T)
- 1
+ TH
TdS =
int
m
MF
m
m
ct
MF
F
F
F
ct
ct
E
E
([
1
-
T
H - H H
H] E + [
1
-
T
H - H H
H]) +
([
1
-
T
H - H H
H] E + [
1
-
T
H - H H
H])
1
+ TH -
dS
T
m
m
ct
m
MF
m
ct
F
MF
F
ct
m
F
F
ct
F
ct
E
That is to say still:
We
=
E
(H E + H) + (T
H E + H)
1
+ TH -
TdS
int
m
MF
MF
F
ct
E
where:
1
-
T
H
H
H H H
m =
m -
m
ct
m
1
-
T
H
H
H H H
MF =
MF -
m
ct
F
1
-
T
H
H
H H H
F =
F -
F
ct
F
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
From where:
E
W
(U B H B U
U B H B U
U B H B U
U B H B U
int
=
T T
m
m
m
m
m +
T
T
m
m
MF
F
F +
T
T
T
F
F
MF
m
m +
T
T
F
F
F
F
F
E
T
T
- 1
T
T
- 1
T
T
- 1
T
T
-
+ B H B
B H
B U
U B H
B
U B H
B U
C
ct
C +
C
ct
cm
m +
m
cm
ct
C +
1
m
cm
ct
cm
m
T
T
- 1
T
T
- 1
T
T
-
+ B H B U
U B H
B
U B H
B U
C
ct
C
F
+ F C ct
C +
1
F
C
ct
C
F
T
T
- 1
T
T
-
+ U B H B U
U B H
B U) dS
m
cm
ct
C
F
+
1
F
C
ct
cm
m
= T
T
U (B H B
)
dS U
U (B H B
)
dS U
U (B H B
)
dS U
U (B H B
)
dS U
m
m
m
m
m +
T
T
F
F
F
F
F +
T
T
m
m
MF
F
F +
T
T
F
T
F
MF
m
m
E
E
E
E
T
T
- 1
T
T
- 1
T
T
- 1
T
T
-
+ (B H B
)
dS
(B H
B
)
dS U
U (B H
B
)
dS
U (B H
B
)
dS U
C
ct
C
+
C
ct
cm
m +
m
cm
ct
C
+
1
m cm
ct
cm
m
E
E
E
E
T
T
- 1
T
T
- 1
T
T
-
+ (B H B
)
dS U
U (B H
B
)
dS
U (B H
B
)
dS U
C
ct
C
F
+ F
C
ct
C
+
1
F
C ct C
F
E
E
E
T
T
- 1
T
T
-
+ U (B H B
)
dS U
U (B H
B
)
dS U
m
cm
ct
C
F
+
1
F
C ct cm
m
E
E
= T
THE U.K.U
THE U.K.U
THE U.K.U
THE U.K.U
K
The U.K.
K U
m
m
m +
T
F
F
F +
T
m
MF
F +
T
F
Fm
+ T
m
+ Tm m + T T
m
m
+ T
The U.K.
K U
THE U.K.U
THE U.K.U
THE U.K.U
F
+ T T F + Tm m F + Tf
m
m +
T
F
F
with:
K
[B H B
B H B
dS
]
m =
T
T
-
m
m
m +
1
cm
ct
cm
S
It is also known that U = (U,
) from where it results that:
F
F
K
B H B dS
F 11 = T
F
F
F
S
K
K
F 11
F 12
K
with: K
B H B dS
F 12 =
T
F =
T
K
K
F
F
F
F 12
22
S
K
B H B dS
F 22 = T
F
F
F
S
K
B H B dS
mf11 = T
m
MF
F
K
= K
K
with:
S
MF
(mf11
mf12)
K
B H B dS
mf12 = T
m
MF
F
S
T
K
= K
Fm
MF
Using the fact that = P U
+ P U one deduces from it that:
F
m
m
T
T
T
T
W
= U
K U + U
K U
+ U
K U + U
K U
int
m
m
m
F
F
F
m
MF
F
F
Fm
m
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
where:
T
T
T
T
K = K + P (K
+ K) P
+ (K
+ K) P
+ P (K
+ K)
m
m
m
F 22
m
mf12
m
m
m
mf12
m
T
T
T
T
K = K
+ K + P (K
+ K) P
+ (K
+ K) P
+ P (K
+ K)
F
F 11
F 22
F 12
F 12
T
T
T
T
K = K
+ K + (K
+ K) P
+ P (K
+ K)
+ P (K
+ K) P
MF
mf11
m
mf12
m
m
F 12
m
F 22
T
K = K
Fm
MF
This is still written:
U
E
W
U, THE U.K.
int = (
m
m
F)
U F
K
K
m [2N×2N]
MF [2N×3N]
where: K
is the elementary matrix of rigidity for an element of
[5N×5N] =
T
K
K
MF [3N×2N]
F [3N×3N]
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:
d+h/2
W ac
=
U
& udzdS
farmhouse
d-h/2 S
= (U & U
+ v & v
+ W & W
) + (+ D) (U
&
+ v
&
+ & U
+ & v
) + (+ D
2 + D 2) (& + &) dS
m
MF
m
X
y
X
y
F
MF
m
X
X
y
y
S
+h/2
+h/2
+h/2
with
dz,
zdz and
,
Z dz.
m =
MF =
= 2
F
- H/2
- H/2
- H/2
Note:
If the plate is homogeneous or symmetrical compared to its average layer then mf=0.
4.5.1 Stamp of traditional elementary mass
4.5.1.1 Element
Q4
The discretization of displacement for this isoparametric element is:
U
K
v
NR
K
U = NR W
=
K
K
K
,…,
1
NR
K 1
=
xk
yk
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
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:
M
0
0
M
0
m
MF
0
M
0
0
M
m
MF
M = 0
0
M
0
0
m
T
M
0
0
M
0
MF
F
T
0
M
0
0
M
MF
F
with:
T
M =
NR
dS
NR
m
m
S
M
= (+ D) T
NR
dS
NR
MF
MF
m
S
M = (+ D
2 + d2
) T
NR
dS
NR
F
F
MF
m
S
where: NR = (NR
NR
1
L
K).
For the continuation, one poses =
+ D and
2
= + D
2 + D.
MF
MF
m
F
F
MF
m
4.5.1.2 Elements of the type DKT, DST
Like:
W
W
0
NR
K
2N
= NR (,)
+ P (,)
X
K
xk
xk
K
K 1=
k=N 1+
P (,
)
y
yk
yk
where: = P U +
P U
m
m
F
one deduces from it that:
the U.K.
W
0
0
NR (,)
0
0
v
K
K
NR
NR
(,) NR
(,) NR
(,) NR
(,) NR
(,) W
.
X =
kxu
kxv
kxw
kxx
kxy
K
k=1
NR
(,) NR
(,) NR
(,) NR
(,) NR
(,)
y
kyu
kyv
kyw
kyx
kyy
xk
yk
The matrix of mass has then as an expression:
M
M
m
MF
M =
M
M
Fm
F
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
The part membrane M of the elementary matrix of mass is composed of the blocks kp (kth line
m
and pième column) following:
NR NR
0
NR NR
NR
NR
NR NR
NR NR
K
p
K pxu + kxu p
K
pxv +
kyu
p
m
+
MF
0
NR NR
NR NR
NR NR
NR NR
NR NR
K
p
K pyu + kxv p
K
pyv +
kyv
p
NR NR
NR NR
NR
NR
NR NR
kxu
pxu +
kyu
pyu
kxu
pxv +
kyu
pyv
+ F
NR NR
NR NR
NR NR
NR NR
pxu
kxv +
pyu
kyv
kxv
pxv +
kyv
pyv
The part inflection M is composed of the blocks kp (kth line and pième column) following:
F
NR NR
0 0
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
K
p
kxw pxw + kyw pyw
kxw
pxx +
kyw
pyx
kxw
pxy +
kyw
pyy
0
0 0
NR
NR
NR NR
NR
NR
NR NR
NR
NR
NR NR
m
+ F kxx pxw + kyx pyw
kxx
pxx +
kyx
pyx
kxx
pxy +
kyx
pyy
0
0 0
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxy
pxw +
kyy
pyw
kxy
pxx +
kyy
pyx
kxy
pxy +
kyy
pyy
The coupling part between the membrane and the inflection M is composed of the blocks kp (kth line and
MF
pième column) following:
NR NR
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
K
pxw
K
pxx
K
pxy
kxu pxw + kyu pyw
kxu
pxx +
kyu
pyx
kxu
pxy +
kyu
pyy
MF
+
F
NR NR
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
K
pyw
K
pyx
K
pyy
kxv pxw + kyv pyw
kxv
pxx +
kyv
pyx
kxv
pxy +
kyv
pyy
The coupling part between the inflection and the membrane M is composed of the blocks kp (kth line and
Fm
pième column) following:
NR
NR
NR
NR
NR NR
NR
NR
NR
NR
NR
NR
kxw
p
kyw
p
kxw
pxu +
kyw
pyu
kxw
pxv +
kyw
pyv
NR NR
NR NR
NR
NR
NR NR
NR
NR
NR NR
MF
kxx
p
kyx
p + F
kxx
pxu +
kyx
pyu
kxx
pxv +
kyx
pyv
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxy
p
kyy
p
kxy pxu + kyy pyu
kxy
pxv +
kyy
pyv
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 inflection. This
approach is used in Code_Aster for the elements of type DKT, DST and Q4G where functions
of form used in the calculation of the matrix of mass of inflection are of command 3. The interpolation for
W is written as follows:
NR
W = NR
(,) W
NR
(,) W
NR
(,) W
3 (K)
1 1
K +
3 (K)
1 2
, K +
3 (K)
1 3
- +
- +
- +
, K
k=1
4.5.2.1 Elements of type DKT
It is known that in the approximation of one Coils-Kirchhoff has = - W and = - W in any point of
X
, X
y
, y
the element.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
Because of discretization stated above one a:
NR
W = NR
(,) W
(J NR
(,) J NR
(,))W
(J NR
(,) J NR
(,))W
3 (K)
1 1
K +
11
3 (K)
1 2 +
21
3 (K)
1 3
, xk +
12
3 (K)
1 2 +
22
3 (K)
1 3
- +
- +
- +
- +
- +
, yk
k=1
since:
W
J
J
W
, K
11
12
, xk
=
W
J
J
W
, K
21
22
, yk
This is still written:
NR
W = NR
(,) W
NR
(,)
NR
(,)
3 (K)
1 1
K +
3 (K) 1 2 xk + 3 (K) 1 3
- +
- +
- +
yk
k=1
NR
= NR (,) W NR (,)
NR
(,)
kww
K +
kwx xk +
kwy yk
k=1
where:
NR
(,
- +
)
= NR
(,
- +
)
3 (K)
1 1
3 (K)
1 1
NR
(,
- +
)
= - J NR
(,
- +
)
- J NR
(,
- +
)
.
3 (K)
1 2
11
3 (K)
1 2
21
3 (K)
1 3
NR
(,
- +
)
= - J NR
(,
- +
)
- J NR
(,
- +
)
3 (K)
1 3
12
3 (K)
1 2
22
3 (K)
1 3
As follows:
the U.K.
W
0
0
NR
(,) NR
(,) NR
(,)
v
kww
kwx
kwy
K
NR
NR
(,) NR
(,) NR
(,)
NR
(,) NR
(,) W
X =
kxu
kxv
kxw
kxx
kxy
K
k=1
NR
(,) NR
(,) NR
(,)
NR
(,)
NR
(,)
y
kyu
kyv
kyw
kyx
kyy
xk
yk
By not taking account of the effects of inertia, the matrix of mass has the following form thus:
M
M
m
MF
M =
M
M
Fm
F
The part membrane M of the elementary matrix of mass is composed of the blocks kp (kth line
m
and pième column) following:
NR NR
0
NR NR
NR
NR
NR NR
NR NR
K
p
K pxu + kxu p
K
pxv +
kyu
p
m
+
MF
0
NR NR
NR NR
NR NR
NR NR
NR NR
K
p
K pyu + kxv p
K
pyv +
kyv
p
NR NR
NR NR
NR
NR
NR NR
kxu
pxu +
kyu
pyu
kxu
pxv +
kyu
pyv
+ F
NR NR
NR NR
NR NR
NR NR
pxu
kxv +
pyu
kyv
kxv
pxv +
kyv
pyv
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
The part membrane-inflection M is composed of the blocks kp (kth line and pième column) following:
MF
NR NR
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
K
pxw
K
pxx
K
pxy
kxu pxw + kyu pyw
kxu
pxx +
kyu
pyx
kxu
pxy +
kyu
pyy
MF
+
F
NR NR
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
K
pyw
K
pyx
K
pyy
kxv pxw + kyv pyw
kxv
pxx +
kyv
pyx
kxv
pxy +
kyv
pyy
The part inflection-membrane M is composed of the blocks kp (kth line and pième column) following:
Fm
NR
NR
NR
NR
NR NR
NR
NR
NR
NR
NR
NR
kxw
p
kyw
p
kxw
pxu +
kyw
pyu
kxw
pxv +
kyw
pyv
NR NR
NR NR
NR
NR
NR NR
NR
NR
NR NR
MF
kxx
p
kyx
p + F
kxx
pxu +
kyx
pyu
kxx
pxv +
kyx
pyv
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxy
p
kyy
p
kxy pxu + kyy pyu
kxy
pxv +
kyy
pyv
The term M of inflection is composed of the blocks kp (kth line and pième column) following:
F
NR
NR
NR
NR
NR
NR
kww pww
kww
pwx
kww
pwy
NR NR
NR
NR
NR
NR
m
kwx
pww
kwx
pwx
kwx
pwy +
NR
NR
NR
NR
NR
NR
kwy
pww
kwy
pwx
kwy
pwy
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
kxw
pxw +
kyw
pyw
kxw
pxx +
kyw
pyx
kxw
pxy +
kyw
pyy
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
F
kxx
pxw +
kyx
pyw
kxx
pxx +
kyx
pyx
kxx
pxy +
kyx
pyy
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxy
pxw +
kyy
pyw
kxy
pxx +
kyy
pyx
kxy
pxy +
kyy
pyy
4.5.2.2 Elements of the DST type
It is known that for these elements one has = - W and = - W where the distortion is constant
X
X
, X
y
y
, y
on the element.
Like:
NR
W = NR
(,) W
(J NR
(,) J NR
(,))W
(J NR
(,) J NR
(,))W
3 (K)
1 1
K +
11
3 (K)
1 2 +
21
3 (K)
1 3
, xk +
12
3 (K)
1 2 +
22
3 (K)
1 3
- +
- +
- +
- +
- +
, yk
k=1
one can also write:
NR
W = NR
(,
- +
)
W + NR
(,
- +
)
+ NR
(,
- +
)
3 (K)
1 1
K
3 (K)
1 2
xk
3 (K)
1 3
yk
K 1
=
+ (J + J) NR
(,
- +
)
+ (J + J) NR
(,
- +
)
11 X
12 y
3 (K)
1 2
21 X
22 y
3 (K)
1 3
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
where:
NR
(,
- +
)
= NR
(,
- +
)
3 (K)
1 1
3 (K)
1 1
NR
(,
- +
)
= - J NR
(,
- +
)
- J NR
(,
- +
)
3 (K)
1 2
11
3 (K)
1 2
21
3 (K)
1 3
NR
(,
- +
)
= - J NR
(,
- +
)
- J NR
(,
- +
)
3 (K)
1 3
12
3 (K)
1 2
22
3 (K)
1 3
NR
NR
(,)
NR
(,)
3 (K)
1 1 =
3 (K)
1 1
- +
- +
k=1
NR
NR
(,)
NR
(,)
3 (K)
1 2 =
3 (K)
1 2
- +
- +
k=1
NR
NR
(,)
NR
(,)
3 (K)
1 3 =
3 (K)
1 3
- +
- +
k=1
U
W
U
W
1
1
1
1
v
v
1
x1
1
x1
X
1y
1y
=
- 1
H
[(B
B P)
(B
B P)
m
m
M
M] T
T
ct
C
+ C
+
C
+
C
= U M
+
M
W
y
W
W
NR
NR
U
U
NR
xN
NR
xN
v
v
NR
yN
NR
yN
One obtains the interpolation for W then:
NR
NR
W = NR
(,) U
NR
(,) v
NR
(,) W
NR
(,)
NR
(,)
5 (K)
1 1
K +
5 (K) 1 2 K +
5 (K) 1 3 K + 5 (K) 1 4 xk + 5 (K) 1 5
- +
- +
- +
- +
- +
yk
k=1
k=1
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:
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where:
NR
(,
- +
)
= (J T
,
1
((
2 K
-)
1 +)
1 + J T (,
2 (
2 K
-)
1 +))
1
NR
(,
- +
)
5 (K)
1 1
11
U
12
U
3 (J)
1 2
+ (J T
,
1
((
2 K
-)
1 +)
1 + J T (,
2 (
2 K
-)
1 +))
1 NR
(,
- +
)
21
U
22
U
3 (J)
1 3
NR
(,
- +
)
= (J T
,
1
((
2 K
-)
1 +)
2 + J T (,
2 (
2 K
-)
1 +))
2
NR
(,
- +
)
5 (K)
1 2
11
U
12
U
3 (J)
1 2
+ (J T
,
1
((
2 K
-)
1 +)
2 + J T (,
2 2 (K
-)
1 +))
2
NR
(,
- +
)
21
U
22
U
3 (J)
1 3
NR
(,
- +
)
= NR
(,
- +
)
5 (K)
1 3
3 (K)
1 1
+ (J T
(
3
,
1
(
K
-)
1 +)
1 + J T (
(
3
,
2 K
-)
1 +))
1 NR
(,
- +
)
11
W
12
W
3 (J)
1 2
+ (J T
(
3
,
1
(
K
-)
1 +)
1 + J T (
(
3
,
2 K
-)
1 +))
1 NR
(,
- +
)
21
W
22
W
3 (J)
1 3
NR
(,
- +
)
= NR
(,
- +
)
5 (K)
1 4
3 (K)
1 2
+ (J T
(
3
,
1
(
K
-)
1 +)
2 + J T (
(
3
,
2 K
-)
1 + 2))NR
(,
- +
)
11
W
12
W
3 (J)
1 2
+ (J T
(
3
,
1
(
K
-)
1 +)
2 + J T (
(
3
,
2 K
-)
1 + 2))NR
(,
- +
)
21
W
22
W
3 (J)
1 3
NR
(,
- +
)
= NR
(,
- +
)
5 (K)
1 5
3 (K)
1 3
+ (J T
(
3
,
1
(
K
-)
1 +)
3 + J T (
(
3
,
2 K
-)
1 +))
3 NR
(,
- +
)
11
W
12
W
3 (J)
1 2
+ (J T
(
3
,
1
(
K
-)
1 +)
3 + J T (
(
3
,
2 K
-)
1 +))
3 NR
(,
- +
)
21
W
22
W
3 (J)
1 3
This can be still written in the following way:
the U.K.
W
NR
(,) NR
(,) NR
(,) NR
(,) NR
(,)
v
kwu
kwv
kww
kwx
kwy
K
NR
NR
(,)
NR
(,)
NR
(,)
NR
(,) NR
(,) W
X =
kxu
kxv
kxw
kxx
kxy
K
k=1
NR
(,)
NR
(,)
NR
(,)
NR
(,)
NR
(,)
y
kyu
kyv
kyw
kyx
kyy
xk
yk
The matrix of mass has the following form thus:
M
M
m
MF
M =
M
M
Fm
F
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:
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The part membrane M of the elementary matrix of mass is composed of the blocks kp (kth line
m
and pième column) following:
NR NR
NR
NR
NR
NR
NR NR
NR
NR
NR NR
NR NR
K
p +
kwu
pwu
kwu
pwv
K pxu + kxu p
K
pxv +
kyu
p
m
+
MF
NR
NR
NR NR
NR
NR
NR NR
NR NR
NR NR
NR NR
kwv
pwu
K
p +
kwv
pwv
K pyu + kxv p
K
pyv +
kyv
p
NR NR
NR NR
NR
NR
NR NR
kxu
pxu +
kyu
pyu
kxu
pxv +
kyu
pyv
+ F
NR NR
NR NR
NR NR
NR NR
pxu
kxv +
pyu
kyv
kxv
pxv +
kyv
pyv
The part membrane-inflection M is composed of the blocks kp (kth line and pième column) following:
MF
NR
NR
NR
NR
NR
NR
NR NR
NR NR
NR NR
kwu
pww
kwu
pwx
kwu
pwy
K pxw
K
pxx
K
pxy
m
+
MF
NR
NR
NR
NR
NR
NR
NR NR
NR NR
NR NR
kwv
pww
kwv
pwx
kwv
pwy
K pyw
K
pyx
K
pyy
NR NR
NR NR
NR
NR
NR NR
NR
NR
NR NR
kxu
pxw +
kyu
pyw
kxu
pxx +
kyu
pyx
kxu
pxy +
kyu
pyy
+ F
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxv
pxw +
kyv
pyw
kxv
pxx +
kyv
pyx
kxv
pxy +
kyv
pyy
The part inflection-membrane M is composed of the blocks kp (kth line and pième column) following:
Fm
NR
NR
NR
NR
NR NR
NR
NR
kww pwu
kww
pwv
kxw p
kyw
p
NR NR
NR
NR
NR
NR
NR NR
m
kwx
pwu
kwx
pwv + MF
kxx
p
kyx
p
NR
NR
NR
NR
NR NR
NR NR
kwy
pwu
kwy
pwv
kxy p
kyy
p
NR NR
NR
NR
NR
NR
NR
NR
kxw
pxu +
kyw
pyu
kxw
pxv +
kyw
pyv
+ NR NR
NR NR
NR
NR
NR NR
F
kxx
pxu +
kyx
pyu
kxx
pxv +
kyx
pyv
NR NR
NR NR
NR NR
NR NR
kxy
pxu +
kyy
pyu
kxy
pxv +
kyy
pyv
The term M of inflection is composed of the blocks kp (kth line and pième column) following:
F
NR
NR
NR
NR
NR
NR
kww pww
kww
pwx
kww
pwy
NR NR
NR
NR
NR
NR
m
kwx
pww
kwx
pwx
kwx
pwy +
NR
NR
NR
NR
NR
NR
kwy
pww
kwy
pwx
kwy
pwy
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
kxw
pxw +
kyw
pyw
kxw
pxx +
kyw
pyx
kxw
pxy +
kyw
pyy
NR NR
NR NR
NR
NR
NR
NR
NR
NR
NR
NR
F
kxx
pxw +
kyx
pyw
kxx
pxx +
kyx
pyx
kxx
pxy +
kyx
pyy
NR NR
NR NR
NR NR
NR NR
NR NR
NR NR
kxy
pxw +
kyy
pyw
kxy
pxx +
kyy
pyx
kxy
pxy +
kyy
pyy
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Processing of offsetting for the elements of plate
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:
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4.5.2.3 Elements of the Q4 type
One proceeds in the same way that for the elements of the DST type but with:
w1
x1
X
1y
= B M
C
y
W NR
xN
yN
where: Bc is the matrix established with [§4.3.1].
One deduces from it that:
the U.K.
W
0 0 NR
(,) NR
(,) NR
(,)
v
kww
kwx
kwy
K
NR
0 0
0
NR (,)
0
W
X =
K
K
k=1
0 0
0
0
NR (,)
y
K
xk
yk
The matrix of mass has the following form thus:
M
0
m
M =
0
MF
The part membrane M of the elementary matrix of mass is composed of the blocks kp (kth line
m
and pième column) following:
NR NR
0
K
p
m
0
NR NR
K
p
The term M of inflection is composed of the blocks kp (kth line and pième column) following:
F
NR
NR
NR
NR
NR
NR
0
0
0
kww pww
kww
pwx
kww
pwy
NR NR
NR
NR
NR
NR
0 NR NR
0
m
kwx
pww
kwx
pwx
kwx
pwy + F
K
p
NR NR
NR
NR
NR
NR
0
0
NR NR
kwy
pww
kwy
pwx
kwy
pwy
K
p
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Processing of offsetting for the elements of plate
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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
)
because the latter are negligible compared to
F
X
X
y
y
S
others. Indeed a multiplicative H2 factor/12 the dregs with the other terms and they becomes negligible
for a thickness report/ratio over characteristic length lower than 1/10. When offsetting is
introduced, these terms of the form (+ D
2
+ d2) (& + &
dS
)
are not any more
F
MF
m
X
X
y
y
S
negligible and are introduced into the form of the matrix of mass.
5
Implementation and postprocessings
Offsetting is introduced by optional key word EXCENTREMENT 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 grid). 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 grid. 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 thus introduces into AFFE_CHAR_MECA the 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 grid 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. “MAIL”
mean that one applies the efforts to the level of the datum-line or plan of the grid. By defect them
efforts will be applied to the plan of the grid of the plate. The efforts of the type are concerned
FORCE_COQUE of the TE0032.
In local reference mark with the element, when the forces and the couples are brackets on “MOY” one uses
simple relation of passage:
C = C - df
X
X
y
C = C + df
y
y
X
to bring back the efforts and the couples in the reference mark of the grid 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:
C = C - (D + H/2) F
X
X
y
C = C + (D + H/2) F
y
y
X
In local reference mark with the element, when the forces and the couples are applied to “INF” one uses
simple relation of passage:
C = C - (D - H/2) F
X
X
y
C = C + (D - H/2) F
y
y
X
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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 + (D + H
/2) N F where C is defined compared to reference mark “INF” “MOY” “SUP” with equal
with - 1, 0 and 1, respectively. When there is no offsetting, the preceding formula is reduced to
C = C + H/N
2 F.
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 grid. 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 grid.
The relation to be used is C = C + DNN F 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 grid, 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 grid. Relations of passage compared to
conditions given on the average layer are as follows:
=
ref.
moy
U
= U
-
DNN
ref.
moy
moy
5.3 Postprocessings
For postprocessings, the results owing to lack of generalized efforts type are given in
locate 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 the TE0033, one will define the four possibilities
following of postprocessing of the efforts by key word PLAN: “INF” “MOY” “SUP” “MAIL” of
commands CALC_ELEM and CALC_CHAM_ELEM with the same direction as previously. The defect is
put at “MAIL”. All calculations are made in plan “MAIL” of the grid (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 “MAIL” to
“MOY” one uses the simple relation of passage:
NR = NR
M = M - NR
D
T = T
To pass from the efforts results generalized of “MAIL” to “SUP” one uses the simple relation of
passage:
NR = NR
M = M - (D + H/)
2 NR
T = T
To pass from the efforts results generalized of “MAIL” to “INF” one uses the simple relation of
passage:
NR = NR
M = M - (D - H/)
2 NR
T = T
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Processing of offsetting for the elements of plate
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:
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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 grid 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
M
F
F
M+dn^F
D
M+dn^F
N
F
F
Displacements are in the following way dependant for a point located at a height Z compared to
average layer:
U = U
+
N
Z = U + (Z + D N
)
moy
moy
ref.
ref.
what is still written:
=
moy
ref.
U
= U + DNN
moy
ref.
ref.
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 inflection of double-layered made up of two different isotropic materials. One studies
the coupling membrane-inflection. The calculation of reference is that of double-layered defined by
DEFI_COQU_MULT made up 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 inflection 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.
Handbook of Référence
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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 grid (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
grid. For the generalized efforts, one can however define a reference mark of postprocessing -
locate 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 and O.C. ZIENKIEWICZ: “A notes one Mass 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 MMN/97/091.
[8]
P. MASSIN, F. VOLDOIRE, S. ANDRIEUX: “Model of thermics for the thin hulls”,
Handbook of Référence of Code_Aster [R3.11.01].
[9]
F. VOLDOIRE: “Thermoelastic Hollow roll”, Manuel de 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.
Handbook of Référence
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Processing of offsetting for the elements of plate
Date:
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Author (S):
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:
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Appendix 1 Facteurs of transverse correction of shearing
for orthotropic or laminated plates
offset
The Hct matrix is defined so that the surface density of transverse energy of shearing obtained
in the case of the three-dimensional distribution of the constraints 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 Hct such as:
1 +h/2
1
1
+h/2
xz
H-1
- 1
= TH T
ct
= Hct with =
and T =
dz = H
.
2
2
2
ct
yz
- H/2
- H/2
To obtain Hct one uses the distribution of following Z obtained starting from the resolution of the equations
of balance 3D without external couples:
Z
Z
= - (
with xz =
= 0 for z=±h/2.
, +
) D;
,
= - (, +
) D
xz
xx X xy y yz
xy X yy, y
yz
- H/2
- H/2
If there is no coupling membrane inflection (symmetry compared to z=0), constraints in
plan of element xx, yy, xy in the case of have as an expression a behavior of pure inflection:
= zA (Z) M with A Z = H Z H-1
()
() F.
If (
H Z) and Hf do not depend on X and y one can determine Hct. Indeed:
M xx, X - Mxy, y
Tx Mxx, X + Mxy, y
M xy, X - M
yy, y
(Z) = D (Z) T
1
+ D (Z)
2
where T =
=
and =
T
M
y
M xy, X + M yy, y
yy, X
M xx, y
like:
Z
With
With
With
With
11 +
33
13 +
32
D = -
D,
1
2 A
With
With
With
H/2
31 +
23
22 +
33
-
Z
With
With
With
With
With
With
11 -
33
13 -
2
2
32
12
31
D = -
D.
2
2 A
With
With
With
With
With
H/2
31 -
23
33 -
2
2
22
32
21
-
+h/2
C =
DTH 1
- D
11
1 1dz;
- H/2
1 +h/2
1
T C
C
T
+h/2
It results from it that
-
H 1
11
12
T
- 1
=
T
with: C =
D H D
dz;
2
2 C
C
12
1
2
12
22
- H/2
- H/2
+h/2
C
=
DTH 1
- D
22
2 2dz
- H/2
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
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
1 +h/2
1
As in addition
- 1
-
H
1
- 1
=
TH T one proposes to take H = C to satisfy them as well as possible
2
2
ct
ct
11
- H/2
two equations whatever T and.
+h/2
By comparing Hct thus calculated with H
H
ct = dz one reveals the coefficients of correction of
- H/2
following transverse shearing: K = H11/H 11;K
= H12/H 12;K = H22/H 22
1
ct
ct
12
ct
ct
2
ct
ct.
For a homogeneous, isotropic or anisotropic plate, one finds as follows: Hct =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:
NR
i-1
I
H
1
1
1
C
= I (H AT
2
2
p p
- Z AT) H-1
I
(
H With
p p
- Z A
11
p
I
p
I
I) +
4
2
2
i=1
p=1
p=1
i-1
I
1
1
3
3
-
1
1
2
2
-
(zi+1 - Z) [ATH 1
I
(
H With
p p
- Z A
I
) + (H AT
p p
- Z AT) H 1A
I
p
I
p
I
I
I]
24
2
2
p=1
p=1
1
+
(z5
5
T
- 1
i+1 - zi) A H
With
80
I
I
1
WITH + A
WITH + A
11
33
13
32
where: H = Z +1 - Z, = (Z +1 + Z
I
I
I
I
I
I) and A
for
2
I represents the matrix A + A
WITH + A
31
23
22
33
layer I.
· Validity of the choice H
= C 1
-
ct
11 can be examined a posteriori when one has an estimate of
the solution (fields of displacements and plane constraints, 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 Hct diagonal and two coefficients k1 and k2) 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 D2 = 0. The choice makes above is then valid and no examination has
posteriori is not necessary.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HT-66/03/005/A
Outline document