Shear Stress Distribution – Rectangular Section For a prismatic member with a rectangular cross-section under transverse shear, what is the ratio of maximum shear stress to average shear stress over the section?

Introduction / Context:
The shear stress distribution in beams depends on cross-section shape. For rectangles, the distribution is parabolic with a maximum at the neutral axis and zero at the top and bottom fibers. Engineers frequently use the ratio of maximum to average shear to relate nominal design values to peak stresses.

Given Data / Assumptions:

• Cross-section: rectangle, width b, depth h.
• Transverse shear force = V.
• Linear elastic behavior, small deformations, Euler–Bernoulli kinematics.
• Average shear stress defined as V divided by area b * h.

Concept / Approach:

Exact shear stress distribution comes from the elastic formula tau = V * Q / (I * b), where Q is the first moment of area about the neutral axis of the portion above or below the point, I is the second moment of area about the neutral axis, and b is the local width. For a rectangle, this distribution is parabolic with a maximum at mid-depth.

Step-by-Step Solution:

Verification / Alternative check:

Integrating the parabolic distribution over the depth reproduces the total shear force V, confirming that the average equals V / (b * h) and the peak equals 1.5 times this value.

Why Other Options Are Wrong:

1 implies a uniform distribution, which is not valid for rectangles under elastic theory. 1.25 and 2.0 are typical of other contexts but not for rectangular sections. 2.5 is excessively high and contradicts the derived parabolic profile.

Common Pitfalls:

Confusing rectangular results with those for circular or I-sections, using shear formula with incorrect Q, or evaluating b at the wrong location. Another mistake is to average the parabolic expression incorrectly, leading to wrong peak factors.

Final Answer:

1.5