4-Axial Members, Beams & Frames (1) (1).pdf

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1. 4. Axial Members, Beams & Frames
ME 411
Introduction to FEA
1

2. 4. Axial Members, Beams & Frames
2
Introduction

3. 4. Axial Members, Beams & Frames
3
Introduction

4. 4. Axial Members, Beams & Frames
4
Interpolation function

5. 4. Axial Members, Beams & Frames
5
Interpolation function

6. 4. Axial Members, Beams & Frames
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Shape functions

7. 4. Axial Members, Beams & Frames
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Shape functionsx
x
1 x
2
El #112
2
1
xx
x-x
(x)N
−
= 12
1
2
xx
x-x
(x)N
−
=
1 1

8. 4. Axial Members, Beams & Frames
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Shape functionsx
x
1 x
2
El #12x
12
1
1x
12
2(1)
d
xx
x-x
d
xx
x-x
(x)w
−
+
−
= 3x
23
2
2x
23
3(2)
d
xx
x-x
d
xx
x-x
(x)w
−
+
−
= x
3El #2

9. 4. Axial Members, Beams & Frames
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Shape functions

10. 4. Axial Members, Beams & Frames
Problem
•In the accompanying figure, the deflection of nodes 2 and 3 are 0.02 mm
and 0.025 mm, respectively. What are the deflections at point A and point
B, provided that linear elements were used in the analysis?
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11. 4. Axial Members, Beams & Frames
Problem
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12. 4. Axial Members, Beams & Frames
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Beam Introduction

13. 4. Axial Members, Beams & Frames
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Beam Introduction

14. 4. Axial Members, Beams & Frames
Beam Introduction
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15. 4. Axial Members, Beams & Frames
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Beam Element

16. 4. Axial Members, Beams & Frames
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Shape Functions

17. 4. Axial Members, Beams & Frames
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Shape Functions

18. 4. Axial Members, Beams & Frames
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19. 4. Axial Members, Beams & Frames
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Quadratic Form U dkd
T
= m atrixsquarek
v ectord
=
= 





=






=
2221
1211
2
1
kk
kk
k
d
d
d  
 
2
2222112
2
111
22211222121111
222112
212111
21
2
1
2212
1211
21
2
)()(
U
dkddkdk
dkdkddkdkd
dkdk
dkdk
dd
d
d
kk
kk
dddkd
T
++=
+++=






+
+
=












==

20. 4. Axial Members, Beams & Frames
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Quadratic Form212111
1
22
U
dkdk
d
+=

 222112
2
22
U
dkdk
d
+=

 dk
d
d
kk
kk
d
d
d
2
2
U
U
U
2
1
2212
1211
2
1
=












=






















21. 4. Axial Members, Beams & Frames
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Stiffness Matrix

22. 4. Axial Members, Beams & Frames
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Load Matrix

23. 4. Axial Members, Beams & Frames
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Load Matrix

24. 4. Axial Members, Beams & Frames
Problem
•Compare the finite element solution to the exact classical beam theory
solution for the cantilever beam shown. Let ??????= 210 GPa,
??????= 4 ×10
−5
m
4
, ??????= 2.5 m, and uniform load ??????= 4 kN/m.
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25. 4. Axial Members, Beams & Frames
Problem
25

26. 4. Axial Members, Beams & Frames
Problem
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27. 4. Axial Members, Beams & Frames
Problem
27

28. 4. Axial Members, Beams & Frames
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Frame Introduction

29. 4. Axial Members, Beams & Frames
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Frame Element

30. 4. Axial Members, Beams & Frames
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Transformation matrix

31. 4. Axial Members, Beams & Frames
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Stiffness matrix

32. 4. Axial Members, Beams & Frames
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Stiffness matrix

33. 4. Axial Members, Beams & Frames
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Stiffness matrix

34. 4. Axial Members, Beams & Frames
Problem
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