Laminated (Leaf) Springs — Maximum Bending Stress For a semi-elliptic leaf spring carrying a central load W with span l and n plates (each of width b and thickness t), what is the expression for the maximum bending stress in the plates?

Introduction:
Laminated springs distribute bending stresses across several thin plates. Designers need formulas for maximum bending stress and tip deflection to proportion b and t for a target load W and allowable stress, balancing weight and ride comfort in vehicles.

Given Data / Assumptions:

• Semi-elliptic leaf spring of span l with n identical plates.
• Each plate has width b and thickness t; plates are in contact and share load.
• Central load W; linear elastic behavior; friction between leaves neglected in stress formula.

Concept / Approach:
The bending stress in each plate is derived from beam theory using the composite section. The standard result for a semi-elliptic spring under mid-load is sigma_max = (3 * W * l) / (2 * n * b * t^2). Deflection is often y = (3 * W * l^3) / (8 * E * n * b * t^3), showing the strong influence of thickness t.

Step-by-Step Solution:
Model each plate as a beam; total bending moment is shared by n plates.Use flexure: sigma = M * y / I; for a rectangular plate I_plate = b * t^3 / 12 and y = t / 2.Combine for n plates under central load W on span l to obtain sigma_max.Final result: sigma_max = (3 * W * l) / (2 * n * b * t^2).

Verification / Alternative check:
Dimensional check: numerator forcelength; denominator arealength^2 yields stress units as required.

Why Other Options Are Wrong:
(6 * W * l) / (n * b * t^2): incorrect factor; overestimates stress by 4 times.(W * l) / (n * b * t): wrong thickness power; stress scales with 1 / t^2.(3 * W * l) / (8 * n * b * t^3): mixes deflection formula into stress expression with wrong t power.

Common Pitfalls:
Confusing stress and deflection formulae; note t^2 vs t^3 dependence.Ignoring number of plates n when sizing leaf thickness.

Final Answer:
sigma_max = (3 * W * l) / (2 * n * b * t^2)