We wish to use FEM for solving the following problems:
δ = 2 x 10-2 mm
1. To develop a system of linear equations for one-dimensional
2. To apply FE method for solving general problems involving
bar structures with different support conditions.
General Loading Condition
Consider a non-uniform bar subjected
to a general loading condition, as
Note: The bar is constrained by a fix support at
the top and is free at the other end. The positive
x-direction is taken downward.
Types of Loading
a) Body force, f
Distributed force per unit volume (N/m3)
Example: self-weight due to gravity
b) Traction force, T
Force per unit area (N/m2)
For a 1-D problem,
Examples: Frictional forces, Viscous drag,
and Surface shear.
c) Point load, Pi
Concentrated load (in Newton) acting at any point i.
Finite element Modeling
The first step is to subdivide the bar into several sections – a
process called discretization.
Note: The bar is discretized
into 4 sections, each has a
uniform cross-sectional area.
The non-uniform bar is
transformed into a stepped
We will use the stepped bar
as a basis for developing a
finite element model of the
original non-uniform bar.
To analyze the stepped bar systematically, a global numbering
scheme is assigned as shown. The x-direction is considered as
the global coordinate direction.
F1, …, F5 represent global
forces acting on the points
connecting all sections of
the stepped bar.
Q1, …, Q5 represent global
displacements of the points
resulting from the forces
acting on th