2. Benchmark solution

2.1. Reference quantities and results

2.1.1. Analytical model of the plate

We consider that the response of the plate corresponds to the response of a console beam (because \(\nu \mathrm{=}0\)). In this case, the movement of the plate is given by the movements of each plate section, in the direction \(x(u)\), \(z(w)\) and their rotation with respect to the axis \(y({\theta }_{y})\).

\({U}_{x}(x,z)\mathrm{=}u(x)+z{\theta }_{y}(x)\) \({U}_{z}(x,z)\mathrm{=}w(x)\)

At all points, the deformations are:

\({\varepsilon }_{xx}(x,z)\mathrm{=}u\text{'}+z{\theta }_{y}\text{'}\) and \(2{\varepsilon }_{xz}(x,z)\mathrm{=}{\theta }_{y}+w\text{'}\mathrm{=}{\gamma }_{z}\)

The beam is embedded in \(x\mathrm{=}0\) and subjected to a vertical force \(F\) in \(x\mathrm{=}L\).

The resolution (principle of virtual work and the balance of forces) gives the expression of generalized efforts in the case of an elastic linear response: \(N\) is the axial force, \({M}_{y}\) the moment along the \(y\) axis and \(T\) the shear force.

\(N(x)\mathrm{=}EAu\text{'}(x)\) and \(>\)

\({M}_{y}(x)\mathrm{=}EI{\theta }_{y}\text{'}(x)\) \(>\)

\(T(x)\mathrm{=}GkA{\gamma }_{z}\) \(>\)

where parameter \(k\) is the shear correction factor

with the boundary conditions, we deduce the movements \(u,w\mathit{et}{\theta }_{y}\):

\(>\), \(>\), \({\theta }_{y}(x)\mathrm{=}\frac{F}{EI}\frac{x(x\mathrm{-}2L)}{2}\)

The deformations and stresses are:

\({\varepsilon }_{xx}\mathrm{=}z\frac{\mathrm{-}F}{EI}(L\mathrm{-}x)\) and \({\sigma }_{xx}\mathrm{=}z\frac{\mathrm{-}F}{I}(L\mathrm{-}x)\)

\(2{\varepsilon }_{xz}\mathrm{=}\frac{F}{GkA}\mathrm{=}{\gamma }_{z}\) and \({\sigma }_{\mathit{xz}}\mathrm{=}\frac{\mathrm{-}F}{A}6({z}^{2}\mathrm{/}{h}^{2}\mathrm{-}1\mathrm{/}4)\)

where \(h\) is the height of the plate. The distribution of shear stresses is parabolic in the thickness of the plate to respect free surface conditions.

(\({\sigma }_{\mathit{xz}}(z\mathrm{=}h\mathrm{/}2)\mathrm{=}{\sigma }_{\mathit{xz}}(z\mathrm{=}\mathrm{-}h\mathrm{/}2)\mathrm{=}0\)).

The vertical response when moving point \(x\mathrm{=}L\) is:

\({U}_{z}(L)\mathrm{=}\frac{F}{3EI}{L}^{3}+\frac{FL}{GkA}\)

2.2. Uncertainties about the solution

The reference solution is analytical. So there is no uncertainty.