Example Problem
using the
One-Dimensional Beam Element
Problem Statement: Traffic Light Pole
In this example problem we will use FEM to analyze the
horizontal beam that supports the traffic lights in the figure
below. We will determine the vertical displacement and rotation
of the horizontal beam at the traffic light mounts, the support
reactions and the internal forces at the light mount near the
center of the beam.
The homogeneous beam shown below, is made of steel with a
modulus of elasticity (
) of 
psi and is tapered
slightly with a round, hollow cross section where L1
is 96 inches and 
is 116 inches. The two, three segment lights that
are at the left end of the beam weigh 170 lb each, for a total of
340 lb and the four segment light at the center of the beam
weighs 220 lb.
Finite Element Model
There are several factors that must be considered when
evaluating a problem for suitability for analysis as an FEM
model. In order to use one-dimensional beam elements:
- Traverse forces - The external forces must be
applied perpendicular to the longitudinal axis of the
beam, otherwise axial forces will be applied to the beam
and the problem will then become a two dimensional one.
When handling a two dimensional beam problem with axial
loading, a column and a row must be added to our
stiffness matrix in order to represent the axial
displacement and force, the element stiffness matrix is
now of order 6 x 6. In this problem the beam is
horizontally oriented with the applied forces (weight of
the traffic signals) acting vertically, therefore there
is no axial force or resultant axial displacement and our
stiffness matrix is of order 4 x 4 .
- Assume rigid support - We will assume that the
upright support (pole) at the right, deflects a
negligible amount (treating it as a rigid support), so we
will not consider the displacement and rotation, that the
support pole's deflection and rotation would add to our
horizontal beam problem. If we were to include the
deflection of the upright support, we would have to model
the support using a stiffness matrix of order 6 x 6 in
order to include the axial force and deflection of the
support pole, and then the displacements and rotation at
the top of the vertical support pole would be used as
boundary conditions for our horizontal beam problem.
- Prismatic member - Finite elements are prismatic
members, a constant cross-section and moment of inertia
must be assumed for each element. When modeling a tapered
beam, an approximate solution can be obtained by dividing
the beam up into elements of constant geometric
properties. For our example we will assume a moment of
inertia equal to that of the center of each section. The
accuracy of the solution can be increased by increasing
the number of elements used to model the beam.
When we define our finite element mesh, we must number the
nodes with respect to the total structure. There is no set way
that we must number the nodes, for this example we will number
element 1 node I as global node 3, element 1 node J, which is
also element 2 node I as global node 1 and element 2 node J as
global node 2, node numbering will be discussed further when we
develop the stiffness matrix. Our FEM mesh along with the values
we will need for our analysis, are shown in the figure below.
