## Benchmark UP1

### Pressure Loaded Beam (Small Strains)

##### Description

A cantilever of length $l$, depth and width ${z}_{0}$, loaded by a uniform pressure load ${p}_{1}$ from above. The cantilever is supposed to be fixed at $x=0$ and $x=l$. The depicted boundary conditions do not restrict transverse deformation which occurs due to the nonzero Poissonâ€™s number. The maximum deflection of the structure is tested.

Two cantilever materials are used:

• Neo-Hookean
• Gent

##### Parameters

$\begin{array}{ccc}\hfill l& \hfill =\hfill & 0.11\mathrm{m}\hfill \\ \hfill {z}_{0}& \hfill =\hfill & 0.00275\mathrm{m}\hfill \\ \hfill {p}_{1}& \hfill =\hfill & 5\mathrm{Pa}\hfill \\ \hfill \kappa & \hfill =\hfill & {10}^{8}\mathrm{Pa}\hfill \\ \hfill \mu & \hfill =\hfill & 3.\overline{33}·{10}^{6}\mathrm{Pa}\hfill \end{array}$

where $\kappa$ is the bulk modulus and $\mu$ is the shear modulus.

##### Theory

The maximum deflection is given by ${u}_{\mathrm{max}}=\frac{1}{64}\frac{{p}_{1}{z}_{0}}{\mu \left(1+\nu \right)}{\left(\frac{l}{{z}_{0}}\right)}^{4}$, where the Poisson's number is given by $\nu =\frac{3\kappa -2\mu }{6\kappa +2\mu }$.

##### FE Model

In ANSYS the problem is solved using $2×2×80=320$ quadratic elements (SOLID186). In femCalc the same number of BRICK60 elements with linear pressure interpolation was used.

##### Results
Material Theory ANSYS (SOLID186) femCalc (BRICK60)
Type $\kappa \text{\hspace{0.17em}}\left[\mathrm{Pa}\right]$ $\nu \text{\hspace{0.17em}}\left[\mathrm{-}\right]$ ${u}_{}\text{\hspace{0.17em}}\left[\mathrm{m}\right]$ ${u}_{}\text{\hspace{0.17em}}\left[\mathrm{m}\right]$ Ratio ${u}_{}\text{\hspace{0.17em}}\left[\mathrm{m}\right]$ Ratio
Neo-Hookean ${10}^{8}$ $0.483516$ $0.000111$ $0.000112$ $1.009$ $0.000112$ $1.009$
Gent ${10}^{8}$ $0.483516$ $0.000111$ $0.000112$ $1.009$ $0.000112$ $1.009$