Rectangular beam under shear: The maximum shear stress q_max in a rectangular cross-section is what multiple of the average shear stress V/(bd)?

Introduction / Context:Designers often compute the average shear stress V/(bd). However, the true distribution is not uniform; it is parabolic for rectangles, making the maximum at the neutral axis larger than the average. Knowing the amplification factor is critical when comparing to permissible or design shear stresses.

Given Data / Assumptions:

• Prismatic rectangular cross-section of breadth b and overall depth d (effective depth used for average shear in RC).
• Elastic theory assumptions apply for shear distribution.

Concept / Approach:For a rectangular section, tau(y) varies parabolically, with tau_max = 1.5 * V / (b * d). This arises directly from tau = VQ/(Ib) and the geometry of the rectangle's first moment area about the neutral axis.

Step-by-Step Solution:Average shear = V / (b * d).Using elastic beam theory, tau_max at the neutral axis = 3/2 * average shear.Therefore q_max = 1.50 times the average.

Verification / Alternative check:The zero shear at the outer fibers and maximum at mid-depth (parabolic law) confirm that average must be lower than peak, with the ratio 1.5 for rectangles.

Why Other Options Are Wrong:1.25, 1.75, 2.0, 2.5: Do not match the exact parabolic result for rectangles.

Common Pitfalls:

• Using average shear directly against code limits intended for nominal/maximum values.
• Confusing the rectangular factor (1.5) with other sections (e.g., I-sections have different factors in webs).

Final Answer:1.50 times the average