Normal Stresses for Beams in Bending

For a straight beam that is being acted upon by a positive bending moment in one axis:

• Top is in compression
• Bottom is in tension
• $x$ is neutral axis
  • Not in tension or compression, strain is zero
  • Coincides with centroidal axis
• $xz$ is neutral plane – contains neutral axes for all cross-sections

Bending stress varies linearly with distance from neutral axis – further away from neutral axis means more stress. Graphically:

Mathematically, this is given by:

$${\sigma }_x=-\frac{M_{z}y}{I_z}$$

where $y$ is the distance from the neutral axis. Note that the negative sign makes it so that positive $y$ is compression, and negative $y$ is tension, just like we want.

• If the beam is bending downward, don’t use the negative!

$I$ is the moment of inertia or second moment of area about the $z$-axis:

$$I=\int y^{2}dA$$

Obviously, the maximum magnitude of bending stress then occurs where $y$ has the greatest magnitude, such that:

$${\sigma }_{\text{max}}=\frac{Mc}{I}$$

where $c$ is the maximum magnitude of $y$.

The above is sometimes written as:

$${\sigma }_{\text{max}}=\frac{M}{Z}$$

where $Z=I/c$ is called the section modulus. For rectangular beams, $Z=I/c=\frac{1}{12}b{d}^3/\frac{d}{2}=\frac{1}{6}b{d}^3$.

Bending due to Axial Loading

Normal stress can occur due to axial loading and loading, such that:

$${\sigma }_{\text{total}}=\pm \frac{F}{A}\pm \frac{My}{I}$$

where $M$ is the bending moment.

$F$ is the axial loading, and it is:

• Positive if force is tensile
• Negative if force is compressive