Finite Element Method

1
Finite Element Method
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
• Equations in local coordinate system
• Equations in global coordinate system
• 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 (contd)

20
Equations in global coordinate system

• Vectors for defining location and orientation of
  frame element in space (contd)

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
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
9.354 x 105 Pa
-6.657 x 105 Pa
-1.214 x 106 Pa
-5.665 x 105 Pa