An analysis of leaf springs with triangular, rectangular, and trapezoidal endings of leaves

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An analysis of elastic multi-leaf springs with triangular, rectangular, and trapezoidal leaf endings is presented, offering valuable insights from both research and educational points of view. Heuristic arguments are first provided explaining why the maximum stress and maximum deflection in a multi-leaf spring of uniform strength are the same as in the triangular cantilever from which the spring is created by dissection, complementing an intuitive deduction of this result commonly used in engineering design textbooks. A rigorous proof based on Euler–Bernoulli beam theory is then presented. It is shown by a matrix analysis that for springs with uniform length decrement and constant leaf thickness, the contact forces between all leaves are equal and that all leaves are of uniform strength. In contrast, for multi-leaf springs with rectangular leaves, the contact forces between different leaves are different, and the leaves are not of uniform strength. New expressions are derived for the spring rate and maximum stresses in springs consisting of graduated-length rectangular leaves, strengthened by additional full-length leaves. The results are compared with the results obtained for multi-leaf springs consisting of graduated-length leaves of uniform strength and added full-length leaves. Two alternative analyzes are then presented, dealing with nipping of springs that results in equal maximum stress across all leaves. The analysis of springs with trapezoidal leaf endings, from which the results for triangular and rectangular leaf endings can be recovered as particular cases, is presented in the Appendix. The presentation throughout the paper is accompanied by a variety of problems intended for student exercise in undergraduate mechanical engineering design courses.