Derivation and solution of the nonlinear finite element model of a beam under thermal loading. #WikiCourses https://wikicourses.wikispaces.com/TopicX+Nonlinear+Solid+Mechanics https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

1. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Introduction to the Finite
Element Method
Nonlinear Beam Deflection

2. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Nonlinear Strain Displacement
Relation

3. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Nonlinear Strain Displacement
Relation
• The von Karman large deflection strain-
displacement relation for the deflections u,
and w can be written as follows
2
22
2
1
x
w
z
x
w
x
u
x














         zm 

4. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Using the relations …
  
  }{
xx1 32
aH
axw
w
   
  }{
1
bH
bxu
u

       bwbbw wNwTHw 
1
       mummu wNwTHu 
1

5. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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For the derivatives …
      bbb
w
wBwT
dx
dH
dx
dw







1
      bbbb
w
wBwT
dx
Hd
dx
wd







1
2
2
2
2
      mmmm
u
wBwT
dx
dH
dx
du







1

6. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Giving the strain relation …
• Where:
• Is a scalar function of x
     
         bbbmm
m
wzww
z
BBB 





2
1
}{
 
x
w




7. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Inplane Forces and Bending
Moments

8. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Inplane Forces and Bending
Moments
• For homogeneous symmetric cross sections:


























T
T
M
N
D
A
M
N


0
0
EAQhA  EIQ
h
D 
12
3
   TEAdzzyxTQN
h
h
T   
2/
2/
),,(
   0),,(
2/
2/
 
h
h
T zdzzyxTQM 

9. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Expanding equation
     
          
     Tm
Tbmm
Tm
NNN
NwAwA
NAN











BB
2
1
][
    bb wDDM B][}]{[  

10. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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The Potential Energy
      
Volume height
TT
ijij dzMNdVU }{}{ 
          
          
         
dz
wDw
NwAwA
ww
U
height
bb
T
b
T
b
Tbmm
TTT
b
T
m
T
m


































 
BB
BB
BB
][
2
1
*



 


11. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Skipping the details …
• We get the element equation as:
 
 
 
   
 
 
 
 











































 
m
b
mb
bmnm
TN
m
b
W
W
n
n
nn
k
T
k
k
00
02
3
1
01
11
2
1
00
0
0
0
 
 













Tm
b
p
p 0
0

12. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Where …
• The linear stiffness matreces
   dzDk
height
b
T
bb  BB ][][
    dzAk
height
m
T
mm  BB][

13. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Where …
• The nonlinear stiffness matreces
     dzAnn
height
T
m
T
bmmb   BB]1[]1[
      dzBNn
height
m
T
nm  B1
        dzAn
height
TT
   BB
2
3
]2[

14. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Where …
• The thermal effect terms
    dzBNk
height
T
T
TN  B][
     dzNp
height
T
T
mTm  B

15. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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In a compact form …
             TTN PPWNNKK  





 2
3
1
1
2
1

16. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Note
• For details for the derivation, follow the link
on the pages:
• https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechani
cs
• https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

17. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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http://WikiCourses.WikiSpaces.com
Solving the Equations for
Post-Buckling Deflection

18. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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When no external loading exists
• We may write the equations in the form:
• Introducing a new function:
           TTN PWNNKK  





 2
3
1
1
2
1
               02
3
1
1
2
1






  TTN PWNNKKW

19. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Using truncated Taylor
expansion
• Linear approximation:
• Where:
         W
dW
Wd
WWW 


             tan21 KNNKK
dW
Wd
TN 



20. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Using Newton Algorithm:
Repeat, until:
              TiiiTNi PWNNKKW  





 2
3
1
1
2
1
     iii
WWK 1tan 
      ii WKW i



1
tan1
      11   iii WWW 
  toliW  1max

21. FEM: Nonlinear Beam Deflection
Mohammad Tawfik
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Maximum deflection vs. Temp.