MMechClassroom.com

Formula reference

Engineering formulas

A compact reference of engineering mechanics and strength of materials formulas with calculators.

Membrane Theory

Laplace Equilibrium Equation for a Membrane Shellp = N₁/R₁ + N₂/R₂Membrane Stresses in Thin-Walled Cylinders and Spheresσθ = pr / t

Tension and Compression

Hooke's Law in Uniaxial Tension or Compressionσ = Eε

Mechanical Properties of Materials

Hooke's Law in Uniaxial Tension or Compressionσ = Eε

Pure Shear and Design of Joints

Hooke's Law in Shearτ = Gγ

Stresses in Bending

Normal Stress in Simple Bending — Navier Formulaσ = M / WBeam Shear Stress — Zhuravsky Formulaτ = QS / (Ib)

Eccentric Compression

Section Kern under Eccentric Compressionrk = d / 8

Three-Dimensional Force Systems

Force Magnitude from Three ComponentsF = √(Fₓ² + Fᵧ² + F_z²)

Sliding Friction. Coulomb's Laws

Limiting Dry Friction ForceFfr,max = μN

Shear and Direct Shear

Hooke's Law in Shearτ = Gγ

Torsion of Solid and Hollow Circular Shafts

Polar Properties of Solid and Hollow Circular SectionsJp = πd⁴ / 32Stress and Angle of Twist of a Circular Shaftφ = TL / (GJp)

Deflections in Bending

Beam Curvature, Bending Moment, and Flexural RigidityM = EI / ρ

Combined Bending and Torsion

Resultant Bending MomentMb = √(Mx² + My²)Equivalent Stress under Combined Bending and Torsionσeq = √(σb² + 3τt²)

Force Projections

Projection of a Force onto an AxisFₓ = F cos α

Elastic Properties of Materials

Hooke's Law in Uniaxial Tension or Compressionσ = Eε

Power Transmission by Shafts

Stress and Angle of Twist of a Circular Shaftφ = TL / (GJp)

Parallel-Axis Theorem and Composite Sections

Parallel-Axis Theorem for Second Moments of AreaI = Ic + Aa²

Moment of a Force About a Point and an Axis

Moment of a Force About a PointM = Fd

Rolling Resistance

Rolling-Resistance MomentMrr = Nδ

Column Slenderness and Weak Buckling Axis

Radius of Gyration and Column Slendernessi = √(I / A)

Torsion

Stress and Angle of Twist of a Circular Shaftφ = TL / (GJp)

Euler Buckling Formula and Critical Buckling Load

Euler Critical Load for a Compressed ColumnNcr = π²EI / leff²

Geometric Properties of Plane Areas

Parallel-Axis Theorem for Second Moments of AreaI = Ic + Aa²

Critical Buckling Stress, Euler Limits and Inelastic Column Formulas

Applicability Limit of Euler's Buckling Formulaλlim = π√(E / σp)

Addition of Forces. Resultant of a Force System

Magnitude of a Resultant from Two ComponentsR = √(Rₓ² + Rᵧ²)

Design of Helical Springs

Polar Properties of Solid and Hollow Circular SectionsJp = πd⁴ / 32

Bending

Normal Stress in Simple Bending — Navier Formulaσ = M / WBeam Shear Stress — Zhuravsky Formulaτ = QS / (Ib)Beam Curvature, Bending Moment, and Flexural RigidityM = EI / ρ

Combined Loading

Equivalent Stress under Combined Bending and Torsionσeq = √(σb² + 3τt²)

Shell Analysis

Laplace Equilibrium Equation for a Membrane Shellp = N₁/R₁ + N₂/R₂Membrane Stresses in Thin-Walled Cylinders and Spheresσθ = pr / t

Contact of Two Bodies: Types and Geometry

Reduced Radius of Curvature in Hertz Contact1/R* = 1/R1 + 1/R2

Dynamic and Cyclic Loading

Dynamic Factor for Vertical Impactδmax = δst + √(δst² + 2hδst)

Contact Stresses

Reduced Radius of Curvature in Hertz Contact1/R* = 1/R1 + 1/R2Hertz Contact Pressure Distributionp₀ = 3F / (2πa²)

Differential Equation of the Elastic Curve

Beam Curvature, Bending Moment, and Flexural RigidityM = EI / ρ

Initial Parameter Method (Macaulay's Method)

Beam Curvature, Bending Moment, and Flexural RigidityM = EI / ρ

Contact of Spheres and Spherical Surfaces

Reduced Radius of Curvature in Hertz Contact1/R* = 1/R1 + 1/R2Hertz Contact of Spherical Surfacesa = (3FR* / 4E*)^(1/3)

Impact Loading Calculations

Dynamic Factor for Vertical Impactδmax = δst + √(δst² + 2hδst)

Other formulas

Moment of a Force CoupleM = FdResultant of a Uniformly Distributed LoadR = qL