Bending Stress in Beam Sections
How to calculate a Bending Stress of a Beam Section
May 24, 2022
MJ
Longinos
Julianna
Widlund, P.E
The bending moment, M, along the length of the beam can be determined from the moment diagram. The bending moment at any location along the beam can then be used to calculate the bending stress over the beam's cross section at that location.
Before you’ll go through this article to learn how to calculate the bending stress in beam, we want to remember that Civils.ai has a free tool where you can easily calculate it. Visit the page later for discovering more!
In pure bending the neutral axis is the one where the stress and strain are zero. But, the stress is not uniform along the structure’s cross section.
In the following case we show how a moment about the z-axis bends a prismatic cross section.
Is clear from the example that the bending moment compresses the top of the beam and elongates the bottom. This is considered a “positive moment” or “smiling beam” - versus the opposite effect of “negative moment” or “frown beam”. At the same time, in the center of the beam, the neutral axis length isn’t changed.
We can also observe that the displacement of the beam varies linearly from the top to the bottom – passing through zero at the neutral axis, from the center to the center. The following image explains it more explicitly:
The question now is: how can we calculate the relation between the applied moment and the stress within the beam?
The formula for calculating bending or normal stress on a beam section is the following:
Where:
M = internal bending moment about the section’s neutral axis
y = perpendicular distance from the neutral axis to a point on the section
I = moment of inertia of the section area about the neutral axis
As we described before, the stress will be maximized by the distance from the neutral axis (y). So, the equation to calculate the maximum bending stress (negative or positive) can be expressed like:
Where:
c = perpendicular distance from the neutral axis to the farthest point on the section
We can summarize by saying that the stress is equal to a couple/moment (M) times the location along the cross section, because the stress isn't uniform along the cross section (Cartesian coordinates), all divided by the second moment of area of the cross section.
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