jn reddy - 1 lecture notes on nonlinear fem meen 673...

JN Reddy Beams 1

Read: Chapter 5

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

MEEN 673Nonlinear Finite Element Analysis

JN Reddy - 1 Lecture Notes on NONLINEAR FEM

2

THE EULER-BERNOULLI BEAM THEORY(development of governing equations)

Undeformed Beam

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

JN Reddy

z,

x

x

z

dwdx−

dwdx−

w

u

Deformed Beam

( )q x

( )f x


3

z

xw

dwdx−z

dwdx−

z

u

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

1

2

3

0

( , )

,( , ) ( )

dwu x z u zdx

uu x z w x

Displacement field

x

z yxzσ

zzσyzσ

yyσ zyσ

xyσxxσ

zxσyxσ

Notation for stress components

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1 3ˆ ˆ) ,( x

x

u z wdwdx

u e e


JN Reddy Beams 4

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) : 5

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)


JN Reddy Beams 6


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


JN Reddy Beams 7


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

11 1 1 1 4

2

2 2

2 222

0

0

( )[ ( )] ( ) ( )

( ) ( )

b b

a a

b

a

b

a

b

a

x x

a a b bx x

x

a bx

x

x

x

x

dvdNv f dx N v f dx v x N x v x N xdx dx

dv N v f dx v x Q v x Qdx

d M d dwv N q dxdx dxdx

d v d wEIdx

2 2 22 2 2 3 2 5 62 ( ) ( )

a b

a bx x

dv dv dvdwN v q dx v x Q Q v x Q Qdx dx dx dxdx

Weak forms

212

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 8


9

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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10

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

JN Reddy 11

( )i x

1( )x

2( )x

3( )x

4( )x


12

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) - 14


JN Reddy 15

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


JN Reddy 16

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) - 17


JN Reddy Beams 18

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


19

Undeformed Beam

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

Timoshenko Beam Theory (TBT)Straightness and inextensibility

JN Reddy

z, w

x, u

x

z

dwdx−

dwdx−

dwdx−

φx

u

Deformed Beams

( )q x

( )f x

THE TIMOSHENKO BEAM THEORY


20

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


21

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 22

11 1

1 1

11 1 1 1 4

11

212

0

0

( )[ ( )] ( ) ( )

( ) ( )

b b

a a

b

a

a

x x

x x

a a b b

x

a bx

x

dvdNv f dx N v f dxdx dx

v x N x v x N xdv N v f dx v x Q v x Qdx

dv duEA v fd

dwdxx dx

1 1 1 4( ) ( )

bx

a b

dx

v x Q v x Q

WEAK FORMS OF TBT

Weak Form of Eq. (1) 1v u


23

Weak Form of Eq. (2)

2

2 22

2

0 b

a

b

a

x

s xx

x

s xx

s x

d dw d dwv GAK q dxdx dx dx dx

dv dvdw dwGAK v q dxdx dx dx dx

dw dwv GAK

N

N

dN

dx x

2 22

2 2 2 5

0

( ) ( )

b

a

b

a

x

x

x

s xx

a b

dv dvdw dwGAK v q dxdx dx dx dx

v x Q v x

N

Q

WEAK FORMS OF TBT(continued)

JN Reddy

2v w

212

dduN EA wdxdx


24

Weak Form of Eq. (3)

3

33 3

33

0

0

b

a

bb

aa

b

a

xx

s xx

xxx x

s xx x

xx

s xx

dd dwv EI GAK dxdx dx dx

dv d ddwEI GAK v dx v EIdx dx dx dx

dv d dwEI GAK vdx dx dx

3 3 3 6( ) ( )a b

dx

v x Q v x Q

JN Reddy

WEAK FORMS OF TBT(continued)

3 xv


25

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


26

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

JN Reddy


27

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

JN Reddy


JN Reddy Beams 28

GENERAL LOGIC IN A COMPUTER PROGRAM for the nonlinear analysis

Logic in the MAIN program

NLS = no. of load stepsInitialize global Kij, fi

Iter = Iter + 1

DO 1 to N

Iter = 0

Impose boundary conditionsand solve the equations

CALL ELKF to calculate Kij(N)

and fi(n), and assemble to form global Kij and Fi

Transfer global information(material properties, geometry and solution)

to element

Iter < Itmax

Error < ε yesno

STOP

Print Solution

Write a message

Yes No

NL=1,NLS

F=F+∆F


Beams 29

IF(MODEL.GE.2)THENC C Define the beam stiffness coefficients, EA, EI, GAKs, from the C geometric and material parameters read in the main programC (should be passed to this subroutine)CC Initialize arraysC

DO 20 I=1,NPE ELF1(I)=0.0ELF2(I)=0.0ELF3(I)=0.0

DO 20 J=1,NPEELK11(I,J)=0.0ELK12(I,J)=0.0ELK13(I,J)=0.0ELK21(I,J)=0.0ELK22(I,J)=0.0ELK23(I,J)=0.0ELK31(I,J)=0.0ELK32(I,J)=0.0ELK33(I,J)=0.0

CALCULATION OF BEAM PARAMETERS AND INITIALIZATIONS


Beams 30

IF(NONLIN.GT.1)THENTAN12(I,J)=0.0TAN13(I,J)=0.0TAN22(I,J)=0.0TAN23(I,J)=0.0TAN32(I,J)=0.0TAN33(I,J)=0.0

ENDIF20 CONTINUE

ENDIFC C Full integration of the coefficientsC

DO 100 NI=1,NGP XI=GAUSPT(NI,NGP) CALL INTERPLN1D(ELX,GJ,IEL,MODEL,NPE,XI)X=ELX(1)+0.5*(1.0+XI)*ELCNST=GJ*GAUSWT(NI,NGP)

C DEFINE AXX, BXX, CXX, DXX, FX, and so on as needed to defineC the element force and stiffness coefficients

CALCULATION OF BEAM PARAMETERS AND INITIALIZATIONS


C The EULER-BERNOULLI beam element (MODEL=2) - LINEARC

IF(MODEL.EQ.2)THENDO 50 I=1,NPEI0=2*I-1ELF1(I)=ELF1(I)+F0*FX*SFL(I)*CNSTELF2(I)=ELF2(I)+F0*QX*SFH(I0)*CNSTELF3(I)=ELF3(I)+F0*QX*SFH(I0+1)*CNSTDO 50 J=1,NPEJ0=2*J-1S11=GDSFL(I)*GDSFL(J)*CNSTH22=GDDSFH(I0)*GDDSFH(J0)*CNSTH23=GDDSFH(I0)*GDDSFH(J0+1)*CNSTH32=GDDSFH(I0+1)*GDDSFH(J0)*CNSTH33=GDDSFH(I0+1)*GDDSFH(J0+1)*CNSTELK11(I,J)=ELK11(I,J)+AXX*S11ELK22(I,J)=ELK22(I,J)+DXX*H22ELK23(I,J)=ELK23(I,J)+DXX*H23ELK32(I,J)=ELK32(I,J)+DXX*H32ELK33(I,J)=ELK33(I,J)+DXX*H33

50 CONTINUEENDIF

Beams 31

iddx 2

2id

dx

i i

CALCULATION OF ELEMENT MATRICES(see Box 5.2.2 of the textbook)

MODEL = Type of physical problem=1, 2nd order eqn. in 1 variable=2, EBT>2, TBT

11

2222

2 2

,

b

a

b

a

x jiij xxx

x jiij xxx

ddK A dx

dx dxdd

K D dxdx dx

=

=

∫

∫

1

2

( )

( )

b

a

b

a

x

i ix

x

i ix

F f x dx

F q x dx

=

=

∫

∫


Beams 32

CALCULATION OF ELEMENT MATRICES

CC The TIMOSHENKO beam element (MODEL=3) - LINEARC

IF(MODEL.GT.2)THENDO 60 I=1,NPEELF1(I)=ELF1(I)+F0*FX*SFL(I)*CNSTELF2(I)=ELF2(I)+F0*QX*SFL(I)*CNSTDO 60 J=1,NPES11=GDSFL(I)*GDSFL(J)*CNSTELK11(I,J)=ELK11(I,J)+AXX*S11ELK33(I,J)=ELK33(I,J)+DXX*S11

60 CONTINUEENDIF

100 CONTINUE ! (loop on NI =1, NGP ends here)

CC Define shear and nonlinear coefficients for the two beam theories asC appropriate in the reduced integration do-loop; define ELK and TAN coefficientsC


Beams 33

IF(MODEL.GT.1)THENII=1DO 220 I=1,NPE

ELF(II) =ELF1(I)ELF(II+1)=ELF2(I)ELF(II+2)=ELF3(I)JJ=1DO 210 J=1,NPE

ELK(II,JJ) = ELK11(I,J)ELK(II,JJ+1) = ELK12(I,J)ELK(II,JJ+2) = ELK13(I,J)ELK(II+1,JJ) = ELK21(I,J)ELK(II+2,JJ) = ELK31(I,J)ELK(II+1,JJ+1) = ELK22(I,J)ELK(II+1,JJ+2) = ELK23(I,J)ELK(II+2,JJ+1) = ELK32(I,J)ELK(II+2,JJ+2) = ELK33(I,J)

210 JJ=NDF*J+1220 II=NDF*I+1ENDIF

REARRANGE ELEMENT COEFFICIENTSJN Reddy - 33 Lecture Notes on NONLINEAR FEM

Beams 34

C Compute the residual vector and tangent coefficient matrix for the Newton iteration method (only)

C IF(NONLIN.GT.1)THEN

DO 230 I=1,NET DO 230 J=1,NET

230 ELF(I)=ELF(I)-ELK(I,J)*ELU(J)II=1DO 260 I=1,NPEJJ=1DO 250 J=1,NPEELK(II,JJ+1) = ELK(II,JJ+1) +TAN12(I,J)ELK(II+1,JJ+1) = ELK(II+1,JJ+1)+TAN22(I,J)IF(MODEL.EQ.2)THEN

ELK(II,JJ+2) = ELK(II,JJ+2) +TAN13(I,J)ELK(II+1,JJ+2) = ELK(II+1,JJ+2)+TAN23(I,J)ELK(II+2,JJ+1) = ELK(II+2,JJ+1)+TAN32(I,J)ELK(II+2,JJ+2) = ELK(II+2,JJ+2)+TAN33(I,J)

ENDIF250 JJ=NDF*J+1260 II=NDF*I+1

ENDIF

COMPUTATION OF RESIDUAL VECTOR AND TANGENT MATRIX


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 37

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


39

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


jn reddy - 1 lecture notes on nonlinear fem meen 673...

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