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3 Rod and beam elements

How to analyze structures shown in pictures using

FEM:

2

3 Rod and beam elements

Rod (spar, truss element) and beam, definitions:

Rod elements can only extend or compress

axially (two-force system!)

Beam elements can carry bending moments

and (and torsion in 3D) in additio to axial

loads

rod

rod

beam

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3.1 Rod

The stiffness matrix of a rod element has already been

defined (spring!):

Thus the basic equation is

General definition: columns of stiffness matrix are

nodal loads imposed on the elements resulting unit

displacement in corresponding degree of freedom

and kee in other de rees of freedom zero.

1 1 11

1 1 2

,k k U

K uk k U

N1 N2

A,E

L

U1 U2

2

1

2

1

11

11

F

F

u

u

L

AE

,i

i ekv

i

A Ek

l

4

3.1 Rod

More formal definition comes from integral

where B is strain-displacement matrix, E on

material matrix or constitutive matrix. Matrix B is

obtained by defining the displacement field in the

rod using linear interpolation

where N is so-called shape function matrix.

N1 N2

A,E

L

u1 u2

Nd

u

u

u

L

x

L

xLxu or)(

2

1

V

dVk EBBT

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3.1 Rod

Strain is the gradient of axial displacement, i.e.

Notice that strain-displacement matrix is derivative of

the shape function vector.

Thus the stiffness matrix can be derived as

2

1,

11,

u

u

LLdx

d

dx

dux

dBBdN

111111

/1/1

0 L

AEAdxLL

EL

LdVkL

V

EBBT

6

3.2 2D Beam

2 dimensional Beam element has for degrees of

freedom, i.e. rotations at the end of the beam as

well as transverse displacements:

In the development of the stiffness matrix the

assumption is a prismatic and materially

homogenous beam.

Notice that axial deformations are excluded from

the definition but they can be readily incorporated

in stiffness matrix from rod element previously

defined.

2

2

1

1

v

v

d

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3.2 2D Beam

The stiffness matrix of a two-dimensional prismatic

beam made of homogenous material can be

derived using integral

in which the strain-displacement matrix B is

obtained from curvatured2v/dx2 using third order

polynomial, which describes the displacement field

as:

V

dVEIk BBT

3

4

2

321 xxxv

8

3.2 2D Beam

Using four shape functions, i.e. each for

corresponding dof, leads to formulation

and strain-displacement function B is obtained

from curvature d2v/dx2

2

2

1

1

4321)(

v

v

NNNNxv

BdN

2

2

2

2

dx

d

dx

vd

EI

M

dx

d

2

2Engineering beam theory:

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3.2 2D Beam

The shape functions are obtained from beam theory:

jossa

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3.2 2D beam

Thus the derivatives of the shape functions shown in

the previous slide result the strain-displacement matrix

And the integration

results

22

22

3

4626

612612

2646

612612

LLLL

LL

LLLL

LL

L

EIdVEIk

V

BBT

T2211

vvd

232232

6212664126

L

x

LL

x

LL

x

LL

x

LB

BdN

2

2

2

2

dx

d

dx

vd

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3.3 Transformation

Stiffness matrix of a rod in local coordinate system (a)

Transformation from local to global system (b)

3.3 Transformation

The global stiffness matrix of a rod can be derived as

where

cos,sin,

22

22

22

22

cs

scsscs

csccsc

scsscs

csccsc

L

AETk'Tk

T

sincos00

00sincosT

11

11

L

AEk'

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3.3 Transformation

Example: calculate the displacements and stresses of

the structure shown in figure, whenA = 300 mm2 for all

rods and E = 200 GPa.

14

3.3 Transformation

Element numbering, directions of local coordinates,

stiffness matrices and degrees of freedom:

1:C->B

2:B->A

3:C->A

uC vC uB vB

uB vB uA vA

uC vC uA vA

uCvCuBvB

uBvBuAvA

uCvCuAvA

cos,sin,

22

22

22

22

cs

scsscs

csccsc

scsscs

csccsc

L

AETk'Tk

T

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3.3 Transformation

Global stiffness matrix and force vector when loading

and boundary conditions are taken into account:

1:C->B

2:B->A

3:C->A

uC vC uB vB uA vA

A

A

B

B

C

C

v

u

v

u

vu

U

16

3.3 Transformation

Solution and results [mm]:

uB =-1,50 mm

vB =-11,3 mm

vA= - 4,00 mm

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3.3 Transformation

Formulate the stiffness matrix of the structure shown in

figure.E = 210 GPa and cross-section is RHS 100x100x4.

Axial stiffness of beamBC =

The stiffnes matrix of beamAB in global coordinate

system is

and of beamsBD and DC:

B

B

A

A

AB

v

u

v

u

scsscs

csccsc

scsscs

csccsc

L

AE1

22

22

22

22

1 , uk

D

D

B

B

BDBDBDBD

DBBD

BDBDBDBD

BDBD

BDv

v

LLLL

LL

LLLL

LL

L

EI

2

22

22

32 ,

4626

612612

2646

612612

uk

C

C

D

D

DCDCDCDC

DCDC

DCBCDCDC

DCDC

DCv

v

LLLL

LL

LLLL

LL

L

EI

3

22

22

33 ,

4626

612612

2646

612612

uk

D

D

D

B

B

v

v

D

D

B

B

v

v

B

B

A

A

v

u

v

u

1u

18

3.3 Transformation

Taking into account the boundary conditions and the axial

stiffness of beamBC (infinite!) results

22

22

22

22

31

scsscs

csccsc

scsscs

csccsc

L

AE

AB

k

22

22

32

4626

612612

2646

612612

BDBDBDBD

BDBD

BDBDBDBD

BDBD

Bd

LLLL

LL

LLLL

LL

L

EIk

D

22

22

33

4626

612612

2646

612612

DCDCDCDC

DCDC

DCDCDCDC

DCDC

DC

LLLL

LL

LLLL

LL

L

EIk

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3.3 Transformation

Global stiffness matrix is then

C

D

D

B

B

DCDCDC

DCDCBDDCBDBDBD

DCDCBDDCBDBDBD

BDBDBDBD

BDBDBDBDAB

Gv

v

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EI

L

EIs

L

AE

dk ,

42600

2446626

6661212612

02646

0612612

2

222

2223323

22

2323

2

D

20

3.3 Transformation

The solution results

Backsubstitution gives element forces, example is

elementBD:

D

Displacementvector

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3.3 Transformation of 2D beam element

Global stiffness matrix of a 2D beam element, when also

axial stiffness is taken into account is

Tk'Tk T

2

2

2

1

1

1

v

u

v

u

U

100000

0cossin000

0sincos000

000100

0000cossin

0000sincos

T

v1

1

u1

v2 u22

Y

X

22

3.2 2D beam: example

Calculate the maximum displacement and maximum

normal stress in the frame shown in picture. Crane beam

is IPE 300 and column is HEA200. Neglect self-weight.

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3.2 2D beam: exampleElementmodel and degrees of freedom (notice assumptions of model!):

E = 210 GPa

AIPE300 = 5380 mm2

IIPE300 = 83,6E6 mm4

AHEA20 = 5380 mm2

IHEA200 = 36,9E6 mm2

1 2

3

1 2 3

4

mmmmmm

mmmmmm

mmmmmm

mmmmlmlmllll

mmmmlmlmllll

mmmmlmlmllll

llllklklkkkk

llllklklkkkk

llllklklkkkk

kkkkkk

kkkkkk

kkkkkk

G

000000

000000

000000

000

000

000

000

000

000

000000

000000

000000

k

4

4

4

3

3

3

2

2

2

1

1

1

v

u

v

u

v

u

v

u

U

u

v

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3.2 2D beam: example

Distributed loading, e.g. self-weght in a

beam element

is changed to nodal loads and moments at

nodes:

2

wL

2

wL

12

2wL

12

2wL

w