In mechanical design of suspension systems, a semi-elliptic leaf spring (laminated spring) is ordinarily arranged as a simply supported member. Identify how it is supported and where the service load is applied in standard automotive and machine applications.

Introduction:
Leaf springs (semi-elliptic/laminated) are widely used in vehicle suspensions to absorb shocks and support loads. Understanding the support and loading configuration is fundamental to predicting bending moments, stresses, and deflections for safe design.

Given Data / Assumptions:

• A semi-elliptic leaf spring consisting of stacked leaves.
• Standard road vehicle arrangement.
• Small deflection theory and linear elastic behavior for analysis.
• Load comes from the axle or shackle assembly.

Concept / Approach:
The spring acts like a simply supported beam. The two eyes at the ends are connected to the chassis via hangers/shackles (supports), while the axle load is transferred near the mid-span through a U-bolt or seat. This support–load configuration dictates a triangular bending-moment diagram with maximum moment at mid-span.

Step-by-Step Solution:
1) Identify supports: the two eyes form the supports at the ends.2) Identify load application: axle clamp/U-bolt applies load near the geometric centre.3) Structural model: simply supported beam with a central point load.4) Result: maximum bending at the mid-span, consistent with standard leaf spring stress formulae.

Verification / Alternative check:
Vehicle manuals and standard machine design texts model the semi-elliptic leaf spring as simply supported at the eyes with the axle load at the centre; derived formulas for stress and deflection assume this configuration.

Why Other Options Are Wrong:
Centre and loaded at the ends: reverses the actual arrangement.
Ends and loaded anywhere: design assumes a defined mid-span load for predictable stress.
Centre and loaded anywhere: physically incorrect for standard mounts.
Cantilevered at one end: describes quarter-elliptic springs, not semi-elliptic.

Common Pitfalls:
Confusing semi-elliptic with quarter-elliptic/cantilever arrangements; assuming distributed loads instead of the central axle load in first-order models.

Final Answer:
ends and loaded at the centre