Formulas

Engineering

Mechanical Engineering: Mechanics & Materials

Stress (Engineering Stress)
σ = F / A₀
σ Stress
F Applied Force
A₀ Original Cross Sectional Area

Strain (Engineering Strain)
ε = ΔL / L₀
ε Strain
ΔL Change in Length
L₀ Original Length

Young’s Modulus (Modulus of Elasticity)
E = σ / ε
E Young’s Modulus
σ Stress
ε Strain

Shear Modulus (Modulus of Rigidity)
G = τ / γ
G Shear Modulus
τ Shear Stress
γ Shear Strain

Bulk Modulus
K = -V (ΔP / ΔV)
K Bulk Modulus
V Original Volume
ΔP Pressure Change
ΔV Volume Change

Poisson’s Ratio
ν = -ε_lateral / ε_axial
ν Poisson’s Ratio
ε_lateral Lateral Strain
ε_axial Axial Strain

Strain Energy (Axial Loading)
U = (σ² / 2E) * V
U Strain Energy
σ Stress
E Young’s Modulus
V Volume

Torsion Formula (Shear Stress in a Shaft)
τ = T * r / J
τ Shear Stress
T Applied Torque
r Radial Distance from the Center
J Polar Moment of Inertia

Polar Moment of Inertia (Solid Shaft)
J = π * d⁴ / 32
J Polar Moment of Inertia
d Diameter of the Shaft

Area Moment of Inertia (Rectangular Beam)
I = b * h³ / 12
I Area Moment of Inertia
b Base Width
h Height of the Beam

Bending Stress (Euler-Bernoulli Beam Theory)
σ = M * y / I
σ Bending Stress
M Bending Moment
y Distance from the Neutral Axis
I Area Moment of Inertia

Deflection of a Simply Supported Beam (Central Point Load)
δ = (P * L³) / (48 * E * I)
δ Deflection
P Point Load
L Length of the Beam
E Young’s Modulus
I Area Moment of Inertia

Critical Buckling Load (Euler’s Formula)
P_cr = π² * E * I / (K * L)²
P_cr Critical Buckling Load
E Young’s Modulus
I Area Moment of Inertia
K Column Effective Length Factor
L Actual Length

Mohr’s Circle for Stress (Radius)
R = √[((σ_x – σ_y)/2)² + τ_xy²]
R Radius of Mohr’s Circle
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress

Von Mises Stress (Ductile Materials)
σ’ = √[σ_x² – σ_xσ_y + σ_y² + 3τ_xy²]
σ’ Von Mises Stress
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress

Mechanical Engineering: Thermodynamics

First Law of Thermodynamics (Closed System)
ΔU = Q – W
ΔU Change in Internal Energy
Q Heat Added to the System
W Work Done by the System

Enthalpy
h = u + P * v
h Specific Enthalpy
u Specific Internal Energy
P Pressure
v Specific Volume

Heat Transfer by Conduction (Fourier’s Law)
Q = -k * A * (dT/dx)
Q Heat Transfer Rate
k Thermal Conductivity
A Cross Sectional Area
dT/dx Temperature Gradient

Heat Transfer by Convection (Newton’s Law of Cooling)
Q = h * A * ΔT
Q Heat Transfer Rate
h Convective Heat Transfer Coefficient
A Surface Area
ΔT Temperature Difference

Heat Exchanger Effectiveness (NTU Method)
ε = (1 – exp[-NTU(1-C_r)]) / (1 – C_r * exp[-NTU(1-C_r)])
ε Effectiveness
NTU Number of Transfer Units
C_r Heat Capacity Rate Ratio C_min C_max

Log Mean Temperature Difference (LMTD)
ΔT_lm = (ΔT₁ – ΔT₂) / ln(ΔT₁ / ΔT₂)
ΔT_lm Log Mean Temperature Difference
ΔT₁ Temperature Difference at End One
ΔT₂ Temperature Difference at End Two

Isentropic Efficiency (Turbine)
η_t = (h₁ – h₂) / (h₁ – h₂s)
η_t Isentropic Efficiency
h₁ Inlet Enthalpy
h₂ Actual Outlet Enthalpy
h₂s Isentropic Outlet Enthalpy

Coefficient of Performance (Refrigerator)
COP_R = Q_L / W_in
COP_R Coefficient of Performance
Q_L Heat Removed from the Cold Reservoir
W_in Work Input

Carnot Efficiency (Heat Engine)
η_th = 1 – T_C / T_H
η_th Theoretical Carnot Efficiency
T_C Cold Reservoir Temperature
T_H Hot Reservoir Temperature

Mass Flow Rate
ṁ = ρ * A * V
ṁ Mass Flow Rate
ρ Density
A Cross Sectional Area
V Flow Velocity

Mechanical Engineering: Fluid Mechanics

Reynolds Number
Re = (ρ * V * L) / μ
Re Reynolds Number
ρ Density
V Velocity
L Characteristic Length
μ Dynamic Viscosity

Darcy-Weisbach Equation (Head Loss)
h_f = f * (L/D) * (V² / (2g))
h_f Head Loss Due to Friction
f Darcy Friction Factor
L Pipe Length
D Pipe Diameter
V Velocity
g Gravity

Drag Force
F_D = (1/2) * ρ * V² * A * C_D
F_D Drag Force
ρ Density
V Velocity
A Reference Area
C_D Drag Coefficient

Lift Force
F_L = (1/2) * ρ * V² * A * C_L
F_L Lift Force
ρ Density
V Velocity
A Reference Area e g Wing Area
C_L Lift Coefficient

Bernoulli’s Equation (Incompressible Flow)
P₁ + (1/2)ρV₁² + ρgh₁ = P₂ + (1/2)ρV₂² + ρgh₂
P Pressure
ρ Density
V Velocity
g Gravity
h Elevation Height

Dynamic Pressure
q = (1/2) * ρ * V²
q Dynamic Pressure
ρ Density
V Velocity

Viscous Shear Stress (Newton’s Law of Viscosity)
τ = μ * (du/dy)
τ Shear Stress
μ Dynamic Viscosity
du/dy Velocity Gradient Perpendicular to the Flow

Hagen–Poiseuille Equation (Laminar Flow in a Pipe)
Q = (π * ΔP * r⁴) / (8 * μ * L)
Q Volumetric Flow Rate
ΔP Pressure Drop
r Pipe Radius
μ Dynamic Viscosity
L Pipe Length

Pump Hydraulic Power
W_pump = ṁ * g * H
W_pump Hydraulic Power
ṁ Mass Flow Rate
g Gravity
H Pump Head

Mach Number
M = V / a
M Mach Number
V Velocity of the Object
a Speed of Sound in the Medium

Electrical Engineering: Circuits & Power

Ohm’s Law
V = I * R
V Voltage
I Current
R Resistance

Electrical Power (DC)
P = V * I
P Power
V Voltage
I Current

Joule’s Law (Resistive Heating)
P = I² * R
P Power Dissipated as Heat
I Current
R Resistance

Capacitance
C = Q / V
C Capacitance
Q Charge Stored
V Voltage

Inductance
V = L * (di/dt)
V Voltage Across the Inductor
L Inductance
di/dt Rate of Change of Current

Impedance (RL Series Circuit)
Z = √(R² + (ωL)²)
Z Impedance
R Resistance
ω Angular Frequency
L Inductance

Resonant Frequency (LC Circuit)
f_r = 1 / (2π √(L C))
f_r Resonant Frequency
L Inductance
C Capacitance

Real Power (AC Circuit)
P = V_rms * I_rms * cos(θ)
P Real Power
V_rms RMS Voltage
I_rms RMS Current
θ Phase Angle Between Voltage and Current

Reactive Power
Q = V_rms * I_rms * sin(θ)
Q Reactive Power
V_rms RMS Voltage
I_rms RMS Current
θ Phase Angle Between Voltage and Current

Apparent Power
S = V_rms * I_rms
S Apparent Power
V_rms RMS Voltage
I_rms RMS Current

Electrical Engineering: Electromagnetics & Machines

Force on a Current Carrying Conductor
F = B * I * L * sin(θ)
F Force
B Magnetic Flux Density
I Current
L Conductor Length
θ Angle Between B and I

Magnetic Flux
Φ_B = B * A * cos(θ)
Φ_B Magnetic Flux
B Magnetic Flux Density
A Area
θ Angle Between B and the Normal to A

Transformer Equation (Turns Ratio)
V_s / V_p = N_s / N_p
V_s Secondary Voltage
V_p Primary Voltage
N_s Number of Secondary Turns
N_p Number of Primary Turns

Back EMF in a DC Motor
E_b = V – I_a * R_a
E_b Back Electromotive Force
V Terminal Voltage
I_a Armature Current
R_a Armature Resistance

Synchronous Speed (AC Motor)
N_s = (120 * f) / P
N_s Synchronous Speed in RPM
f Line Frequency
P Number of Magnetic Poles

Motor Slip
s = (N_s – N_r) / N_s
s Slip
N_s Synchronous Speed
N_r Rotor Speed

Power Factor
PF = cos(θ) = P / S
PF Power Factor
θ Phase Angle
P Real Power
S Apparent Power

Energy Consumption (kWh)
E = P * t / 1000
E Energy in Kilowatt Hours
P Power in Watts
t Time in Hours

Charge and Current
I = dQ / dt
I Current
dQ Change in Charge
dt Change in Time

Skin Depth
δ = √(2 / (ω * μ * σ))
δ Skin Depth
ω Angular Frequency
μ Permeability
σ Conductivity

Civil & Structural Engineering

Allowable Stress (Factor of Safety)
σ_allow = σ_yield / FoS
σ_allow Allowable Stress
σ_yield Yield Stress
FoS Factor of Safety

Bearing Capacity (Terzaghi’s Equation)
q_u = c*N_c + q*N_q + 0.5*γ*B*N_γ
q_u Ultimate Bearing Capacity
c Cohesion
q Surcharge
γ Soil Unit Weight
B Footing Width
N_c N_q N_γ Bearing Capacity Factors

Vertical Stress Under a Point Load (Boussinesq)
Δσ_z = (3P * z³) / (2π * R⁵)
Δσ_z Vertical Stress Increase
P Point Load
z Depth
R Radial Distance from the Load Point

Flow Rate (Manning’s Equation)
V = (1/n) * R_h^(2/3) * S^(1/2)
V Flow Velocity
n Manning’s Roughness Coefficient
R_h Hydraulic Radius
S Channel Slope

Hydraulic Radius
R_h = A / P_w
R_h Hydraulic Radius
A Cross Sectional Area of Flow
P_w Wetted Perimeter

Euler’s Critical Buckling Stress
σ_cr = π² * E / (L_e / r)²
σ_cr Critical Buckling Stress
E Young’s Modulus
L_e Effective Column Length
r Radius of Gyration

Section Modulus
S = I / y_max
S Section Modulus
I Area Moment of Inertia
y_max Distance to the Outermost Fiber

Deflection of a Cantilever Beam (End Load)
δ = (P * L³) / (3 * E * I)
δ Deflection at the End
P Point Load
L Length of the Beam
E Young’s Modulus
I Area Moment of Inertia

Chemical Engineering

Reynolds Number (Pipe Flow)
Re = (ρ * u * d) / μ
Re Reynolds Number
ρ Fluid Density
u Fluid Velocity
d Pipe Diameter
μ Dynamic Viscosity

Friction Factor (Colebrook Equation)
1/√f = -2 log₁₀[(ε/D)/3.7 + 2.51/(Re √f)]
f Darcy Friction Factor
ε Pipe Roughness
D Pipe Diameter
Re Reynolds Number

Ideal Gas Law
P * V = n * R * T
P Absolute Pressure
V Volume
n Number of Moles
R Ideal Gas Constant
T Absolute Temperature

Raoult’s Law (Vapor-Liquid Equilibrium)
P_i = x_i * P_i^*
P_i Partial Pressure of Component i
x_i Mole Fraction in the Liquid
P_i^* Pure Component Vapor Pressure

Henry’s Law
C_i = P_i * H_i
C_i Concentration of Dissolved Gas
P_i Partial Pressure
H_i Henry’s Law Constant

Fick’s First Law (Diffusion)
J = -D * (dc/dx)
J Diffusion Flux
D Diffusion Coefficient
dc/dx Concentration Gradient

Arrhenius Equation (Reaction Rate)
k = A * exp(-E_a / (R T))
k Reaction Rate Constant
A Pre Exponential Factor
E_a Activation Energy
R Gas Constant
T Temperature

Reynolds Transport Theorem
dB_sys/dt = ∂/∂t ∫_cv ρ b dV + ∫_cs ρ b (v · n) dA
B_sys Extensive System Property
b Corresponding Intensive Property
ρ Density
v Velocity
n Normal Vector

Control Systems & Robotics

Transfer Function (General Form)
G(s) = Y(s) / U(s)
G(s) Transfer Function
Y(s) Laplace Transform of the Output
U(s) Laplace Transform of the Input

Damping Ratio (Second-Order System)
ζ = c / (2 √(m k))
ζ Damping Ratio
c Damping Coefficient
m Mass
k Spring Constant

Natural Frequency
ω_n = √(k / m)
ω_n Natural Frequency
k Spring Constant
m Mass

Gain Margin
GM = 1 / |G(jω_pc)|
GM Gain Margin
|G(jω_pc)| Magnitude of the Open Loop Transfer Function at the Phase Crossover Frequency

Phase Margin
PM = 180° + ∠G(jω_gc)
PM Phase Margin
∠G(jω_gc) Phase Angle of the Open Loop Transfer Function at the Gain Crossover Frequency

Engineering Economics

Present Worth Analysis
PW = Σ [F_t / (1 + i)^t]
PW Present Worth
F_t Net Cash Flow in Period t
i Discount Rate
t Time Period

Future Worth Analysis
FW = PW * (1 + i)^n
FW Future Worth
PW Present Worth
i Interest Rate
n Number of Periods

Annual Worth Analysis
AW = PW * [i(1+i)^n / ((1+i)^n – 1)]
AW Annual Worth
PW Present Worth
i Interest Rate
n Number of Periods

Internal Rate of Return (IRR)
0 = Σ [CF_t / (1 + IRR)^t]
CF_t Cash Flow in Time Period t
IRR Internal Rate of Return

Benefit-Cost Ratio
BCR = Σ (Benefits) / Σ (Costs)
BCR Benefit Cost Ratio

General & Interdisciplinary

Root Mean Square (RMS) Value
X_rms = √(1/T ∫_0^T x(t)² dt)
X_rms Root Mean Square Value of the Signal x(t) Over a Period T

Signal-to-Noise Ratio (SNR)
SNR = P_signal / P_noise
SNR Signal to Noise Ratio
P_signal Power of the Signal
P_noise Power of the Noise

Weibull Distribution (Failure Rate)
f(t) = (β/η) (t/η)^{β-1} exp[-(t/η)^β]
f(t) Probability Density Function
t Time
β Shape Parameter
η Scale Parameter

Binomial Coefficient
C(n, k) = n! / (k! (n-k)!)
C(n, k) Number of Combinations
n Total Number of Items
k Number of Items to Choose

Bayes’ Theorem
P(A|B) = [P(B|A) * P(A)] / P(B)
P(A|B) Conditional Probability of A Given B
P(B|A) Probability of B Given A
P(A) Probability of A
P(B) Probability of B

Information Entropy (Shannon)
H = -Σ [p_i * log₂(p_i)]
H Information Entropy
p_i Probability of the i th Possible Value of the Source Symbol

Euler’s Formula
e^(iθ) = cos(θ) + i sin(θ)
e Base of the Natural Logarithm
i Imaginary Unit
θ Angle in Radians
cos Cosine Function
sin Sine Function

Euler’s Identity
e^(iπ) + 1 = 0
e Base of the Natural Logarithm
i Imaginary Unit
π Pi
1 Integer One
0 Integer Zero