Area Moment of Inertia – Rectangle about a Centroidal Axis Parallel to depth d For a rectangular section of width b and depth d, what is the second moment of area (area moment of inertia) about an axis that passes through the centroid and is parallel to the depth d (i.e., the vertical centroidal axis)?
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AI = (b^3 * d) / 12
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BI = (b * d^3) / 12
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CI = (b * d^3) / 3
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DI = (b^3 * d) / 3
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EI = (b * d) / 12
Answer
Correct Answer: I = (b^3 * d) / 12
Explanation
Introduction / Context: In strength of materials and structural analysis, standard formulas for second moments of area (also called area moments of inertia) are used to evaluate bending stiffness about different centroidal axes. For a rectangle, the values differ depending on whether the reference axis is parallel to the width or the depth.
Given Data / Assumptions:
- Plane figure: rectangle with width b (horizontal) and depth d (vertical).
- Axis: centroidal axis parallel to the depth d (vertical centroidal axis through the center).
- Homogeneous, thin area; standard definitions apply.
Concept / Approach: The two principal centroidal area moments for a rectangle are: Ix (about the centroidal horizontal axis, parallel to width) Ix = (b * d^3) / 12 and Iy (about the centroidal vertical axis, parallel to depth) Iy = (d * b^3) / 12. The requested axis is parallel to d, so we need Iy.
Step-by-Step Solution: Identify axis: centroidal vertical axis ⇒ use Iy. Write formula: Iy = (b^3 * d) / 12. Check dimensions: length^4 (b^3 * d) ⇒ correct for area moments.
Verification / Alternative check: By symmetry, swapping b and d swaps the roles of Ix and Iy. If the axis were parallel to width, the answer would be (b * d^3) / 12 instead.
Why Other Options Are Wrong: (b) corresponds to Ix, not the asked axis. (c) and (d) have the correct variables but wrong denominator (3 instead of 12). (e) has wrong power (dimensionally incorrect).
Common Pitfalls: Confusing which axis (x or y) is parallel to width or depth.
Final Answer: I = (b^3 * d) / 12