.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
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.. "api/gallery/reliability/structural/plot_01_freudenthal_interference.py"
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.. only:: html

    .. note::
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        :ref:`Go to the end <sphx_glr_download_api_gallery_reliability_structural_plot_01_freudenthal_interference.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_reliability_structural_plot_01_freudenthal_interference.py:


Freudenthal's stress-strength interference (1947)
=================================================

Alfred Freudenthal (1947) argued that a safety factor, the ratio of mean
strength to mean load, says nothing about safety on its own: two designs
with the same factor can fail at very different rates if their loads and
strengths scatter differently. What matters is the overlap of the two
distributions. For independent normal strength :math:`S` and stress
:math:`s`, :math:`R = P(S > s) = \Phi(\beta)` with
:math:`\beta = (\mu_S - \mu_s)/\sqrt{\sigma_S^2 + \sigma_s^2}`
(:func:`~engineeringkit.reliability.stress_strength_reliability`).

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.stats import norm

    from engineeringkit.reliability import reliability_index_from_probability, stress_strength_reliability








.. GENERATED FROM PYTHON SOURCE LINES 23-28

Same safety factor, different scatter
-------------------------------------

A steel tie with mean strength 450 MPa carries a mean stress of 300 MPa:
a central safety factor of 1.5 in both designs. Stresses in pascals.

.. GENERATED FROM PYTHON SOURCE LINES 28-47

.. code-block:: Python


    designs = {"tight control": (450e6, 20e6, 300e6, 25e6), "loose control": (450e6, 60e6, 300e6, 70e6)}
    rng = np.random.default_rng(1947)
    x = np.linspace(100e6, 700e6, 400)
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.6), sharey=True)
    for ax, (name, (mS, sS, ms, ss)) in zip(axes, designs.items()):
        R = float(stress_strength_reliability(mS, sS, ms, ss))
        simulated = np.mean(rng.normal(mS, sS, 1_000_000) > rng.normal(ms, ss, 1_000_000))
        beta = reliability_index_from_probability(1 - R)
        print(f"{name}: safety factor {mS / ms:.2f}, beta = {beta:.2f}, P(failure) = {1 - R:.2e} (simulated {1 - simulated:.2e})")
        assert np.isclose(R, simulated, atol=5e-4)
        ax.plot(x / 1e6, norm.pdf(x, ms, ss) * 1e6, label="stress")
        ax.plot(x / 1e6, norm.pdf(x, mS, sS) * 1e6, label="strength")
        ax.fill_between(x / 1e6, np.minimum(norm.pdf(x, ms, ss), norm.pdf(x, mS, sS)) * 1e6, color="C3", alpha=0.4, label="interference")
        ax.set(title=f"{name}: P(fail) = {1 - R:.1e}", xlabel="stress [MPa]")
    axes[0].set_ylabel("probability density [1/MPa]")
    axes[0].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/reliability/structural/images/sphx_glr_plot_01_freudenthal_interference_001.png
   :alt: tight control: P(fail) = 1.4e-06, loose control: P(fail) = 5.2e-02
   :srcset: /api/gallery/reliability/structural/images/sphx_glr_plot_01_freudenthal_interference_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    tight control: safety factor 1.50, beta = 4.69, P(failure) = 1.40e-06 (simulated 2.00e-06)
    loose control: safety factor 1.50, beta = 1.63, P(failure) = 5.19e-02 (simulated 5.17e-02)




.. GENERATED FROM PYTHON SOURCE LINES 48-49

The probability of failure differs by orders of magnitude.

.. GENERATED FROM PYTHON SOURCE LINES 49-55

.. code-block:: Python


    p_tight = 1 - stress_strength_reliability(*designs["tight control"])
    p_loose = 1 - stress_strength_reliability(*designs["loose control"])
    assert p_loose > 1000 * p_tight

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.069 seconds)


.. _sphx_glr_download_api_gallery_reliability_structural_plot_01_freudenthal_interference.py:

.. only:: html

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