4. Reference solution: Multifiber beams

4.1. Single-material calculation

4.1.1. Reference quantities and results

The reference solution is that given by the results of the 3D calculation. We test the DZ component of the displacement of the N_ MIL node for 3D and P3 for beams.

4.1.2. Uncertainty about the solution

Uncertainties related to 3D modeling.

4.2. First bi-material calculation

4.2.1. Reference quantities and results

The \(\alpha\) coefficient values of the two materials and of the temperature fields assigned to the various fibers result in the same thermal deformation value on each fiber.

If the temperature at the sub-points is correctly taken into account, this loading does not induce a bending of the beam.

It is therefore verified that the component DZ of the movement of the node P 3 is zero.

4.2.2. Uncertainty about the solution

None.

4.3. Second bi-material calculation

4.3.1. Reference quantities and results

The thermal loading is the same as in the previous calculation, but this time the end nodes are embedded. The total deformation is zero and the thermal deformation is the same as in the previous calculation. In addition, mechanical deformation is equal to thermal deformation.

We have: \({\varepsilon }_{\mathit{th}}\mathrm{=}8.E\mathrm{-}4\)

In the general case we have for each fiber \(i\):

\({\sigma }_{\mathit{xx}}\mathrm{=}{E}_{i}({\varepsilon }_{\mathit{xx}}\mathrm{-}{\varepsilon }_{\mathit{th}})\)

so in this case:

\({\sigma }_{\mathit{xx}}\mathrm{=}\mathrm{-}{E}_{i}{\varepsilon }_{\mathit{th}}\)

Hence for concrete fibers:

\({\sigma }_{\mathit{xx}}\mathrm{=}\mathrm{-}2.4E7\)

and for steel fibers:

\({\sigma }_{\mathit{xx}}\mathrm{=}\mathrm{-}1.6E8\)

4.3.2. Uncertainty about the solution

None.