Download Bending Stresses

The first set of data to
be studied consists of the 5
Web gages, the Top gage, and
the Bottom gage. These 7
gages will be used to examine
at the bending stress
equation from Mechanics of
Materials.
Structural Laboratory
Steel Beam
Bending Stresses
Bending Stress
Theoretical Moment
• Here is the theoretical moment diagram
P
2
V
P
2
P
2
P
2
a
−
a
P
2
Pa
2
−
Pa
2
Pa
2
P
2
M
M=0 at the end (assumed)
M=0 at the end
2
This is the theoretical
moment diagram. It assumes
zero moments at the ends.
Your theoretical moment
diagram will be slightly
different than this one
because the two measured
distances, shown as distance
“a”, may not be exactly the
same for your beam.
Therefore, the reactions may
be different.
Bending Stress
3
Bending Stresses
• Here is the bending stress equation:
• ε is the experimental
My
I
= Eε exp
σ theor =
σ exp
measured strain at the
gages
From the experimental
measured strain, the
experimental bending
stress, σ, can be calculated
We want to compare the
theoretical and
experimental bending
stresses
•
•
Bending Stress
Bending Stress Distribution
y
x
Trendline: slope=mexp
x
Actual NA
σ
x
Theoretical NA
Actual NA
x
x
x
x
Equation of a line:
y=a+mx
•Actual NA is where the strain (or stress) equals zero
•This is where the trendline goes through zero
•Move the NA to this point
4
Here are the equations to
use to calculate the bending
moment through the depth of
the beam.
From each gage we can
calculate the bending stress
using Hooke’s law. The
location of each gage was
measured earlier. Determine
the y distance for each
strain gage, remembering
that y is measured from the
Neutral Axis. Plot the stress
at each gage as a function of
the distance y. Put a
trendline through the data
points. This line shows us
where the Actual Neutral
Axis is located. It is where
the trendline passes through
zero stress. This is what we
will use for the real NA.
Bending Stress
5
Theoretical Bending Stress
y
x
Trendline: slope=mexp
Theoretical line:slope=I/M
Actual NA
My
σ
x
x
x
σ =
x
x
I
I
M
•The slope of the theoretical bending stress equation
is I/M
x
m theor =
y
σ
=
From the theoretical
equation of the bending
stress, we can plot the
theoretical prediction line.
Make sure you pass this line
through the Actual Neutral
Axis that was previously
located.
•We can plot this line
•It passes through the actual Neutral Axis
Bending Stress
6
Equal Slopes
y
x
Trendline: slope=mexp
Theoretical line:slope=I/M
Actual NA
My
σ σ theor =
x
x
x
I
x
x
x
theor. slope =
m exp = m theor =
y
σ theor
I
M
•To have the same slopes to the lines
•Examine the values of I and/or M that would make
the slopes equal
•Are these values reasonable?
=
I
M
The slopes of the
experimental trendline and
the theoretical stress line
will be different. How
different are they? If the
slopes are to be the same,
what values of I or M would
be required? Could these
values reasonably be
expected?