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

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.

FEM EQUATIONS FOR PLANAR FRAMES Consider a planar frame element ?

Equations in local coordinate system Truss + beam 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 for the force vector,

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

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

Equations in global coordinate system Finally, we have

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 Truss + beam

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 Finally, we have

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