Strength of Materials Formula Sheet
All Key Equations in One Place — For Quick Revision & Exam Preparation
Last Updated: March 2026
How to Use This Sheet 📌
- Every important SOM formula organised by topic with all variables defined.
- Use for last-minute revision before GATE, ESE, or university exams.
- Click topic links for detailed explanations and worked examples.
1. Stress & Strain
Normal stress: σ = F/A (Pa or MPa)
Shear stress: τ = F/A (force parallel to section)
Normal strain: ε = ΔL/L₀ (dimensionless)
Shear strain: γ = tan(φ) ≈ φ (radians)
Volumetric strain: εv = ΔV/V = εx + εy + εz
Thermal strain: εT = αΔT
Thermal stress (constrained): σ = EαΔT
2. Hooke’s Law & Elastic Constants
Hooke’s law: σ = Eε (normal), τ = Gγ (shear)
Deformation: ΔL = FL/(AE)
Poisson’s ratio: ν = −εlateral/εaxial (0 ≤ ν ≤ 0.5)
Relationships Between Constants
E = 2G(1 + ν)
E = 3K(1 − 2ν)
E = 9KG/(3K + G)
| Material | E (GPa) | G (GPa) | ν |
|---|---|---|---|
| Mild steel | 200 | 80 | 0.30 |
| Aluminium | 70 | 26 | 0.33 |
| Cast iron | 100–170 | 40–65 | 0.26 |
3. SFD & BMD — Standard Cases
| Beam & Loading | Vmax | Mmax |
|---|---|---|
| SS, central point load P | P/2 | PL/4 |
| SS, UDL w over full span | wL/2 | wL²/8 |
| Cantilever, point load P at tip | P | PL |
| Cantilever, UDL w full span | wL | wL²/2 |
| SS, point load P at distance a | Pb/L or Pa/L | Pab/L |
Key Relationships
dV/dx = −w(x), dM/dx = V(x)
Mmax occurs where V = 0.
4. Bending Stress
Bending equation: M/I = σ/y = E/R
Flexure formula: σ = My/I
Maximum bending stress: σmax = M/Z, where Z = I/ymax
| Section | I | Z = I/ymax |
|---|---|---|
| Rectangle (b × d) | bd³/12 | bd²/6 |
| Circle (D) | πD⁴/64 | πD³/32 |
| Hollow circle | π(Do⁴−Di⁴)/64 | π(Do⁴−Di⁴)/(32Do) |
Parallel axis theorem: I = Icentroid + Ad²
5. Torsion
Torsion equation: T/J = τ/r = Gθ/L
Max shear stress: τmax = TR/J = 16T/(πD³) (solid shaft)
Angle of twist: θ = TL/(GJ)
Power: P = 2πNT/60 (P in watts, N in rpm, T in N·m)
| Section | J (Polar Moment) | Zp = J/R |
|---|---|---|
| Solid circle (D) | πD⁴/32 | πD³/16 |
| Hollow circle | π(Do⁴−Di⁴)/32 | π(Do⁴−Di⁴)/(16Do) |
6. Mohr’s Circle & Stress Transformation
Principal stresses: σ₁, σ₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]
Centre: C = (σx+σy)/2
Radius: R = √[((σx−σy)/2)² + τxy²]
Max shear stress: τmax = R = (σ₁−σ₂)/2
Principal angle: tan(2θp) = 2τxy/(σx−σy)
7. Column Buckling
Euler’s formula: Pcr = π²EI/(Leff)²
Euler stress: σcr = π²E/λ²
Slenderness ratio: λ = Leff/rmin
Radius of gyration: r = √(I/A)
Rankine: PR = σyA/[1 + a(Leff/r)²], a = σy/(π²E)
| End Condition | K | Leff |
|---|---|---|
| Pinned–Pinned | 1.0 | L |
| Fixed–Fixed | 0.5 | 0.5L |
| Fixed–Pinned | 0.7 | 0.7L |
| Fixed–Free | 2.0 | 2L |
8. Failure Theories
For Ductile Materials
Tresca: (σmax − σmin) ≤ σy. τy = 0.5σy
Von Mises (2D): √(σ₁² − σ₁σ₂ + σ₂²) ≤ σy. τy = 0.577σy
For Brittle Materials
Max Normal Stress: |σmax| ≤ σu
9. Thin-Walled Pressure Vessels
Thin Cylinder (t/D < 1/20)
Hoop (circumferential) stress: σh = PD/(2t)
Longitudinal stress: σL = PD/(4t)
σh = 2σL — hoop stress is always double the longitudinal stress.
Where: P = internal gauge pressure, D = internal diameter, t = wall thickness.
Thin Sphere
σ = PD/(4t) — uniform in all directions (equal biaxial stress).
10. Beam Deflections — Standard Cases
| Beam & Loading | Max Deflection (ymax) |
|---|---|
| SS, central point load P | PL³/(48EI) |
| SS, UDL w | 5wL⁴/(384EI) |
| Cantilever, tip load P | PL³/(3EI) |
| Cantilever, UDL w | wL⁴/(8EI) |
General Deflection Equation
EI(d²y/dx²) = M(x)
Integrate twice with boundary conditions to find slope and deflection.