2. Benchmark solution

2.1. Formal solution

The aim is to define deformation \(\varepsilon\) according to the main direction of a reinforcement sheet located in the plane \(({x}_{\mathrm{1 }};{y}_{1})\).

Taking into account the boundary conditions chosen, we can write:

with \(({u}_{\mathrm{x1}},{u}_{\mathrm{y1}})\) the components of the displacement vector in the \(({x}_{\mathrm{1 }};{y}_{1})\) plane and \(\theta\) the angle between the main direction of the reinforcing sheet and \({x}_{1}\).

To define the main direction of the water table, we use the nautical angles \((\alpha ;\beta )\) given by the keyword ANGL_REP. They define a vector \(v\) whose projection \({x}_{p}\) on the tangential plane of the sheet fixes the main direction.

with \((x,y,z)\) the initial coordinate system. For our application, the displacement vector \(U\) is written as:

For tablecloth \(\mathrm{GEOX}\) (plan \((y;z)\)):

The main direction of the tablecloth is

(\(\theta =90°\) ). The deformation is then written:

For tablecloth \(\mathrm{GEOY}\) (plan \((x;z)\)):

The main direction of the tablecloth therefore makes an angle of \(40°\) with the plane of the tablecloth. The deformation is then written:

For tablecloth \(\mathrm{GEOZ}\) (plan \((x;y)\)):

The main direction of the tablecloth therefore makes an angle of \(15°\) with the plane of the tablecloth. The deformation is then written:

These three values will be the analytical reference values for the validation of the calculations.