1

1

3 Rod and beam elementsHow to analyze structures shown in pictures using FEM:

2

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

• Rod elements can only extend or compressaxially (two-force system!)

• Beam elements can carry bending momentsand (and torsion in 3D) in additio to axialloads

rod

rod

beam

2

3

3.1 RodThe stiffness matrix of a rod element has already been

defined (spring!):

Thus the basic equation is

General definition: columns of stiffness matrix arenodal loads imposed on the elements resulting unitdisplacement in corresponding degree of freedomand keeping other degrees of freedom zero.

1 1 11

1 1 2

,k k U

K uk k U

N1 N2

A,EL

U1 U2

2

1

2

1

1111

FF

uu

LAE

,i

i ekvi

A Ekl

4

3.1 RodMore 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,EL

u1 u2

Nduuu

Lx

LxLxu or)(

2

1

V

dVk EBBT

3

5

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,uu

LLdxd

dxdu

x dBBdN

111111

/1/1

0 LAEAdx

LLE

LL

dVkL

V

EBBT

6

3.2 2D Beam2 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 materiallyhomogenous beam.Notice that axial deformations are excluded fromthe definition but they can be readily incorporatedin stiffness matrix from rod element previouslydefined.

2

2

1

1

v

v

d

4

7

3.2 2D BeamThe stiffness matrix of a two-dimensional prismaticbeam made of homogenous material can bederived using integral

in which the strain-displacement matrix B is obtained from curvature d2v/dx2 using third orderpolynomial, which describes the displacement fieldas:

V

dVEIk BBT

34

2321 xxxv

8

3.2 2D BeamUsing four shape functions, i.e. each for corresponding dof, leads to formulation

and strain-displacement function B is obtainedfrom curvature d2v/dx2

2

2

1

1

4321)(v

v

NNNNxv

BdN2

2

2

2

dxd

dxvd

EIM

dxd

2

2

Engineering beam theory:

5

9

3.2 2D BeamThe shape functions are obtained from beam theory:

jossa

10

3.2 2D beamThus the derivatives of the shape functions shown in the previous slide result the strain-displacement matrix

And the integration

results

22

22

3

4626612612

2646612612

LLLLLL

LLLLLL

LEIdVEIk

V

BBT

T2211 vvd

2322326212664126Lx

LLx

LLx

LLx

LB

BdN2

2

2

2

dxd

dxvd

6

11

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

scsscscsccscscsscscsccsc

LAETk'Tk T

sincos0000sincos

T

1111

LAEk'

7

13

3.3 Transformation

Example: calculate the displacements and stresses of the structure shown in figure, when A = 300 mm2 for allrods 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

uCvC uB vB

uBvB uA vA

uCvC uA vA

cos,sin,

22

22

22

22

cs

scsscscsccscscsscscsccsc

LAETk'Tk T

8

15

3.3 Transformation

Global stiffness matrix and force vector when loadingand 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

vuvuvu

U

16

3.3 Transformation

Solution and results [mm]:

uB = -1,50 mmvB = -11,3 mmvA= - 4,00 mm

9

17

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 beam BC = The stiffnes matrix of beam AB in global coordinatesystem is

and of beams BD and DC:

B

B

A

A

AB

vuvu

scsscscsccscscsscscsccsc

LAE

1

22

22

22

22

1 , uk

D

D

B

B

BDBDBDBD

DBBD

BDBDBDBD

BDBD

BD v

v

LLLLLL

LLLLLL

LEI

2

22´

22

32 ,

4626612612

2646612612

uk

C

C

D

D

DCDCDCDC

DCDC

DCBCDCDC

DCDC

DC v

v

LLLLLL

LLLLLL

LEI

3

22

22

33 ,

4626612612

2646612612

uk

D

D

D

B

B

v

v

D

D

B

B

v

v

B

B

A

A

vuvu

1u

18

3.3 Transformation

Taking into account the boundary conditions and the axialstiffness of beam BC (infinite!) results

22

22

22

22

31

scsscscsccscscsscscsccsc

LAE

AB

k

22

22

32

4626612612

2646612612

BDBDBDBD

BDBD

BDBDBDBD

BDBD

Bd

LLLLLL

LLLLLL

LEIk

D

22

22

33

4626612612

2646612612

DCDCDCDC

DCDC

DCDCDCDC

DCDC

DC

LLLLLL

LLLLLL

LEIk

10

19

3.3 Transformation

Global stiffness matrix is then

C

D

D

B

B

DCDCDC

DCDCBDDCBDBDBD

DCDCBDDCBDBDBD

BDBDBDBD

BDBDBDBDAB

G v

v

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEI

LEIs

LAE

dk ,

42600

2446626

6661212612

02646

0612612

2

222

2223323

22

23232

D

20

3.3 Transformation

The solution results

Backsubstitution gives element forces, example is element BD:

D

Displacement vector

11

21

3.3 Transformation of 2D beam element

Global stiffness matrix of a 2D beam element, when alsoaxial stiffness is taken into account is

Tk'Tk T

2

2

2

1

1

1

v

u

v

u

U

1000000cossin0000sincos0000001000000cossin0000sincos

T

v1

1

u1

v2 u22

Y

X

22

3.2 2D beam: exampleCalculate the maximum displacement and maximumnormal stress in the frame shown in picture. Crane beamis IPE 300 and column is HEA200. Neglect self-weight.

12

23

3.2 2D beam: exampleElement model and degrees of freedom (notice assumptions of model!):E = 210 GPaAIPE300 = 5380 mm2

IIPE300 = 83,6E6 mm4

AHEA20 = 5380 mm2

IHEA200 = 36,9E6 mm21 2

3

1 2 3

4

mmmmmmmmmmmmmmmmmmmmmmlmlmllllmmmmlmlmllllmmmmlmlmllll

llllklklkkkkllllklklkkkkllllklklkkkk

kkkkkkkkkkkkkkkkkk

G

000000000000000000

000000000

000000000000000000000000000

k

4

4

4

3

3

3

2

2

2

1

1

1

vu

vu

vu

vu

U

u

v

24

3.2 2D beam: exampleDistributed loading, e.g. self-weght in a beam element

is changed to nodal loads and moments at nodes:

2wL

2wL

12

2wL12

2wL

w