# Where does the maximum deflection occur in the beam

## Calculation of the bending stress of a beam section

**Calculation of the bending stress in beams?**

In this tutorial, we will examine 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 cross section of the beam. We assume that this is the material of the carrier **linear-elastic** (i.e. Hooke's law applies). Bending stress is important, and since the bending of the beam is often the determining outcome in beam construction, it is important to understand.

**1. Calculation of the bending stress by hand**

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

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

For a given beam section, it is evident that the bending stress is maximized by the distance from the neutral axis (and). So, the maximum bending stress occurs either at the top or at the bottom of the beam section, whichever is greater:

Let's consider the example of our I-Ray shown above. In our previous tutorial on the moment of inertia, we already found that the moment of inertia about the neutral axis is I = 4.74 × 10^{8} mm^{4}. Additionally, in the Center of Gravity tutorial, we found the center of gravity, and therefore the location of the neutral axis, 216.29 mm from the bottom of the section. This is shown below:

Obviously, it is very common to ask for the maximum bending stress that the section will experience. For example, suppose we know from our bending moment diagram that the beam experiences a maximum bending moment of 50 kN-m or 50,000 Nm (convert bending moment units).

Then we need to find out if the top or bottom of the section is furthest from the neutral axis. Clearly, the lower part 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 formula above:

Similarly, we could find the bending stress in the section above, as we know it is y = 159.71 mm from the neutral axis (N / A):

The last thing to worry about is whether the tension is causing compression or tension on the fibers of the section. When the beam sags like a "U." then the upper fibers are compressed (negative stress) while the lower fibers are under tension (positive stress). When the beam sags down like a head "U." then it is the other way round: the lower fibers are under pressure and the upper fibers are under tension.

**2. Calculation of the bending stress with SkyCiv Beam**

Of course, you don't have to do these calculations by hand as you can use the SkyCiv Beam - Bending Stress Calculator to find shear and bending stress in a beam! Just start by modeling the beam, using supports and applying loads. Once you click solve, the software will display the maximum stresses from this bending stress calculator. The picture below shows an example of an I-beam subjected to bending stress:

**Start by calculating the bending stress with the SkyCiv Beam Calculator:**

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