1. Reference problem#

1.1. Geometry#

Beam in three-point bending, defined by:

_images/Object_1.svg

With a duplicate section \(T\):

_images/Object_2.svg

In this diagram, \(O\) is located halfway up the section.

The total cross section of the upper steels is \({3.10}^{\mathrm{-}4}{m}^{2}\) and that of the lower steels is \({4.10}^{\mathrm{-}4}{m}^{2}\).

1.2. Material properties#

  • concrete: \(E\mathrm{=}2.{10}^{10}\mathit{Pa}\); \(\nu \mathrm{=}0.2\); \(\rho \mathrm{=}2400{\mathit{kg.m}}^{\mathrm{-}3}\); \(\alpha \mathrm{=}10\mathrm{-}5{K}^{\mathrm{-}1}\)

  • steel: \(E\mathrm{=}\mathrm{2,1}\mathrm{.}{10}^{11}\mathit{Pa}\); \(\nu \mathrm{=}0.33\); \(\rho \mathrm{=}7800{\mathit{kg.m}}^{\mathrm{-}3}\); \(\alpha \mathrm{=}10\mathrm{-}5{K}^{\mathrm{-}1}\)

1.3. Boundary conditions#

Simple press in \(B\): \(\mathit{dy}\mathrm{=}0\)

Press « double » in \(A\): \(\mathit{dx}\mathrm{=}\mathit{dy}\mathrm{=}\mathit{dz}\mathrm{=}0\) as well as \(\mathit{rx}\mathrm{=}\mathit{ry}\mathrm{=}0\).

1.4. Loads#

Three load cases are tested in succession:

  • Load 1: concentrated effort in the middle of the beam, \(F\mathrm{=}10000N\)

  • Loading 2: self-weight of the beam, \(g\mathrm{=}\mathrm{9,8}{\mathit{m.s}}^{\mathrm{-}2}\)

  • Load 3: homogeneous heating of the \(\Delta T\mathrm{=}100K\) beam