Punching a plate: maximum permissible hole diameter from shear versus punch crushing A plate of thickness t is punched. The plate’s maximum shear stress is one-fourth of the punch’s maximum crushing stress. Find the maximum hole diameter that can be punched without exceeding either limit. Choose the correct expression in terms of t.

Given limitsLet allowable shear of plate = τ and allowable crushing of punch = σ_c, with τ = σ_c/4.

For a punched hole of diameter dShear force capacity (plate): P = τ (π d t)Crushing stress on punch: σ_cr = P / (π d² / 4)

Set crushing at limitσ_c = [τ (π d t)] / (π d² / 4) = τ (4 t / d)Using τ = σ_c/4 ⇒ σ_c = (σ_c/4)(4 t / d) = σ_c (t / d)⇒ d = t

Final Answert