## How to Calculate Bending Stress in Beams?

* Calculating Bending Stress:* In this tutorial, we will look at how to calculate the bending stress of a beam using a bending stress formula that relates the longitudinal stress distribution in a beam to the internal bending moment acting on the beam’s cross-section. We assume that the beam’s material is linear-elastic (i.e. Hooke’s Law is applicable). Bending stress is important and since beam bending is often the governing result in beam design, it’s important to understand.

- Calculating Bending Stress by Hand (using a formula)

Let’s look at an example. Consider the I-beam shown below:

At some distance along the beam’s length (the x-axis), it is experiencing an internal bending moment (M) which you would normally find using a bending moment diagram. The general formula for bending or normal stress on the section is given by:

Given a particular beam section, it is obvious to see that the bending stress will be maximized by the distance from the neutral axis (y). Thus, the maximum bending stress will occur either at the TOP or the BOTTOM of the beam section depending on which distance is larger:

Lets’s consider the real example of our I-beam shown above. In our previous moment of inertia tutorial, we already found the moment of inertia about the neutral axis to be I = 4.74×108 mm4. Additionally, in the centroid tutorial, we found the centroid and hence the location of the neutral axis to be 216.29 mm from the bottom of the section. This is shown below:

Obviously, it is very common to require the MAXIMUM bending stress that the section experiences. For example, say we know from our bending moment diagram that the beam experiences a maximum bending moment of 50 kN-m or 50,000 Nm (converting bending moment units).

Then we need to find whether the top or the bottom of the section is furthest from the neutral axis. Clearly, the bottom of the section is further away with a distance of c = 216.29 mm. We now have enough information to find the maximum stress using the bending stress equation above:

Similarly, we could find the bending stress at the top of the section, as we know that it is y = 159.71 mm from the neutral axis (NA):

The last thing to worry about is whether the stress is causing compression or tension of the section’s fibers. If the beam is sagging like a “U” then the top fibers are in compression (negative stress) while the bottom fibers are in tension (positive stress). If the beam is sagging like an upside-down “U” then it is the other way around: the bottom fibers are in compression and the top fibers are in tension.