nonlinear bending of strait beams - tamu...

JN Reddy Beams 1

Nonlinear Bending of Strait Beams

CONTENTS

The Euler-Bernoulli beam theory The Timoshenko beam theory

Governing Equations Weak Forms Finite element models Computer Implementation:

calculation of element matrices

Numerical examples

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Von Kármán NONLINEAR STRAINS

12

2 2 2

31 21 1 12 2 2

1

1

1 1 1

12

m m

i

jiij

j i

x

j

x

uuEx x

uE

u

x

ux x

uu ux x x

Green-Lagrange Strain Tensor Components

2

312

31

1 1

1 1

1

( ), ( )

xx xx

uu O Ox

ux x

x

E u

Order-of-magnitude assumption

Nonlinear Problems (1-D) : 3

NONLINEAR ANALYSIS OF EULER-BERNOULLI BEAMS

2 231

1 2 3

21 1

2 22

0( , ) , , ( )

,xx

dwu x z u z u u w xdx

u du d wzx dx dx

u dwx dx

Displacements and strain-displacement relations

•

z

y

Beam cross section

x

q(x) F0

L

z, w

M0

• •fc w

1 3ˆ ˆ) ,( x xu z w dwdx

u e e

M

V

q(x)

V

M•N N

f(x)

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PRINCIPLE OF VIRTUAL DISPLACEMENTSfor the Euler-Bernoulli beams

2

2

2

2

6

1

[( ) ( )]

[( ) ]

e

b

ea

b

a

b b

a a

eI xx xxV

x

xxx A

x

xx xxx

x xe e eE i ix x i

W dV

d u dw d w d wz dAdxdx dx dx dx

d u dw d w d wN M dxdx dx dx dx

W q wdx f udx Q

2

20 0, d dwNdN ddx

Mf qdx dx dx

Equilibrium equations

0 0 0, , d dwNdN dM dVf V qdx dx dx dx dx

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2

20 0, d dwNdN ddx

Mf qdx dx dx

Equilibrium equations

Stress resultants in terms of deflection2 2

1 12 2

212

2

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

dw dwdx dx

dwdx

dx dx

σ

σ

= = − =

+ +

= × = + − = −

= = −

∫ ∫

∫ ∫

0 0 0, , d dwNdN dM dVf V qdx dx dx dx dx

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Equilibrium equations in terms of displacements(u,w) 2

12

22 2122 2

0

0

d du dwEA fdx dx dx

d d w d dw du dwEI EA qdx dx dx dxdx dx

FF

( )u L

( )w L,x u

,z w

Clearly, transverse load induces both axial displacement u and transverse displacement w.

EULER-BERNOULLI BEAM THEORY(continued)

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11 1 1 1 4

2 22 2

22 2

2 22 2 3 2 5 6

0

0

( ) ( )

( ) ( )

b

a

b

a

a b

x

a bx

x

x

a bx x

dv N v f dx v x Q v x Qdx

d v dvd w dwEI N v q dxdx dx dx dx

dv dvv x Q Q v x Q Qdx dx

Weak forms2

12

du dwN EAdx dx

5 ( )ebQ V x2 ( )e

aQ V x

1 2

eh

3 ( )eaQ M x

6 ( )ebQ M x

1 ( )eaQ N x 4 ( )e

bQ N x

Beams 7

1 2,v u v w

8

2 ( )aQ V x

1 2eh

5 ( )bQ V x

6 ( )bQ M x3 ( )aQ M x

1 ( )aQ N x 4 ( )bQ N x

2 ( )aw x∆

1 2eh

5 ( )bw x∆

6 ( )bx∆ 3 ( )ax∆

1 ( )au x∆ 4 ( )bu x∆

Generalized displacements

Generalized forces

BEAM ELEMENT DEGREES OF FREEDOM

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9

FINITE ELEMENT APPROXIMATION

4

1 1

( ) ( ), ( ) ( ),n

j j j jj j

w x x u x u x ∆

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

dxθ = −

Áe1 = 1 ¡ 3

µx ¡ xa

he

¶2

+ 2

µx ¡ xa

he

¶3

Áe2 = ¡(x ¡ xa)

µ1 ¡ x ¡ xa

he

¶2

Áe3 = 3

µx ¡ xa

he

¶2

¡ 2

µx ¡ xa

he

¶3

Áe4 = ¡(x ¡ xa)

"µx ¡ xa

he

¶2

¡ x ¡ xa

he

#

Hermite cubic polynomials

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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 x

1( )x

2( )x

3( )x

4( )x

11

FINITE ELEMENT MODEL

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2 4

1 1

11 12 1

21 22 2

11 12 12

21 1

( ) ( ), ( ) ( )

{ }[ ] [ ] { }{ }[ ] [ ] { }

, ,

,

b b

a a

b

a

j j j jj j

x xj ji iij ijx x

x jiij ix

u x u x w x x

uK K FK K F

d dd dK EA dx K EA dxdx dx dx dx

ddK EA

dwdx

d dx F fdx dx

wdx

∆

∆

1 4

2222

2 2

22 5 3 6

2

( ) ( )

,

( ) ( )

b

a

b b

a a

b

aa b

x

i i a i bx

x xj ji iij x x

xi i

i i i a i bxx x

dx x Q x Q

d dd dK EI dx EA dxdx dx dx dx

d dF q dx x Q x Q Q Qdx d

dwdx

x

Finite Element Equations5eQ2

eQ

1 23eQ 6

eQ1eQ 4

eQ

5e∆2

e∆

1 23e∆ 6

e∆1e∆ 4

e∆

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MEMBRANE LOCKING

20 1

2xxdu dwdx dx

( )q x

20 1

2

212

0xxdu dwdx dx

du dwdx dx

Membrane strain Beam on roller supports

2

Remedy

make to behave like a constantdwdx

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SOLUTION OF NONLINEAR EQUATIONSDirect Iteration

Direct Iteration Method

Non-Linear Finite Element Model [ ( )] assembled [ ( )]e e e eK F K U U F

th 1

1

Solution { } at iteration is known and solve for{ }

[ ({ } )]{ } { }

r r

r r

U r U

K U U F

K(U)U ≡ F(U)F

U

FC

UCU0

K(U0)

U1

K(U1)

U2

K(U2)

•

•• •

U3

°UC - Converged

solution

U0 - Initial guesssolution

Nonlinear Problems: (1-D) - 13

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SOLUTION OF NONLINEAR EQUATIONS(continued)

Direct Iteration Method

Convergence Criterion

Possible convergence

21

1

21

1

specified tolerance

NEQr rI I

INEQ

rI

I

U U

U

th 1

1

Solution { } at iteration is known and solve for{ }

[ ({ } )]{ } { }

r r

r r

U r U

K U U F

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SOLUTION OF NONLINEAR EQUATIONSNewton’s Iteration Method

Taylor’s series

21 1 1 2

2

1 2 1

1{ ( )} { ( )} ( ) ( )

2 !

{ ( )} ( ) ( ) ,

rrr r r r r r

rr r r r r

R RR U R U U U U U

U U

RR U U U O U U U U

U

1 st

tan

2tan

1 1

Requiring the residual { } to be zero at the 1 iteration, we have

[ ({ } )]{ } { } { } [ ( )] { }

The tangent matrix at the element level is

r

r r r r r r

ni

ij ip p ij j p

R r

K U U R F K U U

RK K F

1Residual, { } [ ({ } )]{ } { }r r rR K U U F

SOLUTION OF NONLINEAR EQUATIONSNewton’s Iteration (continued)

1[ ({ } )]{ } { } [ ( )] { } , { } { } { }r r r r r r rT F K

2 2

1 1 1 1

n nipi

ij ip p i ij p ijp pj j j

KRT K F K T

K(∆) ∆ − F ≡ R(∆)F

∆

FC

∆0

T(∆0)

T(∆1)

T(∆2)

•

••

∆C = ∆3

°∆C - Converged

solution

∆0 - Initial guesssolution

δ ∆1 δ ∆2

∆1 = δ ∆1 + ∆0 ∆2 = δ ∆2 + ∆0Nonlinear Problems: (1-D) - 16

JN Reddy Beams 17

R®i =

2X

°=1

X

p=1

K®°ip ¢°

p ¡ F®i =

nX

p=1

K®1ip up +

4X

P=1

K®2iP ¢P ¡ F®

i

T®¯ij =

Ã@R®

i

@¢¯j

!= K®¯

ij +nX

p=1

@

@¢¯j

¡K®1

ip

¢up+

4X

P=1

@

@¢¯j

¡K®2

iP

¢¢P

T 11ij = K11

ij +nX

p=1

@K11ip

@ujup +

4X

P=1

@K12iP

@uj¢P

= K11ij +

nX

p=1

0 ¢ up +4X

P=1

0 ¢ ¢P

Summary of the N-R Method

Computation of tangent stiffness matrix

[T (f¢g(r¡1)]f¢gr = ¡fR(f¢g(r¡1))g

f¢gr = f¢g(r¡1) + f±¢g

18

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−

dwdx−

dwdx−

φx

u

Deformed Beams

( )q x

( )f x

THE TIMOSHENKO BEAM THEORY

19

KINEMATICS OF THE TIMOSHENKO BEAM THEORY

z

xw

dwdx−

z

φ

zφ

u

Constitutive Equations

,xx xx xz xzE G

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Displacement field

1

2 3

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

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

1 3ˆ ˆ)( xu z w u e e

231 1

21

212

31

3 12 2

xx xx

x

xz xz xz

x

uuEx x

ddu dw zdx dx dx

uuEx x

dwdx

20

Equilibrium Equations

Beam Constitutive Equations

212

21

212

2

xxx

A A

x xxx

A A

s xz s x s xA A

ddu dw duN dA E z dA EAdx dx dx dx

d ddu dwM z dA E z z dA EIdx dx dx dx

dwV K dA GK dA GAKx

wdx

d

d

dwdx

TIMOSHENKO BEAM THEORY (continued)

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0 0 0, ,dN dM dV d dwf V N qdx dx dx dx dx

JN Reddy Beams 21

11 1 1 1 4

11 1 12 1 4

21

0 ( ) ( )

( ) ( )

b

a

b

a

x

a bx

x

a bx

dv N v f dx v x Q v x Qdx

dv duEA v f dx v x Q v x Qdx d

wdxxd

WEAK FORMS OF TBT

2 22

2 2 2 5

0

( ) ( )

b

a

x

s xx

a b

dv dvdw dwGAK v q dxdx dx dx dx

v x Q v x

N

Q

33

3 3 3 6

0

( ) ( )

b

a

xx

s xx

a b

dv d dwEI GAK v dxdx dx dx

v x Q v x Q

22

Finite Element Approximation

11 12 13 1

21 22 23 2

331 32 33

K K K FuK K K w F

S FK K K

(1) (2) (3)

1 1 1( ), ( ), ( )

pm n

j j j j j jj j j

u u x w w x S x

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= =JN Reddy

23

SHEAR LOCKING IN TIMOSHENKO BEAMS

2he

1he

1 2

Linear interpolation of both , xw

1 1 2 2 1 1 2 2, xw( x ) w ( x ) w ( x ) ( x ) S ( x ) S ( x )

1w 2w1S 2S

(1) Thick beam experiences shear deformation,

(2) Shear deformation is negligible in thin beams,

xdwdx

xdwdx

In the thin beam limit it is not possible for the element to realize the requirement

xdwdx

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24

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”.

(2)(2)

(2)(

22

23 32

(3)(3

3)

(3))

(333 )

b

a

b

a

b

a

x

ij x

x

ij jix

x jii

jis

is j

s ij jx

K ... dx

K dx K

ddK EI dxd

ddGAKdx dx

dGAKd

x dK

x

x

GA

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Load,

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20D

efle

ctio

n,

Clamped-clamped

Pinned-pinned

q0

w0

NUMERICAL EXAMPLESPinned-pinned beam (EBT)


• •

• •

q0

q0

JN Reddy

q

w L = Length, H = Height of the beam

= 100LH

= 50LH

= 10LH

(in.

)

(psi.)

Pinned-pinned beam (TBT)


• •

q0

JN Reddy Beams 27

Pinned-pinned beam (EBT, TBT)

0 1 2 3 4 5 6 7 8 9 10Load (lb/in),

0.0

0.2

0.4

0.6

0.8

1.0De

flect

ion

3

4(0.5 ) EHw w LL

=

0q

/ 100(TBT,EBT)L H =

/ 80(TBT,EBT)L H =

/ 50(TBT,EBT)L H =

w/ 10(TBT)L H =

/ 10(EBT)L H =

H = beam heightL = beam length

• • 0q

JN Reddy0 2 4 6 8 10

Load, q0 (psi)0.0

0.1

0.2

0.3

L/H=100

L/H=10

EBT = Euler−Bernoulli beam theoryTBT = Timoshenko beam theory

EBTTBT

TBT

w =

wEH

3 /qL

4

Non

dim

ensi

onal

def

lect

ion


Hinged-Hinged beam (EBT and TBT)

• •

q0

29

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 the

Timoshenko beam theory• Developed Weak forms of EBT and TBT• Developed Finite element models of EBT

and TBT• Discussed membrane locking (due to the

geometric nonlinearity)• Discussed shear locking in Timoshenko beam

finite element• Discussed examples

JN Reddy

nonlinear bending of strait beams - tamu...

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