Chan-An Introduction to Timoshenko Beam Formulation and Its FEM Implementation
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Transcript of Chan-An Introduction to Timoshenko Beam Formulation and Its FEM Implementation
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An Introduction to Timoshenko Beam Formulation and its FEM implementationChan Yum JiCOME, Technische UniversitätMünchen
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Content of presentation
IntroductionFormulation of Timoshenko Beam ElementsFEM implementation
ExampleProblem with FEM implementation
Reasonp-version FEM implementation
ExampleQuestions and Answers
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References
Bathe, K.-J.: Finite Element Procedures(Prentice Hall, Englewood Cliffs, 1996)
Bischoff, M.: Lecture Notes on course Advanced Finite Methods, TUM
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0.1 Introduction: Review of Euler-Bernoulli Beam Theory
Beam is condensed to an 1-D continuumAssumptions
Mid-surface plane remains in mid-surface after bendingCross sections remain straight and perpendicular to mid-surface
One variable (displacement) at each pointApplicable to thin beams
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0.2 How about thick beams?
Shearing force exists inside beam
Assumption “Cross sections remain perpendicular to centroidal plane” no longer valids
Timoshenko theory
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0.3 Timoshenko beam theory
Beam is condensed to an 1-D continuumAssumption
Mid-surface plane remains in mid-surface after bendingCross sections remain straight and perpendicular to mid-surface
Two independent variables (displacement and rotation) at each pointDistributive moments taken into account
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1.1 Governing equations
Kinematic equationsEquilibriumConstitutive equations (Material Laws)
Displacements
Strains Stresses
Forces
Kinematic equations
Material Laws
Equilibrium
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1.2 Kinematic equations
Remember the equations for Euler-Bernoulli beams……
dxdw
=β
2
2
dxwd
dxd
−=−=βκ
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1.2 Kinematic equations
… and here comes the equations for Timoshenko beams!
We still assume cross section remains straight at the moment
γβ −=dxdw
dxdβκ −=
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1.3 Equilibrium
Consider a part of the beam
QMQdxdMm
QdxdQq
+−=+−=
−=−=
'
'
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1.4 Constitutive equations(Material Laws)
Bending part
Shearing part
α takes into account of non-straight cross sections
κEIM =
γαGAQ =
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1.5 Summary of all equations
Kinematic relations
Equilibrium
Material Laws
γβ −=dxdw
dxdβκ −=
γακGAQEIM
==
QMQdxdMm
QdxdQq
+−=+−=
−=−=
'
'
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1.6 Boundary conditions
Displacement / Essential / Dirichlet
Force / Neumann
0
0
)0(
)0(
MM
=
=
l
l
MlM
QlQ
−=
−=
)(
)(
0
0
ˆ)0(
ˆ)0(
ββ =
= ww
l
l
l
wlw
ββ ˆ)(
ˆ)(
=
=
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2.1 Finite Element Method – Weak formulation
FEM is a numerical method of finding approximate solutions“Weak” formulation
The three equations are not satisfied at each point, but only in general sense
Virtual work principle: 0int =+ extWW δδ
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2.2 Virtual work principle
External virtual work
Internal virtual work
As ,
( ) llll
lext MMwQwQdxmwqW δβδβδδδβδδ +++++= ∫ 0000
0
( )∫ +=−l
dxMQW0
int δκδγδ
0int =+ extWW δδ
( ) llll
l
MMwQwQdxMQmwq δβδβδδδκδγδβδ ++++−−+= ∫ 00000
0
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2.3 Virtual work principle– in Matrices
=
βw
u
∂∂
−∂∂
=
x
x0
1*L
=
EIGA0
0αC
=MQ
σ
=
κγ
ε
∂∂
∂∂
=
x
x0
1L
( ) 0d 00
0
=⋅−⋅−⋅−⋅∫ lTl
Tl
T x δuPδuPδupδεσ
=mqp
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2.4 Discretisation
FEM cannot deal with continuous functionsUnknown coefficients (d) with pre-assigned shape functions (N)
nodal values as unknownstwo nodes makes up an elementtwo linear shape functions for an element
Matrix form: u = N · d
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2.4 Discretisation
Because and
and suppose dNuu ⋅=≈ h
( ) 0 00
0
=⋅−⋅−⋅−⋅∫ l
Tl
Tl
T dx δuPδuPδupδεσ
( ) [ ] 0d 0
0
=⋅−⋅−⋅∫ bl
l
x δuPPδupδuCLLu TT
εCσ ⋅= uLε ⋅=
( ) [ ] 0d 0
0
=⋅−⋅⋅−⋅∫ δdPPδdNpδdCBBd TTl
l
x
+⋅=⋅ ∫∫
l
0T
PPNpdCBB xx
ll
d d 00
Stiffness Matrix Load VectorUnknown
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2.5 Implementation
Maple exampleComparison: With Euler-Bernoulli Beam
L
P=t3
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3.1 Problem with FEM implementation
Displacement much smaller than expectedExtremely slow converging rateAdding elements does not helpResult depends on one critical parameter
Displacement = 0 when parameter reaches infinity
Locking
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3.1 Locking behaviour exhibitsslow converging rate
Converging behaviour of FE solution
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30
Number of elements
Rel
ativ
e di
spla
cem
en
Euler Bernoulli (Analytical) Timoshenko (FE approximation)
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3.1 Locking behaviour depends onslenderness
Change of estimated displacement against slenderness
0.01
0.1
1
10
0 2 4 6 8 10 12 14 16 18 20
Slenderness
Rel
ativ
e di
spla
cem
ent
Euler Bernoulli (Analytical) Timoshenko (FE approximation)
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3.2 Reasons of locking
First ReasonEquilibrium:
When t is small, shear dominates if w’ and β do not balance
( )
( )
+′+
′′=
+′+′′−=+′−=
βαββαβ
wGbtEbt
wGAEIQMm
2
12
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3.2 Reasons of locking
Second reasonKinematic equation:
Here, w is linear (set by N1 and N2)Then w’ becomes constantThe only solution for β = constantZero shear if slenderness is towards infinity
γβ −=dxdw
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4.1 Solving problem
The processFormulationFEM ImplementationDiscretisation
Methods on implementationMethods on discretisation
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4.2 High Order functions
Change the discretisation schemeAllow higher order terms in shape functionsβ needs not to be constant
Hierarchic shape functionsNodal modesBubble modesAdvantages
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4.3 Example
Maple sheet
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4.4 Graph showing convergence ofp-method
Shapes of deflection with different orders considered
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
Length
Def
lect
ion 1st order
2nd order3rd orderExact
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5 Conclusion
Timoshenko beam theory is applicable for both thick and thin beamsIt suffers from severe locking behaviourwhen linear shape functions are applied directlyEmploying high order functions can solve the problem
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6 Questions and Answers
Your comments are also welcomed