.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/yield/plot_01_tresca_criterion.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_materials_yield_plot_01_tresca_criterion.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_materials_yield_plot_01_tresca_criterion.py:


Tresca's maximum-shear yield criterion (1864)
=============================================

Henri Tresca (1864) punched and extruded lead, copper and iron through dies
at the Conservatoire des Arts et Métiers and found that metals flow like a
viscous fluid once the largest shear stress reaches a constant. With
principal stresses :math:`\sigma_1 \ge \sigma_2 \ge \sigma_3` the criterion
is :math:`\sigma_1 - \sigma_3 = \sigma_y`
(:func:`~engineeringkit.materials.tresca_equivalent`). It is the first
criterion for yielding under combined stresses, and in the plane-stress
principal plane it draws a hexagon. The example applies it to a shaft in
bending and torsion and to biaxial stress states.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import factor_of_safety, tresca_equivalent
    from engineeringkit.materials.visualizers import plot_yield_surfaces








.. GENERATED FROM PYTHON SOURCE LINES 24-31

A shaft in bending and torsion
------------------------------

At the surface of a shaft the bending stress :math:`\sigma` and the
torsional shear :math:`\tau` give principal stresses
:math:`\sigma/2 \pm \sqrt{(\sigma/2)^2 + \tau^2}` and zero. Tresca's
equivalent stress is their spread, :math:`\sqrt{\sigma^2 + 4\tau^2}`.

.. GENERATED FROM PYTHON SOURCE LINES 31-41

.. code-block:: Python


    sy = 350e6
    sigma, tau = 120e6, 90e6
    center, radius = sigma / 2, np.hypot(sigma / 2, tau)
    principal = [center + radius, center - radius, 0.0]
    equivalent = tresca_equivalent(principal)
    print(f"principal stresses {np.round(np.array(principal) / 1e6, 1)} MPa")
    print(f"Tresca equivalent {equivalent / 1e6:.1f} MPa, safety factor {factor_of_safety(sy, equivalent):.2f}")
    assert np.isclose(equivalent, np.sqrt(sigma**2 + 4 * tau**2))





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

 .. code-block:: none

    principal stresses [168.2 -48.2   0. ] MPa
    Tresca equivalent 216.3 MPa, safety factor 1.62




.. GENERATED FROM PYTHON SOURCE LINES 42-48

Where the zero stress matters
-----------------------------

When both in-plane principal stresses have the same sign, the largest
shear acts out of the plane and the zero third stress sets the spread.
Equal biaxial tension yields at :math:`\sigma_y`, not at infinity.

.. GENERATED FROM PYTHON SOURCE LINES 48-60

.. code-block:: Python


    states = np.array([[1.0, 1.0, 0.0], [1.0, 0.5, 0.0], [1.0, -1.0, 0.0], [0.5, -0.5, 0.0]]) * sy
    for state, eq in zip(states, tresca_equivalent(states)):
        print(f"sigma_1, sigma_2 = {state[0] / sy:+.1f}, {state[1] / sy:+.1f} sigma_y -> Tresca {eq / sy:.2f} sigma_y")
    np.testing.assert_allclose(tresca_equivalent(states) / sy, [1, 1, 2, 1])

    ax = plot_yield_surfaces(sy)
    ax.plot(states[:, 0] / sy, states[:, 1] / sy, "ko")
    ax.set_title("Tresca's hexagon")
    ax.figure.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/yield/images/sphx_glr_plot_01_tresca_criterion_001.png
   :alt: Tresca's hexagon
   :srcset: /api/gallery/materials/yield/images/sphx_glr_plot_01_tresca_criterion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    sigma_1, sigma_2 = +1.0, +1.0 sigma_y -> Tresca 1.00 sigma_y
    sigma_1, sigma_2 = +1.0, +0.5 sigma_y -> Tresca 1.00 sigma_y
    sigma_1, sigma_2 = +1.0, -1.0 sigma_y -> Tresca 2.00 sigma_y
    sigma_1, sigma_2 = +0.5, -0.5 sigma_y -> Tresca 1.00 sigma_y





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

   **Total running time of the script:** (0 minutes 0.027 seconds)


.. _sphx_glr_download_api_gallery_materials_yield_plot_01_tresca_criterion.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/materials/yield/plot_01_tresca_criterion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_tresca_criterion.ipynb <plot_01_tresca_criterion.ipynb>`

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

      :download:`Download Python source code: plot_01_tresca_criterion.py <plot_01_tresca_criterion.py>`

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

      :download:`Download zipped: plot_01_tresca_criterion.zip <plot_01_tresca_criterion.zip>`


.. only:: html

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

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