Impact loading on an elastic member – stress under sudden application A load of magnitude W is applied suddenly (without impact velocity) to an axially loaded elastic member. The maximum stress intensity produced is x times the stress that would result if the same load were applied gradually. What is x?

Introduction / Context:
Elastic members respond differently to loads depending on how the load is applied. A suddenly applied load creates transient elastic oscillations before settling, and the initial maximum stress can exceed the static value. This is crucial for safety factors in lifting, hoisting, and structural detailing.

Given Data / Assumptions:

• Axially loaded prismatic member, linear elasticity.
• Load W applied suddenly from zero to full value (no initial kinetic energy).
• No damping during the first response peak.

Concept / Approach:
For a linear spring–mass analogy, a suddenly applied step load produces a displacement twice the static displacement at the first peak. Stress is proportional to strain in elasticity, so the maximum stress is twice the static (gradually applied) stress. This factor is often referred to as the “sudden load factor”.

Step-by-Step Solution:
Static case: σ_static = W / A (for axial load).Sudden step load: peak displacement = 2 * δ_static.Therefore peak stress: σ_peak = 2 * σ_static.Hence x = 2.

Verification / Alternative check:
Energy method: Work done by the load equals increase in strain energy. For sudden load, W * δ_peak = (1/2) * k * δ_peak^2, with δ_peak = 2δ_static.

Why Other Options Are Wrong:

• 1 corresponds to gradual application.
• 3 and 4 require impact with velocity (additional kinetic energy) or different dynamics.
• 1.5 has no basis for the step-load case.

Common Pitfalls:
Confusing sudden application (step) with impact (drop). A falling load can cause factors higher than 2.

Final Answer:
2