Simple stresses and strains

• Elasticity
  • 🔹 Elasticity is the property of a material that allows it to return to its original shape and size after the removal of external forces.
  • 🔹 If a material does not return to its original shape, it has plastic deformation.

• Stress (σ)
• 🔹 Stress is the internal resistance per unit area induced in a material due to an external force.
  🔹 It is given by: \[ \sigma = \frac{F}{A} \]
  • where,
  • $\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">\sigma \; =\; <strong>Stress</strong>\; (N/m²\; or\; Pascal)</span>}$
  • $\mathrm{<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">F</span></span></span></span>\; =\; <strong>Applied\; Force</strong>\; (N)</span>}$
  • $A$ = Cross-sectional Area (m²)
• Types of Stress
  • 1️⃣ Tensile Stress – When a material is stretched, increasing its length.
  • 2️⃣ Compressive Stress – When a material is compressed, decreasing its length.
  • 3️⃣ Shear Stress – When a material experiences parallel forces that cause sliding.
  • 4️⃣ Bending Stress – When a material bends due to external loads.
  • 5️⃣ Torsional Stress – When a material is twisted by torque.

• Strain (ε)
  • 🔹 Strain is the deformation per unit length due to applied stress.
  • 🔹 It is a dimensionless quantity given by:

$$\epsilon =\frac{\mathrm{\Delta }L}{\mathrm{L<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

• • where,
  • $\epsilon$ = Strain
  • $\Delta L$ = Change in length
  • $L$ = Original length
• Types of Strain
  • 1️⃣ Tensile Strain – Increase in length due to tensile stress.
  • 2️⃣ Compressive Strain – Decrease in length due to compressive stress.
  • 3️⃣ Shear Strain – Angular distortion due to shear stress.

• Elastic Limit
  • 🔹 The elastic limit is the maximum stress a material can withstand while still returning to its original shape.
  • 🔹 If stress exceeds this limit, permanent deformation occurs.

• Hooke’s Law
  • 🔹 Hooke’s law states that stress is directly proportional to strain within the elastic limit.

$$\sigma =E\epsilon$$

• • where,
  • $E$ = Modulus of Elasticity (Young’s Modulus)
  • $\sigma$ = Stress
  • $\epsilon$ = Strain
• 🔹 Beyond the elastic limit, Hooke’s law is no longer valid, and the material undergoes plastic deformation.

• Modulus of Elasticity (E)
  • 🔹 Also called Young’s Modulus, it measures a material’s stiffness.
  • 🔹 Higher modulus = stiffer material.
  • 🔹 Given by:

$$E=\frac{\sigma }{\mathrm{\epsilon <span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

• • 🔹 Common values:
  • Steel: 200 GPa
  • Aluminum: 70 GPa
  • Rubber: 0.01 GPa (very elastic)

• 7. Deformation of a Body Due to Force
  • 🔹 When a force is applied, a body changes its shape or size.
  • 🔹 Elastic Deformation: Temporary, reversible change.
  • 🔹 Plastic Deformation: Permanent, irreversible change.
  • 🔹 Factors affecting deformation:
    • Material properties
    • Magnitude of force
    • Shape and size of the object

8. Deformation Due to Self-Weight

🔹 A material can deform under its own weight, especially in long structures like beams, cables, and towers.
🔹 For a vertical rod under its own weight, the elongation is given by:

$$\mathrm{\Delta }L=\frac{\rho g{L}^2}{2\mathrm{E<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

where,

• $\rho$ = Density of material
• 
  $g$$g$ = Acceleration due to gravity
• 
  $L$$L$ = Length of the rod
• 
  $E$$E$ = Young’s modulus

9. Principle of Superposition

🔹 The Principle of Superposition states that the total deformation of a body due to multiple forces is equal to the sum of deformations due to individual forces.

🔹 Mathematically,

$$\mathrm{\Delta }{L}_{\text{total}}=\mathrm{\Delta }{L}_1+\mathrm{\Delta }{L}_2+\mathrm{.}\mathrm{.}\mathrm{.}+\mathrm{\Delta }{L}_{\mathrm{n<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;\; font-size:\; 16px;"></span>}}$$

🔹 This principle applies only in the elastic range where Hooke’s law holds.

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