In this article we solve in closed form a nonlinear differential equation modelling the planar, non-inflectional elastica (deflection \$y\$) of a thin, flexible, simply supported, \$x\$-straight rod, which withdraws bending and doesn't retain its original length \$L\$ when deformed under a compressive thrust. We solve a Dirichlet two-point boundary value problem providing an explicit inverse function \$x(y)\$ through elliptic integrals of I and II kind. Integration is performed twice via the ``shooting'' getting both branches of elastica. In such a way \$y(x)\$ is required to be invertible, and then our solution is found within each of two monotonicity \$x\$-ranges for \$y\$% , before and beyond the bifurcation value \$x^{*}=L/2\$. An auxiliary unknown is then introduced and successively computed through a suitable welding condition. In such a way \$x=x(y)\$ is completely known in its two symmetrical branches.

### Elliptic integrals solution to elastica's boundary value problem of a rod bent by axial compression

#### Abstract

In this article we solve in closed form a nonlinear differential equation modelling the planar, non-inflectional elastica (deflection \$y\$) of a thin, flexible, simply supported, \$x\$-straight rod, which withdraws bending and doesn't retain its original length \$L\$ when deformed under a compressive thrust. We solve a Dirichlet two-point boundary value problem providing an explicit inverse function \$x(y)\$ through elliptic integrals of I and II kind. Integration is performed twice via the ``shooting'' getting both branches of elastica. In such a way \$y(x)\$ is required to be invertible, and then our solution is found within each of two monotonicity \$x\$-ranges for \$y\$% , before and beyond the bifurcation value \$x^{*}=L/2\$. An auxiliary unknown is then introduced and successively computed through a suitable welding condition. In such a way \$x=x(y)\$ is completely known in its two symmetrical branches.
##### Scheda breve Scheda completa Scheda completa (DC)
G. Mingari Scarpello; D. Ritelli
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/45869`
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• ND
• ND