2. Benchmark solution#

2.1. Calculation method used for the reference solution#

With a beam kinematics and a resultant force model, the curvature (in large rotations) of the cantilever subjected to the bending moment \(M=\text{mb}\) is, with the previous numerical data:

\(\frac{d\theta }{\text{dx}}\mathrm{=}\frac{\text{mb}}{{\text{EI}}_{y}}\mathrm{=}\frac{t}{L}\)

It’s Euler’s solution.

2.2. Benchmark results#

_images/Object_15.svg

According to Euler’s solution, the deformation is an arc. In section \({P}_{2}{P}_{3}\) \((x=L)\), the rotation is equal to:

\(\theta (x\mathrm{=}L)\mathrm{=}t\text{.}\)

In the absence of normal force, the mean surface remains inextensible and the radius of curvature is given by:

\(R\mathrm{=}{(\frac{d\theta }{\text{dx}})}^{\mathrm{-}1}\mathrm{=}\frac{L}{t}\)

The horizontal displacement is then

\(u\mathrm{=}R\text{sin}\theta \mathrm{-}L\mathrm{=}L(\frac{\text{sin}t}{t}\mathrm{-}1)\)

and the vertical displacement is

\(v\mathrm{=}R(1\mathrm{-}\text{cos}\theta )\mathrm{=}\frac{L}{t}(1\mathrm{-}\text{cos}t)\)

2.3. Bibliographical references#

    1. AL MIKDAD: Statics and Dynamics of Beams in Large Rotations and Resolution of Nonlinear Instability Problems. Doctoral thesis, Compiègne University of Technology (1998).

  2. J.C. SIMO and L. VU QUOC: A Three-dimensional Finite Strain Rod Model. Part II: Computational Aspects. Compute. Meth. Call. Mech. Engrg.58, 79‑116 (1986).

  3. J.C. SIMO, D.D. FOX and M.S. RIFAI: We Have a Stress Resulting Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory. Compute. Meth. Call. Mech. Engrg.79, 21-70 (1990).