.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/members/plot_01_hooke_axial.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_members_plot_01_hooke_axial.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_members_plot_01_hooke_axial.py:


Hooke's law: ut tensio, sic vis (1678)
======================================

In 1676 Robert Hooke published an anagram, *ceiiinosssttuu*, and in 1678 its
solution, *ut tensio, sic vis*: "as the extension, so the force". Springs,
wires and wooden rods all stretched in proportion to the weight hung on
them. Written for a bar of length :math:`L`, area :math:`A` and modulus
:math:`E` (Young's, 1807), the extension is :math:`\delta = PL/(AE)`
(:func:`~engineeringkit.structural_engineering.axial_deformation`) and the
stored energy is the triangle under the load-extension line,
:math:`U = P\delta/2 = P^2L/(2AE)`
(:func:`~engineeringkit.structural_engineering.strain_energy_axial`). The
example repeats Hooke's experiment on a steel wire and checks both.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

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

    from engineeringkit.structural_engineering import axial_deformation, strain_energy_axial








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Weights on a wire
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 27-38

.. code-block:: Python


    L, d, E = 2.0, 1e-3, 200e9  # a 2 m steel wire, 1 mm diameter
    A = np.pi * d**2 / 4
    loads = 9.81 * np.arange(0, 11) * 2.0  # 0 to 20 kg in 2 kg steps [N]
    stretch = axial_deformation(loads, L, A, E)
    for P, dl in list(zip(loads, stretch))[::5]:
        print(f"load {P:6.1f} N: extension {1e3 * dl:.3f} mm")
    stiffness = np.polyfit(stretch, loads, 1)[0]
    print(f"measured stiffness {stiffness / 1e3:.1f} kN/m = AE/L = {A * E / L / 1e3:.1f} kN/m")
    assert np.isclose(stiffness, A * E / L)





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

 .. code-block:: none

    load    0.0 N: extension 0.000 mm
    load   98.1 N: extension 1.249 mm
    load  196.2 N: extension 2.498 mm
    measured stiffness 78.5 kN/m = AE/L = 78.5 kN/m




.. GENERATED FROM PYTHON SOURCE LINES 39-41

Energy is the area under the line
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-56

.. code-block:: Python


    P_max = loads[-1]
    area = simpson(loads, x=stretch)
    U = strain_energy_axial(P_max, L, A, E)
    print(f"area under the load-extension line {1e3 * area:.3f} mJ, P^2 L/(2AE) = {1e3 * U:.3f} mJ")
    assert np.isclose(area, U) and np.isclose(U, P_max * stretch[-1] / 2)

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(1e3 * stretch, loads, "o-")
    ax.fill_between(1e3 * stretch, loads, alpha=0.2, label=f"U = {1e3 * U:.1f} mJ")
    ax.set(xlabel="extension [mm]", ylabel="load [N]", title="Ut tensio, sic vis")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/members/images/sphx_glr_plot_01_hooke_axial_001.png
   :alt: Ut tensio, sic vis
   :srcset: /api/gallery/structural_engineering/members/images/sphx_glr_plot_01_hooke_axial_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    area under the load-extension line 245.063 mJ, P^2 L/(2AE) = 245.063 mJ





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

   **Total running time of the script:** (0 minutes 0.029 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_members_plot_01_hooke_axial.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/members/plot_01_hooke_axial.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hooke_axial.ipynb <plot_01_hooke_axial.ipynb>`

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

      :download:`Download Python source code: plot_01_hooke_axial.py <plot_01_hooke_axial.py>`

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

      :download:`Download zipped: plot_01_hooke_axial.zip <plot_01_hooke_axial.zip>`


.. only:: html

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

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