Bending stress distribution in beams For pure bending of a prismatic member, the direct longitudinal stress σ at a point a distance r from the neutral axis is proportional to which power of r?
-
Ar
-
Br^0 (independent of r)
-
Cr^2
-
Dr^(-1)
-
Er^3
Answer
Correct Answer: r
Explanation
Introduction / Context:Understanding how bending stress varies across the depth of a beam is fundamental for sizing sections and placing reinforcement.
Given Data / Assumptions:
- Prismatic beam under pure bending moment M.
- Linear elastic (Hookean) material; plane sections remain plane.
- Neutral axis passes through the centroid.
Concept / Approach:The flexure formula states σ = M * y / I, where y is the perpendicular distance from the neutral axis and I is the second moment of area. Hence, σ varies linearly with distance from the neutral axis.
Step-by-Step Solution:
Start from σ = M * y / I.Replace y by r (given notation): σ ∝ r.Therefore, proportional to the first power of r.Verification / Alternative check:At r = 0 (neutral axis), σ = 0; at the extreme fibre (maximum r), σ is maximum—consistent with linear variation.
Why Other Options Are Wrong:
- r^2 or r^3 would imply nonlinear distributions inconsistent with beam theory.
- Independence of r or inverse dependence contradicts observed bending stress profiles.
Common Pitfalls:Confusing shear stress distribution (parabolic in rectangles) with bending normal stress (linear).
Final Answer:r