2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Integration of the equilibrium equation in a deformed configuration by assuming no extension of the mean axis.
2.2. Benchmark results#
The balance is described by:
(2.1)#\[ {w} _ {x}\ text {'}\ text {'}\ text {'} (z) =\ frac {-M (z)} {{\ mathit {EI}}} _ {y}} =\ frac {- {F} _ {F} _ {F} _ {F} _ {F}} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {{\ mathit {EI}} _ {y}}\]
The \({w}_{x}(z)\) move is therefore worth:
(2.2)#\[ {w} _ {x} (z) =\ frac {{F}} _ {f} _ {x}} {{F} _ {z}}\ left\ {-z+\ frac {1} {\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left (1-\ mathrm {cos} (\ mathrm {\ alpha} z)\ right)\ right]\ right\ right\}\]
with \(\mathrm{\alpha }=\sqrt{\frac{{F}_{z}}{{\mathit{EI}}_{y}}}\) and \(l\approx {l}_{0}\). This last lever arm approximation is acceptable for moderate \({F}_{x}\) efforts.
2.3. Uncertainties about the solution#
Analytical solution
2.4. Bibliographical references#
VOLDOIRE, Y. BAMBERGER: Structural mechanics, 2008.