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LingoScholar / Equation and applicability library
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This first library covers uniform cantilevers under a tip force and simply supported beams under a central point force. These are fundamental mechanics references. No opening reduction, VOL/HOL strength, material properties or design-code compliance is inferred.
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Public method catalogue
Published equations. Named boundaries.
H is cantilever height; L is support span; F is the signed point force; EI is effective bending rigidity; GAₛ is effective shear rigidity, including the researcher’s shear correction. Displacement is evaluated at the force application point.
Cantilever · Euler–Bernoulli
u = F H³ / (3EI)
- Point force and displacement at the top of a uniform cantilever.
- Plane-section bending reference; transverse shear is omitted.
MIT OpenCourseWare, 1.050 Solid Mechanics (Fall 2004)Chapter 10 — Deflections due to Bending. Printed pp. 268–269 (PDF pp. 6–7), End-loaded Cantilever and Point Load, Simply-Supported Beam; pp. 269–270 superposition.Cantilever tip bending compliance. The central-load beam coefficient follows the displayed point-load solution at b = L/2 and x = L/2. Checked 2026-10-17.
Cantilever · Timoshenko
u = F [H³ / (3EI) + H / GAₛ]
- Point force and displacement at the top of a uniform cantilever.
- Constant effective shear rigidity GAₛ already includes the researcher’s shear correction.
Colomés, Rocha, Van der Meer and Lesueur, TU Delft (2023)PyJive workshop: TimoshenkoModel. Cantilever beam example — displayed analytical solution w(L); Slenderness and shear locking.Exact uniform-cantilever bending-plus-shear expression. GAₛ is an effective shear rigidity. Checked 2026-10-17.
Three-point beam · Euler–Bernoulli
u = F L³ / (48EI)
- Simply supported beam; one point force at midspan; displacement at midspan.
- Zero support displacement; transverse shear is omitted.
Nils P. Vedvik, NTNU, TMM4175 (2021)Thick laminates and sandwich structures. Three-point bending: equations (2) and (3), preceding equation (4).Simply supported central-load expressions. This implementation substitutes the researcher’s effective EI and GAₛ for EI and κGbh; it does not calculate a laminate stiffness. Checked 2026-10-17.
MIT OpenCourseWare, 1.050 Solid Mechanics (Fall 2004)Chapter 10 — Deflections due to Bending. Printed pp. 268–269 (PDF pp. 6–7), End-loaded Cantilever and Point Load, Simply-Supported Beam; pp. 269–270 superposition.Cantilever tip bending compliance. The central-load beam coefficient follows the displayed point-load solution at b = L/2 and x = L/2. Checked 2026-10-17.
Three-point beam · Timoshenko
u = F [L³ / (48EI) + L / (4GAₛ)]
- Simply supported beam; one point force at midspan; displacement at midspan.
- Constant effective shear rigidity GAₛ; no local indentation or support settlement.
Nils P. Vedvik, NTNU, TMM4175 (2021)Thick laminates and sandwich structures. Three-point bending: equations (2) and (3), preceding equation (4).Simply supported central-load expressions. This implementation substitutes the researcher’s effective EI and GAₛ for EI and κGbh; it does not calculate a laminate stiffness. Checked 2026-10-17.
Optional boundary extension
Keep support movement separate.
For cantilevers, independent linear sliding and rocking springs can be included in both methods. This is a documented application derivation, rather than a published CLT wall design method.
us = F/Ks · M = FH · θ = M/Kθ · uθ = Hθ = FH²/Kθ
Ks uses kN/mm; Kθ uses kN·mm/rad. Add F/Ks and FH²/Kθ to the member displacement. The model excludes spring coupling, changing uplift/contact, friction and nonlinear connection response.
MIT OpenCourseWare, 1.050 Solid Mechanics (Fall 2004) · Linear torsional spring lawPrinted p. 273 (PDF p. 11), linear torsional spring relation Mᵀ = Kᵀφ; pp. 269–270 superposition. Spring law only. The optional wall extension is derived here from M = FH, θ = M/Kθ, uθ = Hθ and us = F/Ks, not presented as a published CLT wall design equation.