Reference problem
=====================


Geometry
---------


.. image:: images/10000000000003F4000001BA94DE6E4356733B05.jpg
    :width: 2.3937in
    :height: 2.252in


.. _RefImage_10000000000003F4000001BA94DE6E4356733B05.jpg:

The concrete veil has a semi-cylindrical shape: the height is :math:`H=10m` and the average radius is :math:`R\mathrm{=}\mathrm{10 }m`.

The thickness of the veil is :math:`e\mathrm{=}\mathrm{0,6}m`. The "equivalent mean radius", in the sense of BPEL on the vertical section of the veil is therefore equal to :math:`{r}_{m}\mathrm{=}\mathrm{0,283}m`, knowing that:

:math:`{r}_{m}\mathrm{=}\frac{\mathit{eH}}{2(e+H)}` (reference BPEL 2.1.5)

The cables each describe a semicircle in a horizontal plane, and thus travel along the length of the veil. The dimensions of the plans containing the cables are:


* for cable 1: :math:`{z}_{1}\mathrm{=}1m`;


* for cable no. 2: :math:`{z}_{2}=\mathrm{3,5}m`;


* for cable no. 3: :math:`{z}_{3}=6m`;


* for cable no. 4: :math:`{z}_{4}=\mathrm{8,5}m`.

The cables 3 and 4 have an eccentricity with respect to the mean radius of the veil, equal respectively to:


* :math:`{\mathrm{ex}}_{3}=\mathrm{0,05}m`;


* :math:`{\mathrm{ex}}_{4}=\mathrm{0,1}m`.

The cross-sectional area of each cable is :math:`{S}_{a}=\mathrm{1,5}{.10}^{-4}{m}^{2}`.


Material properties
------------------------


Material: concrete constituting the veil
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Elastic properties:


.. csv-table::

    "Young's module", ":math:`{E}_{b}={3.10}^{10}\mathrm{Pa}`"
    "Poisson's Ratio", ":math:`{\nu }_{b}=\mathrm{0,2}`"


Characteristic parameters for estimating voltage losses:


.. csv-table::

    "Flat rate of tension loss due to concrete creep", ":math:`{x}_{\mathrm{flu}}=\mathrm{0,07}`"
    "Flat rate of stress loss due to concrete shrinkage", ":math:`{x}_{\mathrm{ret}}=\mathrm{0,08}`"



Material: steel constituting the cables
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Elastic properties:


.. csv-table::

    "Young's module", ":math:`{E}_{a}=\mathrm{2,1}{.10}^{11}\mathrm{Pa}`"
    "Poisson's Ratio", ":math:`{\nu }_{a}\mathrm{=}\mathrm{0,3}`"



+--------------------------------------------------------+---------------------------------------------------------------------+
|Characteristic parameters for estimating voltage losses:                                                                      |
+--------------------------------------------------------+---------------------------------------------------------------------+
|Relaxation of steel at 1000 hours                       |:math:`{\rho }_{1000}\mathrm{=}2\text{\%}`                           |
+--------------------------------------------------------+---------------------------------------------------------------------+
|Dimensional relaxation coefficient of prestressed steel |:math:`{\mu }_{0}\mathrm{=}\mathrm{0,3}`                             |
+--------------------------------------------------------+---------------------------------------------------------------------+
|Elastic limit stress of steel                           |:math:`{f}_{\mathit{prg}}\mathrm{=}\mathrm{1,77}{.10}^{9}\mathit{Pa}`|
+--------------------------------------------------------+---------------------------------------------------------------------+
|Curved friction coefficient                             |:math:`f=\mathrm{0,2}{\mathrm{rad}}^{-1}`                            |
+--------------------------------------------------------+---------------------------------------------------------------------+
|Tension loss coefficient per unit length                |:math:`\varphi \mathrm{=}{3.10}^{\mathrm{-}3}{m}^{\mathrm{-}1}`      |
+--------------------------------------------------------+---------------------------------------------------------------------+


Loading
----------

A normal force of traction is applied to both ends of each cable. The value of the voltage applied is :math:`{F}_{0}\mathrm{=}{2.10}^{5}N`.

To evaluate the voltage losses due to cable relaxation over time, the following relationships are used:


.. csv-table::

    ":math:`\Delta {\sigma }_{\mathit{pj}}\mathrm{=}\Delta {\sigma }_{p}(x)r(j)` ", "(reference BPEL 3.3,24)"
    ":math:`\Delta {\sigma }_{p}\mathrm{=}\frac{6}{100}{\rho }_{1000}(\frac{{\sigma }_{\mathit{pi}}(x)}{{f}_{\mathit{prg}}}\mathrm{-}{\mu }_{0}){\sigma }_{\mathit{pi}}(x)` ", "(reference BPEL 3.3,23)"
    ":math:`r(j)\mathrm{=}\frac{j}{j+9\mathrm{\times }{r}_{m}}` ", "(reference BPEL 3.3,24 and 2.1,51)"



.. csv-table::

    ":math:`{\sigma }_{\mathit{pi}}(x)` ", "called initial voltage, the voltage at the :math:`x` abscissa point, after instantaneous voltage losses."
    ":math:`j` ", "evaluation moment, in days"
    ":math:`{r}_{m}` ", "equivalent mean radius, in :math:`\mathit{cm}`"


The characteristics are evaluated on day :math:`j\mathrm{=}10`.

To assess the voltage losses in the vicinity of the anchorages, a setback at anchors :math:`\Delta \mathrm{=}{5.10}^{\mathrm{-}4}m` is taken into account.

**Note:**

*This problem ignores the resolution of the balance of the complete steel-concrete structure and is limited to the determination according to the BPEL of the prestress in the cables.*