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Calculating the Centroid of a Beam Section

Centroid Calculation: A Simple Guide

The centroid or center of mass of beam sections is useful for beam analysis when the moment of inertia is required for calculations such as shear/bending stress and deflection. This article guides you through a simple process of how to calculate centroid.

How to Find Centroid

First of all, you need to know how to find centroid. Beam sections are usually made up of one or more shapes. So to find the centroid of an entire beam section area, it first needs to be split into appropriate segments. After this, the area and centroid of each individual segment need to be considered to find the centroid of the entire section.

Alternatively, try our free Centroid Calculator:

How to Calculate Centroid (Centroid Equation):

Consider the I-beam section shown below. To calculate the vertical centroid (in the y-direction) it can be split into 3 segments as illustrated:

How to Calculate Centroid, centroid calculator, how to find centroid

Now we simply need to use the centroid equation for calculating the vertical (y) centroid of a multi-segment shape:

Calculate the Centroid of a Beam Section, centroid calculation formula

We will take the datum or reference line from the bottom fo the beam section. Now let’s find Ai and yi for each segment of the I-beam section shown above so that the vertical or y centroid can be found.

[math]
\text{Segment 1:}\\
\begin{align}
{A}_{1} &= 250\times38 = 9500 {\text{ mm}}^{2}\\
{y}_{1} &= 38 + 300 + \tfrac{38}{2} = 357 \text{ mm}\\\\
\end{align}
[math]

[math]
\text{Segment 2:}\\
\begin{align}
{A}_{2} &= 300\times25 = 7500 {\text{ mm}}^{2}\\
{y}_{2} &= 38 + \tfrac{300}{2} = 188 \text{ mm}\\\\
\end{align}
[math]

[math]
\text{Segment 3:}\\
\begin{align}
{A}_{3} &= 38\times150 = 5700 {\text{ mm}}^{2}\\
{y}_{3} &= \tfrac{38}{2} =19 \text{ mm}\\\\
\end{align}
[math]

Calculate the Centroid of a Beam Section - 2

In the event that the cross-section is composed of two materials or a composite material, then one of the materials will have to be multiplied by the modular ratio such that the equation entire section becomes uniform.

[math]
n=\frac{E_{1}}{E_{2}}
[math]

Typically, E1 is the modulus of elasticity of the non-prevailing material, and E2 is the modulus of elasticity of the prevailing material, although whichever order that is preferred will not affect the solution of the centroid. Adjusting for the second material, the centroid equation becomes the following.

[math]
\bar{y}=\frac{\sum{A}_{i}{y}_{i}+\sum {n}{A}_{i}{y}_{i}}{\sum{A}_{i}+\sum {n}{A}_{i}}
[math]

Of course, manually computing for the moment of inertia is not necessary with the use of our fantastic Centroid Calculator to find the vertical (y) and horizontal (x) centroids of beam sections.

For a more complete list of sections and functionality, you can try out our Section Builder

Visit the next step: How to Calculate the Moment of Inertia of a Beam Section.

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