.. 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_01_galileo_cantilever.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_01_galileo_cantilever.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_01_galileo_cantilever.py:


Galileo's cantilever (1638)
===========================

In the *Two New Sciences* (1638) Galileo asked how strong a cantilever is,
and answered that its breaking moment grows with the width :math:`b` and the
*square* of the depth :math:`h`, so a plank on edge is stronger than one laid
flat by the ratio :math:`h/b`. He assumed the whole section pulls with the
same stress and pivots about its bottom edge, which gives
:math:`M = \sigma bh^2/2`. Linear elasticity (Navier, 1826) puts the pivot at
the centroid with a linear stress profile, :math:`M = \sigma bh^2/6`
(:func:`~engineeringkit.structural_engineering.rectangular_section_modulus`):
Galileo's scaling law was right, his constant three times too large.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import simpson

    from engineeringkit.structural_engineering import rectangular_second_moment, rectangular_section_modulus








.. GENERATED FROM PYTHON SOURCE LINES 24-26

A plank on edge and laid flat
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-36

.. code-block:: Python


    b, h, sigma = 0.05, 0.20, 10e6  # 50 x 200 mm timber, 10 MPa
    for name, (width, depth) in {"on edge": (b, h), "flat": (h, b)}.items():
        Z = rectangular_section_modulus(width, depth)
        print(f"{name:8s}: Galileo M = {sigma * width * depth**2 / 2 / 1e3:5.1f} kN m, elastic M = {sigma * Z / 1e3:5.2f} kN m")
    ratio = rectangular_section_modulus(b, h) / rectangular_section_modulus(h, b)
    stiffness_ratio = rectangular_second_moment(b, h) / rectangular_second_moment(h, b)
    print(f"strength ratio edge/flat = {ratio:.1f} (h/b = {h / b:.1f}), stiffness ratio = {stiffness_ratio:.0f}")
    assert np.isclose(ratio, h / b) and np.isclose(stiffness_ratio, (h / b) ** 2)





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

 .. code-block:: none

    on edge : Galileo M =  10.0 kN m, elastic M =  3.33 kN m
    flat    : Galileo M =   2.5 kN m, elastic M =  0.83 kN m
    strength ratio edge/flat = 4.0 (h/b = 4.0), stiffness ratio = 16




.. GENERATED FROM PYTHON SOURCE LINES 37-41

Where Galileo's factor of three comes from
------------------------------------------

Integrate each assumed stress profile over the depth to get the moment.

.. GENERATED FROM PYTHON SOURCE LINES 41-60

.. code-block:: Python


    y = np.linspace(-h / 2, h / 2, 2001)
    galileo = np.full_like(y, sigma)  # uniform tension, pivot at the bottom edge
    elastic = sigma * y / (h / 2)  # linear, pivot at the centroid
    M_galileo = simpson(galileo * b * (y + h / 2), x=y)
    M_elastic = simpson(elastic * b * y, x=y)
    print(f"Galileo / elastic = {M_galileo / M_elastic:.3f}")
    assert np.isclose(M_galileo / M_elastic, 3, rtol=1e-4)
    assert np.isclose(M_elastic, sigma * rectangular_section_modulus(b, h), rtol=1e-4)

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(galileo / 1e6, y * 1e3, label="Galileo: uniform tension")
    ax.plot(elastic / 1e6, y * 1e3, label="Navier: linear about centroid")
    ax.axvline(0, color="gray", lw=0.5)
    ax.set(xlabel="stress at failure [MPa]", ylabel="height in section [mm]", title="Stress across the root section")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_01_galileo_cantilever_001.png
   :alt: Stress across the root section
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_01_galileo_cantilever_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Galileo / elastic = 3.000





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

   **Total running time of the script:** (0 minutes 0.066 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_beams_plot_01_galileo_cantilever.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_01_galileo_cantilever.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_galileo_cantilever.ipynb <plot_01_galileo_cantilever.ipynb>`

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

      :download:`Download Python source code: plot_01_galileo_cantilever.py <plot_01_galileo_cantilever.py>`

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

      :download:`Download zipped: plot_01_galileo_cantilever.zip <plot_01_galileo_cantilever.zip>`


.. only:: html

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

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