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The Finite Element Method

Read: Chapter 5

Euler-Bernoulli beam theory Governing Equations Finite element model Numerical examples

Timoshenko beam theory Governing Equations Finite element model Shear locking Numerical example

Euler-Bernoulli and Timoshenko Beams

CONTENTS

2

KINEMATICS OF THE LINEARIZED EULER-BERNOULLI BEAM THEORY

Undeformed Beam

Euler-Bernoulli Beam Theory (EBT)is based on the assumptions of(1)straightness, (2)inextensibility, and(3)normality

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z,

x

x

z

dwdx−

dwdx−

w

u

Deformed Beam

( )q x

( )f x

Strains, displacements, and rotations are small

90

3

z

xw

dwdx−z

dwdx−

z

u

Kinematics of Deformation in the Euler-Bernoulli Beam Theory (EBT)

1 2 30= − = =( , ) , , ( )dwu x z u z u u w xdx

Displacement field (constructed using the hypothesis)

21

11 21

31

3 1

0

∂= = = −∂∂∂= + = − + =

∂ ∂

ε ε

γ

,

.

xx

xz

u du d wzx dx dxuu dw dw

x x dx dx

Linear strains

2

2

0

= = −

= =

σ ε

σ γ

,xx xx

xz xz

du d wE E Ezdx dx

G

Constitutive relationsx

z yxzσ

zzσyzσ

yyσ zyσ

xyσxxσ

zxσyxσ

Notation for stress components

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4

Euler-Bernoulli Beam Theory

, , .σ σ σ= = ⋅ = xx xx xzA A A

N dA M z dA V dA

Definition of stress resultants

x

q(x) F0

L

z, w

M0

• •

z

y

Beamcross section

cf

+ MV

q(x)

VM •

fc w

( )q x

( )f xfc w

xxσ xx xxσ σΔ+xzσ

xz xzσ σΔ+

xΔ

( )q x

( )f xfc w

N N NΔ+

V V VΔ+

xΔ

M MΔ+M

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5

Equilibrium equations

0, 0, 0fdN dM dVf V q c wdx dx dx

+ = − = + − =

Euler-Bernoulli Beam Theory (Continued)

Stress resultants in terms of deflection2

2

2 2

2 2

2

2

xxA A

xxA A

du d w duN dA E Ez dA EAdx dxdx

du d w d wM z dA E Ez z dA EIdx dx dx

dM d d wV EIdx dx dx

= = − =

= × = − = −

= = −

σ

σ

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20,1 ,A AA

z ddA A zA dA I⋅ = ⋅= =⋅

x

q(x) F0

L

z, w

M0

Bending of a beam

Beams

cf

w

Axial deformation of a bar

Barsu

6

Governing equations in terms of the displacements

2 2

2 2

0 0

0 0

,

,f

d duEA f x Ldx dx

d d wEI c w q x Ldx dx

− − = < <

+ − = < <

Euler-Bernoulli Beam Theory (Continued)

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Axial displacement is uncoupled from transverse displacement

7

Weak Form of the EB Beam Theory2 2

2 2 0 0,fd d wEI c w q x Ldx dx

+ − = < <

Weak form

22

2 2

2 2

2 2

0 b

a

bb

aa

xh

i f hx

xx

i h hf i h i ix

x

d wdv EI c w q dxdx dx

dv d w d wd dEI c v w v q dx v EIdx dx dxdx dx

é ùæ ö÷çê ú÷ç= + -÷ê úç ÷÷çè øê úë ûé ù é ùæ ö æ ö÷ ÷ç çê ú ê ú÷ ÷ç ç= - + - + ⋅÷ ÷ê ú ê úç ç÷ ÷÷ ÷ç çè ø è øê ú ê úë û ë û

ò

ò

Secondary variable (shear force)

Implies that the primary variable is w(displacement)

2

1 320 ( ) ( )b

a

x

f a bx

dv d d wEI c vw vq dx v x Q v x Qdx dx dx

é ùæ ö÷çê ú÷= - + - - -ç ÷ê úç ÷çè øê úë ûò

{ } set of weight functionsiv −

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Governing equation

8

2 2

1 32 2

2

2

0 ( ) ( )b

a

b

a

xi h

f i h i i a i bx

x

i h

x

d v d wEI c v w v q dx v x Q v x Q

dx dx

dv d wEI

dx dx

é ùê ú= + - - -ê úê úë û

é ùæ ö÷çê ú÷+ - ⋅ç ÷ê úç ÷çè øê úë û

ò

Primary Variable, θSlope/rotation

Secondary variable(Bending Moment)

2 2

1 32 2

2 4

0 ( ) ( )b

a

a b

xi h

f i h i i a i bx

i i

x x

d v d wEI c v w v q dx v x Q v x Q

dx dxdv dv

Q Qdx dx

é ùê ú= + - - -ê úê úë û

æ ö æ ö÷ ÷ç ç÷ ÷- - ⋅ - - ⋅ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

ò

2 2

1 32 2

2 2

2 42 2

( ), ( )

( ) , ( )

a b

a b

h hh a h b

x x

h hh a h b

x x

d w d wd dQ EI V x Q EI V xdx dxdx dx

d w d wQ EI M x Q EI M x

dx dx

é ù é ùæ ö æ ö÷ ÷ç çê ú ê ú÷ ÷ç ç= =- = - =÷ ÷ê ú ê úç ç÷ ÷÷ ÷ç çè ø è øê ú ê úë û ë ûæ ö æ ö÷ ÷ç ç÷ ÷ç ç= =- = - =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

Weak Form (Continued)

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9

1 ( )=− aQ V x

1 2eh

3 ( )= bQ V x

4 ( )= bQ M x2 ( )=− aQ M x

1 ( )aw xΔ =

1 2eh

3 ( )bw xΔ =

4 ( )bxθΔ =2 ( )axθΔ =

Generalized displacements

Generalized forces

Beam Element Degrees of Freedom

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FINITE ELEMENT APPROXIMATION:Some Remarks

Continuity requirement based on the weak form, which requires that the second derivative of w exists and square-integrable.

Continuity based on the primary variables, which requires carrying w and its first derivative as the nodal variables, requires cubic approximation w.

Post-computation of secondary variables (bending moment and shear force) requiresthe third derivative of w to exist.

11

FINITE ELEMENT APPROXIMATION

2 30 1 2 3( )w x c c x c x c x» + + +

2 30 1 2 3 1

2 30 1 2 3 3

21 2 3 2

21 2 3 4

2 3

2 3

( )( )( )( )

a a a a

b b b b

a a a

b b b

w x c c x c x c xw x c c x c x c x

x c c x c xx c c x c x

q

q

Δ

Δ

Δ

Δ

» + + + º

» + + + º

»- - - º

»- - - º

42 3

0 1 2 31

( ) ( )j jj

w x c c x c x c x x=

» + + + = Då f

Primary variables (serve as the nodal variables that must becontinuous across elements) , dww

dxθ = −

φe1 = 1− 3x− xahe

2

+ 2x− xahe

3

φe2 = −(x− xa) 1− x− xahe

2

φe3 = 3x− xahe

2

− 2 x− xahe

3

φe4 = −(x− xa)x− xahe

2

− x− xahe

Hermite cubic polynomials

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● ● ● ●● ●,w q w ×

HERMITE CUBIC INTERPOLATION FUNCTIONS

he

he

he

1

1

xhe

x

xx

x x

x x

slope = 1

slope = 0

slope = 0

slope = 0

slope = 1

slope = 0

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( )i xf1( )xf

2( )xf

3( )xf

4( )xf

13

1 2ee21 Δ≡θ ee

42 Δ≡θ

eew 11 Δ≡ eew 32 Δ≡

eh

ee q,Q 22 ee q,Q 44

ee q,Q 11ee q,Q 33

1 2ee21 Δ≡θ

eh

4

1

( ) ( )j jj

w x xfΔ=

Ȍ

FINITE ELEMENT MODEL

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4

j=1

Keij∆

ej − F ei = 0 or [Ke]{∆e} = {F e}

⎡⎢⎣Ke11 Ke

12 Ke13 Ke

14

Ke21 Ke

22 Ke23 Ke

24

Ke31 Ke

32 Ke33 Ke

34

Ke41 Ke

42 Ke43 Ke

44

⎤⎥⎦⎧⎪⎨⎪⎩∆e1∆e2∆e3∆e4

⎫⎪⎬⎪⎭ =

⎧⎪⎨⎪⎩qe1qe2qe3qe4

⎫⎪⎬⎪⎭+⎧⎪⎨⎪⎩Qe1Qe2Qe3Qe4

⎫⎪⎬⎪⎭Keij =

xb

xa

EId2φeidx2

d2φejdx2

+ cf φeiφej dx F ei =

xb

xa

φei q dx+Qei

14

For element-wise constant values of EeIe and qe:

[Ke] =2EeIeh3e

⎡⎢⎣6 −3he −6 −3he−3he 2h2e 3he h2e−6 3he 6 3he−3he h2e 3he 2h2e

⎤⎥⎦ {F e} = qehe12

⎧⎪⎨⎪⎩6−he6he

⎫⎪⎬⎪⎭+⎧⎪⎨⎪⎩Q1Q2Q3Q4

⎫⎪⎬⎪⎭

Finite Element Model (Continued)

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M(x) = −EI d2w

dx2= −EI

4

j=1

∆ejd2φejdx2

V (x) =dM

dx= − d

dxEId2w

dx2= −EI

4

j=1

∆ejd3φejdx3

σx(x, z) = −M(x)zI

= Ezd2w

dx2= Ez

4

j=1

∆ejd2φej(x)

dx2

Postprocessing

(and cf = 0):

15

2EI

h3

⎡⎢⎢⎢⎢⎢⎣6 −3h −6 −3h 0 0−3h 2h2 3h h2 0 0−6 3h 6 + 6 3h− 3h −6 −3h−3h h2 3h− 3h 2h2 + 2h2 3h h2

0 0 −6 3h 6 3h0 0 −3h h2 3h 2h2

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

U1U2U3U4U5U6

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

+

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Q11Q12

Q13 +Q21

Q14 +Q22

Q23Q24

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=q0L

48

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

12−L24012L

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭+

•1 21 2

11

11, qQ

12

12, qQ

14

14, qQ

13

13, qQ

2 3

21

21 , qQ

22

22 , qQ

23

23 , qQ

24

24 , qQ

21• • •

Q13 +Q21 = 0, Q14 +Q

22 = 0

1

ASSEMBLY OF TWO BEAM ELEMENTSconnected end-to-end

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2/h L=

Given problem

,EA EI

L

P

3

3( ) PLw L

EI=

Exact solution (according to the Euler-Bernoullibeam theory)

,EA EI

L

P

• •

1 2,U U 3 4,U U

{F e} = qehe12

⎧⎪⎨⎪⎩6−he6he

⎫⎪⎬⎪⎭+⎧⎪⎨⎪⎩Q1Q2Q3Q4

⎫⎪⎬⎪⎭

[Ke] =2EeIeh3e

⎡⎢⎣6 −3he −6 −3he−3he 2h2e 3he h2e−6 3he 6 3he−3he h2e 3he 2h2e

⎤⎥⎦

eh L=

0

Boundary conditions:

1 2 3 40 0, ,U U Q P Q= = = =

A SIMPLE EXAMPLE - 1

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[ ]{ } { } { }e e e eK q QD = +One element discretization

A SIMPLE EXAMPLE – 1 (continued)

2

3 33 2

3 2 4

3 22 4

2

6

12 6 40 412 6 36 4 12

0

6 4

( / )( ) /

EIPL

EI EI EIU PPEI L PLL L LU

EI EI EIEI EI EI LU L LL LEI EIL L

é ù ì ü ì üï ï ï ïê ú ï ï ï ïï ïê ú ï ïï ï ï ï= = = =í ý í ýê ú é ùï ï ï ïê ú ï ï ï ï ê úë ûï ï ï ïê ú ï ïî þï ïî þë û

3

2 22

4 2 4

3 2

2

12

6 0 612 6 212

6 4

( / )( ) /

EI PLEI

PEI L PLLUEI EI EIEI L

L LEI EIL L

-= = =-

é ùê úë û

,EA EI

L 4 x LdwUdx ==-

3U

2

2PLEI

3

3( ) PLw L

EI=

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Solution using Cramer’s rule

EXAMPLE – 2: A determinate frame structure

Given structure

a

b

F

A B

C P

ab

F

A B

C

2

1

12

3

1 2

2

1

P

Finite element discretization

• •

•

PP bx

1A B

F

••

F

PP b

x

P

2

C

B•

•

F

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EXAMPLE – 2 (continued)

Bar element, AB1 1

1 1

1 11 1

0

A A

B B

A B B,

u QE Au Qa

Pau Q P uE A

ì ü ì üé ù- ï ï ï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï-ë ûï ï ï ïî þ î þ

= =- =-

Beam element, AB1

2 221 1

31

2 22

1 2

6 3 6 33 2 326 3 6 3

3 3 2

0 0

AA

AA

BB

BB

B BA A, , ,

wa a Qa a a a QE I

wa a a Qa a a a Q

w Q F Q Pb

q

q

q

ì üì üé ù ï ï- - - ï ï ï ïï ïê ú ï ïï ï ï ïê ú ï ï- ï ïï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï- ï ï ï ïê ú ï ï ï ïê ú ï ï ï ï- ï ï ï ïë û ï ïî þ ï ïî þ= = =- =

PP b

x

1A B

F

••a

F

PP b

x

P

2

C

B•

•

F

Bar element, BC2 2

CFbu

E A=

12 2

22 23

12 2

2

1 2

6 3 6 33 2 326 3 6 33 3 2

0 0 0

BB

BB

CC

CC

C CB B, , ,

wb b Qb b b b QE I

wb b b Qb b b b Q

w Q P Q

q

q

q

ì üì üé ù ï ï- - - ï ï ï ïï ïê ú ï ïï ï ï ïê ú ï ï- ï ïï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï- ï ï ï ïê ú ï ï ï ïê ú ï ï ï ï- ï ï ï ïë û ï ïî þ ï ïî þ= = =- =

Beam element, BC

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Displacements at C relative to point B

xk

L

q0

z, w

k w (L )

k

k w (L )

1 2• •1

U4 0¹

14 0Q =

11Q 1

2Q

1 2 0U U= =

3U

=- =-13 3

( )Q kw L kU

=3

( )kw L kU

13Q

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ì ü ì üé ù ï ï ï ï- ï ï ï ïï ï ï ïê ú = = = =í ý í ýê ú ï ï ï ï-ê ú ï ï ï ïë û ï ï ï ïî þ î þ

1 11 2 3 2 3

2 2

1 1, 0,

1 1

s ss s s

s s

u Qk u u U Q kU

u Q

Alternatively,

EXAMPLE – 3: Handling of a vertical spring

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SOLUTION TO THE SPRING-SUPPORTED BEAM

1 1 12 2

2 1 203

3 2 32 2

4 2 4

6 3 6 3 63 2 32

126 3 6 3 63 3 2

Boundary conditio

U w QL LU QL L L L Lq LEIU w QL L LU QL L L L L

q

q

ì ü ì üé ù ì ü=- - - ï ï ï ïï ïï ï ï ïï ïê ú ï ï ï ïï ïê ú ï ï ï ïï ï=- -ï ï ï ïï ïï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï ï=- ï ï ï ï ï ïê ú ï ï ï ï ï ïê ú ï ï ï ï ï ï=- ï ï ï ï ï ïë û î þï ï ï ïî þ î þ

1 1 3 3 40 0 0ns, , ,w Q kU Qq= = =- =

3kU-

0

00

33 20

2 4

12 6 6

126 4

EI EI Uk q LL LEI EI

LUL L

é ù ì ü ì üï ï ï ïê ú+ ï ï ï ïï ïê ú ï ïï ï ï ï=ê ú í ý í ýï ï ï ïê ú ï ï ï ïê ú ï ï ï ï-ï ïî þï ïî þê úë û

Condensed equations for the unknown generalized nodal displacements

HANDLING OF A POINT SOURCES INSIDE AN ELEMENT

2 3 2

1 2

2 3 2

3 4

( ) 1 3 2 , ( ) 1

( ) 3 2 , ( )

s s ss s s

h h h

s s s ss s s

h h h h

f f

f f

æ ö æ ö æ ö÷ ÷ ÷ç ç ç= - + = - -÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è ø è øé ùæ ö æ ö æ ö÷ ÷ ÷ê úç ç ç= - = - -÷ ÷ ÷ç ç ç÷ ÷ ÷ê úè ø è ø è øë û

f= ò0( ) ( )

h

i iq q s s ds

f f= = =ò 0 00( ) ( ) ( ), 1,2, 3, 4

h

i i iq q s s ds F s i

1 2

0F= 01 2

Fq = 0

3 2

Fq

=- 02 8

F hq = 0

4 8

F hq

=0

0

for placedat 0.5Fs h

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ff

=

== =-ò 00

0

1, 2, 3, 4( ) ( ) ,h

ii i

s s

id

q q s s ds Mds

1 2

0F0s

hs

●

d= -0 0

( ) ( )q s F s s

1 20M

0s

hs

●

d= -0 0

( ) '( )q s M s s

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xL

0F

EXAMPLE – 4: A simply-supported beam

(a) Find the center deflection using oneEuler-Bernoulli element in full beam

1 1 1 12 2

2 1 2 23

3 2 3 32 2

4 2 4 4

6 3 6 33 2 326 3 6 3

3 3 2

U w q QL LU q QL L L LEIU w q QL L LU q QL L L L

q

q

ì ü ì ü ì üé ù =- - - ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïê ú ï ï ï ï ï ï=- ï ï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï ï=- ï ï ï ï ï ïê ú ï ï ï ï ï ïê ú ï ï ï ï ï ï=- ï ï ï ï ï ïë û î þ î þ î þ

0

0= 0

1 2

Fq = 0

3 2

Fq

= 04 8

F Lq

1 2= 0

2 8

F Lq

0

8

F L

- 0

8

F L

2 22 0

3 2 24

1228 12

UL L F LEIUL L L

é ù ì ü ì üï ï ï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï-ï ïî þï ïî þë û

Condensed equations

2 20 0

2 416 16,F L F LU U

EI EI= =-

1 1 2 2 3 3 4 4

2 2 4 4

2 220

2 30 0

116

0 516 8 8 64

( ) ( ) ( ) ( ) ( )( ) ( )

( . )

w x U x U x U x U xU x U x

F L x x xx xEI L L L

F L F LL Lw LEI EI

f f f f

f f

= + + +

= +

ì üé ù é ùï ïæ ö æ ö æ öï ïï ïê ú ê ú÷ ÷ ÷ç ç ç= - - + -÷ ÷ ÷í ýç ç çê ú ê ú÷ ÷ ÷ç ç çï ïè ø è ø è øê ú ê úï ïë û ë ûï ïî þæ ö÷ç= - - =-÷ç ÷çè ø

0

0

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xL

0F

EXAMPLE – 4: A simply-supported beam

(b) Find the center deflection using oneEuler-Bernoulli element in half beam

1 1 12 2

2 1 23

3 2 32 2

4 2 4

6 1 5 6 1 51 5 0 5 1 5 0 2516

6 1 5 6 1 53 0 25 1 5 0 5

. .. . . .

. .. . .

U w QL LU QL L L LEIU w QL L LU QL L L L

q

q

ì ü ì üé ù =- - - ï ï ï ïï ï ï ïê ú ï ï ï ïê ú ï ï ï ï=- ï ï ï ïê ú =í ý í ýê ú ï ï ï ï=- ï ï ï ïê ú ï ï ï ïê ú ï ï ï ï=- ï ï ï ïë û î þ î þ

0

0

0

2

F

==-

4

3 0

0,

0.5

U

Q F

1==

1

2

0,

0

U

Q

0

-0

0.5F

22

033

00 5 1 516 0 511 5 6

. . ..

UL LEI FUL L

é ù ì ü ì üï ï ï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï-ï ïî þï ïî þë û

Condensed equations

3 20 0

2 2

3 320 0

3 2

4 1 532 3 1 16

4 0 532 3 1 48

. ,

.

F L F LLUEI L EI

F L F LLUEI L EI

= =

= =

2

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EXERCISE PROBLEM

2 2 2

2 2 2 0 0,d d w d wEI P x Ldx dx dx

+ = < <

Problem: Develop weak form and the finite element model of the following equation, where w and P are unknowns:

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EXERCISE PROBLEM

dF0

hh k

Linear elasticspring,

Rigid loading frame

q0

2EI EI

Problem: Use the minimum number of EBT elements to find the compression in the spring, reactions at the fixed support, and spring force.

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Governing Equations Finite element model Shear locking Numerical example

TIMOSHENKO BEAM THEORYand its Finite Element Model

28

Kinematics of Timoshenko Beam Theory

Undeformed Beam

Euler-Bernoulli Beam Theory (EBT)Straightness, inextensibility, and normality

Timoshenko Beam Theory (TBT)Straightness and inextensibility

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z, w

x, u

x

z

dwdx−

φx

u

dwdx−

dwdx−

Deformed Beams

( )q x

( )f x

90

29

Timoshenko Beam Theory

Kinematic Relations

1

2 3

1

1

31

3 1

( , ) ( ) ( ),0, ( , ) ( )

,

x

xxx

xz x

u x z u x z xu u x z w x

du du zx dx dx

uu dwx x dx

f

fe

g f

= +

= =

¶= = +

¶

¶¶= + = +

¶ ¶

z

x

w

dwdx−

z

u

Constitutive Equations

xxx xx

xz xz x

dduE E zdx dx

dwG Gdx

fs e

s g f

æ ö÷ç= = + ÷ç ÷÷çè øæ ö÷ç= = + ÷ç ÷÷çè ø

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xzf

xf

30

Equilibrium Equations

Beam Constitutive Equations

0 0 0fdN dV dMf , q c w , V .dx dx dx

+ = − − + = − + =

xxx

A A

xxA A

s xz s sA A

ddu duN dA E z dA EAdx dx dx

du d dM z dA E z z dA EIdx dx dx

dw dwV K dA GK dA GAKdx dx

fs

f fs

s f f

æ ö÷ç= = + =÷ç ÷ç ÷è ø

æ ö÷ç= = + =÷ç ÷÷çè ø

æ ö æ ö÷ ÷ç ç= = + = +÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

ò ò

ò ò

ò ò

(1)

0 (2)

s f

s

d dwGAK c w qdx dx

d d dwEI GAKdx dx dx

f

ff

é ùæ ö÷çê ú- + + =÷ç ÷÷ê úçè øë ûæ ö æ ö÷ ÷ç ç- + + =÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø

Governing Equations in terms of the displacements

Timoshenko Beam Theory (Continued)

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31

Weak Form of Eq. (1)

1

11 1 1

1

0b

a

bb

aa

b

a

x

s fx

xx

s f sxx

x

sx

d dwv GAK c w q dxdx dx

dv dw dwGAK c v w v q dx v GAKdx dx dx

dv dwGAKdx dx

f

f f

f

ì üé ùï ïæ öï ï÷çê ú= - + + -÷í ýç ÷÷ê úçï ïè øï ïë ûî þì üé ù é ùï ïæ ö æ öï ï÷ ÷ç çê ú ê ú= + + - - ⋅ +÷ ÷í ýç ç÷ ÷÷ ÷ê ú ê úç çï ïè ø è øï ïë û ë ûî þ

é æ ö÷ç= + ÷ç ÷÷çè ø

ò

ò

ò 1 1

1 1

11 1 1 1 1 3

( ) ( )

( ) ( )

a b

b

a

f

a s b sx x

x

s f a bx

c v w v q dx

dw dwv x GAK v x GAKdx dx

dv dwGAK c v w v q dx v x Q v x Qdx dx

f f

f

ì üùï ïï ïê ú + -í ýê úï ïï ïë ûî þé ù é ùæ ö æ ö÷ ÷ç çê ú ê ú- ⋅ - + - ⋅ +÷ ÷ç ç÷ ÷÷ ÷ê ú ê úç çè ø è øë û ë û

ì üæ öï ïï ï÷ç= + + - - ⋅ - ⋅÷í ýç ÷÷çï ïè øï ïî þò

WEAK FORMS OF TBT

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32

Weak Form of Eq. (2)

2

22 2

22

0b

a

bb

aa

b

a

x

sx

xx

sx x

x

sx

d d dwv EI GAK dxdx dx dx

dv d dw dEI GAK v dx v EIdx dx dx dx

dv d dwEI GAK vdx dx dx

ff

f ff

ff

é ùæ ö æ ö÷ ÷ç çê ú= - + +÷ ÷ç ç÷ ÷÷ ÷ê úç çè ø è øë ûé ùæ ö æ ö é ù÷ ÷ç çê ú ê ú= + + - ⋅÷ ÷ç ç÷ ÷÷ ÷ê úç ç ê úè ø è ø ë ûë ûé æ ö æ ö÷ ÷ç ç= + +÷ ÷ç ç÷ ÷÷ ÷ç çè ø è øë

ò

ò

ò 2 2

22 2 2 2 4

( ) ( )

0 ( ) ( )

a b

b

a

a bx x

x

s a bx

d ddx v x EI v x EIdx dx

dv d dwEI GAK v dx v x Q v x Qdx dx dx

f f

ff

ù æ ö æ ö÷ ÷ç çê ú - ⋅ - - ⋅÷ ÷ç ç÷ ÷÷ ÷ê ú ç çè ø è øûé ùæ ö æ ö÷ ÷ç çê ú= + + - ⋅ - ⋅÷ ÷ç ç÷ ÷÷ ÷ê úç çè ø è øë û

ò

Total Potential Energy2 2

2

1 3 2 4

( )2 2 2

( ) ( ) ( ) ( )

b

a

b

a

x fsx x

x

a b a bx

cG A KE I d dww , w dxdx dx

w q dx w x Q w x Q x Q x Q

φφ φ

φ φ

Π = + + +

− + + + +

Weak Forms of TBT (continued)

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33

Finite Element Approximation

{ }{ }

{ }{ }

11 12 1

21 22 2

K K FwSK K F

é ù ì üé ù é ù ï ïì ü ï ïï ïê úê ú ê úë û ë û ï ïï ïê ú =í ý í ýï ï ï ïê úé ù é ù ï ï ï ïî þê úê ú ê ú ï ïë û ë û î þë û

11 12 21

22 21

1 21 3 2( ) ( ) ( ) ( )

b b

a a

b b

a a

b

a

x xji iij s f i j ij s j jix x

x xj jiij s i j ij s ix x

x

i i i a i b i i a i bx

dd dK GAK c dx, K GAK dx Kdx dx dx

d ddK EI GAK dx, K GAK dxdx dx dx

F q dx x Q x Q , F x Q x

yy yy y j

j yjj j j

y y y j j

æ ö÷ç ÷ç= + = =÷ç ÷ç ÷çè øé ùê ú= + =ê úê úë û

= + + = +

ò ò

ò ò

ò 4Q

1 1( ), ( )

m n

j j j jj j

w w x S xy f j= =

» »å å

FINITE ELEMENT MODELS OFTIMOSHENKO BEAMS

2he

1 2he

1

w w s s221 1

3

he

1 3he

1

w1

2 2

w w s ss2 23 312m n= =

3m n= =

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Shear Locking in Timoshenko Beams

Linear interpolation of both , :xw f

1 1 2 2 1 1 2 2, xw( x ) w ( x ) w ( x ) ( x ) S ( x ) S ( x )y y f y y» + » +

2he

1he

1 21w 2w

1S 2S

(1) Thick beam experiences shear deformation,

(2) Shear deformation is negligible in thin beams,

xdwdx

f ¹-

xdwdx

f =-

Thus, in the thin beam limit it is not possible for the element to realize the requirement

xdwdx

f =-JN Reddy

2 2

1 1( ), ( )e e e e

j j x j jj j

w w x S xy f y= =

» »å å

35

SHEAR LOCKING - REMEDY

In the thin beam limit, φ should become constant so that it matches dw/dx. However, if φ is a constant then the bending energy becomes zero. If we can mimic the two states (constant and linear) in the formulation, we can overcome the problem. Numerical integration of the coefficients allows us to evaluate both φ and dφ/dx as constants. The terms highlighted should be evaluated using “reduced integration”.

(1)(1)11 (1) (1)

(1)12 (2) 21

(2)(2)22 (2) (2)

b

a

b

a

b

a

x jiij s f i jx

xi

ij s j jix

x jiij s i jx

ddK GAK c dxdx dx

dK GAK dx Kdx

ddK EI GAK dxdx dx

yyy y

yy

yyy y

æ ö÷ç ÷ç= + ÷ç ÷ç ÷çè ø

= =

é ùê ú= +ê úê úë û

ò

ò

ò

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JN Reddy Beams 36

STIFFNESS MATRICES OF TIMOSHENKO BEAM ELEMENT

(for constant EI and GA)

1 1 12 2

1 2 23

20 3 32 2

2 4 4

6 3 6 33 326 3 6 3

3 3

e ee e

e ee e e e e ee e

e e ee ee

e ee e e e e e

h h w q Qh h h h q QE I

h h wh q Qh h h h q Q

x z fm

z x f

ì ü ì üé ù ì ü- - - ï ï ï ïï ï ï ï ï ïï ïê ú ï ï ï ïï ï ï ï ï ïê ú ï ï- ï ï ï ïï ïï ï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï- ï ï ï ï ïê ú ï ï ï ï ïê ú ï ï ï ï ï- ï ï ï ï ïê ú ï ïî þë û ï ï ïî þ î

021 5 6 1 5 6 12. , . , , ee ee e e e e e

e e s e

E IG A K h

x z m

ïïïïïïþ

= + L = - L L = = L

Reduced integration linear element (RIE)

1 1 12 2

1 2 23

2 3 32 2

2 4 4

6 3 6 33 32

2

26 3 6 3

3 3

e ee e

e ee e e e e ee e

e ee ee e

e ee e e e e e

h h w q Qh h h h q QE I

h h wh q Qh h h h q Q

fm

f

ì ü ì üé ù ì ü- - - ï ï ï ïï ï ï ï ï ïï ïê ú ï ï ï ïï ï ï ï ï ïê ú ï ï- S Q ï ï ï ïï ïï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï- ï ï ï ï ïê ú ï ï ï ï ïê ú ï ï ï ï ï- Q S ï ï ï ï ïê ú ï ïî þë û ï ï ïî þ î

21 0 1 0 6 1 13 2. , . , ,e ee e e e e e e

e e s e

E IG A K h

m

ïïïïïïïþ

S = + L Q = - L L = = + L

Consistent interelement element (CIE)

Linear approximation of both w and f

of w and dependent quadratic approximation of Hermite cubic approximation

f

3

3( ) FLw L

EI=

Exact solution (according to the E-B beam theory)

One element discretization using the RIE element

,EA EI

L

F

• •

1 2,U U 3 4,U U

Boundary conditions:

1 2 3 40 0, ,U U Q F Q= = = =

AN EXAMPLE of TBT

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1 1 12 2

1 2 23

20 3 32 2

2 4 4

6 3 6 33 326 3 6 3

3 3

e ee e

e ee e e e e ee e

e e ee ee

e ee e e e e e

h h w q Qh h h h q QE I

h h wh q Qh h h h q Q

x z fm

z x f

ì ü ì üé ù ì ü- - - ï ï ï ïï ï ï ï ï ïï ïê ú ï ï ï ïï ïê ú ï ï ï ïï ï- ï ï ï ïï ï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï ï- ï ï ï ï ï ïê ú ï ï ï ï ï ïê ú ï ï ï ï ï- ï ï ï ï ïê ú î þë û ï ï ïî þ î þ

021 5 6 1 5 6 12. , . , , ee ee e e e e e

e e s e

E IG A K h

x z m

ïïï

= + L = - L L = = L

AN EXAMPLE (TBT) (continued)

( )

( )

3 2 33 0

33 2 2 240

3 3

3

32

3

3

6 32 12 1 5 62 6 123 0 6 9

1 506 4

1 5 036

75 36

( . )( )

.When too stiff

( . )When 0, then ( . )

s

UL F LEI FL FLUUL EI EIL L L L

EI FL FLUGAK L EI EI

FLUEI

EI

FLE

GA

I

m xm x x

ì üé ù ì üï ï ï ï L + Lï ï ï ïê ú = = =í ý í ýê úï ï ï ï L-ï ïë û î þï ïî þ

L = = = =

+ LL ¹ = =

L =

+ L

2 2 22

2 2

2 1 1 1 3 0 2612 6 5( ) ( ) . .

s s s

H H H HK L L K K L L L

n n æ ö æ ö æ ö+ + ÷ ÷ ÷ç ç ç= = = =÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø

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One element discretization using the CIE element

AN EXAMPLE of TBT

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1 12 2

1 23

2 32 2

2 4

2

6 3 6 33 2 326 3 6 3

3 3 2

1 0 3 1 0 6 1 12. , . , ,s

L L w QL L L L QEI

L L wL QL L L L Q

E IG A K L

fm

f

m

ì üé ù ì ü ï ï- - - ï ï ï ïï ïê ú ï ïï ï ï ïê ú ï ï- S Q ï ïï ïï ï ï ïê ú =í ý í ýê ú ï ï ï ï- ï ï ï ïê ú ï ï ï ïê ú ï ï ï ï- Q S ï ï ï ïê ú ï ïë û î þ ï ïî þ

S = + L Q = - L L = = + L

( )3 2 3

333 2 2 2

4

6 32 22 12 93 2 0 12 9 ( )

UL FEI L FL FLUUL EI EIL L L L

m mm

ì üé ù ì üï ï ï ï S Sï ï ï ïê ú = = =í ý í ýê úï ï ï ï S-S S-ï ïë û î þï ïî þ

Condensed equations for the unknown displacements

AN EXAMPLE (TBT) (continued)

3 3

3

3 3

3

2

3

2 22

2 2

0 1 112 9 3

1 3 1 1212 9 3 1 12

2 1 1 1 3 0 261

1 3

2 6

3

5

When and ; then( )

( )( )When 0, ( ) ( )

( ) ( ) .

( )

.s s s

FL FLUEI EI

FL FLUEI EI

EI H H H HGAK L L K K L L L

FLEI

mm

m

n n

SL = S= = = =

S-

S + L + LL ¹ = = =

S- + L

æ ö æ ö æ ö+ + ÷ ÷ ÷ç ç çL = = = = =÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè è

L

ø ø è ø

+

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SUMMARY

In this lecture we have covered the following topics:

• Derived the governing equations of the Euler-Bernoulli beam theory

• Derived the governing equations of theTimoshenko beam theory

• Developed Weak forms of EBT and TBT• Developed Finite element models of EBT

and TBT• Discussed shear locking in Timoshenko beam

finite element• Discussed assembly of beam elements• Discussed examples

JN Reddy