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Axisymmetric thermoelastic hulls and 1D
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Organization (S): EDF/MTI/MN, RNE/AMV
Handbook of Référence
R3.07 booklet: Machine elements on average surface
R3.07.02 document
Numerical modeling of the mean structures:
axisymmetric thermoelastoplastic hulls
and 1D
Summary:
One presents a numerical formulation for the modeling of the structures at average surface of geometry
particular:
· hulls with symmetry of revolution around the axis 0y,
· invariant hulls with unspecified section along the axis 0z.
One describes the isotropic thermoelastoplastic case completely, within the framework of the theories of
LOVE-KIRCHHOFF and of HENCKY-MINDLIN-REISSNER, as well as the various studied loadings, for
the selected isoparametric finite element.
The examples of validation suggested show qualities of the finite element.
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Axisymmetric thermoelastic hulls and 1D
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Contents
1 Introduction ............................................................................................................................................ 3
2 continuous Problem ................................................................................................................................... 3
2.1 Description of the geometry, kinematics .............................................................................. 4
2.2 Thermoelastoplastic balance ..................................................................................................... 8
3 Formulation of the finite element. Discretization ........................................................................................ 13
3.1 Description of the selected finite element ................................................................................................ 13
3.1.1 Motivations ........................................................................................................................... 13
3.1.2 General presentation of the element ...................................................................................... 14
3.1.3 Transformations finite element/finite element of reference ..................................................... 14
3.1.4 Surface numerical integration ......................................................................................... 15
3.1.5 Numerical integration in the thickness ............................................................................... 16
3.2 Formulation of the elementary terms ........................................................................................... 17
3.2.1 Mass, center of gravity, stamps inertia .......................................................................... 17
3.2.2 Stamp of mass ................................................................................................................. 18
3.2.3 Second member of centrifugal force .................................................................................... 19
3.2.4 Second member of gravity ............................................................................................. 19
3.2.5 Second member of distributed loads ................................................................................. 19
3.3 Calculation of the strains and the stresses .................................................................................. 20
4 Validation - Cas test ............................................................................................................................. 21
4.1 Roll under pressure interns ...................................................................................................... 21
4.2 Plate circular embedded under uniform pressure [V3.03.100] ................................................. 25
4.3 Axisymmetric modal analysis of a thin spherical envelope [V2.03.007] ............................ 30
5 Conclusion ........................................................................................................................................... 31
6 Bibliography ........................................................................................................................................ 32
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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1 Introduction
One is interested in what follows to the mechanical modeling of mean structures to average surface
of particular geometry:
· hulls with symmetry of revolution around the axis 0y,
· hulls with invariant unspecified sections along the axis 0z.
More particularly, one limits oneself if mechanical parameters (materials, loadings)
are independent of a direction of space (the circumference for the hulls of revolution, the axis 0z
for hulls C_PLAN and D_PLAN).
For the resolution of chained thermomechanical problems, one must use before the finite element
of thermal hull describes in [R3.11.01] according to case's in its axisymmetric version, or its version
plane invariant according to 0z.
One gives hereafter first of all a progress report on the description of the mechanical model: kinematics, law of
thermoelastoplastic behavior. Then one presents the selected finite element, the interpolation and
method of integration.
One gives finally some numerical results of application, by comparison with solutions
analytical.
2 Problem
continuous
The geometry is defined in a unidimensional way:
· by the meridian line in the plan (0xy) for a hull of revolution,
· by the section of the hull in the plan (0xy) for an invariant hull in Z.
In this last case, by analogy with the two-dimensional problems, one considers two cases:
· the case “forced plane”, i.e. that of a free hull according to the direction 0z, or
that of an arc in the 0xy plan,
· the case “plane deformations”, i.e. when displacements according to the direction 0z
are null.
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Axisymmetric thermoelastic hulls and 1D
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2.1
Description of the geometry, kinematics
One considers a hull of revolution of axis 0y, or an invariant hull according to the axis 0z. For all
two, average surface is defined by the curve = AB in the 0xy plan: is a meridian line
for the hull of revolution, or the section for the invariant hull according to 0z.
y
O
X
Z
Appear 2.1-a: Coque of revolution
y
T
B
N
m

!!

E
S
y O
·
E
X
With
Z
ex
Appear Meridian 2.1-b:
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Key:
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y
O
X
Z
Appear 2.1-c: Hull with invariant section according to 0z

The curve = AB is parameterized by the curvilinear X-coordinate S. One will note the derivative partial S
by: , S.
In a point m of one defines the local reference mark (N, T, ez) by:
Om
T =
, S; N T = ez.
Om, S
One notes also the angle such as:
N = cos E


X + sin E Y.
The curvature of is defined by:
1 = - n.t =
R
, S
, S
In the case of the hull of revolution, the position on the parallel passing by m is noted.
tangent vector on this parallel is E. For the meridian line located in the 0xy plan, = 0 and
E
= - E


Z. The radius of curvature of the parallel in m is:
R
R =
where R is X-coordinate X of the point m of.
cos
On the other hand, for an invariant hull according to Z this parallel is a right generator, directed according to ez,
of null curvature.
The transformations kinematics of the hull are defined by displacement U of the point m of
surface average, as by rotation S of normal N at the point Mr. the vector U can be
expressed in local base:
U () = U () .T () + (
W) .n
S
S
S
S
(S).
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Axisymmetric thermoelastic hulls and 1D
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Or in Cartesian base:
U () = U (S) E + U (S) E
S
X
X
y
y.
The deformations of the hull associated with this transformation (U, S) are determined by:
· a membrane tensor of deformation E,
· a tensor of variation of curvature K,
· a vector of deformation of distortion tranverse.
This last appears in the theory of hulls of HENCKY-MINDLIN-NAGHDI and not in that of
LOVE. According to displacement U and rotation S, these sizes are expressed (cf [bib1]):
Case
Hull of revolution
Invariant hull according to 0z
U expressed in
W
W
base local
E = U
S
S +
,
E = U +
R
S
, S
R
(N, T, ez)
1
E
= (- U sin + W cos)
R
Kss = S
, S
Kss = S
, S
sin
K = -

R
S
U
U
S = S + Ws -
,
S = + W -
R
S
, S
R
U expressed in
E = U
S
y, S cos - ux, S sin
E = U
S
y, S cos - ux, S sin
base total
(
U
E, E
, E
E
X
=
X
y
Z
)


R
Kss = S
, S
Kss = S
, S
sin
K = -

R
S
S = S + ux, S cos + uy, S sin
S = S + ux, S cos + uy, S sin
Note:
The change of direction of the curvilinear X-coordinate S does not modify the values of:
S, Ess
E
,
,
but the sign changes of,
U,
W,
R,
K,
K
S
.
Under the framework of the theory of LOVE, the condition S = the 0 (normals with the hull remain it afterwards
deformation) results in a direct relation between rotations S and the slope W, S. Les
components of the tensor variation of curvature are according to displacement in the local base:
U
R
K = - W
, S
+
- U
, S
S
S

,
R
R2
sin
U
K

=
W

-
R
, S



R
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Axisymmetric thermoelastic hulls and 1D
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Key:
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If displacement is expressed in total base:
1
K
=
U

sin - U
cos - U

cos
- U
S

sin
R (X, S
y, S
) X, S
y, S
sin
K =
(U cos + U
X S
y
S sin
,
,
)
R
It is noticed that the expression of the variations of curvature according to displacement in theory of
LOVE are rather complicated and that it utilizes derivative second. If one is required
interpolation conforms i.e. here C1, this requires the use of finite elements of high degree.
The tensors E, K
,
allow to express the three-dimensional deformation in the thickness.
H H
On [Figure 2.1-d], one indicates by x3 the position in the thickness -,
compared to fiber

2 2
average, at the point m, of curvilinear X-coordinate S on.
S
!!

T
x3
N
!!

m
H
+
2
R
H
- 2

Appear 2.1-d
In a point thickness, displacement is expressed in total reference mark:
U (S, X)
3
= (U (S)
-
(S) .x
sin
3
(S)
) .e + (
U (S)
+ (S) .x cos
3
(S
X
S
X
y
S
)).ey
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Axisymmetric thermoelastic hulls and 1D
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In order to take account of the variation of metric in the thickness (due to the curvature of surface
average), one defines the functions:
X
X

3
3
S (x3) = 1 +
; (x3) = 1 +
.cos
R
R
For a sufficiently thin hull, this correction is negligible:
S
1


;


1
In practice this correction carried out if MODI_METRIQUE:“OUI” in AFFE_CARA_ELEM
[U4.42.01] is useless if the reports/ratios H R and H R, when they exist, are lower than 1

15.
In theory of HENCKY-MINDLIN-NAGHDI, the components of the tensor of deformation are:

1

S (S, x3) =
(Es + X K
3 S)
S


1

(S, x3) =
(E + x3
K)

(only in the case hull of revolution)



1
sx
=
3 (S, x3)


2
S

S
2.2 Balance
thermoelastoplastic
It is considered that the material constitutive of the hull is thermoelastoplastic isotropic. One makes
the usually allowed assumption that the transverse normal constraint is null: X X 0. The law of
3 3
behavior most general is written then:

HT



11
11
C 11
11
C 22
0
11 - 11





HT
22 = C2211
C2222
0
22 - 22




1
0
0
X

11
C X X 1

3
3 3
3
X
where C (, µ) of Cijkl components is the local matrix of behavior in plane constraints and µ
represent the whole of the internal variables when the behavior is nonlinear. In the continuation
index 1 makes reference to the curvilinear X-coordinate and 2 with or Z. With the three-dimensional deformations
defined above, one associates the components of the tensor then forced:
· in the case of a hull of revolution:

=

HT
HT
S
Cssss (S -
+
S)
Cs (-
)


=

HT
HT


C S (S - S +
)

C
(-)

sx = C
ssx
X sx
3
3 3
3
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· in the case hull invariant according to direction Z and free in Z (“forced plane”):

C
C


sszz zzss
HT
S = (Cssss -
) (S - S)
Czzzz


zz = 0

sx = C
ssx
X sx
3
3 3
3

· in the case hull invariant according to direction Z and blocked in Z (“plane deformations”):

=

HT
S
Cssss (S - S)


=

HT
zz
Czzss (S - S)

sx = Cssx X sx
3
3 3
3
One draws the expression from it from the elastic energy of deformation, which one will deduce the matrix from rigidity in
function of the kinematics of hull seen in the paragraph [§2.1]:
· in the case hull of revolution:
1
2 H

/2
W él =
2
2



(
)




2
2





1.


2

0
-/2 [Cssss S + C
+ Cs + C S S
+ Cssx X sx
+
- rdsd dx
H
3 3
3] (S
)
3

· in the case invariant hull according to Z, in “plane constraints”:
H/2
1

C
C

=

W él
sszz zzss

(C


2
2
ssss -
) S + 2Cssx X
.dsdx

3 3
sx
S
3
2
C
3

- 2
zzzz
H/

· in the case invariant hull according to Z, in “plane deformations”:
H
1
/2
W él =
2




2
2

.
2

-/2 [Cssss S + Cssx X
3 3
sx
S dsdx3
H
3]
Note:
In thermoelasticity, if one notes E the modulus YOUNG and the Poisson's ratio, one a:
E
E
E
C
=
; C
=
I (, J
iiii
iijj
) {1,}
2; C
=
1 - v2
1 - v
X X
2
11 3 3
1 + v
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The following sizes are defined:
· the membrane rigidity of a hull of revolution:
[
H/
2 + - 1
C
C
C]
S
ssss
S
=
.
dx
ij

; who is worth:



C
C

3

- H/2
I
J

S


Eh 1


1
2
-
in elasticity and absence of correction of metric in the thickness;
1


· the rigidity of coupling membrane-inflection of a hull of revolution:
H
[
2
+ - 1 C
C
B] =
X
S
ssss
S
.
.
dx
ij
3


, which is null in elasticity and in
H

C
C
3


I
J
-

S


2
absence of correction of metric in the thickness;
· the rigidity of inflection of a hull of revolution:
H
[
2
+ - 1 C
C
D]
=
x2
S
ssss
S
.
.
dx
ij
3


, which is worth:
H

C
C
3


I
J
-

S


2
Eh3
1
2

12 1
(-)
in elasticity and absence of correction of metric in the thickness;
1


· the transverse rigidity of distortion of a hull of revolution:
H
2
+ - 1
G
S

=


. C
dx
sx
, which is worth:
H
ssx X
3
3
2
3 3

-
S
2
Eh
1
+ in elasticity and absence of correction of metric in the thickness.
For an invariant hull according to direction Z, one considers in these expressions only the terms
ij = S; moreover one must replace there (S + -)
1 by S: the coefficients thus are defined
C D, B D
, D D
C
C
C
S
S
S
and C, B
, D
S
S
S
for the case, respectively, plane deformations or of
plane constraints. In elasticity, the coefficients DC, B C
, Cd.
S
S
S
, are the products of the coefficients
C D, B D
, D D
S
S
S
by 1
2
-. Lastly, the coefficient of transverse rigidity of distortion Gsx is
3
identical for three modelings to the correction of metric near.
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Axisymmetric thermoelastic hulls and 1D
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One can thus express elastic energy according to the tensors of deformations of hull:
E, K
,
by:
· for a hull of revolution:
1
2
W él =
2

2
2
2
2
2
2 0 [Cs Es + Bss Es Kss + Dss Kss + C E
+ B E
K
+ D K


+ 2 (Cs E
S.E. + Bs (E

S K + E
K
S) + Ds K
S.K)
Gsx3 2
+
S R. ds.d
2


· for an invariant hull according to Z in “plane constraints”:
1
Gsx

W él
C
2
C
C

=
C

2
3
2
S.E. S + 2Bss E S. Kss + Dss Kss +
S. ds
2
2

· for an invariant hull according to Z in “plane deformations”:
1
Gsx

W él
D
2
D
D

=
C

2
3
2
S.E. S + 2Bss E S. Kss + Dss Kss +
S. ds
2
2

For these expressions, it is necessary to add the potential associated with the thermal stresses, which will be one
contribution to the second member (whom one will express below in total reference mark):
· in the case hull of revolution:
2 H

/2
(
Lth
ref.
) =








T - T
C
+ C
+ C
+ C
rd dx ds
V
3
0
- H/[(
) (ssss S) S (S
))]

2
expression which for an isotropic elastic behavior becomes:
2
H
/2
sin

HT
E
ref. vx


(
L



3


V) =


sin
cos


(T - T) - vx, S
+ vy, S
+ X
S, S -
S
rd dx ds
1 -

R



R


3

0
- H/2


· in the case invariant according to Z in “plane constraints”:
H
/2



HT
ref.
CsszzCzzss
(
L

V) =

(T - T) Cssss -
S dx ds


C
3


-/2
zzzz

H

expression which for an isotropic elastic behavior becomes:
H/2
HT

ref.
(
L V) =

sin cos 3



E T - T
- v,
+ v,
+ X
dx ds
3
-
,
H/2 [
(
) (X S
y S
S S)]
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· in the case invariant according to Z in “plane deformations”:
H

/2
Lth
ref.
(
=

T - T
C
dx ds
V)


3
-/2 [(
) ssss S
H
]
expression which for an isotropic elastic behavior becomes:
H

/2
HT

E
ref.

(
L


3
V) =

(T - T)
1
(- vx, ssin +vy, scos + X S, S) dx
ds


3

-

/2 -
H


In these three expressions, one deliberately neglected the correction of metric in the thickness
(terms in S, seen for rigidity). Moreover the temperature T which appears is defined by
thermal model of hull with three fields (cf [R3.11.01]):

X 2

X
X
X
X
T (S X) = T m (S)
3
1

T S (S) 3
3
1
T I
,
.


(S) 3

1 3
3


H +
+
2h
H +
-
+
2h
H




From the whole of these expressions, one deduces the tensors from generalized efforts NR and M (efforts
normal and bending moments) associated the generalized deformations E and K by the principle of
virtual work. They are related to the tensor of the three-dimensional constraints by:
H/2
NR
=
dx

- H
/
3
2
H/2
M
=
X.
3
dx

- H

/
3
2
(where one neglected the variations of metric in the thickness).
Note:
Transverse energy of shearing
The model of hull presented above, said HENCKY-MINDLIN-NAGHDI, rests on one
kinematic assumption: the parameters W and S indicate the normal displacement of the point
m of average surface and the rotation of normal vector N.
One also frequently finds the model known as of REISSNER which rests on an assumption
statics of the distribution of stresses shear transverse. Parameters
kinematics deduced W and S in this model are weighted averages in
the thickness of normal displacement and local rotations. If one wishes to place oneself in it
tally, it is enough to affect the coefficient = 5/6 at the end of transverse energy of shearing
(in 2s). (cf [bib7], [bib9]).
Lastly, if one wants, for a thin hull, to be located within the framework of the model of
LOVE-KIRCHHOFF, one can neutralize the energy of shearing with a great value of
(which penalizes the condition S = 0), for example 106 H/R, where H is the thickness and R
a characteristic radius of curvature or a distance characteristic of the loadings:
(cf [bib 2]). In practice the user can inform the value of under key word A_CIS of
command AFFE_CARA_ELEM [U4.42.01].
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Axisymmetric thermoelastic hulls and 1D
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Key:
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3
Formulation of the finite element. Discretization
3.1
Description of the selected finite element
3.1.1 Motivations
The choice of framework HENCKY-MINDLIN-NAGHDI to describe the kinematics of hull, presented to
paragraph [§2], led to expressions of the deformations where the derivative are limited to command 1,
contrary to the model of LOVE-KIRCHHOFF. This offers the advantage of being able to use an element
finished of a nature limited while ensuring conformity. The natural choice is the element of LAGRANGE P2,
isoparametric, which makes it possible to have a fine representation of a curved geometry and the maid
estimates of the constraints.
The degrees of freedom are of course displacements and rotations.
As it is known as previously, the model of LOVE-KIRCHHOFF can be recovered by penalization
for a very large parameter affecting the transverse energy of shearing.
This formulation joined the category of the finite elements of hulls known as “degenerated”, i.e. built
by injecting the kinematics of hull in elements of three-dimensional continuous mediums:
cf [bib10].
As for all the finite elements of hulls, of the particular aspects must be analyzed: the catch
in account of the rigid modes and risks of blocking of membrane or shearing.
In the case of the axisymmetric hull of revolution, there is only one rigid mode: translation according to
the axis of symmetry 0y.
On the other hand, in the case of the invariant hull according to the direction 0z, there are three rigid modes: two
translations in the plan (X 0y) and rotation around 0z.
So that the finite element is powerful, it is necessary that the approximations retained for
description of displacement ensure an exact representation of the state of null deformations (mode
rigid). In practice, as the concept of rigid mode is defined compared to the total reference mark one decides
thus to describe displacements in total base (E, E
X
y), in which rigid modes
(functions closely connected) are represented by the selected interpolation.
As for the risks of blocking out of membrane and transverse shearing, usual processing
consist in a selective numerical integration (cf [bib2]), but the practice reveals that these
phenomena seldom appear for the hulls of revolution.
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3.1.2 General presentation of the element
The selected element of reference is quadratic, isoparametric with three nodes and three degrees of freedom
by node. These degrees of freedom are:
U, U
X
y:
components of displacement U in total reference mark,
S:
rotation around ez of normal N.
See [Figure 3.1.2-a].
This element is a generalization of the element of plane beam curved. It is well adapted to
discretization of the hulls with meridian curvature R variable, cf [bib2].
U
y

U X
N.1
N.3
N.2
·
·
·
T-1
0
+1
!!


N
Appear 3.1.2-a: Elément of reference
The functions of form (basic) are the polynomials of LAGRANGE:
- 1 +
+

1
NR
2
1 () =
; “
NR
2 () =
; “
NR
3 () = 1 -
2
2
3.1.3 Transformations finite element/finite element of reference
y
y
- 1
0
+1
2
N2
·
·
·

y
N3
N1
N3
N2
3
y
N1
1
X
X
X
2
x3 1
The geometry is interpolated using the co-ordinates (X, y
I
I) of the three nodes N1, NR 3, N2:
3
3
X () = X NR "
I
I ()

y

; (
)
=
y NR
I
I

“()
I =1
I =1
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In the same way using the ddl (U, U
X
y
,
S

on the nodes, one a:
I
I
I)
3
3
U
=

X ()

U NR
X
I;
=

I
() uy ()
U
NR
y
I
I
()
I =1
I =1
3
= “
S ()

NR
S
I
I
()
I =1
One also needs the jacobien of the transformation:
(
ds
2
2
m) =
() = (X,) + (y,)
D
And of the vectors of the local base:
1
T () =
X
y
m (


X +

) (
E
E

,

,
Z)
(
1
N) =
y
X
m (


X -

) (
E
E

,

,
Z)
Finally:
y,
- X,
cos
=
m (
=
)
;
sin
m ()
The meridian curvature is obtained by:
1

= - (n.t,) D
1
.
=
X. y


- y. X
3
R
ds
m () (,
,
,
,)
Because of the P2 interpolation, the derivative second which appears below express with the assistance
co-ordinates of the three nodes by:
X
= X
1 + X
2 - 2. X y

3
= y
1 + y
2 - 2. y
,
,
3
3.1.4 Surface numerical integration
For numerical integrations along the element one uses a numerical formula of integration with
four points of GAUSS, single for all the terms to be integrated. This formula reveals them
blockings mentioned in the paragraph [§3.1.1] in the event of extremely localized plasticization. One
thus advise to avoid the use of these elements in plasticity for the moment. The formula
of numerical integration at four points of Gauss will be replaced later on by a formula with
two points of supposed Gauss to avoid these nuisances.
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3.1.5 Numerical integration in the thickness
For an elastic behavior, insofar as it is admitted that one limits oneself to characteristics
uniform rubber bands in the thickness, rigidities [C], [B]
, [D
ij
ij
ij]
and Gsx defined in the paragraph
3
[§2.2] are calculated exactly.
For a non-linear behavior, one subdivides the initial thickness in NR layers thicknesses
identical numbered in the direction of the normal to the average surface of the element. For each
sleep one uses three points of integration. The points of integration are located in higher skin of
sleep, 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 from 3 to 5 layers in the thickness for a number of
points of integration being worth 7, 9 and 11 respectively.
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

1/3
1 = - 1

4/3
2 = 0

1/3
3 = +1
1
N
y () D =

I y (I)
- 1
i=1
Formulate numerical integration for a layer in the thickness in plasticity
For a thermoelastic behavior, one uses integration, by layer in the thickness
H
H
-, + described previously in the non-linear field, of the thermomechanical terms seen

2
2
in the paragraph [§2.2]. It is then necessary to use STAT_NON_LINE with a behavior
rubber band.
Note:
One already mentioned with [§2.2]. and in [R3.07.04] that the value of the coefficient of correction in
transverse shearing for the elements of 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.
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3.2
Formulation of the elementary terms
3.2.1 Mass, center of gravity, stamps inertia
In the case of the hulls of revolution, the mass is worth:
2 H/2 (+ - dx rd ds
2
2
3
=
hrd ds =
H R ds
-
1
2
H
S



)


0


/




0




where is the presumedly constant density of the element.
The position of the center of inertia is given in the Oxyz reference mark of [§2.1] by:
xG = 0

H2
1 cos
yr ds +
sin
rds


+
12






R
R
y

G =
R ds

zG = 0
The terms of the matrix of inertia compared to O in the Oxyz reference mark of [§2.1] have then for
expression:
x2
h3
cos2

I
= 2 H (
+ y2) +
(sin2 +
+ X
cos + 2 y
sin) rds
xx/O
2
12
2



h3

1
cos
I
= 2 hx2 +
(cos2 + 2 X
cos) rds
yy/O
where =
+
.

R
R

12


x2
h3
cos2

I
= 2 H (
+ y2) +
(sin2 +
+ X
cos + 2 y
sin) rds
zz/O
2
12
2


In the case of the invariant hulls according to 0z, the mass is worth:
H/2
-
=
H/2
sdx ds
3
H ds.


The position of the center of inertia is defined in the Oxy reference mark of [§2.1] by:

H2
cos
X ds +

ds


12

R
X


G =
ds


H2
sin
y ds +

ds


12

R
y


G =
ds

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The terms of the matrix of inertia compared to O in the Oxyz reference mark of [§2.1] have then for
expression:

h3

I
= hy2 +
(sin2 + 2 y
sin) ds
xx/O

12




h3

I
= I
= hxy +
(sin cos + X
sin + y
cos) ds
xy/O
yx/O

12



1
where =
.

h3

R
I
= H
x2 +
(cos2 + 2 X
cos) ds
yy/O

12




h3

I
= H (x2 + y2) +
(1 + 2 X
cos + 2 y
zz/O
sin) ds

12



H2
h3
Terms in
for the centers of inertia and
for the matrices of inertia are not taken in
12
12
count in the programming. That amounts neglecting the variation of metric with the curvature in
the calculation of these terms.
3.2.2 Stamp of mass
2 H/2

The term: v. ,
, of kinetic energy is treated while considering
- H/2
(S x3) v (S x3) rdx D
3 ds
0
constant density in the thickness and the correction of metric due to the curvature
negligible. The intégrande is burst in three terms:
·
H (U .u + U .u
kinetic energy of
X
X
y
y)
translation
h3
·


kinetic energy of
S.S
12
rotation
h3
·

sin
-
+

+
cos
+
kinetic energy of
12
((U U)
(U U
X S
X S
y S
y S)
coupling, with:
1
cos
= +
R
R
for the case axisymmetric hull of revolution.
1
= for the case invariant hull according to 0z (moreover in this case the integral disappears
R
2 rd

).
0
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3.2.3 Second member of centrifugal force
In the case of the hulls of revolution, a vector rotation is considered: =.
2nd y, carried by
the axis of revolution. The term of the second corresponding member is:
2

H/2
2
2 .r U -
X
.
X
S
3 sin
dx3rd ds



- H/2
(
)

0
= 2 H2 2
2
r. U D
X


ds
0
(one neglects the correction of metric in the thickness).
In the case of the invariant hulls according to 0z, a vector rotation is considered: =.
3rd Z,
perpendicular in the plan of the section.
The second member is then:
H
2.
+.


3 (X U
y U) ds
X
y
3.2.4 Second member of gravity
In the case of the hulls of revolution, gravity is directed according to E Y.
The second member is:
2 gh U R

y

D ds
0
In the case of the invariant hulls according to 0z, this one is directed in the plan
X 0y: G = G E + G
X
X
y E Y.
The second member is:
H G .e + G .e D S

(X X
y
y)
3.2.5 Second member of distributed loads
These distributed loads can be two forces in the plan (X 0y) and couples it M Z carried by the axis
0z. Two forces, which one considers that they are applied to average surface, will be able
to be provided in total reference mark (E, E
X
y) or room (T, N). The second member is:
2 (F U + F U + M) rdds
X
X
y
y
Z
S
0
2
(in invariant hull according to Z, the integral
rd

disappears).
0
Note:
The specific actions are treated as nodal forces where they are applied,
since they work in the ddl of the finite element.
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3.3
Calculation of the strains and the stresses
After resolution, one with the possibility with operator CALC_ELEM [U4.81.01] of calculating with the nodes them
elementary fields according to:
· generalized deformations E
, K

: option DEGE_ELNO_DEPL,
· three-dimensional deformations on average fiber and in skins internal and external
(with or without correction of curvature): option EPSI_ELNO_DEPL,
· three-dimensional constraints on average fiber and in skins internal and external
(with or without correction of curvature): option SIGM_ELNO_DEPL in linear elasticity,
· generalized efforts NR
, M

(with or without correction of curvature): option
EFGE_ELNO_DEPL in linear elasticity.
These values with the nodes are obtained by extrapolation starting from the values at the points of GAUSS of
the element, according to the exposed method in [bib4] [R3.06.03].
Lastly, one can have also the values NR
, M

at the points of GAUSS of the element: option
SIEF_ELGA_DEPL in linear elasticity.
No postprocessing of constraints or generalized efforts is for the moment available for
nonlinear behaviors materials.
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4
Validation - Cas test
One considers hereafter, to judge capacities of this formulation, some examples of application
(cf [bib10]).
4.1
Roll under internal pressure
One studies a vertical roll subjected to a pressure interns p constant on the part y < 0, and null
on y > 0: to see [Figure 4.1-a].
L/2
C
R
+ L/10
B2
B
X
- L/10
B1
p
- L/2
With
Figure 4.1 - a: Cylindre under axisymmetric pressure
The radius is: R = 4 m, the thickness T = 0.25 m, the length L = 10 Mr. That-Ci is selected so that
the effects edge free in y = ± L/2 are negligible on the solution (into axisymmetric, L must
1
to check:
L > 3 Rt = 3 m here).
2
The material is elastic (E
=
1 Pa
,

=.
0)
3.
The boundary conditions are: p = 1N/m2, vertical displacement of A null.
One chooses the solution obtained by model LOVE-KIRCHHOFF.
To reach it numerically, one takes as coefficient of shearing: = 106, to inhibit
the distortions S. the analytical solution is:
P
P
for y
0: U (y)

(2nd y
cos y
y

=
-
),


cos sin
4
S (y) =
E
3
(
y -
y
X
)
8 D
8 D
P
P
for y
0: U (y)

E y
cos y
- y

=
,


cos
sin
4
S (y) =
E
3
(
y +
y
X
)
8 D
8 D
And 3
And
with D = 12 1-2
(), 44=DR2.
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The efforts generalized are (sin =)

0:
And
p
NR

= U

- y

X
(y) M

;
= Of
S
X (y)
= E

sin y
R

4 2
The three-dimensional constraints are:
NR
M
X
M X




= +
3


; S
=
S 3
12
12
, D
'where:
T
T3
T3

Pr
E y


X
3

(y, X) = 1 -
cos y
3
+
2
sin y

3

T
2
T
1 - 2

for y 0:


Pr X
S
3
(y, X
3
) =.

E y
sin y
3


T
T

1 - 2

Pr E y


X
3

(y, X) =
cos y
3
-
2

sin y
3


T
2

T
1 - 2

for y 0:
Pr X
S

3
(y, X
3
) =.

E y
sin y
3


T
T

1 - 2
For a regular grid of one hundred meshs and two hundred nodes, one finds:
Reference
Aster
% difference
Ux displacement
Not A
63.9488
63.922
­ 0.042
Not B
32.000
32.005
0.015
Not C
0.05120
0.08755
Rotation S
Not A
0.06583
0.04057
Not B
41.133
41.165
0.078
Normal effort NR
Not B
2.0000
2.0003
0.015
Not B1 (to ­ L/10)
3.84429
3.8442
0.002
Moment ms
Not B1
4.01497 10­2
4.013 10­2
0.05
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Appear 4.1-b: Flèche of the cylinder under pressure
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Appear 4.1-c: Rotation of the cylinder under pressure.
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Appear 4.1-d: Bending moments axial cylinder under pressure
4.2
Plate circular embedded under uniform pressure [V3.03.100]
One considers the plate of radius R = 1 m, thickness T = 0,1 m (see [Figure 4.2-a] below)
embedded on its circumference.
y
R
p
p
0
X
D
With
R/2
Appear 4.2-a
The material is elastic (E
= 1 P
. has
,

=.
0)
3. The pressure is: p
= 1 NR
.
/m2.
The boundary conditions are: in 0:
. ,
S
= 0 in
A: U
= U
X
y = 0. , S = 0.
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5
One is interested in the solutions of the models of REISSNER =


and of LOVE-KIRCHHOFF (one
6
will take = 106).
The analytical solution is for the arrow:
pR4
X 2
X 2

U
1
1

y (X) = -
-



.
64D


R
-



R
+
And 3
16 T 2
1
5
with
D =



0 for the solution
12 (
=

=
=
1 - 2
);

if
;

5
R
1
6
LOVE-KIRCHHOFF.
pR2 X
The distortion is indeed: S (X) = -
.
16D 2
pR2

X 2
Rotation
1

S is: S (X) =
X
-

.
16D



R
The variations of curvature are (sin
= +)

1:
pR2
X 2
K
X
1 3

S () =
-
-
16D


R
pR2
X 2
K (X) = -

1

-

16D


R
The bending moments are (sin
= +)

1:
pR2
X 2

M
X
3
1
S () =
(
+)
(
)

16


R -
+
pR2
X 2

M (X)

=
(
1 + 3)
(1)


16


R -
+
The constraints are written:

E
S (X, X) =
X [K
3
+
2
3
S (X)
K (X)]
1 -


E
(X, X) =
X [
3
+
2
3

K (X) K (X)]
1 -
S
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One notices independence in rotation, variations of curvature and moments
bending. In center 0 of the plate:
pR4
pR2
uy ()
0 = -
(1 +), M () 0 = M
S

()
0

= -
(1 +),
64D
16
pR2
K ()
0 = K
S

()
0

= -
.
16D
2

E T Pr
S (,
0 ±t/2) =
(,
0 ±t/2) = #
.
1 - 2 16D
It is noticed that one is in compression in higher skin of plate.
pR2
pR2
With embedding a: M S (R) =
; M (R) =
.
8
8
S
uy
Appear 4.2-b: Flèche, rotation of an embedded circular plate
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For a regular grid of 10 meshs (21 nodes) one finds:
Reference
Aster
% difference
Uy displacement

5
­ 101.827
­ 101.7769
0.049
Not D =



6
LOVEKIRCHHOFF
­ 95.9765
­ 95.0395
0.978

5
­ 178.424
­ 178.368
0.031
Point 0 =



6
LOVEKIRCHHOFF
­ 170.625
­ 169.761
0.507
Rotation S

5
256.001
0.024
Not D =



6
LOVEKIRCHHOFF
255.94
257.123
0.462
Variation of Kss curvature

5
173.406
1.60
Not D =

6
LOVEKIRCHHOFF
170.625
162.765
4.61
Variation of curvature K

5
514.001
0.024
Not D =

6
LOVEKIRCHHOFF
511.875
512.242
0.46
Moment ms

5
­ 0.081751
+0.617
Point 0 =

6
­ 0.08125
LOVEKIRCHHOFF
­ 0.081394
­ 0.18

5
0.12373
­ 1.02
Not A =

6
0.125
LOVEKIRCHHOFF
0.10717
­ 14.3
Moment M

5
­ 0.081751
0.617
Point 0 =

6
­ 0.08125
LOVEKIRCHHOFF
­ 0.081394
­ 0.18

5
0.037121
­ 1.01
Not A =

6
0.03750
LOVEKIRCHHOFF
0.032146
­ 14.3
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A

Code_Aster ®
Version
5.0
Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
06/12/00
Author (S):
P. MASSIN, F. VOLDOIRE, C.SEVIN
Key:
R3.07.02-B
Page:
29/32
It is noticed that solution LOVE-KIRCHHOFF (= 106) is less quite approximate than that by

5
REISSNER =


at the variations of curvature and the time bending. On the other hand, them
6
displacements and rotations are well calculated.
These differences are due to the relative thickness of this plate, with respect to the coarseness of the grid
chosen. The figures hereafter show the comparison of the solutions analytical and numerical, in
case LOVE-KIRCHHOFF, on grids of 10 and 100 elements.
- K
Kss
Figure 4.2 - C: Variations of curvature of an embedded circular plate
The layout of the variations of curvature K
and K
S
illustrate the fact that these two components are not
not approximate in the same manner: first is linear since derived from a function of form
P2, while second is constant per pieces.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A

Code_Aster ®
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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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Author (S):
P. MASSIN, F. VOLDOIRE, C.SEVIN
Key:
R3.07.02-B
Page:
30/32
4.3 Axisymmetric modal analysis of a thin spherical envelope
[V2.03.007]
One considers a sphere, of average radius Rm = 2.5 m, thickness T = 0.10 Mr.
The material is elastic (E = 200000 MPa, = 0,3), of density = 7800 kg/m3.
y
R
X
Appear 4.3-a: Sphere
One studies his axisymmetric free vibrations within framework LOVE-KIRCHHOFF (= 106).
One uses a grid made up of 40 meshs and 81 nodes. One is interested in the frequencies included/understood
between 220 and 375 Hz. Compared to the reference solution [V2.03.007] one finds like 5 first
frequencies:

1
2
3
4
5
Reference
237.25
282.85
305.2
324.2
346.8
Aster
237.32
282.78
304.95
323.7
346.2
Table 4.3-a: Fréquences of the axisymmetric modes
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A

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Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
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P. MASSIN, F. VOLDOIRE, C.SEVIN
Key:
R3.07.02-B
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5 Conclusion
The finite elements that we propose were selected with a quite particular aim: structural analysis
thin axisymmetric, or of orthogonal sections of infinite hulls with independence in
direction Z, with the concern of obtaining a good precision on the membrane and flexional solution all
by having a simple element of establishment and not too expensive.
The choice of the degrees of freedom allows a good representation of the boundary conditions. Moreover,
this displacement formulation and rotation lead to elements of smaller degree: elements
are P2 out of membrane and P2 in inflection. It appears that they are easy to handle and that their formulation
allows to use a structure of pre and post simple processor, favors considerable for
to carry out rather fine grids (unidimensional) and to easily display the results (on
a simple curve). Selected kinematics: formulation of HENCKY-MINDLIN-NAGHDI, in
displacements and rotations of average surface makes it possible to utilize the energy of shearing
transverse (interesting for the hulls average thickness).
This energy can be affected of a factor of correction: if one wants to place oneself in theory of
REISSNER, it is enough to choose = 5/6 instead of 1 (but of course, the arrow W and rotations
in this theory only weighted averages in the thickness are). Moreover, the formulation of
hull of LOVE-KIRCHHOFF (for the very mean structures) can be simulated by penalization of
condition of nullity of the transverse distortion, by choosing a factor = 106 × H L, H being
the thickness and L a characteristic distance (radius of curvature, zone of application of the loads…).
The non-linear behaviors in plane constraints are available for these elements. One announces
however that the constraints generated by the transverse distortion are treated elastically, fault
of better. Indeed, the taking into account of a transverse shearing constant not no one on the thickness and
determination of the correction associated on rigidity with shearing compared to a model
satisfying the boundary conditions are not possible and thus return the use of these
elements, when transverse shearing is nonnull, rigorously impossible in plasticity. In
any rigor, for nonlinear behaviors, it would thus be necessary to use these elements in
tally of the theory of Coils-Kirchhoff.
Elements corresponding to the machine elements exist in thermics; chainings
thermomechanical are thus available with finite elements of thermal hulls to three nodes
described in [R3.11.01] according to case's in its axisymmetric version, or its invariant plane version according to
0z.
In the case-test treated, the phenomena of blocking did not appear. Decomposition of
the deformation energy will make it possible, where necessary, to integrate in a selective way the terms
persons in charge for blocking, such a modification not having to raise particular difficulties. One
more detailed study must of course be undertaken on this subject, as for the numerical methods to use
to avoid this blocking when the thickness becomes low.
The possible developments are:
· anisotropy in order to be able to treat the multi-layer hulls,
· problems of buckling,
· decomposition in Fourier series to study nonaxisymmetric problems of
hulls of revolution,
· the taking into account a variable thickness…
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A

Code_Aster ®
Version
5.0
Titrate:
Axisymmetric thermoelastic hulls and 1D
Date:
06/12/00
Author (S):
P. MASSIN, F. VOLDOIRE, C.SEVIN
Key:
R3.07.02-B
Page:
32/32
6 Bibliography
[1]
B. ALMROTH - D. BRUSH: Buckling off bars, punts and shells. Mc Graw-Hill 1975.
[2]
J.L. BATOZ - G. DHATT: Modeling of the structures by finite elements. Volume 3 Hulls.
Hermès 1992.
[3]
D. BUI - F. VOLDOIRE: Presentation of a finite element of cylindrical hull P2 out of membrane
and Morley in inflection. Note EDF-DER-MMN, HI 71/6715, of the 10.10.90.
[4]
X. DESROCHES: Calculation of the constraints to the nodes by a local method of smoothing by
least squares. Note EDF-DER-MMN of the 20.01.92 [R3.06.03].
[5]
G. DHATT - G. TOUZOT: A presentation of the method of the elements finis.2ème edition.
Maloine SA 1984.
[6]
GREEN - ZERNA: Theoretical elasticity. Univ. Oxford 1954.
[7]
TIMOSHENKO and WOINOWSKY-KRIEGER: Plates and hulls. Béranger 1961.
[8]
F. VOLDOIRE: Formulation and numerical evaluation of an elastoplastic model of hull
axisymmetric enriched. Note EDF-DER-MMN, HI-73/7518, of the 04.02.92.
[9]
D. BUI: Shearing in the plates and the hulls: modeling and calculation. Note
EDF-DER-MMN, HI-71/7784, of the 20.02.92.
[10]
S. ANDRIEUX - F. VOLDOIRE: Models of hulls. Applications in linear statics. School
of Eté CEA-EDF-INRIA of Analyze Numérique 1992.
Handbook of Référence
R3.07 booklet: Machine elements on average surface
HI-75/00/006/A

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