Sudden vs. gradual loading on a bar (impact factor concept): For the same final load W on a linear elastic bar, compare the maximum stress produced when the load is applied suddenly (without initial velocity) versus when it is applied gradually from zero to W. What is the ratio (sudden : gradual)?

Introduction / Context:
Load application rate strongly affects peak stresses in elastic members. Under impact or sudden loading, the bar stores more strain energy at the same final load level compared with a slowly applied (quasi-static) load. This concept is often introduced using energy methods in strength of materials.

Given Data / Assumptions:

• Linear elastic, homogeneous bar obeying Hooke’’s law.
• Axial loading only; no damping.
• Sudden load means load W is applied instantaneously from zero to W with no initial kinetic energy.
• Gradual load means load increases slowly from 0 to W (static application).

Concept / Approach:

Use strain energy equivalence. Under gradual loading, the strain energy stored is (1/2) * W * δ, where δ is the static extension under W. Under sudden loading, the bar oscillates and its first peak force equals 2W, producing a peak stress twice the static stress for the same stiffness.

Step-by-Step Solution:

Verification / Alternative check:

From vibration solution of a single-degree-of-freedom system with step load, the displacement overshoots to 2δ_static at first peak, confirming a factor of 2 in stress.

Why Other Options Are Wrong:

• 0.5 or 1 underestimates dynamic amplification.
• 3 or 4 overestimates; requires additional kinetic energy (true impact with velocity), not a mere sudden step.

Common Pitfalls:

• Confusing “sudden” with “impact with velocity”; the factor exceeds 2 only when kinetic energy is added.
• Comparing average stresses instead of peak stresses.

Final Answer:

2.