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The theorem shifts a second moment of area from a centroidal axis to a parallel axis.
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Formula
I and Ic in m4; A in m2; d in m
| Symbol | Meaning | Unit |
|---|---|---|
| Area moment about the shifted axis | m4 | |
| Area moment about the parallel centroidal axis | m4 | |
| Area of the section | m2 | |
| Distance between parallel axes | m |
I=I_c+A d^2It follows by expanding I=∫y2dA about the centroid: the first-moment cross term vanishes. The axes must be parallel and have the same bending orientation. This is not the mass parallel-axis theorem, which uses mass rather than area. For a composite section, apply the formula to each part.
It follows by expanding I=∫y2dA about the centroid: the first-moment cross term vanishes. The axes must be parallel and have the same bending orientation. This is not the mass parallel-axis theorem, which uses mass rather than area. For a composite section, apply the formula to each part.
The theorem shifts a second moment of area from a centroidal axis to a parallel axis.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Parallel-axis theorem for area moments” and check its unit and magnitude.
For A=0.02 m2, Ic=1×10−5 m4 and d=0.10 m, I=Ic+Ad2=2.1×10−4 m4.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: I and Ic in m4; A in m2; d in m.
Formula cluster
The theorem shifts a second moment of area from a centroidal axis to a parallel axis.
For A=0.02 m2, Ic=1×10−5 m4 and d=0.10 m, I=Ic+Ad2=2.1×10−4 m4.
Calculate section moments for built-up beams.
Here is how to work through a typical Parallel-axis theorem for area moments (I = Ic + Ad²) task step by step:
Task
Work through the numerical example for “Parallel-axis theorem for area moments” and check its unit and magnitude.
Solution path
For A=0.02 m2, Ic=1×10−5 m4 and d=0.10 m, I=Ic+Ad2=2.1×10−4 m4.
Shifting the integration coordinate expands (y+d)2. The term containing y vanishes because the first area moment about the centroidal axis is zero. The remaining terms are centroidal area moment Ic and the addition Ad2. This also explains why a farther axis has a larger I. Use the area version only for parallel axes; d is their perpendicular separation. Ic must be about the centroidal axis with the same orientation. The theorem is useful for composite beam sections, but do not confuse it with the mass-moment theorem.
The target axis must be parallel to the centroidal axis, and both must use the same area orientation. Switching from horizontal to vertical is not a parallel shift. Measure d perpendicularly between axes. For skew axes or product moments, use an appropriate transformation. Use the area version only for parallel axes; d is their perpendicular separation. Ic must be about the centroidal axis with the same orientation. The theorem is useful for composite beam sections, but do not confuse it with the mass-moment theorem.
The area theorem adds A d2; the mass-moment theorem adds m d2. The former describes section geometry in m4 and is used for bending rigidity; the latter describes mass distribution in kg·m2 and appears in rotational dynamics. Similar form, different quantities and applications. Use the area version only for parallel axes; d is their perpendicular separation. Ic must be about the centroidal axis with the same orientation. The theorem is useful for composite beam sections, but do not confuse it with the mass-moment theorem.
Write the quantity and unit from the variable list next to every value: I and Ic in m4; A in m2; d in m. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For A=0.02 m2, Ic=1×10−5 m4 and d=0.10 m, I=Ic+Ad2=2.1×10−4 m4. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Use the area version only for parallel axes; d is their perpendicular separation. Ic must be about the centroidal axis with the same orientation. The theorem is useful for composite beam sections, but do not confuse it with the mass-moment theorem.
The background states this starting condition: It follows by expanding I=∫y2dA about the centroid: the first-moment cross term vanishes. The axes must be parallel. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Use the area version only for parallel axes; d is their perpendicular separation. Ic must be about the centroidal axis with the same orientation. The theorem is useful for composite beam sections, but do not confuse it with the mass-moment theorem.
Quanta exam set
Curated exam set for "Parallel-axis theorem for area moments":
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Which formula describes Parallel-axis theorem for area moments?
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How do you rearrange Parallel-axis theorem for area moments for Base equation for the target quantity?
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Which common mistake happens with Parallel-axis theorem for area moments?
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