Formula sheet

Every key formula from the finished topics, grouped by branch. Use the Print button for a handout.

Conventions used. Tension and tensile stress are positive. A positive (sagging) bending moment makes a beam curve like a smile. Unless stated otherwise, SI units are used: forces in N or kN, lengths in m or mm, stresses in Pa, MPa or GPa.

Engineering Mechanics

Resultant of a force system

Components of a force: $F_x = F\cos\theta, \quad F_y = F\sin\theta$
Components of the resultant: $R_x = \sum F_x, \quad R_y = \sum F_y$
Magnitude of the resultant: $R = \sqrt{R_x^2 + R_y^2}$
Direction of the resultant: $\theta = \tan^{-1}\frac{R_y}{R_x}$

Equilibrium and support reactions

Force equilibrium: $\sum F_x = 0, \quad \sum F_y = 0$
Moment equilibrium: $\sum M_A = 0$
Equivalent load of a UDL: $W = wL, \quad \text{acting at mid-length}$
Reaction of a simply supported beam: $R_B = \frac{\sum (\text{load} \times \text{distance from A})}{L}$

Centroid and second moment of area

Centroid of a composite area: $\bar{y} = \frac{\sum A_i y_i}{\sum A_i}$
Second moment of a rectangle (about its own centroid): $I = \frac{bh^3}{12}$
Parallel-axis theorem: $I = \sum \left( I_i + A_i d_i^2 \right)$

Dry friction and the friction angle

Limiting static friction: $F_{\max} = \mu_s N$
Kinetic friction: $F = \mu_k N$
Angle of repose: $\tan\phi_s = \mu_s$
Acceleration down a rough slope: $a = g\left(\sin\theta - \mu_k\cos\theta\right)$

Strength of Materials

Normal and shear stress, strain

Normal stress: $\sigma = \frac{P}{A}$
Shear stress (average): $\tau = \frac{V}{A}$
Normal strain: $\varepsilon = \frac{\delta}{L}$
Hooke's law: $\sigma = E\varepsilon$
Factor of safety: $\text{FoS} = \frac{\sigma_{\text{y}}}{\sigma_{\text{actual}}}$

The stress-strain diagram

Elastic modulus (slope of the straight part): $E = \frac{\sigma}{\varepsilon}$
Percent elongation: $\%\,\text{elongation} = \frac{L_f - L_0}{L_0}\times 100$
Permanent strain after unloading: $\varepsilon_p = \varepsilon - \frac{\sigma}{E}$

Axial deformation of bars

Axial deformation: $\delta = \frac{PL}{AE}$
Free thermal expansion: $\delta_T = \alpha L \,\Delta T$
Thermal stress (fully restrained bar): $\sigma = -E\alpha\,\Delta T$

Torsion of circular shafts

Polar moment, solid shaft: $J = \frac{\pi d^4}{32}$
Polar moment, hollow shaft: $J = \frac{\pi}{32}\left(d_o^4 - d_i^4\right)$
Torsion formula: $\tau = \frac{T\rho}{J}, \quad \tau_{\text{max}} = \frac{Tr}{J}$
Angle of twist: $\phi = \frac{TL}{GJ}$

Shear force and bending moment diagrams

Shear force at a section: $V = \sum F_{\text{up, left}} - \sum F_{\text{down, left}}$
Load–shear relation: $\frac{\mathrm{d}V}{\mathrm{d}x} = -w$
Shear–moment relation: $\frac{\mathrm{d}M}{\mathrm{d}x} = V$
Simply supported beam, central point load: $M_{\text{max}} = \frac{PL}{4}$
Simply supported beam, full-span UDL: $M_{\text{max}} = \frac{wL^2}{8}$
Cantilever, end load or full UDL: $M_{\text{max}} = PL \;\text{or}\; \frac{wL^2}{2}$

Bending stress: the flexure formula

Flexure formula: $\sigma = \frac{My}{I}$
Maximum bending stress: $\sigma_{\text{max}} = \frac{Mc}{I} = \frac{M}{Z}$
Section modulus: $Z = \frac{I}{c}$
Rectangular section: $Z = \frac{bh^2}{6}$

Beam deflection

Moment–curvature: $EI\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = M$
Simply supported, central point load: $\delta_{\text{max}} = \frac{PL^3}{48EI}$
Simply supported, full-span UDL: $\delta_{\text{max}} = \frac{5wL^4}{384EI}$
Cantilever, end load: $\delta_{\text{max}} = \frac{PL^3}{3EI}$
Cantilever, full-span UDL: $\delta_{\text{max}} = \frac{wL^4}{8EI}$

Stress transformation and Mohr's circle

Stress on a plane at angle θ: $\sigma_\theta = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta$
Shear stress on that plane: $\tau_\theta = -\frac{\sigma_x - \sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta$
Principal stresses: $\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$
Principal direction: $\tan 2\theta_p = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$
Maximum shear stress: $\tau_{\text{max}} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

Thin-walled pressure vessels

Cylinder, hoop stress: $\sigma_h = \frac{pd}{2t}$
Cylinder, longitudinal stress: $\sigma_L = \frac{pd}{4t}$
Sphere (any direction): $\sigma = \frac{pd}{4t}$
Thin-wall condition: $\frac{d}{t} \ge 20$

Buckling of columns (Euler)

Euler critical load: $P_{\text{cr}} = \frac{\pi^2 EI}{(KL)^2}$
Radius of gyration: $r = \sqrt{\frac{I}{A}}$
Slenderness ratio: $\lambda = \frac{KL}{r}$
Critical stress: $\sigma_{\text{cr}} = \frac{\pi^2 E}{\lambda^2}$

Structural Analysis

Trusses: method of joints

Joint equilibrium: $\sum F_x = 0, \quad \sum F_y = 0$
Determinacy check (m members, r reactions, j joints): $m + r = 2j$
Force components of a member: $F_x = F\cos\theta, \quad F_y = F\sin\theta$

Trusses: method of sections

Moment equilibrium about a point on the section: $\sum M_{\text{point}} = 0$
Vertical force equilibrium of the cut part: $\sum F_y = 0$
Number of unknowns on the cut: $\text{at most } 3$

Fluid Mechanics

Hydrostatic pressure and force on surfaces

Pressure at depth h: $p = \rho g h$
Resultant force on a vertical rectangle: $F = \frac{1}{2}\rho g h^2 b$
Centre of pressure (above the base): $y_b = \frac{h}{3}$
Overturning moment about the base: $M = F\cdot\frac{h}{3}$

Continuity and Bernoulli's equation

Continuity (incompressible): $A_1 v_1 = A_2 v_2 = Q$
Bernoulli's equation: $\frac{p_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{p_2}{\rho g} + \frac{v_2^2}{2g} + z_2$
Venturi pressure drop: $p_1 - p_2 = \frac{1}{2}\rho\left(v_2^2 - v_1^2\right)$

Thermodynamics

The first law and ideal gas processes

First law (closed system): $Q = \Delta U + W$
Ideal gas law: $PV = nRT$
Isothermal work: $W = P_1 V_1 \ln\frac{V_2}{V_1}$
Adiabatic relation: $P V^{\gamma} = \text{constant}, \quad \gamma = \frac{c_p}{c_v}$
Adiabatic work: $W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$

The Otto cycle

Compression ratio: $r = \frac{v_1}{v_2}$
Temperatures after adiabatic processes: $T_2 = T_1 r^{\gamma - 1}, \quad T_4 = T_3 r^{1 - \gamma}$
Heat added at constant volume: $q_{\text{in}} = c_v (T_3 - T_2)$
Thermal efficiency: $\eta = 1 - \frac{1}{r^{\gamma - 1}}$
Net work: $w_{\text{net}} = \eta\, q_{\text{in}}$

Electrical Circuits

Ohm's law and resistor networks

Ohm's law: $V = IR$
Power: $P = VI = I^2 R = \frac{V^2}{R}$
Series resistance: $R_{\text{eq}} = R_1 + R_2 + R_3$
Parallel resistance: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

RC circuit transient response

Time constant: $\tau = RC$
Charging voltage: $v_C(t) = V\left(1 - e^{-t/\tau}\right)$
Discharging voltage: $v_C(t) = V\,e^{-t/\tau}$
Current while charging: $i(t) = \frac{V}{R}\,e^{-t/\tau}$
Energy stored: $E = \tfrac{1}{2} C v_C^2$

Series RLC resonance

Inductive reactance: $X_L = 2\pi f L$
Capacitive reactance: $X_C = \frac{1}{2\pi f C}$
Impedance: $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$
Resonant frequency: $f_0 = \frac{1}{2\pi\sqrt{LC}}$
Quality factor and bandwidth: $Q = \frac{1}{R}\sqrt{\frac{L}{C}}, \quad \text{BW} = \frac{f_0}{Q}$
Phase angle: $\varphi = \tan^{-1}\frac{X_L - X_C}{R}$