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Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
1/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
Organization (S)
: EDF/AMA, IAT St CYR, CNAM
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
Document
V
V
3
3
.
.
0
0
1
1
.
.
4
4
0
0
0
0
SSLL400 - Non-prismatic beam, subjected
with efforts specific or distributed
Summary:
This test be resulting from the validation independent of version 4 of the models of gates.
This test allows the checking of calculations of beam straight lines in the linear static field.(a modeling
with elements of beams POU_D_E, right beam of EULER).
One calculates simultaneously 3 beams of the different sections: section rings, right-angled, and general. These
beams are subjected to efforts specific or distributed.
The values tested are displacements and rotations, the efforts generalized, and the stresses.
Code_Aster
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Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
2/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
S
1
S
1
S
1
S
1
S
1
1
Problem of reference
1.1 Geometry
1.1.1 Right beam of variable circular section
Z
y
X
Appear 1.1.1-A
Length
: 1 m
Radius with embedding: 0,1 m
Radius at the end
free
: 0,05 m
1.1.2 Right beam of variable rectangular section
y
Z
X
Appear 1.1.2-A
Length
: 1 m
with embedding: Hy = 0,05 m
Hz = 0,10 m
at the loose lead: Hy = 0,05 m
Hz = 0,05 m
1.1.3 Right beam of variable general section
S
1
S
2
y
Z
X
Appear 1.1.3-A
Length
: 1 m
with embedding: To = 10
- 2
m ²
Iy = 8,3333 10
- 6
m
4
at the loose lead: To = 2,510
- 3
m ²
Iy = 5,20833 10
- 7
m
4
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
3/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
1.2
Properties of materials
Young modulus:
E = 2. 10
11
AP
Poisson's ratio:
= 0,3
Density:
= 7800 Kg.m
- 3
1.3
Boundary conditions and loading
Boundary condition:
Embedded end: DX = DY = DZ = DRX = DRY = DRZ = 0
Loading:
On the right beam of variable circular section and on the right beam of rectangular section
variable, one applies successively:
Loading case
Nature
1
a specific effort following X at the loose lead, Fx = 100 NR
2
a specific effort following Y at the loose lead, Fy = 100 NR
3
one specific moment around axis X at the loose lead, MX = 100 Mr. N
4
one specific moment around axis Z at the loose lead, Mz = 100 Mr. N
5
a distributed load on the whole of the beam, fx = 100 N.m-1
6
a distributed load on the whole of the beam, fy = 100 N.m-1
To the right beam of variable general section, one applies:
Loading case
Nature
7
an effort of gravity according to Z with G = 9,81m. S
2
Code_Aster
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Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
4/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
2
Reference solutions
2.1
Method of calculation used for the reference solutions
2.1.1 Section
circular
2.1.1.1 Beam subjected to a specific tensile load Fx
The equilibrium equation is:
()
()
X EA X
U
X
With X
With
C xL
C
With
With
=
=
+
=
-
0
1
1
1
2
1
with
and
,
()
NR L
F
X
=
While integrating twice [R3.08.01], we obtain displacements according to the force applied,
that is to say:
U (X) = L F
E WITH
X
L + C X
X
1
,
and thus at the end L of the beam:
U (L) =
L
E WITH
With
F
X
1
2
The efforts intern are given by:
NR X
EA X
U
X X
F
X
()
()
()
=
=
and stresses by:
xx
NR X
With X
=
()
()
2.1.1.2 Beam subjected to a specific bending load Fy
The equilibrium equation, under the assumption of Euler, is given by the equation:
()
2
2
2
2
4
1
4
0
1
1
1
2
1
X I.E.(internal excitation) (X)
v
X
=
I X
I
C xL
C
I
I
Z
Z
Z
Z
Z
=
+
= -
with
and
,
V
L
F
y
y
()
.
=
We solve the equation by integration by taking account of the law of behavior modified
MF = I.E.(internal excitation) v
X
Z
Z
2
2
and the equilibrium equation
MF
X
V
Z
y
+
= 0
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
5/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
Four successive integrations, by holding account for the calculation of the constants of integration that:
()
()
()
()
X I.E.(internal excitation) (X)
v
X
(L)
V L
F
I.E.(INTERNAL EXCITATION) (L)
v
X
L
v
X
v
Z
y
y
Z
2
2
2
2
0
0
0
0
0
= -
= -
=
=
=
lead to the expression of
v X
()
:
v (X) + F L
E I
X (L X + cx)
(L + cx)
y
Z
=
-
2
2
2
6
3
2
1
and with the expression of
Z
(X)
(
)
(
)
Z
y
Z
(X) = + F L
E I
X L Lx L cx cx C X
L
cx
2
2
2
2
2
3
6
6
3
6
2
1
-
+
-
+
+
.
The efforts intern are given by:
()
(
)
V X
F
MF X
F L X
y
y
Z
y
()
=
=
-
and stresses by:
()
()
xx
Z
Z
xy
y
X
MF X
R X
I X
V X
With X
()
()
()
()
=
=
(not of coefficient of correction dcisaillement in assumption of Euler)
2.1.1.3 Beam subjected to one specific torque MX
The movement is given by the equation:
()
X G I
X
X
I X
I
C xL
C
I
I
p
X
p
p
p
p
()
,
=
=
+
= -
0
1
1
1
2
4
1
1
4
with
and
M
L
M
X
X
()
=
Code_Aster
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Version
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Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
6/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
After integration, and by taking account of the fact that:
G I
L
X (L) = M
p
X
X
()
,
and
()
X
0
0
=
we obtain the expression of
X
X
()
:
(
)
(
)
X
X
p
X
L M
G I
X
L
L cx
C X
L cx
()
=
+
+
+
3
3
3
1
2
2
2
3
.
We must also have for the internal efforts and the stresses:
()
()
()
()
()
()
()
()
M X
M
X
M X
I X R X
X
M X
I X R X
X
X
xy
X
p
T
xz
X
p
T
()
=
=
=
2.1.1.4 Beam subjected to one specific bending moment My
The reasoning to find the solution analytical is the same one as previously. We use
law of behavior
M
X
I.E.(internal excitation)
X
W
X
y
y
()
()
= -
2
2
and the equilibrium equation
MF
X
V
y
Z
-
= 0.
Calculation
constants of integration differs: one has
V L
Z
()
= 0
and
M
L
M
F
y
y
()
=
.
The expression of W (X) is obtained:
(
)
(
)
W X
L M
E I
X
cx
L + cx
y
y
()
=
+
6
1
2
3 L 2
2
,
and the expression of
y
(X):
(
)
(
)
y
X
L M
E I
L
L cx C X
L cx
() = 3
X 3
y
y
2
1
+
+
+
3
2 2
3
.
One must also have for the internal efforts and the stresses:
()
()
() ()
()
V X
MF X
My
X
MF X R X
I X
Z
y
xx
y
y
()
=
=
=
0
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
7/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
2.1.1.5 Beam subjected to a tensile load regularly distributed fx
Balance is described by the equation
()
X
EA (X) ux
F
With X
With
C xL
C
With
With
X
=
with
and
-
=
+
= -
1
2
1
1
2
2
1
1.
By integrating first once this equation, we obtain:
E With X
U
X
F X C
X
()
= -
+
1
.
The limiting condition
NR L
()
= 0
imply
C
F L
X
1
=
. We thus have:
(
)
U
X
F
L X
E With X
X
= -
-
()
that is to say:
(
)
U X
F
L X
E With X dx C
X
()
()
=
-
+
2
C
2
is given so that
U ()
.
0
0
=
Taking everything into account, we have:
(
)
U X
L F
E With C
C X C X
L C X Log
L
L C X
L C X
X
()
=
+
+
+
+
+
2
1 2
2
.
The efforts intern are deduced from the law of behavior
NR X
E With X
U
X
()
()
=
:
NR X
F
L X
X
()
(
),
=
-
and the stresses are given by:
(
)
(
)
xx
X
X
NR X
With X
F
L X
With
With
With
X
L
()
()
()
=
=
-
+
-
1
2
1
2
Code_Aster
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Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
8/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
2.1.1.6 Beam subjected to a bending load regularly distributed fy
On the basis of the equilibrium equation:
()
2
2
2
2
4
1
4
1
2
1
1
1
X
E I X
v
X
F
I X
I
C xL
c= II
Z
y
Z
Z
Z
Z
()
,
= -
=
+
-
with
and
we carry out four successive integrations. The determination of the constants of integration is
made starting from the following limiting conditions:
()
()
V
L
M
L
v
X
v
y
Z
()
()
=
=
=
=
0
0
0
0
0
0
The analytical expression for
()
()
v X
Z
and
in the presence of a loading distributed is, taking everything into account:
(
)
[
(
)
(
)
(
)
()
(
)
(
)
[
]
() =
v X
F L
I.E.(internal excitation) C
L cx
L cx
X
L C
L C
X
C
C
C
Log
C xL L
L cx
LLC X
X
L F X
I.E.(internal excitation)
L cx
L
Lx
Lcx X
C C
y
Z
Z
y
Z
-
+
-
+
-
-
+
-
+
-
+
+
+
+
=
+
+
-
+
+
- +
3
4
2
2
2
2
4
3
3
4
5
3
2
2 2
3
3
2
2
2
12
6
9
3
2
2
2
1
6
12
6
6
3
3
3
1
1
1
The efforts intern are given by:
()
(
)
V X
F
L X
MF X
F L X
y
y
Z
y
()
(
)
=
-
=
-
and
1
2
2
stresses by:
()
xy
y
xx
Z
Z
X
V
X
With X
X
MF X R X
I X
()
()
()
()
()
()
=
=
Code_Aster
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Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
9/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
2.1.2 Section
rectangular
2.1.2.1 Beam subjected to a specific tensile load Fx
The equilibrium equation is:
()
(
)
()
X
EA (X) ux =
With X
With
With
With xL
NR L
F
X
with
=
+
-
=
0
1
2
1
While integrating twice, and by taking account of the fact that:
()
E WITH L
U
X L
F
U
X
()
()
,
,
=
=
0
0
for the determination of the constants of integration, we obtain the analytical expression of U (X), that is to say
:
U X
F L
WITH E C Log
C xL
X
()
.
=
+
1
1
For the internal and forced efforts, we have:
()
()
NR X
F
NR X
With X
X
xx
()
=
=
2.1.2.2 Beam subjected to a specific bending load Fy
The movement is given by the equation:
()
()
2
2
2
2
3
1
3
0
1
1
1
2
1
X
E I X
v
X
I X
I
C xL
C
I
I
V L
F
Z
Z
Z
Z
Z
y
y
()
,
.
=
=
+
= -
=
with
and
The same reasoning that for the circular section leads to the following result:
(
)
(
)
()
(
)
(
)
v X
F L
E I C
L cx
C X
C X
L L cx Log
L
L cx
L
cx
X
F L
I.E.(internal excitation)
X L X cx
L cx
y
Z
Z
y
Z
()
.
= -
+
-
+
+
+
+
=
- +
+
2
3
2
2
3 2
2
2
2
2
2
2
2
1
1
Code_Aster
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Version
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Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
10/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
We must have for the internal efforts and the stresses:
()
(
)
()
()
()
()
()
()
()
V X
F
M
X
F L X
X
H X M
X
I X
X
V X
With X
y
y
F
y
xx
y
F
Z
xy
y
Z
Z
()
=
=
-
=
=
2
2.1.2.3 Beam subjected to one specific torque MX
The movement is given by the equation:
()
()
X G I
X
X
I X
I
C xL
c= II
MR. L
M
p
X
p
p
p
p
X
X
()
,
.
=
=
+
-
=
0
1
1
1
2
1
3
1
3
with
and
By the same reasoning as the beam with circular section, we obtain the analytical expression
of
()
X
X
:
()
(
)
(
)
X
X
p
X
L M X
L
cx
I
G L
cx
=
+
+
2
2
1
2.
I
p
1
and
I
p
2
are calculated according to formulas' given in the reference material [R3.08.01].
The internal efforts and the stresses are given by:
()
()
()
()
M
X
M
X
M X
I X R X
X
X
xy
X
p
T
xz
()
=
=
=
2.1.2.4 Beam subjected to one specific bending moment My
The same reasoning is taken again that previously, the analytical expressions are obtained
following for
()
W X
X
y
()
and
:
(
)
()
(
)
(
)
W X
L M X
E I
L
cx
X
L M X L cx
E I
L
cx
y
y
y
y
y
()
,
= -
+
=
+
+
2
2
2
2
2
1
1
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
11/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
for the efforts:
()
V
X
MF X
M
Z
y
y
()
=
=
0
and for the stresses:
xx
Z
y
y
X
H X MF X
I
X
()
()
()
()
=
2
2.1.2.5 Beam subjected to a tensile load regularly distributed fx
The equilibrium equation is:
()
X EA X
U
X
F
With X
With
C xL
c= AA
X
()
.
= -
=
+
-
with
and
1
2
1
1
1
After two integrations and by taking account of the fact that:
NR L
()
= 0
to determine the first constant of integration,
and
U
()
0
0
=
to determine the second,
we obtain the analytical expression of
U X
()
:
(
)
U X
L F
E With C
C X
L LLC Log
L
L C X
X
()
= -
+
+
+
1 2
The efforts intern are known by the following expression:
NR X
F
L X
X
()
(
)
=
-
and stresses by:
xx
X
X
F
L X
With X
()
(
)
()
=
-
2.1.2.6 Beam subjected to a bending load regularly distributed fy
The equilibrium equation is:
()
2
2
2
2
1
3
2
1
1
3
1
1
X
I.E.(internal excitation) X
v
X
F
I X
I
C xL
c= II
Z
y
Z
Z
Z
Z
()
.
= -
=
+
-
with
and
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
12/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
We integrate successively four times this equation. The constants of integration are calculated
by taking account of the fact that:
()
()
V
L
MF L
v
X
v
y
Z
()
()
=
=
=
=
0
0
0
0
0
0
The analytical result for the arrow and rotation in L is as follows:
(
)
(
)
[
(
)
(
)
()
(
)
(
) (
)
[
]
(
)
v
L F
E I C
L C X
X
L C
L C
X
C
C
C
L
L C
Lcx
LLC X
C X
Log
L
L cx
X
L F
I.E.(internal excitation) C L cx
X LLC
LLC
X
C
C
C
L
Lcx
C X
Log
L
L cx
y
Z
Z
y
Z
(X) =
3
4
2
2
2
3
4
2
2
2
2 2
3
3
2
3
2
2
3
4
2
2 2
4
6
4
5
2
6
4
8
4
2
4
2
2
3
2
2
4
2
1
1
+
+
+
+
-
+
+
+
+
+
+
=
+
+
+
+
-
+
+
+
+
.
The efforts intern are given by the following expressions:
()
(
)
V
X
F
L X
MF X
F L X
y
y
Z
y
()
(
)
,
=
-
=
-
1
2
2
stresses by:
()
()
()
xy
y
xx
Z
y
Z
X
V
X
With X
X
MF X H
I X
()
()
()
=
=
2
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
13/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
2.1.3 Section
general
2.1.3.1 Beam subjected to the forces of gravity
The efforts of gravity are applied along axis Z. The movement of the beam induced by these
efforts is thus a movement of bending in the plan (X O Z).
The equilibrium equation is given by the expression:
()
()
2
2
2
2
1
2
2
1
1
2
1
4
2
1
1
4
1
1
1
1
X
E I
X
W
X
With X G
With X
With
C xL
c= AA
and I X
I
D xL
D
I
I
y
y
y
y
y
()
()
.
=
=
+
-
=
+
= -
linear weight
with
1 2
4 3
4
By integrating first once, we obtain the shearing action intern:
V X
With X G dx C
Z
()
()
= -
+
1
C
1
is given so that
()
V L
Z
= 0
.
We obtain:
()
V
X
L WITH
G
C
C xL
C
Z
()
.
=
-
+
+ +
1
3
3
3
1
1
By integrating second once, we obtain the internal bending moment:
M
X
V
X dx
C
y
Z
()
()
.
=
+
2
C
2
is calculated so that
()
MR. L
y
= 0
.
We obtain:
(
)
M
X
With
G
L
L X
L
L C
L C
Lcx
Lcx
L C X
C X
y
()
=
-
+
+
+
+
+ +
1
2
2
2
2
2 2
2
2
2
12
6
8
3
4
2
2
.
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
14/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
We calculate then rotation starting from the law of behavior
= E I X
X
y
y
()
.
We thus have
()
X
M X
E I
X dx
C
y
y
=
+
()
()
3
with such as (0) = 0.
The arrow
W X
()
is given starting from the relation of Euler:
y
W
X
= -
.
We calculate
W X
()
by integration of
y
X
()
:
W X
X
C
y
()
()
= -
+
4
with
C
4
such as
W ()
0
0
=
.
Analytical expressions of
y
X
()
and
W X
()
are not retranscribed here because they are much
too much heavy. They were calculated, like the preceding ones, by the formal computation software
MATHEMATICA.
2.2
Results of reference
· Displacements and rotations at the loose lead
· Interior efforts at the two ends
· Stresses at the two ends
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Report/ratio n° 2314/A of the Institute Aerotechnics “Proposal and realization for new cases
tests missing with the validation beams ASTER “
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
15/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
The model is composed of 10 elements right beam of Euler.
S1 section: variable circular section
with embedding,
R1 = 0.1 m (full section)
in the loose lead, R2 = 0.05 m (full section)
S2 section: variable rectangular section
with embedding,
H
y1
= 0.05 m
H
z1
= 0.10 m
with the loose lead, H
y2
= 0.05 m
H
z2
= 0.05 m
S3 section: variable general section
with embedding,
With
1
= 10
2
m ²
I
y1
= 8.3333 10
6
m
4
at the loose lead,
With
2
= 2.5 10
3
m ²
I
y2
= 5.20833 10
7
m
4
3.2
Characteristics of the mesh
3 sections X 10 elements
POU_D_E
3.3
Functionalities tested
Controls
AFFE_CARA_ELEM
BEAM
SECTION
RING
RECTANGLE
GENERAL
MECA_STATIQUE
OPTION
EFGE_ELNO_DEPL
SIGM_ELNO_DEPL
AFFE_CHAR_MECA FORCE_NODALE
FX
FY
MX
MY
FORCE_POUTRE
FX
FY
GRAVITY
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
16/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Loading case
Section
Identification
Reference
Aster Variation
%
1 S1
U (L)
3.1831E08
3.1831E08
0.00E+00
N (0)
1.0000E+02
1.0000E+02
0.00E+00
N (L)
1.0000E+02
1.0000E+02
0.00E+00
xx
(0)
3.1831E+03 3.1831E+03
0.00E+00
xx
(L)
1.2732E+04 1.2732E+04
0.00E+00
2 S1
v (L)
4.2441E06
4.2441E06
0.00E+00
Z
(L)
8.4882E06 8.4882E06
0.00E+00
v
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
v
y
(L) 1.0000E+02
1.0000E+02 0.00E+00
MF
Z
(0) 1.0000E+02
1.0000E+02 0.00E+00
MF
Z
(L) 0.0000E+00
2.0008E11 0.00E+00
xx
(0)
1.2732E+05 1.2732E+05
0.00E+00
xx
(L)
0.0000E+00 2.0380E07
0.00E+00
xy
(0)
3.1831E+03 3.1831E+03
0.00E+00
xy
(L)
1.2732E+04 1.2732E+04
0.00E+00
3 S1
X
(L)
3.8621E05 3.8621E05
0.00E+00
m
X
(0) 1.0000E+02
1.0000E+02 0.00E+00
m
X
(L) 1.0000E+02
1.0000E+02 0.00E+00
xy
(0)
6.3661E+04 6.3661E+04
0.00E+00
xy
(L)
5.0929E+05 5.0929E+05
0.00E+00
xz
(0)
6.3661E+04 6.3661E+04
0.00E+00
xz
(L)
5.0929E+05 5.0929E+05
0.00E+00
4 S1
W (L)
8.4882E06
8.4882E06
0.00E+00
y
(L)
2.9708E05 2.9708E05
0.00E+00
v
Z
(0) 0.0000E+00
2.9103E10 0.00E+00
v
Z
(L) 0.0000E+00
0.0000E+00 0.00E+00
MF
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
MF
y
(L) 1.0000E+02
1.0000E+02 0.00E+00
xx
(0)
1.2732E+05 1.2732E+05
0.00E+00
xx
(L)
1.0185E+06 1.0185E+06
0.00E+00
5 S1
U (L)
1.2296E08
1.2335E08
0.323
N (0)
1.0000E+02
1.0000E+02
0.00E+00
N (L)
0.0000E+00
9.6633E13
0.00E+00
xx
(0)
3.1831E+03 3.1831E+03
0.00E+00
xx
(L)
0.0000E+00 1.2303E10
0.00E+00
6 S1
v (L)
1.3486E06
1.3486E06
0.001
Z
(L)
2.1220E06 2.1220E06 0.003
v
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
v
y
(L) 0.0000E+00
1.8195E10 0.00E+00
MF
Z
(0) 5.0000E+01
5.0000E+01 0.00E+00
MF
Z
(L) 0.0000E+00
2.1245E12 0.00E+00
xx
(0)
6.3662E+04 6.3662E+04
0.00E+00
xy
(0)
3.1831E+03 3.1830E+03
0.
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
17/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
Loading case
Section
Identification
Reference
Aster Variation
%
1 S2
U (L)
1.3862E07
1.3865E07
0.022
N (0)
1.0000E+02
1.0000E+02
0.00E+00
N (L)
1.0000E+02
1.0000E+02
0.00E+00
xx
(0)
2.0000E+04 2.0000E+04
0.00E+00
xx
(L)
4.0000E+04 4.0000E+04
0.00E+00
2 S2
v (L)
1.8969E04
1.8546E04
2.232
Z
(L)
3.0238E04 2.9465E04 2.556
v
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
v
y
(L) 1.0000E+02
1.0000E+02 0.00E+00
MF
Z
(0) 1.0000E+02
1.0000E+02 0.00E+00
MF
Z
(L) 0.0000E+00
8.0035E11 0.00E+00
xx
(0)
2.4000E+06 2.4000E+06
0.00E+00
xx
(L)
0.0000E+00 3.8417E06
0.00E+00
xy
(0)
2.0000E+04 2.0000E+04
0.00E+00
xy
(L)
4.0000E+04 4.0000E+04
0.00E+00
3 S2
X
(L)
8.3506E04 7.8827E04 5.603
m
X
(0) 1.0000E+02
1.0000E+02 0.00E+00
m
X
(L) 1.0000E+02
1.0000E+02 0.00E+00
xy
(0)
1.5600E+06 1.5600E+06
0.00E+00
xy
(L)
4.0371E+06 3.8400E+06 4.882
xz
(0)
1.5600E+06 1.5600E+06
0.00E+00
xz
(L)
4.0371E+06 3.8400E+06 4.882
4 S2
W (L)
1.2000E04
1.2001E04
0.014
y
(L)
3.600E04 3.6012E04 0.034
v
Z
(0) 0.0000E+00
3.2014E10 0.00E+00
v
Z
(L) 0.0000E+00
0.0000E+00 0.00E+00
MF
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
MF
y
(L) 1.0000E+02
1.0000E+02 0.00E+00
xx
(0)
1.2000E+06 1.2000E+06
0.00E+00
xx
(L)
4.8000E+06 4.8000E+06
0.00E+00
5 S2
U (L)
6.1370E08
6.1463E08
0.151
N (0)
1.0000E+02
1.0000E+02
0.00E+00
N (L)
0.0000E+00
1.8758E12
0.00E+00
xx
(0)
2.0000E+04 2.0000E+04
0.00E+00
xx
(L/2)
1.3333E+04 1.3333E+04
0.00E+00
xx
(L)
0.0000E+00 7.5033E10
0.00E+00
6 S2
v (L)
6.8626E05
6.7302E05
1.929
Z
(L)
9.4847E05 9.2730E05 2.232
v
y
(0) 1.0000E+02
1.0000E+02 0.00E+00
v
y
(L) 0.0000E+00
4.3661E10 0.00E+00
MF
Z
(0) 5.0000E+01
5.0000E+01 0.00E+00
MF
Z
(L) 0.0000E+00
2.3042E11 0.00E+00
xx
(0)
1.2000E+06 1.2000E+06
0.00E+00
xx
(L)
0.0000E+00 1.1060E06
0.00E+00
xy
(0)
2.0000E+04 2.0000E+04
0.00E+00
xy
(L)
0.0000E+00 1.7464E07
0.00E+00
7 S3
W (L)
3.8259E05
3.8259E05
0.00E+00
y
(L)
5.7388E05 5.7387E05 0.003
v
Z
(0) 4.4633E+02
4.4635E+02 0.004
MF
y
(0) 1.7535E+02
1.7535E+02 0.00E+00
4.2 Remarks
Modeling being made in beams of Euler, coefficients of shearing ky = kz = 1.
Code_Aster
®
Version
5.0
Titrate:
SSLL400 - Non-prismatic beam, subjected to specific efforts
Date:
03/05/02
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.400-A
Page:
18/18
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A
5
Summary of the results
The results obtained confirm that elements POU_D_E with variable section present a good
degree of reliability.
For the circular section, the results all are exact with the nodes (one finds the properties of
the element with constant section) except for the efforts distributed where the effect of the smoothness of discretization
fact of feeling.
For a rectangular section and a general section, it is necessary to discretize finely to have one
correct solution.