Code_Aster ®
Version
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Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 1/8

Organization (S): EDF/RNE/AMV

Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.006 document

SDLL06 - Transitory Réponse of a post
embed-free

Summary

In this case test, one analyzes the transitory response of a not deadened embed-free beam, modelled by one
system masses - arises and subjected to an unspecified dynamic loading.

One tests the discrete element in inflection, the calculation of the clean modes by the method of Lanczos and calculation of
transitory response by modal recombination of the subjected structure is with a accélérogramme (modeling
A) maybe with an equivalent imposed force (modeling B).

The diagram of Euler is used.

The results obtained are in concord with the results of reference (analytical results).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 2/8

1
Problem of reference

1.1 Geometry

It is a problem suggested initially in the reference [bib1] and contained in [bib2].

xr (T)
Co-ordinates
Fx (T)
B
m
points (m)
0
0
To 0
B 10
Iz
0
0
I
y
y
X
With
X
(T)
(T)


· beam AB: beam hurled without mass length AB, L = 10 m and of moment of inertia
IZ = 0,3285 m4.
·
3
specific mass in b: m = 43,8 10 kg

1.2
Properties of materials

Young modulus:
10
E = 4. 10 Pa
Density:
= 0 kg/m3

1.3
Boundary conditions and loadings

Boundary conditions:
Only authorized displacements are the translations according to axis X.
Point A is embedded: dx = Dy = dz = drx = dry = drz = 0.

Loadings:
· modeling a: transverse acceleration at point a: (T)


Time (S)


0 0.025 0.05
(T)
(m/s2)

Acceleration according to X (ms2)
0 9.81 0
P0 = 9,81
T (S)
0
t0 = 0.025 0.05


· modeling b: forces transverse at point b: Fx (T) with Fx (T) = ­ Mr. (T)

1.4 Conditions
initial

The system is at rest: with T = 0, dx (0) =0, dx/dt (0) = 0 in any point.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 3/8

2
Reference solution

2.1
Method of calculation used for the reference solution

The problem is dealt with by model with degree of freedom. The post is regarded as one
slim not deadened and nonheavy beam of rigidity k= 3EIZ/l3 = 3,942.107 NR/Mr. the superstructure
3
located at the node of the post by a specific mass m = 43,8 10 kg are modelled.

The two loading cases lead to the calculation of the response of a system to a degree of freedom subjected
with an acceleration (T) of an unspecified form:

K
3e.I
&
X + 2 X = - (T
Z
R
R
) with =
=
the Eigen frequency of the system and X
m
Mr. l3
R it
relative displacement of the point B compared to point A. the solution is obtained by integration of
the integral of Duhamel [bib3]:

T
m
X (T) = -
(T) sin (T -) D
R



0

2.2
Results of reference

Displacement relating to the point B.

For a triangular imposed acceleration, one can calculate the integral of Duhamel analytically
[bib3]:


P0
sin T

T
< T
: X
0
= -
T
R
-
2
T


0


P
2 sin
0
(T - t0) sin T

T
0 < T < 2t: X
0
= -
2t0 - T
R
-
-



2
T


0



P
T
> 2t
: X
0
0
R = -
2 sin
sin
2
sin
3
[
(T - t0) -
(T - t0) -
T]

T

0

2.3
Uncertainty on the solution

No if one calculates the integral of Duhamel analytically [bib3]. About the precision of
method of integration numerical employed to calculate the integral of Duhamel ([bib1], [bib2]):
method of Simpson with 40 points per period.

2.4 References
bibliographical

[1]
R.W. Clough and J. Penzien: Dynamics off structures New York, Mac Graw-Hill, 1975,
p. 102-105
[2]
Guide Technical VPCS AFNOR - 1990
[3]
J.S. Przemieniecki: Theory off matrix structural analysis New York, Mac Graw-Hill, 1968,
p. 351-357
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 4/8

3 Modeling
With

3.1
Characteristics of modeling

The elements are modelled by discrete elements with 6 degrees of freedom “DIS_TR”.

y
y
NO2
m
K
NO1
X
X
(T)
(T)


Node NO1 is subjected to an imposed acceleration (T). One calculates the relative displacement of the node
NO2 compared to the displacement of node NO1 and one compares it with calculated displacement
analytically.
Temporal integration is carried out with the algorithm of Euler (not of time: 5. 10­4 S).

3.2
Characteristics of the grid

The grid consists of 2 nodes and a discrete element (DIS_TR).

3.3 Functionalities
tested

Commands
AFFE_MODELE
GROUP_MA
“MECANIQUE”
“DIS_TR”
AFFE_CARA_ELEM
DISCRET
NOEUD
M_TR_D_N
MAILLE
K_TR_D_L
AFFE_CHAR_MECA
DDL_IMPO
MODE_ITER_INV
CALC_FREQ
PROCHE
CALC_CHAR_SEISME
MONO_APPUI
MACRO_PROJ_BASE
DYNA_TRAN_MODAL
METHODE
EULER
REST_BASE_PHYS
RECU_FONCTION
RESU_GENE
FORMULE
CALC_FONC_INTERP

Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 5/8

4
Results of modeling A

4.1 Values
tested

Relative displacement of node NO1 (in meters).

Time (S)
Analytical calculation
Code_Aster Error
(%)
0,010 ­ 6,511E05
­ 6,495E05
0
0,015 ­ 2,185E04
­ 2,183E04
0
0,020 ­ 5,139E04
­ 5,136E04
­ 0,058
0,024 ­ 8,809E04
­ 8,806E04
­ 0,039
0,026 ­ 1,115E03
­ 1,115E03
­ 0,041
0,030 ­ 1,679E03
­ 1,679E03
­ 0,014
0,035 ­ 2,523E03
­ 2,523E03
­ 0,004
0,040 ­ 3,457E03
­ 3,457E03
0
0,045 ­ 4,412E03
­ 4,412E03
0,004
0,049 ­ 5,143E03
­ 5,143E03
0,005
0,051 ­ 5,485E03
­ 5,485E03
0,005
0,055 ­ 6,109E03
­ 6,109E03
0,005
0,060 ­ 6,765E03
­ 6,765E03
0,005
0,065 ­ 7,269E03
­ 7,269E03
0,005
0,070 ­ 7,610E03
­ 7,610E03
0,005
0,075 ­ 7,779E03
­ 7,780E03
0,005
0,080 ­ 7,774E03
­ 7,775E03
0,004
0,085 ­ 7,595E03
­ 7,595E03
0,004

4.2 Parameters
of execution

Version:
STA 5.02


Machine:
SGI ORIGIN 2000

Time CPU To use:
3,16 seconds



Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 6/8

5 Modeling
B

5.1
Characteristics of modeling

The elements are modelled by discrete elements with 6 degrees of freedom “DIS_TR”.

y
y
Fx (T)
Fx (T)
m
NO2
K
NO1
X
X


Node NO2 is subjected to an imposed force Fx (T). One calculates the relative displacement of node NO2
compared to the displacement of node NO1 and one compares it with the displacement calculated in
references [bib1] and [bib2].
Temporal integration is carried out with the algorithm of Euler (not of time: 10­3 S).

5.2 Functionalities
tested

Commands


AFFE_MODELE
GROUP_MA
“MECANIQUE”
“DIS_TR”
AFFE_CARA_ELEM
DISCRET
NOEUD
M_TR_D_N


MAILLE
K_TR_D_L
AFFE_CHAR_MECA
DDL_IMPO
FORCE_NODALE
MODE_ITER_INV
CALC_FREQ
PROCHE
CALC_CHAR_SEISME
MONO_APPUI
MACRO_PROJ_BASE
DYNA_TRAN_MODAL
METHODE
EULER
REST_BASE_PHYS
RECU_FONCTION
RESULTAT

5.3
Characteristics of the grid

It is the same grid as for modeling A.

Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 7/8

6
Results of modeling B

6.1 Values
tested

Relative displacement of node NO1 (in meters).

Time (S)
References
Code_Aster Error
(%)
[bib1], [bib2]
0,01
­ 6,500E05 ­ 6,447E05 ­ 0,82
0,02
­ 5,130E04 ­ 5,127E04 ­ 0,064
0,03
­ 1,679E03 ­ 1,678E03 ­ 0,037
0,04 ­ 3,457E03
­ 3,457E03
0,013
0,05 ­ 5,316E03
­ 5,317E03
0,022
0,06 ­ 6,764E03
­ 6,766E03
0,035
0,07 ­ 7,609E03
­ 7,611E03
0,027
0,08 ­ 7,774E03
­ 7,776E03
0,024
0,09 ­ 7,244E03
­ 7,246E03
0,028
0,1 ­ 6,068E03
­ 6,069E03
0,014
0,12
­ 2,242E03 ­ 2,242E03 ­ 0,017
0,14
2,367E03 2,369E03 0,071
0,16
6,149E03 6,152E03 0,041
0,18
7,783E03 7,785E03 0,029
0,2
6,698E03 6,699E03 0,018

6.2 Parameters
of execution

Version:
STA 5.02


Machine:
SGI ORIGIN 2000

Time CPU To use:
1,9 seconds


Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Code_Aster ®
Version
5.0
Titrate:
Transitory SDLL06 Réponse of a embed-free post

Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 8/8

7
Summary of the results and remarks general

The simplified model presented in this case test makes it possible to validate the method of numerical resolution.
To deal with the real physical problem, it would be necessary to take into account the effects of inertia (mass of
post, effect of inertia of rotation around B of the superstructure) and of compression of the post
(actual weight).

For modeling A, the error made with a step of time of 5. 10­4 S is about 0,01%;
for modeling B (not of time of 10­3 S) it is about 0,6%.

One will be able to supplement this case test by checking the convergence of the results for other step values
time and by comparing the results obtained with other diagrams of integration.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A

Outline document