Engineering Tools — Analytical Calculation Pipelines for Engineers on Engineering Tools

Derivation of the Euler-Bernoulli Beam Equation EI·y″ = M(x)

Tue, 17 Mar 2026 00:00:00 +0000

Step-by-step derivation of the moment-curvature relation EI·y″ = M(x), from plane sections assumption to the Euler-Bernoulli bending equation. All hypotheses stated explicitly.

Hertz Contact Stress in Deep Groove Ball Bearings — Analytical Pipeline

Wed, 19 Mar 2025 00:00:00 +0000

5-node closed-form pipeline for maximum contact pressure and subsurface shear stress in deep groove ball bearings. Hamrock-Brewe approximation, Python notebook, verified on bearing 6206.

Derivation of the Euler Critical Buckling Load Pcr = π²EI/L²

Tue, 17 Mar 2026 00:00:00 +0000

Step-by-step derivation of the Euler buckling formula from equilibrium on the deflected geometry, boundary conditions, and eigenvalue problem. Pinned-pinned column, all assumptions explicit.

Derivation of the Curvature Formula κ = y″/(1+y′²)^(3/2)

Wed, 18 Mar 2026 00:00:00 +0000

Step-by-step derivation of the curvature of a plane curve from its geometric definition, with the small-angle simplification to κ ≈ y″ used in beam theory.

Derivation of Cantilever Tip Deflection δ = PL³/(3EI)

Wed, 18 Mar 2026 00:00:00 +0000

Step-by-step derivation of the cantilever beam tip deflection formula from the moment-curvature relation, double integration, and boundary conditions.

Concrete Penetration Depth — Analytical Pipeline (Li & Chen 2003)

Thu, 02 Apr 2026 00:00:00 +0000

Closed-form model for rigid projectile penetration into semi-infinite concrete targets, based on Li & Chen (2003). Cavity expansion theory, 7-node pipeline, Python notebook verified against Forrestal et al. (1994) test data.

Fatigue Life of Notched Plates — Analytical Pipeline (Hazizi 2023)

Thu, 19 Mar 2026 00:00:00 +0000

10-node closed-form pipeline for fatigue life prediction in plates with semicircular notches, based on Hazizi et al. (2023). Includes Kt correction, mean stress, and Python notebook with FEM verification via PrePoMax.

Thick-Walled Cylinder Stress Analysis — Analytical Pipeline (Croccolo 2009)

Thu, 19 Mar 2026 00:00:00 +0000

Closed-form Lamé equations for radial and hoop stress in thick-walled cylinders under internal pressure, based on Croccolo & Vincenzi (2009). Python notebook with interference fit verification case.

About — Engineering Tools

Mon, 01 Jan 0001 00:00:00 +0000

Engineering Tools publishes transparent calculation pipelines extracted from peer-reviewed papers, with Python notebooks and full intermediate values. By G. Ganz, mechanical engineer.

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