Strength of Materials formula solvers

Categories — 73 of them, 1787 solvers in all

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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.

ASME Required Wall Thickness (t = PR/(SE − 0.6P))

t = \frac{P R}{S E - 0.6 P}

Strength of MaterialsMechanicsCivil & SurveyingMinimum required wall thickness of a cylindrical shell under internal pressure, from ASME Section VIII Division 1 UG-27(c)(1): pressure times inside radius over allowable stress times joint efficiency, less the 0.6P correction.

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.

Beam Reaction by Moment Equilibrium — Two Point Loads

R_A = \frac{P_1 (L - a_1) + P_2 (L - a_2)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at support A of a simply supported beam carrying two point loads, from ΣM = 0 taken about support B. Each load is weighted by its distance from the FAR support, which is the whole content of the moment equation.

Bearing Stress on a Pin or Bolt (σ = P/dt)

\sigma_{b} = \frac{P}{d \, t}

Strength of MaterialsMechanicsCivil & SurveyingBearing stress where a pin or bolt presses on the side of its hole: the load divided by the PROJECTED area, bolt diameter times plate thickness. The third failure mode of a bolted lap joint, after net-section tension and bolt shear.

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.

Cable Arc Length (Parabolic Series)

s = L \left( 1 + \frac{8 d^{2}}{3 L^{2}} - \frac{32 d^{4}}{5 L^{4}} \right)

Strength of MaterialsHow much cable is actually strung between two supports, which is always more than the span. The series says the excess grows with the square of the sag ratio, so a shallow cable is barely longer than its span and a deep one is noticeably longer. It is what you order cable by, what a stringing chart is built on, and what tells you how much slack a temperature change has to absorb.

Cable Horizontal Tension (Parabolic)

H = \frac{w L^{2}}{8 d}

Strength of MaterialsThe one relation every suspended cable problem starts from. A flexible cable carrying a load spread evenly along the horizontal hangs in a parabola, and the horizontal pull it needs is the load times the span squared over eight times the sag. Read it backwards and it is the most useful statement in the subject: sag and tension trade against each other, and you cannot have both small.

Cable Sag Ratio

\frac{d}{L}

Strength of MaterialsSag divided by span. It is the single number that decides everything else about a suspended cable: how far above H the support tension runs, how much longer the cable is than the span, and — the reason this page exists — whether the parabola may be used at all. Below about 1 in 10 the parabola and the catenary agree within a percent; above it they part company.

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.

Catenary Arc Length

s = 2 a \sinh \frac{L}{2a}

Strength of MaterialsThe exact length of cable hanging between two level supports under its own weight. No series, no truncation — the arc length of a catenary is a hyperbolic sine, and that is the closed form. Compare it against the parabolic series on the same span to see how little the difference is when the cable is taut and how quickly it grows when it is not.

Catenary Parameter

a = \frac{H}{w}

Strength of MaterialsOne length that fixes the entire shape of a hanging chain: the horizontal tension divided by the weight per unit length of cable. Every catenary is the same curve, y = a cosh(x/a), scaled by this number — a small a gives a sharply drooping rope, a large a gives a nearly straight one. It is also a real distance on the drawing, the height of the low point above the directrix.

Catenary Sag

d = a \left( \cosh \frac{L}{2a} - 1 \right)

Strength of MaterialsThe exact sag of a cable that carries only its own weight — a bare conductor, a chain, a slack guy wire, an unloaded haul rope. The parabola is not an approximation to this; it is the answer to a different loading. This is the answer when every metre of CABLE weighs the same, rather than every metre of ground beneath it.

Catenary Tension at a Point

T = H \cosh \frac{x}{a}

Strength of MaterialsThe tension anywhere along a free-hanging cable, from the horizontal distance to its lowest point. Written the other way it is the prettiest result in the subject: T = w·y, the tension at any point equals the weight per metre times the height of that point above the directrix. The cable holds itself up by hanging from a line that is not there.

Centroid of a Composite Area (ȳ = ΣAȳ / ΣA)

\bar{y} = \frac{A_1 y_1 + A_2 y_2}{A_1 + A_2}

Strength of MaterialsCivil & SurveyingGeometryCentroid of an area built from two parts: the area-weighted average of the parts' own centroids, measured from any convenient datum. This is the step that must come before the parallel axis theorem, because it fixes the axis everything else is measured from.

Coffin-Manson Strain-Life Relation

\dfrac{\Delta\varepsilon_p}{2} = \varepsilon_f' \left( 2 N_f \right)^{c}

Strength of MaterialsManson's and Coffin's independent 1953-54 discovery that plastic strain amplitude against reversals to failure is a straight line on log-log axes. It is the tool for low-cycle fatigue — thermal cycling, seismic demand, forming operations, solder joints — where the part yields every cycle and a stress-based method has nothing to say.

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.

Compactive Viscosity of Snow (Kojima)

\eta = \eta_0 \, e^{f \rho_s}

Snow & IceSoil MechanicsStrength of MaterialsThe compactive viscosity of snow, rising exponentially with its own density — Kojima's 1967 relation, in the form Anderson's SNTHERM and most land-surface schemes have used since. It is why snow settles fast on the first night and then almost stops: the act of compacting is what makes it resist compacting.

Composite Density from the Rule of Mixtures

\rho_c = \rho_f V_f + \rho_m (1 - V_f)

Strength of MaterialsThe one rule of mixtures that is not a bound but an identity: mass adds whichever way the load runs, so a void-free composite's density is exactly the volume-weighted average of its constituents. That makes it the standard check on a laminate — weigh the panel, measure it, and the shortfall against this number is porosity.

Compressive Strength of Sintered Snow

\sigma_c = \sigma_i \left( \frac{\rho_s}{\rho_i} \right)^{\! n}

Snow & IceStrength of MaterialsSoil MechanicsUnconfined compressive strength of sintered snow as a power law in density, normalised to solid ice. It is an empirical fit with very wide scatter, not a law, and it leaves out the variable that matters as much as density does: how long the snow has been sitting undisturbed.

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.

Critical Fibre Length

l_c = \frac{\sigma_f \, d}{2 \, \tau_i}

Strength of MaterialsThe shortest fibre that can be loaded to its own breaking stress before it pulls out of the matrix. Load enters a discontinuous fibre only through shear at its surface, so a short fibre simply slides; below this length it never breaks, never carries its full share, and the composite around it is weaker for it.

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.

Fibre Volume Fraction from Weight Fraction

V_f = \frac{W_f / \rho_f}{W_f / \rho_f + (1 - W_f) / \rho_m}

Strength of MaterialsThe conversion between what a burn-off or acid-digestion test measures and what every micromechanics equation wants. They are not the same number and they are not close: carbon fibre is half again as dense as cured epoxy, so a 70% fibre laminate by weight is only about 61% by volume. Getting this backwards is the most common large error in composite calculations, and it always runs in the unsafe direction.

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.

Griffith Critical Stress

\sigma_c = \sqrt{\dfrac{2 E \gamma}{\pi a}}

Strength of MaterialsA. A. Griffith's 1921 answer to why glass breaks at a thousandth of the strength its atomic bonds promise: cracks. A crack only grows if the elastic energy released pays for the two new surfaces it creates, and that balance gives a fracture stress falling as the square root of the flaw size. The foundation the whole subject is built on.

Halpin–Tsai Transverse Modulus

E_2 = E_m \, \frac{1 + \xi \eta V_f}{1 - \eta V_f}, \quad \eta = \frac{E_f/E_m - 1}{E_f/E_m + \xi}

Strength of MaterialsThe semi-empirical form that actually predicts the transverse stiffness of a lamina, sitting above the inverse rule of mixtures and far below the longitudinal one. One fitted parameter, the reinforcing factor ξ, carries everything the two bounds cannot see: the shape of the fibre cross-section and how the fibres are packed against one another.

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.

Inverse Rule of Mixtures — Transverse Modulus

\frac{1}{E_2} = \frac{V_f}{E_f} + \frac{1 - V_f}{E_m}

Strength of MaterialsStiffness across the fibres, where the two phases share the same stress instead of the same strain and their compliances add. The answer is dominated by whichever phase is softer, which for a fibre composite is always the resin — and that is why the transverse modulus of a lamina is a small fraction of the longitudinal one rather than a modest reduction from it.

Irwin Plastic Zone Size

r_p = \dfrac{1}{\beta \pi} \left( \dfrac{K}{\sigma_y} \right)^{2}

Strength of MaterialsThe elastic solution says the stress at a crack tip is infinite, which no material tolerates: it yields instead, over a small zone whose size this estimates. The zone has to be small compared with the crack and the remaining ligament, or linear elastic fracture mechanics does not apply at all — which makes this the validity check for every other page in this set.

Lamé Hoop Stress in a Thick-Walled Cylinder

\sigma_{\theta} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 + \frac{r_o^{2}}{r^{2}}\right)

Strength of MaterialsMechanicsPhysicsCircumferential (hoop) stress at any radius r in a thick-walled cylinder with internal pressure only — the Lamé solution. Unlike the thin-wall formula it shows the stress varying through the wall, peaking at the bore.

Lamé Radial Stress in a Thick-Walled Cylinder

\sigma_{r} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 - \frac{r_o^{2}}{r^{2}}\right)

Strength of MaterialsMechanicsPhysicsRadial stress at any radius r in a thick-walled cylinder with internal pressure only. It is compressive throughout, equal to −p at the bore and exactly zero at the free outside surface.

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.

Major Poisson's Ratio of a Lamina

\nu_{12} = \nu_f V_f + \nu_m (1 - V_f)

Strength of MaterialsThe sideways contraction of a lamina pulled along its fibres, taken as the same volume-weighted average that gives the longitudinal modulus. It is the third of the four constants classical lamination theory needs, and it comes with a trap the isotropic world never prepares anyone for: the ratio measured across the fibres is a different, much smaller number.

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 Cable Tension at the Support (Parabolic)

T_{\max} = \sqrt{H^{2} + \left( \dfrac{w L}{2} \right)^{2}}

Strength of MaterialsAt the support the cable is pulling in two directions at once: horizontally with H, which is the same everywhere along it, and vertically with half the total load, which is what the tower has to hold up. The tension in the cable is the resultant of those two, and it is the largest tension anywhere in the span. This is the number the fitting, the insulator, the anchorage and the factor of safety are all sized on.

Maximum Cable Tension from the Sag Ratio (Parabolic)

T_{\max} = H \sqrt{1 + \frac{16 d^{2}}{L^{2}}}

Strength of MaterialsThe same support tension as the load-based form, written the way it is actually used: the horizontal tension multiplied by a factor that depends on nothing but the sag ratio. It shows in one line why a shallow cable can be designed on H alone and a deep one cannot — at 1 in 20 the multiplier is 1.005, at 1 in 5 it is 1.077.

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.

Membrane Stress in a Thin-Walled Sphere (σ = pr/2t)

\sigma = \frac{p r}{2 t}

Strength of MaterialsMechanicsPhysicsMembrane stress in the wall of a thin-walled sphere under internal pressure, σ = pr/2t. A sphere is stressed equally in every direction, which is why it is the most efficient pressure vessel shape there is.

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.

Paris Law Crack Growth Rate

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

Strength of MaterialsParis and Erdogan's 1963 observation that fatigue crack growth plots as a straight line on log-log axes against the range of stress intensity. It is the equation damage-tolerant design is built on — the reason a cracked aircraft structure can be flown, inspected and flown again rather than scrapped.

Paris Law Cycles to Failure

N = \dfrac{a_f^{\,1 - m/2} - a_i^{\,1 - m/2}}{\left( 1 - \tfrac{m}{2} \right) C \left( Y \, \Delta\sigma \sqrt{\pi} \right)^{m}}

Strength of MaterialsThe Paris rate integrated from the flaw you found to the flaw that fails: how many more cycles the part has. This is the arithmetic behind every damage-tolerance inspection interval in aviation and every fitness-for-service remaining-life assessment on a pressure vessel. It assumes constant amplitude loading and a geometry factor Y that does not change as the crack grows.

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.

Rule of Mixtures — Longitudinal Modulus

E_1 = E_f V_f + E_m (1 - V_f)

Strength of MaterialsStiffness along the fibres: each phase carries its own volume share of the load, so the moduli simply add in proportion. It is the upper bound on what any fibre and matrix can give you, it is the one direction where a real lamina nearly reaches the bound, and it is the number people wrongly use for every other direction.

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.

Specific Stiffness (E / ρ)

E_s = \frac{E}{\rho}

Strength of MaterialsModulus divided by density — the number that explains why an aircraft is worth building out of carbon fibre and a bridge usually is not. Steel, aluminium, magnesium and titanium all sit within a few percent of the same value, which is one of the quiet surprises of materials engineering; a unidirectional carbon laminate sits three or four times higher along its fibres and no better than the resin across them.

Specific Strength (σ / ρ)

\sigma_s = \frac{\sigma}{\rho}

Strength of MaterialsStrength divided by density: how much load a material carries for each kilogram of itself. Unlike specific stiffness, where all the structural metals tie, this one separates them sharply — and it is where fibre composites win by a margin big enough to change what an aeroplane is made of.

Strain Energy Release Rate

G = \dfrac{K^{2} \left( 1 - \nu^{2} \right)}{E}

Strength of MaterialsThe bridge Irwin built between Griffith's energy picture and his own stress picture: the energy a crack releases per unit of new area is the square of the stress intensity divided by an effective modulus. Written here in the PLANE STRAIN form. For plane stress — a thin sheet — enter Poisson's ratio as 0, which makes E prime equal E exactly.

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.

Stress Intensity Factor

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

Strength of MaterialsIrwin's 1957 result, and the single most useful equation in fracture mechanics: the entire elastic stress field around a crack tip is set by one number. Compare K against the material's fracture toughness K_IC and you have the answer to whether a cracked part will break — or, turned around, the crack size that part can tolerate.

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.

Support Reaction — Simple Beam, Uniform Load (R = wL/2)

R = \frac{w L}{2}

Strength of MaterialsCivil & SurveyingMechanicsReaction at each support of a simply supported beam under a uniformly distributed load: half the total load wL. It is also the maximum shear in the beam, which is why it sizes the bearing and the end connection.

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.

Tsai–Hill Failure Index for a Lamina

FI = \left( \frac{\sigma_1}{X} \right)^{2} - \frac{\sigma_1 \sigma_2}{X^{2}} + \left( \frac{\sigma_2}{Y} \right)^{2} + \left( \frac{\tau_{12}}{S} \right)^{2}

Strength of MaterialsHill's anisotropic yield condition reduced to plane stress, which is the standard way to ask whether a single ply survives a combined stress state. It is interactive: the longitudinal, transverse and shear stresses are not checked one at a time but together, so a ply can fail with every individual stress comfortably below its own allowable. What it gives back is an index, not a margin, and it will not tell you how the ply failed.

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.

Von Mises Equivalent Stress (Plane Stress)

\sigma_v = \sqrt{\sigma_x^{2} - \sigma_x \sigma_y + \sigma_y^{2} + 3\tau_{xy}^{2}}

Strength of MaterialsMechanicsCivil & SurveyingVon Mises equivalent stress for a plane stress state: the single tensile stress that would do the same distortion damage as the combination of σx, σy and τxy. Compare it directly against the yield strength.

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.