Engineering Formulas

Master engineering with formulas for mechanical, electrical, and civil engineering, including statics and dynamics.

📌 Mechanical: Statics

Equilibrium of Forces:
\( \sum F_x = 0, \sum F_y = 0 \)
Equilibrium of Moments:
\( \sum M = 0 \)
Force Magnitude:
\( F = \sqrt{F_x^2 + F_y^2} \)
Angle of Force:
\( \theta = \tan^{-1} \left( \frac{F_y}{F_x} \right) \)
Moment (Torque):
\( M = F d \sin \theta \)
Centroid (x-coordinate):
\( \bar{x} = \frac{\sum (x_i A_i)}{\sum A_i} \)
Centroid (y-coordinate):
\( \bar{y} = \frac{\sum (y_i A_i)}{\sum A_i} \)
Friction Force:
\( F_f = \mu N \)
Resultant Force (Vector Sum):
\( F_R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2} \)
Truss Reaction (Method of Joints):
\( \sum F = 0 \) at each joint
Beam Shear Force:
\( V = \sum F_{\text{vertical}} \)

📌 Mechanical: Dynamics

Velocity:
\( v = \frac{\Delta x}{\Delta t} \)
Acceleration:
\( a = \frac{\Delta v}{\Delta t} \)
Newton’s Second Law:
\( F = m a \)
Kinematic Equation (Constant \( a \)):
\( v = v_0 + a t \)
Displacement (Constant \( a \)):
\( x = x_0 + v_0 t + \frac{1}{2} a t^2 \)
Velocity Squared:
\( v^2 = v_0^2 + 2 a (x - x_0) \)
Work:
\( W = F d \cos \theta \)
Kinetic Energy:
\( KE = \frac{1}{2} m v^2 \)
Potential Energy (Gravity):
\( PE = m g h \)
Power:
\( P = \frac{W}{t} = F v \cos \theta \)
Angular Velocity:
\( \omega = \frac{\Delta \theta}{\Delta t} \)
Centripetal Acceleration:
\( a_c = \frac{v^2}{r} = \omega^2 r \)

📌 Mechanical: Thermodynamics

Ideal Gas Law:
\( PV = nRT \)
First Law of Thermodynamics:
\( \Delta U = Q - W \)
Work (Constant Pressure):
\( W = P \Delta V \)
Enthalpy:
\( H = U + PV \)
Specific Heat (Constant Volume):
\( Q = m c_v \Delta T \)
Specific Heat (Constant Pressure):
\( Q = m c_p \Delta T \)
Efficiency (Carnot):
\( \eta = 1 - \frac{T_C}{T_H} \)
Heat Transfer (Conduction):
\( Q = k A \frac{\Delta T}{L} t \)
Entropy Change:
\( \Delta S = \frac{Q_{\text{rev}}}{T} \)
Thermal Expansion:
\( \Delta L = L_0 \alpha \Delta T \)
Gas Work (Isothermal):
\( W = nRT \ln \left( \frac{V_2}{V_1} \right) \)

📌 Mechanical: Materials

Stress:
\( \sigma = \frac{F}{A} \)
Strain:
\( \epsilon = \frac{\Delta L}{L_0} \)
Young’s Modulus:
\( E = \frac{\sigma}{\epsilon} \)
Shear Stress:
\( \tau = \frac{F}{A} \)
Shear Strain:
\( \gamma = \tan \theta \approx \theta \) (small angles)
Shear Modulus:
\( G = \frac{\tau}{\gamma} \)
Poisson’s Ratio:
\( \nu = -\frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{longitudinal}}} \)
Bulk Modulus:
\( K = -\frac{\Delta P}{\Delta V / V} \)
Torsional Stress:
\( \tau = \frac{T r}{J} \) (where \( J \) is polar moment)
Bending Stress:
\( \sigma = \frac{M c}{I} \) (where \( I \) is moment of inertia)
Moment of Inertia (Rectangle):
\( I = \frac{b h^3}{12} \) (about centroidal axis)

📌 Electrical: Circuits

Ohm’s Law:
\( V = I R \)
Power (Electrical):
\( P = V I = I^2 R = \frac{V^2}{R} \)
Resistors in Series:
\( R_{\text{eq}} = R_1 + R_2 + \cdots + R_n \)
Resistors in Parallel:
\( \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \)
Kirchhoff’s Current Law (KCL):
\( \sum I_{\text{in}} = \sum I_{\text{out}} \)
Kirchhoff’s Voltage Law (KVL):
\( \sum V = 0 \) (around a loop)
Capacitance:
\( C = \frac{Q}{V} \)
Capacitors in Parallel:
\( C_{\text{eq}} = C_1 + C_2 + \cdots + C_n \)
Capacitors in Series:
\( \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n} \)
RC Time Constant:
\( \tau = R C \)
Current in RC Circuit (Charging):
\( I(t) = \frac{V}{R} e^{-t / \tau} \)

📌 Electrical: Power

AC Power (Average):
\( P = V_{\text{rms}} I_{\text{rms}} \cos \phi \)
RMS Voltage:
\( V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \)
RMS Current:
\( I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} \)
Power Factor:
\( \cos \phi = \frac{R}{Z} \) (where \( Z \) is impedance)
Impedance (RLC Circuit):
\( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
Inductive Reactance:
\( X_L = \omega L = 2\pi f L \)
Capacitive Reactance:
\( X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \)
Resonant Frequency:
\( f_0 = \frac{1}{2\pi \sqrt{L C}} \)
Transformer Voltage Ratio:
\( \frac{V_{\text{secondary}}}{V_{\text{primary}}} = \frac{N_{\text{secondary}}}{N_{\text{primary}}} \)
Three-Phase Power:
\( P = \sqrt{3} V_L I_L \cos \phi \) (line values)
Efficiency:
\( \eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% \)

📌 Civil: Structural Analysis

Beam Deflection (Cantilever, Point Load):
\( \delta = \frac{F L^3}{3 E I} \)
Beam Deflection (Simply Supported, Point Load):
\( \delta = \frac{F L^3}{48 E I} \) (at center)
Moment of Inertia (Circle):
\( I = \frac{\pi r^4}{4} \)
Shear Stress in Beam:
\( \tau = \frac{V Q}{I b} \) (where \( Q \) is first moment)
Buckling Load (Euler):
\( P_{cr} = \frac{\pi^2 E I}{L_e^2} \) (where \( L_e \) is effective length)
Slope of Beam (Point Load):
\( \theta = \frac{F L^2}{2 E I} \) (cantilever)
Reaction Force (Statically Determinate):
\( R = \sum F / \text{supports} \)
Truss Member Force (Method of Sections):
\( F = \frac{M}{d} \) (moment about cut point)
Column Slenderness Ratio:
\( \lambda = \frac{L_e}{r} \) (where \( r = \sqrt{I/A} \))
Stress Concentration Factor:
\( K_t = \frac{\sigma_{\text{max}}}{\sigma_{\text{nominal}}} \)

📌 Civil: Fluid Mechanics

Continuity Equation:
\( A_1 v_1 = A_2 v_2 \)
Bernoulli’s Equation:
\( P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \)
Hydrostatic Pressure:
\( P = \rho g h \)
Buoyant Force:
\( F_b = \rho g V_{\text{submerged}} \)
Reynolds Number:
\( Re = \frac{\rho v D}{\mu} \) (pipe flow)
Head Loss (Darcy-Weisbach):
\( h_L = f \frac{L}{D} \frac{v^2}{2g} \)
Friction Factor (Laminar):
\( f = \frac{64}{Re} \) (\( Re < 2000 \))
Pipe Flow Rate:
\( Q = A v \)
Drag Force:
\( F_D = \frac{1}{2} \rho v^2 C_D A \)
Lift Force:
\( F_L = \frac{1}{2} \rho v^2 C_L A \)
Manning’s Equation (Open Channel):
\( v = \frac{1}{n} R_h^{2/3} S^{1/2} \)

📌 Control Systems

Transfer Function:
\( G(s) = \frac{Y(s)}{U(s)} \)
First-Order System Response:
\( y(t) = y_{\text{ss}} (1 - e^{-t/\tau}) \) (step response)
Time Constant:
\( \tau = \frac{1}{a} \) (for \( \dot{y} + a y = b u \))
Second-Order System:
\( G(s) = \frac{\omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2} \)
Natural Frequency:
\( \omega_n = \sqrt{\frac{k}{m}} \) (mass spring system)
Damping Ratio:
\( \zeta = \frac{c}{2 \sqrt{k m}} \) (where \( c \) is damping coefficient)
Overshoot:
\( MP = e^{-\frac{\zeta \pi}{\sqrt{1 - \zeta^2}}} \times 100\% \)
Settling Time (2%):
\( t_s = \frac{4}{\zeta \omega_n} \)
Peak Time:
\( t_p = \frac{\pi}{\omega_n \sqrt{1 - \zeta^2}} \)
Steady-State Error (Step Input):
\( e_{ss} = \frac{1}{1 + K_p} \) (where \( K_p \) is position constant)
Laplace Transform (Derivative):
\( \mathcal{L}\{ \dot{y}(t) \} = s Y(s) - y(0) \)