# 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 – user32882 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

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).