.. 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_02_navier_beam.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_02_navier_beam.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_02_navier_beam.py:


Navier's beam theory (1826)
===========================

Navier's 1826 lectures at the École des ponts et chaussées put beam theory
in its modern form: plane sections stay plane, the neutral axis passes
through the centroid, the stress is :math:`\sigma = My/I`
(:func:`~engineeringkit.structural_engineering.bending_stress`) and the
curvature is :math:`y'' = M/(EI)`. The example loads a steel beam
(:class:`~engineeringkit.structural_engineering.SimplySupportedBeam`), checks
the midspan deflection :math:`5wL^4/(384EI)`, and checks that the stress
profile carries no net force and exactly the applied moment.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

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

    from engineeringkit.structural_engineering import SimplySupportedBeam, bending_stress, rectangular_second_moment








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Shear, moment and deflection
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-40

.. code-block:: Python


    L, E, w = 6.0, 200e9, 8e3
    b, h = 0.10, 0.30
    I = rectangular_second_moment(b, h)
    result = SimplySupportedBeam(L, E, I).solve_uniform_load(w, n=601)
    summary = result.summary()
    print(f"max moment {summary['max_abs_moment'] / 1e3:.1f} kN m, midspan deflection {1e3 * summary['max_abs_deflection']:.2f} mm")
    assert np.isclose(summary["max_abs_moment"], w * L**2 / 8)
    assert np.isclose(summary["max_abs_deflection"], 5 * w * L**4 / (384 * E * I))

    fig, axes = result.plot()
    for ax, label in zip(axes, ["shear [N]", "moment [N m]", "deflection [m]"]):
        ax.set_ylabel(label)
    fig.tight_layout()




.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_02_navier_beam_001.png
   :alt: plot 02 navier beam
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_02_navier_beam_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max moment 36.0 kN m, midspan deflection 3.00 mm




.. GENERATED FROM PYTHON SOURCE LINES 41-45

The stress across the midspan section
-------------------------------------

With ``y`` measured downward, a sagging moment puts the bottom in tension.

.. GENERATED FROM PYTHON SOURCE LINES 45-63

.. code-block:: Python


    M = summary["max_abs_moment"]
    y = np.linspace(-h / 2, h / 2, 401)
    sigma = bending_stress(M, y, I)
    net_force = simpson(sigma * b, x=y)
    resisting_moment = simpson(sigma * b * y, x=y)
    print(f"peak stress {sigma.max() / 1e6:.1f} MPa, net force {net_force:.2e} N, moment {resisting_moment / 1e3:.1f} kN m")
    assert abs(net_force) < 1e-6 * M / h
    assert np.isclose(resisting_moment, M)

    fig, ax = plt.subplots(figsize=(4, 4))
    ax.plot(sigma / 1e6, y * 1e3)
    ax.axvline(0, color="gray", lw=0.5)
    ax.invert_yaxis()
    ax.set(xlabel="stress [MPa]", ylabel="depth below neutral axis [mm]", title="Midspan section")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_02_navier_beam_002.png
   :alt: Midspan section
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_02_navier_beam_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    peak stress 24.0 MPa, net force 0.00e+00 N, moment 36.0 kN m





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

   **Total running time of the script:** (0 minutes 0.063 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_beams_plot_02_navier_beam.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_02_navier_beam.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_navier_beam.ipynb <plot_02_navier_beam.ipynb>`

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

      :download:`Download Python source code: plot_02_navier_beam.py <plot_02_navier_beam.py>`

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

      :download:`Download zipped: plot_02_navier_beam.zip <plot_02_navier_beam.zip>`


.. only:: html

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

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