MECHANICS  THEORY



Introduction

Cantilever Beam with Force,
Distributed Load, and Point Moment
Causing Moments about Both the
x and y Axes


Unsymmetric beam bending is really just two problems added together using the
principle of superposition. Normally, a beam is loaded in the ydirection causing
a moment about the z axis. But the beam can also be loaded in the zdirection
causing a moment about the y axis. Both y and zdirections are perpendicular
to the beam as shown in the diagram. Each direction can be solved separately
for bending stress, and then add the results together.
If the load is at an angle to the beam, but is in the yz plane, then the load
can be reduced into two forces in the direction of the y and z axes.
In this section, all loads are assumed to act through the beam shear center
(generally the centroid) so that there is no rotation or twisting about the xaxes.
This helps simplify the calculations.






Unsymmetric Beam Bending

Bending Loading in ydirection
Produces Moment About z axis 

First, consider a beam that is loaded only in the ydirection as shown at the
left. These loads cause a bending moment about the z axes. This has been analyzed
previously in the
Bending Stress section. At any given location aa, the bending stress will be
where M_{z} is the internal moment and I_{z} is the moment
of inertia, both about the z axis.
Notice that a positive y produces a negative stress which indicates a compression
stress. Also, the doubleheaded arrow represents a moment rotating about the
vector direction. Using the
righthand rule, the thumb points in the direction
of the doubleheaded arrow and the fingers are in the direction of the moment.




Bending Loading in zdirection
Produces Moment About y axis


Similarly, loads in the zdirection will cause a moment about the y axes as
shown at the left. The bending stress is
where M_{y} is the internal moment and I_{y} is the moment
of inertia, both about the y axis. The equation does not have a negative sign
since a positive z value will produce a positive tension stress.
Again, the double headed arrow is a convenient method to represent a moment
vector when drawing in 2D. The moment rotates about the direction of the doubleheaded
arrow.




Unsymmetric Bending Stress is the
Total of the
Bending Stress about the
z and y axis 

The two bending stresses, σ_{bz} and σ_{by} can
be added together using the principle of superposition. This gives the final
unsymmetric bending stress as
The maximum tension and compression bending stress is not obvious. As shown
in the diagram at the left, the bending stress is a tetrahedral shape and increases
as the distance from the neutral increases.
The neutral axis in not horizontal but is rotated depending on the ratio of
the moments, M_{y} and M_{z}, and the moment of inertias, I_{y} and
I_{z}. More
details about the NA is given in the next section below.






Neutral Axis (NA)

Neutral Axis Rotation Angle 

It is helpful in understanding the bending stress if the location of the neutral
axis is known, especially for unsymmetric bending, The definition of the
NA is where the bending stress is zero. Thus, if the unsymmetric bending stress
equation is forced to equal zero, the result will be the line equation for the
NA, or
This is an equation for a line that goes through the origin. The maximum
bending stress is located at the point which is furthest from the neutral axis
(perpendicular direction).
The angle between the NA and the z axis is



