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

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

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.

FEM EQUATIONS FOR PLANAR FRAMES Consider a planar frame element

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

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

Equations in local coordinate system + 

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

Equations in global coordinate system Coordinate transformation where ,

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

Equations in global coordinate system Therefore,

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

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

Equations in local coordinate system where

Equations in global coordinate system

Equations in global coordinate system Coordinate transformation where ,

Equations in global coordinate system Direction cosines in T3

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

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

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

Equations in global coordinate system Therefore,

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.

CASE STUDY Finite element analysis of bicycle frame

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

CASE STUDY Horizontal load Constraints in all directions

CASE STUDY M = 20X

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