Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The results considered to be reference were obtained by SAMCEF using 60 increments for each loading step. The characteristics of the various resolution methods are as follows:


.. csv-table::

    "A", "Using the elastic matrix."
    "B", "Using the tangent matrix; this is only reevaluated at the first iteration of each increment (REAC_INCR = 1 and REAC_ITER = 0). It's the modified Newton method."
    "C", "Using the tangent matrix; this is reevaluated at each iteration of each increment (REAC_INCR = 1 and REAC_ITER = 1). It's the classical Newton method."



Benchmark results
----------------------


+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|Results achieved with 60 increments for each loading step                                                                                                                                                                                                |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|                                                         |Element 1                                                      |Element 2                                                      |Element 3                                                      |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|Force                                                    |:math:`{\sigma }^{\mathrm{xx}}`|:math:`{\sigma }^{\mathrm{yy}}`|:math:`{\sigma }^{\mathrm{xx}}`|:math:`{\sigma }^{\mathrm{yy}}`|:math:`{\sigma }^{\mathrm{xx}}`|:math:`{\sigma }^{\mathrm{yy}}`|
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|3.00                                                     |1.500000D+00                   |6.938894D—18                   |9.000000D—01                   |—1.040834D—17                  |6.000000D—01                   |—6.938894D—18                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|6.00                                                     |3.000000D+00                   |4.861944D—13                   |1.800000D+00                   |—2.081668D—17                  |1.200000D+00                   |—3.469447D—18                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|9.00                                                     |3.147155D+00                   |3.199571D—01                   |3.511707D+00                   |—1.900098D—01                  |2.341138D+00                   |—1.279828D—01                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|12.95                                                    |3.252919D+00                   |5.950074D—01                   |5.814267D+00                   |—3.523377D—01                  |3.878832D+00                   |—2.380030D—01                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|15.00                                                    |3.213822D+00                   |4.873069D—01                   |6.017834D+00                   |3.174572D—02                   |5.768340D+00                   |—5.231355D—01                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
|16.93                                                    |3.2092970D+00                  |4.753345D—01                   |6.149462D+00                   |3.048490D—01                   |7.571241D+00                   |—7.863557D—01                  |
+---------------------------------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+


Uncertainty about the solution
---------------------------

Uncertainty less than :math:`\text{1 \%}`.


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


1. Fundamental tests for two and three dimensional, small strain, elastoplastic finite element analysis, 1987, NAFEMS