Exact determination of the global tip deflection of both close-coiled and open-coiled cylindrical helical compression springs having arbitrary doubly-symmetric cross-sections

• Compact deflection formulas for cylindrical helical springs are derived for seven solid sections.
• One may use them without the need for any extra information, some charts, and correction factors.
• Frequently used formulas are obtained readily from the present formulas.
• Present formulas are also verified with recent experiments and finite element solutions.

A general formulation of the static deflection under an axial force is required for accurate static, buckling and dynamic analyses. Even today, however, the helical spring formulas derived in 1960s still continue to be used in the spring design. So designers maintain to design helical springs with limited options in making a change in both the helix pitch angles and cross-section types. In this study, in order to carry out such formulation and get closed-form solutions for the vertical tip deflection, Castigliano's first theorem is directly employed to the linear elastic problem of cylindrical helical springs with large pitch angles. Derivation takes into account for the whole effect of the stress resultants such as axial and shearing forces, bending and torsional moments on the deformations. Cylindrical helical springs having doubly symmetric cross-sections such as a solid/hollow circle, a square, a horizontal/vertical rectangle, and a horizontal/vertical ellipse made of isotropic and homogeneous linear elastic materials are all handled in this work. For each shape of cross-section considered in the study, a closed form global formula in a compact form is offered for users with the common notations and common design parameters as currently used. These formulas may be directly used without hesitation for both closed-coiled (CC), α≤10°α≤10°, and open-coiled (OC), α≥10°α≥10°, cylindrical helical springs. That is one may use those formulas without the need for any extra information than he already has and without involving any design chart and correction factor. Some of formulas derived in this study are compared to the commonly used formulas in the available literature. It is verified that those formulas may be obtained readily from the present formulas by considering their certain assumptions. Benefits of this study related to previous ones are also discussed. Present global formulas are also verified with available recent experiments and finite element solutions. The groundwork for the data to be used directly in Castigliano's first theorem was obtained by using differential geometry of a helical structure from a general spatial rod, and governing equations derived by using equilibrium equations, constitutive equations and geometrical compatibility relations in vector forms. As a result, the author expects that a designer is to be free to design more accurate springs by using the global analytical formulas presented in this study.