Mechanics of Materials formula sheet — printable

Normal (Axial) Stress

\sigma = \frac{P}{A}

Normal Strain (ε = δ/L)

\varepsilon = \frac{\delta}{L}

Young's Modulus (E = σ/ε)

E = \frac{\sigma}{\varepsilon}

Axial Deformation (δ = PL/AE)

\delta = \frac{P L}{A E}

Average Shear Stress (τ = V/A)

\tau = \frac{V}{A}

Shear Modulus (G = τ/γ)

G = \frac{\tau}{\gamma}

Poisson's Ratio

\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Relation Between E, G and ν

E = 2G(1 + \nu)

Factor of Safety

FS = \frac{\sigma_{u}}{\sigma_{allow}}

Thermal Stress in a Restrained Member

\sigma = E \alpha \Delta T

Polar Moment of Inertia — Solid Shaft

J = \frac{\pi d^{4}}{32}

Torsional Shear Stress (τ = Tr/J)

\tau = \frac{T r}{J}

Angle of Twist (φ = TL/JG)

\varphi = \frac{T L}{J G}

Shaft Torque from Power and Angular Speed

T = \frac{P}{\omega}

Rotational Power (P = τω)

P = \tau \omega

Shaft Diameter from Allowable Torsional Shear

d = \sqrt[3]{\frac{16 T}{\pi \tau}}

Shear Stress in a Parallel Key

\tau = \frac{2T}{d \, w \, L}

Area Moment of Inertia — Rectangle

I = \frac{b h^{3}}{12}

Area Moment of Inertia — Solid Round Bar

I = \frac{\pi d^{4}}{64}

Parallel Axis Theorem (I = I_c + Ad²)

I = I_c + A d^{2}

Moment of Inertia — I-Beam or Built-Up Section

I = \frac{B H^{3} - (B - t_w)(H - 2t_f)^{3}}{12}

Elastic Section Modulus (S = I/c)

S = \frac{I}{c}

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Max Bending Moment — Uniform Load

M = \frac{w L^{2}}{8}

Bending Stress (σ = Mc/I)

\sigma = \frac{M c}{I}

Bending Stress from Section Modulus (σ = M/S)

\sigma = \frac{M}{S}

Transverse Shear Stress (τ = VQ/Ib)

\tau = \frac{V Q}{I b}

Shear Flow (q = VQ/I)

q = \frac{V Q}{I}

Beam Deflection — Simply Supported, Centre Load

\delta = \frac{P L^{3}}{48 E I}

Beam Deflection — Simply Supported, Uniform Load

\delta = \frac{5 w L^{4}}{384 E I}

Cantilever Deflection — End Load

\delta = \frac{P L^{3}}{3 E I}

Cantilever Deflection — Uniform Load

\delta = \frac{w L^{4}}{8 E I}

Radius of Gyration (r = √(I/A))

r = \sqrt{\frac{I}{A}}

Slenderness Ratio (KL/r)

\lambda = \frac{K L}{r}

Euler Critical Buckling Load

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

Hoop Stress in a Thin-Walled Cylinder

\sigma_{h} = \frac{p d}{2 t}

Longitudinal Stress in a Thin-Walled Cylinder

\sigma_{l} = \frac{p d}{4 t}

Combined Axial and Bending Stress

\sigma = \frac{P}{A} + \frac{M c}{I}

Maximum Principal Stress (Mohr's Circle)

\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Minimum Principal Stress (Mohr's Circle)

\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Maximum In-Plane Shear Stress

\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Stress Concentration (σmax = Kt σnom)

\sigma_{max} = K_t \, \sigma_{nom}

Thread Tensile Stress Area

A_{t} = \frac{\pi}{4} \left( d - k_{t} p \right)^{2}

Bolt Preload from Torque (T = KDF)

T = K D F

Joint Stiffness Ratio of a Bolted Joint

C = \frac{k_{b}}{k_{b} + k_{m}}

External Load That Separates a Preloaded Joint

P_{0} = \frac{F_{i}}{1 - C}

Fillet Weld Effective Throat

a = 0.707 \, z

Fillet Weld Capacity from Throat Area

F = \tau \, A_t

Fillet Weld Size for a Load per Unit Length

w = \frac{f}{0.707 \, \tau_{a}}

Shear Stress on a Fillet Weld Throat

\tau = \frac{F}{0.707 \, w L}

Welding Heat Input

H = \frac{\eta \, V \, I}{S}

Carbon Equivalent (IIW)

CE = C + \frac{Mn}{6} + \frac{Cr + Mo + V}{5} + \frac{Ni + Cu}{15}

Cutting Speed and Spindle Speed

V = \pi \, D \, N

Milling Table Feed Rate

v_f = N \, z \, f_z

Material Removal Rate — Turning

Q = V \, f \, a_p

Machining Time — Turning Pass

t_m = \frac{L}{f \, N}

Taylor Tool Life Equation

V \, T^{\,n} = C

Vickers Hardness

HV = \dfrac{2F \sin(136^\circ/2)}{d^{2}} = \dfrac{1.8544\,F}{d^{2}}

Hall–Petch Relation

\sigma_y = \sigma_0 + k_y \, d^{-1/2}

True Stress from Engineering Stress

\sigma_t = \sigma_e \, (1 + e), \qquad \varepsilon_t = \ln(1 + e)

Goodman Fatigue Criterion

\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}

Basquin S-N Relation

\sigma_a = \sigma_f' \, (2N_f)^{b}

Miner's Cumulative Damage Rule (Three Blocks)

D = \frac{n_1}{N_1} + \frac{n_2}{N_2} + \frac{n_3}{N_3}

Stress Intensity Factor

K = Y \, \sigma \, \sqrt{\pi a}

Paris Law Crack Growth Rate

\dfrac{da}{dN} = C \left( \Delta K \right)^{m}

Plastic Section Modulus — Rectangle

Z = \frac{b h^{2}}{4}

Plastic Moment Capacity (Mp = Z fy)

M_p = Z f_y

LRFD Load Combination (1.2D + 1.6L)

U = 1.2 D + 1.6 L

Elastic Modulus from Compressive Strength

E_c = k \sqrt{f'_c}