Section II.2
**
Section
II.2 Elastic
Bending of
Homogeneous
Beams
**

The general bending stress equation for **elastic, homogeneous** beams is
given as

(II.1)

where Mx and My are the bending moments about the x and y centroidal axes,
respectively. Ix and Iy are the second moments of area (also known as moments
of inertia) about the x and y axes, respectively, and Ixy is
the product of inertia. Using this equation it would be possible to calculate
the bending stress at any point on the beam cross section regardless of moment
orientation or cross-sectional shape. Note that Mx, My, Ix, Iy, and Ixy are
all unique for a given **section** along the length of the beam. In other
words, they will not change from one point to another on the cross section.
However, the x and y variables shown in the equation correspond to the
coordinates of a point on the cross section at which the stress is to be
determined.
**Sign Convention on Bending Moment Components Mx and My:**

As far as the general bending stress equation is concerned,
if a moment component puts the first quadrant of the beam cross section in
compression, it is treated as positive (see the examples shown below). Notice
that this is just a sign
convention for the moment components and should not be confused with the sign associated
with the bending stress.

**Neutral Axis:**

When a homogeneous beam is subjected to elastic bending, the neutral
axis (NA) will pass through the centroid of its cross section, but the
orientation of the NA
depends on the orientation of the moment vector and the cross sectional
shape of the beam.

When the loading is unsymmetrical (at an angle) as seen in the
figure below, the NA will also be at some angle - **NOT**
necessarily the same angle as the bending moment.

Realizing that at any point on the neutral axis, the bending strain and stress
are zero, we can use the general bending stress equation to find its
orientation. Setting the stress to zero and solving for the slope y/x gives

(II.2)
A positive angle is
defined as counter clockwise from the horizontal centroidal axis.

Notice that we can use the equation for orientation of NA to examine special
cases. For example, if the cross section has an axis of symmetry, Ixy = 0. In
addition if only Mx is applied, then NA will have angle of zero which is
consistent with what we would expect from mechanics of materials.

From this equation, we see that the orientation of NA is a function of both
loading condition as well as cross sectional geometry.

EXAMPLE
PROBLEMS

- Example 1* Thin-walled beam with
horizontally symmetric
cross section under a horizontal bending moment
- Example 2* Thin-walled beam with
horizontally symmetric cross
section under an oblique bending moment
- Example 3* Thin-walled beam with
unsymmetric
cross section under an oblique bending moment
- Example 4 Skin-stringer beam with
unsymmetric cross section under horizontal bending moment

* Interactive Examples

To Section
II.3
To Section
II.1

To Index Page of
Pure Bending