Strain Energy and Impact Loading

When a body deforms elastically, external work is stored as strain energy $U$. Release the load within the elastic range and that energy returns (ideal spring). Impact and suddenly applied loads are analysed by equating energy, not by static $\frac{P}{A}$ alone (Bansal Ch. 4).

What we cover
- Gradual, sudden, and impact axial cases
- Resilience definitions
- Shear and torsion energy
- Problem strategy

Load rises slowly from $0$ to $P$; average force during loading is $\frac{P}{2}$:

Strain energy density (resilience) . Modulus of resilience uses $\sigma=\sigma_y$ (or elastic limit).

**Suddenly applied load $P$** (load appears at once, bar initially unstressed, remains elastic):
energy ⇒ and

Impact / falling weight of weight $P$ from height $h$ onto a collar:

If $h\gg\delta$, approximate . Instantaneous stress (axial bar).

Assumptions are idealised (no energy loss, mass of bar neglected, linear spring) — state them in answers.

Uniform shear: .

Circular shaft torsion: .

Castigliano’s theorem (linear elastic): displacement in the direction of force $P_{i}$ is . Useful for indeterminate structures and deflection checks later in the course.

- Using $U=P\delta$ for gradual loading (missing $\frac{1}{2}$).
- Applying static $\sigma=\frac{P}{A}$ to sudden/impact problems.
- Forgetting to include $\delta$ in the potential drop $P(h+\delta)$.
- Mixing $E$ and $G$ in shear-energy formulas.