.. 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_02_strain_gauge_rosette.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_02_strain_gauge_rosette.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_02_strain_gauge_rosette.py:


Simmons and Ruge's bonded strain gauge (1938)
=============================================

Stress cannot be measured, only strain. In 1936 Edward Simmons at Caltech,
and in April 1938 Arthur Ruge at MIT, independently cemented fine
resistance wire to a specimen, so that its resistance changed with the
strain of the surface beneath; their names became the "SR-4" gauge. One
gauge measures strain in one direction; a *rosette* of three at known
angles gives the whole plane strain state
(:func:`~engineeringkit.structural_engineering.strain_rosette`), and
Hooke's law turns it into stress
(:func:`~engineeringkit.structural_engineering.plane_stress_from_strains`).
The example simulates a rosette on a shaft under bending and torsion and
recovers the torque from the three readings.

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import mohr_circle, plane_stress_from_strains, strain_rosette
    from engineeringkit.structural_engineering.visualizers import plot_mohr_circle








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Readings of a 0-45-90 rosette on a shaft surface
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-39

.. code-block:: Python


    E, nu, G = 200e9, 0.3, 200e9 / 2.6
    d, M, T = 0.05, 1.2e3, 2.0e3
    sx, txy = 32 * M / (np.pi * d**3), 16 * T / (np.pi * d**3)
    ex, ey, gxy = sx / E, -nu * sx / E, txy / G
    angles = np.radians([0.0, 45.0, 90.0])
    readings = ex * np.cos(angles) ** 2 + ey * np.sin(angles) ** 2 + gxy * np.sin(angles) * np.cos(angles)
    rng = np.random.default_rng(1938)
    noisy = readings + rng.normal(0, 2e-6, 3)  # a few microstrain of noise
    print("gauge readings [microstrain]:", np.round(noisy * 1e6, 1))





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

 .. code-block:: none

    gauge readings [microstrain]: [ 490.   702.8 -146.5]




.. GENERATED FROM PYTHON SOURCE LINES 40-42

Back to stress and torque
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-58

.. code-block:: Python


    strains = strain_rosette(noisy, angles)
    stress = plane_stress_from_strains(*strains, E, nu)
    torque = stress[2] * np.pi * d**3 / 16
    print(f"recovered: sigma_x {stress[0] / 1e6:.1f} MPa (true {sx / 1e6:.1f}), tau_xy {stress[2] / 1e6:.1f} MPa (true {txy / 1e6:.1f})")
    print(f"torque {torque:.0f} N m (applied {T:.0f} N m)")
    exact = plane_stress_from_strains(*strain_rosette(readings, angles), E, nu)
    assert np.allclose(exact, (sx, 0.0, txy), atol=1e-3 * sx)
    assert abs(torque - T) < 0.05 * T

    _, radius, s1, s2 = mohr_circle(*stress)
    print(f"principal stresses {s1 / 1e6:.1f} and {s2 / 1e6:.1f} MPa")
    ax = plot_mohr_circle(*stress)
    ax.set_title("Stress from a strain-gauge rosette")

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/stress/images/sphx_glr_plot_02_strain_gauge_rosette_001.png
   :alt: Stress from a strain-gauge rosette
   :srcset: /api/gallery/structural_engineering/stress/images/sphx_glr_plot_02_strain_gauge_rosette_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    recovered: sigma_x 98.0 MPa (true 97.8), tau_xy 81.7 MPa (true 81.5)
    torque 2005 N m (applied 2000 N m)
    principal stresses 144.3 and -46.2 MPa





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

   **Total running time of the script:** (0 minutes 0.055 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_stress_plot_02_strain_gauge_rosette.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_02_strain_gauge_rosette.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_strain_gauge_rosette.ipynb <plot_02_strain_gauge_rosette.ipynb>`

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

      :download:`Download Python source code: plot_02_strain_gauge_rosette.py <plot_02_strain_gauge_rosette.py>`

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

      :download:`Download zipped: plot_02_strain_gauge_rosette.zip <plot_02_strain_gauge_rosette.zip>`


.. only:: html

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

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