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Calculating Bending Stress of a Beam Section

How to Calculate Bending Stress in Beams?

In this tutorial, we will look at how to calculate the bending stress in 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.

1. Calculate Bending Stress by Hand with Bending Stress Formula (Equations)

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

Beam Bending Stress, stress equation, beam stress, bending stress in a beam, bending stress formula, bending stress equation

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:

Calculate Bending Stress in a Beam Section, stress equation, bending moment formula, bending stress equation, beam stress,bending stress formula

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:

Calculate Bending Stress in a Beam Section, stress equation, bending moment formula, bending stress equation, beam stress,bending stress formula

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 = 4.74×108 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:

Neutral Axis of I-beam, stress equation, bending moment formula, beam stress, bending stress equation, bending stress in a beam,bending stress formula

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:

Calculate Bending Stress in a Beam Section, stress equation, bending moment formula, beam stress, bending stress equation, bending stress formula

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):

Calculate Bending Stress in aBeam Section, stress equation, bending moment formula, bending stress equation, beam stress, bending stress formula

The last thing to worry about is whether the beam 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.

Tensile or Compressive Bending Stress, stress equation, bending moment formula, beam stress, bending stress equation, bending stress in a beam, bending stress formula

2. Calculate Bending Stress using Software

Through this article, you have learned the bending stress formula for calculation. Of course, you don’t need to do these calculations by hand because you can use the SkyCiv Beam Calculator to find shear and bending stress in a beam! Simply start by modeling the beam, with supports and applying loads. Once you hit solve, the software will show the max stresses from this bending stress calculator. The image below shows an example of an I-beam experiencing bending stress:

SkyCiv Beam Calculator Software Stress Analysis, beam stress, bending stress in a beam, bending stress equation,bending stress formula

If you want to give it a try first, Free Beam Bending Moment Calculator is a great way to start, or simply sign up for free today!

Users can also use the following Beam Stress Software to calculate the bending stress and other beam stresses, using a simple section building tool:

beam stress how to calculate bending stress in a beam, using bending stress formula and software,bending stress equation

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