.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/stability/plot_04_johnson_parabola.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_stability_plot_04_johnson_parabola.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_stability_plot_04_johnson_parabola.py:


Johnson's parabola (1893)
=========================

Euler's hyperbola overestimates the strength of stocky columns, which
yield or buckle inelastically, and tests scattered below it. J. B. Johnson
and his co-authors, in *The Theory and Practice of Modern Framed
Structures* (1893), fitted a parabola that starts at the yield stress and
meets Euler's curve tangentially at half of it
(:func:`~engineeringkit.structural_engineering.johnson_critical_stress`),
at the transition slenderness :math:`C_c = \sqrt{2\pi^2E/\sigma_y}`. The
AISC column formula used the same parabola for decades. End conditions
enter through the effective length :math:`KL`
(:func:`~engineeringkit.structural_engineering.effective_length_factor`).
The example draws the design curve and checks that the two branches meet
with the same slope.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import effective_length_factor, johnson_critical_stress








.. GENERATED FROM PYTHON SOURCE LINES 26-28

The design curve
----------------

.. GENERATED FROM PYTHON SOURCE LINES 28-47

.. code-block:: Python


    E, sy = 200e9, 250e6
    Cc = np.sqrt(2 * np.pi**2 * E / sy)
    slenderness = np.linspace(1, 250, 500)
    sigma = johnson_critical_stress(sy, E, slenderness)
    print(f"transition slenderness Cc = {Cc:.1f}, where the critical stress is {johnson_critical_stress(sy, E, Cc) / 1e6:.0f} MPa")
    h = 1e-4
    left = (johnson_critical_stress(sy, E, Cc - h) - johnson_critical_stress(sy, E, Cc - 2 * h)) / h
    right = (johnson_critical_stress(sy, E, Cc + 2 * h) - johnson_critical_stress(sy, E, Cc + h)) / h
    assert np.isclose(left, right, rtol=1e-4)

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(slenderness, sigma / 1e6, label="Johnson + Euler")
    ax.plot(slenderness, np.pi**2 * E / slenderness**2 / 1e6, "--", label="Euler alone")
    ax.axvline(Cc, color="gray", lw=0.5)
    ax.set(xlabel="slenderness KL/r", ylabel="critical stress [MPa]", ylim=(0, 300))
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/structural_engineering/stability/images/sphx_glr_plot_04_johnson_parabola_001.png
   :alt: plot 04 johnson parabola
   :srcset: /api/gallery/structural_engineering/stability/images/sphx_glr_plot_04_johnson_parabola_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    transition slenderness Cc = 125.7, where the critical stress is 125 MPa




.. GENERATED FROM PYTHON SOURCE LINES 48-50

The same strut with four end conditions
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 50-58

.. code-block:: Python


    L, r = 3.0, 0.03
    for ends in ("fixed-fixed", "fixed-pinned", "pinned-pinned", "fixed-free"):
        K = effective_length_factor(ends)
        print(f"{ends:14s} K = {K:.3f}  KL/r = {K * L / r:6.1f}  sigma_cr = {johnson_critical_stress(sy, E, K * L / r) / 1e6:6.1f} MPa")
    assert np.isclose(np.tan(np.pi / effective_length_factor("fixed-pinned")), np.pi / effective_length_factor("fixed-pinned"))

    plt.show()




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

 .. code-block:: none

    fixed-fixed    K = 0.500  KL/r =   50.0  sigma_cr =  230.2 MPa
    fixed-pinned   K = 0.699  KL/r =   69.9  sigma_cr =  211.3 MPa
    pinned-pinned  K = 1.000  KL/r =  100.0  sigma_cr =  170.8 MPa
    fixed-free     K = 2.000  KL/r =  200.0  sigma_cr =   49.3 MPa





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

   **Total running time of the script:** (0 minutes 0.084 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_stability_plot_04_johnson_parabola.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/stability/plot_04_johnson_parabola.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_johnson_parabola.ipynb <plot_04_johnson_parabola.ipynb>`

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

      :download:`Download Python source code: plot_04_johnson_parabola.py <plot_04_johnson_parabola.py>`

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

      :download:`Download zipped: plot_04_johnson_parabola.zip <plot_04_johnson_parabola.zip>`


.. only:: html

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

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