Formula solver index — formula.expert

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.