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




Geometry
---------

This non-regression test case represents a three-point test of a thick beam. The beam is represented by a rectangular element 7.5 meters long and 4.7 meters high. The element contains a square hopper with a side of 1.5 meters located according to Figure 1.1.-a. A fixed support is placed at the left end of the base of the element and a vertical support is considered to be placed at the right end. A vertical load equal to 3 MN is applied in the direction of gravity at 4.7 meters from the left edge. The geometry, boundary conditions, and material definitions were taken directly from the reference [:ref:`Shlaich et al., 1987 <Shlaich et al., 1987>`].

P=3 MIN


.. image:: images/10000000000001FF0000018415022DC70A8F7BD4.png
    :width: 5.3228in
    :height: 4.0417in


.. _RefImage_10000000000001FF0000018415022DC70A8F7BD4.png:

Image 1.1-1: Sail girder with hopper




Materials
---------

The reinforced concrete used for plane modeling corresponds to a linear elastic material with the following characteristics:

Concrete:

.. csv-table::

    "Young's modulus", ":math:`E` ", "21 GPa"
    "Poisson's Ratio", ":math:`\nu` ", "0.2"
    "Elastic limit", ":math:`{f}_{\mathit{cm}}` ", "36 MPa"


**Table** 1.2-1

Reinforcing steel:


.. csv-table::

    "Young's modulus", ":math:`E` ", "210 GPa"
    "Elastic limit", ":math:`{f}_{y}` ", "500 MPa"


**Table** 1.2-2




Boundary conditions and loading
------------------------------------

The beam is assumed to have a fixed support at the level of the left side and a carriage-type support (locked in the vertical direction) at the level of the level of the right side. The load and boundary conditions are taken as shown in Figure 1.1-1.




Characteristics of the mesh
----------------------------

Number of knots: 13504

Number of meshes and types: 13200 meshes QUAD4




Benchmark solution
---------------------




Diagram of the Bielles-Tirants truss
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

One of the solutions proposed in reference [:ref:`Schlaich et al., 1987 <Schlaich et al., 1987>`] is shown in Figure 2.1-a. The author approaches the problem via an *ad hoc* definition of the Bielles-Tirants model.


.. image:: images/100000000000020D0000017E1C570449EE3CD749.png
    :width: 5.4689in
    :height: 3.9791in


.. _RefImage_100000000000020D0000017E1C570449EE3CD749.png:

Figure 1.5-1: BT reference model [:ref:`Shclaich et al., 1987 <Shclaich et al., 1987>`]




Benchmark results
~~~~~~~~~~~~~~~~~~~~~~~~

The tie rod sections are shown in the following table:


.. csv-table::

    "Pulling", "Knot i", "Knot j", "Force [MN]", "Length [m] *", "As [:ref:`m2 <m2>`]*"
    "T2", "5", "6", "1.33", "1.33", "5.6", "0.0024"
    "T1", "0", "5", "1.07", "1.07", "3", "0.0027"


**Table** 1.5-1: Tie rod sections [:ref:`Shclaich et al. 1897 <Shclaich et al. 1897>`]