CHAPTER 4

FEM FOR FRAMES

The FThe Finite Element Methodinite Element MethodA Practical CourseA Practical Course

CONTENTSCONTENTS INTRODUCTION FEM EQUATIONS FOR PLANAR FRAMES

– Equations in local coordinate system– Equations in global coordinate system

FEM EQUATIONS FOR SPATIAL FRAMES– Equations in local coordinate system– Equations in global coordinate system

CASE STUDY REMARKS

INTRODUCTIOINTRODUCTIONN

Frame members are loaded axially and transversely.

It is capable of carrying, axial, transverse forces, as well as moments.

Frame elements are applicable for the analysis of skeletal type systems of both planar frames (2D frames) and space frames (3D frames).

Known generally as the beam element or general beam element in most commercial software.

FEM EQUATIONS FOR PLANAR FEM EQUATIONS FOR PLANAR FRAMESFRAMES

Consider a planar frame element

Y, V

X, U

node 1 (u1, v1, z1)

x, , u y, v

z

z

l=2a

node 2 (u2, v2, z2)

1 1

2 1

3 1

4 2

5 2

6 2

diplacement components at node 1

diplacement components at node 2

ze

z

d u

d v

d

d u

d v

d

d

??

Equations in local coordinate Equations in local coordinate systemsystemTruss +

beam

Truss

Beam

From the truss element,

1 1 4 2

1 12 2

4 22

0 0 0 0

0 0 0 0 0

0 0 0 0

0 0

. 0 0

0

AE AEa a

trusse AE

a

d u d u

d u

d u

sy

k

(Expand to 6x6)

node 1 (u1, v1, z1)

x, , u y, v

z

l=2a

node 2 (u2, v2, z2)

1 1

2 1

3 1

4 2

5 2

6 2

ze

z

d u

d v

d

d u

d v

d

d

Equations in local coordinate Equations in local coordinate systemsystem

From the beam element

3 2 3 2

2

3 2

3 1 5 2 6 22 1

3 3 3 3

2 2 2 2 2 12 3

2 3 1

3 35 22 2

26 2

( ) ( ) ( )( )

0 0 0 0 0 0

0

0

0 0 0

.

z z z z

z z z

z z

z

z z

EI EI EI EI

a a a aEI EI EI

a a a zbeame

EI EI

a aEI

za

d d v dd v

d v

d

d vsy

d

k

(Expand to 6x6)

node 1 (u1, v1, z1)

x, , u y, v

z

l=2a

node 2 (u2, v2, z2)

Equations in local coordinate systemEquations in local coordinate system

0

00.

00

0000

00000

0000

2

22

sya

AE

aAE

aAE

trussek

3 2 3 2

2

3 2

3 3 3 3

2 2 2 22 3

2

3 3

2 22

0 0 0 0 0 0

0

0

0 0 0

.

z z z z

z z z

z z

z

EI EI EI EI

a a a aEI EI EI

a a abeame

EI EI

a aEI

a

sy

k+

a

EIa

EI

a

EIa

AEa

EI

a

EI

a

EIa

EI

a

EI

a

EI

a

EIa

AEa

AE

e

z

zz

zzz

zzzz

sy2

2

3

2

32

2

322

3

2

3

2

3

2

322

23

2

2323

.

00

0

0

0000

k

Equations in local coordinate systemEquations in local coordinate system

Similarly so for the mass matrix

2

22

8

2278.

0070

61308

132702278

00350070

105

a

asy

aaa

aa

Aae

m

And for the force vector,2

2

1

1

13

2

2

23

y

y

x sx

y sy

f a

se

x sx

y sy

f a

s

f a f

f a f

m

f a f

f a f

m

f

Equations in global coordinate Equations in global coordinate systemsystem Coordinate transformation

Similar to trusses

ee TDd

where

j

j

j

i

i

i

e

D

D

D

D

D

D

3

13

23

3

13

23

D ,

100000

0000

0000

000100

0000

0000

yy

xx

yy

xx

ml

ml

ml

ml

T

D3i -2

D3i-1

D3j -2

D2j

2a

x

u1

u2

global node j local node 2

global node i local node 1

0

X

Y

o

x

y

v1

v2

D3j

D3j - 1

z2

z1

D3i

Equations in global coordinate Equations in global coordinate systemsystem

D3i -2

D3i-1

D3j -2

D2j

2a

x

u1

u2

global node j local node 2

global node i local node 1

0

X

Y

o

x

y

v1

v2

D3j

D3j - 1

z2

z1

D3i

cos( , ) cos

cos( , ) sin

j ix

e

j ix

e

X Xl x X

l

Y Ym x Y

l

cos( , ) cos(90 ) sin

cos( , ) cos

j iy

e

j iy

e

Y Yl y X

l

X Xm y Y

l

2 2( ) ( )e j i j il X X Y Y

Direction cosines in T:

(Length of element)

Equations in global coordinate systemEquations in global coordinate system

Finally, we have

TkTK eT

e

TmTM eT

e

eT

e fTF

100000

0000

0000

000100

0000

0000

yy

xx

yy

xx

ml

ml

ml

ml

T

FEM EQUATIONS FOR FEM EQUATIONS FOR SPATIAL FRAMESSPATIAL FRAMES

Consider a spatial frame element

v1

u1 w1

x

y

z

1

2

z2

w2

x2

y2

z1

x1

y1

u2

v2

1 1

2 1

3 1

4 1

5 1

6 1

27

28

29

210

211

212

x

y

ze

x

y

z

d u

d v

d w

d

d

d

ud

vd

wd

d

d

d

d

Displacement components at node 1

Displacement components at node 2

??

Equations in local coordinate systemEquations in local coordinate system

3 2 3 2

3 2 3 2

1 21 21 1 1 1 2 2 2 2

2 23 3 3 3

2 2 2 2

3 3 3 3

2 2 2 2

2

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

z z z z

y y y y

y yx xz z

AE AEa a

EI EI EI EI

a a a a

EI EI EI EI

a a a a

GJa

e

u v w u v w

k

2

2

3 2

3 2

2

2 3

22 3

2

23 3

2 2

3 3

2 2

2

2

2

.

0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0

0 0

0 0

0

y y y

z z z

z z

y y

y

z

GJa

EI EI EI

a a aEI EI EI

a a a

AEa

EI EI

a a

EI EI

a a

GJa

EI

aEI

a

sy

v1

u1 w1

x

y

z

1

2

z2

w2

x2

y2

z1

x1

y1

u2

v2

Truss + beam

Equations in local coordinate Equations in local coordinate systemsystem

2

2

2

22

22

22

8080070.0220782200078000007060001308060130008003500000700130270002207813000270220007800000350000070

105

aa

rsya

a

aaaaaa

rraa

aa

Aa

x

xx

e

m

whereA

Ir x

x 2

Equations in global coordinate systemEquations in global coordinate system

Y

X

Z

y x

z

D6i-1

D6i-2

D6i-3

D6i-4

D6i-5

D6j-2

D6j-1

D6j-3

D6j-4

D6j

D6i

d6

d5

d4

d3

d2 d1

d12

d11

d10

d9

d8

d7

D6j-5 y

x

z

1

2 3

Equations in global coordinate systemEquations in global coordinate system

Coordinate transformation

ee TDd where

j

j

j

j

j

j

i

i

i

i

i

i

e

D

D

D

D

D

D

D

D

D

D

D

D

6

16

26

36

46

56

6

16

26

36

46

56

D ,

3

3

3

3

T000

0T00

00T0

000T

T

zzz

yyy

xxx

nml

nml

nml

3T

Equations in global coordinate systemEquations in global coordinate system

cos( , ), cos( , ), cos( , )

cos( , ), cos( , ), cos( , )

cos( , ), cos( , ), cos( , )

x x x

y y y

z z z

l x X m x Y n x Z

l y X m y Y n y Z

l z X m z Y n z Z

Direction cosines in T3

zzz

yyy

xxx

nml

nml

nml

3T

Equations in global coordinate Equations in global coordinate systemsystem Vectors for defining location and orientation of

frame element in space

Y

X

Z

x

1

2 3

12 VV

13 VV

y

y

z

)()( 1312 VVVV

2V

1V

3V

ZZYYXXV

1111

ZZYYXXV

2222

ZZYYXXV

3333

lkkl

lkkl

lkkl

ZZZ

YYY

XXX

k, l = 1, 2, 3

221

221

221122 ZYXVVal

ZZYYXXVV

21212112

ZZYYXXVV

31313113

Equations in global coordinate systemEquations in global coordinate system

Vectors for defining location and orientation of frame element in space (cont’d)

a

ZZxZxn

a

YYxYxm

a

XXxXxl

x

x

x

2),cos(

2),cos(

2),cos(

21

21

21

)()(

)()(

1312

1312

VVVV

VVVVz

Za

ZY

a

YX

a

X

VV

VVx

222

)( 212121

12

12

})()(){(2

1213131212131312121313121

123

ZYXYXYXZXZXZYZYA

z

221313121

221313121

221313121123 )()()( YXYXXZXZZYZYA

Y

X

Z

1

2 3

12 VV

13 VV

y

z

)()( 1312 VVVV

2V

1V

3V

x

Equations in global coordinate systemEquations in global coordinate system Vectors for defining location and orientation of

frame element in space (cont’d)

21 31 31 21123

21 31 31 21123

21 31 31 21123

1( )

2

1( )

2

1( )

2

z

z

z

l z X Y Z Y ZA

m z Y Z X Z XA

n z Z X Y X YA

xzy

xzxzy

xzxzy

xzxzy

lmmln

nllnm

mnnml

Y

X

Z

1

2 3

12 VV

13 VV

y

z

)()( 1312 VVVV

2V

1V

3V

x

Equations in global coordinate systemEquations in global coordinate system

Finally, we have

TkTK eT

e

TmTM eT

e

eT

e fTF

3

3

3

3

T000

0T00

00T0

000T

T

zzz

yyy

xxx

nml

nml

nml

3T

CASE CASE STUDYSTUDY

Finite element analysis of bicycle frame

CASE STUDYCASE STUDYYoung’s modulus,

E GPaPoisson’s ratio,

69.0 0.33

74 elements (71 nodes)

Ensure connectivity

CASE STUDYCASE STUDY

Constraints in all directions

Horizontal load

CASE STUDYCASE STUDY

M = 20X

CASE CASE STUDYSTUDY

-9.68 x 105 Pa

-1.214 x 106 Pa

-6.34 x 105 Pa

-6.657 x 105 Pa

9.354 x 105 Pa

-5.665 x 105 Pa

-6.264 x 105 Pa

Axial stress