Slide_chpt05-FEM for Beams有限元课件

v( x)
Px
2
(3L x )
3
6 EI y v( x L)
6

Step 1: Element matrices
1 12 bh
3
PL
Iz
1 12
0.1 0.06
3
3EI y
m
4
1.8 10
0.75 0.25 0.75 0.125
The Finite Element Method
A Practical Course
CHAPTER 5: FEM FOR BEAMS
1
CONTENTS

INTRODUCTION FEM EQUATIONS – Shape functions construction – Strain matrix – Element matrices
P=1000 N 0.1 m
E=69 GPa
=0.33
0.5 m
0.06
m
The Finite Element Method by G. R. Liu and S. S. Quek
16
EXAMPLE
Exact solution:
EI y v
4 4
x Eq. (2.59)
fy 0
fy 0
3 3a 3
3a 2 2a 3a 2 4a
12
The Finite Element Method by G. R. Liu and S. S. Quek
Element matrices
me
N NdV dA
T V A
a
a
N Ndx A
15
EXAMPLE
Consider the cantilever beam as shown in the figure. The beam is fixed at one end and it has a uniform cross-sectional area as shown. The beam undergoes static deflection by a downward load of P=1000N applied at the free end. The dimensions and properties of the beam are shown in the figure.
Element matrices
f s1 N1 1 ms1 N2 d f ya 1 N 3 fs2 m N4 s2 f y a f s1 f a2 y 3 ms1 f y a fs2 2 fya 3 ms 2
d1 = v1 d4 = 2 d2 = 1 0 2a x= - a
= 1
d3 = v2
x,
x= a
=1
To obtain constant coefficients – four conditions
(1) (2) v( 1) v1 dv dx
(3) (4)
1
1
At x= a or = 1
4
The Finite Element Method by G. R. Liu and S. S. Quek
FEM EQUATIONS

Shape functions construction Strain matrix Element matrices
The Finite Element Method by G. R. Liu and S. S. Quek
2 3
The Finite Element Method by G. R. Liu and S. S. Quek
10
Strain matrix
xx u x
y v
2
x

2 2
2
yLv
Eq. (2-47)
Therefore,
B yLN y x N y
Evaluate integrals
78 Aa me 105 sy.
22a 8a
2
27 13a 78
13a 2 6a 22a 2 8a
13
The Finite Element Method by G. R. Liu and S. S. Quek
or
v( ) p ( )α
T

v x

v x

1 v a

1 a
( 1 2 2
3 3 )
2
The Finite Element Method by G. R. Liu and S. S. Quek
7
Shape functions construction
Ae
1
or
α Ae de
1
The Finite Element Method by G. R. Liu and S. S. Quek
9
Shape functions construction
Therefore,
v N( )d e
where
N( ) PA e N1 ( )
1

N 2 ( )
3
N 3 ( )
N 4 ( )

in which
N1 ( ) N 2 ( ) N 3 ( ) N 4 ( )
1 4 a 4 1 4 a 4
( 2 3 ) (1 )
2 3
( 2 3 )
3
( 1 )
1 a
1
2 a
1
1 a
1
2 a
1 1 3 2 a 1 3 3 a 4
or
d e A eα

2 1 3 4 0 1 a a a a 2 3 0 1 a a a a
11
The Finite Element Method by G. R. Liu and S. S. Quek
Element matrices
ke

V
B cBdV E y dA
T 2 A 1
a
a 2 2
(

2 2
x
N) (
T

2 2
x
N)dx
EI z
1 a
4
1
[

2 2
The Finite Element Method by G. R. Liu and S. S. Quek
3
INTRODUCTION

In beam structures, the beams are joined together by welding (not by pins or hinges). Uniform cross-section is assumed. FE matrices for beams with varying cross-sectional area can also be developed without difficulty.
fe
N fb dV
T
k ed e m ed e fe
The Finite Element Method by G. R. Liu and S. S. Quek
14
Remarks

Theoretically, coordinate transformation can also be used to transform the beam element matrices from the local coordinate system to the global coordinate system.
5
Shape functions construction

Consider a beam element
d1 = v1 d4 = 2 d2 = 1 0 2a x= - a
= 1
d3 = v2
x,
x= a
=1
Natural coordinate system:
The Finite Element Method by G. R. Liu and S. S. Quek

The transformation is necessary only if there is more than one beam element in the beam structure, and of which there are at least two beam elements of different orientations.

x a
6
Shape functions construction
Assume that v( ) 2 3 0 1 2 3
In matrix form:
v( ) 1



2

3

0 1 2 3

N] [
T


N]ad
EI z a
3

1
1
N N d
T
ke
EI z a
3

1
1
N1 N1 N N 2 1 N 3 N1 N 4 N1
N1 N 2 N2 N2 N3 N 2 N4 N2