engineering fundamentals Extension Springs
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Spring Geometry
Extension springs typically appear as follows,

Extension springs are typically manufactured with an initial tension Fi which presses the coils together in the free state. This fabrication method allows consistent free lengths to be produced, but since the initial tension is not zero, the spring rate is not truly linear when measured from the resting state. However once the initial tension is overcome, the spring does behave linearly.

Shear Stress
Since extension springs have an initial tension in their resting state, they also have a shear stress in their coils while at rest. The maximum shear stress (at rest) ti occurs on the inner face of the coils, and is given by the equation,

where D is the nominal diameter of the spring, d is the wire diameter, and W is the Wahl Correction Factor.

After the initial tension is overcome, the extension spring can be analyzed as a compression spring with a negative force. The maximum shear stress (tmax) in the spring increases with the load and is given by,

The spring extension D is given by,

where G is the shear modulus and nt is the total number of coils.

End Stress Concentrations
Consider a regular hook we typically see on an extension spring. The geometry of the hook often causes stress concentration which leads to failure. The following illustration shows this geometry and defines the radial parameters r1 to r4,

The maximum bending stress at point A and the maximum shear stress at point B can be expressed as follows,

Extension Spring Factor of Safety
If a compression spring fails, catastrophic failure of the supported assembly is often prevented by the fact that the parts containing the ends of the compression spring will at worst compress the remains of the spring.

With an extension spring, there is no such safety geometry since the spring is in tension. For this and other reasons, extension spring maximum working stresses are typically limited to three-fourths (3/4) of those for compression springs of similar geometry and material.

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