Hoe to model Beams using the finite element method? #WikiCourses #Num001 https://wikicourses.wikispaces.com/TopicX+Beams

1. FEM: Beams
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Introduction to the Finite
Element Method
Beams

2. FEM: Beams
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Beams
• Beams are the most-used structural
elements.
• Many real structures may be approximated
as beam elements
• Two main beam theories:
– Euler-Bernoulli beam theory
– Timoshenko beam theory

3. FEM: Beams
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Euler-Bernoulli Beam Theory
• The main assumption in the Euler-
Bernoulli beam theory is that the beam’s
thickness is too small compared to the
beam length
• That assumption resulted in that the sheer
deformation of the beam may be
neglected without much error in the
analysis

4. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Governing Equation
• The equation governing the deformation of
and E-B beam under transverse loading
may be written in the form:
  )(2
2
2
2
xF
dx
wd
xEI
dx
d







5. FEM: Beams
Mohammad Tawfik
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The Thin-Beam Elements
• The thin beam element has a special feature,
namely, the two degrees of freedom at each
node are related.

6. FEM: Beams
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Beam Interpolation Function
     axHxw 
3
4
2
321)( xaxaxaaxw 
    axxxxw 32
1
            axxaxHa
dx
xdH
dx
xdw
x
2
3210







7. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Interpolation Function
     aHww 00 1 
 
 
 
 
  aT
a
a
a
a
lH
lH
H
H
w
w
w
w
x
x










































4
3
2
1
2
2
1
1
0
0
'
'
     aHww x 0'0' 1 
     alHwlw  2
     alHwlw x 2''

8. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Interpolation Function









































4
3
2
1
2
32
2
2
1
1
3210
1
0010
0001
'
'
a
a
a
a
ll
lll
w
w
w
w















































2
2
1
1
2323
22
4
3
2
1
'
'
1212
1323
0010
0001
w
w
w
w
llll
llll
a
a
a
a

9. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Interpolation Function
               ee
wxNwTxHaxHxw 
1
     e
wTa
1

   

4
1i
ii wxNxw

10. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Interpolation Function
     



























2
32
3
3
2
2
3
32
3
3
2
2
23
2
23
1
l
x
l
x
l
x
l
x
l
x
l
x
x
l
x
l
x
xNxN
T

11. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Interpolation Functions
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
N(x)
X
N1
N2
N3
N4

12. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Stiffness Matrix
• The governing equation is:
• Using the series solution
  )(2
2
2
2
xF
dx
wd
xEI
dx
d






   

4
1i
ii wxNxw

13. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Stiffness Matrix
• The governing equation becomes
• Applying Galerkin method:
  )()(
4
1
2
2
2
2
xRxFw
dx
Nd
xEI
dx
d
i
i
i







   















ee l
j
i
i
i
l
j dxNxFw
dx
Nd
xEI
dx
d
dxNxR
0
4
1
2
2
2
2
0
)()(

14. FEM: Beams
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Beam Stiffness Matrix
• Using integration by parts, twice, and
ignoring the boundary terms, we get:
• In matrix form:
  0)(
0
4
1
2
2
2
2


















 
el
j
i
i
ji
dxNxFw
dx
Nd
dx
Nd
xEI
        
ee l
e
l
xxxx dxNxFwdxNNxEI
00
)(