.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/cables/plot_01_hooke_inverted_chain.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_cables_plot_01_hooke_inverted_chain.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_cables_plot_01_hooke_inverted_chain.py:


Hooke's inverted chain (1675)
=============================

Robert Hooke hid his solution to the shape of an arch in a Latin anagram
at the end of *A Description of Helioscopes* (1675); decoded after his
death, it read *ut pendet continuum flexile, sic stabit contiguum rigidum
inversum*: as hangs the flexible line, so but inverted will stand the rigid
arch. A hanging cable can only pull, so it takes the shape of its loads'
moment diagram; turned upside down, the same shape is an arch that only
pushes. Christopher Wren used the idea for the dome of St Paul's. For a
three-hinged arch (:func:`~engineeringkit.structural_engineering.three_hinged_arch`)
the moment is the simple-beam moment less the thrust times the rise,
:math:`M = M_0 - Hy`, and on a parabolic arch under a load uniform along
the span it vanishes everywhere. The example checks this, then moves the
load off the crown to show the bending that an arch of the wrong shape must
carry.

.. GENERATED FROM PYTHON SOURCE LINES 21-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import parabolic_cable, three_hinged_arch








.. GENERATED FROM PYTHON SOURCE LINES 27-29

The cable and the arch carry the same load with the same force
--------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    L, f, w = 40.0, 8.0, 20e3
    arch = three_hinged_arch(L, f, w=w)
    H_cable, T_cable, _ = parabolic_cable(w, L, f)
    print(f"arch thrust {arch.thrust / 1e3:.0f} kN, cable tension {H_cable / 1e3:.0f} kN")
    print(f"largest arch moment {np.max(np.abs(arch.moment)):.1e} N m")
    print(f"springing compression {-arch.axial_force[0] / 1e3:.0f} kN, cable support tension {T_cable / 1e3:.0f} kN")
    assert np.isclose(arch.thrust, H_cable)
    assert np.isclose(-arch.axial_force[0], T_cable)
    assert np.max(np.abs(arch.moment)) < 1e-6 * w * L**2





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

 .. code-block:: none

    arch thrust 500 kN, cable tension 500 kN
    largest arch moment 1.9e-09 N m
    springing compression 640 kN, cable support tension 640 kN




.. GENERATED FROM PYTHON SOURCE LINES 41-43

A point load is not funicular for a parabola
--------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-67

.. code-block:: Python


    P = 400e3
    patterns = {
        "uniform load": {"w": w},
        "point load at quarter span": {"point_loads": [(L / 4, P)]},
        "both": {"w": w, "point_loads": [(L / 4, P)]},
    }
    fig, axes = plt.subplots(2, 1, sharex=True, figsize=(7, 5))
    axes[0].plot(arch.x, arch.y, color="k")
    axes[0].plot([0, L / 2, L], [0, f, 0], "o", mfc="white", color="k")
    axes[0].plot(arch.x, f - 4 * f * arch.x * (L - arch.x) / L**2 - f, ":", color="gray", label="hanging cable, inverted")
    axes[0].set(ylabel="y [m]", title="Three-hinged parabolic arch")
    axes[0].set_aspect("equal")
    axes[0].legend(loc="lower right")
    for label, kwargs in patterns.items():
        r = three_hinged_arch(L, f, **kwargs)
        axes[1].plot(r.x, r.moment / 1e3, label=label)
        assert np.allclose(r.moment[[0, -1]], 0) and np.isclose(np.interp(L / 2, r.x, r.moment), 0, atol=1e-6)
    axes[1].axhline(0, color="gray", lw=0.5)
    axes[1].set(xlabel="x [m]", ylabel="moment [kN m]")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/cables/images/sphx_glr_plot_01_hooke_inverted_chain_001.png
   :alt: Three-hinged parabolic arch
   :srcset: /api/gallery/structural_engineering/cables/images/sphx_glr_plot_01_hooke_inverted_chain_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.098 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_cables_plot_01_hooke_inverted_chain.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/cables/plot_01_hooke_inverted_chain.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hooke_inverted_chain.ipynb <plot_01_hooke_inverted_chain.ipynb>`

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

      :download:`Download Python source code: plot_01_hooke_inverted_chain.py <plot_01_hooke_inverted_chain.py>`

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

      :download:`Download zipped: plot_01_hooke_inverted_chain.zip <plot_01_hooke_inverted_chain.zip>`


.. only:: html

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

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