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An Introduction to Timoshenko Beam Formulation and its FEM implementationChan Yum JiCOME, Technische UniversitätMünchen
Content of presentation
IntroductionFormulation of Timoshenko Beam ElementsFEM implementation
ExampleProblem with FEM implementation
Reasonp-version FEM implementation
ExampleQuestions and Answers
References
Bathe, K.-J.: Finite Element Procedures(Prentice Hall, Englewood Cliffs, 1996)
Bischoff, M.: Lecture Notes on course Advanced Finite Methods, TUM
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
0.2 How about thick beams?
Shearing force exists inside beam
Assumption “Cross sections remain perpendicular to centroidal plane” no longer valids
Timoshenko theory
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
1.1 Governing equations
Kinematic equationsEquilibriumConstitutive equations (Material Laws)
Displacements
Strains Stresses
Forces
Kinematic equations
Material Laws
Equilibrium
1.2 Kinematic equations
Remember the equations for Euler-Bernoulli beams……
dxdw
=β
2
2
dxwd
dxd
−=−=βκ
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 −=
1.3 Equilibrium
Consider a part of the beam
QMQdxdMm
QdxdQq
+−=+−=
−=−=
'
'
1.4 Constitutive equations(Material Laws)
Bending part
Shearing part
α takes into account of non-straight cross sections
κEIM =
γαGAQ =
1.5 Summary of all equations
Kinematic relations
Equilibrium
Material Laws
γβ −=dxdw
dxdβκ −=
γακGAQEIM
==
QMQdxdMm
QdxdQq
+−=+−=
−=−=
'
'
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)(
=
=
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
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
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
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
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
2.5 Implementation
Maple exampleComparison: With Euler-Bernoulli Beam
L
P=t3
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
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)
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)
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
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
4.1 Solving problem
The processFormulationFEM ImplementationDiscretisation
Methods on implementationMethods on discretisation
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
4.3 Example
Maple sheet
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
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
6 Questions and Answers
Your comments are also welcomed