.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/fatigue/plot_01_goodman_diagram.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_fatigue_plot_01_goodman_diagram.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_fatigue_plot_01_goodman_diagram.py:


Goodman's mean-stress diagram (1899)
====================================

Wöhler's endurance tests reversed the stress fully, but most machine parts
carry a steady stress with an alternating one on top. John Goodman's
*Mechanics Applied to Engineering* (1899) drew the line that design still
uses: the alternating stress a part can carry falls linearly from the
endurance limit :math:`S_e` at zero mean to nothing at the ultimate strength
:math:`S_{ut}`,

.. math:: \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n}

(:func:`~engineeringkit.materials.goodman_safety_factor`). The example
compares it with Gerber's parabola (1874) and Soderberg's line (1930) and
reads safety factors along a load line.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import goodman_safety_factor








.. GENERATED FROM PYTHON SOURCE LINES 26-33

A bolt with a preload
---------------------

A steel bolt (:math:`S_{ut} = 830` MPa, :math:`S_e = 130` MPa for a
rolled thread) carries a mean stress of 400 MPa from its preload and an
alternating 40 MPa from the external load. Scaling both stresses by
:math:`n` puts the point on the Goodman line.

.. GENERATED FROM PYTHON SOURCE LINES 33-40

.. code-block:: Python


    Sut, Sy, Se = 830e6, 660e6, 130e6
    sa, sm = 40e6, 400e6
    n = goodman_safety_factor(sa, sm, Se, Sut)
    print(f"Goodman safety factor {n:.3f}")
    assert np.isclose(n * sa / Se + n * sm / Sut, 1)





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

 .. code-block:: none

    Goodman safety factor 1.266




.. GENERATED FROM PYTHON SOURCE LINES 41-47

Three failure lines
-------------------

Gerber's parabola :math:`\sigma_a/S_e + (\sigma_m/S_{ut})^2 = 1` follows
test data through the middle; Goodman's line lies inside it, and
Soderberg's, drawn to the yield strength, is the most conservative.

.. GENERATED FROM PYTHON SOURCE LINES 47-69

.. code-block:: Python


    mean = np.linspace(0, Sut, 200)
    goodman = Se * (1 - mean / Sut)
    gerber = Se * (1 - (mean / Sut) ** 2)
    soderberg = Se * np.clip(1 - mean / Sy, 0, None)
    assert np.all(soderberg <= goodman + 1e-6) and np.all(goodman <= gerber + 1e-6)
    external = np.linspace(10e6, 80e6, 8)
    n_external = goodman_safety_factor(external, sm, Se, Sut)
    print("safety factor as the alternating stress grows:", np.round(n_external, 2).tolist())
    assert np.all(np.diff(n_external) < 0)

    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.plot(mean / 1e6, gerber / 1e6, label="Gerber (1874)")
    ax.plot(mean / 1e6, goodman / 1e6, label="Goodman (1899)")
    ax.plot(mean / 1e6, soderberg / 1e6, label="Soderberg (1930)")
    ax.plot([0, n * sm / 1e6], [0, n * sa / 1e6], "k:", lw=1, label="load line")
    ax.plot(sm / 1e6, sa / 1e6, "ko", label=f"bolt, n = {n:.2f}")
    ax.set(xlabel="mean stress [MPa]", ylabel="alternating stress [MPa]", title="Mean-stress diagram", ylim=(0, 1.2 * Se / 1e6))
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/fatigue/images/sphx_glr_plot_01_goodman_diagram_001.png
   :alt: Mean-stress diagram
   :srcset: /api/gallery/materials/fatigue/images/sphx_glr_plot_01_goodman_diagram_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    safety factor as the alternating stress grows: [1.79, 1.57, 1.4, 1.27, 1.15, 1.06, 0.98, 0.91]





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

   **Total running time of the script:** (0 minutes 0.039 seconds)


.. _sphx_glr_download_api_gallery_materials_fatigue_plot_01_goodman_diagram.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/fatigue/plot_01_goodman_diagram.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_goodman_diagram.ipynb <plot_01_goodman_diagram.ipynb>`

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

      :download:`Download Python source code: plot_01_goodman_diagram.py <plot_01_goodman_diagram.py>`

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

      :download:`Download zipped: plot_01_goodman_diagram.zip <plot_01_goodman_diagram.zip>`


.. only:: html

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

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