1. The Finite Element Method A Practical Course
CHAPTER 4
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
CASE STUDY
REMARKS

3. INTRODUCTION Frame members are loaded axially and transversely.
It is capable of carrying, axial, transverse forces, as well as moments.
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
Truss + beam
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 for the force vector,

9. Equations in global coordinate system
Coordinate transformation
Similar to trusses
where ,

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

11. Equations in global coordinate system
Finally, we have

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
Truss + beam

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
Finally, we have

22. CASE STUDY
Finite element analysis of bicycle frame

23. 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

24. CASE STUDY
Horizontal load
Constraints in all directions

25. CASE STUDY
M = 20X

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