Benchmark solution
=====================

The aim is to validate the use of linear elements to represent the steels present in concrete represented by solid elements.

Two models are possible:


* Modeling A: A quadratic solid element contains a linear bar element. The middle nodes of quadratic solid elements are not connected to the members (:math:`\text{L}`).


* B modeling: A quadratic volume element contains two linear bar elements. All volume nodes are connected to steel nodes (:math:`\text{LL}`).

The reference solution was obtained with CASTEM 2000. The mesh used contains the same number of cells but the bar elements modeling the reinforcements are quadratic (:math:`\text{Q }`).

The results in terms of stress in concrete, displacement and force in steel are compared.


.. image:: images/10000201000005520000036A4F8E10F425C59469.png
    :width: 5.1181in
    :height: 3.2756in


.. _RefImage_10000201000005520000036A4F8E10F425C59469.png:

Figure 1: Movement along the beam


.. image:: images/10000201000005180000036EA894512B9698E80B.png
    :width: 5.1181in
    :height: 3.4366in


.. _RefImage_10000201000005180000036EA894512B9698E80B.png:

Figure 2: Stresses in concrete


.. image:: images/100002010000051A0000035CE009FAC419FB5C15.png
    :width: 5.1181in
    :height: 3.3661in


.. _RefImage_100002010000051A0000035CE009FAC419FB5C15.png:

Figure 3: Effort in the bar

As a result, modeling B (:math:`\text{LL}`) is very close to the quadratic Castem model :math:`\text{Q}`. Modeling A (:math:`\text{L}`) is not sufficient and we can observe stress oscillations in concrete.

The model (:math:`\text{LL}`) is therefore recommended for modeling bar elements in conjunction with solid quadratic elements.


Bibliographical references
---------------------------


1. NECS. NOTE D'ETUDE: Reinforced concrete modeling study: quadratic concrete mesh with linear bar element. N001_A301_2012_ ET_EDF. 2012.