Engineering Formulas Reference

Universal Constants & Properties

🔬 Physical Constants & Material Properties

Universal Constants

Gravitational acceleration (g)	9.80665 m/s²
Avogadro's number (N_A)	6.022 × 10²³ mol⁻¹
Gas constant (R)	8.314 J/(mol·K)
Boltzmann constant (k)	1.381 × 10⁻²³ J/K
Stefan-Boltzmann constant (σ)	5.670 × 10⁻⁸ W/(m²·K⁴)
Speed of light (c)	2.998 × 10⁸ m/s
Permittivity of free space (ε₀)	8.854 × 10⁻¹² F/m
Permeability of free space (μ₀)	4π × 10⁻⁷ H/m

Material Properties (20°C)

Material	Density	Young's Modulus
Steel	7850 kg/m³	200 GPa
Aluminum	2700 kg/m³	70 GPa
Concrete	2400 kg/m³	25-40 GPa
Water	998 kg/m³	-
Air (STP)	1.225 kg/m³	-
Copper	8960 kg/m³	110-130 GPa
Wood (oak)	700-900 kg/m³	7-15 GPa

📊 Structural & Mechanical

Structural Mechanics

Beam Stress and Deflection

$$\sigma = \frac{M \cdot y}{I}$$

Bending stress where M = moment, y = distance from neutral axis, I = moment of inertia

$$\delta = \frac{F \cdot L^3}{48 \cdot E \cdot I}$$

Point load deflection (simple beam) where F = load, L = span, E = Young's modulus

Column Buckling (Euler's Formula)

$$P_{cr} = \frac{\pi^2 \cdot E \cdot I}{(K \cdot L)^2}$$

Critical buckling load where K = effective length factor (0.5, 0.7, 1.0, or 2.0)

Axial Stress and Strain

$$\sigma = \frac{F}{A} \qquad \varepsilon = \frac{\sigma}{E}$$

Stress and strain where F = force, A = area, E = Young's modulus

$$\delta = \frac{F \cdot L}{A \cdot E}$$

Axial deformation

Section Properties

\begin{aligned} \text{Rectangular:} \quad I &= \frac{b \cdot h^3}{12} \\ \text{Circular:} \quad I &= \frac{\pi \cdot d^4}{64} \\ \text{Area:} \quad A &= \frac{\pi \cdot d^2}{4} \end{aligned}

Stress Transformations

$$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$$

Principal stresses

⚡ Electrical Engineering

Electrical Engineering

Ohm's Law & Power

$$V = I \cdot R \qquad P = V \cdot I = I^2 \cdot R = \frac{V^2}{R}$$

Voltage, current, resistance, and power relationships

AC Circuits

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

Impedance where $X_L = 2\pi fL$, $X_C = \frac{1}{2\pi fC}$

$$\text{Power Factor} = \frac{P}{S} = \frac{R}{Z}$$

Transformers

$$\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1}$$

Transformer voltage and current ratios

Three-Phase Power

$$P = \sqrt{3} \cdot V_L \cdot I_L \cdot PF$$

Three-phase power where $V_L$ = line voltage, $I_L$ = line current

💧 Fluid Mechanics

Fluid Mechanics & Hydraulics

Continuity Equation

$$Q = A_1 \cdot v_1 = A_2 \cdot v_2$$

Conservation of mass where Q = flow rate, A = cross-sectional area, v = velocity

Bernoulli's Equation

$$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L$$

Energy conservation including head loss

Darcy-Weisbach Equation

$$\Delta P = f \cdot \frac{L}{D} \cdot \frac{\rho \cdot v^2}{2}$$

Pressure drop where f = friction factor, L = length, D = diameter

Reynolds Number

$$Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{v \cdot D}{\nu}$$

Flow regime classification (laminar: Re < 2300, turbulent: Re> 4000)

Hagen-Poiseuille (Laminar Flow)

$$Q = \frac{\pi \cdot \Delta P \cdot r^4}{8 \cdot \mu \cdot L}$$

Flow rate for laminar flow in circular pipes

🌡️ Thermal Engineering

Thermal Engineering

Heat Transfer (Conduction)

$$q = \frac{k \cdot A \cdot \Delta T}{L}$$

Fourier's law where k = thermal conductivity, A = area, L = thickness

Heat Transfer (Convection)

$$q = h \cdot A \cdot \Delta T$$

Newton's law of cooling where h = heat transfer coefficient

Heat Transfer (Radiation)

$$q = \epsilon \cdot \sigma \cdot A \cdot (T_1^4 - T_2^4)$$

Stefan-Boltzmann law where ε = emissivity, σ = Stefan-Boltzmann constant

Combined Heat Transfer

$$\frac{1}{U} = \frac{1}{h_1} + \frac{L}{k} + \frac{1}{h_2}$$

Overall heat transfer coefficient for composite wall

Thermodynamic Relations

$$Q = m \cdot c \cdot \Delta T$$

Sensible heat where m = mass, c = specific heat

$$Q = m \cdot h_{fg}$$

Latent heat where $h_{fg}$ = latent heat of vaporization

⚙️ Mechanical Engineering

Mechanical Engineering

Power Transmission

$$P = T \cdot \omega = \frac{T \cdot 2\pi n}{60}$$

Power = Torque × Angular velocity where n = RPM, ω = rad/s

Gear Ratios

$$GR = \frac{N_2}{N_1} = \frac{\omega_1}{\omega_2} = \frac{T_2}{T_1}$$

Gear ratio where N = number of teeth, ω = angular speed

Springs

$$F = k \cdot x \qquad k = \frac{G \cdot d^4}{8 \cdot D^3 \cdot N_a}$$

Spring force and stiffness where k = spring constant, G = shear modulus

Bearings

$$L_{10} = \left(\frac{C}{P}\right)^3 \times 10^6 \text{ revolutions}$$

Bearing life where C = basic dynamic load rating, P = equivalent load

📐 Mathematics & Units

Mathematics & Unit Conversions

Common Unit Conversions

Length	1 m = 1000 mm = 3.281 ft
Force	1 kN = 224.8 lbf = 1000 N
Pressure	1 MPa = 145 psi = 10 bar
Power	1 kW = 1.341 HP = 1000 W
Energy	1 kWh = 3412 BTU = 3.6 MJ
Torque	1 N·m = 0.738 lb·ft

Geometric Formulas

\begin{aligned} \text{Circle:} \quad A &= \pi r^2 = \frac{\pi d^2}{4} \\ \text{Rectangle:} \quad A &= l \cdot w \\ \text{Sphere:} \quad V &= \frac{4\pi r^3}{3} \\ \text{Cylinder:} \quad V &= \pi r^2 h \end{aligned}

Trigonometry

\begin{aligned} \sin^2\theta + \cos^2\theta &= 1 \\ \tan\theta &= \frac{\sin\theta}{\cos\theta} \end{aligned}

\begin{aligned} \text{Law of Sines:} \quad \frac{a}{\sin A} &= \frac{b}{\sin B} = \frac{c}{\sin C} \\ \text{Law of Cosines:} \quad c^2 &= a^2 + b^2 - 2ab \cdot \cos C \end{aligned}

Statistics & Error Analysis

\begin{aligned} \text{Mean:} \quad \bar{x} &= \frac{\sum x_i}{n} \\ \text{Standard Deviation:} \quad \sigma &= \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \\ \text{Propagation of Error:} \quad \delta f &= \sqrt{\left(\frac{\partial f}{\partial x} \cdot \delta x\right)^2 + \left(\frac{\partial f}{\partial y} \cdot \delta y\right)^2} \end{aligned}

📐 Engineering Formulas Reference

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