1. Strain Energy

Definition

Strain Energy is the energy stored in a body when it is deformed due to an applied load. When the load is removed, this energy is released and the body returns to its original shape (if the material is within the elastic limit).

In simple words:

Strain energy is the work done by external forces in deforming a body, which is stored as internal energy.

Basic Formula

For a gradually applied load:$$U = \frac{1}{2} P \delta$$Where

• $U$U = Strain energy (Joule)
• $P$P = Applied load (N)
• $\delta$δ = Deformation or extension (m)

Strain Energy in Terms of Stress and Strain

$$U = \frac{\sigma^2}{2E} \times V$$Where

• $\sigma$σ = Stress
• $E$E = Young’s Modulus
• $V$V = Volume of material

Strain Energy per Unit Volume (Strain Energy Density)

$$u = \frac{\sigma^2}{2E}$$This represents energy stored per unit volume of material.

Strain Energy in a Bar

For a bar subjected to axial load:$$U = \frac{P^2 L}{2AE}$$Where

• $L$L = Length of bar
• $A$A = Cross-sectional area
• $E$E = Young’s modulus

Strain Energy in Different Types of Loading

Type of Loading	Strain Energy Formula
Axial loading	$U = \frac{P^2L}{2AE}$U=2AEP2L​
Bending	$U = \int \frac{M^2}{2EI} dx$U=∫2EIM2​dx
Torsion	$U = \frac{T^2L}{2GJ}$U=2GJT2L​

Where

• $M$M = Bending moment
• $T$T = Torque
• $I$I = Moment of inertia
• $G$G = Shear modulus
• $J$J = Polar moment of inertia

Applications of Strain Energy

1. Design of springs
2. Shock absorbing systems
3. Impact load calculations
4. Structural analysis
5. Energy methods like Castigliano’s theorem

2. Impact Loading

Definition

Impact loading occurs when a load is applied suddenly or by collision, such as a falling weight striking a structure.

Unlike gradual loading, impact loading produces much higher stress because energy is transferred quickly.

Examples:

• Hammer striking metal
• Drop test in structures
• Railway wheel hitting rail joints
• Punching and forging operations

Case 1: Suddenly Applied Load

If a load P is suddenly applied to a bar:

Maximum stress produced is:$$\sigma_{max} = 2\sigma$$This means stress becomes twice the static stress.

Maximum deflection:$$\delta_{max} = 2\delta$$Thus, sudden loading causes double deformation compared to gradual loading.

Case 2: Falling Load (Impact of Falling Weight)

Consider a weight W falling from height h on a bar.

Energy balance principle:

Potential Energy = Strain Energy$$W(h + \delta) = \frac{\sigma^2}{2E} \times V$$From this relation, the maximum stress in the bar can be calculated.

Maximum Stress Formula

$$\sigma_{max} = \frac{W}{A} \left(1 + \sqrt{1 + \frac{2AEh}{WL}} \right)$$Where

• $W$W = Falling load
• $h$h = Height of fall
• $A$A = Area of bar
• $L$L = Length of bar
• $E$E = Young’s modulus

Comparison of Loading Types

Loading Type	Stress Produced
Gradual load	$\sigma = P/A$σ=P/A
Suddenly applied load	$2\sigma$2σ
Impact load	Much greater than $2\sigma$2σ

Practical Engineering Examples

1. Drop forging
2. Pile driving
3. Railway track loading
4. Machine hammering
5. Crash and collision design

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