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Beam Rotation from a Linear Curvature

Easymoment-areacurvature
A prismatic beam segment AB is 4.0 m long, with x measured from A toward B. Positive curvature is defined as dθ/dx. The curvature κ = M/EI varies linearly from 0 at A to +0.003 m⁻¹ at B. Small-deflection theory applies and the rotation at A is not given. Find the change in rotation θB − θA, in radians.
  1. A+0.01200 rad
  2. B+0.00800 rad
  3. C+0.00600 rad
  4. D+0.00300 rad
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Answer: C. +0.00600 rad

First moment-area theorem: θB − θA = ∫κ dx = area of the curvature diagram. The diagram is a triangle, base 4.0 m, height 0.003 m⁻¹, so area = ½(4.0)(0.003) = 0.006 rad. θA is not needed because only the difference is asked. Wrong: 0.012 uses base × height and forgets the ½ of the triangle; 0.008 is the tangent deviation (a displacement-type quantity, 0.006 × 4/3 in m), confusing slope with displacement; 0.003 just reads the end curvature as the rotation.
rad
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