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FORMULA REFERENCE

Essential Engineering Formula Reference

Key formulas from mechanics of materials, fluid mechanics, thermal engineering, and vibration — each with an explanation and a link to an interactive calculator.

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Contents

Mechanics of Materials & Structural Analysis

Hooke's Law (Tension / Compression)
\(\sigma = E \varepsilon\)
σ: stress [Pa] — E: Young's modulus [Pa] — ε: strain [–]. The fundamental relationship for elastic deformation.
→ Mohr's Circle Tool
Beam Maximum Deflection (Fixed-Fixed, Central Point Load)
\(\delta_{max} = \dfrac{PL^3}{192EI}\)
P: load [N] — L: span [m] — E: Young's modulus [Pa] — I: second moment of area [m⁴].
→ Beam Deflection Calculator
Simply Supported Beam — Central Point Load Deflection
\(\delta_{max} = \dfrac{PL^3}{48EI}\)
Maximum deflection of a simply supported beam (pinned at both ends) under a central point load.
→ Beam Deflection Calculator
Euler Buckling Load
\(P_{cr} = \dfrac{\pi^2 EI}{(KL)^2}\)
K: effective length factor (pin-pin = 1.0, fixed-fixed = 0.5). Critical load for a column under axial compression.
→ Buckling Calculator
Slenderness Ratio
\(\lambda = \dfrac{KL}{r}, \quad r = \sqrt{\dfrac{I}{A}}\)
r: radius of gyration. Higher λ means greater susceptibility to buckling (λ > 100 indicates a slender column).
→ Buckling Calculator
Thin-Walled Pressure Vessel — Hoop Stress (Cylinder)
\(\sigma_\theta = \dfrac{pr}{t}\)
p: internal pressure [Pa] — r: inner radius [m] — t: wall thickness [m]. Hoop stress is twice the axial stress.
→ Pressure Vessel Calculator
Hertz Contact Stress (Sphere on Flat)
\(p_0 = \dfrac{3F}{2\pi a^2}, \quad a = \left(\dfrac{3FR}{4E^*}\right)^{1/3}\)
F: load — R: sphere radius — E*: equivalent elastic modulus. Used for contact stress in rolling bearings and gears.
→ Hertz Contact Calculator
Thermal Stress (Uniform Temperature Change)
\(\sigma_{th} = E \alpha \Delta T\)
α: coefficient of thermal expansion [1/K] — ΔT: temperature change [K]. Stress induced in a constrained member.
→ Thermal Stress Calculator

Fluid Mechanics & CFD

Reynolds Number
\(Re = \dfrac{\rho V L}{\mu} = \dfrac{VL}{\nu}\)
ρ: density — V: characteristic velocity — L: characteristic length — μ: dynamic viscosity — ν: kinematic viscosity. Re < 2300: laminar; Re > 4000: turbulent.
→ Reynolds Number Calculator
Darcy-Weisbach Equation (Pipe Pressure Drop)
\(\Delta p = f \dfrac{L}{D} \dfrac{\rho V^2}{2}\)
f: friction factor (laminar: 64/Re) — L: pipe length — D: pipe diameter. The fundamental formula for pipe system design.
→ Pipe Flow Calculator
Bernoulli's Equation
\(p + \dfrac{1}{2}\rho V^2 + \rho g z = \text{const}\)
Energy conservation along a streamline for inviscid, steady, incompressible flow.
→ Pipe Flow Calculator
Continuity Equation (Incompressible Flow)
\(\nabla \cdot \mathbf{V} = 0 \quad \Rightarrow \quad A_1 V_1 = A_2 V_2\)
Mass conservation for incompressible flow. A smaller cross-sectional area means higher velocity.

Thermal Engineering

Fourier's Law of Heat Conduction
\(q = -\lambda \dfrac{dT}{dx}\)
q: heat flux [W/m²] — λ: thermal conductivity [W/(m·K)]. Applied to steady-state conduction through a flat wall.
→ Heat Diffusion Simulation
Convective Heat Transfer (Newton's Law of Cooling)
\(q = h (T_s - T_\infty)\)
h: convective heat transfer coefficient [W/(m²·K)]. Also known as Newton's law of cooling.
→ Heat Exchanger NTU Calculator
Thermal Resistance (Series Connection)
\(R_{total} = \sum_i \dfrac{L_i}{\lambda_i A_i}\)
Total thermal resistance of a multi-layer wall in series. Analogous to Ohm's law in electrical circuits.
→ Heat Diffusion Simulation
Heat Exchanger — NTU Method
\(\varepsilon = \dfrac{Q}{Q_{max}}, \quad NTU = \dfrac{UA}{C_{min}}\)
ε: heat exchanger effectiveness — U: overall heat transfer coefficient — A: heat transfer area — C_min: minimum heat capacity rate.
→ Heat Exchanger NTU Calculator
Heat Diffusion Equation (1D)
\(\dfrac{\partial T}{\partial t} = \alpha \dfrac{\partial^2 T}{\partial x^2}, \quad \alpha = \dfrac{\lambda}{\rho c_p}\)
α: thermal diffusivity [m²/s]. Governing equation for transient heat conduction. Non-dimensionalized using the Fourier number Fo = αt/L².
→ Heat Diffusion Simulation

Vibration & Dynamics

Natural Frequency of a 1-DOF System
\(\omega_n = \sqrt{\dfrac{k}{m}}, \quad f_n = \dfrac{\omega_n}{2\pi}\)
k: spring stiffness [N/m] — m: mass [kg]. The fundamental formula for resonant frequency.
→ Eigenvalue & Vibration Analysis Tool
Damping Ratio and Critical Damping
\(\zeta = \dfrac{c}{c_{cr}}, \quad c_{cr} = 2\sqrt{km}\)
ζ < 1: underdamped (oscillatory decay) — ζ = 1: critically damped — ζ > 1: overdamped.
Thin Plate Buckling (Simply Supported on All Four Edges, In-Plane Compression)
\(N_{cr} = k_c \dfrac{\pi^2 D}{b^2}, \quad D = \dfrac{Et^3}{12(1-\nu^2)}\)
D: flexural rigidity — b: plate width in the load direction — k_c: buckling coefficient (depends on aspect ratio).
→ Plate Buckling Calculator

Fracture & Fatigue Mechanics

Stress Intensity Factor (Mode I)
\(K_I = \sigma \sqrt{\pi a} \cdot F(a/W)\)
σ: far-field stress — a: half crack length — F: geometry factor. Fracture occurs when K_I > K_IC.
→ J-Integral & Fracture Mechanics Tool
Paris Law (Crack Growth)
\(\dfrac{da}{dN} = C (\Delta K)^m\)
C, m: material constants — ΔK: stress intensity factor range. Predicts fatigue crack growth rate.
→ Paris Law Calculator
Goodman Diagram (Fatigue Life Assessment)
\(\dfrac{\sigma_a}{\sigma_e} + \dfrac{\sigma_m}{\sigma_u} = 1\)
σ_a: stress amplitude — σ_m: mean stress — σ_e: fully reversed fatigue limit — σ_u: ultimate tensile strength.
→ Goodman Diagram Tool

→ CAE · FEM · CFD Glossary (100+ terms)
→ Full tool index (300+ interactive calculators)

How to Use

  1. Select a discipline tab (Mechanics, Fluids, Thermal, or Vibration) to filter formulas by domain.
  2. Enter material properties (e.g., Young's modulus E=210 GPa for mild steel, kinematic viscosity ν=1.0×10⁻⁶ m²/s for water at 20°C) into the input fields.
  3. Input geometry and load parameters, then click Calculate to solve for stress, deflection, flow rate, heat transfer coefficient, or natural frequency with unit conversion handled automatically.

Worked Example

A cantilever steel beam (E=200 GPa, length L=1.5 m, rectangular section 50 mm × 80 mm) supports a concentrated load P=5 kN at the free end. Maximum deflection δ = PL³/(3EI). With I = 1.067×10⁻⁶ m⁴, δ = (5000 × 1.5³)/(3 × 200×10⁹ × 1.067×10⁻⁶) = 8.4 mm. Maximum bending stress σ = M×y/I = (7500 × 0.04)/1.067×10⁻⁶ = 281 MPa, acceptable for yield strength 250 MPa design limit.

Practical Notes

  1. Verify Reynolds number (ρVD/μ) before applying laminar correlations; turbulent flow in pipes requires Darcy-Weisbach friction factor iteration.
  2. Temperature-dependent material properties (aluminum modulus drops 3% from 20°C to 100°C) must be updated for thermal transient analysis.
  3. Modal analysis requires precise moment of inertia calculation; use parallel-axis theorem for composite sections in vibration problems.
  4. Cross-reference SI base units throughout; pressure in Pa (not bar) prevents calculation cascade errors.