How to find the centroid and moment of inertia of a composite section

Key concept: Composite-section problems are really about bookkeeping: split the shape into simple pieces, keep signs consistent for holes, and move every piece to the same reference axes before summing. If one dimension is missing, the setup still matters because the method is what gets graded.

Step by step

Step 1: Break the section into simple shapes

For a composite section, separate the figure into rectangles, triangles, circles, or cutouts.

For each part, write down:

• area $A_i$
• centroid coordinates $(x_i,y_i)$
• local second moments of area $I_{x,i}$ and $I_{y,i}$

Step 2: Find the centroid

Use the area-weighted average:

$\bar{x}=\frac{\sum A_i x_i}{\sum A_i},\qquad \bar{y}=\frac{\sum A_i y_i}{\sum A_i}$

If there is a void, treat it as a negative area.

Step 3: Shift each part to the centroidal axes

Apply the parallel-axis theorem:

$I_x=\sum\left(I_{x,i}+A_i d_{y,i}^2\right),\qquad I_y=\sum\left(I_{y,i}+A_i d_{x,i}^2\right)$

where $d_{x,i}$ and $d_{y,i}$ are the distances from each part’s centroid to the composite centroid.

Step 4: Final check

Make sure your answer is in consistent units, usually mm$^4$ or m$^4$.

If you upload the actual section dimensions, I can compute the centroid and both centroidal moments directly.

Pitfall alert

The most common mistake is mixing up the centroid of each sub-shape with the centroid of the whole section. Another easy miss is forgetting that holes subtract area and subtract inertia. Also, never use the parallel-axis theorem before you know the composite centroid.

Try different conditions

If the reference axes are not centroidal, first compute moments about the given axes and then shift to the centroidal axes. If the shape is symmetric about one axis, that centroid coordinate is immediately known, which cuts the work in half.

Further reading

centroid, parallel-axis theorem, second moment of area