.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/vessels/plot_01_lame_thick_cylinder.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_vessels_plot_01_lame_thick_cylinder.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_vessels_plot_01_lame_thick_cylinder.py:


Lamé's thick-walled cylinder (1833)
===================================

The hoop stress :math:`pr/t` of a boiler shell
(:func:`~engineeringkit.structural_engineering.thin_wall_vessel_stresses`)
assumes the stress is uniform through the wall. Gabriel Lamé and Émile
Clapeyron, in their memoir on the equilibrium of elastic solids (written
in Russia, published in Paris in 1833), solved the hollow cylinder exactly:
:math:`\sigma_r = A - B/r^2` and :math:`\sigma_\theta = A + B/r^2`
(:func:`~engineeringkit.structural_engineering.thick_wall_cylinder_stresses`).
The hoop stress is highest at the bore, and no thickness of wall can carry
an internal pressure above the material's strength. The example plots the
stresses through a gun-barrel wall and finds the radius ratio at which the
thin-wall formula is 5% low.

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq

    from engineeringkit.structural_engineering import thick_wall_cylinder_stresses, thin_wall_vessel_stresses








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Stresses through a thick wall
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-45

.. code-block:: Python


    ri, ro, p = 0.05, 0.1, 200e6
    r = np.linspace(ri, ro, 201)
    radial, hoop, axial = thick_wall_cylinder_stresses(r, ri, ro, p)
    print(f"hoop stress {hoop[0] / 1e6:.0f} MPa at the bore, {hoop[-1] / 1e6:.0f} MPa outside")
    print(f"thin-wall formula {p * ri / (ro - ri) / 1e6:.0f} MPa")
    assert np.isclose(hoop[0], p * (ro**2 + ri**2) / (ro**2 - ri**2))
    assert np.isclose(np.trapezoid(hoop, r), p * ri)  # the wall carries the bursting force p * 2 ri per unit length

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(r * 1e3, hoop / 1e6, label="hoop")
    ax.plot(r * 1e3, radial / 1e6, label="radial")
    ax.plot(r * 1e3, axial / 1e6, label="axial (closed ends)")
    ax.set(xlabel="radius [mm]", ylabel="stress [MPa]", title=f"Lamé cylinder, p = {p / 1e6:.0f} MPa")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/structural_engineering/vessels/images/sphx_glr_plot_01_lame_thick_cylinder_001.png
   :alt: Lamé cylinder, p = 200 MPa
   :srcset: /api/gallery/structural_engineering/vessels/images/sphx_glr_plot_01_lame_thick_cylinder_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    hoop stress 333 MPa at the bore, 133 MPa outside
    thin-wall formula 200 MPa




.. GENERATED FROM PYTHON SOURCE LINES 46-48

When is a wall thin?
--------------------

.. GENERATED FROM PYTHON SOURCE LINES 48-68

.. code-block:: Python



    def error(ratio, t=1.0):
        """Shortfall of the thin-wall hoop stress against Lamé's peak, for inner radius / thickness."""
        exact = thick_wall_cylinder_stresses(ratio * t, ratio * t, (ratio + 1) * t, 1.0)[1]
        return 1 - thin_wall_vessel_stresses(1.0, ratio * t, t)[0] / exact


    ratios = np.linspace(1, 40, 200)
    limit = brentq(lambda k: error(k) - 0.05, 1, 40)
    print(f"the thin-wall hoop stress is 5% low at r/t = {limit:.1f}")
    assert 9 < limit < 11

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(ratios, 100 * np.array([error(k) for k in ratios]))
    ax.axvline(limit, color="gray", ls="--")
    ax.set(xlabel="inner radius / wall thickness", ylabel="thin-wall error [%]")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/vessels/images/sphx_glr_plot_01_lame_thick_cylinder_002.png
   :alt: plot 01 lame thick cylinder
   :srcset: /api/gallery/structural_engineering/vessels/images/sphx_glr_plot_01_lame_thick_cylinder_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    the thin-wall hoop stress is 5% low at r/t = 10.0





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

   **Total running time of the script:** (0 minutes 0.121 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_vessels_plot_01_lame_thick_cylinder.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/vessels/plot_01_lame_thick_cylinder.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_lame_thick_cylinder.ipynb <plot_01_lame_thick_cylinder.ipynb>`

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

      :download:`Download Python source code: plot_01_lame_thick_cylinder.py <plot_01_lame_thick_cylinder.py>`

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

      :download:`Download zipped: plot_01_lame_thick_cylinder.zip <plot_01_lame_thick_cylinder.zip>`


.. only:: html

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

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