O metodě konečných prvků
Lect_5.ppt
Pitfalls of FE Computing
M. Okrouhlík
Ústav termomechaniky, AV ČR, Praha
Plzeň, 2010
N u m erica l m eth od s in F E A
E qu ilib rium p ro b lem s
K (q ) q  Q
so lu tion o f alg eb raic eq u ation s
S tead y -state v ib ratio n p rob lem s
K  Ω
2

M q  0
g en eralized eig en v alu e p rob lem
P rop ag ation p ro b lem s
  F
Mq
int
F
ext
,
F
int

 B σ dV
T
V
step b y step in teg ratio n in tim e
In lin ear cases w e h av e
  C q  Kq  P (t )
Mq
Now a few examples
from
• Statics
• Steady-state vibration
• Transient dynamics
using
• Rod elements
• Beam elements
• Bilinear (L) and biquadratic (Q) plane elements
S tatic loa d in g o f a ca n tilev er b ea m b y a v ertica l
fo rce a ctin g at th e free en d .
B eam 4 -dof elem en ts (E u ler-B erno u lli) are u sed.
2
4
1
3
P
1
2
3
k max
 6

2 EI 
k  3
l 

 sym.
3l
2l
2
6
 3l
6
3l 
2 
l

 3l 
2 
2l 
Kq  P
v tip 
1
10
PL
3
‘Exact’ formula for thin beams
3 EI
1 .9 047 619 047 619 0 5 e-00 3 1 .9 047 61 904 761 90 4e-003
1 .90 476 19 047 6 0 43 e-00 3
1.9 04 761 904 76 1 904 e-0 03
A thin cantilever beam,
vertical point load at the free end,
four-node plane stress elements
FE technology
Default in ANSYS
L
-3
6
Reduced integration in Marc
maximum displacements for kmax = 10
x 10
5
4
3
"EXACT"
2
Data obtained with
10 elements
1
KHGS
0
0
Exact integration
1
2
0
1
2
3
4
5
6
3
4
5
case
6
7
8
9
k 
B
T
10
EB d V
V
full integration
1 … anal, 2 … Gauss q.,
different types of underintegration
Exact integration
does not yield
‘exact’ results
Gauss quadrature and
underintegrated elements
Summary_2
Isoparametric approach
Four-node bilinear element, plane stress, full integration
Four-node bilinear element, plane stress, full integration
Comparison of beam and L elements
used for modelling of a static loading
of a cantilever beam
• Beam … even one element gives a
negligible error
• L … too stiff in bending, tricks have to be
employed to get “correct” results
A single four-node plane stress elem ent
G eneralized eigenvalueproblem
F ull integration
( K   M )q  0
1
2
3
1
2
3
4
5
6
4
5
6
7
8
7
8
4
3
x 10 GEIVP
GEIVP
15000
2
10000
1
5000
0
0
5
10
consistent
0
0
5
diagonal
10
Free transverse vibration of a thin elastic cantilever beam
FE vs. continuum approach (Bernoulli_Euler theory)
Generalized eigenvalue problem
( K   M )q  0
1
2
3
Two cases will be studied
- L (bilinear, 4-node, plane stress elements)
- Beam elements
kmax
ax ial, bending
and F Eversus
frequenccounter
ies [H z ]
Frequencies
in [Hz]
10000
Analytical axial
8000
L
Analytical bending
6000
4000
FE frequencies
2000
0
0
1
2
3
4
5
6
7
8
9
10
diagonal
Natural frequencies and modes of a cantilever beam … diagonal mass m.
Four-node plane stress elements, full integration
1, 100.506
2, 586.4532
3, 1299.7804
4, 1501.8821
5, 2651.6171
6, 3862.638
7, 3931.881
8, 5258.5331
9, 6313.8768
10, 6567.2502
11, 7802.0473
12, 8571.0469
13, 8907.7934
14, 9804.6267
15, 10394.9981
16, 10520.9827
Sixth FE frequency is the second axial
Seventh FE frequency is the fifth bending
This we cannot say without
looking at eigenmodes
Higher frequencies are useless
due to discretization errors
axial, bending and FE frequencies [Hz]
10000
8000
6000
4000
2000
0
0
1
2
3
4
5
6
consistent
7
8
9
10
Natural frequencies and modes of a cantilever beam … consistent mass m.
Four-node plane stress elements, full integration
1, 101.2675
2, 615.4501
3, 1302.6992
4, 1663.1767
5, 3135.596
6, 3941.6886
7, 4996.7233
8, 6682.2631
9, 7215.1641
10, 9593.4882
11, 9739.1512
12, 12430.9913
13, 12740.3253
14, 14990.9545
15, 16164.1614
16, 16950.4893
Eigenfrequencies of a cantilever beam
Four-node bilinear element, plane strain
Relative errors [%] of FE frequencies
x … axial – continuum, o … bending – continuum, * … FE frequencies
frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*) frekvence analyticke axialni (x), ohybove (o) a MKP frekvence (*)
10000
10000
8000
8000
6000
6000
4000
4000
2000
2000
0
0
1
2
3
4
5
6
7
8
9
10
relativni chyba v % pro axialni (x) a ohybove (o) frekvence
25
0
0
1
2
3
4
5
6
7
8
9
10
relativni chyba v % pro axialni (x) a ohybove (o) frekvence
20
Relative errors for axial (x) and bending (o) freq.
20
10
15
0
10
-10
5
0
Relative errors for axial (x) and bending (o) freq.
1
1.5
2
2.5
3
3.5
konsistentni formulace
4
4.5
5
-20
1
1.5
2
2.5
3
3.5
diagonalni formulace
4
4.5
5
Free transverse vibration of a thin elastic cantilever beam
FE (beam element) vs. continuum approach (Bernoulli_Euler theory)
1
2
2
3
kmax
4
1
 156

 lA 
m 
420 

 sym.
22 l
4l
2
54
13 l
156
 13 l 
2 
 3l

2
 22 l 

2
4l 
3
 6

2 EI 
k  3
l 

 sym.
( K   M )q  0
3l
2l
2
6
 3l
6
3l 
2 
l

 3l 
2 
2l 
Free transverse vibration of a thin elastic beam
ANALYTICAL APPROACH
The equation of motion of a long thin beam considered as continuum
undergoing transverse vibration is derived under Bernoulli-Euler assumptions,
namely
- there is an axis, say x, of the beam that undergoes no extension,
- the x-axis is located along the neutral axis of the beam,
- cross sections perpendicular to the neutral axis remain planar during the
deformation – transverse shear deformation is neglected,
- material is linearly elastic and homogeneous,
- the y-axis, perpendicular to the x-axis, together with x-axis form a principal plane of
the beam.
These assumptions are acceptable for thin beams – the model ignores shear
deformations of a beam element and rotary inertia forces.
For more details see
Craig, R.R.: Structural Dynamics. John Wiley, New York, 1981 or
Clough, R.W. and Penzien, J.: Dynamics of Structures, McGraw-Hill, New York,
1993.
T he equation is usually presented in the form

 v
 v


EI


A

p
(
x
,
t
)
2 
2 
2
x 
x 
t

2
2
2
w here x is a longitudinal coordinate, v is a transversal displacem ent of the beam in y
direction, w hich is perpendicular to x, t is tim e, E is the Y oung’s m odulus, I is the planar
m om ent of inertia of the cross section, A is the cross sectional area and  is the density.
O n the right hand side of the equation there is the loading p ( x , t ) - generally a function
of space and tim e - acting in the xy plane.For free transverse vibrations w e have zero on
the right-hand side of E q. (1). If the bending stiffness E I is independent of tim e and
space coordinates w e can w rite
 v
4
x
A  v
2

4
EI  t
2
 0
.
(4 a)
A ssum ing the steady state vibration in a harm onic form
v ( x , t )  V ( x ) cos(  t   )
w e get
4
d V (x)
dx
4
  V (x)  0
4
(4 b )
w here w e hav e introduced an aux iliary variable b y
   A  /( EI )
4
2
(5)
T he gen eral solution of E q. (4) can be assum ed (see K re ysig, E .: A dvanced
E ngineerin g M athem atics, John W iley & S ons, N ew Y ork, 1993 ) in the form
V ( x )  C 1 sinh  x  C 2 cosh  x  C 3 sin  x  C 4 cos  x
(6)
w here constants C 1 to C 4 depend on boun dary conditions.
A pplyin g boundary cond itions for a thin cantilever beam (clam ped – free) w e get a
freq u en cy d eterm in an t
[0,
[lam,
[sinh(lam*L)*lam^2,
[cosh(lam*L)*lam^3,
1,
0,
cosh(lam*L)*lam^2,
sinh(lam*L)*lam^3,
0,
1
]
lam,
0
]
-sin(lam*L)*lam^2, -cos(lam*L)*lam^2]
-cos(lam*L)*lam^3,
sin(lam*L)*lam^3]
F ro m th e co nd itio n th at th e freq u en cy d eterm in an t is
eq u al to zero w e g et th e frequen cy equ ation in h e
form
cosh  L cos  L  1  0
R oots of this equation can only be found num erically, D enoting x i   i L
w e get th e
natural frequencies in the form
i 
xi
L
2
I
E
A

i  1, 2 , 3 , ...
C om p arison of an alytical an d F E resu lts
counter
1
2
3
4
5
6
7
8
9
continuum frequencies
5.26650 4690912090e+002
3.300 462151726965e+003
9.24 1389593048039e+003
1.81 0943523875022e+004
2.99 3619402962561e+004
4.4 71949023233439e+004
6. 245945376065551e+004
8. 315607746908118e+004
1.0 68093617279631e+005
5
9
x 10
o - FEM, x - analytical
FE frequencies
5.26650 9194371887e+002
3.300 571391657554e+003
9.24 3742518773286e+003
1.81 2669270993247e+004
3.00 1165614576545e+004
4.4 96087393371327e+004
6. 308228786109306e+004
8. 451287572802173e+004
1.0 92740977881639e+005
relative errors for FE frequencies [%]
2.5
8
2
angular frequencies
7
6
FE computation with
10 beam elements
Consistent mass matrix
Full integration
Are analytically computed
frequencies exact to be used
as an etalon for error analysis?
1.5
5
To answer this you have to recall
the assumptions used for the
thin beam theory
4
1
3
2
0.5
1
0
0
5
10
counter
15
20
0
0
5
counter
10
Comparison of Beam and Bilinear Elements
Used for Cantilever Beam Vibration
• 10 beam elements … the ninth bending
frequency with 2.5% error
• 10 beam elements … this element does not
yield axial frequencies
• 10 bilinear elements … the first bending
frequency with 20% error, the errors goes
down with increasing frequency counter
• 10 bilinear elements … the errors of axial
frequencies are positive for consistent mass
matrix, negative for diagonal mass matrix
• Where is the truth?
T ran sien t p ro b lem s in lin ear dyn am ics, no d am p in g
  Kq  P t 
Mq
M o d elling th e 1 D w av e equ atio n
 u
2
t
2
 u
2
 c
2
0
x
2
eps
t = 1.6
dis
1.5
1.4
1.2
1
1
0.5
0.8
0
-0.5
0.6
0
0.5
1
0.4
vel
100
1
50
0.5
0
0
-50
0
0.5
1
L1 cons 100 elem Houbolt (red)
0.5
1
acc
1.5
-0.5
0
-100
0
0.5
1
Newmark (green), h= 0.005, gamma=0.5
Rod elements used here, the results depend on the method of integration
Example of a transient problem
C lassical L am b’s problem
L or Q
1 m
ra d ia l
Loading
a point force
equiv. pressure
1 m
B
Elements
L, Q, full int.
Consistent mass
axisymmetric
Mesh
Coarse 20x20
Medium 40x40
Fine
80x80
Newmark
a xia l
C
A
p0
t
T im p
Axial displacements for point force loading
Fine, Q
Fine L and medium Q
Medium L and coarse Q
Coarse, L
Axial displacements for pressure loading
Point and pressure loading of the solid cylinder by a rectangular pulse in time
12000
P-L
coarse
P-Lmedium
P-Lfine
P-Q
coarse
P-Qmedium
P-Qfine
Total energy
10000
point Q fine
Total energy [J]
8000
FEM-models with the concentrated point-load
6000
point Q medium
4000
point L fine
point Q coarse
2000
point L medium
point L coarse
For all FEM-models with distributed loads
0
0
1
Time [s]
2
x 10
-4
Pollution-free energy production by a proper misuse of FE analysis