Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.008 document
SDLL08 - Plane Grillage of beams
(metal sections)
Summary:
This three-dimensional problem first of all consists in carrying out a modal analysis and then to study
harmonic response of a mechanical structure of a plane netting of beams. This test of Mécanique of
Structures corresponds to a dynamic analysis of a linear model having a linear behavior. It
only one modeling includes/understands.
This problem thus makes it possible to test the element of beam of Euler Bernouilli in transverse inflection, the calculation of
frequencies and of the modes of vibration by the method of Lanczos and the use of linear relations enters
displacements of two points in modal analysis and harmonic answer.
The results are in agreement with the analytical results of guide VPCS.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
2/6
1
Problem of reference
1.1 Geometry
Z, W
F
C
FG
y, v
L1
B = H
G
E = I
X, U
With
D
L2
Length: L1 = L2 = 5 m
Cross-section (section out of I): IPE 200
surface
With = 2.872 10­3 m2
moment of inertia
Iz = 1.943 10­5 m4
(other parameters of beam not used)
Co-ordinates of the points (in meters):
With
B = H
C
D
E = I
F
G
X
­ 2.5
­ 2.5
­ 2.5
2.5
2.5
2.5
0
y
­ 2.5
0.
2.5
­ 2.5
0.
2.5
0.
Z
0.
0.
0.
0.
0.
0.
0.
1.2
Material properties
E = 2.1011 Pa
= 7.800. kg/m3
1.3
Boundary conditions and loadings
Points A, C, D, F: (U = v = W = 0.)
Points B, E: rotulée connection (continuity of U, v, W)
Sinusoidal force at the point G
F
T
() = F
G
0 sin T
F =
0
­ 1105 NR
= 80 rad/S
1.4 Conditions
initial
With T = 0, structure at rest.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
3/6
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is that given in card SDLL08/89 of the guide VPCS which presents
method of calculation in the following way:
A method of Rayleigh-Ritz makes it possible to make calculation with two degrees of freedom from
assumptions of following symmetrical deformations:
· for the point of X-coordinate there of the members AC and DF L1 length


y + L1

2
W
= W
AB
B sin
L1
· for the point of X-coordinate X of the cross-piece BE L2 length


X + L2

2
W
=
+
BE
WB
WG sin
L2
L1
L2
y
X
With
B
C
B
G
E
WB
WB
WAC
W
W
G
BE
2.2
Results of reference
The first two Eigen frequencies and symmetrical clean modes (other frequencies
clean of this system are not studied). For the clean modes, one with the value
following: WB/WG
In harmonic answer one a:
·
WB max and WG max,
+
·
W
W
B
G max at the point G.
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
J.M. BIGGS. Introduction to Structural Dynamics. New York: Mc Graw Hill, p.184 (1964).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
One uses the element of beam of Euler Bernouilli POU_D_E
y
C
F
B H
G
I
E
X
With
D
3 beams:
ABC, DEF, cut out HGI each one in 10 meshs SEG2
The nodes (B, H) and (E, I) have the same co-ordinates.
Limiting conditions:
beams ABC and DEF
DDL_IMPO:
(GROUP_NO: (PABC, PDEF)
DX: 0., DY: 0., DRY: 0. )
beam HGI
(GROUP_NO: (PHGI)
DX: 0., DY: 0., DRX: 0. )
nodes ends
(GROUP_NO: (NACDF)
DZ: 0. )
Liaison_ddl:
DZB ­ DZH = 0. and DZE ­ DZI = 0.
Force_nodale:
Node: G Fz: ­ 1.E5
Names of the nodes:
With = N1
B = N6
C = N11
D = N21
E = N26
F = N31
H = N41
G = N46
I = N51
3.2
Characteristics of the grid
A number of nodes:
33
A number of meshs and types:
3 * 10 = 30 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
TOUT
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
GROUP_NO
[U4.25.01]
LIAISON_DDL
FORCE_NODALE
NOEUD
AFFE_MATERIAU
TOUT
[U4.23.02]
AFFE_MODELE
“MECANIQUE”
'POU_D_E
TOUT
[U4.22.01]
DEFI_MATERIAU
ELAS
[U4.23.01]
MODE_ITER_SIMULT
METHODE
“TRI_DIAG”
[U4.52.02]
CALC_FREQ
OPTION
“PLUS_PETITE”
COMB_MATR_ASSE
[U4.53.01]
3.4 Remarks
The blocking of ddl DX and DY in all the nodes makes it possible to select only the modes of inflection
transverse (in the “vertical” plane).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
5/6
4
Results of modeling A
4.1 Values
tested
Frequency (Hz)
Command of the clean mode
Reference
Aster
% difference
1
16.456
16.4190
­ 0.22
2
38.165
38.0468
­ 0.31
Clean mode: value of WB/WG
Command of the clean mode
Reference
Aster *
% difference
symmetrical
1
1.213
1.213
0.
2
­ 0.412
­ 0.412
0.
*
WB = DZ out of B (N6)
WG + WB = DZ in G (N46)
mode 1:
WB = 0.5480
WG + WB = 1.
mode 2:
WB = ­ 0.6698
WG + WB = 0.9559
Harmonic answer:
Not
Type of value
Reference
Aster
% difference
(m)
B, E
WB max
­ 0.098
­ 0.1003
2.45
G
W
­ 0.125
­ 0.1271
1.60
G max *
G
W
­ 0.227
­ 0.2274
0.18
B + WG max
4.2 Remarks
Calculations carried out by:
MODE_ITER_SIMULT METHOD: “TRI_DIAG”
OPTION: “PLUS_PETITE”
NMAX_FREQ: 3
One obtains an antisymmetric mode for a frequency F = 22.5676 Hz. This Eigen frequency
depends on the constant of provided torsion; this one is not defined in the bench-mark data.
Values WB/WG are not checked in the test but are obtained manually starting from WB
and WG + WB.
Value (WG) max is not checked in the test. One has only access to WB max and (WB + WG)
max. WG max is obtained manually by difference.
Contents of the file results:
the first 3 Eigen frequencies, displacement of the nodes B, E, G in harmonic answer.
4.3 Parameters
of execution
Version: 3.02.21
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
5 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL08 plane Grillage of beams (metal sections)
Date:
07/01/98
Author (S):
B. QUINNEZ
Key:
V2.02.008-C Page:
6/6
5
Summary of the results
The values of the Eigen frequencies and the clean vectors are obtained with a precision < 0.3%.
The variation of 2.5% on the maximum arrows at the points B and E would deserve to check the solution of
reference, to supplement the validation of the harmonic answer.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/96/035 - Ind A