For a circular beam in transverse shear, what is the ratio of maximum shear stress to average shear stress (τ_max / τ_avg)?

Introduction / Context:Transverse shear stress distribution in beams depends on the cross-sectional shape. Designers often need the ratio between the maximum shear stress and the average shear stress to convert between nominal and peak values. This ratio is section-dependent and is used in checking shear capacity and service behavior.

Given Data / Assumptions:

• Prismatic beam with a solid circular cross-section of diameter D.
• Subjected to a transverse shear force V.
• Linear elastic behavior; Saint-Venant assumptions apply.

Concept / Approach:Average shear stress τ_avg = V / A = V / (pi * D^2 / 4). The exact shear distribution for a solid circle yields a parabolic variation with the maximum value at the neutral axis equal to τ_max = (4/3) * τ_avg. This contrasts with a solid rectangle where τ_max = (3/2) * τ_avg.

Step-by-Step Solution:Step 1: Compute τ_avg = V / A.Step 2: Use the known closed-form result for a circular section: τ_max = 4/3 * τ_avg.Step 3: Hence, the ratio τ_max / τ_avg = 4/3.

Verification / Alternative check:Deriving from τ = V * Q / (I * b(y)) with the circular-section expression for first moment Q confirms the 4/3 multiplier at the neutral axis.

Why Other Options Are Wrong:

• 3/2: Applies to a solid rectangular section, not a circle.
• 5/4, 2, 1: Do not match the exact solution for a circular section.

Common Pitfalls:Memorizing the rectangular ratio (3/2) and applying it to all shapes; using τ_avg directly to check peak stresses.

Final Answer:4/3