A closely coiled helical compression spring of mean coil diameter \(D\) and wire diameter \(d\), is loaded by an axial force \(F\).
The maximum shear stress developed in the wire is
GATE ME - 2026,
Answer: A
The maximum shear stress is \(\frac{8FD}{\pi d^{3}} + \frac{4F}{\pi d^{2}}\).
A section of the wire carries a torque \(T = F\cdot\frac{D}{2}\) and a direct shear force \(F\). Torsion gives a surface shear stress of
\[ \begin{aligned} \tau_{t} &= \frac{16T}{\pi d^{3}} = \frac{16F(D/2)}{\pi d^{3}} \\ &= \frac{8FD}{\pi d^{3}}, \end{aligned} \]
and the direct shear, taken as uniform over the wire area \(\pi d^{2}/4\), is \(4F/\pi d^{2}\). On the inner side of the coil the two add.
\(2F/\pi d^{2}\) halves the direct shear, while \(16FD/\pi d^{3}\) and \(32FD/\pi d^{3}\) use the full diameter \(D\) as the lever arm or a bending formula in place of torsion.
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