.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/beams/plot_05_macaulay_singularity_functions.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_structural_engineering_beams_plot_05_macaulay_singularity_functions.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_structural_engineering_beams_plot_05_macaulay_singularity_functions.py:


Macaulay's singularity functions (1919)
=======================================

Integrating :math:`EIy'' = M` for a beam with several loads used to mean a
separate moment equation, and two constants of integration, for every
stretch between loads, all joined by continuity conditions. William Herrick
Macaulay (1919), following Clebsch (1862), wrote the moment once for the
whole beam with brackets :math:`\langle x - a\rangle^n` that vanish to the
left of :math:`a`, so only two constants remain.
:class:`~engineeringkit.structural_engineering.Beam` adds the unknown
reactions to those two constants and solves equilibrium and the support
conditions together, so statically indeterminate beams need no special
treatment. The example solves a propped cantilever under a point load, a
couple and a triangular load, and checks it against the older method of
superposition (:func:`~engineeringkit.structural_engineering.superpose_beam_results`).

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import Beam, superpose_beam_results








.. GENERATED FROM PYTHON SOURCE LINES 26-28

A propped cantilever under three kinds of load
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-39

.. code-block:: Python


    L, E, I = 8.0, 200e9, 1.2e-4
    loads = {"point_loads": [(3.0, 40e3)], "moments": [(6.0, 30e3)], "distributed_loads": [(4.0, 8.0, 0.0, 15e3)]}
    propped = Beam(L, E, I, ((0.0, "fixed"), (L, "roller")))
    result = propped.solve(**loads, n=801)
    (RA, RB), (MA,) = result.reactions, result.support_moments
    print(f"reactions {RA / 1e3:.2f} kN and {RB / 1e3:.2f} kN, wall moment {MA / 1e3:.2f} kN m")
    print(f"largest deflection {result.deflection.min() * 1e3:.2f} mm at x = {result.x[np.argmin(result.deflection)]:.2f} m")
    assert np.isclose(RA + RB, 40e3 + 15e3 * 4 / 2)
    assert abs(result.deflection[0]) < 1e-15 and abs(result.slope[0]) < 1e-15 and abs(result.deflection[-1]) < 1e-15





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    reactions 34.66 kN and 35.34 kN, wall moment -67.25 kN m
    largest deflection -7.39 mm at x = 4.39 m




.. GENERATED FROM PYTHON SOURCE LINES 40-45

The same beam by superposition
------------------------------

Remove the roller: the cantilever under the loads deflects at its tip,
and the redundant reaction is the tip force that pushes it back.

.. GENERATED FROM PYTHON SOURCE LINES 45-63

.. code-block:: Python


    cantilever = Beam(L, E, I, ((0.0, "fixed"),))
    free = cantilever.solve(**loads, n=801)
    unit = cantilever.solve(point_loads=[(L, -1.0)], n=801)  # an upward unit force at the tip
    R = -free.deflection[-1] / unit.deflection[-1]
    combined = superpose_beam_results(free, cantilever.solve(point_loads=[(L, -R)], n=801))
    print(f"redundant by superposition: {R / 1e3:.2f} kN")
    assert np.isclose(R, RB)
    assert np.allclose(combined.deflection, result.deflection, atol=1e-12)

    fig, axes = result.plot()
    axes[0].set_ylabel("shear [N]")
    axes[1].set_ylabel("moment [N m]")
    axes[2].set_ylabel("deflection [m]")
    axes[0].set_title("Propped cantilever by singularity functions")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_05_macaulay_singularity_functions_001.png
   :alt: Propped cantilever by singularity functions
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_05_macaulay_singularity_functions_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    redundant by superposition: 35.34 kN





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.125 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_beams_plot_05_macaulay_singularity_functions.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/structural_engineering/beams/plot_05_macaulay_singularity_functions.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_05_macaulay_singularity_functions.ipynb <plot_05_macaulay_singularity_functions.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_05_macaulay_singularity_functions.py <plot_05_macaulay_singularity_functions.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_05_macaulay_singularity_functions.zip <plot_05_macaulay_singularity_functions.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_