Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 1/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
Organization (S):
EDF/RNE/AMV
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.006 document
SDLL06 - Transitory response of a post
embed-free
Summary
In this case test, one analyzes the transitory response of a not deadened embed-free beam, modelized by one
system masses - arises and subjected to an unspecified dynamic loading.
One tests the discrete element in bending, 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).
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 2/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
1
Problem of reference
1.1 Geometry
It is a problem suggested initially in the reference [bib1] and contained in [bib2].
m
Iz
I
y
X
X
y
Fx (T)
B
With
To 000
0
10
0
B
(T)
(T)
Co-ordinates
points (m)
xr (T)
·
beam AB: beam hurled without mass length AB, L = 10 m and of moment of inertia
I
Z
= 0,3285 m
4
.
·
specific mass in b: m = 43,8 10
3
kg
1.2
Properties of materials
Young modulus:
E = 4. 10
10
AP
Density:
= 0 kg/m
3
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)
T (S)
(T)
t0 = 0.025 0.05
0
(m/s2)
P0 = 9,81
Time (S)
0 0.025 0.05
Acceleration according to X (ms
2
)
0 9.81 0
·
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.
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 3/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
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= 3EI
Z
/L
3
= 3,942.10
7
NR/Mr. the superstructure
located at the node of the post is modelized by a specific mass m = 43,8 10
3
kg.
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:
()
&&
X
X
T
R
R
+
= -
2
with
=
=
K
m
E.I
Mr. L
Z
3
3
the Eigen frequency of the system and
X
R
relative displacement of the point B compared to point A. the solution is obtained by integration of
the integral of Duhamel [bib3]:
()
()
(
)
X T
T
T
D
R
T
= -
-
m
sin
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]:
(
)
(
)
(
)
[
]
T
T
X
P
T
T
T
T
T
T
X
P
T
T
T
T T
T
T
T
X
P
T
T T
T
T
T
R
R
R
<
= -
-
< <
= -
- -
-
-
>
= -
-
-
-
-
0
0
2 0
0
0
0
2 0
0
0
0
0
3 0
0
0
2
2
2
2
2
2
:
sin
:
sin
sin
:
sin
sin
sin
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
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 4/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
3 Modeling
With
3.1
Characteristics of modeling
The elements are modelized by discrete elements with 6 degrees of freedom
“DIS_TR”
.
m
y
NO2
NO1
X
y
X
K
(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. 10
4
S).
3.2
Characteristics of the mesh
The mesh consists of 2 nodes and a discrete element (
DIS_TR)
.
3.3 Functionalities
tested
Controls
AFFE_MODELE
GROUP_MA
“MECHANICAL”
“DIS_TR”
AFFE_CARA_ELEM
DISCRETE
NODE
M_TR_D_N
NET
K_TR_D_L
AFFE_CHAR_MECA
DDL_IMPO
MODE_ITER_INV
CALC_FREQ
NEAR
CALC_CHAR_SEISME
MONO_APPUI
MACRO_PROJ_BASE
DYNA_TRAN_MODAL
METHOD
EULER
REST_BASE_PHYS
RECU_FONCTION
RESU_GENE
FORMULATE
CALC_FONC_INTERP
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 5/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
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
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 6/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
5 Modeling
B
5.1
Characteristics of modeling
The elements are modelized by discrete elements with 6 degrees of freedom
“DIS_TR”
.
m
y
NO2
NO1
X
y
X
K
Fx (T)
Fx (T)
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: 10
3
S).
5.2 Functionalities
tested
Controls
AFFE_MODELE
GROUP_MA
“MECHANICAL”
“DIS_TR”
AFFE_CARA_ELEM
DISCRETE
NODE
M_TR_D_N
NET
K_TR_D_L
AFFE_CHAR_MECA
DDL_IMPO
FORCE_NODALE
MODE_ITER_INV
CALC_FREQ
NEAR
CALC_CHAR_SEISME
MONO_APPUI
MACRO_PROJ_BASE
DYNA_TRAN_MODAL
METHOD
EULER
REST_BASE_PHYS
RECU_FONCTION
RESULT
5.3
Characteristics of the mesh
It is the same mesh as for modeling A.
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 7/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
6
Results of modeling B
6.1 Values
tested
Relative displacement of node NO1 (in meters).
Time (S)
References
[bib1], [bib2]
Code_Aster Error
(%)
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
Code_Aster
®
Version
5.0
Titrate:
Transitory SDLL06 Response of a embed-free post
Date: 14/09/01
Author (S): Fe WAECKEL
Key: V2.02.006-C Page: 8/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-62/01/012/A
7
Summary of the results and remarks general
The simplified model presented in this case test makes it possible to validate the method of resolution numerical.
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 pitch of time of 5. 10
4
S is about 0,01%;
for modeling B (not of time of 10
3
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.