**How to Calculate Bending Stress in Beams?**

Understanding bending stress is important because beam bending plays a crucial role in beam design. This tutorial will look at how to calculate bending stress in a beam with a formula. This formula 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).

**1. Calculate Bending Stress by Hand with Bending Stress Formulas (Equations)**

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

At a specific point along the beam’s length (the x-axis), there exists an internal bending moment (M), normally determined using a bending moment diagram. The general formula for bending or normal stress on the section is:

When considering a specific section of a beam, it becomes clear that the bending stress will reach its maximum value at a specific distance from the neutral axis (y). Thus, the maximum bending stress will occur either at the top or bottom of the beam section, depending on which distance is greater:

Let’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 = 4.74×10^{8} mm^{4}. 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:

It’s usually necessary to determine the maximum bending stress experienced by a section. For instance, let’s assume we have determined, from the bending moment diagram, that the beam encounters a maximum bending moment of 50 kN-m or 50,000 Nm (after converting the bending moment units).

Then we need to find whether the top or bottom of the section is farther away from the neutral axis. Clearly, the bottom of the section has a greater distance, measuring c = 216.29 mm. With this information, we can proceed to calculate the maximum stress by employing the bending stress equation provided 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 final consideration involves determining whether the beam stress is causing compression or tension of the section’s fibers.

- If the beam sagging like a “U” shape, the top fibers experience compression (negative stress), while the bottom fibers undergo tension (positive stress).
- If the beam sags in an upside-down “U” shape, the situation is reversed: the bottom fibers are subjected to compression, while the top fibers experience tension.

**2. Calculate Bending Stress using Software**

Through this article, you have learned the bending stress formula for calculation. However, hand calculation isn’t necessary as you can use the SkyCiv Beam Calculator to find shear and bending stress in a beam. By simply modeling the beam, incorporating supports, and applying loads, you can get the max stresses using this bending stress calculator. The image below shows an example of an I-beam experiencing bending stress:

Users can also use the following Beam Stress Software to calculate the bending stress and other beam stresses, using a simple section-building tool. So check out our beam tool above or sign up to experience the software for free today!