Shear deflection of a cantilever (Timoshenko effect ignored) For a cantilever of length L, cross-sectional area A, and shear modulus G carrying a concentrated load W at its free end, what is the tip deflection contribution due to shear alone (ignoring shape factor corrections)?

Introduction / Context:
Total deflection of a beam comprises bending and shear components. For short, deep members or materials with low shear modulus, the shear component can be significant. Recognizing and estimating the shear deflection helps avoid serviceability issues and informs when Timoshenko (shear-deformable) theory must be used instead of Euler–Bernoulli (bending-dominant) theory.

Given Data / Assumptions:

• Cantilever of length L, uniform area A, shear modulus G.
• End load W applied at the free tip.
• Shear deflection only is requested; bending deflection is not part of this specific answer.
• Uniform shear strain distribution is assumed (no shear correction factor k applied).

Concept / Approach:
From elementary shear deformation theory, the average shear strain gamma is shear stress divided by shear modulus: gamma = tau / G. For a prismatic member with end shear V = W (constant along the length), the average shear stress is tau_avg = V / A. The shear rotation per unit length is then gamma = V / (A G), leading to a tip shear deflection equal to this rotation integrated over length L.

Step-by-Step Solution:
Shear force along the cantilever: V = W (constant).Average shear strain: gamma = V / (A G) = W / (A G).Assuming uniform gamma → shear rotation is constant = W / (A G).Tip deflection due to shear: delta_s = (W / (A G)) * L = W L / (A G).

Verification / Alternative check:
In Timoshenko theory, a shear correction factor k (≈ 1.2 for rectangles) modifies the formula to delta_s = W L / (k A G). With k = 1, we retrieve the elementary expression asked in the problem.

Why Other Options Are Wrong:
W L^3 / (3 E I) and W L^2 / (2 E I) are bending deflection forms, not shear deflection. W / (A G) omits length. 2 W L / (A G) overestimates by a factor of 2 without theoretical basis.

Common Pitfalls:
Mixing bending and shear deflections; forgetting the role of shear modulus G; or applying a shear correction factor unintentionally when the problem asks for the basic expression.

Final Answer:
W L / (A G)