.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/stress/plot_01_mohr_circle.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_stress_plot_01_mohr_circle.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_stress_plot_01_mohr_circle.py:


Mohr's circle (1882)
====================

Otto Mohr (1882) showed that the normal and shear stresses on every plane
through a point, plotted against each other, fall on a single circle. Its
centre is the mean normal stress, its radius the largest in-plane shear
stress, and its ends on the axis are the principal stresses
(:func:`~engineeringkit.structural_engineering.mohr_circle`). Turning the
element by :math:`\theta` moves the point by :math:`2\theta` around the
circle (:func:`~engineeringkit.structural_engineering.stress_transformation`).
The example traces the circle by rotating an element, checks that the
principal stresses are the eigenvalues of the stress tensor, and finds the
plane of maximum shear.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import mohr_circle, stress_transformation
    from engineeringkit.structural_engineering.visualizers import plot_mohr_circle








.. GENERATED FROM PYTHON SOURCE LINES 25-30

A shaft under bending and torsion
---------------------------------

At the surface of a shaft, bending gives :math:`\sigma_x` and torsion
gives :math:`\tau_{xy}`; there is no stress across the shaft.

.. GENERATED FROM PYTHON SOURCE LINES 30-42

.. code-block:: Python


    sx, sy, txy = 120e6, 0.0, 50e6
    center, radius, s1, s2 = mohr_circle(sx, sy, txy)
    print(f"centre {center / 1e6:.1f} MPa, radius {radius / 1e6:.1f} MPa, principal {s1 / 1e6:.1f} and {s2 / 1e6:.1f} MPa")
    eigen = np.linalg.eigvalsh([[sx, txy], [txy, sy]])
    assert np.allclose([s2, s1], eigen)

    theta = np.linspace(0, np.pi, 181)
    sxp, syp, txyp = stress_transformation(sx, sy, txy, theta)
    assert np.allclose(np.hypot(sxp - center, txyp), radius)
    assert np.allclose(sxp + syp, sx + sy)  # the first invariant





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

 .. code-block:: none

    centre 60.0 MPa, radius 78.1 MPa, principal 138.1 and -18.1 MPa




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Principal planes and the plane of maximum shear
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-66

.. code-block:: Python


    theta_p = 0.5 * np.arctan2(2 * txy, sx - sy)
    on_principal = stress_transformation(sx, sy, txy, theta_p)
    on_shear = stress_transformation(sx, sy, txy, theta_p + np.pi / 4)
    print(f"principal plane at {np.degrees(theta_p):.1f} deg: shear {on_principal[2]:.1e} Pa")
    print(f"max-shear plane at {np.degrees(theta_p) + 45:.1f} deg: shear {on_shear[2] / 1e6:.1f} MPa")
    assert np.isclose(on_principal[0], s1) and abs(on_principal[2]) < 1e-6 * radius
    assert np.isclose(abs(on_shear[2]), radius) and np.isclose(on_shear[0], center)

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    plot_mohr_circle(sx, sy, txy, ax=axes[0])
    axes[0].plot(sxp[::15], txyp[::15], ".", color="k", label="rotated element")
    axes[0].set_title("Mohr's circle")
    axes[1].plot(np.degrees(theta), sxp / 1e6, label=r"$\sigma_{x'}$")
    axes[1].plot(np.degrees(theta), txyp / 1e6, label=r"$\tau_{x'y'}$")
    axes[1].axvline(np.degrees(theta_p), color="gray", ls="--")
    axes[1].set(xlabel="element rotation [deg]", ylabel="stress [MPa]", title="Stress transformation")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/stress/images/sphx_glr_plot_01_mohr_circle_001.png
   :alt: Mohr's circle, Stress transformation
   :srcset: /api/gallery/structural_engineering/stress/images/sphx_glr_plot_01_mohr_circle_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    principal plane at 19.9 deg: shear 0.0e+00 Pa
    max-shear plane at 64.9 deg: shear -78.1 MPa





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

   **Total running time of the script:** (0 minutes 0.059 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_stress_plot_01_mohr_circle.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/stress/plot_01_mohr_circle.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_mohr_circle.ipynb <plot_01_mohr_circle.ipynb>`

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

      :download:`Download Python source code: plot_01_mohr_circle.py <plot_01_mohr_circle.py>`

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

      :download:`Download zipped: plot_01_mohr_circle.zip <plot_01_mohr_circle.zip>`


.. only:: html

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

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