Strength of Materials

Stress, strain, stress strain diagram, factor of safety, thermal stresses, strain energy, proof resilience and  modulus of resilience. Shear force and bending moment diagram – cantilever beam, simply supported beam,  continuous beam, fixed beam. Torsion in shafts and springs, thin cylinder shells.

• Stress ($\sigma$): Force per unit area ($\sigma =F/A$). Types include tensile, compressive, and shear stress.
• Strain ($\epsilon$): Deformation per unit length (ε=ΔL/L).
• Hooke’s Law: In the elastic region, σ=Eε, where $E$ is Young’s modulus

Stress-Strain Diagram Characteristics

1. Elastic Region: Linear slope (Young’s modulus  E) where deformation is reversible.
2. Yield Point: Transition to plastic deformation (upper/lower yield points in steel)2.
3. Plastic Region: Permanent deformation occurs.
4. Tensile Strength: Maximum stress before necking.
5. Breaking Elongation: Strain at fracture.

Factor of Safety

$$\text{Safety Factor}=\frac{\text{Ultimate Strength}}{\text{Design Strength}}$$

• Example: A bridge designed for 100 tons with a safety factor of 2 can safely support 200 tons3.
• Accounts for material uncertainties, manufacturing defects, and environmental factors3.

Thermal Stresses

• Caused by temperature changes (
  $\mathrm{\Delta }T$) in constrained materials:

$${\sigma }_{\text{thermal}}=E\cdot \alpha \cdot \mathrm{\Delta }T$$

• • Where
    $\alpha$ = thermal expansion coefficient4.
• Thermal Shock: Rapid temperature changes inducing surface tension (e.g., glass quenching)4.

Strain Energy and Resilience

• Strain Energy: Area under stress-strain curve
  •  (
    $U=\frac{1}{2}\sigma \epsilon$ 

    $U$).

• Proof Resilience: Maximum elastic energy storage (area up to yield point).
• Modulus of Resilience: Energy per unit volume .

$U_R=\frac{{\sigma }_y^2}{2E}$

Shear Force (SF) and Bending Moment (BM) Diagrams

Key steps for analysis:

• Calculate support reactions.
• Determine shear forces at critical sections.
• Compute bending moments.

Torsion and Thin Cylinders

• Torsion in Shafts: 
  • Shear stress ($\tau =\frac{Tr}{J}$),
    • where $T$ = torque, $r$ = radius, $J$ = polar moment.
• Thin-Walled Cylinders:
  • Hoop stress: ${\sigma }_h=\frac{PD}{2t}$
  • Longitudinal stress: ${\sigma }_l=\frac{PD}{4t}$
    • (where $P$ = pressure, $D$ = diameter, $t$ = thickness).

These principles enable engineers to predict material behavior, design safe structures, and optimize components under mechanical and thermal loads.