Chan-An Introduction to Timoshenko Beam Formulation and Its FEM Implementation

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

QQ

=

=

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