Consider the close-coiled spring shown in Fig. P.14.24. Show...

Consider the close-coiled spring shown in Fig. P.14.24. Show that if the shear force in the coil is sesumed to give a uniform stress, the maximum shear stress in the coil is given as $$ \tau_{\operatorname{man}}=\frac{P}{A}+\frac{P R r}{J} $$ where $A$ is the cross-sectional area of the wire making up the spring and $J$ is its polar moment of aren. Next, show that for $n$ complete coils of the spring, the elongation $\delta$ of the spring resulting from twisting of the wire is given as $$ \delta=\frac{4 P R^1 n}{G r^4} $$ What other deformation contributions give rise to further deflection?

Consider the close-coiled spring shown in Fig. P.14.24. Show that if the shear force in the coil is sesumed to give a uniform stress, the maximum shear stress in the coil is given as
$$
\tau_{\operatorname{man}}=\frac{P}{A}+\frac{P R r}{J}
$$
where $A$ is the cross-sectional area of the wire making up the spring and $J$ is its polar moment of aren. Next, show that for $n$ complete coils of the spring, the elongation $\delta$ of the spring resulting from twisting of the wire is given as
$$
\delta=\frac{4 P R^1 n}{G r^4}
$$

What other deformation contributions give rise to further deflection?

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Key Concepts

Torsional Deformation

This concept deals with the twisting of the wire under an applied load. In a close?coiled spring the load produces a torque in the wire that causes it to twist. The resulting deformation can be calculated using the polar moment of inertia and the shear modulus, and it is responsible for one component of the overall spring deflection. The formula for the elongation ? derived in the question is based on this twisting effect.

Bending Deformation

Because the wire in a close?coiled spring is curved, the application of load also results in bending of the coil. Each curved segment experiences a bending moment due to the load, and the curvature of the coil means that the wire undergoes an additional bending deformation which is not captured in a pure torsion analysis. This bending contribution leads to extra deflection of the spring beyond that calculated from torsional twisting.

Axial (Extensional) Deformation

In addition to twisting and bending, the wire may also stretch along its length under tension. Although in many spring designs this axial stretching is relatively small compared to the twisting and bending effects, it can still contribute to the overall deflection. The direct tensile strain in the material, due to the load, results in an axial elongation which adds further to the spring’s total deflection.

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