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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 1/18

Organization (S): EDF/AMA, IAT St CYR, CNAM
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
V3.01.400 document

SSLL400 - Variable Poutre of section, 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 constraints.
Handbook of Validation
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SSLL400 - Variable Poutre of section, subjected to specific efforts
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Key: V3.01.400-A Page: 2/18


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
: 0,05 m
free

1.1.2 Right beam of variable rectangular section

y
X
Z
S1
S
1
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

Z
y
S1
S2
X

Appear 1.1.3-A

Length
: 1 m

with embedding: To = 10-2 m ²
Iy = 8,3333 10-6 m4
at the loose lead: To = 2,510-3 m ²
Iy = 5,20833 10-7 m4
Handbook of Validation
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 3/18


1.2
Properties of materials

Young modulus:
E = 2. 1011 Pa
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. s2

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SSLL400 - Variable Poutre of section, subjected to specific efforts
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 4/18


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 tractive effort Fx

The equilibrium equation is:

(U

X
EA X)
0 with
(
With X)

To 1 C
X
X =
=
+
1
L
With

and C =
2 - 1,
A1
NR (L) = Fx

While integrating twice [R3.08.01], we obtain displacements according to the force applied,
that is to say:

L F
X

U (X) =
X
E WITH L + C X,
1

and thus at the end L of the beam:

L
U (L) =
F
E WITH
With X
1
2

The efforts intern are given by:

U
NR (X) = EA (X)
(X) = F


X
X

and constraints by:

NR (X)
xx =

With (X)

2.1.1.2 Beam subjected to a specific bending stress Fy

The equilibrium equation, under the assumption of Euler, is given by the equation:

2
2
4
v

X
I.E.(internal excitation)

(X)
= 0 with I (X) = I 1+ C
2
X
Z


2
Z
1
X
Z



L



1

I 4
2
Z
and C =

1,

I -
1
Z
V (L) = F
y
y.

We solve the equation by integration by taking account of the law of behavior modified
2v
MF
MF = I.E.(INTERNAL EXCITATION)
Z
Z
Z
+ V = 0
x2 and the equilibrium equation X
y
Handbook of Validation
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 5/18


Four successive integrations, by holding account for the calculation of the constants of integration that:


2v
I.E.(internal excitation)

(X)
(L) = V
- (L) = -

F
X
Z


x2
y
y



2v
I.E.(INTERNAL EXCITATION) (L)
(L)
Z
= 0

x2

v () =
0
0
X
v ()
0 = 0
lead to the expression of v (X):

F L2
2
y
X (L
3
- X + 2 cx)
v (X) = +


E I
2
6
Z
(L + cx)
1

and with the expression of Z (X)

2
2
2
2
2
-
+
-
+
y
F L
X (6 L Lx
3
6 L cx cx 2 C X)
Z (X) = +
.
E I
3
6
Z
(L
+ cx)
1

The efforts intern are given by:

V (X) = F
y
y
MF (X) = F (L -

X)
Z
y
and constraints by:

R (X)
xx (X) = M Z
F (X) Iz (X)

V (X)

y
xy =
(
(not of coefficient of correction dcisaillement in assumption of Euler)
With X)

2.1.1.3 Beam subjected to one specific torque MX

The movement is given by the equation:




X 4
G I (X
X
)
0 with I (X)

I
1 C
X
p


X
p
p

=
=
+
1
L
1
I p 4
and C =
2
,
1


I -
p1
M (L) = M
X
X
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 6/18


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):

2
2
2
L M
X 3 L + 3 L cx + C X
X
(
)

.
X (X) = 3G I
3
p1
(L+cx)

We must also have for the internal efforts and the constraints:

M (X) = M
X
X
M (X)
(X)
X
=
R X
xy
I (X)
()
T
p
M (X)
(X)
X
=
R X
xz
I (X)
()
T
p

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
2w
MFy
law of behavior M (X) = - I.E.(internal excitation) (X
y
y
)
- V = 0. Calculation
x2 and the equilibrium equation X
Z
constants of integration differs: there are V
L
Z () = 0 and M
(L) = M
F
y.
y

The expression of W (X) is obtained:

L M
2
y
X (3 L + 2cx)
W (X) =

,
6th I
2
y
(L + cx)
1

and the expression of y (X):
L M
X
2
2 2
y
(3 L + 3Lcx + C X)
y (X) =

3rd I
3
.
y1
(L + cx)

One must also have for the internal efforts and the constraints:

V (X
Z
) = 0
MF (X) = My
y

R (X)
(X) = MF (X)
xx
y
I (X)
y
Handbook of Validation
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HT-66/02/001/A

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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 7/18


2.1.1.5 Beam subjected to a tractive effort regularly distributed fx

Balance is described by the equation


U

X 2
EA (X)
=
F
with
(
With X)

To 1 C
X
X
X


-
=
+
1
L
1

To 2
and C = 2
- 1.
A1

By integrating first once this equation, we obtain:

U
E With (X)
= - F X + C
X
X
1.

The condition limits NR (L) = 0 implies C
F L
1 =
X
. We thus have:

U
(L-x)
= -

F
X
X E With (X)

that is to say:

(L X)
U (X) = F
dx + C
X E With (X)
2

c2 is given so that U ()
0 =.
0

Taking everything into account, we have:

L
2
2
C X + C X + (L + C X) Log
L F
L + C X
U X
X
() =
.
E WITH C2
L + C X
1

U
The efforts intern are deduced from the law of behavior NR (X) = E A (X):
X
NR (X) = F (L - X
X
),

and the constraints are given by:

()
F (L - X)

NR X
X
xx (X) =
=
With (X)

X 2
With +
1
(A - WITH
2
1)


L
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Key: V3.01.400-A Page: 8/18


2.1.1.6 Beam subjected to a bending stress regularly distributed fy

On the basis of the equilibrium equation:

2
2
4
v

X
E I (X)
= - F
with I (X) = I
1+ C
2
X
Z

2
y
Z
1
X
Z



L




1

I 4
2
Z
and c=

1,

I -
1
Z
we carry out four successive integrations. The determination of the constants of integration is
made starting from the following limiting conditions:

V (L
y
) = 0
M (L
Z
) = 0
v ()

0 = 0
X
v ()
0 = 0

The analytical expression for v (X) and (Z) in the presence of a loading distributed is, taking everything into account:

- F L3
v (X) =
y
[- 6L2 cx + x2
2
4
3
3
4
5
9
3
2
2
2
4
2
(- LLC - LLC) + X (- C + C - c) +
12 I.E.(internal excitation) C (L + cx)
z1

X

Log 1 + C
(6L3 12L2cx 6Lc2x2)


L
+
+



+ 3
(
L F X
X)
y
2
2
2
Z
=
3
3
3
1
3 [L - Lx + Lcx + X (- C + c)]
6EI (L + cx)
z1

The efforts intern are given by:

V (X) = F (L - X
y
y
)
1
and
MF (X) =
F (L - X) 2
Z
y
2

constraints by:

V (X)

y
xy (X) = A (X)

R (X)
(X) = MF (X)
xx
Z
Iz (X)
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SSLL400 - Variable Poutre of section, subjected to specific efforts
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Key: V3.01.400-A Page: 9/18


2.1.2 Section
rectangular

2.1.2.1 Beam subjected to a specific tractive effort Fx

The equilibrium equation is:


U
X
EA (X)
= 0 with
(
With X)
A1 (A
With
2
1)
X
X
=
+
-
L
NR (L) = F

X
While integrating twice, and by taking account of the fact that:

U
E WITH (L)
(L) = F,
X
X
(
U)
0 =,
0

for the determination of the constants of integration, we obtain the analytical expression of U (X), that is to say
:

F L

X
U X
X
() =
Log 1 + C
.
WITH E C


L
1

For the internal and forced efforts, we have:

NR (X) = Fx
NR (X)


xx =
(
With X)

2.1.2.2 Beam subjected to a specific bending stress Fy

The movement is given by the equation:

2
2
3
v

X
E I (X)
= 0 with I (X) = I 1+ C
2
X
Z


2
Z
1
X
Z



L



1
I 3
2
Z
and C =

1,

I -

1
Z
V (L) = F
y
y.

The same reasoning that for the circular section leads to the following result:


L
2
2
3 2
2
2 L cx + C X - C X + 2 L (L + cx) Log

F L
L + cx
y



v (X) = - the 2nd I c3
Z
(L + cx)
1

F L2
y
X (2L - X + cx)
(X)
Z
=
.
2EI
2
Z
(L + cx)
1
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SSLL400 - Variable Poutre of section, subjected to specific efforts
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Key: V3.01.400-A Page: 10/18


We must have for the internal efforts and the constraints:

V (X) = F
y
y
M (X) = F (L - X)
F
y
Z
H (X) M (X)
(X)
y
Fz

xx
=
2I (X)
Z
V (X)
(X)
y
xy
= (Ax)

2.1.2.3 Beam subjected to one specific torque MX

The movement is given by the equation:




X 3
G I (X
X
)
0 with I (X)

I
1 C
X
p


X
p
p

=
=
+
1
L
1
I
p
3

and c=
2
,
1

I -
p1
M (L) = M
X
X.

By the same reasoning as the beam with circular section, we obtain the analytical expression
of ()
X X:
L M X X (2 L + cx)
(X)
.
X
= 2 I
2
p G (L + cx)
1

I
p and I
are calculated according to formulas' given in the reference material [R3.08.01].
1
p2

The internal efforts and the constraints are given by:

M (X) = M
X
X
M (X)
(X)
X
=
R X
xy
=
I (X)
()
T
xz
p

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) and (X)

y
:

L M X2
y
W (X) = - 2nd Iy (L + cx)
1

L M X (2L + cx)
(
y
X)
y
=
,
2
2nd I y (L + cx)
1
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 11/18


for the efforts:

V (X
Z
) = 0

MF (X) = M
y
y

and for the constraints:

H (X) MF (X)

Z
y
xx (X) =

2 I y (X)

2.1.2.5 Beam subjected to a tractive effort regularly distributed fx

The equilibrium equation is:


U

X
EA (X)
F
with A (X)

To 1 C
X
X
X

= -
=
+
1
L
With

and c= 2 -.
1
A1

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):

- L F

L

U X
X
() =
C X +
2
(L+ LLC) Log

E With C
L + C X
1



The efforts intern are known by the following expression:

NR (X) = F (L - X
X
)
and constraints by:
F (L - X)

X
xx (X) =

With (X)

2.1.2.6 Beam subjected to a bending stress regularly distributed fy

The equilibrium equation is:

2
2
3
v

X
I.E.(internal excitation) (X)
= - F
with I (X) = I
1 + C
2
X
Z


2
y
Z
z1
X


L



1

I 3
Z 2
and c=

1.

I -
z1
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 12/18


We integrate successively four times this equation. The constants of integration are calculated
by taking account of the fact that:

V (L
y
) = 0
MF (L
Z
) = 0
v
()

0 = 0
X

v ()
0 = 0

The analytical result for the arrow and rotation in L is as follows:

L3 F
v
y
(X) =
X 6 L C + 4 L c2 + x2 5c2 + 2 c3 - c4
4th I c4
Z
(L + cx) [(
)
(
)
1
+ (
L
6L2 + 4L2c + 8Lcx + 4Lc2 X + 2c2 x2) Log L + cx

3
(
L F
X)
y
3
2
2
3
4
Z
=
2
+ 2
+
3
+ 2 -
3
2 [X (LLC
LLC) X (C
C
c)]
4EI C (L + cx)
z1
+ (
L
2L2 + 4Lcx + 2c2 x2) Log
.
L + cx

The efforts intern are given by the following expressions:

V (X) = F (L - X
y
y
)
1
MF (X) =
F (L - X) 2
Z
y
,
2

constraints by:

V (X)

y
xy (X) = A (X)

MF Z (X) H

y
xx (X) =
Iz (X) 2
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SSLL400 - Variable Poutre of section, subjected to specific efforts
Date:
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Key: V3.01.400-A Page: 13/18


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 inflection in the plan (X O Z).

The equilibrium equation is given by the expression:

2
2w
E I (X)
= A (X) G
2
X
y

2
X
1 2
4 3
4
linear weight
2
with
(

X
With X) = 1
With 1 + C


L
1


2
With
2
c=
- 1

1
With
4

X
and I (X) = I
y
y1 1 + D


L
1
I
y2 4
D =

1.

I -
y1

By integrating first once, we obtain the shearing action intern:

V X = -
With X G dx + C
Z ()
()

1

C1 is given so that V (L)
Z
= 0.

We obtain:
L With G
3
1

X

3
V
X =
- 1 + C
(1 C
Z ()
).
C
3



L +
+




By integrating second once, we obtain the internal bending moment:

M (X) = V (X) dx + C
y
Z
.
2

C2 is calculated so that M (L)
y
= 0.

We obtain:

With G

2
2
2 L C X
M
X
1
2
2
2 2
6
8
3
4
2

y () =
(L X) L + L C + L C + Lcx + Lcx +
.
2

2
2
12 L

+ C X
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J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key: V3.01.400-A Page: 14/18


y
We calculate then rotation starting from the law of behavior = E I (X
y
).
X

M (X
y
)
We thus have (X) =
dx + C
E I (X
y
)
3


with such as (0) = 0.

W
The arrow W (X) is given starting from the relation of Euler: y = -.
X

We calculate W (X) by integration of y (X):

W (X) = -
(X) + C
y
4


with C4 such as W ()
0 = 0.

The 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
· Constraints at the two ends

2.3
Uncertainty on the solution

Analytical solution.

2.4 References
bibliographical

[1]
Report/ratio n° 2314/A of Institut Aérotechnique “Proposition and realization of new cases
tests missing with the validation beams ASTER “

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5.0
Titrate:
SSLL400 - Variable Poutre of section, 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


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,
Hy1 = 0.05 m
Hz1 = 0.10 m
in the loose lead, Hy2 = 0.05 m
Hz2 = 0.05 m

S3 section: variable general section

with embedding,
A1 = 102 m ²
Iy1 = 8.3333 106 m4
at the loose lead,
A2 = 2.5.103 m ²
Iy2 = 5.20833 107 m4

3.2
Characteristics of the grid

3 sections X 10 elements POU_D_E

3.3
Functionalities tested

Commands


AFFE_CARA_ELEM
POUTRE
SECTION
CERCLE



RECTANGLE



GENERALE

MECA_STATIQUE
OPTION
EFGE_ELNO_DEPL



SIGM_ELNO_DEPL

AFFE_CHAR_MECA FORCE_NODALE
FX



FY


MX


MY

FORCE_POUTRE
FX


FY

PESANTEUR



Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL400 - Variable Poutre of section, 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


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


vy (0) 1.0000E+02
1.0000E+02 0.00E+00


vy (L) 1.0000E+02
1.0000E+02 0.00E+00

mfz (0) 1.0000E+02
1.0000E+02 0.00E+00

mfz (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


MX (0) 1.0000E+02
1.0000E+02 0.00E+00


MX (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


vz (0) 0.0000E+00
­ 2.9103E10 0.00E+00


vz (L) 0.0000E+00
0.0000E+00 0.00E+00

mfy (0) 1.0000E+02
1.0000E+02 0.00E+00

mfy (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


vy (0) 1.0000E+02
1.0000E+02 0.00E+00


vy (L) 0.0000E+00
­ 1.8195E10 0.00E+00

mfz (0) 5.0000E+01
5.0000E+01 0.00E+00

mfz (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.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL400 - Variable Poutre of section, 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


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


vy (0) 1.0000E+02
1.0000E+02 0.00E+00


vy (L) 1.0000E+02
1.0000E+02 0.00E+00

mfz (0) 1.0000E+02
1.0000E+02 0.00E+00

mfz (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


MX (0) 1.0000E+02
1.0000E+02 0.00E+00


MX (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


vz (0) 0.0000E+00
­ 3.2014E10 0.00E+00


vz (L) 0.0000E+00
0.0000E+00 0.00E+00

mfy (0) 1.0000E+02
1.0000E+02 0.00E+00

mfy (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


vy (0) 1.0000E+02
1.0000E+02 0.00E+00


vy (L) 0.0000E+00
­ 4.3661E10 0.00E+00

mfz (0) 5.0000E+01
5.0000E+01 0.00E+00

mfz (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


vz (0) ­ 4.4633E+02
­ 4.4635E+02 0.004

mfy (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.
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL400 - Variable Poutre of section, 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


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.
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Outline document