1. Finite Element Method CHAPTER 6: FEM FOR FRAMES
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 6:
FEM FOR FRAMES

2. CONTENTS INTRODUCTION FEM EQUATIONS FOR PLANAR FRAMES
Equations in local coordinate system
Equations in global coordinate system
FEM EQUATIONS FOR SPATIAL FRAMES
REMARKS

3. INTRODUCTION Deform axially and transversely.
It is capable of carrying both axial and transverse forces, as well as moments.
Hence combination of truss and beam elements.
Frame elements are applicable for the analysis of skeletal type systems of both planar frames (2D frames) and space frames (3D frames).
Known generally as the beam element or general beam element in most commercial software.

4. FEM EQUATIONS FOR PLANAR FRAMES
Consider a planar frame element

5. Equations in local coordinate system
Combination of the element matrices of truss and beam elements
From the truss element,
Truss
Beam
(Expand to 6x6)

6. Equations in local coordinate system
From the beam element,
(Expand to 6x6)

7. Equations in local coordinate system+ 

8. Equations in local coordinate system
Similarly so for the mass matrix and we get
And for the force vector,

9. Equations in global coordinate system
Coordinate transformation
where ,

10. Equations in global coordinate system
Direction cosines in T:
(Length of element)

11. Equations in global coordinate system
Therefore,

12. FEM EQUATIONS FOR SPATIAL FRAMES
Consider a spatial frame element
Displacement components at node 1
Displacement components at node 2

13. Equations in local coordinate system
Combination of the element matrices of truss and beam elements

14. Equations in local coordinate system
where

15. Equations in global coordinate system

16. Equations in global coordinate system
Coordinate transformation
where ,

17. Equations in global coordinate system
Direction cosines in T3

18. Equations in global coordinate system
Vectors for defining location and orientation of frame element in space
k, l = 1, 2, 3

19. Equations in global coordinate system
Vectors for defining location and orientation of frame element in space (cont’d)

20. Equations in global coordinate system
Vectors for defining location and orientation of frame element in space (cont’d)

21. Equations in global coordinate system
Therefore,

22. REMARKS
In practical structures, it is very rare to have beam structure subjected only to transversal loading.
Most skeletal structures are either trusses or frames that carry both axial and transversal loads.
A beam element is actually a very special case of a frame element.
The frame element is often conveniently called the beam element.

23. CASE STUDY
Finite element analysis of bicycle frame

24. CASE STUDY 74 elements (71 nodes) Ensure connectivity Young’s modulus,
E GPa
Poisson’s ratio, 
69.0
0.33
74 elements (71 nodes)
Ensure connectivity

25. CASE STUDY
Horizontal load
Constraints in all directions

26. CASE STUDY
M = 20X

27. CASE STUDY Axial stress -9.68 x 105 Pa -6.264 x 105 Pa -6.34 x 105 Pa