Impact vs Gradual Loading — Strain Energy For a linear elastic member, the strain energy stored when a load is applied suddenly (without initial velocity) is how many times the strain energy stored when the same load is applied gradually?

Introduction:
Energy methods compare how loading history influences internal energy and maximum stresses. A classic result contrasts gradual (static) loading with a sudden application of the same load in linear elasticity.

Given Data / Assumptions:

• Linearly elastic member (Hooke's law).
• Load P applied either gradually (from 0 to P) or suddenly (instantaneous but without impact velocity).
• No damping; mass effects only through equilibrium of the final state for the sudden case.

Concept / Approach:
Under gradual loading, maximum deflection is delta, and strain energy U_gradual = (1/2) * P * delta. For sudden loading, the system oscillates and the maximum static-equivalent deformation reaches 2 * delta, giving double the strain energy.

Step-by-Step Solution:
Gradual: U_g = (1/2) * P * deltaSudden: peak force equals P but deflection doubles: delta_max = 2 * deltaU_sudden = (1/2) * P * (2 * delta) = P * delta = 2 * U_g

Verification / Alternative check:
Equivalent spring model with stiffness k: delta = P / k; sudden case yields maximum extension 2 * (P / k).

Why Other Options Are Wrong:
equal to: ignores dynamic amplification.one-half: opposite of the correct amplification.four times: would require additional kinetic impact (drop), not a mere sudden application without velocity.

Common Pitfalls:
Confusing sudden loading (no drop height) with impact loading from a height, which can produce larger factors.

Final Answer:
twice