Tangential Deviation by Moment-Area
Mediummoment-areadeflection
A 4.0 m beam segment AB has curvature κ(x) increasing linearly from 0 at A to +0.003 m⁻¹ at B, with x measured from A and y″ = κ (small deflections). Find the deviation tB/A = yB − yA − θA·L of point B from the tangent drawn at A, in millimeters.
- A+8.00 mm
- B+16.0 mm
- C+12.0 mm
- D+6.00 mm
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Answer: A. +8.00 mm
Second moment-area theorem: tB/A = (curvature area) × (centroid distance from B). Area = ½(4)(0.003) = 0.006 rad. The triangle is tallest at B, so its centroid is L/3 = 4/3 m from B. tB/A = 0.006(4/3) = 0.008 m = 8.00 mm. Wrong: 16 mm uses 2L/3 from B (centroid measured from the wrong end); 12 mm uses L/2 as if the diagram were a rectangle; 6 mm just reads the area (0.006 rad) as millimeters, skipping the moment arm.
m mm
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