# Biaxial Bending Sign convention

I did a couple of quick calcs to better understand the phenomenon of biaxial bending. One thing that is currently tripping me up is the sign convention. For example, if we have a rectangular beam as follows

with $b = 15''$ and $h = 30''$ and a coordinate system established as show with origin at exactly the middle. $I_x = 33750\text{ in}^2$ and $I_y = 8437.5\text{ in}^2$. Furthermore we have a moment $M_x = 100\text{ lb.in}$ occuring about the +x axis and a moment $M_y = 100\text{ lb.in}$ occuring about the +y axis. Intuitively, we can tell $M_x$ is causing compression at the bottom and tension at the top of this section. Likewise we can also say $M_y$ is causing compression at the right and tension at the left of the section. However, when we run the numbers:

\begin{alignat}{2} \sigma_{top} &= \frac{M_xy}{I_x} = \frac{(100\text{ lb.in})*(15\text{ in})}{33750\text{ in}^4} &&= 0.044\text{ psi} \\ \sigma_{bot} &= \frac{M_xy}{I_x} = \frac{(100\text{ lb.in})*(-15\text{ in})}{33750\text{ in}^4} &&= -0.044\text{ psi} \\ \sigma_{right} &= \frac{M_yx}{I_y} = \frac{(100\text{ lb.in})*(7.5\text{ in})}{8437.5\text{ in}^4} &&= 0.088\text{ psi} \\ \sigma_{left} &= \frac{M_yx}{I_y} = \frac{(100\text{ lb.in})*(-7.5\text{ in})}{8437.5\text{ in}^4} &&= -0.088\text{ psi} \\ \end{alignat}

It is highly confusing to me that $\sigma_{bot}$ and $\sigma_{right}$, which are both in compression, have opposing signs. Equally confusing is the fact that $\sigma_{top}$ and $\sigma_{left}$, which are both in tension, have opposing signs. Can anyone bring light to this basic mechanics of materials issue?

• Your "numbers" assume a sign convention. Where do these numbers (or the formulas beyond them) come from?
– Pere
Aug 30 '16 at 20:07
• the numbers are given from the problem. The formulas are your regular Euler-Bernoulli bending stress equation from mechanics of materials. See under "Euler-Bernoulli" bending theory in en.wikipedia.org/wiki/Bending Aug 30 '16 at 20:15
• As far as I can see, the source you pointed just shows ${\sigma}= \frac{M y}{I_x}$. For the other equation, where does it come from that it is ${\sigma}= \frac{M x}{I_y}$ instead of ${\sigma}= -\frac{M x}{I_y}$, when you use the sign convention for X and Y you show in your drawing?
– Pere
Aug 30 '16 at 21:55

## 1 Answer

This confusion seems to come from the difference between the sign convention for moments and the arbitrarily chosen coordinate system of the beam.

For the most common sign convention for moments, a positive bending moment causes the beam to bend in a U shape (compression at the top, tension at the bottom):

So the way you have drawn your coordinate system, the bending moment about the x-axis causes tension on the top and compression on the bottom (from the positive bending moment convention this would be a negative moment). The bending moment about the y-axis causes tension on the left, and compression on the right. As you have noticed, this causes tension to be positive for bending about the x axis and negative for bending about the y-axis.

So you have mixed two different bending moment sign conventions.

Since the coordinate system of the beam can be selected arbitrarily, one way to avoid this confusion is to use a coordinate system which matches with the positive bending moment sign convention:

Alternatively, you could take the absolute value of $\frac{My}{I}$ then add the sign afterwards in order to ensure that tensile stresses are positive and compressive stresses are negative (or vice versa as long as you are consistent).