Strength of Materials formulas — 60 solvers

Angle of Twist (φ = TL/JG)

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

Strength of MaterialsMechanicsPhysicsAngle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long drive and torque shafts.

Area Moment of Inertia — Rectangle

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

Strength of MaterialsMechanicsGeometryArea moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Area Moment of Inertia — Solid Round Bar

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

Strength of MaterialsMechanicsGeometryArea moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and entered as a plain number.

Average Shear Stress (τ = V/A)

\tau = \frac{V}{A}

Strength of MaterialsMechanicsPhysicsAverage shear stress on a bolt, pin or weld throat: the transverse force divided by the area resisting it, in Pa or psi.

Axial Deformation (δ = PL/AE)

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

Strength of MaterialsMechanicsPhysicsElongation of an axially loaded bar from load, length, area and Young's modulus — the workhorse δ = PL/AE of hanger design.

Basquin S-N Relation

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

Strength of MaterialsMechanicsThe straight line an S-N curve becomes on log-log paper: alternating stress against reversals to failure, with the fatigue strength coefficient and exponent from a materials database.

Beam Deflection — Simply Supported, Centre Load

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

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

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

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Bending Stress (σ = Mc/I)

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

Strength of MaterialsMechanicsPhysicsBending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

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

\sigma = \frac{M}{S}

Strength of MaterialsMechanicsPhysicsBending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Bolt Preload from Torque (T = KDF)

T = K D F

Strength of MaterialsMechanicsPhysicsBolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every published torque spec.

Bulk Modulus (K = ΔP·V₀/ΔV)

K = \frac{\Delta P \, V_0}{\Delta V}

Strength of MaterialsMechanicsPhysicsBulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

Cantilever Deflection — End Load

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

Strength of MaterialsMechanicsPhysicsTip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered in m⁴ as a plain number.

Cantilever Deflection — Uniform Load

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

Strength of MaterialsCivil & SurveyingMechanicsTip deflection of a cantilever carrying a uniformly distributed load along its whole length, δ = wL⁴/8EI — three-eighths of the sag the same total load would cause at the tip.

Combined Axial and Bending Stress

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

Strength of MaterialsCivil & SurveyingMechanicsExtreme-fibre stress where an axial force and a bending moment act together, as in an eccentrically loaded column or a beam-column. Enter a negative c for the relieved face.

Consolidation Settlement of Normally Consolidated Clay

S_c = \frac{C_c\,H}{1 + e_0}\log_{10}\!\frac{\sigma'_f}{\sigma'_0}

Soil MechanicsStrength of MaterialsPrimary consolidation settlement of a normally consolidated clay layer from its compression index, thickness and the stress increase applied.

Degree of Saturation (Se = wGs)

S = \frac{w\,G_s}{e}

Soil MechanicsStrength of MaterialsDegree of saturation from water content, specific gravity of solids and void ratio, using the phase identity Se = wGs.

Dry Unit Weight from Gs and Void Ratio

\gamma_d = \frac{G_s\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsDry unit weight of a soil from the specific gravity of its solids and its void ratio, the phase-diagram route used to back out e in the lab.

Dry Unit Weight from Moist Unit Weight

\gamma_d = \frac{\gamma}{1 + \dfrac{w}{100}}

Soil MechanicsStrength of MaterialsStrips the pore water out of a measured bulk unit weight to give the dry unit weight used for compaction control and phase work.

Elastic Section Modulus (S = I/c)

S = \frac{I}{c}

Strength of MaterialsMechanicsGeometryElastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into a bending stress.

Euler Critical Buckling Load

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

Strength of MaterialsMechanicsPhysicsEuler's critical buckling load for a slender column, using the end-condition factor K and the area moment of inertia in m⁴.

Factor of Safety

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

Strength of MaterialsMechanicsPhysicsFactor of safety as ultimate or yield strength divided by the allowable working stress, the engineer's declared margin of ignorance.

Fixed-End Moment — Fixed-Fixed Beam, Uniform Load

M_F = \frac{w L^{2}}{12}

Strength of MaterialsCivil & SurveyingMechanicsMoment at each built-in end of a fixed-fixed beam under a uniformly distributed load, wL²/12. Midspan carries only wL²/24, half as much again the other way.

Goodman Fatigue Criterion

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

Strength of MaterialsMechanicsThe modified Goodman line for infinite life under a fluctuating stress: alternating stress over endurance limit plus mean stress over ultimate strength equals one over the factor of safety.

Hoop Stress in a Thin-Walled Cylinder

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

Strength of MaterialsMechanicsPhysicsHoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t — exactly twice the longitudinal stress.

Liquidity Index

LI = \frac{w - PL}{PI}

Soil MechanicsStrength of MaterialsLiquidity index locating the natural water content of a clay between its plastic limit and liquid limit, a direct index of consistency.

Longitudinal Stress in a Thin-Walled Cylinder

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

Strength of MaterialsMechanicsPhysicsLongitudinal (axial) stress in a thin-walled cylinder under internal pressure, σ = pd/4t — exactly half the hoop stress.

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Strength of MaterialsMechanicsPhysicsMaximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

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

Strength of MaterialsMechanicsPhysicsMaximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Max Moment — Simple Beam, Off-Centre Point Load

M = \frac{P a (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsMaximum bending moment in a simply supported beam with a single point load at distance a from one support, M = Pab/L, occurring directly under the load.

Maximum In-Plane Shear Stress

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

Strength of MaterialsCivil & SurveyingMechanicsRadius of Mohr's circle: the largest shear stress on any plane through a plane-stress element, equal to half the difference of the two principal stresses.

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}}

Strength of MaterialsCivil & SurveyingMechanicsLarger of the two principal stresses for a plane-stress element, from the normal stresses σx and σy and the shear τxy — the circle's centre plus its radius.

Miner's Cumulative Damage Rule (Three Blocks)

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

Strength of MaterialsMechanicsLinear damage summation over three blocks of a variable-amplitude load history. Failure is predicted when the damage fraction D reaches 1.

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}}

Strength of MaterialsCivil & SurveyingMechanicsSmaller of the two principal stresses for a plane-stress element — the circle's centre minus its radius. Often the compressive one, and the value brittle materials care about least.

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

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

Strength of MaterialsCivil & SurveyingGeometryStrong-axis second moment of area of a doubly symmetric I-shape, taken as the full bounding rectangle minus the two rectangular voids beside the web.

Normal (Axial) Stress

\sigma = \frac{P}{A}

Strength of MaterialsMechanicsPhysicsAxial stress in a bar or hanger rod — the internal force divided by the cross-sectional area that carries it, in Pa or psi.

Normal Strain (ε = δ/L)

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

Strength of MaterialsMechanicsPhysicsNormal strain as the change in length divided by the original length, a dimensionless ratio usually quoted in microstrain.

Parallel Axis Theorem (I = I_c + Ad²)

I = I_c + A d^{2}

Strength of MaterialsCivil & SurveyingGeometrySecond moment of area of a shape about an axis parallel to its own centroidal axis: add A times the offset squared. The transfer term that builds every plate girder and flitch beam.

Plastic Moment Capacity (Mp = Z fy)

M_p = Z f_y

Strength of MaterialsCivil & SurveyingMoment at which a compact steel section is fully plastic, with every fibre at yield. The nominal flexural strength Mn used in LRFD steel design before the φ factor.

Plastic Section Modulus — Rectangle

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

Strength of MaterialsCivil & SurveyingGeometryPlastic section modulus of a solid rectangle, the first moment of the two half-areas about the equal-area axis. Exactly 1.5 times the elastic section modulus bh²/6.

Plasticity Index (PI = LL − PL)

PI = LL - PL

Soil MechanicsStrength of MaterialsPlasticity index of a fine-grained soil as the liquid limit minus the plastic limit, the width of the moisture range where clay behaves plastically.

Poisson's Ratio

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

Strength of MaterialsMechanicsPhysicsPoisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Polar Moment of Inertia — Solid Shaft

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

Strength of MaterialsMechanicsGeometryPolar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Radius of Gyration (r = √(I/A))

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

Strength of MaterialsMechanicsGeometryRadius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Reinforced Concrete Nominal Moment Capacity

M_n = A_s f_y \left(d - \frac{a}{2}\right)

Civil & SurveyingStrength of MaterialsNominal flexural strength of a singly reinforced concrete section: the tension force in the rebar times the internal lever arm to the centroid of the Whitney compression block.

Relation Between E, G and ν

E = 2G(1 + \nu)

Strength of MaterialsMechanicsPhysicsThe isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Relative Compaction (Percent Proctor)

R = \frac{\gamma_{d,field}}{\gamma_{d,max}}\times 100

Soil MechanicsStrength of MaterialsRelative compaction of placed fill as the field dry unit weight divided by the Proctor maximum dry unit weight, in percent.

Relative Density of a Granular Soil

D_r = \frac{e_{max} - e}{e_{max} - e_{min}}\times 100

Soil MechanicsStrength of MaterialsRelative density of a sand or gravel, placing its in-situ void ratio on the scale between its loosest and densest laboratory states.

Saturated Unit Weight

\gamma_{sat} = \frac{(G_s + e)\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsUnit weight of a soil whose voids are completely full of water, from the specific gravity of the solids and the void ratio.

Shear Flow (q = VQ/I)

q = \frac{V Q}{I}

Strength of MaterialsCivil & SurveyingMechanicsLongitudinal shear force per unit length that must cross a joint in a built-up beam — the number that sets nail spacing, bolt pitch and weld size.

Shear Modulus (G = τ/γ)

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

Strength of MaterialsMechanicsPhysicsShear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

Slenderness Ratio (KL/r)

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

Strength of MaterialsMechanicsPhysicsSlenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

Stress Concentration (σmax = Kt σnom)

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

Strength of MaterialsMechanicsPeak elastic stress at a hole, notch, groove or fillet: the nominal stress on the net section multiplied by a geometric factor read from a Peterson chart.

Support Reaction — Simple Beam, Off-Centre Point Load

R_A = \frac{P (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at the near support of a simply supported beam with one point load at distance a from it. The far reaction is the remainder, P − R_A = Pa/L.

Thermal Stress in a Restrained Member

\sigma = E \alpha \Delta T

Strength of MaterialsMechanicsPhysicsStress raised in a fully restrained member that is heated or cooled — the cause of rail sun kinks and cracked pipe anchors.

Torsional Shear Stress (τ = Tr/J)

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

Strength of MaterialsMechanicsPhysicsTorsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Transverse Shear Stress (τ = VQ/Ib)

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

Strength of MaterialsCivil & SurveyingMechanicsShear stress at any height in a beam cross-section, peaking at the neutral axis. For a rectangle it works out to exactly 1.5 times the average V/A.

Void Ratio and Porosity (e = n/(1 − n))

e = \frac{n}{1 - n}

Soil MechanicsStrength of MaterialsConverts between void ratio, the void volume per unit of solid, and porosity, the void volume per unit of total soil volume.

Water (Moisture) Content

w = \frac{M_w}{M_s}\times 100

Soil MechanicsStrength of MaterialsGravimetric water content of a soil as the mass of pore water divided by the mass of oven-dry solids, expressed as a percentage.

Young's Modulus (E = σ/ε)

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

Strength of MaterialsMechanicsPhysicsYoung's modulus as the ratio of normal stress to normal strain, the stiffness constant of a material in its elastic range.