Download - Lecture 5-beam elements.ppt - West Virginia University

MAE 456 Finite Element Analysis

Beam Elements

Slides from this set are adapted from B.S. Altan, Michigan Technological U.


Bending of Beams – Definition of Problem

Beam with a straight axis (x-axis), with varying cross-section

(A(x)), loaded transversely (q(x) in y direction) with transverse

deflections (v(x) in y direction).

x

MV

q(x)v(x)

x

x=0 x=L

A(x)

y

z

2



ρε

yx −=)(

x

ρ

2

2

dx

vdEyE −== εσ

y

2

21

dx

vd≅

ρ

dAdx

vdEyydA

AA

∫∫ −=2

22σ

3



dAdx

vdEyydA

AA

∫∫ −=2

22σ

2

2

dx

vdEIM −=

2

2

dx

vdEI

dx

d

dx

dMV =−=

2

2

2

2

)(dx

vdEI

dx

d

dx

dVxq == Resulting Field Equation

Continued from previous slide.

4



Summary

)(4

4

xqdx

vdEI =

Slope:dx

dvx =)(θ

Bending Moment:2

2

)(dx

vdEIxM =

Shear Force:3

3

)(dx

vdEIxV −=

Field equation:

5



Boundary Conditions: Four boundary conditions are

necessary to solve a bending problem. Boundary

conditions can be:

deflection:

bending Moment:

shear force:

slope:

)(),0( Lvv

)(),0( Lθθ

)(),0( LMM

)(),0( LVV

6


Beam Element – Shape Functions

• Recall that shape functions are used to interpolate displacements.7



• There are two degrees of freedom (displacements)

at each node: v and θz.

• Each shape function corresponds to one of the

displacements being equal to ‘one’ and all the

other displacements equal to ‘zero’.

• Note that everything we do in this course

assumes that the displacements are small.8



Bd

dNNd

Ey

dx

dEy

dx

dEy

dx

vdEyE

−=

−=

−=

−==

2

2

2

2

2

2

εσ

9• Recall that the B row vector is used to interpolate stresses & strains.


Beam Element – Formal Derivation

• The formal beam element stiffness matrix

derivation is much the same as the bar element

stiffness matrix derivation. From the minim-

ization of potential energy, we get the formula:

• As with the bar element, the strain energy of the

element is given by .kddT

21

10



The result is:

which operates on d = [v1, θz1, v2, θz2]T.

−

−−−

−

−

=

22

22

3

4626

612612

2646

612612

LLLL

LL

LLLL

LL

L

EIk

11



• The moment along the element is given by:

• The stress is given by:

BdyEI

Myx −=−=σ

12


Beam Element w/Axial Stiffness

• If we combine the bar and beam stiffness

matrices, we get a general beam stiffness

matrix with axial stiffness.

13


Uniformly Distributed Loads

Laterally:

14


Equivalent Loadings

15


Orientating Element in 3-D Space

• Transformation matrices are used to transform

the equations in the element coordinate system

to the global coordinate system, as was shown

for the bar element.

16