Code_Aster ®
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Titrate:
Finite elements of voluminal hulls


Date:
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Author (S):
X. DESROCHES Key
:
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: 1/42
Organization (S): EDF-R & D/AMA
Handbook of Référence
R3.07 booklet: Machine elements on average surface
R3.07.04 document

Finite elements of voluminal hulls


Summary:

With an aim of supplementing the library of finite elements of plate plans [R3.07.03] currently available
in Code_Aster (DKT, DST, Q4G…), one proposes to introduce two finite elements of voluminal hull or
three-dimensional [bib1]. This new modeling COQUE_3D [U1.12.03] makes it possible to carry out calculations of
structures hull of an unspecified form with a better approximation of the geometry and
kinematics.

One will limit oneself to the framework of linear kinematics. One thus remains in small displacements and small
deformations. No restriction is made on the type of behavior in plane constraints.

The two elements which are introduced are the quadratic element quadrangle Hétérosis with 9 nodes and sound
triangular equivalent with 7 nodes. The formulation of the continuous problem is done in Cartesian co-ordinates, it
who allows to avoid explicit calculations of the curvatures. These two elements have as a correspondent the element
linear of hull with 3 nodes presented in the document [R3.07.02].

These two new elements are validated on existing case-tests of plate, and on three new cases
tests of hull developed in the documentation of validation and whose principal conclusions are
presented briefly here.

This note also presents in appendix how to take into account the anisotropy of materials.
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Count

matters

1 Introduction ............................................................................................................................................ 4
2 Formulation ............................................................................................................................................ 5
2.1 Geometry of the hull .................................................................................................................... 5
2.1.1 Geometrical description of average surface ................................................................... 6
2.1.2 Description of the geometry of the hull ................................................................................ 7
2.1.3 Notice ............................................................................................................................... 8
2.2 Theory of the plates and the hulls ................................................................................................ 9
2.2.1 Kinematics ........................................................................................................................... 9
2.2.1.1 Field of displacement ............................................................................................. 9
2.2.1.2 Expression of the three-dimensional deformations ..................................................... 10
2.2.2 Law of behavior ............................................................................................................ 11
3 Principle of virtual work ................................................................................................................ 13
3.1 Work of deformation ................................................................................................................... 13
3.1.1 Energy interns elastic hull ...................................................................................... 13
3.1.2 Expression of the resulting efforts ......................................................................................... 14
3.2 Work of the forces and couples external ........................................................................................ 14
3.3 Work of the inertias ............................................................................................................. 16
3.4 Principle of virtual work ................................................................................................................ 16
4 numerical Discretization of the variational formulation resulting from the principle of virtual work ............ 17
4.1 Introduction .................................................................................................................................... 17
4.2 Discretization of the geometrical terms ....................................................................................... 18
4.3 Discretization of the field of displacement ..................................................................................... 19
4.3.1 Element Hétérosis Q9H ........................................................................................................ 19
4.3.2 Element triangle T7H ............................................................................................................ 20
4.3.3 Notice ............................................................................................................................. 20
4.4 Discretization of the field of deformation ....................................................................................... 20
4.5 Stamp rigidity .......................................................................................................................... 21
4.5.1 Decomposition of the elementary matrices .......................................................................... 22
4.5.2 Assembly of the elementary matrices .............................................................................. 22
4.5.2.1 Degrees of freedom ...................................................................................................... 22
4.5.2.2 Fictitious rotations ...................................................................................................... 22
4.6 Stamp of mass .......................................................................................................................... 23
4.6.1 Discretization of displacement for the matrix of mass ................................................... 23
4.6.2 Stamp of elementary mass ............................................................................................. 24
4.6.3 Assembly of the elementary matrices of mass .............................................................. 24
4.7 Numerical integration for elasticity ........................................................................................... 25
4.7.1 Surface integration ........................................................................................................... 25
4.7.2 Integration in the thickness ................................................................................................. 26
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4.8 Numerical integration for plasticity ........................................................................................ 26
4.9 Discretization of elementary work for the loadings ................................................... 27
4.9.1 Elementary discretization of the work of the forces and external couples being exerted on
surface average .................................................................................................................. 27
4.9.1.1 Loads given in the total reference mark ................................................................. 28
4.9.1.2 Loads given in the local reference mark .................................................................... 28
4.9.2 Elementary discretization of the work of the forces and external couples being exerted on
contour 28
4.9.3 Discretization of the term of gravity .................................................................................. 28
4.9.4 Discretization of the term of pressure ..................................................................................... 28
4.9.5 Discretization of the terms of inertia centrifuges ...................................................................... 29
4.9.6 Taking into account of the loadings of thermal dilation ................................................. 30
4.9.7 Assembly ......................................................................................................................... 31
5 Validation ............................................................................................................................................. 31
5.1 Case test in linear statics .......................................................................................................... 32
5.1.1 Static case test n° 1 ........................................................................................................... 32
5.1.2 Static case test n° 2 ........................................................................................................... 32
5.2 Case test in dynamics .................................................................................................................. 32
6 thermomechanical Chaining ............................................................................................................... 32
6.1 Description ..................................................................................................................................... 32
6.2 Case-test ......................................................................................................................................... 33
7 Establishment of the elements of hull in Code_Aster .................................................................. 34
7.1 Description ..................................................................................................................................... 34
7.2 Introduced use and developments ........................................................................................ 34
7.3 Calculation in linear elasticity ............................................................................................................ 35
7.4 Plastic design ........................................................................................................................ 35
8 Conclusion ........................................................................................................................................... 36
9 Bibliography ........................................................................................................................................ 37
Appendix 1
Extension to the anisotropic materials not programmed ......................................... 38
Appendix 2
Functions of form for element Q9H .................................................................. 40
Appendix 3
Functions of form for element T7H .................................................................. 41

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1 Introduction

One introduces into Code_Aster two finite elements of hull voluminal with shearing
transverse (the quadrange with 9 nodes MEC3QU9H and the triangle with 7 nodes MEC3TR7H) in calculation of
structures hull of an unspecified form. To represent this type of structures, one used until
present with Code_Aster of the elements of plate with plane facets which induced inflections
parasites and of the too restrictive hulls of revolution on the type of structure [R3.07.02].
development was carried out for isotropic materials with linear kinematics. They cannot
thus to be used that within the framework of small displacements and small deformations. This
formulation can be extended to anisotropic materials [Annexe 1] and to nonlinear kinematics
[R3.07.05].

For the resolution of chained thermomechanical problems, one must use the elements before
stop thermal hull with 7 and 9 nodes described in [R3.11.01].

One develops hereafter the mechanical continuous problem by describing the kinematics of hull of the type
Hencky-Mindlin-Naghdi (assumption of the cross-sections or plane) supplemented by a distortion
transverse and the law of thermo behavior elastoplastic. Thanks to a parameter of penalization
one can pass from a theory with shearing to a theory without shearing. One presents then
the selected finite elements which are isoparametric quadratic elements making it possible to have one
fine representation of a curved geometry and good estimates of the constraints. The interpolation and
the method of integration are also described.

One validates finally the development on some cases of test.

The nonlinear kinematics of these hulls is treated in the reference material [R3.07.05].
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2 Formulation

2.1
Geometry of the hull

For the elements of voluminal hull one defines a surface of reference, or surfaces
average, left (of curvilinear co-ordinates 1 2 for example) and a thickness (
H 1 2)
measured according to the normal on the average surface. This thickness must be small compared to
other dimensions (extensions, radii of curvature) of the structure to be modelled. The figure [Figure 2.1-
has] below illustrates our matter.

Solid 3D
Z
H
Y
B
X
L
R1
R2
Thickness H < L, B, R1, R2
2
N, 3
H
E, X
3
3
1
E, X
2
2
O

E, X
1
1


Appear 2.1-a

The position of the points of the hull is given by the curvilinear co-ordinates (1 2) of surface
average and rise 3 compared to this surface. (O, ek) is the total Cartesian reference mark,
associated axes (xk).
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2.1.1 Geometrical description of average surface

Local natural base and bases Cartesian local

That is to say P an unspecified point of the average surface of reference, one a:

COp = x0k (1
, 2
) ek

One defines the vectors has natural local base of the tangent plan out of P with, attached to P by:

COp
has =
= COp


,



and one defines unit normal N by:

a1 has
N =
2
a1 a2

3 is the variable of position in the thickness associated with N.

(has, has, has
1
2
3) constitutes the natural base attached to P.

The curvilinear frame of reference (1 2) not being inevitably orthogonal, the base (A) is not
thus not inevitably orthogonal (and even orthonormée). A local base is thus defined
orthonormée T K as follows:

has
T
1
=
, T = N T
T = N
1

has
2
1
3
1

and one notes (S, S
1
2) the frame of reference associated to (T, T
1
2).

Calculation of the tensor of curvature

The tensor of curvature is related to the variation of the normal on. It is defined by its components
mixed:

N = - C.a.
,


or by its components covariantes: C
= - A.N, = n.a




. This tensor is symmetrical
since has
= has

,

. Its trace tr C is the average curvature and its determinant the curvature
Gaussian.
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2.1.2 Description of the geometry of the hull

That is to say Q an unspecified point of, volume of the hull thickness H considered constant, one a:


OQ + COp + PQ = COp + 3 H N
2

where 3 [- 1,]
1.




3
1, 2,
H


2 constitutes a curvilinear frame of reference of.

One can also write OQ according to his components (xk) in the total base (ek):

OQ = X E
K
K

Local natural base, bases orthonormée local and tensor metric

As for P, one defines the natural base of space 3D (gk) attached to Q by:

OQ
H
g1 G
G =
= + N has
G
2
3
,
,
3 =
= N






2
g1 g2

As (gk) is not inevitably orthogonal, one defines a local base orthonormée (Tk) like
follows:

G
T
1
=
, T = N T
, T = N
1

G
2
1
3
1

and one notes (xk) the frame of reference associated with (Tk).

One will call (Tk) the local orthonormée base, and (xk) the co-ordinates in this base
orthonormée local.

By definition, one a:

OQ X J
T =
=
E
J
= T E
K
~
~
J
K
J


xk
xk
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X J
J
with
T components of (Tk) in the total base (E J). (They are also them
X
~ = K
K
components of the matrix of passage of (Tk) with (E J) since the matrix of passage is
J
orthogonal. Thus if T = T E
K
K
K
J one has also E
= T T
K
J
J).

One defines the metric tensor G associated with Q by his components deduced from the scalar products
vectors of the local orthonormée base:

Gij = I
T .Tj

This tensor G is worth the Id identity.

2.1.3 Notice

The figures [Figure 2.1.3-a] and [Figure 2.1.3-b] illustrate the geometrical magnitudes mentioned
above.

2
t2
N
a2 a1, T1
P
1


Appear 2.1.3-a

T1
3
T1

N
1
Q
P


Appear 2.1.3-b

It should be noted that two local orthonormées bases, that associated average surface (tk) and
the other with the volume of the hull (Tk) are confused only when the curvature is null. In it
cases the elements of hull are comparable to elements of plate
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2.2
Theory of the plates and the hulls

These elements are based on the theory of the plates and the hulls according to which:

2.2.1 Kinematics
2.2.1.1 Field of displacement

The cross-sections which are the sections perpendicular to average surface remain right;
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 hull.

If one notes Q', the position of Q after deformation, one a:

OQ' = OQ + QQ' = OQ + U (Q)

where the field of displacement chosen, corresponding to the kinematics of Hencky-Mindlin, is written:


U (Q) = (
U P) + 3 H (P) with (P) .n = 0
2

where U (P) and (P) are respectively the vector displacement and the vector rotation of P, projection
of Q on the average surface of the hull. The fact that (P) .n = 0 indicates that one does not take in
count in this kinematics rotations of the hull around its normal.

Notation:

One notes ~ the quantities expressed in the local Cartesian bases (tk) or (Tk) for the points
P and Q respectively. It results from it that:

· the vector three-dimensional displacement U can be written U = ~
U T
K K or U = U.E.
K K,
where it is expressed respectively in its local orthonormée base or the base
Cartesian total,
· the vector displacement of average surface U can be written U = ~u T
K K or
U = U.E.
K
K according to whether it is expressed in its local orthonormée base or in the base
Cartesian total,
· the vector rotation of average surface is written = ~
T in its orthornormée base
local. being the rotation of normal N (on average surface), one also writes = N
with, vector rotation of average surface, such as = ~
T. The equivalence of both
~
~
~
~
formulations shows that 1 =,
2
2 =
- 1.
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2.2.1.2 Expression of the three-dimensional deformations

The tensor of deformation is calculated in the local orthonormée Cartesian base (Tk). It is defined
like the half-difference of the metric tensors associated the local orthonormées bases afterwards and
before deformation. The metric tensor associated this base in the not-deformed state is simply
OQ'
the Id identity, while the metric tensor of the deformed state is Gij = T'.T
I
'J with You K = ~

.
xk

The components of the tensor of deformation in (Tk) are thus given by:

~
~
U
U
~
1






=
~ + ~
2

X

X
~
~

~
1 U U

3
3 =
~ + ~
2 x3

X

The equations above are linear relations deformation-displacements. Variables of
~
displacement are components the U.K.

The components ~
kl of the tensor ~ can be also expressed according to the components in
U p
total reference mark. Indeed as in the total reference mark
xm
= I
J
I
J
K
L
~
K
L
ije E
= ijT T T T
K L
= klT T one thus deduces from it immediately that:
~

I
J
kl = ijT T
K L. (ek) and (Tk) are the bases contravariantes associated (ek) and (Tk)
respectively such as: I.E.(internal excitation) .e
I
J = ij
and T .T
E
T
J = ij. As the bases (K) and (K) are
orthonormées, their associated bases contravariantes are confused with themselves. Thus of
J
J
even manner that one had T = T E
K
J
K
K
J one finds T
= T E
K
.
If one notes T = T I E
K
~
K
L
~
K I T then T T: = T
kl
T =. For the continuation one indicates by ~

the form of the tensor of the deformations in the local orthonormé reference mark and by the expression of
even tensor in the total reference mark. The relation of passage of the one with the other is given above in
term of tensors.
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Note:

J
The Tk terms contain the terms of curvature of the hull.

One notes in the relations deformation-displacements that the component ~
33 is not
determined by kinematics. This is to be associated the assumption nullity constraints
transverse normals ~
33 = 0 justified by the behavior of the hulls.
In the literature (see for example [bib3]), the modeling of the hulls by the approach based on
curvilinear components ~
U of displacement reveals explicitly the sizes of
K
curvature on the level of the form of the tensor of deformation [bib5]. Like, in general,
geometry of the hull is not known explicitly, one must thus determine
numerically the geometrical characteristics that are the vectors has, G

,… and them
curvatures C. With the finite element method it is necessary to derive them twice
functions of form (see page 20 of [bib5] and [R3.07.02]) to calculate C. This can return
their vague calculation according to the family of the functions of selected form. The made error depends on
these last (linear, quadratic, cubic polynomials….) and becomes independent of
refinement of the grid. A formulation utilizing derivation first functions of
form (calculation of slopes) does not present this disadvantage. Thus the consequent error with calculations
terms of curvature in a formulation based on the curvilinear approach does not decrease
with the refinement of the grid whereas for the formulation described above it becomes small
by increasing the number of finite elements. Within sight of the preceding observations, approach known as
curvilinear was not followed.

2.2.2 Law of behavior

The behavior of the hulls is a behavior 3D in “plane constraints”. It binds the components
constraints and deformations, in the form of vectors, in the local orthonormée base.
transverse constraint ~
33 is null bus regarded as negligible compared to the others
components of the tensor of the constraints (assumption of the plane constraints). The law of behavior
most general is written then as follows:

~
~
~

HT
11
11 - 11
~
~
~


HT
22
22 - 22
~
~



12
= (
C, µ)
~
12


~


~
13



1
~

~


23

2


~
where (
C, µ) is the local matrix of behavior in plane constraints and µ represents the unit
variables intern when the behavior is nonlinear.

For behaviors where the transverse distortions are uncoupled from the deformations from
~
membrane and of inflection, (
C, µ) is put in the form:

~
~ H
0
C =
~
0 H
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~
~
where H (, µ) is a matrix of behavior of membrane-inflection 3x3 and H (,)
µ a matrix of
transverse behavior of distortion 2x2. The two phenomena being uncoupled one can too
to write the behavior in the form:

~
~
MF ~
MF
~ = (
C,) ~ with:

µ


~
~
~

HT
11
11 -
11
~
~
~
~ ~


13
1
HT
~
~
~
MF = 22 = H (, µ) ~
~
22 - 22 = H (, µ) ~

MF and ~ = ~ = H (,)

~ = H (,) ~




~

~

µ
23

µ
2
12


12

One will remain from now on within the framework of this assumption.

For an isotropic homogeneous linear behavior elastic, one has as follows:

1 v
0
0
0



v 1
0
0
0
1


v

~
E
0 0
0
0
C =

2

2
K (1 - v)
1

v 0 0 0
0


2
K (1 - v)
0 0
0
0


2


where K is factor of transverse correction of shearing whose significance is given in
reference material of the elements of plate [R3.07.03], and [bib4] for more details. It
coefficient is worth 5/6 for a theory of the Reissner type and 1 within the framework of the theory of
Hencky-Mindlin. Lastly, if one chooses K very large, one brings back oneself to a theory of the type Coils-Kirchhoff.
One neutralizes the transverse distortion by penalization of associated energy by taking K = 106 H/R
(H being the thickness of the hull and R its average radius of curvature).

Always in the isotropic case, the two only nonnull components of ~
HT are ~th
II for i=1,2,
such as:
~
HT
ref.
II
= (T - T
)

where is the thermal dilation coefficient and T T ref.
-
the difference in supposed temperature
known.

Note:

One does not describe the variation thickness nor that of the transverse deformation ~
33 that one
can however calculate by using the preceding assumption of plane constraints. In addition
no restriction is made on the type of behavior in plane constraints which one can
to represent.
Same manner as T T: = ~
one can deduce some ():
MF =
:
~
T
T
Tmf = MF and
():
=
:
~
T T
T =, which makes it possible to find ~mf and ~ starting from the tensor of
deformations in the total reference mark.
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3
Principle of virtual work

3.1
Work of deformation

In 3D the expression of the work of deformation is written:

H/2
H/2
H/2
~
W
=
(~ ~
D
) V

=
(~ ~
C ~ D
) V

=
(Pr PsC P p Pq
D
) V
def
ij ij
ij ijkl kl
rs I K ijkl K L pq
S - H/2
S - H/2
S - H/2

H/2
H/2
=
(C
D
) V

=
(D
) V
rs rspq pq
ij ij
S - H/2
S - H/2

It is checked that this expression is invariant compared to the base in which the tensors are
expressed. One chooses for the continuation of this document all to express in the local base (Tk) in
knowing that one passes from the local tensor of behavior to the total tensor of behavior by
p
Q
relation C
=
~
P R P Sc
P P
rspq
I
K
ijkl K
L.

The general expression of the work of deformation 3D for the element of hull is worth:

H/2
H/2
H/2
H/2
W
=
(~ ~
D
) V

=
(~~ ~

C D
) V

=
(~ ~ ~
H
D
) V

+
(~ ~ ~
H D
) V
def
MF
MF



S - H/2
S - H/2
S - H/2
S - H/2

where S is average surface and the position in the thickness of the hull varies between ­ H/2 and +h/2. It
in the expression of the work of deformation a contribution of deformation appears in
membrane-inflection and a contribution of transverse shearing strain.

3.1.1 Energy interns elastic hull

It is expressed in the following way:

1
E

2
2
2
2
2
int =
[
(~
~
~ ~

+ 22 + 2)
11 22 + G (~
12 + K (~
~
1 +))]FD
2
1 - 2 11
2

S
E
where K is the factor of correction in transverse shearing defined in paragraph 2 and G = 2 1 (+.
)
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3.1.2 Expression of the resulting efforts

One notes:

NR
~

M
~

11 +h/2 11
11 +h/2 11
T +h/2 ~
1
13
NR = NR
~
~
22 = 22 dz; M = M 22 = 22 zdz; T =
=
dz
T
~
2

.
23





/
~
NR
/2
~
- H2
- H/2
12
H

12
-
M12
12

NR, NR, NR
11
22
12 are the generalized efforts of membrane (in NR/m);
M, M, M
11
22
12 are the generalized efforts of inflection or moments (in NR);
T, T
1
2, are the generalized efforts of shearing or sharp efforts (in NR/m);

The expression of the resulting efforts that one gives here is an approximate expression which does not hold
count curvature of the hull (cf p.316 of [bib3]). The error made on these efforts is then in
H2/R where 1/R is the average curvature. When the hull becomes plane, expressions given
above are exact and the significance of the resulting efforts can be found in [R3.07.03].
We will not develop more this aspect in addition documented well in [bib3] because the theory of
hull used here does not rest on a resulting generalized deformations formulation/efforts but
on three-dimensional/forced a deformations formulation.

3.2
Work of the forces and couples external

The work of the forces being exerted on the voluminal hull is expressed in the following way:

+h/2
+h/2
W
F. D
U V
F. D
U.S.
F. D
U zds
ext. = v
+ S
+ C

S - H/2
S
C - H/2

where F, F, F
v
S
C are the voluminal, surface efforts and of contour being exerted on the hull,
respectively. C is the part of the contour of the hull on which the efforts of contour FC are
applied.

has) Charges given in the total reference mark:

With the kinematics of [§2.2.1], one determines as follows:

~
~
W
(F U

+ c) dS + (U

+) ds = (F U

+ C (T
2 1 - T))dS
ext. =
I I
I I
I I
I I
I I
I
I
1 2i
S
C
S

+
~
~
~
~
(U

+ (T
2 1 - T))ds
1 2
= (F U

+ C (T
2 1 - T))dS
1 2
+ (U +) ds
I I
I
I
I
I I
I
I
I
C
S
C
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· where are present on the hull:

F, F, F
1
2
3:
surface forces acting along the axes of the reference mark
Cartesian total
+h/2

F =
. dz
I
F E + F .e
v
I
S
I
where I.E.(internal excitation) are the vectors of the total Cartesian base.
- H/2
C, C, C
1
2
3:
surface couples acting around the axes of the reference mark
total.
+h/2
H

C =
Z
. dz
I
F E ± F .e
v
I
S
I where E
2
I are the vectors of the total Cartesian base.
- H/2

· and where are present on the contour of the hull:

1,2, 3:
linear forces acting along the axes of the reference mark
Cartesian total.
+h/2


where E
I =
. dz
F E
C
I

I are the vectors of the total Cartesian base.
- H/2
1, 2, 3:
linear couples acting around the axes of the total reference mark.
+h/2


where E
I =
Z. dz
F E
C
I

I are the vectors of the total Cartesian base.
- H/2

Note:

One also notes and the linear distributions of force and moment applied to
contour of the finite element.

b) Charges given in the local reference mark:

One has then:

3
~
~
~
3 ~
~
~
Wext = (F T u.a. ~

c) dS
(
T U


) ds
I I + 1
1 +
2
2
+
I I +
1
1 +
2
2
=
S
i=1
C
i=1

3
3
(~
~
~
~
~
~
F T U
C
~


C
~) dS
(
T U


) ds
I I + 1 2 -
2 1
+
I I +
1 2 -
2 1
S
i=1
C
i=1

~ ~ ~
Forms of F, F, F
1
2
3 and ~, ~, ~
C C C
1 2
3 are the analogues of the expressions obtained for
F, F, F
1
2
3 and C, C, C
1
2
3 by replacing I.E.(internal excitation) by Ti.

Note:

For the couple C, the contribution ~
c3 associated N is null in theory of hull.
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3.3
Work of the inertias

Work due to the quantities of acceleration is written:

.
W ac = OQ'.OQ' FD


where is the density.

.
It is supposed that OQ', the vector of acceleration of the Q' point are following form:

.
OQ' = &
U.E.
0
K
K + [X E
K
K]

where one neglected the forces of Coriolis and the correction of metric in the thickness.

D 2U
One notes &
U
K
K =
O, E (by
dt 2, and is the uniform vector of rotation of the total reference mark (
K)
report/ratio with a Galiléen reference mark which in the same beginning O as the total reference mark).

One expresses in the total base (ek):

= kek

For virtual displacement OQ', one a:

OQ' = U.E.
K
K

Work due to the quantities of acceleration becomes then:

W ac
U
0
ac
ac
= K ek [U&k ek + (xk ek)]FD = W +W
farmhouse
hundred


with:
W ac
=
U U & FD
mass
K K


and:
W ac =
U

0
K E K [(xk ek)]FD
hundred



3.4
Principle of virtual work

For a static loading, he is written in the following way: W
= W
ext.
def where Wext is the sum
various elementary work, corresponding to the various loadings.

In harmonic dynamics (calculations of clean modes), the principle of virtual work
give:W
+ W ac
ext.
farmhouse = 0
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4
Numerical discretization of the variational formulation
exit of the principle of virtual work

4.1 Introduction

This chapter is devoted to the discretization of the various terms of energy introduced into the chapter
precedent. The choice of framework HENCKY-MINDLIN-NAGHDI to describe the kinematics of hull,
presented at the paragraph [§2] led to expressions of the deformations where the derivative are limited
with command 1, contrary to the model of LOVE-KIRCHHOFF. One can thus use a finite element
of a nature limited while ensuring conformity (see p.110 [bib7]).

The degrees of freedom are 3 displacements in the total reference mark and 2 rotations in local reference mark.

The selected elements are isoparametric quadrangles or triangles. The quadrangle is
represented below. The quadrangles give the best results (see p.202 [bib8]).
better choice consists in taking for these elements of the quadratic functions of interpolation (see
p.224 of [bib8]) in order to modelling the effects of membrane correctly, of inflection and shearing.
According to the results based on many case-tests of the literature, the best alternative is it
quadratic isoparametric quadrangle, which makes it possible to have a fine representation of a geometry
curve and of good estimates of the constraints. One chooses among the elements with functions
quadratic the element hétérosis (Q9H) whose displacements are approached by the functions
of interpolation of the Sérendip element and rotations by the functions of the element of Lagrange
(cf Annexe3). This choice is justified hereafter.

2
2
3
4
=
7
3
1
3
3
7
3
8
4
= -
1
P
1
1
1
6
8
6
1
5
2
5
2 = 1
-
1
2

Appear 4.1-a: Représentations of the isoparametric quadrangle
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The figure [Figure 4.1-b] summarizes the three families of elements previously named.

Sérendip element
Element of Lagrange Elément Hétérosis
~
U,
K

~

Appear 4.1-b: Familles of finite elements for the isoparametric quadrangle

Risks of bloquage or locking of membrane or shearing appear when
the thickness of the hull becomes small compared to its radius of curvature and that functions
of interpolation are of a too low nature. To solve them a selective numerical integration is used
[bib6]. For certain types of boundary conditions (embedding) with the Sérendip element it
locking persists in spite of selective integration. Moreover, for the element of Lagrange, this type
of integration leads to singularities in the matrix of rigidity. The element Hétérosis Q9H with
selective integration does not encounter the problems mentioned and seems being more
powerful for the modeling of the very thin hulls (see p.224 [bib8]). It should be noted that this
element has a mode of deformation without associated energy if it is used only. This mode
disappears when one uses more than two elements [bib7].
For the elements triangle, the element Hétérosis T7H is essential for the same reasons but proves
definitely less powerful (see paragraph 5 concerning the validation).

One decides to carry out all calculations of discretization in the total Cartesian base.

4.2
Discretization of the geometrical terms

The co-ordinates x0k of a point P of average surface are interpolated by the functions of form
in the following way:

Nb1
x0 = NR 1 () x0
K
I
ik
i=1


()
where the Nb1 number and functions of form NR 1
0
I
depend on the type of element chosen, and xik is them
co-ordinates with node I of the element.

The vectors covariants has (attached to the point P) are then given by:

Nb1 () 1
= Nor x0 E has

ik
K
i=1




J
The calculation of the Tk vectors is avoided because the Tk components contain the sizes of curvature
whose calculation is often vague Co

Mrs. it was shown in the paragraph [§2.2.1].
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In order to avoid the presence of the terms of curvature, one writes:

Nb1
N = NR () 1 N
I
I
i=1

where nor is the normal vector with the nodes of the element.

4.3
Discretization of the field of displacement

One adopts the following writing for displacement at the point Q:

Nb1
Nb2

U = NR () 1 U.E. + 3 NR (2)
~
~
H
I
ik
I
I (T
i2 I -
T
K
1
i1 i2)
2
i=1
i=1


(2)
where T are evaluated with the nodes, and where it is observed that the functions of interpolation Ni and them
~
Nb2 for rotations numbers are a priori different from those used for displacements the U.K.

By expressing Ti according to their components in the total Cartesian base, one obtains:

Nb1
Nb2

U = NR () 1 U.E. + 3 NR (2)
~
~
H
I
I (T
i2 I K
1 + T
I
ik K
i1 i2k) ek
2
i=1
i=1


One calculates then the various elementary terms, in order to obtain the complete discretized formulation.
In the continuation one uses the convention of summation of Einstein, while having with the spirit that the number
~
interpolations is Nb1 for x0, N, U
K
K, and Nb2 for, T.

4.3.1 Element Hétérosis Q9H

With this element, the number of interpolations for the geometry (x0k, N) and displacements the U.K. are
Nb1=8 (nodes nodes and mediums on the sides), while the number of interpolations for T and
~
rotations is Nb2=9 (nodes nodes and mediums on the sides + barycentre). The number of degrees
of freedom total of the element is thus Nddle=3x8+2x9=42.

Functions of interpolation NR (1)
(2)
I
and Ni respectively for the geometry and displacements, and for
rotations, can be found for example in [bib2] and are quoted in appendix 2.

The elementary vector of displacement can be put in the following form:

~
qe = (
~ ~
~ ~
~
~
U, U, U,…, U, U, U,…,
11
12
13
11
12
i1
i2
i3
i1
i2
91
9
2)
I =,
1 8
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4.3.2 Element triangle T7H

With this element Nb1=6 (nodes nodes and mediums on the sides) and Nb2=7 (nodes nodes and mediums
sides + barycentre). The number of degrees of freedom total of the element is Nddle=3x6+2x7=32.

()
1
The 6 functions of interpolation Nor which are traditional can be found in [bib2] and are
(2)
quoted in appendix 4. On the other hand the 7 Ni are it much less and their expressions are
data in Annexe 3.

The elementary vector of displacement can be put in the following form:

~
qe = (
~ ~
~ ~
~
~
U, U, U,…, U, U, U,…,
11
12
13
11
12
i1
i2
i3
i1
i2
71
7
2)
I =,
1 6

4.3.3 Notice

One notices on the level of the elementary vector ~
qe the presence of terms associated with the local base and
at the total base.

4.4
Discretization of the field of deformation

The field of deformation is expressed like the symmetrized gradient of the field of displacement:


1
=
S U = (U + U T)
2
Like:

U
= [
NR
] qe
()
() ~
X
X

one thus has:


U = NR
qe
()
~
X


where NR gathers the functions of form NR ()
1
(2)
I
and Ni
and matrices of passage Ti K
, is
X
the reverse of the jacobien J and ~
qe is the vector of the degrees of freedom to the nodes (translations the U.K. and
~
rotations).

Taking into account these relations and of ~
= T T, one obtains the components of the tensor of
deformation in the local reference mark:

~
~ ~
= B qe

~
where B is the matrix of interpolation of ~
, such as:

~
B = T T S J -
1
(
NR)
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Note:

If one takes again the expression of
Nb1
Nb2
()
1
3
(2)
~
~
U (X) = NR U.E. +
NR H
I
I (T
i2 I K
1 + T
I
ik
K
i1 i2k) ek = U (X) + U (X)
T
R
one notices
2
i=1
i=1
that the terms of membrane are contained in the first part U T ()
X of U (X) and that them
terms of inflection are contained in the second part U R ()
X of U ()
X. Terms of
~
~ ~

E
m = B m Q
transverse shearing come from the two contributions. One obtains as follows: ~
~
~

E
F = B F Q where
~
~ ~
= B qe


~
B = T SJ-1 NR
m
MF
1 ()
~
H
B = T SJ - 1 [

NR
F
MF
3
2 ()] by simple decomposition of the expression ~
~ ~
= B qe. One calls
2
~
B = T SJ-


1
(
NR)
membrane part of the deformation projection on the membrane-inflection part of the field of
deformation room of the symmetrized gradient of the translations in the total reference mark. Part is called
inflection of the deformation projection on the membrane-inflection part of the field of deformation
room of the symmetrized gradient of rotations in the total reference mark. One calls transverse distortion
projection on the shearing part of the local field of deformation of the gradient symmetrized of
total displacement.

4.5
Stamp rigidity

The principle of virtual work is written in the following way: W
= W
ext.
def is still
UT K U UT
=
F in matric form where K is the matrix of rigidity coming from the assembly
in the total reference mark of the whole of the elementary matrices of rigidity. At the elementary level
discretization of the work of deformation is written with the preceding notations:

1

~ ~ ~
~
W el = ~et
T
Q
B C B det J D

D D ~e
~et
E ~e
def
Q = Q K Q
1
2
3

- 1Ar

where Ar is the area of reference of the element.
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4.5.1 Decomposition of the elementary matrices

This matrix of rigidity includes/understands three contributions due to the deformations of membrane, of inflection
~ E
~ E
~ E
~
and of transverse distortion. One has as follows: K
K + K + K E
= m
F
with:

1
~ E
~ T
~
K =
B H B det J D D
1 D
2
;
m
m
m
3
- 1Ar
1
~ E
~ T
~
K =
B H B det J D D
1 D
2
;
F
F
F
3

- 1Ar
1
~ E
~ T
~
K =
B H B det J D



D
1 D
2
.

3
- 1Ar

4.5.2 Assembly of the elementary matrices

The principle of virtual work for the whole of the elements is written:

nbelem
W
= We
T
def
def = U KU where U is the whole of the degrees of freedom of the discretized structure
e=1
and K comes from the assembly of the elementary matrices.

4.5.2.1 Degrees of freedom

The process of assembly of the elementary matrices implies that all the degrees of freedom are
expressed in the total reference mark. In the total reference mark, the degrees of freedom are the three
displacements compared to the three axes of the total Cartesian reference mark and three rotations compared to
these three axes. One thus uses, for the degrees of freedom of rotation, of the matrices of passage of
locate local orthonormé T with the total reference mark for each element.

4.5.2.2 Rotations
fictitious

Rotation compared to the normal with the hull is not a true degree of freedom. To ensure
compatibility between the passage of the local reference mark to the total reference mark, one thus adds a degree of freedom
additional room of rotation to the hull which is that corresponding to rotation compared to
normal on the average surface of the element. This implies an expansion of the blocks of dimension (5,5)
matrix of local rigidity in blocks of dimension (6,6) by adding a line and a column
correspondent with this rotation. These additional lines and these columns are a priori null. One
then carry out the passage of the matrix of local rigidity extended to the matrix of total rigidity.
In the preceding transformation, one was satisfied to add rotations compared to
normals on the surface of the elements without modifying the deformation energy. The contribution to energy
brought by these additional degrees of freedom is indeed null and no rigidity is to them
associated.
The matrix of total rigidity thus obtained presents the risk however to be noninvertible. For
to avoid this nuisance it is allowed to allot a small rigidity to these degrees of freedom
additional on the level of the matrix of widened local rigidity. Practically, one chooses it between 10­
6 and 10­3 time the diagonal minor term of the matrix of rigidity of local rotation. The user can
to choose this multiplicative coefficient COEF_RIGI_DRZ itself in AFFE_CARA_ELEM; by defect it
10­5 is worth.
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4.6
Stamp of mass

The virtual work of the effects of inertia can be expressed in the form:

W ac = &U (Q) .U (Q
farmhouse
) D


It is supposed that the deformations and displacements remain sufficiently small so that the normal
on the average surface of the hull remains unchanged.
With these assumptions, we can write the field of virtual displacement:


H
U (Q
) (,
1,
2)
3 = U (P
) (,
1)
2 + 3

(,
1) N
2

(,
1)
2
2

and the field of acceleration:

&
H
U (Q) (,
2)
1
3 = &u (P) (,)
2 +
&
3
(,) N (
2
,)
1
1
1 2
2

In this expression, we neglected the gyroscopic terms.

4.6.1 Discretization
displacement for the matrix of mass

At the point Q, one takes as interpolation of the field of displacement:

U

N
N


Nb
I1
0
-
Nb
I 3
I 2

1
2
I1


H





U (

Q) (,
1
2
1,
2)
3 = NR I
(,
1)
2 U I 2 - 3
NOR (, 1) 2 nI3
0
- nI1

I 2
2
I 1
=

I 1
=
U
N
N


I 3
- I2
I1
0 I3

For the field of acceleration, the interpolation is written:

U
0
I
- N
N


1
Nb
&
&
1
Nb
I 3
I 2

2
I1


H


&


U (Q) (,
1
2
2) = NR (
3
I,)
1
1
2 &uI 2 -
NR (
3
I,)
N
0
I
- nor &
1
2
3
1

I2
2
I 1
=



I =


&
1
uI

- N
N
0
I
I
&
3
2
1
I3

We rewrite the two preceding equations in the matric form:

U (Q,)

NR ue
) (1 2 3 =

&U (Q
,) NR
2
3 =
&
ue
) (1



where NR is the matrix of interpolation, whose expression is:




1 0 0
0
- nI3 nI2
0
-


nNb23
nNb22



H


1
2

H


NR = NEITHER 0 1 0 - 3 NOR N
0
I 3
- nI1
-
2


3 NNb2 N
0
Nb23
- nNb21


2


2
0 0 1
- N
N
0





I 2
I1

-


N
N
0
Nb22
Nb21

I =,
1
1
Nb


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The vector ue is the elementary nodal vector of displacements in the total reference mark which is put
in the following form:

E
U = (U, U, U,…, U, U, U,…,
,
,
I = Nb
11
12
13
11
12
13
1
I
i2
i3
1
I
i2
i3
Nb21
Nb22
Nb23)
,
1
1

4.6.2 Stamp of elementary mass

With the preceding notations, the virtual work of the effects of inertia is put in the matric form
following:

Winertie
and
E E
farmhouse
= U M U &

with Me the matrix of coherent mass which can be expressed in the form:

Me =
NTN det (J


())D D D
3
1
2
3
E


It is important to note that because of the curvature, a coupling of the terms of translation with those
of rotation is possible (indeed, det (J ())
3
is not constant in the thickness).

4.6.3 Assembly of the elementary matrices of mass

The assembly of the matrices of mass follows same logic as that of the matrices of rigidity.
degrees of freedom are the same ones and one finds the processing specific to normal rotations to
surface hull. Although the matrix of coherent mass is built in the total reference mark, it
remain singular compared to the rotation of the normal in each node. We need
to supply this matrix on the basis of the variational form:

Nb2
Winertie = m
N
I.E.(internal excitation) (N I N I
) &I
I =1

where is selected constant for me by element and calculated according to the formula:

m
Cm
E =
max

max being the major term due to rotations (in the local reference mark of the element) on the diagonal
matrix Me. It is thus to note that with this intention it was necessary to bring back the contribution
the rotations initially expressed in the total reference mark of the element, the local reference mark of
the element by change of reference mark.

For modal calculations utilizing at the same time the calculation of the matrix of rigidity and that of
stamp of mass, it is necessary to take a mass on the degree of normal rotation on the surface of the hull
being worth C time the diagonal minor term of the matrix of mass for the terms of rotation in
locate local, where C is worth between 10­6 and 10­3. One chooses to confuse the values of this coefficient with
those of the COEF_RIGI_DRZ for the equivalent operation on the matrix of rigidity. By defect C is worth
thus 10­5. That makes it possible to inhibit, during a modal analysis, the modes being able to appear on
additional degree of freedom of rotation around the normal on the surface of the hull.
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4.7
Numerical integration for elasticity

4.7.1 Integration
surface

For the element Hétérosis Q9H the inflection part of the matrix of stiffness is integrated classically
with 9 points of Gauss while the parts membrane and shearing are obtained by integration
reduced with 4 points of Gauss.

For element T7H, by analogy with Q9H, the matrix of stiffness is obtained with 7 points
of integration of Hammer for the inflection part and 3 points of integration of Hammer for the parts
shearing and membrane.

Cordonnées of the points
Weight
1 = 1/3;1 = 1/3
9/80
2 = has;2 = has
155 + 15
With =

6 + 15
2400
has =

21
3 = 1 - 2a;3 = has
With
4 = has;4 = 1 - 2a
With

31/240 - A
5 = B;5 = B
B = 4/7 - has
6 = 1 - 2b;6 = B
31/240 - A
7 = B;7 = 1 - 2b
31/240 - A
1 -
1
N
y (
,) D D
= I y
(I, I)
0 0
i=1

Normal numerical formulas of integration on triangle T7H (Hammer)

X-coordinates of the points
Weight
Ordinates of the points
Weight µ
1 = - 3/5
5/9
1 = - 3/5
5/9

=
2 = 0
8/9
2 0
8/9
3 = + 3/5
5/9
2 = + 3/5
5/9
1 1
N
N
y (
,) dd = iµj y (I, J)
- -
1 1
i=1 j=1

Normal numerical formulas of integration 3x3 on quadrangle Q9H (Gauss)

The principle of reduced integration consists in evaluating the membrane and shearing strains
at the points of reduced integration and to extrapolate them at the points of traditional integration. This returns to
to suppose that these deformations are bilinear on element Q9H and linear on the T7H. Les
functions of form chosen to make this extrapolation are related traditional to form
bilinear of the quadrangle with 4 nodes for the Q9H and linear of the triangle with 3 nodes for the being worth T7H
1 at the points of reduced integration.
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For more details on the principle of reduced or selective integration, one can refer to [bib6].

Cordonnées of the points
Weight
1 = 1/6;1 = 1/6
1/6
2 = 2/3;2 = 1/6
1/6
3 = 1/6;3 = 2/3
1/6
1 -
1
N
y (,) D D
=

I y
(I, I)
0 0
i=1

Numerical formulas of integration reduced on triangle T7H (Hammer)

For the elements quadrangle an integration of Gauss 2x2 is used.

Cordonnées of the points
Weight
1 = 1/3;1 = 1/3
1
2 = 1/3;2 = - 1/3
1
3 = - 1/3;3 = 1/3
1
3 = - 1/3;3 = - 1/3
1
1 1
N
y (
,) dd = I y (I, I)
- -
1 1
i=1

Reduced numerical formulas of integration 2x2 on quadrangle Q9H (Gauss)

4.7.2 Integration in the thickness

Integration in the thickness is made with three points for the two elements.

Cordonnées of the points
Weight
1 = - 1
1/3
2 = 0
4/3
3 = +1
1/3
1
N
y (
) D = I y (I)
- 1
i=1
Formulate numerical integration in the thickness in elasticity

4.8
Numerical integration for plasticity

The principle of surface integration remains the same one as in elasticity, but the initial thickness is
divided into NR identical layers thicknesses. There are three points of integration per layer. Points
of integration are located in higher skin of layer, in the middle of the layer and in lower skin
of layer. For NR layers, the number of points of integration is of 2N+1. One advises to use of 3
with 5 layers in the thickness for a number of points of integration being worth 7, 9 and 11 respectively.
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For rigidity, one calculates for each layer, in plane constraints, the contribution to the matrices of
rigidity of membrane, inflection and transverse distortion. These contributions are added and
assemblies to obtain the matrix of total tangent rigidity.

For each layer, one calculates the state of the constraints (11,22,12) and the whole of the variables
interns, in the middle of the layer and in skins higher and lower of layer, from
local plastic behavior and of the local field of deformation (11,22,12). The positioning of
points of integration enables us to have the rightest estimates, because not extrapolated, in skins
lower and higher of layer, where it is known that the constraints are likely to be maximum.
plastic behavior does not include/understand for the moment the terms of transverse shearing which
are treated in an elastic way, because transverse shearing is uncoupled from the behavior
membrane in plane constraints.

Cordonnées of the points
Weight
1 = - 1
1/3
2 = 0
4/3
3 = +1
1/3
1
N
y (
) D =
I y (I)
- 1
i=1

Formulate numerical integration for a layer in the thickness in plasticity

Note:

One already mentioned with [§2.2.2] that the value of the coefficient of correction in shearing
transverse for the elements of plate and hull was obtained by identification of
elastic complementary energies after resolution of balance 3D. This method is not
more usable in elastoplasticity and the choice of the coefficient of correction in shearing
transverse is posed then. The transverse terms of shearing are thus not affected
by plasticity and are treated elastically, for want of anything better. If one places oneself in
theory of Coils-Kirchhoff for a value of this coefficient of 106 H/R (H being the thickness of
the hull and R its average radius of curvature) transverse terms of shearing
become negligible and the approach is more rigorous.

4.9
Discretization of elementary work for the loadings

4.9.1 Discretization
elementary of the work of the forces and external couples being exerted
on average surface

According to the paragraph [§3.2], one recalls that one has for these efforts and couples:

W = (fu + c) dS
ext.

S

where S is the average surface of the hull.

For the first term of this expression one has as follows:
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4.9.1.1 Loads given in the total reference mark


1
2
~
~
2
W
=
F NR
[
U + C NR
(
T
2 1 - T
)] det
1 2

D 1
D
ext.
K
I
ik
K
J
J
J K
J
J K

H
2
Ar

with det = det J (3 =)
0

4.9.1.2 Loads given in the local reference mark

1
1
2
~
~
2
W
= [
F NR T NR U +
C T NR
(
T
2 1 - T
)] det
1 2

D 1
D
ext.
J J K
I
ik
K
J
J
J K
J
J K

H
2
Ar

4.9.2 Discretization
elementary of the work of the forces and external couples being exerted
on contour

According to the paragraph [§3.2], one recalls that one has for these efforts and couples:

W = (U +) ds
ext.

C

where C is the average contour of the hull. and linear distributions of force and moment
applied to the contour of the hull in the total reference mark.

1
2
~
~
The discretization gives then: W
=
[NR U + NR
(
T
2 1 - T
D
)] S
ext.
K
I
ik
K
J
J
J K
j1 j2k

C

4.9.3 Discretization of the term of gravity

One has for this term:

1


3
2
~
~
Wpes =
G U (Q D
) V = gkU (Q D
) V
K
= G [NR
K
I U +
NR
ik
J
(
T
j2 J K
1 - T
D
)] V
j1 j2k

2
E
E
E

That is to say: W
= G NR
1 U FD
pes
K
I
ik
by supposing negligible the second term of expression Ci
E
above.

4.9.4 Discretization of the term of pressure

It is supposed that the pressure p is applied to the average surface of the hull. One has then:

W
= ep

N U (P D
) S = ep
(has has) U (P)
1
2

D 1
D
near
2
With
With
R
R

where e=±1 according to whether p is applied in internal or external skin.
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Like A
= E has
1

K K, this is still written: W
= epN

U v
D 1
D
near
I
ik K
2
Ar
v J ° J ° - J ° J °
1
12 23
13 22


where v = J ° J ° - J ° J ° J °
,
= J
2
13 21
11 23
ij
ij (3
=)
0.

v
° °
°
°
J J - J J
3
11 22
12 21

4.9.5 Discretization of the terms of centrifugal inertia

One adds with the expression of the field of accelerations of the paragraph [§4.6] the corresponding term
with the accelerative forces centrifuges if the total reference mark (O, ek) is in uniform rotation by
report/ratio with a Galiléen reference mark which in the same beginning O as the total reference mark. The expression of the field
accelerations becomes as follows:

&
H
U (Q) (,
2)
1
3 = &u (P) (,)
2 +
&
3
(,) N (
2
,) + [COp]
1
1
1 2

2

where one neglected the forces of Coriolis and the correction of metric in the thickness.

One expresses in the total base (ek): = K ek.

By taking again the expression of: W inertia = &U (Q) .U (Q D
) the contribution of the terms is identified

of centrifugal inertia: W inertia
0
hundred
= U.E. [(X E) D] V
K K
K K
by neglecting the terms of rotation
E
in virtual displacement. The terms of mass are unchanged compared to [§4.6].

Like one a:

x0
=
x0
= x0
E
E
E
E
K
K
p
p
K
K
p
K
qpk E Q

where eqpk is the permutation of Lévi-Strauss.

One also writes:

(x0 E) = E
E
x0
K
K
qpk
srq
R
p
K E K

From where it results from it that:

1
W inertia =
U
NR () 1 E
E
x0 NR () 1 det J D D
1 D
hundred

is
I
qpk srq
R
p
jk
J
2 3
- 1Ar
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4.9.6 Taking into account of the loadings of thermal dilation

One treats only the case where the characteristics thermo elastic E, depend only on
average temperature T in the thickness. Moreover, the material is thermo isotropic rubber band
homogeneous in the thickness.

The variational formulation of work due to thermal dilations is written:

1
+
1
+
HT
~
~
~th
~T
~ T ~ ~th
~T
~ T ~
W = - (
C
-
) FD = Q
B C
J
= Q
B C
-
Id
J


E
det
D

D

D


E
(T T ref.) det
D

D

D
1
2
3

1
2
3

1
- A
1
- A
E
R
R
The temperature is represented by the model of thermics to three fields according to [R3.11.01]:

T (, 3) = T m (
). P1 (3) + T S (). P2 (3) + Ti





() .P3 (3),

with: Pj (3): three polynomials of LAGRANGE in the thickness: ] - 1, + [
1:

2


P () = 1 - (); P
3
3
2 (3
) = (1+ 3
); P
1 3
3
3 (3
) = - (1 - 3
);
2
2

From the representation of the temperature above, one obtains:

· the average temperature in the thickness:
1 +1
1
T () =
T
(, 3) D 3 = 4
;
2 - 1
(Tm () +Ts () +Ti




()
6
· the average variation in temperature in the thickness:
$
+1
T () = 3 T
(,
3) D
3
3 = T S () - T I



();
- 1

Thus the temperature can be written in the following way:

~
T (,) = T () + T$ (). /2 + T (,
3
3
3) such as:
+1~
1
+
T
(
~
,
3) = 0;
T
3 (,
3) = 0.
1
-
1
-

~
If the temperature is indeed closely connected in the thickness one has, T = 0.

It is necessary to evaluate the three-dimensional thermal stresses, in each point of integration
in the thickness. These constraints of thermal origin withdrawn from the mechanical constraints
usual are calculated at the points of integration in the thickness by:

~ther
.

E
ref.

=
(T - T + $T. /2
2
3
)
1 -

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4.9.7 Assembly

The variational formulation of the work of the efforts external for the unit of the elements is written
then:

nbelem
W = We = T
ext.
ext.
U F where U is the whole of the degrees of freedom of the discretized structure and
e=1
F comes from the assembly of the vectors forces elementary.

As for the matrices of rigidity, the process of assembly of the vectors forces elementary
imply that all the degrees of freedom are expressed in the total reference mark. In the total reference mark,
the degrees of freedom are three displacements compared to the three axes of the total Cartesian reference mark
and three rotations compared to these three axes. Matrices of passage of the reference mark are thus used
room with the total reference mark for rotations of each element.

Note:

The external efforts can also be defined in the reference mark user. One then is used
stamp passage of the reference mark user towards the local reference mark of the element to have the expression
of these efforts in the local reference mark of the element and to deduce the vector from it elementary room forces
corresponding. For the assembly one passes then from the local reference mark of the element to the total reference mark.

5 Validation

To judge relevance of thick the hull formulation, few examples of application
according to relate to as well linear statics as the calculation of clean modes. Three new cases
tests relating to the two finite elements described in the preceding parts were integrated in
Code_Aster. They come to enrich the case-tests by the elements of plate already present in
environment of Code_Aster. Most of these case-tests were indexed in [bib10].

The three new case-tests, two in statics plus one in dynamics, are traditional examples
of validation drawn from [bib3]. Reference solutions, analytical or numerical, resulting from [bib3]
are compared with the numerical results given by Code_Aster. For more information on these
case-tests, one will refer to the documentation of validation indicated in reference.
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5.1
Case test in linear statics

5.1.1 Static case test n° 1

The first case test is that of a cylindrical panel subjected to its own weight [V3.03.107].

This test makes it possible to highlight effects of membrane more important than those of inflection. It
allows to measure the performance of the elements hulls compared to elements DKT or DKQ of which
the interpolation out of membrane is linear.

5.1.2 Static case test n° 2

The second case test is that of a helicoid hull subjected to two concentrated types of loading
[V3.03.108].

The helicoid shape of the hull makes it possible to study the geometrical representation of the finite elements.
The concentrated loadings can be:

· in the plan: the influence due to the effects of membrane is then not important and it
behavior dominating is that due to the inflection,
· except plan: the effects of membrane affect the behavior of the hull.

5.2
Case test in dynamics

This case test is a simplified model of paddle of compressor, which is in fact a cylindrical panel
[V2.03.102].

This test highlights the performances of the elements in dynamic behavior by the data
frequencies and clean modes.

The frequencies and clean modes of the paddle are experimental values which are used as results
of reference.

6 Chaining
thermomechanics

6.1 Description

For the resolution of chained thermomechanical problems, one must use for thermal calculation
finite elements of thermal hull [R3.11.01] whose field of temperature is recovered like
input datum of Code_Aster for mechanical calculation. It is necessary thus that there is compatibility between
thermal field given by the thermal hulls and that recovered by the mechanical hulls. It
the last is defined by the knowledge of 3 fields TEMP_SUP, TEMP and TEMP_INF given in skins
lower, medium and higher of hull.

The table below indicates compatibilities between the elements of mechanical hull and hull
thermics.

Modeling Nets
Finite element
to use with Maille Elément
finished
Modeling
THERMIQUE
MECANIQUE
COQUE QUAD9
THCOQU9
//////////////
QUAD9 MEC3QU9H COQUE_3D
COQUE TRIA7
THCOTR7///////////////
TRIA7 MEC3TR7H COQUE_3D
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Note:
· The nodes of the thermal elements of hulls and mechanical hulls must
to correspond. The grids for thermics and mechanics will thus have the same number
and the same type of meshs.
· The elements of thermal hulls surface are treated like plane elements by
projection of the initial geometry on the level defined by the first 3 nodes.

The thermomechanical chaining is also possible if one knows by experimental measurements
variation of the field of temperature in the thickness of the structure or certain parts of
structure. In this case one works with a card of temperature defined a priori; the field of
temperature is not given any more by three values TEMP_INF, TEMP and TEMP_SUP of thermal calculation
obtained by EVOL_THER. It can be much richer and contain an arbitrary number of points
of discretization in the thickness of the hull. Operator DEFI_NAPPE allows to create of such
profiles of temperatures starting from the data provided by the user. These profiles are affected by
order AFFE_CARTE (cf the case-test HSNS100B). It will be noted that it is not necessary for
mechanical calculation that the number of points of integration in the thickness is equal to the number of points
of discretization of the field of temperature in the thickness. The field of temperature is
automatically interpolated at the points of integration in the thickness of the elements of hulls.

6.2 Case-test

The case-tests for the thermomechanical chaining enters of the thermal elements of hulls and of
mechanical elements of hulls are the HPLA100C (elements MEC3QU9H) and HPLA100D (elements
MEC3TR7H). It is about a heavy thermoelastic hollow roll in uniform rotation [V7.01.100]
subjected to a phenomenon of thermal dilation where the fields of temperature are calculated with
THER_LINEAIRE by a stationary calculation.

Z
IH
Re
Interior radius IH = 19.5 mm
External radius Re = 20.5 mm
Not F
R = 20.0 mm
Thickness
H = 1.0 mm
Height
L = 10.0 mm
R
Z
NR
K
Z
Q
P
J
y
D
C
H
M
+
L
R
With
B
X
F


Thermal dilation is worth: T () - Tref () = 0 5
. (T + T
S
I) + 2.(T + T
S
I) (R - R)/H


with:

·
T = 0 5
. °C, T = - 0 5
. °C, T
= 0. °C
S
I
ref.

·
T = 01
. ° C, T = 01
. ° C, T
= 0. ° C
S
I
ref.


One tests the constraints, the efforts and bending moments in L and Mr. Les results of reference are
analytical. One obtains very good results whatever the type of element considered.
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7
Establishment of the elements of hull in Code_Aster

7.1 Description

These elements (of names MEC3TR7H and MEC3QU9H) are pressed on meshs TRIA7 and curved QUAD9.
These elements are not exact with the nodes and it is necessary to net with several elements to obtain
correct results.

7.2
Introduced use and developments

These elements are used in the following way:

MA = CREA_MAILLAGE (GRID: MAILINI
MODI_MAILLE: (OPTION:“QUAD8_9”
TOUT:“OUI”)…)

One calls upon a routine MODI_MAILLE of modification of the grid to pass from the elements
quadrangles with 8 nodes with the elements quadrangles with 9 nodes or many elements triangles to 6
nodes with the elements triangles with 7 nodes.

AFFE_MODELE (MODELING: “COQUE_3D”…) for the triangle and the quadrangle

One calls upon routine INI080 for the position of the points of Hammer and Gauss on the surface of
the corresponding hull and weights.

AFFE_CARA_ELEM (HULL:(EPAISSEUR:“EP”
ANGL_REP: ('' '')
COEF_RIGI_DRZ: “CTOR”)

To make postprocessings (forced, generalized efforts,…) in a reference mark chosen by
the user who is not the local reference mark of the element, one defines the X1 direction of the reference mark user
like the projection of a direction of reference D on the surface 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 on the surface of the element fixes the second direction at the point of observation concerned.
vector product of two vectors previously definite Y1=N X1 makes it possible to define the local trihedron
in which will be expressed the generalized efforts representing the state of stresses. The user
will have to take care that the selected reference axis is not found parallel with the normal of some
elements of hull. By defect, the direction of reference D is axis X of the total reference mark of definition of
grid.
Value CTOR corresponds to coefficent that the user can introduce for the processing of the terms
of rigidity and mass according to normal rotation on the surface of the hull. This coefficient must be
sufficient small not to disturb the energy balance of the element and not too small so that
the matrices of rigidity and mass are invertible. A value of 10­5 is put by defect.

ELAS: (E:NAKED Young: ALPHA:. RHO:. )
For an elastic thermo behavior isotropic homogeneous in the thickness one uses the key word
ELAS in DEFI_MATERIAU where the coefficients E are defined, Young modulus, coefficient of
Poisson, thermal dilation coefficient and RHO density.

AFFE_CHAR_MECA (DDL_IMPO: (
DX:. DY:. DZ:. DRX:. DRY:. DRZ:. DDL of hull in the total reference mark.
FORCE_COQUE: (FX:. FY:. FZ:. MX:. MY:. MZ:. ). They are the efforts
surface on elements of hull. These efforts can be given in the total reference mark or
in the reference mark user defined by ANGL_REP.

FORCE_NODALE: (FX:. FY:. FZ:. MX:. MY:. MZ:. ). They are the efforts
hull in the total reference mark.
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7.3
Calculation in linear elasticity

The matrix of rigidity and the matrix of mass (respectively options RIGI_MECA and MASS_MECA)
are integrated numerically in the TE0401 and TE0406, respectively. Calculation takes account of
fact that the terms corresponding to the DDL of rotation of hull are expressed in the local reference mark of
the element. A matrix of passage makes it possible to pass from the local DDL to the total DDL.

Elementary calculations (CALC_ELEM) currently available correspond to the options:

· EPSI_ELNO_DEPL and SIGM_ELNO_DEPL which provide the strains and the stresses
with the nodes in the reference mark user of the element in lower skin, with semi thickness and in
higher skin of hull, the position being specified by the user. Calculation is carried out in
the TE0410. One stores these values in the following way: 6 components of deformation or
constraints,
· EPXX EPYY EPZZ EPXY EPXZ EPYZ or SIXX SIYY SIZZ SIXY SIXZ SIYZ,
· EFGE_ELNO_DEPL: who gives the efforts generalize by element with the nodes from
displacements: NXX, NYY, NXY, MXX, MYY, MXY, QX, QY. This option is calculated
in the TE0410,
· SIEF_ELGA_DEPL: who gives the constraints by element to the points of Gauss in
locate local element starting from displacements: SIXX, SIYY, SIZZ, SIXY, SIXZ,
SIYZ. This option is calculated in the TE0410,
· EPOT_ELEM_DEPL: who gives the elastic energy of deformation per element from
displacements. This option is calculated in the TE0401,
· ECIN_ELEM_DEPL: who gives the kinetic energy by element. This option is calculated in
the TE0401,

Finally the TE0416 calculates also option FORC_NODA of calculation of the nodal forces for the operator
CALC_NO.

7.4
Plastic design

The matrix of rigidity is also integrated numerically, by layers, in the TE0414. One calls upon
the option of calculation STAT_NON_LINE in which one defines in the level of the nonlinear behavior it
a number of layers to be used for numerical integration. All laws of plane constraints
available in Code_Aster can be used.

STAT_NON_LINE (….
COMP_INCR: (RELATION:''
COQUE_NCOU:“A NUMBER OF LAYERS”)
….)

Elementary calculations (CALC_ELEM) currently available correspond to the options:

· EPSI_ELNO_DEPL which provides the deformations by element to the nodes in the reference mark
user starting from displacements, in lower skin, with semi thickness and in skin
higher of hull. This option is calculated in the TE0410,
· SIGM_ELNO_COQU which makes it possible to obtain the stress field in the thickness by element
with the nodes for a given layer and a position requested (in lower skin, with
medium or in higher skin of layer). These values are given in the reference mark
user. This option is calculated in the TE0415,
· SIEF_ELNO_ELGA which makes it possible to obtain the efforts generalized by element with the nodes in
the reference mark user. This option is calculated in the TE0415,
· VARI_ELNO_ELGA which calculates the field of internal variables and the constraints by element
with the nodes for all the layers, in the local reference mark of the element. This option is
calculated in the TE0415.
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8 Conclusion

The finite elements of hull curves which we describe here are used in the structural analyzes
thin curves whose thickness report/ratio over characteristic length is lower than 1/10. Two
finite elements of voluminal hull being pressed on quadrangular and triangular meshs were
introduced into Code_Aster. They were selected with a quite particular aim: to be able to represent one
complete behavior of curved structures whereas until now one could only use
elements with plane facets which induced parasitic inflections and required to refine them
grids.

It is elements for which strains and stresses in the plan of the element
vary linearly with the thickness of the hull. Selected kinematics is a kinematics hull
of Hencky-Mindlin-Naghdi type allowing to utilize the transverse energy of shearing.
distortion associated with transverse shearing is constant in the thickness of the element.
variable correction on the coefficient K of shearing transverse offers a flexibility in use
allowing to pass from the theory of HENCKY-MINDLIN-NAGHDI for k=1, with that of REISSNER
for k=5/6 and with that of LOVE_KIRCHHOFF (for very mean structures) if a value is chosen
K equalizes à106 × H/L H being the thickness and L a characteristic distance (radius of curvature
means, zone of application of the loads….). As in this last case, one uses a method of
penalization to make small the terms of shearing transverse, one can, if a value is taken
K too important, to make singular the numerical system. In this case, it is necessary to decrease the value of
k.

The default value of K is 5/6. It is generally used when the structure to be netted has one
thickness report/ratio over characteristic length ranging between 1/20 and 1/10. For thicknesses more
weak where the transverse distortion becomes low one can want to use a value of k=106 × H/L
(to be able to make comparisons with elements of plate DKT for example). When
transverse distortion is nonnull, the elements of hull do not satisfy the equilibrium conditions
3D and boundary conditions on the nullity of stresses shear transverse on the faces
higher and lower of hull, compatible with a constant transverse distortion in
the thickness of the hull. It results from it thus that on the level from the behavior a coefficient from 5/6 for
a homogeneous hull corrects the usual relation between the constraints and the transverse distortion of
way to ensure the equality enters energies of shearing of the model 3D and the model of hull to
constant distortion. In this case, the arrow ~
u3 has as an interpretation average transverse displacement
in the thickness of the hull and not the displacement of the average surface of the hull.

For structures low thickness in order to avoid the phenomena of blocking, one uses
under-integration reduced for the parts membrane and shearing of the matrix of rigidity. The choice
on the finite elements went on the elements quadrangle Hétérosis Q9H and triangle T7H. En effet,
among the finite elements with quadratic functions of interpolation, the performance of the Hétérosis element
Q9H is known. It is in particular higher than that of the elements Sérendip Q9S or the elements
of Lagrange Q9. This performance rests however on the selective integration of the element with
reduced integration of the terms of membrane and shearing on the one hand, and normal integration of
terms of inflection in addition. By analogy with Q9H, one took the finite element T7H like element of
triangular form. However, as far as possible, one will use the Q9H rather than the T7H which is
definitely less powerful.

The non-linear behaviors in plane constraints are available for these elements. One
announce however that the constraints generated by the transverse distortion are treated
elastically, for want of anything better. Indeed the rigorous taking into account of a transverse shearing
constant not no one on the thickness and the determination of the correction associated on rigidity with
shearing compared to a model satisfying the equilibrium conditions and the boundary conditions
are not possible and thus return the use of these elements, when transverse shearing
is nonnull, rigorously impossible in plasticity. Rigorously, for behaviors not
linear, it would thus be necessary to use these elements within the framework of the theory of Coils-Kirchhoff.

Elements corresponding to the machine elements exist in thermics; chainings
thermo mechanics is thus available with finite elements of thermal hulls to 7 and 9
nodes. Extensions of the preceding formulation presented in appendix allow also the catch
in account of the anisotropy of materials and kinematic non-linearity. This second extension
is operational in Code_Aster and is the subject of a reference material [R3.07.05].
Handbook of Référence
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9 Bibliography

[1]
S. Ahmad, Will go B.M., O.C. Zienkiewicz, “Analysis off thick and thin Shell structures by curved
finite elements “, IJNME, Vol.2, p.419-451,1970.
[2]
J.L. Batoz, G. Dhatt, “Modélisation of the structures by finite elements”, Volume 1, Solides
rubber bands - Hermès, Paris, 1990.
[3]
J.L. Batoz, G. Dhatt, “Modélisation of the structures by finite elements”, Volume 3, Coques -
Hermès, Paris, 1992.
[4]
B. Bui, “shearing in the plates and hulls: modeling and calculation”, Note
HI-71/7784, 1992.
[5]
D. Bui, “Modélisation of the hulls thicknesses average by an approach 3D “degenerated””,
Note EDF-DER HI-74/95/013, 1992.
[6]
E.Carnoy, G. Laschet, “isoparametric Eléments of hull”, LTAS, Rapport SF-108,
November 1992.
[7]
T.J.R. Hughes, “The Finite Method Element”, Prentice-Hall, 1987.
[8]
J.F. Imbert, “Analyze of strutures by finite elements”, 3rd edition - Cepaduès Editions,
1992.
[9]
E. Lorentz “Une non-linear relation of behavior hyperelastic”, Note EDF-DER
HI-74/95/011/0.
[10]
P. Massin, “Fonctionnalités available for the elements of hulls and plates in
Code_Aster “, Note EDF-DER HI-74/97/027/0.
[11]
O.C. Zienkiewicz, “The finite elements method”, 3nd edition - Mc Graw-Hill 1977.
[12]
R3.07.02:F. Voldoire, C. Sevin, “Coques thermoelastic axisymmetric and 1D”, Manuel
of reference of Code_Aster.
[13]
R3.07.03: P. Massin, “Eléments of plate DKT, DST, DKQ, DSQ and Q4G”, Manuel of
reference of Code_Aster.
[14]
R3.07.05: P. Massin, Mr. Al Mikdad, “Eléments finished of voluminal hull into nonlinear
geometrical “, Manuel of reference of Code_Aster.
[15]
R3.11.01: P. Massin, F. Voldoire, S. Andrieux, “Modèle of thermics for the hulls
thin ", Manuel of reference of Code_Aster.
[16]
V2.03.102: P. Massin, A. Laulusa, “Vibrations free of a paddle of compression”, Manuel
of validation of Code_Aster.
[17]
V3.03.107: P. Massin, D. Bui, A. Laulusa, “cylindrical Panneau subjected to its own weight”,
Handbook of validation of Code_Aster.
[18]
V3.03.108: P. Helicoid Massin, D. Bui, A. Laulusa, “Coque under concentrated loadings”,
Handbook of validation of Code_Aster.
[19]
V7.01.100: P. Massin, F. Voldoire, “heavy thermoelastic hollow Cylindre in rotation
uniform “, Manuel of validation of Code_Aster.
Handbook of Référence
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Appendix 1 Extension with anisotropic materials not
programmed

It is considered that the hull consists of an orthotropic material, axes of orthotropism ~
~
xk associated the base
kk. The law of behavior in these axes is written:

~
~
~
~
~
~
= Sk
(6x)
1
(6x6) (6x)
1

~
where S is the matrix of flexibility of the component K.

Are ~
and ~, tensors of strain and stresses in the axes ~xk, one a:

~ T ~~
=
Q Q
~ T ~~
= QQ

where Q = [T, T, T
1
2
3]
(Q
.
) is the matrix of the cosine directors of T
/K
ij = T K
I
J
K in the base K K.
K

In vectorial form, one a:

~
~ ~~
=
T
~
~~~
= T

~
where the components of T are defined according to those of Q.

Conversely, one a:

~
~
~ - 1~
= T
~
~
~ - 1~
= T

therefore, one obtains:

~
~ ~~ ~ - 1~
= TS T
K


that one writes:

~
~ ~
= Sk

To be coherent with the assumption of plane constraint ~
33 = 0, one write:

~
~
~
R = Skr R
(5x)
1
(5x)
5 (5x)
1
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with the symbol R like tiny room, which gives:

~
~ ~
~
~

- 1
R = Ck
,
R
C = S
K
Kr


that one récrit by omitting the symbol R,

~
~ ~
= Ck
The elastic deformation energy W el is:

1
1
~ ~ ~
W el
T ~ E
T
Q
B C B det J D D D ~e
=

Q
2
K
1
2
3
- 1Ar

If the hull consists of Nc layers, each layer being regarded as a component K, then:

2e+ H
Nc
K
1
~ ~ ~
W el
T ~ E
T
= Q
B C B det J D D D
~ E

K
Q
2
1
2
3
K =12e- H Ar
K

where E
+
K and ek are the X-coordinates of the limits lower and higher of the layer K thickness
E = e+ - E
-
+
K
K
K, with e1 = - H/2 and E Nc = H/2.

While posing:
+
E
K
K + -
E
E

K
3 =
+
,
- 11,
3
3
[
]
H
H
one a:
Nc
1
E 1
~ ~ ~
W el
T ~ E
K
T
=

det (1, 2,
~
Q
B C B
J
Q
K


3) D
D
D
E
2
H
1
2
3
K =1
- 1Ar

In the same way, for work due to thermal dilations W HT, one a:

~
~k
K
K
K
HT = (T,
1
T,
2
T,
3
0 0)
0

where the ki are the dilation coefficients thermal of the layer K in the axes of orthotropism (~~xk).

With the relation:
~ K
~ ~~

K
HT = T HT
one obtains:
1
T ~
~
W HT
T ~ E
Q B (
~ K
=
- Ck HT) det J D D D
1
2
3

- 1Ar
That is to say:
Nc E 1 T ~ ~
W HT T ~ E
K
~ K
= Q
B C det J D D D


H
K
HT
1
2
3
h=1
- 1Ar
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Appendix 2 Fonctions of form for element Q9H

These functions are given on page 174 of [bib8].

A2.1 Fonctions of form for the translations

8 functions of the shape of incomplete Lagrange of the element quadrangle Q9H [Figure A2.2-a] for
the interpolation of displacements the U.K. are:

1
·
NR ()
1 (, 1 2) = (- 1+ +) (1+) (1+) I
I
I
1 1
2i 2
I
1 1
2i
=,
1 2,
2
3 4
4
1
·
NR ()
1 (,
2
1 2
) = (1 -) (1+) I
I
1
2i
=,
2
5 7
2
1
·
NR ()
1 (,
2
1
2
) = (1 -) (1+) I
I
2
I
=,
1 1
6 8
2

1 = 1
-
I = 1 8
, 4
I
;
2 = - 1 I = 15, 2
I
;
with: 1 = 0 I = 5 7
I
;
and 2 = 0 I = 6 8
I
;
.
1 = +1 I = 2 6, 3
I
.
2 = +1 I = 3 7, 4
I
.

A2.2 Fonctions of form for rotations

9 functions of the shape of Lagrange of the element quadrangle Q9H [Figure A2.2-a] for the interpolation of
~
rotations are:

-
NR (2) (,
P
1
2
) = NR () NR
I
I
1
I ()
2 where Nor ()
Pr
P =
for p=1,2 and where R describes the whole of both
laughed Pr
- Pi
nodes aligned with node I in the direction P.

1 = 1
-
I = 1 8
, 4
I
;
2 = - 1 I = 15, 2
I
;
One a: 1 = 0 I = 5 7
I
;
and 2 = 0 I = 6 8
I
;
.
1 = +1 I = 2 6, 3
I
.
2 = +1 I = 3 7, 4
I
.



2
2
(0,1)
4
(0,1)
7
3
4
7
3
(1,0)
8
9
(1,0)
8
(0,0)
6

(0,0)
6

1
1
1
5
2
1
2
5

Appear A2.2-a: Degrés of freedom for the translations and rotations of the element quadrangle Q9H
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Appendix 3 Fonctions of form for element T7H

A3.1 Fonctions of form for the translations

6 functions of form of triangular element T7H [Figure A3.2-a] for the interpolation of displacements
the U.K. are given on page 175 of [bib8]:
·
NR 1
()
1 (1
, 2
) = 2
(2 2
- 1)
·
NR 1
()
2 (1
, 2
) = (2 - 1)
·
NR 1
()
3 (1
, 2
) = 1 (2 1 - 1)
·
NR 1
()
4 (1
, 2
) = 4 2

·
NR 1
()
5 (1
, 2
) = 4 1
·
NR 1
()
6 (1
, 2
) = 4 1 2

where: = 1 - 1 - 2

A3.2 Fonctions of form for rotations
~
7 functions of form of triangular element T7H [Figure A3.2-a] for the interpolation of rotations
are:
()
1
· N2 (
(2)
1
, 2
) = 2
(2 2
-) 1+ NR
1
7

9
()
1
· N2 (
(2)
1
, 2
) = (1 - 1
- 2
) [(21 - 1 - 2) -] 1+ NR
2
7

9
()
1
· N2 (
(2)
1
, 2
) = 1 (2 1 -) 1 + NR
3
7

9
()
4
· N2 (
(2)
1
, 2
) = 4 2
(1 - 1
- 2
) - NR
4
7

9
()
4
· N2 (
(2)
1
, 2
) = 4 1
(1 - 1
- 2
) - NR
5
7
9
()
4
· N2 (
(2)
1
, 2
) = 4 - NR
6
1 2
7
9
with:
· NR (2)
7 (1
, 2
) = 27 1 2
(1 - 1 - 2
)


2
2
(0,1)
(0,1)
1
1
6
6
4
4
7
(1,0)
(1,0)
2
2
(0,0)
5
3

(0,0)
5
3

1
1

Appear A3.2-a: Degrés of freedom for the translations and rotations of the element triangle T7H
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