Mechanical / Statics
Normal Stress
Stress is force per unit area acting on a material
\sigma = \frac{F}{A}σ (Pa)F (N)A (m²)
Mechanical / Statics
Normal Strain
Strain is the ratio of deformation to original length
\varepsilon = \frac{\Delta L}{L_0}ε ΔL (m)L₀ (m)
Mechanical / Statics
Hooke's Law
Linear relationship between stress and strain in elastic region
\sigma = E \varepsilonσ (Pa)E (Pa)ε
Mechanical / Statics
Shear Stress
Average shear stress from transverse force
\tau = \frac{V}{A}τ (Pa)V (N)A (m²)
Mechanical / Statics
Torque
Torque (moment) from a perpendicular force
\tau = F \cdot rτ (N·m)F (N)r (m)
Mechanical / Statics
Moment of Inertia (Rectangle)
Second moment of area for a rectangular cross-section about centroidal axis
I = \frac{bh^3}{12}I (m⁴)b (m)h (m)
Mechanical / Statics
Moment of Inertia (Circle)
Second moment of area for a circular cross-section
I = \frac{\pi d^4}{64}I (m⁴)d (m)
Mechanical / Dynamics
Newton's Second Law
Force equals mass times acceleration
F = maF (N)m (kg)a (m/s²)
Mechanical / Dynamics
Kinetic Energy
Energy of motion
KE = \frac{1}{2}mv^2KE (J)m (kg)v (m/s)
Mechanical / Dynamics
Gravitational Potential Energy
Energy due to height in a uniform gravitational field
PE = mghPE (J)m (kg)g (m/s²)h (m)
Mechanical / Dynamics
Power (Work/Time)
Average power over a time interval
P = \frac{W}{t}P (W)W (J)t (s)
Mechanical / Dynamics
Power (Rotational)
Mechanical power from torque and angular velocity
P = \tau \omegaP (W)τ (N·m)ω (rad/s)
Mechanical / Dynamics
Linear Momentum
Product of mass and velocity
p = mvp (kg·m/s)m (kg)v (m/s)
Mechanical / Dynamics
Impulse
Change in momentum from force over time
J = F \cdot \Delta tJ (N·s)F (N)Δt (s)
Mechanical / Machine Design
Gear Ratio
Ratio of gear teeth or angular velocities
GR = \frac{N_2}{N_1} = \frac{\omega_1}{\omega_2}GR N₂ N₁
Mechanical / Machine Design
Spring Force (Hooke's Law)
Force exerted by a spring
F = kxF (N)k (N/m)x (m)
Mechanical / Machine Design
Helical Spring Rate
Spring constant for helical compression spring
k = \frac{Gd^4}{8D^3n}k (N/m)G (Pa)d (m)D (m)
Mechanical / Machine Design
Factor of Safety
Ratio of yield strength to working stress
FS = \frac{S_y}{\sigma}FS Sᵧ (Pa)σ (Pa)
Mechanical / Machine Design
V-Belt Length
Length of V-belt for two pulleys
L = 2C + \frac{\pi(D+d)}{2} + \frac{(D-d)^2}{4C}L (m)C (m)D (m)d (m)
Electrical / DC Circuits
Ohm's Law
Voltage equals current times resistance
V = IRV (V)I (A)R (Ω)
Electrical / DC Circuits
Electrical Power (V×I)
Power equals voltage times current
P = VIP (W)V (V)I (A)
Electrical / DC Circuits
Electrical Power (I²R)
Power dissipated in a resistance
P = I^2RP (W)I (A)R (Ω)
Electrical / DC Circuits
Resistors in Series
Total resistance of two resistors in series
R_{total} = R_1 + R_2Rₜₒₜₐₗ (Ω)R₁ (Ω)R₂ (Ω)
Electrical / DC Circuits
Resistors in Parallel
Total resistance of two resistors in parallel
\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}Rₜₒₜₐₗ (Ω)R₁ (Ω)R₂ (Ω)
Electrical / DC Circuits
Capacitance
Capacitance is charge stored per unit voltage
C = \frac{Q}{V}C (F)Q (C)V (V)
Electrical / DC Circuits
Capacitor Energy
Energy stored in a capacitor
E = \frac{1}{2}CV^2E (J)C (F)V (V)
Electrical / DC Circuits
RC Time Constant
Time constant of RC circuit
\tau = RCτ (s)R (Ω)C (F)
Electrical / AC Circuits
Impedance Magnitude
Impedance magnitude of a series RLC circuit
Z = \sqrt{R^2 + (X_L - X_C)^2}Z (Ω)R (Ω)Xₗ (Ω)Xc (Ω)
Electrical / AC Circuits
Inductive Reactance
Opposition to current by inductor
X_L = 2 \pi f L = \omega LXₗ (Ω)f (Hz)L (H)
Electrical / AC Circuits
Capacitive Reactance
Opposition to current by capacitor
X_C = \frac{1}{2 \pi f C} = \frac{1}{\omega C}Xc (Ω)f (Hz)C (F)
Electrical / AC Circuits
Resonant Frequency
Frequency at which XL equals XC
f_0 = \frac{1}{2 \pi \sqrt{LC}}f₀ (Hz)L (H)C (F)
Electrical / AC Circuits
Power Factor
Cosine of the phase angle between voltage and current
PF = \cos(\phi)PF φ (°)
Electrical / AC Circuits
Real Power (AC)
Actual power consumed in AC circuit
P = VI\cos(\phi)P (W)V (V)I (A)φ (°)
Civil / Beams
Cantilever Beam Deflection (Point Load)
Maximum deflection at free end of cantilever with point load
\delta = \frac{FL^3}{3EI}δ (m)F (N)L (m)E (Pa)
Civil / Beams
Simply Supported Beam Deflection
Maximum deflection at center of simply supported beam
\delta_{max} = \frac{FL^3}{48EI}δₘₐₓ (m)F (N)L (m)E (Pa)
Civil / Beams
Bending Stress
Stress due to bending moment
\sigma = \frac{My}{I}σ (Pa)M (N·m)y (m)I (m⁴)
Civil / Beams
Section Modulus
Geometric property for bending strength
S = \frac{I}{c}S (m³)I (m⁴)c (m)
Civil / Beams
Shear Stress in Beams
Shear stress distribution in beams
\tau = \frac{VQ}{It}τ (Pa)V (N)Q (m³)I (m⁴)
Civil / Columns
Euler Buckling Load
Critical buckling load for slender columns
P_{cr} = \frac{\pi^2 EI}{(KL)^2}Pcr (N)E (Pa)I (m⁴)K
Civil / Columns
Slenderness Ratio
Ratio determining column buckling behavior
\lambda = \frac{KL}{r}λ K L (m)r (m)
Civil / Columns
Radius of Gyration
Geometric property for column stability
r = \sqrt{\frac{I}{A}}r (m)I (m⁴)A (m²)
Civil / Hydraulics
Manning's Equation
Velocity in open channel flow
V = \frac{1}{n}R^{2/3}S^{1/2}V (m/s)n R (m)S
Civil / Hydraulics
Hydraulic Radius
Ratio of flow area to wetted perimeter
R = \frac{A}{P}R (m)A (m²)P (m)
Thermodynamics / Laws
First Law of Thermodynamics
Energy balance: change in internal energy equals heat added minus work done
\Delta U = Q - WΔU (J)Q (J)W (J)
Thermodynamics / Laws
Ideal Gas Law
Equation of state for ideal gases
PV = nRTP (Pa)V (m³)n (mol)R (J/(mol·K))
Thermodynamics / Heat Transfer
Heat Transfer (Sensible)
Heat required to change temperature
Q = mc\Delta TQ (J)m (kg)c (J/(kg·K))ΔT (K)
Thermodynamics / Cycles
Carnot Efficiency
Maximum possible efficiency of heat engine
\eta = 1 - \frac{T_C}{T_H}η Tc (K)Tₕ (K)
Thermodynamics / Heat Transfer
Fourier Heat Conduction (1D)
Heat transfer through a solid by conduction
Q = -k A \frac{dT}{dx}Q (W)k (W/(m·K))A (m²)dT/dx (K/m)
Thermodynamics / Heat Transfer
Newton's Law of Cooling
Heat transfer by convection
Q = hA(T_s - T_\infty)Q (W)h (W/(m²·K))A (m²)Tₛ (K)
Thermodynamics / Heat Transfer
Stefan-Boltzmann Radiation
Emitted radiative power from a diffuse gray surface
Q = \varepsilon \sigma A T^4Q (W)ε σ (W/(m²·K⁴))A (m²)
Fluids / Fluid Dynamics
Reynolds Number
Dimensionless number indicating laminar or turbulent flow
Re = \frac{\rho vD}{\mu}Re ρ (kg/m³)v (m/s)D (m)
Fluids / Fluid Dynamics
Continuity Equation
Conservation of mass in fluid flow
A_1 v_1 = A_2 v_2A₁ (m²)v₁ (m/s)A₂ (m²)v₂ (m/s)
Fluids / Pipe Flow
Darcy-Weisbach Equation
Head loss due to friction in pipe flow
h_f = f \frac{L}{D} \frac{v^2}{2g}hf (m)f L (m)D (m)
Fluids / Pipe Flow
Pump Power
Power required by pump
P = \frac{\rho g Q H}{\eta}P (W)ρ (kg/m³)g (m/s²)Q (m³/s)
Fluids / Fluid Statics
Hydrostatic Pressure
Pressure due to fluid column
P = \rho ghP (Pa)ρ (kg/m³)g (m/s²)h (m)
Fluids / Fluid Statics
Buoyancy Force
Upward force on submerged object
F_b = \rho g VFb (N)ρ (kg/m³)g (m/s²)V (m³)
Materials / Stress & Strain
Poisson's Ratio
Ratio of lateral to axial strain
\nu = -\frac{\varepsilon_{transverse}}{\varepsilon_{axial}}ν εₜ εₐ
Materials / Stress & Strain
Shear Modulus
Relationship between elastic modulus and shear modulus
G = \frac{E}{2(1+\nu)}G (Pa)E (Pa)ν
Materials / Stress & Strain
Bulk Modulus
Resistance to uniform compression
K = \frac{E}{3(1-2\nu)}K (Pa)E (Pa)ν
Materials / Stress & Strain
True Stress
Stress based on instantaneous area
\sigma_t = \sigma_e(1 + \varepsilon_e)σₜ (Pa)σₑ (Pa)εₑ
Materials / Failure Theories
Von Mises Stress
Equivalent stress for 2D plane stress
\sigma_v = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}σᵥ (Pa)σ₁ (Pa)σ₂ (Pa)
Materials / Fatigue
Basquin Equation (Fatigue)
High-cycle fatigue life prediction
\sigma_a = \sigma_f^\prime (2N_f)^bσₐ (Pa)σ'f (Pa)Nf b
Mechanical / Statics
Torsional Shear Stress
Shear stress in a shaft due to torsion
\tau = \frac{Tr}{J}τ (Pa)T (N·m)r (m)J (m⁴)
Mechanical / Statics
Angle of Twist
Angular deflection of shaft under torsion
\phi = \frac{TL}{GJ}φ (rad)T (N·m)L (m)G (Pa)
Mechanical / Statics
Polar Moment of Inertia (Solid Circle)
Polar moment of inertia for solid circular shaft
J = \frac{\pi d^4}{32}J (m⁴)d (m)
Mechanical / Statics
Polar Moment of Inertia (Hollow Circle)
Polar moment of inertia for hollow circular shaft
J = \frac{\pi (d_o^4 - d_i^4)}{32}J (m⁴)dₒ (m)dᵢ (m)
Mechanical / Statics
Hoop Stress (Thin-Wall Cylinder)
Circumferential stress in thin-walled pressure vessel
\sigma_h = \frac{pr}{t}σₕ (Pa)p (Pa)r (m)t (m)
Mechanical / Statics
Longitudinal Stress (Thin-Wall Cylinder)
Axial stress in thin-walled pressure vessel
\sigma_l = \frac{pr}{2t}σₗ (Pa)p (Pa)r (m)t (m)
Mechanical / Statics
Stress in Thin-Wall Sphere
Stress in thin-walled spherical pressure vessel
\sigma = \frac{pr}{2t}σ (Pa)p (Pa)r (m)t (m)
Mechanical / Dynamics
Work (Constant Force)
Work done by a constant force
W = Fd\cos(\theta)W (J)F (N)d (m)θ (°)
Mechanical / Dynamics
Centripetal Force
Force required for circular motion
F = \frac{mv^2}{r}F (N)m (kg)v (m/s)r (m)
Mechanical / Dynamics
Angular Momentum
Rotational momentum of a body
L = I\omegaL (kg·m²/s)I (kg·m²)ω (rad/s)
Mechanical / Dynamics
Rotational Kinetic Energy
Kinetic energy of rotating body
KE = \frac{1}{2}I\omega^2KE (J)I (kg·m²)ω (rad/s)
Mechanical / Dynamics
Simple Pendulum Period
Period of small oscillations
T = 2 \pi \sqrt{\frac{L}{g}}T (s)L (m)g (m/s²)
Mechanical / Dynamics
Spring-Mass Period
Period of spring-mass oscillation
T = 2 \pi \sqrt{\frac{m}{k}}T (s)m (kg)k (N/m)
Mechanical / Statics
Friction Force
Friction force between surfaces
F_f = \mu NFᶠ (N)μ N (N)
Mechanical / Statics
Parallel Axis Theorem
Moment of inertia about parallel axis
I = I_c + Ad^2I (m⁴)Ic (m⁴)A (m²)d (m)
Electrical / DC Circuits
Inductor Energy
Energy stored in an inductor
E = \frac{1}{2}LI^2E (J)L (H)I (A)
Electrical / DC Circuits
RL Time Constant
Time constant of RL circuit
\tau = \frac{L}{R}τ (s)L (H)R (Ω)
Electrical / DC Circuits
Capacitors in Series
Total capacitance of capacitors in series
\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2}Cₜₒₜₐₗ (F)C₁ (F)C₂ (F)
Electrical / DC Circuits
Capacitors in Parallel
Total capacitance of capacitors in parallel
C_{total} = C_1 + C_2Cₜₒₜₐₗ (F)C₁ (F)C₂ (F)
Electrical / AC Circuits
Transformer Voltage Ratio
Voltage ratio equals turns ratio
\frac{V_1}{V_2} = \frac{N_1}{N_2}V₁ (V)V₂ (V)N₁ N₂
Electrical / AC Circuits
Three-Phase Power
Total power in three-phase system
P = \sqrt{3} V_L I_L \cos(\phi)P (W)Vₗ (V)Iₗ (A)φ (°)
Electrical / AC Circuits
Apparent Power
Product of RMS voltage and current
S = VIS (VA)V (V)I (A)
Electrical / AC Circuits
Reactive Power
Power stored and returned by reactive elements
Q = VI\sin(\phi)Q (VAR)V (V)I (A)φ (°)
Electrical / DC Circuits
Voltage Divider
Output voltage of resistive voltage divider
V_{out} = V_{in} \frac{R_2}{R_1 + R_2}Vₒᵤₜ (V)Vᵢₙ (V)R₁ (Ω)R₂ (Ω)
Electrical / DC Circuits
Current Divider
Current through one branch of parallel resistors
I_1 = I_{total} \frac{R_2}{R_1 + R_2}I₁ (A)Iₜₒₜₐₗ (A)R₁ (Ω)R₂ (Ω)
Electrical / DC Circuits
Coulomb's Law
Force between two point charges
F = k_e \frac{q_1 q_2}{r^2}F (N)kₑ (N·m²/C²)q₁ (C)q₂ (C)
Electrical / AC Circuits
RMS to Peak Voltage
Relationship between peak and RMS for sinusoids
V_{peak} = V_{rms} \sqrt{2}Vₚₑₐₖ (V)Vᵣₘₛ (V)
Civil / Beams
Simply Supported Beam (Uniform Load)
Maximum deflection under uniform distributed load
\delta_{max} = \frac{5wL^4}{384EI}δₘₐₓ (m)w (N/m)L (m)E (Pa)
Civil / Beams
Cantilever Beam (Uniform Load)
Maximum deflection of cantilever under uniform load
\delta_{max} = \frac{wL^4}{8EI}δₘₐₓ (m)w (N/m)L (m)E (Pa)
Civil / Beams
Maximum Moment (Simply Supported, Uniform)
Maximum bending moment at center
M_{max} = \frac{wL^2}{8}Mₘₐₓ (N·m)w (N/m)L (m)
Civil / Beams
Maximum Moment (Simply Supported, Point Load)
Maximum bending moment at center
M_{max} = \frac{PL}{4}Mₘₐₓ (N·m)P (N)L (m)
Civil / Columns
Euler Critical Stress
Critical buckling stress for slender columns
\sigma_{cr} = \frac{\pi^2 E}{(KL/r)^2}σcr (Pa)E (Pa)K L (m)
Civil / Hydraulics
Chezy Equation
Velocity in open channel flow
V = C\sqrt{RS}V (m/s)C (m^(1/2)/s)R (m)S
Thermodynamics / Laws
Combined Gas Law
Relates pressure, volume, and temperature changes
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}P₁ (Pa)V₁ (m³)T₁ (K)P₂ (Pa)
Thermodynamics / Laws
Specific Heat Ratio
Ratio of specific heats at constant pressure to volume
\gamma = \frac{c_p}{c_v}γ cₚ (J/(kg·K))cᵥ (J/(kg·K))
Thermodynamics / Cycles
Adiabatic Process
Pressure-volume relation for adiabatic process
PV^\gamma = constantP (Pa)V (m³)γ
Thermodynamics / Cycles
Otto Cycle Efficiency
Thermal efficiency of ideal Otto cycle
\eta = 1 - \frac{1}{r^{\gamma-1}}η r γ
Thermodynamics / Cycles
Diesel Cycle Efficiency
Thermal efficiency of ideal Diesel cycle
\eta = 1 - \frac{1}{r^{\gamma-1}} \frac{r_c^\gamma - 1}{\gamma(r_c - 1)}η r rc γ
Thermodynamics / Cycles
Refrigeration COP
Coefficient of performance for refrigeration
COP_R = \frac{Q_L}{W} = \frac{T_L}{T_H - T_L}COPᵣ Qₗ (J)W (J)Tₗ (K)
Thermodynamics / Cycles
Heat Pump COP
Coefficient of performance for heat pump
COP_{HP} = \frac{Q_H}{W} = \frac{T_H}{T_H - T_L}COPₕₚ Qₕ (J)W (J)Tₕ (K)
Thermodynamics / Heat Transfer
Log Mean Temperature Difference
Effective temperature difference for heat exchangers
LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}LMTD (K)ΔT₁ (K)ΔT₂ (K)
Thermodynamics / Heat Transfer
Overall Heat Transfer Coefficient
Combined convection-conduction-convection resistance
\frac{1}{U} = \frac{1}{h_1} + \frac{t}{k} + \frac{1}{h_2}U (W/(m²·K))h₁ (W/(m²·K))h₂ (W/(m²·K))t (m)
Thermodynamics / Heat Transfer
Thermal Resistance (Conduction)
Thermal resistance of a plane wall
R = \frac{L}{kA}R (K/W)L (m)k (W/(m·K))A (m²)
Fluids / Fluid Dynamics
Bernoulli Equation
Conservation of energy along a streamline
P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2P (Pa)ρ (kg/m³)v (m/s)g (m/s²)
Fluids / Fluid Dynamics
Torricelli's Theorem
Velocity of fluid flowing from an orifice
v = \sqrt{2gh}v (m/s)g (m/s²)h (m)
Fluids / Pipe Flow
Minor Head Loss
Head loss through fittings and valves
h_m = K \frac{v^2}{2g}hₘ (m)K v (m/s)g (m/s²)
Fluids / Fluid Dynamics
Drag Force
Aerodynamic or hydrodynamic drag force
F_D = \frac{1}{2} \rho v^2 C_D AFᴅ (N)ρ (kg/m³)v (m/s)Cᴅ
Fluids / Fluid Dynamics
Lift Force
Aerodynamic or hydrodynamic lift force
F_L = \frac{1}{2} \rho v^2 C_L AFₗ (N)ρ (kg/m³)v (m/s)Cₗ
Fluids / Fluid Dynamics
Froude Number
Ratio of inertial to gravitational forces
Fr = \frac{v}{\sqrt{gL}}Fr v (m/s)g (m/s²)L (m)
Fluids / Fluid Dynamics
Mach Number
Ratio of flow velocity to speed of sound
Ma = \frac{v}{a}Ma v (m/s)a (m/s)
Fluids / Pipe Flow
Hagen-Poiseuille Equation
Volumetric flow rate for laminar pipe flow
Q = \frac{\pi \Delta P d^4}{128 \mu L}Q (m³/s)ΔP (Pa)d (m)μ (Pa·s)
Materials / Failure Theories
Tresca Criterion (Max Shear Stress)
Maximum shear stress failure criterion
\tau_{max} = \frac{\sigma_1 - \sigma_3}{2}τₘₐₓ (Pa)σ₁ (Pa)σ₃ (Pa)
Materials / Stress & Strain
Stress Concentration
Maximum stress at geometric discontinuity
\sigma_{max} = K_t \sigma_{nom}σₘₐₓ (Pa)Kₜ σₙₒₘ (Pa)
Materials / Fatigue
Goodman Diagram (Fatigue)
Modified Goodman line for fatigue design
\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1σₐ (Pa)σₘ (Pa)Sₑ (Pa)Sᵤₜ (Pa)
Materials / Fatigue
Miner's Rule (Cumulative Damage)
Linear damage accumulation rule
\sum \frac{n_i}{N_i} = 1nᵢ Nᵢ
Materials / Failure Theories
Fracture Toughness (Griffith)
Mode I stress intensity factor
K_I = Y \sigma \sqrt{\pi a}Kᵢ (MPa·√m)Y σ (Pa)a (m)
Materials / Stress & Strain
True Strain
True strain from engineering strain
\varepsilon_t = \ln(1 + \varepsilon_e)εₜ εₑ