development of continuum mechanics based beam elements...
TRANSCRIPT
Kyungho Yoon
2014
Development of Continuum Mechanics Based Beam Elements for
Linear and Nonlinear Analysis
1
Contents
1. Introduction to Continuum Mechanics Based Beam Elements
2. Warping Model for Discontinuously Varying Cross-section
3. Implementation for General Nonlinear Analysis
4. Improvement of Nonlinear Performance
5. Summary & Future Works
2
1. Introduction to Continuum Mechanics Based Beam Elements
3
Applications of Beam Finite Element Analysis
<Fabric bundle><Offshore platform><Bridge>
<Shell model>462 DOFs
<Beam model>36DOFs
<Solid model>1134 DOFs
<Physical model>4
Literature of Beam Element
• Search word “beam element” limit to exact keyword “finite element method”
5
Various Beam Elements
ABAQUSB21, B31, B22, B32, B23, B33, B34
ADINABEAM, ISOBEAM
ANSYSBEAM3, BEAM4, BEAM23, BEAM24, BEAM44, BEAM54, BEAM161, BEAM188, BEAM189
• In commercial software
• J. Frischkorn, S Reese, A solid-beam finite element and nonlinear constitutive modelling, Comput. Methods Appl. Mech. Eng. 2013.
• R Gonçalves, M Ritto-Corrêa, D Camotim, A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory, Comput. Mech. 2010.
• S.R. Eugster, C. Hesch, P. Betsch, Ch. Glocker, Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates, Int. J. Numer. Meth. Engng. 2013.
• F. Mohri, N. Damil, M.P. Ferry, Large torsion finite element model for thin-walled beams, Comput. Strut. 2008.
• In literature
• Degenerated beam• Z.X. Li, A co-rotational formulation for 3D beam element using vectorial rotational variables, Comput. Mech. 2007.• P.S. Lee, G. McClure, A general three-dimensional L-section beam finite element for Elastoplastic large deformation
analysis, Comput. Struct. 2006.
Pros• 3D geometry• Fully coupled strain states• Simple and Straightforward• Modeling capability
Cons• Cross-sectional shape• Nonlinear performance
6
Problem description #1 – Twisting Action
• Warping effect
Twisting angle
Tors
ion
0 1 2 3 4 50
100
200
300
400
with warpingwithout warping
26078=k 8284.81=k
stiff flexible
7
Problem description #1 – Twisting Action
St. Venant torsion theory
Vlasov thin-walled theory
Benscoter theory Jourawsky theory
Formulation
Uniform torsion
O O O O
Non-uniformtorsion
X O O O
Secondary warping
X X O O
• Mathematical theory
( , )wu y zϕ=x
xzyu xw ∂
∂=
)(),(θ
φ )(),( xzyuw αφ= ),()(
),( zyx
xzyu xw ψ
θφ +
∂∂
=
• Finite element implementation• A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A mixed beam model with non-uniform warping derived from the
Saint Venant rod. Comput. Struct. 2014.• A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A geometrically exact beam model with non-uniform warping
coherently derived from the Saint Venant rod, Engng. Struct 2014.• C.S. Jog, I.S. Mokashi, A finite element method for the Saint-Venant torsion and bending problems for prismatic
beams,. Compt. Struct. 2014.• J. Murin, M. Aminbaghai, V. Kutis, A new Timoshenko finite beam element including non-uniform torsion of open
and closed cross-sections, Engng. Struct. 2014.8
Problem description #2 – Degenerated Element
• Degeneration • Previous approach
• New approach
Vxx t+= 1( )i ih t= ∑x x
Vxxx )(1 ∑+= iih
Embeded node
Square frame
Change Gauss point
9
Degeneration Procedure
degeneration
degeneration
∑∑∑===
++=q
k
kz
tmkk
q
k
ky
tmkk
q
kk
tk
mt zrhyrhrh1
)(
1
)(
1
)( )()()( VVxx
∑=
=p
j
mjkj
mk ytshy
1
)()( ),( ∑=
=p
j
mjkj
mk ztshz
1
)()( ),(
• Geometry interpolation of the sub-beams• Geometry interpolation
of solid model
• Kinematic assumption
∑=
=n
i
mi
ti
mt tsrh1
)()( ),,( xx
kz
tmjk
ky
tmjkk
tmjk
t zy VVxx )()()( ++=
∑ ∑= =
=q
k
mjk
tp
jjk
mt tshrh1
)(
1
)( ),()( xx
10
Degeneration Procedure
∑=
=p
j
mjkj
mk ytshy
1
)()( ),( ∑=
=p
j
mjkj
mk ztshz
1
)()( ),( ∑=
=p
j
mjkj
mk ftshtsf
1
)()( ),(),(
∑∑∑∑====
+++=q
k
kx
tk
tmkk
q
k
kz
tmkk
q
k
ky
tmkk
q
kk
tk
mt tsfrhzrhyrhrh1
)(
1
)(
1
)(
1
)( ),()()()()( VVVxx α
11
Calculation of Warping Function
∑=
=p
j
mjkj
mk ftshtsf
1
)()( ),(),(
• Finite element discretization
• St. Venant equations
02
)(2
2
)(2
=∂
∂+
∂∂
zf
yf m
km
k Ωin
ynznz
fny
fn zy
mk
z
mk
y −=∂
∂+
∂∂ )()(
Γon
• Multiply connected cross-section• M.M.S. Nasser et. al., Boundary integral equations with the generalized Neumann
kernel for Laplace’s equation in multiply connected regions, Applied Mathematics and Computation, 2011.
• R. Barrettam, On stress function in Saint-Venant beams, Meccanica, 2013.• M.D. Bird, C.R. Steele, A solution procedure for laplace’s equation on multiply
connected circular domains, Journal of Applied Mechanics, 1992.
12
Novel Features
The formulation is simple and straightforward.
The formulation can handle all complicated 3-D geometries including curved and twisted geometries, varying cross-sections, and arbitrary cross-sectional shapes (including thin/thick-walled and open/closed cross-sections).
Warping effects fully coupled with bending, shearing, and stretching are automatically included.
Seven degrees of freedom (only one additional degree of freedom for warping) are used at each beam node to ensure inter-elemental continuity of warping displacements.
The pre-calculation of cross-sectional properties (area, second moment of area, etc.) is not required because the beam formulation is based on continuum mechanics.
Analyses of short, long, and deep beams are available, and eccentricities of loadings and displacements on beam cross-sections are naturally considered.
The basic formulation can be easily extended to general nonlinear analyses that consider geometrical and material nonlinearities. 13
Curved Beam Problem
• Problem description • Numerical results
D/Rw
N=4 N=8 Ref.
0.1 1.8613E-05 1.8599E-05 1.9020E-05
0.2 4.0321E-06 4.0296E-06 4.2771E-06
0.5 6.1726E-08 6.1694E-08 6.1724E-08
14
Tapered Beam Problem
• Problem description • Numerical results
Constrained warpingPresent study Ref.
0.025 1.1454E-04 1.16549E-040.095 5.9483E-05 6.01527E-050.225 2.0123E-05 1.91705E-050.475 8.2445E-06 7.11663E-06
θ5.0tan
15
Turbine Blade Problem
• Problem description • Numerical results• Multiply connected cross-section• Curved, twisted and tapered geometry• Eccentricity
Shell : 4680 DOFsBeam : 56 DOFs
16
Journal Paper
17
2. Warping Model for Discontinuously Varying Cross-section
18
Motivation
• Challenge
• Previous models• Free warping• Constrained warping• Varying & curved geometry
• Twisting center• Warping displacement shape
19
[ ] kr
q
k
kR
mR
kL
mLk
mkk
mw tsftsftsfrhtsr Vu ∑
=
++=1
)()()()( ),(),(),()(),,( ββα
Free warping
Interface warpings
New Warping Displacement Model
kr
q
kk
mkk
mw tsfrhtsr Vu ∑
=
=1
)()( ),()(),,( α
• Previous model • New warping displacement
20
Free Warping Function
• Existing method
• Proposed new method
1. Solve St. Venant equation for global coordinates2. Calculate twisting center3. Calculate warping function for twisting center
3 step procedure
Simultaneously calculate the free warping function and the corresponding twisting center
Coordinate transformed St. Venant equation
+ Zero bending moment condition
=
−
00BF
00H00HNNK G
GG
GGG
y
z
z
y
zy
λλ
one step procedure
21
Interface Warping Function
=
−
−−−
0
BB
λ
FF
000LL000HH000HHLNNK0
LNN0K
00
2
1
21
21
222
111(II)
(I)
y
z
(II)
(I)
(II)T(I)T
(II)z
(I)z
(II)y
(I)y
(II)(II)z
(II)y
(II)
(I)(I)z
(I)y
(I)
GG
GGGG
GGGGGG
λλ
Coordinate transformed St. Venant equation with twisting center multiplier+ Zero bending moment condition
+ Constraint condition for interconnected domain
22
Discontinuously Varying Thin-walled Cross-section
• Problem description • Numerical results
107 DOFs
230 DOFs Bending induced by torsion
23
Partially Constrained Warping Problem
• Problem description • Numerical results
24
Curved beam problem
• Problem description • Numerical results
25
Journal Paper
26
3. Implementation for General Nonlinear Analysis
27
Geometric nonlinearity Material nonlinearityStrain-displacement relation Finite rotation Strain-stress relation
Green-Lagrange strain Rodrigues formulation Implicit return mapping scheme with beam state projection
Implementation for nonlinear analysis
∫∫∫∫ −ℜ=++ ∆+
V ijijttt
V ijijt
V ijijt
V rsijijrs VdeSVdSVdSVdeeC0000
000
000
000
000 δκδηδδ
ℜ= ∆+∆+∆+∫ tt
V ijtt
ijtt VdS
0
000 εδ
• Principle of virtual work
12
jiij
j i
uux x
ε ∂∂
= + ∂ ∂
1 12 2
ji k kij
j i i j
uu u ux x x x
ε ∂∂ ∂ ∂
= + + ∂ ∂ ∂ ∂
cos sinsin cos
θ θθ θ
− =
R
11θ
θ−
=
R ij ijkl klS C ε=
[ ]Tkkz
ky
kxkkkk wvu αθθθ 00000000 =Uwith parameter
• Incremental equilibrium equation
eij ijkl klS C ε=e p
ij ij ijε ε ε= +
28
Implementation for nonlinear analysis
∑∑∑∑====
+++=q
k
kx
tk
tmkk
q
k
kz
tmkk
q
k
ky
tmkk
q
kk
tk
mt tsfrhzrhyrhrh1
)(
1
)(
1
)(
1
)( ),()()()()( VVVxx α
( ) ( ) ( ) ( )0 0
1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( , )( )
q q q qm m t t k t k m t t k t k m t t t t k t t k
k k k k y y k k z z k k k x k xk k k k
h r h r y h r z h r f s t α α+∆ +∆ +∆ +∆
= = = == + − + − + −∑ ∑ ∑ ∑u u V V V V V V
202
0
00
0
00 )(ˆcos1
)(ˆsin)( k
k
kk
k
kk θRθRIθR
θ
θθθ −
++=
[ ]Tkz
ky
kx
k θθθ 0000 =θ2
02
02
00kz
ky
kx
k θθθθ ++=
−−
−=
00
0)(ˆ
00
00
00
0kx
ky
kx
kz
ky
kz
k
θθθθθθ
θRwith
kx
tkkx
tt VθRV )(0=∆+ ky
tkky
tt VθRV )(0=∆+ kz
tkkz
tt VθRV )(0=∆+
• Geometry interpolation
• Displacement interpolation
• Finite rotation parameterization
• Finite rotation of director
29
Wagner Strain
• Y.L. Pi, M.A. Bradford, B. Uy, A spatially curved-beam element with warping and Wagner effects, Int. J. Numer. Meth. Engng 2005.
• F. Mohri, N. Damil, M.P. Ferry, Large torsion finite element model for thin-walled beams, Comput. Struct. 2008.
Addtionally introduce Wagner moment and Wagner strain
Additionally introduce three Wagner’s coefficients
<Abaqus> <Consider Wagner effect> To consider fully coupled nonlinear strain terms is very important
Explicitly consideration
Explicitly consideration
30
Wagner strain
=)(
)(
0
)(0
m
mm
zy
xx
+−
+=
xm
xm
xm
xm
tmk
t
mt
zyzy
fx
θθθθ
α
cossinsincos
)()(
)()(
)(
)(x
- Geometry interpolation
- Covariant vectors
∂∂=
00
0
)(1
0
rxmg
−∂∂−−∂∂
∂∂+∂∂=
)sincos()cossin(
)()(
)()(
)(
)(1
xm
xm
x
xm
xm
x
tmk
t
mt
zyrzyr
rfrx
θθθθθθ
αg
22)(2)(2)()(0
)(110 ))((
21)(
21
rzy
rf
rf
rx xmm
tm
k
tm
kmt
∂∂
++∂∂
+∂∂
∂∂
=θααε
- Covariant Green-Lagrange strain
- Local Green-Lagrange strain
22)(2)(2)()(110 ))((
21)(
21
xzy
xf
xf xmm
tm
k
tm
kt
∂∂
++∂∂
+∂∂
=θααε
31
Cross-Shaped Beam Problem
- Problem description
Elasto perfectly plastic material model211 /100.2 mNE ×=
0=ν
0=H
111894 DOFs
42 DOFs
32
Cross-Shaped Beam Problem
- Numerical results
Beam problemsSolid elemen
t model(ADINA)
Shell element model(ADINA)
Beam element models
ADINA BEAM(ADINA)
BEAM188(ANSYS)
Present beam
Cross-shapedbeam problem
X X X
Solid element
Presentbeam element
DOFs 111894 42
Load step
800 1
Twisted Cantilever Beam Problem
- Problem description
211 /100.2 mNE ×=
0=ν
Elastic material model
34
Twisted Cantilever Beam Problem
- Numerical results
Beam problems
Solid element mo
del(ADINA)
Shell element mo
del(ADINA)
Beam element models
ADINA BEAM
(ADINA)
BEAM188(ANSYS)
Present beam
Twisted cantilever beam problem(Load Case I)
X X
Twisted cantilever beam problem(Load Case II)
X X 35
Lateral post buckling problem
- Problem description
36
Lateral post buckling problem
- Numerical results
37
Journal paper
38
4. Improvement of Nonlinear Performance
39
Problem Description
∫∫∫∫ −ℜ=++ ∆+
V ijijttt
V ijijt
V ijijt
V rsijijrs VdeSVdSVdSVdeeC0000
000
000
000
000 δκδηδδ
Shear force problemBending moment problem
1. Upper bound2. Slow convergence3. Solution reliability
• Problems in nonlinear analysis
1. Upper bound means accumulated error.2. There is no upper bound when simulating with linear stiffness matrix only.
The error is accumulated in nonlinear stiffness matrix term.40
Problem Description
“ It was shown that the finite element interpolation of all of theses rotation variables spoils the objectivity of the adopted strain measures with respect to a rigid rotation. The formulations based on the interpolation of iterative and incremental rotations were additionally proved to be dependent on the history of deformation. Interestingly, the co-rotational technique does not necessarily suffer from these drawbacks. ”
• G. Jelenic, M.A. Crisfield, Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics, Comput. Methods Appl. Mech. Engrg. 1999.
“ Under rotations including a significant rigid-body component, many elements produce over-stiff solutions due to ‘self-straining’.”
‘Self-straining’ of the rigid body rotation
♦ Stiffening of rigid body rotation
♦ Deficiency of shear eigenmode
41
Stiffening of rigid body rotation
• Principle of virtual work
∫∫ +=ΠV ijij
t
V klijklij VdSVdeCe00
000
000 ηδδδ
∫ ++=ΠVlinear VdeeGeeEeeE
0
0120120220220110110 4 δδδδ
∫ ++=ΠV
tttnonlinear VdSSS
0
0120120220220110110 2 ηδηδηδδ
• Eigen deformation
[ ] [ ]TTrrr xy−==
42
21 φφφ
• Strain values
01110110 =
∂∂
==x
ee rφδ 02220220 =
∂∂
==y
ee rφδ 0)(21 21
120120 =∂∂
+∂∂
==xy
ee rr φφδ
=∂∂
∂∂
+∂
∂∂∂
+∂∂
∂∂
+∂
∂∂∂
=
=∂
∂∂∂
+∂
∂∂∂
=
=∂
∂∂∂
+∂
∂∂∂
=
0)(21
8181
22221111120
2211220
2211110
yxyxyxyx
yyyy
xxxx
rrrrrrrr
rrrr
rrrr
φδφδφφφδφδφφηδ
δφφδφφηδ
δφφδφφηδ
• Internal virtual work
0=Π= r
linear φuδ ∫ +=Π
= V
ttnonlinear VdSS
r 0
0220110 )(
81
φuδ
42
Deficiency of shear eigenmode
• Eigen deformation
Aφ Bφ Cφ[ ] [ ]TTAAA xy
42
21 == φφφ
[ ]TrAB y 022
=−= φφφ
[ ]TrAC x022
=+= φφφ
• Internal virtual work
=Π=Π
=Π=Π
+=Π=Π
∫∫∫
==
==
==
V
tnonlinearlinear
V
tnonlinearlinear
V
ttnonlinearlinear
VdSG
VdSG
VdSSG
CC
BB
AA
0
0
0
0110
0220
0220110
21,221,2
)(81,2
φuφu
φuφu
φuφu
δδ
δδ
δδ
43
Study on Nonlinear Stiffness
∫∫∫ −=+ ∆+
V ijt
ijtt
V ijt
ijV rsijrs
ij VdSeRVdSVdeCe000
000
00
000 δδηδTotal Lagrangian :
NLKLK
TDDD
TCCC
TBBB
TAAANL
1111111111111111 φφφφφφφφK λλλλ +++=∑=
=8
1i
TiiiL φφK λ
TD
TD
TD
TD
TDDD 366366331111 φφφφφφφφφφ λλλλλ +++= 44
Eigen Recomposition
∫∫∫ −=+ ∆+
V ijt
ijtt
V ijt
ijV rsijrs
ij VdSeRVdSVdeCe000
000
00
000 δδηδTotal Lagrangian :
NLKLK
Eigen decomposition : ∑=i
TiiiNL φφK λ
Estimation of eigenvalue : iNLTiiNL
Tii φKφφKφ ≅=λ
Eigen recomposition : ∑=i
TiiiNL φφK λ
♦ Assumed Eigenvector for 2D beam
iφ( : assumed eigenvectors)
45
Straight cantilever problem
Load(single load
step)
Number of iteration
Without recomposition
With recomposition
2 9 7
4 14 11
6 19 13
8 X 14
10 X 13
20 X 15
30 X 16
40 X 16
50 X 18
46
Straight cantilever problem
Load(single load
step)
Number of iteration
Without recomposition
With recomposition
100 25 24
200 41 37
300 59 49
400 84 66
500 109 86
600 136 103
700 167 135
800 205 165
900 245 201
1000 289 234 19% improved47
Right angle frame problem
48
Offshore jacket problem
49
Summary & Future works
50
Summary
The continuum mechanics based beam finite elements allowingtwisting behavior are proposed.
The new modeling method to construct continuous warping displacement fields for discontinuously varying arbitrary cross-sections is presented.
The nonlinear formulation of the continuum mechanics based beam elements is presented.
A novel numerical method to improve nonlinear performance is proposed.
51
Future works
- Continuum mechanics based beam element- Composite (warping, zigzag, slip …)- Distortional mode - Jourasky type warping- Implementation for dynamic analysis
- Warping for discontinuously varying cross-section- Implementation for nonlinear analysis- Buckling analysis
- Nonlinear implementation- Buckling analysis (Inelastic, twisted structure)- Characteristics of twisting actions (poynting and swift effect)
- Improvement of nonlinear performance- Spurious energy mode (locking treatment)- Extension of eigen recomposition method- Recursive load step, problem dependent tangent stiffness matrix
52
Curriculum Vitae
• Education
• KAIST, Ph. D. Program, Division of Ocean System Engineering• KAIST, Master of Science, Division of Ocean System Engineering• KAIST, Bachelor of Science, Department of Mechanical Engineering
• Publications[1] K Yoon, PS Lee, Nonlinear performance of continuum mechanics based beam elements focusing on large twisting behaviors, Computer Methods in Applied Mechanics and Engineering, 2014.[2] S Yeo, H Chung, K Yoon, PS Lee, Y Hong, J Cha, Ballast planning for accuracy control on offshore floating dock. Ocean Engineering, submitted.[3] K Yoon, PS Lee. Modeling the warping displacement fields for discontinuously varying cross-section beams, Computers and Structures, 2014.[4] K Yoon, Y Lee, PS Lee. A continuum mechanics based 3-D beam finite element with warping displacements and its modeling capabilities, Structural Engineering and Mechanics, 2012.[5] Y Lee, K Yoon, PS Lee. Improving the MITC3 shell finite element by using the Hellinger-Reissner principle, Computers and Structures, 2012.
• Presentations[1] K Yoon, PS Lee, A continuum mechanics based beam finite element for Elastoplastic large displacement analysis, CST2014, 2014.[2] Y Hong, JH Cha, J Kim, K Cho, D Choi, K Yoon, S Yeo, Dimension control method for the erection of structures on offshore floating dock, Oceans’ 14 MTS/IEEE, 2014.[3] 윤경호, 전형민, 이필승, 연속체역학기반빔유한요소, 대한기계학회 CAE 및응용역학부문춘계학술대회, 2014.[4] 전형민, 윤경호, 이필승, Enriched MITC3 쉘요소개발, 대한기계학회 CAE 및응용역학부문춘계학술대회, 2014.[5] S Yeo, K Yoon, PS Lee, Y Hong, JH Cha, H Chung. Ballasting plan optimization for accuracy control on offshore floating dock, 12th International Symposium on Practical Design of Ships and Other Floating (PRADS), 2013.[6] K Yoon, PS Lee, Development of general beam finite elements with warping displacements, Sixth M.I.T. Conference on Computational Fluid & Solid Mechanics, 2011.[7] 윤경호, 이필승, 뒤틀림변위를고려한일반빔유한요소의개발, 한국전산구조공학회정기학술대회, 2011.
53
Q & A
54