Benchmark solution
=====================


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 :math:`M=\text{mb}` is, with the previous numerical data:

:math:`\frac{d\theta }{\text{dx}}\mathrm{=}\frac{\text{mb}}{{\text{EI}}_{y}}\mathrm{=}\frac{t}{L}`

It's Euler's solution.


Benchmark results
----------------------


.. image:: images/Object_15.svg
    :width: 334
    :height: 242


.. _RefImage_Object_15.svg:

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

:math:`\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:

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

The horizontal displacement is then

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

and the vertical displacement is

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


Bibliographical references
---------------------------


1. M. 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).