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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
1/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA, SAMTECH















Manual of Reference
R3.07 booklet: Machine elements on average surface
R3.07.05 document



Voluminal elements of hulls into nonlinear
geometrical




Summary:

We present in this document the theoretical formulation and the numerical establishment of a finite element of
voluminal hull for analyzes into nonlinear geometrical. This approach must make it possible to take in
count great displacements and great rotations of mean structures, of which the thickness report/ratio on
characteristic length is lower than 1/10. One will take care that these rotations remain lower than 2
.

This formulation is based on an approach of continuous medium 3D, degenerated by the introduction of
kinematics of hull in plane stresses in the weak form of balance. The measurement of the deformations
that we retain is that of Green-Lagrange, combined énergétiquement with the stresses of Piola-Kirchhoff
of second species. The formulation of balance is thus Lagrangian total.

The geometrical entirely nonlinear problem is examined in first. The case of linear buckling is
treaty like a borderline case of the first approach.
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Voluminal elements of hulls into nonlinear geometrical
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Count
matters
1
Introduction ............................................................................................................................................ 4
2
Formulation ............................................................................................................................................ 5
2.1
Geometry of the elements of voluminal hull ................................................................................ 5
2.2
Kinematics of the voluminal hulls .............................................................................................. 6
2.3
Law of behavior ....................................................................................................................... 9
2.3.1
Taking into account of transverse shearing ......................................................................... 10
3
Principle of virtual work ...................................................................................................................... 10
3.1
Virtual work interns ..................................................................................................................... 10
3.1.1
Incremental form of virtual work interns ....................................................................... 11
3.1.2
Passage of the total reference mark to the local reference mark ............................................................................ 11
3.1.3
Relation deformation-displacement ...................................................................................... 13
3.1.4
Calculation of the stresses of Cauchy ........................................................................................ 16
3.1.4.1
General case .............................................................................................................. 16
3.1.4.2
Approximation in small deformations .................................................................... 16
4
Numerical discretization of the variational formulation resulting from the principle of virtual work ............. 19
4.1
Finite elements ................................................................................................................................. 19
4.2
Discretization of the field of displacement ...................................................................................... 20
4.3
Discretization of the gradient of displacement ................................................................................... 22
4.3.1
Gradient of total displacement ............................................................................................. 22
4.3.2
Gradient of virtual displacement .......................................................................................... 24
4.3.3
Gradient of iterative displacement .......................................................................................... 25
4.3.4
Gradient of the iterative variation of virtual displacement ..................................................... 26
4.4
Discretization of the variational formulation resulting from the principle of virtual work ......................... 27
4.4.1
Vector of the forces intern .................................................................................................. 27
4.4.2
Stamp tangent rigidity .................................................................................................. 28
4.4.3
Diagrams of integration .......................................................................................................... 30
4.4.3.1
Operators of deformations of substitution ............................................................ 30
4.4.3.2
Substitution of the geometrical part of the tangent matrix of rigidity ................. 32
5
Rigidity around the transform of the normal .................................................................................. 34
5.1
Singularity of the tangent matrix of rigidity ................................................................................ 34
5.2
Principle of virtual work for the terms associated with rotation around the normal ................ 34
5.3
Notice ...................................................................................................................................... 37
5.4
Borderline case analysis geometrically linear ......................................................................... 37
5.5
Determination of the coefficient K ....................................................................................................... 38
6
Linear buckling ............................................................................................................................ 39
7
Establishment of the elements of hull in Code_Aster .................................................................. 42
7.1
Description ..................................................................................................................................... 42
7.2
Use ....................................................................................................................................... 42
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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7.3
Calculation in geometrical nonlinear “elasticity” ........................................................................... 42
7.4
Establishment ................................................................................................................................... 43
7.4.1
Amendment of
TE0414
........................................................................................................ 43
7.4.2
Addition of a routine
VDGNLR
.................................................................................................. 43
7.5
Calculation in linear buckling ....................................................................................................... 43
8
Conclusion ........................................................................................................................................... 44
9
Bibliography ........................................................................................................................................ 45
Appendix 1
: Flow chart of calculation in linear buckling .................................................. 46
Appendix 2
: Flow chart of geometrical nonlinear calculation ............................................... 50
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
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1 Introduction
The great transformations of hull are characterized by great displacements of surface
average and of great rotations of initially normal fibers on this surface. The transformation
thus is represented exactly, at least in the continuous problem. The derivation of the objects
finite elements associated the linearized system of equations resulting from the principle of virtual work is carried out
without any simplifying assumption on displacements or rotations. Moreover, one new
diagram of selective numerical integration is presented in order to solve the problem of blocking in
membrane and in transverse shearing.
The degrees of freedom of rotation retained are the components of the vector of space iterative rotation.
Between two iterations, it is the vector of the infinitesimal rotation superimposed on the configuration
deformation. This choice led to a tangent matrix of rigidity which is not symmetrical. This is due to
nonvectorial character of great rotations which actually belong to the differential variety
SO (3). Rotations must remain lower than 2
because of the choice of update of large
rotations established in Code_Aster, for which there is not bijection between the vector of full slewing
and the orthogonal matrix of rotation.
An important difference compared to the linear analysis is to be announced. The finite elements objects are
directly built in the total reference mark; displacements and rotations nodal are measured
in the total reference mark.
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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2 Formulation
In this chapter, we present the various equations controlling the problem of deformation
hull within the framework of a theory of great transformations.
2.1
Geometry of the elements of voluminal hull
The voluminal hull is represented by volume
(together of the points
(
)
Q
3
0
) built
around average surface
(together of the points
(
)
P
3
0
=
). In any point
Q
of
, one
built a local orthonormé reference mark
(
) (
) (
)
[
]
T
T
N
1 1 2
3
2
1 2
3
1 2
,
:
,
:
,
. The vector
()
N
1 2
,
represent the normal on the surface
.
X
,
E
1
y
,
E
2
Z
,
E
3
H
3
2
1
)
,
(
,
N
)
0
,
,
(
3
2
1
1
T
)
0
,
,
(
3
2
1
2
=
T
2
1
)
0
(
3
Q
)
0
,
,
(
3
2
1
1
=
T
)
0
,
,
(
3
2
1
2
T
)
0
(
3
=
P
·
·
Appear voluminal 2.1-a: Hull. Local reference marks on the configuration of reference
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In the initial configuration, the position of an unspecified point Q normal on the average surface can
to be expressed, according to the position of the revolved center P of normal fiber, the manner
following:
(
)
(
)
(
)
X
X
N
Q
P
H
1 2
3
1 2
3
1 2
2
,
,
,
=
+
2.2
Kinematics of the voluminal hulls
·
X
,
E
1
y
,
E
2
Z
,
E
3
H
)
,
(
2
1
N
=
)
,
(
)
,
(
)
,
(
2
1
2
1
2
1
N
N
)
,
,
(
3
2
1
X
Q
)
0
,
,
(
3
2
1
=
X
P
)
,
,
(
3
2
1
X
Q
)
0
,
,
(
3
2
1
=
X
P
)
,
,
(
3
2
1
U
Q
)
0
,
,
(
3
2
1
=
U
P
)
0
(
3
Q
)
0
(
3
=
P
·
)
0
(
3
Q
·
·
)
0
(
3
=
P
Appear voluminal 2.2-a: Hull.
Great transformations of an initially normal fiber on the average surface
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In the deformed configuration, the position of the point Q can also be expressed according to
position of the point P:
(
)
(
)
(
)
X
X
N
Q
P
H
1 2
3
1 2
3
1 2
2
,
,
,
=
+
where
N
is the unit vector obtained by great rotation of the normal
N
.
The vector
N
is not necessarily normal on the deformed average surface, because of
transverse shearing strain. It is connected to the initial normal vector by the relation:
(
)
N
N
=
1 2
,
is the orthogonal operator of the great rotation around the vector
, of angle
, undergone by fiber
who was initially normal on the average surface whose expression is given by:
=
× =
+
× + -
exp [
] cos [] sin [
]
cos [
]
I
1
2
where
[
]
×
is the antisymmetric operator of the vector of full slewing
of which the matric expression
is:
[
]
× =
-
-
-




0
0
0
Z
y
Z
X
y
X
and
[
]
is the symmetrical operator given by
[
]
=
T
.
More details on great rotations and their digital processing can be found in [bib1]
or [R5.03.40]. One can also write:
(
)
(
)
T
T
T
T
1
1 2
1
2
1 2
2
=
=
,
,
One can express the virtual variation of the operator of great rotation in the form:
[
]
=
×
W
where
[
]
W
×
is the antisymmetric operator of the vector of space virtual rotation
W
who is also
rotation part of the functions tests:
[
]
W B
W B
B
× =
R
3
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Its matric expression is:
[
]
W
× =
-
-
-




0
0
0
W
W
W
W
W
W
Z
y
Z
X
y
X
One can also express the iterative variation of the operator of great rotation in the form:
[
]
=
×
W
where
W
is the vector of space iterative rotation, which is also the rotation part of the solution of
system of linearized equations.
This vector can be connected to the vector of total iterative rotation. There are thus the relations:
W
T
=
()
and
W
T
=
()
where
T ()
is the differential operator of rotation, of which the expression according to the vector of rotation
total is given by:
T ()
sin []
cos []
sin [
]
=
- -
× + -
I
1
2
3
This matrix has the same values and clean vectors that the matrix
and the relation checks:
T
T
()
()
()
=
T
In addition, the iterative variation of the matrix of virtual rotation can be put in the form:
[
] [
]
=
×
×
W
W
The total displacement of the point Q on fiber can be connected to the displacement of the center of gravity P:
(
)
(
)
(
) (
)
(
)
U
U
N
N
Q
P
H
1 2
3
1 2
3
1 2
1 2
2
,
,
,
,
=
+
-
In order to lead to a system of linearized equations, obtained starting from the weak form of balance,
we need to calculate various differential variations of this total displacement.
virtual displacement has as an expression:
(
)
(
)
(
)
(
)
U
U
W
N
N
Q
P
H
1 2
3
1 2
3
1 2
1 2
2
0
,
,
,
,
;
=
+
=
Iterative displacement has as an expression:
(
)
(
)
(
)
(
)
U
U
W
N
N
Q
P
H
1 2
3
1 2
3
1 2
1 2
2
0
,
,
,
,
;
=
+
=
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Voluminal elements of hulls into nonlinear geometrical
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The iterative variation of virtual displacement has as an expression:
(
)
(
)
(
)
(
)
(
)
U
W
W
N
Q
H
1 2
3
3
1 2
1 2
1 2
2
,
,
,
,
=
Note: The formulation suggested remains limited to rotations lower than
2
. This limit is
had with the particular choice of update of the great rotations established in Code_Aster. This is
had with nonthe bijection enters the vector of full slewing and the orthogonal matrix of rotation.
2.3
Law of behavior
We consider a linear law of behavior hyper elastic: local stresses of
Piola-Kirchhoff of second species are proportional to the local deformations of
Green-Lagrange:
~
~
S OF
=
Hereafter, the symbol ~ indicates the quantities expressed in the orthonormé reference mark
(
) (
) (
)
[
]
T
T
N
1 1 2
3
2
1 2
3
1 2
,
:
,
:
,
.
The matrix of elastic behavior linear in plane stresses is written as follows:
(
)
(
)
(
)
D
=
-
-
-
+
+
+




















E
E
E
E
sym
Ek
Ek
1
1
0
0
0
1
0
0
0
2 1
0
0
2 1
0
2 1
2
2
2
E
being the Young modulus,
the Poisson's ratio and
K
the coefficient of correction of
transverse shearing.
In the local reference mark, the state of Piola-Kirchhoff stress of second species is plane
(
)
~
S
N
=
0
and
can be characterized by a vector with 5 components:
~
~
~
~
~
~
S
=














S
S
S
S
S
T T
T T
T T
T N
T N
1 1
2 2
1 2
1
2
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The vector of the deformations of Green-Lagrange is also expressed him in the local reference mark by one
vector with 5 components:
~
~
~
~
~
~
E
=














E
E
T T
T T
T T
T N
T N
1 1
2 2
1 2
1
2
Here, we were unaware of the term
~
E
N
who is normal on the average surface and who is not inevitably
no one. This is a consequence of the assumption of the plane stresses.
2.3.1 Taking into account of transverse shearing
The correction of the transverse shear stress is carried out by extension of equivalences
energy given in the case of small deformations and of small displacements [R3.07.03].

3
Principle of virtual work
The principle of virtual work is the weak formulation of the static balance of the internal forces and
external forces:
int
-
=
ext.
0
The non-linearity of the equilibrium equations leads us to solve the system above way
iterative by a method of Newton. We carry out thus the exact linearization of the principle of
virtual work with each iteration, which leads to the equality:
int
int
-
=
-
ext.
ext.
3.1
Internal virtual work
The virtual work of the internal forces can be written on the initial configuration in the form:
()
int
~.~
=
E S D
where
~
E
and
~
S
are the vectors of deformation of Green-Lagrange and Piola-Kirchhoff stress of
second species respectively, expressed in the local reference mark. Indeed, like the state of stress
is plane for Piola-Kirchhoff of second species, we use the formulation of the principle of work
virtual in the local reference mark. However, to limit the passages of the local reference mark to the total reference mark and
vice versa, the vectors of strains and local stresses are not calculated explicitly
in the local reference mark but they are obtained by the rotation of their representation in the total reference mark.
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3.1.1 Form
incremental
internal virtual work
The iterative variation of the work of virtual work interns is written:
(
)
int
~. ~
~.~
=
+
E S
E S D
In this equality, iterative variation of the vector of local stresses of Piola-Kirchhoff of
second species is calculated by the iterative discrete form of the relation of behavior:
~
~
S
D E
=
3.1.2 Passage of the total reference mark to the local reference mark
In tensorial form one passes from the tensor of the total stresses to the tensor of the stresses
local
3 3
×
(see [bib4] p. 111 for the stresses of Cauchy, the same relations applying to
stresses of Piola-Kirchhoff of second species) while using:
[~]
[]
S
P S P
T
=
and of the tensor of the local stresses to the tensor of the total stresses by the inversion of the relation
the preceding one:
[]
[~]
S
P S P
T
=
In the two preceding expressions, the matrix of passage of the local reference mark to the total reference mark is
an orthogonal matrix
P
P
1
T
-
=
, and its expression clarifies according to the unit vectors of
identify orthonormé local is:
(
)
(
)
(
)
(
)
P
T
T
N

1 2
3
1
1 2
3
2
1 2
3
1 2
,
,
,
,
=






T
T
T
Within the framework of the conventional notation, one will be able to note:
(
)
(
)
(
) ()
T
E
T
E
T
N
E
1 1 2 3
0 1
2 1 2 3
0 2
3 1 2 3
1 2
0 3
,
,
,
,
=
=
=
=
with the orthogonal matrix of passage (initial rotation):
(
)
(
)
(
)
(
)
[
]
0 1 2
3
1 1 2
3
2 1 2
3
3 1 2
3
,
,
:
,
:
,
=
T
T
T
It will be noticed that:
0
=
P
T
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The two relations of rotation of the stresses are also valid for the tensors of the deformations
of Green-Lagrange. Nevertheless, a writing which connects the vectors of local and total deformation is
necessary. This relation makes it possible to pass from the vector
6 1
×
total deformations with the vector
6 1
×
local deformations:
~
E
H E
6 1
6.6 6 1
×
× ×
=
with the form of the matrix of transformation of the vectors
6 1
×
of deformation (see [bib2]
p. 258):
H
6.6
12
12
12
1 1
1 1
1 1
22
22
22
2 2
2 2
2 2
32
32
32
3 3
3 3
3 3
1 2
1 2
1 2
1 2
2 1
1 2
2 1
1 2
2 1
2 3
2 3
2 3
2 3
3 2
2 3
3 2
2 3
3 2
2
2
2
2
2
2
2
×
=
+
+
+
+
+
+
L
m
N
L m
m N
N L
L
m
N
L m
m N
N L
L
m
N
L m
m N
N L
L L
m m
N N
L m
L m
m N
m N
N L
N L
L L
m m
N N
L m
L m
m N
m N
N L
N L
L L
m m
N N
L m
L m
m N
m N
N L
N L
3 1
3 1
3 1
3 1
1 3
3 1
1 3
3 1
1 3
2
2
+
+
+














and components of the unit vectors of the local reference mark:
L
m
N
L
m
N
L
m
N
1
1
1
1
1
2
1
1
3
2
2
1
2
2
2
2
2
3
3
3
1
3
3
2
3
3
3
=
=
=
=
=
=
=
=
=
T E
T E
T E
T E
T E
T E
T E
T E
T E
.
.
.
.
.
.
.
.
.

These expressions are general for the curvilinear reference marks. In the Cartesian total reference mark
[
]
E
E
E
1
2
3
:
:
, these components are:
()
()
()
()
()
()
()
()
()
L
T
m
T
N
T
L
T
m
T
N
T
L
T
m
T
N
T
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
1
2
3
1
2
3
1
2
3
=
=
=
=
=
=
=
=
=
We have, actually, need for a writing which connects the vector of local deformation
5 1
×
and it
vector of total deformation
6 1
×
:
~
E
H E
5 1
5 6.6 1
×
× ×
=
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For that, one forgets the third line of the expression of
H
6.6
×
(line associated with
S
N
):
()
()
()
()
()
()
()
()
()
()
()
()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
()
H
5 6
1
2
1
2
1
2
2
2
2
2
2
2
1
2
1
2
1
2
2
3
2
3
2
3
3
1
3
1
3
1
1
1
1
1
1
1
2
2
2
2
2
2
1
2
2
1
1
1
2
3
1
2
3
2 1
1
2 2
2
2 3
3
2
1
1
2
2
2
2
3
3
2 1
1
2
2
2
2
3
3
1
2
2
3
3
1
1
2
2
3
3
1
1
2
1
2
2
×
=






+
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
2
2
1
1
2
2
1
2
3
3
2
2
3
3
2
2
3
3
2
3
1
1
3
3
1
1
3
3
1
1
3
3
2
3
3
1
3
1
1
2
1
2
2
3
2
3
3
1
3
1
1
2
1
2
2
3
2
3
3
1
3
1
+
+
+
+
+
+
+
+





The same preceding relations can be applied for the passage of the vectors of
total deformation with the local deformation.
3.1.3 Relation
deformation-displacement
The tensor
3 3
×
total deformations of Green-Lagrange is defined by (see for example
[bib2]):
(
)
[]
E
U
U
U
U
=
+
+
1
2
T
T
with the tensor of the gradient of displacements:
=










<
> =










U





X
y
Z
U v W
U
X
v
X
W
X
U
y
v
y
W
y
U
Z
v
Z
W
Z

The tensor of deformation of Green-Lagrange can be also written:
[]
(
)
E
F F I
=
-
1
2
T
with F the tensor gradient of the deformations
3 3
×
who is not symmetrical:
F
X
I
U
=
= +
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and I the tensor identity:
I
=




1 0 0
0 1 0
0 0 1
The vector
6 1
×
total deformations of Green-Lagrange is ordered as follows (see [bib4]
p 117):
(
)
(
)
(
)
E
=














=
+
+
+














+
+
+
+
+
+
+
+
+
E
E
E
U
v
W
U
v
U
W
v
W
U
v
W
U
v
W
U
v
W
U U
v v
W W
xx
yy
zz
xy
xz
yz
X
y
Z
y
X
Z
X
Z
y
X
X
X
y
y
y
Z
Z
Z
X
y
X y
X

,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
21
21
2
2
2
2
2
2
2
2
2
2
y
X Z
X Z
X
Z
y Z
y Z
y
Z
U U
v v
W W
U U
v v
W W
,
,
,
,
,
,
,
,
,
,
,
,
+
+
+
+


















It as follows is calculated:
E
Q
With ux ux
=
+




1
2 (
)
with:
Q
=












1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1
0 1 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0
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and the vector of the gradient of displacements:
U
X
=






















U
U
U
v
v
v
W
W
W
X
y
Z
X
y
Z
X
y
Z
,
,
,
,
,
,
,
,
,
and tensor A depend on the gradient of displacements:
With ux




=














U
v
W
U
v
W
U
v
W
U
U
v
v
W
W
U
U
v
v
W
W
U
U
v
v
W
W
X
X
X
y
y
y
Z
Z
Z
y
X
y
X
y
X
Z
X
Z
X
Z
X
Z
y
Z
y
Z
y
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Virtual variation, noted
, from the deformations of Green-Lagrange is obtained by a calculation
differential:
E
Q
U
X
U
X
=
+








With
In this expression and that which follows, we took account of the following property (see
[bib4] p 141):
1
2
1
2
With
U
X
U
X
With ux ux




=




Iterative variation
is it also obtained by a differential calculus:
E
Q With ux
U
X
=
+








The iterative variation of the virtual deformation of Green-Lagrange is put thus in the form:
E
With
U
X
U
X
Q With ux
U
X
=




+
+








conventional term
nonconventional term
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Whereas the first term of this expression is conventional for the continuous mediums 3D, it
second, which translates the taking into account of great rotations, is less.

3.1.4 Calculation of the stresses of Cauchy

3.1.4.1 Case
General
The tensor
3 3
×
total stresses of Piola-Kirchhoff of second species is connected to the tensor
3 3
×
total stresses of Cauchy by the relation:
[]
()
[]
S
F F
F
=
-
-
det
1
T
Thus, knowing the state of the stresses of Piola-Kirchhoff of second species, one can calculate the state
stresses of Cauchy by the relation:
[]
()
[]
=
1
det F F S F
T
It should be noted that the state of stresses of Cauchy is not plane, in general, contrary to the state of
stresses of Piola-Kirchhoff of second species. In addition, the choice of a local reference mark in which
to represent this tensor is not at all obvious. It will be however shown, in the following paragraph, that
within the framework of the small deformations, there is a local reference mark, easily identifiable, in which
the state of stresses of Cauchy is him-also plane.
In the case of completely general laws, a detailed attention will have to relate to the diagrams
of numerical integration allowing to calculate the values of substitution of the gradient
F
at the points
of normal numerical integration.

3.1.4.2 Approximation in small deformations
It is reminded the meeting [bib4] that the gradient F can be written thanks to the polar decomposition under two
forms:
F RU
VR
=
=
where
R
R
=
-
T
is an orthogonal tensor, and where
U
and
V
are symmetrical matrices of strain
defined positive.
Into the geometrical nonlinear field, we can introduce an important simplification
in the polar decomposition of the gradient of the deformations if the deformations remain small. This
simplification is not introduced into nonlinear calculation but in postprocessing of the stresses.
Strain at the point Q being minor in front of the great rotation of the section:
U
V
I
One can then write:
F R
=
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where
is the tensor of great rotation which transforms the normal
N
in
N
:
N N
=
Simplification translates the fact that on a section, the transformation is reduced to a great rotation.
With this approximation of the gradient of the deformations, one can write:
F R
=
and thus, by exploiting the orthogonality of
one obtains:
F
-
1
T
and:
()
det F
1
.
These simplifications lead to the final relation:
[]
[]
S
T
This relation translates the fact that the stresses of Cauchy are quite simply obtained by
great rotation of the stresses of Piola-Kirchhoff of second species.
One can now rewrite the property of plane stresses of the tensor of Piola-Kirchhoff of
second species
[]
N S N
.
=
0
in the new form:
[]
N
N
.
T
=
0
who leads in addition to the property:
[]
N
N
.
=
0
That is to say still:
~
N N
=
0
Stresses of Cauchy
(
)
[
]
1 2
3
,
are also plane in the local reference mark
(
) (
)
(
)
[
]
T
T
N
1
2
1 2
3
1 2
3
1 2
,
:
,
:
,
obtained by great rotation of the local reference mark on
initial configuration:
[
] [
]
T
T
N
T
T
N
1
2
1
2
:
:
:
:
=
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In this reference mark, we can write all the components of the tensor
[]
as follows:
[]
[]
[]
[]
[]
[]
[]
[]
[]
~
~
~
~
~
~
~
~
.
.
.
.
.
.
.
.
.
T T
T T
T N
T T
T T
T N
T N
T N
T
T
T
T
T
N
T
T
T
T
T
N
N
T
N
T
N
N
1 1
1 2
1
2 1
2 1
2
1
1
0
1
1
1
2
1
2
1
2
2
2
1
2






=





By taking again the relation
[]
[]
S
T
, one obtains:
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
T
T
T
T
T
N
T
T
T
T
T
N
N
T
N
T
N
N
T S T
T S T
T S N
T
S T
T
S T
T
S N
N S T
N S T
N S N
1
1
1
2
1
2
1
2
2
2
1
2
1
1
1
2
1
2
1
2
2
2
1
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.




=





from where the final result:
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
T T
T T
T N
T T
T T
T N
T N
T N
T T
T T
T N
T T
T T
T N
T N
T N
S
S
S
S
S
S
S
S
1 1
1 2
1
2 1
2 1
2
1
1
1 1
1 2
1
2 2
2 2
2
1
2
0
0






=





In so far as the deformation remains small, components of the tensor of the stresses of
Cauchy in the local reference mark attached to the deformed configuration are identical to the components
tensor of the stresses of Piola-Kirchhoff of second species in the local reference mark attached to
initial configuration.
We take the party in the continuation, to consider only the stresses of Piola-Kirchhoff of
second species. We must note that within the framework of a more general constitutive law, one
will be able to pass from a stress measurement to another as indicated in the preceding paragraph.
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4
Numerical discretization of the variational formulation
exit of the principle of virtual work
4.1 Elements
finished
The three figures below summarize the finite elements choices concerning the voluminal hulls
[R3.07.04]. The selected finite elements are isoparametric quadrangles or triangles.
quadrangle is represented below. One chooses among the elements with functions of interpolation
quadratic, the element hétérosis whose displacements are approached by the functions
of interpolation of the Sérendip element and rotations by the functions of the element of Lagrange.
All the justifications as for these choices are given in [R3.07.04].
Sérendip element
Hétérosis element
Element of Lagrange
NB1 = 8
NB2 = 9
U
K
, ~
~
Appear 4.1-a: Families of finite elements for the isoparametric quadrangle
1
3
2
3
1
2
4
5
6
7
8
1
1
= -
2
1
= -
3
1
=
Appear voluminal 4.1-b: Element of reference
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3
3
2
1
P
1
2
4
5
6
7
8
Appear 41-c: Real voluminal element
4.2
Discretization of the field of displacement
With an aim of avoiding the explicit calculation of the curvatures, which becomes extremely heavy in the case of them
great rotations, we choose to interpolate the normal on the initial average surface with the place
to interpolate rotations:
(
)
()
(
)
N
N
1 2
2
1
2
1 2
,
,
=
=
NR
I
I
NB
I
where
()
(
)
NR
I2
1 2
,
indicate the function of interpolation to the node
I
among
NB2
nodes of Lagrange.
The same interpolations are adopted for the transform of the initial normal:
(
)
()
(
)
N
N
1
2
12
1
2
1
2
,
,
=
=
NR
I
NB
I
The interpolation of the initial position of a point on the average surface of the hull (not P) is given
by:
(
)
()
(
)
X
1 2
1
1
1
1 2
,
,
=




=
NR
X
y
Z
I
I
NB
I
where
()
(
)
NR
I1
1 2
,
indicate the function of interpolation to the node
I
among
NB
NB
1
2 1
=
-
nodes of
Serendip.
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The interpolation of the initial position of an unspecified point of the hull (not Q) can then be written
in the form:
(
)
()
(
)
()
(
)
X
1 2
3
1
1
1
1 2
3
2
1 2
1
2
2
,
,
,
=




+




=
=
NR
X
y
Z
H
NR
N
N
N
I
I
NB
I
I
X
y
Z
I
NB
The same interpolations are adopted for the deformed position of an unspecified point of fiber:
(
)
()
(
)
()
(
)
X


1 2
3
1
1
1
1 2
3
2
1
2
1
2
2
,
,
,
=




+




=
=
NR
X
y
Z
H
NR
N
N
N
I
I
NB
I
I
X
y
Z
I
NB
The interpolations for the positions initial and deformation being the same ones, we can adopt them
for the real displacement of an unspecified point of the hull:
(
)
()
(
)
()
(
)
U

1 2
3
1
1
1
1 2
3
2
1 2
1
2
2
,
,
,
=




+




-










=
=
NR
U
v
W
H
NR
N
N
N
N
N
N
I
I
NB
I
I
X
y
Z
I
X
y
Z I
I
NB
Thus, the interpolation of virtual displacement becomes:
(
)
()
(
)
()
(
)


U
1 2
3
1
1
1
1 2
3
2
1 2
1
2
2
0
0
0
,
,
,
=




-
-
-
-








=
=
NR
U
v
W
H
NR
N
N
N
N
N
N
W
W
W
I
I
NB
I
I
Z
y
Z
X
y
X
I
X
y
Z I
I
NB
In the same way, the interpolation of iterative displacement becomes:
(
)
()
(
)
()
(
)


U
1 2
3
1
1
1
1 2
3
2
1 2
1
2
2
0
0
0
,
,
,
=




-
-
-
-








=
=
NR
U
v
W
H
NR
N
N
N
N
N
N
W
W
W
I
I
NB
I
I
Z
y
Z
X
y
X
I
X
y
Z I
I
NB
Moreover, the interpolation of the iterative variation of virtual displacement is:
(
)
()
(
)
(
)
(
)
U
W
W N
1 2
3
3
2
1
2
1
2
2
,
,
=
=
H
NR
I
I
NB
I
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4.3
Discretization of the gradient of displacement
4.3.1 Gradient of total displacement
The vector of the gradient of real displacement can be connected to the isoparametric gradient of
real displacement by the following relation:

U
J
U
X
=
-
~
1
The isoparametric gradient of displacement is organized as follows:




U
=






















U
U
U
v
v
v
W
W
W
,
,
,
,
,
,
,
,
,
1
2
3
1
2
3
1
2
3
The matrix jacobienne generalized
9 9
×
~
J
-
1
can be expressed according to the matrix jacobienne
isoparametric transformation
3 3
×
as follows:
~
J
J
J
J
-
-
-
-
=




1
1
1
1
0
0
0
0
0
0
The isoparametric gradient of real displacement can be calculated as follows:

U
p
=




NR
E
1
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with the first matrix of derived from the functions of form:
()
()
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
NR
NR
NR
NR
NR
I
I
I
I
I
I
I
I
I
I
I
I




=


























1
1
1
1
1
1
1
3
2
3
2
2
3
2
3
2
2
1
2
1
2
1
2
1
2
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
0
0
0
0
0
0
1
1
2
0
0
0
0
0
0
0
0
0
0
0
3
2
3
2
2
3
2
2
3
2
2
2
2
3
2
2
3
2
2
1
2
1
2
1
2
NR
NR
NR
I
NB
H
NR
NR
NR
NR
NR
I
I
I
NB
NB
NB
NB
NB
,
,
,
,
,
,
,
























































=
L
()
()
()
()
NR
NR
NR
NR
NB
NB
NB
NB
2
2
3
2
2
3
2
2
2
2
0
0
0
0
0
0
0
1
2
,
,


























































































and the vector of “generalized nodal real displacement”:
p
E
X
X
y
y
Z
Z
I
X
X
y
y
Z
Z NB
U
v
W
N
N
N
N
N
N
I
NB
N
N
N
N
N
N
=
-
-
-












=
-
-
-






































M
M
M


1
1
2
,
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Finally, one will be able to write the gradient of real displacement in the form:
U
J
p
X
NR
E
=




-
~
1
1
4.3.2 Gradient of virtual displacement
While proceeding similarly to the gradient of real displacement, one can connect the two gradients of
virtual displacement:
U
J
U
X
=
-
~
1
The isoparametric gradient of virtual displacement can be calculated as follows:
U
U
=




NR
E
2
with the second matrix of derived from the functions of form:
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
N
NR
N
NR
N
NR
N
NR
I
I
I
I
I
I
I
Z
I
y
I
Z
I
y




=


























-
-
2
1
1
1
1
1
1
3
2
3
2
3
2
3
2
1
2
1
2
1
2
1
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
()
()
()
()
12
12
3
2
3
2
3
2
3
2
12
12
3
2
3
2
3
2
3
2
12
12
1
1
2
2
1
1
2
2
0
0
0
0
0
0
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
Z
y
I
Z
I
X
I
Z
I
X
Z
X
I
y
I
X
I
y
I
X
y
X
-
-
-
-
-
-
-














,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()










































=
-
-
-
-
LI
NB
H
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NB
Z
NB
y
NB
Z
NB
y
NB
Z
NB
y
NB
Z
NB
X
1
1
2
0
0
0
0
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
3
2
2
3
2
2
1
1
2
2
1
1
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
-
-
-
-
-

































3
2
2
3
2
2
2
2
2
2
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
2
2
1
1
2
2
0
0
0
0
0
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NB
Z
NB
X
NB
Z
NB
X
NB
y
NB
X
NB
y
NB
X
NB
y
NB
X
,
,
,
,
,
,


























































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and the vector of the virtual nodal variables:



U
E
X
y
Z I
X
y
Z NB
U
v
W
W
W
W
I
NB
W
W
W
=












=




































M
M
M
1
1
2
,
Finally, one will be able to write the gradient of virtual displacement in the form:
U
J
U
X
NR
E
=




-
~
1
2
4.3.3 Gradient of iterative displacement
The step here is similar to virtual calculation. It is enough to replace
by
:
U
J
U
X
NR
E
=




-
~
1
2
with the vector of the iterative nodal variables:



U
E
X
y
Z I
X
y
Z NB
U
v
W
W
W
W
I
NB
W
W
W
=












=






































M
M
M
1
1
2
,
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4.3.4 Gradient of the iterative variation of virtual displacement
U
J
Q
X
NR
E
=




-
~
1
3
with:
()
()
()
()
()
()
()
()
()
NR
H
NR
NR
NR
NR
NR
NR
NR
NR
NR
I
NB
I
I
I
I
I
I
I
I
I




=




























=

















3
3
2
3
2
2
3
2
3
2
2
3
2
3
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
1
2
1
2
1
2
L
L
,
,
,
,
,
,
,










and the vector of the iterative variation “nodal virtual displacement” generalized:
(
)
(
)
Q
W
W N
E
I
I
NB
=
=




















.
.
.
.
.
.
,
1
2
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4.4 Discretization of the variational formulation resulting from the principle of
virtual work
We take again the iterative variation (between two iterations) of internal virtual work:
(
)
int
~. ~
~.~
=
+
E S
E S D
and iterative variation of the vector of local stresses of Piola-Kirchhoff of second:
~
~
S
D E
=
Then, the linearized form of the principle of virtual work of the §3 can be written for the finite element described
above in the following matric form:
(
)
the U.K.
U
U
F
R
E
and E
E
E
E
.
.
=
-
where
U
E
is the nodal vector of the functions tests. One deduces the system from it from equations:
K
U
F
R
and E
E
E
=
-
where:
K
and
is the tangent matrix of rigidity
U
E
is the elementary vector of the solution of the linearized system of equations (nodal vector
between two iterations)
is the external level of load
F
E
is the nodal vector of the external forces (associate with
=
1
)
R
E
is the nodal vector of the internal forces
4.4.1 Vector of the internal forces
It is a vector
(
)
6
1 3
1
×
+ ×
Nb
entirely expressed in the total reference mark and which must be
evaluated with each iteration by the relation:
R
B S
J
E
T
D D D
=
~ ~ det
2
1
2
3
with the vector of the local stresses Piola-Kirchhoff of second species:
~
~
S
OF
=
It is reminded the meeting that the symbol ~ indicates an object expressed in the local reference mark.
The local deformations of Green-Lagrange are updated to each iteration:
~
~
E B p
=
1 E
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where the operator of the total deflections (first operator of the deformations) is written:
~
~
B
H Q
With U J
1
1
1
1
2
=
+












-
X
NR
with the gradient of real displacement:
U
J
p
X
NR
E
=




-
~
1
1
The operator of the virtual deformations (second operator of the deformations):
~
~
B
H Q With U J
2
1
2
=
+












-
X
NR
is highlighted by the relations:
~
~
~
~
E B U
E B
U
=
=
2
2
E
E
4.4.2 Stamp tangent rigidity
The tangent matrix of rigidity which is serious
(
) (
)
6
1 3
6
1 3
×
+ × ×
+
NB
NB
express yourself too
entirely in the total reference mark. One must be able to evaluate it with each iteration if it is wanted that
convergence of the method of Newton is quadratic. In a conventional way into nonlinear
geometrical, it takes the form:
K
K
K
T
E
m
E
G
E
=
+
where the first part represents the material part:
K
B DB
J
m
E
T
D D D
=
~
~
det
2
2
1
2
3
and the second part represents the geometrical part, it even made up of two parts:
K
K
K
G
E
G
E
G
E
=
+
conventional
nonconventional
with the conventional part of the geometrical part (see [bib4] p. 141):
K
J
S J
J
G
E
T
NR
NR
D D D
conventional
=
















-
-
~
~
det
1
2
1
2
1
2
3
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where
S
the generalized tensor of the stresses expressed in the total reference mark is written:
[] []
S
S
S
S
9 9
3 3
0
0
0
0
0
0
×
×
=






[]
The nonconventional part of the geometrical part not represents terms uncoupled from rotation
symmetrical which has as a form:
()
K
Z
N
G
E
I
I
I I
3 3
×
=
×
×
nonconventional
,
[
] [
]
where
N
I
is the transform of the initial normal to the node
I
and
Z
I
a vector
3 1
×
with the node
I
NB
=
1
2
,
nodal vector
(
)
3
2 1
×
×
NB
I
Z
Z
Z
I
I
I
NB
=
=




















.
.
.
.
.
.
,
1
2
The nodal vector
Z
I
is similar to a vector of internal force and its expression is:
Z
B S
J
I
T
D D D
=
~ ~ det
3
1
2
3
with the operator of the iterative variation of the virtual deformations (third operator of
deformations):
~
~
B
H Q With U J
3
1
3
=
+












-
X
NR
who is highlighted by the relation:
~ ~ det
~
.~
B S
J
E
S
3
1
2
3
T
D D D
D
=
nonconventional
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4.4.3 Diagrams
of integration
The integration of the terms of rigidity in the thickness of the hull is identical to the method used in
analyze linear geometrical [R3.07.04] for nonlinear behaviors. 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. A layer in the thickness of the hull appears sufficient in the majority of the cases.
In order to be able bearing with the problem of blocking out of membrane of the curved hulls and to solve it
problem of blocking in transverse shearing, it is necessary to modify the diagram of integration
on average surface. If the technique is completely known in linear analysis, it is it less
in geometrical nonlinear analysis.
The procedure is presented in the form of a generalization of the separation of the effects of membrane, of
bending and of transverse shearing if one uses the deformations of Green-Lagrange:
~
~
~
E
E
E
=




m
S
where
~
~
~
~
E
m
T T
T T
T T
E
E
=




1 1
2 2
1 2
represent the deformation of membrane-bending and
~
~
~
E
S
T N
T N
=




1
2
deformation of
transverse shearing.
During the numerical evaluation of the deformations at the points of normal numerical integration of Gauss
(9 points for the quadrilateral and 7 points for the triangle), the relation is used
~
~
E
p
=
B
E
1
.
amendment is introduced on the level of the first operator of the deformations:
~
~
~
B
B
B
1
1
1
=


MF
S
substitution
substitution
~
B
MF 1substitution
and
~
B
s1substitution
are the first operators of the deformations of substitution of
membrane-bending and of transverse shearing, respectively.
During the calculation of the nodal vector of the internal forces and material part of the tangent matrix of
rigidity, the amendment is introduced in a way similar to the level of the second operator of
deformations:
~
~
~
B
B
B
2
2
2
=


MF
S
known
known

4.4.3.1 Operators of deformations of substitution
In what follows the points of normal and reduced numerical integration of Gauss, on surface
average, are
NPGSN
NPGSR
=
=
9
4
and
, respectively, for the element
quadrilateral, and
NPGSN
=
7
and
NPGSR
=
3
, respectively, for the triangular element.
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Membrane-bending part
At the point
INTSN
among
NPGSN
points of normal numerical integration of Gauss of surface
average, one will calculate:
(
)
(
)
(
)
(
)
(
)
~
~
~
~
,
B
B
B
B
MF
MF
MF
I
INTSR
NPGSR
MF
INTSN
INTSN
INTSN
NR INTSN
INTSR
1
1
1
1
1
substitution
normal
complete
normal
incomplete
reduced
incomplete
=
-
+
=
where
INTSR
is a point among
NPGSR
points of numerical integration reduced by Gauss of
surface average.
In the expression above,
~
B
MF 1
normal
complete
represent the first three lines of
~
B
1
calculated with
points of normal numerical integration by considering the complete matrix
NR




1
. The operator
~
B
MF 1
normal
incomplete
represent the first three lines of
~
B
1
calculated at the points of numerical integration
normal by considering a matrix
NR




1
incomplete where the columns of rotation are cancelled:
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
I
NB
Inc
I
I
I
I
I
I




=
























=
1
1
1
1
1
1
1
1
2
1
2
1
2
0
0
0 0 0
0
0
0 0 0
0
0
0
0 0 0
0
0
0 0 0
0
0
0 0 0
0
0
0
0 0 0
0
0
0 0 0
0
0
0 0 0
0
0
0
0 0 0
1
1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
L
L
,
,
,
,
,
,
,
0 0
0 0 0
0 0 0
















































(
)
~
B
MF
INTSR
1
reduced
incomplete
the first three lines represent of
~
B
1
calculated at the points
of reduced numerical integration with the matrix
NR
Inc




1
incomplete above definite. They are thus
stored to be extrapolated at each point of normal numerical integration.
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Transverse shearing part
For the transverse shearing part, one will calculate:
(
)
(
)
(
)
~
~
,
B
B
S
I
INTSR
NPGSR
S
INTSN
NR INTSN
INTSR
1
1
1
substitution
reduced
complete
=
=
where
(
)
~
B
S
INTSR
1
reduced
complete
represent the two last lines of
~
B
1
calculated at the points of integration
numerical reduced with a matrix
NR




1
complete. They are also stored to be extrapolated
at each point of normal numerical integration.

4.4.3.2 Substitution of the geometrical part of the tangent matrix of rigidity
The nonconventional part of the tangent matrix of rigidity
K
G
E
nonconventional
is numerically
integrated into the points of normal integration of Gauss. No operation of substitution is necessary.
For the conventional part of the tangent matrix of rigidity, we use substitution:
K
K
K
K
K
G
E
G
E
G
E
G
E
G
E
conventional
substitution
conventional
normal
complete
membrane
bending
conventional
normal
incomplete
membrane
bending
conventional
reduced
incomplete
membrane
bending
conventional
reduced
complete
shearing
transverse
=
-
+
+
where:
K
G
E
conventional
normal
complete
membrane
bending
is numerically integrated on the points of normal integration with a matrix
NR




2
supplements, and the local stresses of membrane bending only;
K
G
E
conventional
normal
incomplete
membrane
bending
is numerically integrated on the points of normal integration with a matrix
NR




2
incomplete, and local stresses of membrane bending only;
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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Key
:
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K
G
E
conventional
reduced
incomplete
membrane
bending
is numerically summoned on the points of integration reduced with a matrix
NR




2
incomplete, and integrated local stresses of membrane bending only;
K
G
E
conventional
reduced
complete
shearing
transverse
is numerically summoned on the points of integration reduced with a matrix
NR




2
supplements, and the integrated local stresses of transverse shearing only;
To be able to calculate the two last tangent matrices in the preceding equation, us
let us carry out the numerical integration of the local stresses on
NPGSN
points of integration
normal:
(
)
(
) (
)
~
~
det
,
S
S
J
INTSR
NR INTSR
INTSN
D
D
D
I
INTSN
NPGSN
=
=
1
2
3
1
This equation contains the terms of weight of the points of Gauss.
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Key
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5
Rigidity around the transform of the normal
5.1
Singularity of the tangent matrix of rigidity
Although the finite elements objects of the hull are expressed directly in the total reference mark
[
]
E
E
E
1
2
3
:
:
(the degrees of freedom are displacements and rotations in the total reference mark),
the tangent matrix of rigidity presents a singularity compared to the component of rotation
around the transform of the normal in each node:
K N
T
E
W W
I
NB
=




=
0
1
2
,
Contributions
(
)
W N
=
I
NB
1
2
,
are null.
In the preceding equation, this matrix represents the rigidity of rotation in the total reference mark. Its
structure is full:
[]
K
T
E
I
W W
I
K
K
K
K
K
K
K
K
K
=




11
12
13
12
22
23
31
32
33
it is a nonsymmetrical matrix.

This singularity is a direct consequence of the kinematics of hull. It is due to the product
vectorial appearing in linearized displacements (virtual and incremental). Thus displacement
between two iterations is given by:
(
)
(
)
(
)
(
)
U
U
W
N
Q
p
H
1 2
3
1 2
3
1 2
1 2
2
,
,
,
,
=
+
It is noticed that the contribution
(
)
(
)
3
1 2
1 2
2
H
W
N
,
,
is perpendicular to
N
. One
interpret this singularity in the following way: the rotation of an initially normal fiber on the surface
average does not lead to a strain of the aforementioned, and consequently does not induce deformation.
5.2 Principle of virtual work for the terms associated with rotation
around the normal
We propose to define the full slewing around the transform of the normal in the hull like
the projection of the vector of full slewing on the transform of the normal:
N
N
=
.
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Voluminal elements of hulls into nonlinear geometrical
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Key
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It is reminded the meeting that the vector of rotation
is an invariant of the matrix of rotation
=
×
exp [
]
=
The vector of rotation is a clean vector of the matrix of rotation associated with the eigenvalue
identity. So the first relation is rewritten:
N
N
N
N
=
=
=
(
).(
)
.
This relation translates an important result:
The projection of the vector of full slewing on the transform of the normal is equal to
projection of the vector of full slewing on the initial normal
In discrete form, one defines a deformation energy associated with this rotation:
()
N
N
I
I
NB
K
=
=
1
2
2
1
2
,
where
K
is a rigidity of torsion of which the determination of the value will be discussed further. One supposes
that this rigidity remains constant and undergoes neither virtual variation nor incremental variation.
The existence of the potential is supposed:
N
N
N
=
=
1
2
1
2
K
I
I
NB
((.
) (.
))
,
that one can rewrite in a more elegant form:
N
N
N
=
=
1
2
1
2
K
I
I
NB
([
])
,
By exploiting the property of orthogonality
-
=
1
T
matrix of rotation:
N
N
N N
N
N
N
N
N
=
=
=
=
T
T
T
(
) (
)
This property will be exploited in the double linearization of the potential energy.
One rewrites the potential in the form:
N
N
N
=
=
1
2
1
2
K
I
I
NB
([
])
,
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Key
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First linearization of
N
, allows to obtain the virtual variation:
N
N
N
N
N
N
N
=
+
=
=
=
1
2
1
2
1
2
K
K
I
I
NB
I
I
NB
(
[
]
[
]
)
(
[
])
,
,
It is necessary to express this form according to the function tests of rotations retained in
variational form.
=
-
T
W
1
()
with the form of the matrix reverses of the differential operator of rotation:
T
-
=
-
× +
-




1
2
2
2
1
2
1 1 2
2
()
tan
[]
[
]
tan
[
]
I
From where the final form of the virtual work which makes it possible to deduce the vector from the interior forces:
N
wT
N
N
=
-
=
K
T
I
I
NB
(
() [
])
,
1
2
One carries out the second linearization of
N
:
(
)
(
)
N
W T
N
N
T
N
N
=
+
-
-
=
K
T
T
I
I
NB
.
() [
]
() [
]
,
1
2
with the particular choice of the ddls of rotation
W
=
0
, and owing to the fact that the initial normal “does not move
not “during the iterations
N
=
0
.
The expression of the tangent operator who gives rise to the terms corresponding to the ddls of
rotation around the transform of the normal of tangent matrix is as follows:
(
)
(
)
(
)
(
)
N
W T
N
N T
W
T
N
N
=
+
-
-
=
-
=
K
W
K
T
I
I
NB
T
I
I
NB
.
() [
]
()
.
() [
]
,
,
1
1
2
1
2
In this relation, the last term is a differential term due to the nonlinear relation between
parameters of rotation. Its linearization is heavy to carry out and its contribution will be neglected in
the expression of the tangent operator.
With the property:
N
N
N
N
=
, we give the final expression:
(
)
(
)
N
W T
N
N T
W
-
-
=
K
T
I
I
NB
.
() [
]
()
,
1
1
2
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Key
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The contribution of the operator is noted
T
-
1
() []
I
in the tangent matrix of rigidity.
5.3 Notice
The potential energy brought by the terms of rotation around the transform of the normal is
nonnull even for a rigid rotational movement. This energy does not correspond to one
deformation. For this reason it must be nonsignificant. The default value of
COEF_RIGI_DRZ
must guarantee that.
5.4
Borderline case analysis geometrically linear
In the case of the analysis geometrically linear, initial configuration and configuration
deformation are confused what leads us to confuse the initial normal
N
with its transform
N
:
N
N
Rotations become small in this case and the operator of great rotation becomes:
=
×
+ ×
exp [
] [] [
]
I
The differential operator of rotations becomes:
T
() []
I
and the parameters of rotations become confused:
W
and
W
All these approximations introduced into virtual work lead to its simplification:
N
N
N
=
K
I
K
NB
(
[
])
,
1
2
The same approximations introduced into the differential part of virtual work also lead to
its simplification:
(
)
N
N
N
=
K
I
K
NB
[
]
,
1
2
The two last equations are those of the analysis geometrically linear. They show that
contributions in the vector of the interior forces and the tangent matrix of rigidity
cover the borderline case well with [R3.07.04].
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5.5
Determination of the coefficient K
The coefficient K is calculated with each iteration of each pitch of time. With each iteration of Newton
of each pitch of time, one builds with
NB2
nodes the matrix of passage
[
]
I
I
I
NB
=
=
T
T
N
1
2
1
2
:
:
;
,
who allows to pass from the vector
W
I
, vector of iterative rotation to the node expressed in the reference mark
total
[
]
E
E
E
1
2
3
:
:
with the vector
~
W
I
expressed in local reference mark
[
]
T
T
N
1
2
:
:
I
:
~
;
,
W
W
I
I
I
I
NB
=
=
1
2
One can build in each node, the matrix
[]
K
T
E
I
W W
~
of size
3 3
×
[
]
[
]
~ ~
K
K
T
E
W W I
I T
T
E
W W I
I
=
This matrix represents the rigidity of rotation in the local reference mark. Its structure, nonsymmetrical, is:
[
]
~ ~
K
T
E
W W I
T T
T T
T T
T T
N T
N T
I
K
K
K
K
K
K
=






1 1
1 2
2 1
2 2
1
2
0
0
0
The coefficient
K
is then calculated according to the relation:
K
COEFF RIGI DRZ KMIN
=
×
_
_
where COEF_RIGI_DRZ is a coefficient introduced like a mechanical characteristic of hull by
the user. In conventional linear analysis of the hulls or plates, this coefficient is selected enters
0.001 and 0.000001. By defect it is worth 0.00001. In the case of great rotations calculated with
great pitches of load, one advise to use value 0.001.
KMIN
is the minimum of the nonnull terms of rotation on the diagonal of
~
K
T
E
.
KMIN
MIN
K
K
I
NB
T T
T T
I
=




=
1
2
1 1
2 2
,
,
Note:
It would be undoubtedly more rigorous to calculate
K
with the first iteration of the first pitch of time
and to store this value like invariant information during the iterations and the pitches of
following times. The experiment shows that this way of proceeding is often not optimal in
the measurement where the values of the coefficients of the matrices of rigidity can evolve/move in way
important during a calculation in great displacements. The value of
K
determined initially
can then become too small and the matrix rigidity to end up being singular.
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Date
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Key
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6 Buckling
linear
Linear buckling is presented in the form of a particular case of the geometrical nonlinear problem. It
is based on the assumption of a linear dependence of the fields of displacements, deformations and
stresses compared to the level of load, and this, before the critical load is not reached.
In a total Lagrangian formulation, one recalls that linearized balance can be written under
variational form:
int
int
-
=
-
ext.
ext.
maybe in matric form after discretization:
(
)
U
U
U
F R
=
-
K
T
where dependence of the tangent matrix of rigidity
K
T
is nonlinear compared to the vector of
generalized nodal displacement
p
p
=
U
e=, Nel
E
1
.
If we suppose the linear dependence of displacement compared to the level of load, one can
to write:
U
U
=
0
where
U
0
is the solution obtained following a linear analysis for
=
1
by:
K U
F
0 0
=
where
K
0
is the tangent matrix of initial rigidity. One can then develop the tangent matrix of
rigidity in a linear way compared to the level of load:
(
)
U
E
Nel
T
E
E
U
E
E
=
=
+
+
+
1
0
,
….
K
K
K
K
where
K
U
E
is the matrix of initial displacements depending on
p
0
E
, traditionally neglected in
Code_Aster, and
K
E
the matrix of the initial stresses depending on the total tensor of the stresses
of Piola-Kirchhoff of second species
[]
S
0
and of the local vector
~
S
0
. These stresses are
voluntarily confused with the stresses of Cauchy. They are obtained by a postprocessing
linear analysis.
For the rotation part of
p
E
, the assumption of linearity of the deformations according to the level of
load results in the equality of:
N
N
I
I
=
who leads us to confuse the initial normals
N
I
with their transforms
N
I
.
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The matrix of the initial stresses
K
E
represent the constant part in
geometrical part
tangent matrix of rigidity. It is evaluated at the point
p
0
1
E
and
=
with a transform of
normal replaced by the initial normal:
K
K
K
E
E
E
=
+
conventional
nonconventional
with the conventional part of the geometrical part (see [bib4] volume 1 p. 141):
K
J
S J
J
E
conventional
T
NR
NR
D D D
=
















-
-
~
~
det
1
2
1
2
1
2
3
where the second matrix of derived from the functions of form becomes:
()
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
N
NR
N
NR
N
NR
N
NR
N
I
I
I
I
I
I
I
Z
I
y
I
Z
I
y
I
Z




=


























-
-
2
1
1
1
1
1
1
3
2
3
2
3
2
3
2
2
1
2
1
2
1
2
1
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
()
()
()
-
-
-
-
-
-
-




























NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
I
y
I
Z
I
X
I
Z
I
X
I
Z
I
X
I
y
I
X
I
y
I
X
I
y
I
X
2
3
2
3
2
3
2
3
2
2
2
3
2
3
2
3
2
3
2
2
2
1
1
2
2
1
1
2
2
0
0
0
0
0
0
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()




























=
-
-
-
-
-
-
LI
NB
H
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
NB
Z
NB
y
NB
Z
NB
y
NB
Z
y
NB
Z
NB
X
NB
Z
NB
X
1
1
2
0
0
0
0
0
3
2
2
3
2
2
3
2
2
3
2
2
2
2
12
3
2
2
3
2
2
3
2
2
3
2
2
1
1
2
2
1
1
2
2
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
NB
Z
NB
X
NB
y
NB
X
NB
y
NB
X
NB
y
NB
X
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
2
2
2
2
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
0
0
0
0
1
1
2
2
,
,
,
,
-
-
-




























































































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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
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Key
:
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HT-66/05/002/A
and the generalized tensor of the total stresses:
[] []
S
S
S
S
9 9
3 3
0
0
0
0
0
0
×
×
=






[
]
The nonconventional part of the geometrical part represents the terms uncoupled from rotation, not
symmetrical. Since the current algorithm of resolution of the problem to the eigenvalues [R5.06.01]
(
)
[
]
K
K
K
0
0
+
+
=
U
(
being the level of critical load) considers only matrices
symmetrical, one makes symmetrical, while dividing by two the sum with its transposed, the matrix:
()
[] []
(
)
K
Z
N
E
I
I
I I
nonconventional
,
=
×
×
1
2
where
N
I
is the normal with the node
I
and
Z
I
a vector
3 3
×
with the node
I
NB
=
1
2
,
nodal vector
(
)
3
3
2
× ×
NB
I
Z
:
Z
Z
I
I
I
NB
=
=




















.
.
.
.
.
.
,
1
2
The nodal vector
Z
I
is similar to a vector of internal force and its expression is:
Z
B S
J
I
T
D D D
=
~ ~ det
3
1
2
3
with the operator of the iterative variation of the virtual deformations (third operator of
deformations):
~
~
B
HQJ
3
1
3
=




-
NR
who is highlighted by the relation:
~ ~ det
~
~
B S
J
E
S
3
1
2
3
T
D D D
D
=
nonconventional
Note:
For numerical integration in the thickness of the various terms of rigidity, we retain one
diagram of Gauss at two points just like in elasticity for the geometrical linear hulls
[R3.07.04].
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HT-66/05/002/A
7
Establishment of the elements of hull in Code_Aster
7.1 Description
These elements (of names
MEC3QU9H
and
MEC3TR7H
) are pressed on meshs
QUAD9
and
TRIA7
who are
of geometry curves [R3.07.04].
7.2 Use
These elements are used in the following way:
AFFE_MODELE (MODELING: “COQUE_3D”…)
for the triangle and the quadrangle.
One calls upon the routine
INI080
for standard calculations of numerical integration.
AFFE_CARA_ELEM (HULL:(THICKNESS:“EP”
ANGL_REP
:
(
'
''
')
COEF_RIGI_DRZ
:
“CTOR”)
to introduce the characteristics of hull.
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 are defined
E
, Young modulus,
, coefficient of
Poisson,
, thermal expansion factor and
density.
AFFE_CHAR_MECA (DDL_IMPO: (
DX:. DY:. DZ:. DRX:. DRY:. DRZ:.
DDL of plate in the total reference mark.
FORCE_COQUE: (FX:. FY:. FZ:. MX:. MY:. MZ:. )
They are the surface efforts on elements of plate. These efforts can be given in
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 of hull in the total reference mark.

7.3
Calculation in geometrical nonlinear “elasticity”
Calculation imposes the following instructions user:
COMP_ELAS: (RELATION: “elas”
COQUE_NCOU: 1 (or more)
DEFORMATION: “green_gr”)
Numerical integration in the thickness is based on an approach multi-layer with 3 points
of integration by layer. It is about the approach currently used in nonlinear hardware
[R3.07.04]. Options of postprocessing
SIEF_ELNO_ELGA
stresses and
VARI_ELNO_ELGA
variables intern (here null) by defect are activated with the convergence of each filed pitch.
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Version
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
43/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
7.4 Establishment
Options
FULL_MECA
,
RIGI_MECA_TANG
, and
RAPH_MECA
are already active in the catalogs
elementary
mec3qu9h. cata
and
mec3tr7h. cata
for material non-linearity. They direct it
calculation towards
/extremely/te0414.f
, then towards
/extremely/vdxnlr.f
to calculate and store, inter alia,
stamp tangent symmetrical rigidity in the address corresponding to the mode
MMATUUR
PMATUUR
.
For the geometrical nonlinear analysis, the calculation of the tangent matrix of rigidity is directed towards
new routine
VDGNLR
. This matrix is not symmetrical and must be stored in the address
corresponding to the mode
MMATUNS PMATUNS
.
One defines the two local modes at the same time symmetrical and nonsymmetrical, at exit of the catalogs
elementary. The tangent matrix of nonsymmetrical rigidity into nonlinear geometrical is stored
with the address reserved for a nonsymmetrical matrix. On the other hand, if it is about nonmaterial linearity in
small deformations, all the tangent matrix of rigidity is stored with the address corresponding to
nonsymmetrical mode. The lower triangular part is duplicated starting from the triangular part
higher. Material nonlinear calculation in small deformations thus proceeds also in not
symmetrical.
7.4.1 Amendment
TE0414
Calculation is directed towards
/extremely/vdgnlr.f
when the type of behavior
COMP_ELAS
is checked,
i.e. when the problem is nonlinear geometrical.
7.4.2 Addition of a routine
VDGNLR
According to the option, the routine
/extremely/vdgnlr.f
has as a role of
:
Options:
FULL_MECA
and
RAPH_MECA
:
To calculate the 6 components of the state of the local stresses of Cauchy (confused with the state of
stresses of Piola-Kirchhoff of second species) at the points of normal numerical integration and it
nodal vector of the internal forces. They are stored in the local modes
ECONTPG PCONTPR
and
MVECTUR PVECTUR
respectively.
Options:
FULL_MECA
and
RIGI_MECA_TANG
:
To calculate and store the tangent matrix of nonsymmetrical rigidity in the mode
MMATUNS PMATUNS
.
7.5
Calculation in linear buckling
The option
RIGI_MECA_GE
, inactive until now, is activated in the elementary catalogs
mec3qu9h. cata
and
mec3tr7h. cata
.
The new one
TE0402
is dedicated to the calculation of the matrix of geometrical rigidity due to the stresses
initial for the buckling of Euler. One recovers the plane states of the local stresses of Cauchy
(component
S
N
null) at the points of normal numerical integration of Gauss. These states of
stresses must be obtained by postprocessing with the option of calculation
SIEF_ELGA_DEPL
following
a linear analysis (mode
ECONTPG PCONTRR
).
In analysis of buckling of Euler, stresses
[]
of Cauchy can be confused with
stresses
[]
S
of Piola-Kirchhoff of second species. Therefore we will keep
notation
[]
S
.
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
The matrix of rigidity of the initial stresses can be broken up into a conventional part
symmetrical and a nonsymmetrical nonconventional part. First is calculated according to
total tensor of the stresses
3 3
×
, contrary to the second which, it, is calculated according to
vector of the local stresses
5 1
×
.
Since the current algorithm of resolution of the problem to the eigenvalues [R5.06.01] does not consider
that symmetrical matrices, we force the symmetry of the nonconventional part of the matrix
geometrical before storing the higher triangular part of all the matrix in the mode
MMATUUR PMATUUR
.

8 Conclusion
The formulation that we have just described applies to the curved mean structural analyzes in
great displacements, whose thickness report/ratio over characteristic length is lower than 1/10. It
comes in direct object from the finite elements described in the preceding reference material
[R3.07.04] and used within the framework of small displacements and deformations. They rest on
same meshs quadrangle and triangle.
Their formulation rests on an approach of continuous medium 3D into which one introduces one
kinematics of hull of the Hencky-Mindlin-Naghdi type, in plane stresses, in the formulation
weak of balance. The measurement of the deformations retained is that of Green-Lagrange,
combined énergétiquement with the stresses of Piola-Kirchhoff of second species. The formulation
balance is thus Lagrangian total. The transverse distortion is treated same manner
that in [R3.07.04]. Rotations must remain lower than 2
because of the choice of update of
great rotations established in Code_Aster for which there is not bijection between the vector of
full slewing and the orthogonal matrix of rotation.
Because of singularity of the tangent matrix of rigidity compared to the component of rotation
around the transform of the normal, one defines a fictitious deformation energy associated this
rotation. With this rotation, one associates a rigidity of constant torsion. Interior efforts associated
this potential energy are taken into account. This potential energy, nonnull, does not correspond
with a physical deformation. One thus needs that it remains negligible, which the user can control in
imposing a value of
COEF_RIGI_DRZ
being worth of 10
­ 3
to 10
­ 5
.
For the postprocessing of the stresses, one limits oneself to the framework of the small deformations. One then could
to prove the identity enters the tensor of the stresses of Piola-Kirchhoff observed the initial geometry
and the tensor of the stresses of Cauchy in the deformed geometry. Moreover, the state of the stresses
being plane for the tensor of Piola-Kirchhoff, one finds this property for the state of stresses of
Cauchy. It should be noted that in more general contexts, this property is not preserved.
Linear buckling is treated like a particular case of the geometrical nonlinear problem. It
rest on the assumption of a linear dependence of the fields of displacements, deformations and
stresses compared to the level of load, before the critical load. It results from it that one can
to linearly develop the tangent matrix of rigidity compared to the level of load. One finds
then the geometrical part of the matrices of nonlinear the geometrical General obtained while identifying
deformed normal on the average surface and the initial normal, which is coherent with
linearity of the deformations according to the level of load.
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
9 Bibliography
[1]
Al Mikdad Mr., “Statics and Dynamics of the Beams in Great Rotations and Resolution of
Problems of Nonlinear Instability “, thesis of doctorate, University of Technology of
Compiegne, 1998.
[2]
Bathe K.J., “Finite Element Proceedings in Analysis Engineering”, Prentice Hall, New Jersey,
1982.
[3]
Cardonna A., “Year Integrated Approach to Mechanism Analysis”, thesis of doctorate,
University of Liege, 1989.
[4]
Crisfield Mr. A., “Non-linear Finite Element Analysis off Solids and Structures”, Volume 1:
Essentials, John Wiley, Chichester, 1994.
[5]
Crisfield Mr. A., “Non-linear Finite Element Analysis off Solids and Structures”, Volume 2:
Advanced topics, John Wiley, Chichester, 1994.
[6]
Jettor pH., “
Nonlinear kinematics of the Hulls
“, report/ratio SAMTECH, contract
PP/GC-134/96, 1998.
[7]
Simo J.C., “
The (symmetric) Hessian
for Geometrically Nonlinear Models in Solid
Mechanics: Intrinsic Definition and Geometric Interpretation ", comp. Methods Appl. Mech. 96
: 189-200, 1992.
[8]
Vautier I., “Implemented of
STAT_NON_LINE
“, manual of Code_Aster development
[D9.05.01].
[9]
Massin P., “Elements of plate DKT, DST, DKQ, DSQ and Q4g”, Manual of Reference of
Code_Aster [R3.07.03].
[10]
Massin P., Laulusa A., Al Mikdad Mr., Bui D., Voldoire F., “Numerical Modeling of
Voluminal hulls “, Manual of Reference of Code_Aster [R3.07.04].
[11]
Lorentz E., “Relation of Nonlinear elastic behavior”, Manual of Reference of
Code_Aster [R5.03.20].
[12]
Jacquart G., “Method of Ritz in linear and nonlinear dynamics”, Manual of Reference
of Code_Aster [R5.06.01].
[13]
Aufaure Mr., “Static and Dynamic Modeling of the Beams in Great Rotations”,
Manual of Reference of Code_Aster [R5.03.40].
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Code_Aster
®
Version
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
46/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
Appendix 1: Flow chart of calculation in linear buckling
Local reference marks with the NB2 nodes
[
]
T
T
N
1
2
:
:
I

Loop on the points of normal numerical integration of Gauss
·
recovery of the vector of the local stresses
~
~
~
~
~
~
~
~
~
~
~
S
=














=














S
S
S
S
S
S
S
T T
T T
T T
T N
T N
T T
T T
T T
T N
T N
1 1
2 2
1 2
1
2
1 1
2 2
1 2
1
2
2
2
2
starting from the 6 components tensors stored in mode PCONTRR
~
~
~
~
~
S
S
S
S
S
T T
T T
T T
T N
T N
1 1
2 2
1 2
1
2
0














·
formation of the symmetrical tensor
3 3
×
local stresses
[~]
S
·
construction of the matrix of transformation
(
)
(
)
(
)
(
)
(
) ()
P
T
T
T
T
N


1
2
3
1
1
2
3
2
1
2
3
3
1
2
3
3 1 2 3
1 2
,
,
,
,
=






=
T
T
T
where
·
calculation of the symmetrical tensor
3 3
×
total stresses
[]
[~]
S
P S P
T
=
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Code_Aster
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
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X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
·
for the nonconventional term, calculation of
HQ
=




×
×
[
]
[
]
HSFM
HSS
3 9
2 9
()
()
()
()
()
()
() ()
() ()
() ()
()
()
()
(
)
()
()
() ()
() ()
() ()
()
()
()
()
()
()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
HQ
=
+
+
+
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
1
2
1
2
1
2
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
2
3
2
3
2
3
3
3
3
3
3
2
3
2
3
2
3
2
3
3
2
2
3
3
2
2
3
3
3
1
2
3
1
2
2
3
3
1
1
2
3
1
2
2
3
3
1
1
2
3
1
2
2
3
3
1
2
1
1
2
2
2
2
3
3
1
2
1
2
2
3
2
3
3
1
3
1
2
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
3
1
3
1
3
1
3
1
1
3
3
1
1
3
3
1
1
3
1
1
2
2
2
2
3
3
1
2
1
2
2
3
2
3
3
1
3
1
1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1
0 1 0 1 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 1 0
+
+
+
























·
calculation of the opposite matrix jacobienne
J
-
1
and of the determinant
det J
·
calculation of
~
J
-




1
3
NR
with:
()
()
()
()
()
()
()
()
()
~
;
,
,
,
,
,
,
J
J
J
J
-
-
-
-
=








=


























1
1
1
1
3
3
2
3
2
2
3
2
3
2
2
3
2
3
2
2
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
1
2
1
2
NR
H
NR
NR
NR
NR
NR
NR
NR
NR
NR
I
I
I
I
I
I
I
I
I
L
LI
NB
=




























1
2
,
·
calculation of the third operator of the deformations
~
~
B
HQJ
3
1
3
=




-
NR
·
calculation and numerical integration
Z
B S
J
I
T
D D D
=
~ ~ det
3
1
2
3
·
calculation of the generalized tensor of the total stresses
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Code_Aster
®
Version
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
[] []
S
S
S
S
9 9
3 3
0
0
0
0
0
0
×
×
=






[
]
·
calculation of
~
J
-




1
2
NR
with
:
()
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
N
NR
N
NR
N
NR
N
NR
N
I
I
I
I
I
I
I
Z
I
y
I
Z
I
y
Z




=


























-
-
2
1
1
1
1
1
1
3
2
3
2
3
2
3
2
12
1
2
1
2
1
2
1
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
()
()
()
-
-
-
-
-
-
-




























NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
y
I
Z
I
X
I
Z
I
X
Z
X
I
y
I
X
I
y
I
X
y
X
12
3
2
3
2
3
2
3
2
12
12
3
2
3
2
3
2
3
2
12
12
1
1
2
2
1
1
2
2
0
0
0
0
0
0
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()




























=
-
-
-
-
-
-
LI
NB
H
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
NB
Z
NB
y
NB
Z
NB
y
NB
Z
y
NB
Z
NB
X
NB
Z
NB
X
1
1
2
0
0
0
0
0
3
2
2
3
2
2
3
2
2
3
2
2
2
2
12
3
2
2
3
2
2
3
2
2
3
2
2
1
1
2
2
1
1
2
2
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
NB
Z
NB
X
NB
y
NB
X
NB
y
NB
X
NB
y
NB
X
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
2
2
2
2
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
0
0
0
0
1
1
2
2
,
,
,
,
-
-
-




























































































·
calculation and numerical integration of the matrix of geometrical rigidity conventional
K
J
S J
J
E
T
NR
NR
D D D
conventional
=
















-
-
~
~
det
1
2
1
2
1
2
3
Fine loops on the points of integration
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Voluminal elements of hulls into nonlinear geometrical
Date
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X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
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:
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R3.07 booklet: Machine elements on average surface
HT-66/05/002/A

Loop on all the nodes of Lagrange with distinction of the super node
·
calculation of
[]
Z
I
×
knowing that
Z
Z
I
I
I
NB
=
=
















.
.
.
.
.
.
,
1
2
·
calculation of the vector normal
N
I
and of its antisymmetric matrix
[]
N
I
×
·
calculation of the nonconventional matrix of geometrical rigidity
()
[] []
K
Z
N
E
I
I
I I
I
NB
3 3
1
2
×
=
×
×
=
nonconventional
,
,
·
addition of
()
K
E
I I
3 3
×
nonconventional
,
with distinction of the super node

Fine loops on the nodes
Storage of the higher triangular part of
K
E

END

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Voluminal elements of hulls into nonlinear geometrical
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X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
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R3.07.05-B
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HT-66/05/002/A
Appendix 2: Flow chart of geometrical nonlinear calculation
Local reference marks with the NB2 nodes
[
]
T
T
N
1
2
:
:
I

Beginning Loops JN on the NB2 nodes

IF JN
NB1
·
recovery of the vector of total displacement already updated by MAJOUR:
()
(
)
(
)
(
)
U II
ZR IDEPLP IDEPLM
JN
II
II
JN
NB
I
=
+
- +
- +
=
=
1 6
1
1 3
1
1
*
;
,
;
,
·
recovery of the vector of full slewing already updated by MAJOUR
()
(
)
(
)
(
)
I
II
ZR IDEPLP
JN
II
II
JN
NB
=
- +
- + +
=
=
1 6
1
3
1 3
1
1
*
;
,
;
,
ELSE JN
·
recovery of the vector of full slewing
()
(
)
(
)
I
II
ZR IDEPLP
NB
II
II
JN
NB
=
- +
+
=
=
1 6
1
1 3
2
*
;
,
;

END IF JN
·
calculation of the matrix of rotation
[]
I
I
=
×
exp
·
transform of the normal
N
N
I
I
I
=

Fine Loops on the NB2 nodes
Calculation of
p
E
X
X
y
y
Z
Z I
X
X
y
y
Z
Z NB
U
v
W
N
N
N
N
N
N
I
NB
N
N
N
N
N
N
=
-
-
-












=
-
-
-




































M
M
M


1
1
2
,
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Voluminal elements of hulls into nonlinear geometrical
Date
:
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Key
:
R3.07.05-B
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R3.07 booklet: Machine elements on average surface
HT-66/05/002/A

Beginning Loops INTSR on the points of normal reduced integration of Gauss
·
construction of part of the operators
~, ~
B
B
1
2
with the J = 1, INTSR points of integrations to be able to extrapolate them

Fine INTSR Loops on the points of normal reduced integration of Gauss

Beginning Loops INTSN on the points of normal numerical integration of Gauss
·
construction of the matrix of transformation:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
P
T
T
T
T
T
T


1 2
3
1 1 2
3
2 1 2
3
3 1 2
3
1
1 2
3
2
1 2
3
3
1 2
3
,
,
,
,
,
,
,
=
<
>
<
>
<
>






=






T
T
T
where
(
) (
)
T
N
3 1 2
3
1 2
,
,
=
·
calculation of the opposite matrix jacobienne
J
-
1
and of the determinant
det J
·
calculation of
~
J
J
J
J
-
-
-
-
=




1
1
1
1
0
0
0
0
0
0
·
calculation of the second matrix of derived from the functions of form
NR




1
background image
Code_Aster
®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
52/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
()
()
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
NR
NR
NR
NR
NR
I
I
I
I
I
I
I
I
I
I
I
I




=


























1
1
1
1
1
1
1
3
2
3
2
2
3
2
3
2
2
1
2
1
2
1
2
1
2
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
0
0
0
0
0
0
1
1
2
0
0
0
0
0
0
0
0
0
0
0
3
2
3
2
2
3
2
2
3
2
2
2
2
3
2
2
3
2
2
1
2
1
2
1
2
NR
NR
NR
I
NB
H
NR
NR
NR
NR
NR
I
I
I
NB
NB
NB
NB
NB
,
,
,
,
,
,
,
























































=
L
()
()
()
()
NR
NR
NR
NR
NB
NB
NB
NB
2
2
3
2
2
3
2
2
2
2
0
0
0
0
0
0
0
1
2
,
,


























































































·
calculation of
U
J
p
X
NR
E
=




-
~
1
1
·
calculation of
With U
X
U
v
W
U
v
W
U
v
W
U
U
v
v
W
W
U
U
v
v
W
W
U
U
v
v
W
W
X
X
X
y
y
y
Z
Z
Z
y
X
y
X
y
X
Z
X
Z
X
Z
X
Z
y
Z
y
Z
y




=














,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
·
calculation of
H Q
With U
+








1
2
X
background image
Code_Aster
®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
53/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
()
()
()
()
()
()
() ()
() ()
() ()
()
()
()
(
)
()
()
() ()
() ()
() ()
()
()
()
()
()
()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
H
=
+
+
+
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
1
2
1
2
1
2
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
2
3
2
3
2
3
3
3
3
3
3
2
3
2
3
2
3
2
3
3
2
2
3
3
2
2
3
3
3
1
2
3
1
2
2
3
3
1
1
2
3
1
2
2
3
3
1
1
2
3
1
2
2
3
3
1
2
1
1
2
2
2
2
3
3
1
2
1
2
2
3
2
3
3
1
3
1
2
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
() ()
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
3
1
3
1
3
1
3
1
1
3
3
1
1
3
3
1
1
3
1
1
2
2
2
2
3
3
1
2
1
2
2
3
2
3
3
1
3
1
+
+
+












·
calculation of the first operator of the deformations
~
~
B
H Q
With U J
1
1
1
1
2
=
+












-
X
NR
·
calculation of the vector of the local deformations
~
~
E B p
=
1 E
·
calculation of the second matrix of derived from the functions of form
NR




2
()
()
()
()
()
()
()
()
()
()
NR
NR
NR
NR
NR
NR
NR
H
NR
N
NR
N
NR
N
NR
N
NR
I
I
I
I
I
I
I
Z
I
y
I
Z
I
y




=


























-
-
2
1
1
1
1
1
1
3
2
3
2
3
2
3
2
1
2
1
2
1
2
1
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
L
,
,
,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
()
()
()
()
12
12
3
2
3
2
3
2
3
2
12
12
3
2
3
2
3
2
3
2
12
12
1
1
2
2
1
1
2
2
0
0
0
0
0
0
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
Z
y
I
Z
I
X
I
Z
I
X
Z
X
I
y
I
X
I
y
I
X
y
X
-
-
-
-
-
-
-














,
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()










































=
-
-
-
-
LI
NB
H
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NB
Z
NB
y
NB
Z
NB
y
NB
Z
NB
y
NB
Z
NB
X
1
1
2
0
0
0
0
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
3
2
2
3
2
2
1
1
2
2
1
1
,
,
,
,
,
,
,
()
()
()
()
()
()
()
()
()
()
-
-
-
-
-

































3
2
2
3
2
2
2
2
2
2
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
2
2
1
1
2
2
0
0
0
0
0
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NR
N
NB
Z
NB
X
NB
Z
NB
X
NB
y
NB
X
NB
y
NB
X
NB
y
NB
X
,
,
,
,
,
,


























































background image
Code_Aster
®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
54/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A
·
calculation of the second operator of the deformations
~
~
B
H Q With U J
2
1
2
=
+












-
X
NR
·
calculation and numerical integration
R
B S
J
E
T
D D D
=
~ ~ det
2
1
2
3
·
calculation of the matrix of behavior
D
·
calculation and numerical integration
K
B DB
J
m
E
T
D D D
=
~
~
det
2
2
1
2
3
·
construction of the symmetrical tensor
3 3
×
local stresses
[~]
S
·
calculation of the symmetrical tensor
3 3
×
total stresses
[]
~
S
P [S] P
T
=
·
calculation of
[]
~
~
B
H S J
3
1
3
T
NR
=




-
calculation and numerical integration
Z
B S
J
I
T
D D D
=
~ ~ det
3
1
2
3
·
calculation of the generalized tensor of the total stresses
[] []
S
S
S
S
9 9
3 3
0
0
0
0
0
0
×
×
=






[
]
·
calculation and numerical integration of conventional rigidity
K
J
S J
J
G
E
T
NR
NR
D D D
conventional
=
















-
-
~
~
det
1
2
1
2
1
2
3

Fine INTSN loops on the points of integration
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Code_Aster
®
Version
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Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
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Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A

Beginning Loops JN on the NB2 nodes
·
calculation of
[]
Z
I
×
knowing that
Z
Z
I
I
I
NB
=
=




















.
.
.
.
.
.
,
1
2
·
calculation of
[]
N
I
×
·
calculation of the nonsymmetrical matrix
()
[] []
K
Z
N
G
E
I
I
I I
3 3
×
=
×
×
nonconventional
,


IF JN
NB1
·
addition of
()
K
G
E
I I
3 3
×
nonconventional
,
with distinction of the extra-node
ELSE JN
·
assignment of
()
K
G
E
I I
3 3
×
nonconventional
,
with distinction of the extra-node

END IF JN
Storage of all the nonsymmetrical matrix
K
and
background image
Code_Aster
®
Version
7.4
Titrate:
Voluminal elements of hulls into nonlinear geometrical
Date
:
05/04/05
Author (S):
X. DESROCHES, P. MASSIN, Mr. Al MIKDAD
Key
:
R3.07.05-B
Page
:
56/56
Manual of Reference
R3.07 booklet: Machine elements on average surface
HT-66/05/002/A


























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