Shear in a prismatic square beam under longitudinal loading: Where is the shear stress maximum and along which plane does it act most critically?
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AOn the middle fibre along the horizontal (neutral axis) plane
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BOn the lower extreme fibre along a horizontal plane
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COn the top extreme fibre along a vertical plane
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DEqually on every fibre along a horizontal plane
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EOnly at the corners along a diagonal plane
Answer
Correct Answer: On the middle fibre along the horizontal (neutral axis) plane
Explanation
Introduction / Context:In beams with rectangular or square cross-sections, transverse shear stress varies parabolically over the depth. Understanding where it peaks is important for web sizing and checking shear capacity.
Given Data / Assumptions:
- Square (rectangular) prismatic beam.
- Transverse shear due to bending action.
- Elastic behavior; Saint-Venant shear distribution applies.
Concept / Approach:For a rectangular section, τ(y) = (3/2) * V / (b * h) * (1 − (2y/h)^2), which is a parabola with maximum at the neutral axis (y = 0) and zero at the top and bottom fibres.
Step-by-Step Solution:Recognize the parabolic distribution across depth.At the neutral axis (middle fibre), τ_max = 1.5 * V / (b * h).At extreme fibres, τ = 0.Therefore, the critical shear occurs along the horizontal plane through the neutral axis.
Verification / Alternative check:Shear flow q = V * Q / I b is maximum where first moment Q is maximum, i.e., at the neutral axis.
Why Other Options Are Wrong:Lower/top fibres: shear is zero there.Equal at every fibre: false; distribution is parabolic.Corners/diagonals: not where τ is maximum in rectangular sections.
Common Pitfalls:Confusing shear with bending stress (which is maximum at extreme fibres).
Final Answer:On the middle fibre along the horizontal (neutral axis) plane.