48610 Chap.11 Centroids and Second Moment of Area
Centroids and Second Moment of Area
Weeks 6 to 8 asked whether a structure holds still. Week 9 asks how big the members have to be, and both answers start from the centroid of the cross section, the point at which the area balances. It is where the weight of a uniform body acts in every free body diagram, and it is where the bending axis passes, so an error here propagates silently into every stress computed afterwards.
The subject prints the procedure as a boxed sequence: draw the shape relative to a chosen origin, decompose it into simple shapes, look up each shape's own centroid and area, work out its offset from the origin, and combine with the area weighted average.
The second moment of area, which the guide notes is the same thing as the area moment of inertia, follows the same four steps with one addition: each part's tabulated value must be moved to the axis you actually want, using the parallel axis theorem, which adds the area multiplied by the square of the distance between axes.
The transfer term usually dominates, and forgetting it understates a section by a factor of two or three. Two traps recur. Reference tables state values about a particular axis, which may not be the one you measured from, and the guide warns explicitly to check which axis each list uses. And the centroid must be computed before the second moment, because the value bending needs is the one about the centroidal axis.
What this chapter covers
- 01
Why the centroid matters: weight acts there and bending happens about it
- 02
The boxed centroid procedure and the parts table
- 03
The area weighted average for both coordinates
- 04
Symmetry as the cheapest simplification
- 05
Additive and subtractive routes, and why they must agree
- 06
Negative areas for removed material, in both sums
- 07
Reference tables and the corner they use as origin
- 08
The second moment of area and its synonym
- 09
The parallel axis theorem and the transfer term
- 10
Rectangle about its centroid against about its base
- 11
Reading the split between own value and transfer term
Centroid and second moment of a T section
- +1Tabulate the two parts. The web has an area of 720 square millimetres with its own centroid 30.0 mm up; the flange has an area of 960 square millimetres with its centroid at 66.0 mm.
- +1Form the first moments and divide by the total area. The products are 21 600 and 63 360, summing to 84 960, and the total area is 1680, so the centroid sits 50.57 mm above the base. Check it against the geometry: mid height is 36 mm, and the centroid is well above that because the wide flange sits high.
- +1Work out each part's own second moment. The web gives 12 times 60 cubed over 12, which is 216 000; the flange gives 80 times 12 cubed over 12, which is 11 520. The flange is wide but shallow, and because depth is cubed its own contribution is small.
- +1Compute the transfer terms. The web centroid is 20.57 mm from the section centroid, giving 720 times 20.57 squared, that is 304 692. The flange centroid is 15.43 mm away, giving 960 times 15.43 squared, that is 228 519.
- +1Add all four numbers: 216 000 plus 304 692 plus 11 520 plus 228 519 gives 760 731, about 7.61 times ten to the fifth. The transfer terms supply about seventy per cent of the total, which is normal for a T section.
Key terms
- Centroid
- The point at which an area balances, found as the area weighted average of the centroids of its parts. A uniform body's weight acts there, and a beam bends about the axis through it.
- Geometric decomposition
- Splitting an awkward shape into rectangles, triangles and circles whose own centroids and areas come from a reference table, so that no integration is needed.
- First moment of area
- The product of an area and the distance of its centroid from a reference axis. Summing these and dividing by the total area gives the centroid of a composite shape.
- Second moment of area
- A section property measuring how far material lies from an axis, also called the area moment of inertia, with units of millimetres to the fourth power. It governs bending stiffness.
- Transfer term
- The area multiplied by the square of the distance from a part's own centroid to the axis of interest, added to that part's own second moment by the parallel axis theorem.
- Subtractive approach
- Handling a hole or notch by taking a larger simple shape and entering the removed piece with a negative area in both the area sum and the moment sum.
- Centroidal axis
- An axis passing through the centroid of a section. It is the axis about which bending stress is calculated, which is why the centroid must be found first.
Centroids and Second Moment of Area FAQ
Why does the transfer term matter so much?
Because it depends on the square of the distance, while a part's own second moment depends only on its shape. A flange 12 mm deep has a tiny second moment about its own centre, yet moving it 15 mm away from the neutral axis multiplies its contribution by about twenty.
In the worked T section the transfer terms supply about seventy per cent of the total, so omitting them would understate the section by a factor of roughly three.
How do I know which value a reference table is giving me?
Read the diagram beside it rather than the formula. The same rectangle has a second moment of b h cubed over twelve about its own centroidal axis and b h cubed over three about its base, differing by exactly the transfer term. For a right angled triangle, tables commonly give the centroid relative to the corner containing the right angle, which may not be the corner you measured from.
The guide warns about this explicitly, and it is the commonest source of an answer that is wrong by a fixed offset.
What is the difference between the two moments of inertia?
The second moment of area is about a cross section, has units of millimetres to the fourth power and governs bending stiffness. The mass moment of inertia is about a solid body, has units of kilogram metres squared and governs angular acceleration. Both obey parallel axis theorems of the same form, and tables sometimes call the first an area moment of inertia, which is where the confusion begins.
The units decide which one a given equation needs.
Should I use the additive or the subtractive route?
Whichever has fewer chances to go wrong for that shape. A section with a small circular hole is faster subtractively, because a circle has a tabulated area and centroid and cutting it out is one negative row. A stepped profile with several ledges is usually faster additively, since the subtractive version needs a bounding rectangle plus several negative pieces.
When it is genuinely borderline, do both: the subject sets the same U profile both ways and notes the answers must agree, which makes the second route a free check rather than repetition.
Assessment move
Set the work out as a table with one row per part and one column each for area, distance and their product, because that is the format the subject's own solutions use and it is the only layout in which a marker can see which term went wrong. Practise on sections you can find rather than only on the set exercises: a length of aluminium angle, a piece of channel, a flat bar with a hole.
For each, compute the centroid, then the second moment about the centroidal axis, then check the split between own values and transfer terms and see whether it matches your intuition about where the material is.
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