taueng עברית

Strength of materials formula sheet

The formulas of a first mechanics of materials course in one place, ready to print. Each row is a quantity and its formula, grouped by type of load.

Axial load and material relations

Quantity	Formula
Normal stress	$σ = \frac{F}{A}$
Strain	$ε = \frac{δ}{L}$
Hooke's law	$σ = E ε$
Axial elongation	$δ = \frac{F L}{E A}$
Thermal elongation	$δ_{T} = α Δ T L$
Poisson's ratio	$ν = - \frac{ε _{lat}}{ε _{axial}}$
Shear modulus	$G = \frac{E}{2 ( 1 + ν )}$

Torsion

Quantity	Formula
Torsional shear stress	$τ = \frac{T r}{J}$
Angle of twist	$φ = \frac{T L}{G J}$
J, solid shaft	$J = \frac{π d ^{4}}{32}$
J, hollow shaft	$J = \frac{π ( D ^{4} - d ^{4} )}{32}$
Power	$P = T ω$

Bending

Quantity	Formula
Bending stress	$σ = - \frac{M y}{I}$
Maximum bending stress	$σ_{m a x} = \frac{M c}{I} = \frac{M}{S}$
I, rectangle	$I = \frac{b h ^{3}}{12}$
I, circle	$I = \frac{π d ^{4}}{64}$
Shear stress in a beam	$τ = \frac{V Q}{I t}$
Elastic curve	$E I \frac{d ^{2} y}{d x ^{2}} = M (x)$

Beam deflections

Quantity	Formula
Cantilever, end load	$δ_{m a x} = \frac{P L ^{3}}{3 E I}$
Cantilever, uniform load	$δ_{m a x} = \frac{w L ^{4}}{8 E I}$
Simply supported, centre load	$δ_{m a x} = \frac{P L ^{3}}{48 E I}$
Simply supported, uniform load	$δ_{m a x} = \frac{5 w L ^{4}}{384 E I}$

Combined stress, failure and buckling

Quantity	Formula
Principal stresses	$σ_{1, 2} = \frac{σ _{x} + σ _{y}}{2} \pm (\frac{σ _{x} - σ _{y}}{2})^{2} + τ_{x y}^{2}$
Maximum in-plane shear	$τ_{m a x} = (\frac{σ _{x} - σ _{y}}{2})^{2} + τ_{x y}^{2}$
Principal angle	$tan 2 θ_{p} = \frac{2 τ _{x y}}{σ _{x} - σ _{y}}$
von Mises (plane stress)	$σ_{v} = σ_{1}^{2} - σ_{1} σ_{2} + σ_{2}^{2}$
Tresca	$τ_{m a x} = \frac{σ _{y}}{2}$
Euler buckling	$P_{cr} = \frac{π ^{2} E I}{( K L ) ^{2}}$
Thin-walled vessel, hoop	$σ_{h} = \frac{p r}{t}$
Thin-walled vessel, axial	$σ_{a} = \frac{p r}{2 t}$

Notes

In Euler buckling, K is the effective-length factor: 1 for pinned–pinned, 2 for a cantilever, 0.5 for fixed–fixed.

The sign of bending stress depends on convention: with a positive moment, fibres above the neutral axis are in compression.

Before an exam, check your course syllabus for whether a formula sheet is allowed and which.