Mechanics & design

Strength of Materials (SOM)

10 self-contained study topics — notes, diagrams, formulas, and worked examples for exams and GATE.

Topics

Fig 10.1 — Normal stress σ acts perpendicular to the cut area; shear stress τ acts parallel. Bending σ is linear in y; torsion τ is zero at the centre and maximum at the outer fibre.
Schematic diagram for study — aligned with standard B.Tech / GATE syllabus.
Loading types and corresponding stresses. Reference table — axial, shear, bending, and torsion with stress distributions (RK Bansal / standard SOM convention).
Stress and Strain Basics
Stress is internal force intensity $\sigma=P/A$ (normal) or $\tau=V/A$ (shear). Strain is relative deformation $\varepsilon=\delta/L$ or $\gamma=\delta_s/L$ . The engineering stress–strain curve (RK Bansal / GATE SOM) defines elastic limit, yield, UTS, and fracture; true stress–strain accounts for instantaneous area.
$\sigma=\dfrac{P}{A}$
Fig 1.3 — Blue: ductile (mild steel). Red: brittle (cast iron). Ductile: large plastic strain. Brittle: fracture near σᵤ with minimal elongation. % elongation and cup-cone fracture distinguish ductile break in lab tests.
Schematic diagram for study — aligned with standard B.Tech / GATE syllabus.
Comparison of ductile and brittle stress–strain curves. Ductile materials absorb more energy and show warning before fracture; brittle materials fail suddenly.
Hooke's Law & Elastic Constants
Hooke’s law states that within the proportional limit, stress is proportional to strain: $\sigma=E\varepsilon$ and $\tau=G\gamma$ . The elastic constants $E$ , $G$ , $K$ , and $\nu$ are related by $E=2G(1+\nu)=3K(1-2\nu)$ (RK Bansal / GATE SOM).
$\sigma=E\varepsilon$
Axial Loading & Thermal Stress
Axial bars deform by $\delta=PL/(AE)$ (elastic). Stepped/composite bars use series/parallel stiffness. Free thermal expansion is $\delta_T=\alpha L\Delta T$ ; when constrained, thermal stress $\sigma_T=E\alpha\Delta T$ develops (RK Bansal). Combined mechanical + thermal problems superpose displacements with compatibility.
$\delta=\dfrac{PL}{AE}$
Bending Moment Diagram
$Key relations for this topic are being expanded - refer to your subject textbook meanwhile.$
Bending Stress in Beams
Pure bending of elastic beams gives the flexure formula $\dfrac{M}{I}=\dfrac{\sigma}{y}=\dfrac{E}{R}$ . Bending stress $\sigma=My/I$ is zero at the neutral axis and maximum at the extreme fibres. Section modulus $Z=I/y_{\max}$ governs strength design (RK Bansal).
$\dfrac{M}{I}=\dfrac{\sigma}{y}=\dfrac{E}{R}$
Shear Stress in Beams
$Key relations for this topic are being expanded - refer to your subject textbook meanwhile.$
Deflection of Beams
$Key relations for this topic are being expanded - refer to your subject textbook meanwhile.$
Torsion of Circular Shafts
For elastic circular shafts, shear stress and twist follow $\dfrac{T}{J}=\dfrac{\tau}{r}=\dfrac{G\theta}{L}$ . Maximum shear stress is at the outer radius: $\tau_{\max}=Tr/J$ . Power transmitted $P=T\omega=2\pi NT/60$ (RK Bansal / VB Bhandari shaft design).
$\dfrac{T}{J}=\dfrac{\tau}{r}=\dfrac{G\theta}{L}$
Euler and Rankine Column Buckling
$Key relations for this topic are being expanded - refer to your subject textbook meanwhile.$
Theories of Failure
$Key relations for this topic are being expanded - refer to your subject textbook meanwhile.$