.. 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_03_young_secant_formula.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_03_young_secant_formula.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_03_young_secant_formula.py:


Young's eccentric column (1807)
===============================

No column is loaded exactly on its axis. Thomas Young (1807) analysed a
column loaded at an eccentricity :math:`e`: the load bends it, the bending
adds to the eccentricity, and the peak stress follows the *secant formula*
(:func:`~engineeringkit.structural_engineering.secant_formula_stress`),

.. math::

    \sigma_\max = \frac{P}{A}\left[1 + \frac{ec}{r^2}\sec\left(\frac{KL}{2r}\sqrt{\frac{P}{EA}}\right)\right].

The stress grows faster than the load, so a factor of safety must be put
on the load, not on the stress. Solved for the load at first yield
(:func:`~engineeringkit.structural_engineering.secant_formula_load`), it
gives a family of column curves that start below the squash load and
approach Euler's hyperbola. The attribution to Young follows the
secondary literature; the original lectures were not consulted.

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import euler_buckling_load, secant_formula_load, secant_formula_stress








.. GENERATED FROM PYTHON SOURCE LINES 29-31

Stress against load: not proportional
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-40

.. code-block:: Python


    E, sy, A, r, c, L = 200e9, 250e6, 5e-3, 0.05, 0.1, 6.0
    P_euler = euler_buckling_load(E, A * r**2, L)
    P = np.linspace(0, 0.95, 200) * P_euler
    stress = secant_formula_stress(np.maximum(P, 1.0), A, 0.02, c, r, L, E)
    half = secant_formula_stress(P_euler / 2, A, 0.02, c, r, L, E)
    print(f"Euler load {P_euler / 1e3:.0f} kN; at half of it the stress is {half / 1e6:.0f} MPa")
    assert secant_formula_stress(P_euler / 2, A, 0.02, c, r, L, E) > 2 * secant_formula_stress(P_euler / 4, A, 0.02, c, r, L, E)





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

 .. code-block:: none

    Euler load 685 kN; at half of it the stress is 192 MPa




.. GENERATED FROM PYTHON SOURCE LINES 41-43

Column curves for several eccentricity ratios
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-60

.. code-block:: Python


    slenderness = np.linspace(10, 250, 120)
    fig, ax = plt.subplots(figsize=(6, 4))
    for ratio in (0.0, 0.1, 0.3, 0.6, 1.0):
        e = ratio * r**2 / c
        curve = [secant_formula_load(sy, A, e, c, r, k * r, E) / A for k in slenderness]
        ax.plot(slenderness, np.array(curve) / 1e6, label=f"ec/r² = {ratio}")
    ax.plot(slenderness, np.minimum(np.pi**2 * E / slenderness**2, sy) / 1e6, "k--", lw=0.8, label="Euler / yield")
    ax.set(xlabel="slenderness KL/r", ylabel="average stress P/A at first yield [MPa]", ylim=(0, 270))
    ax.legend()
    fig.tight_layout()

    P_yield = secant_formula_load(sy, A, 0.02, c, r, L, E)
    print(f"e = 20 mm: first yield at {P_yield / 1e3:.0f} kN, {P_yield / min(sy * A, P_euler):.0%} of the ideal column")
    assert np.isclose(secant_formula_stress(P_yield, A, 0.02, c, r, L, E), sy)

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/stability/images/sphx_glr_plot_03_young_secant_formula_001.png
   :alt: plot 03 young secant formula
   :srcset: /api/gallery/structural_engineering/stability/images/sphx_glr_plot_03_young_secant_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    e = 20 mm: first yield at 395 kN, 58% of the ideal column





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

   **Total running time of the script:** (0 minutes 0.205 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_stability_plot_03_young_secant_formula.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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