.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/energy/plot_02_mohr_unit_load.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_energy_plot_02_mohr_unit_load.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_energy_plot_02_mohr_unit_load.py:


Mohr's unit-load method (1874)
==============================

To find how far one joint of a truss moves, Otto Mohr (1874) applied the
principle of virtual work with an imaginary unit load at that joint. The
internal forces :math:`n` it produces, working through the real member
extensions :math:`NL/(AE)`, must equal the work of the unit load through
the wanted displacement:

.. math::

    1\cdot\delta = \sum \frac{nNL}{AE}

(:func:`~engineeringkit.structural_engineering.unit_load_truss_deflection`).
Both sets of forces come from statics alone, so a determinate truss needs
no stiffness matrix. For beams the sum becomes :math:`\int mM/(EI)\,dx`
(:func:`~engineeringkit.structural_engineering.unit_load_beam_deflection`).
The example finds the midspan deflection of a Warren truss member by
member and checks it against the direct stiffness method.

.. GENERATED FROM PYTHON SOURCE LINES 24-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import PlanarTruss, method_of_joints, unit_load_beam_deflection, unit_load_truss_deflection








.. GENERATED FROM PYTHON SOURCE LINES 30-32

A Warren truss carrying two panel loads
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-58

.. code-block:: Python


    nodes = np.array([[0, 0], [4, 0], [8, 0], [12, 0], [2, 3], [6, 3], [10, 3]], float)
    members = ((0, 1), (1, 2), (2, 3), (0, 4), (4, 1), (1, 5), (5, 2), (2, 6), (6, 3), (4, 5), (5, 6))
    fixed = [0, 1, 7]
    E, A = 200e9, 2e-3
    loads = np.zeros(14)
    loads[[3, 5]] = -60e3
    unit = np.zeros(14)
    unit[5] = -1.0  # downward unit load at the midspan joint 2

    N = method_of_joints(nodes, members, loads, fixed).member_forces
    n = method_of_joints(nodes, members, unit, fixed).member_forces
    lengths = np.array([np.hypot(*(nodes[j] - nodes[i])) for i, j in members])
    contributions = n * N * lengths / (A * E)
    delta = unit_load_truss_deflection(N, n, lengths, A, E)
    print(f"midspan deflection by virtual work: {delta * 1e3:.3f} mm")

    stiffness = PlanarTruss(nodes, members, E, A).solve(loads, fixed)
    print(f"by the direct stiffness method:     {-stiffness.displacements[5] * 1e3:.3f} mm")
    assert np.isclose(delta, -stiffness.displacements[5])

    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.bar([f"{i}-{j}" for i, j in members], contributions * 1e3)
    ax.set(xlabel="member", ylabel="share of deflection [mm]", title="Where the midspan deflection comes from")
    fig.tight_layout()




.. image-sg:: /api/gallery/structural_engineering/energy/images/sphx_glr_plot_02_mohr_unit_load_001.png
   :alt: Where the midspan deflection comes from
   :srcset: /api/gallery/structural_engineering/energy/images/sphx_glr_plot_02_mohr_unit_load_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    midspan deflection by virtual work: 3.429 mm
    by the direct stiffness method:     3.429 mm




.. GENERATED FROM PYTHON SOURCE LINES 59-64

The same idea for a beam
------------------------

Midspan deflection of a simply supported beam under a uniform load: the
unit load at midspan gives the triangle :math:`m = x/2` up to midspan.

.. GENERATED FROM PYTHON SOURCE LINES 64-74

.. code-block:: Python


    L, w, I = 6.0, 10e3, 5e-5
    x = np.linspace(0, L, 601)
    M = w * x * (L - x) / 2
    m = np.minimum(x, L - x) / 2
    beam = unit_load_beam_deflection(x, M, m, E, I)
    print(f"beam midspan deflection {beam * 1e3:.3f} mm, 5wL^4/384EI = {5 * w * L**4 / (384 * E * I) * 1e3:.3f} mm")
    assert np.isclose(beam, 5 * w * L**4 / (384 * E * I))

    plt.show()




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

 .. code-block:: none

    beam midspan deflection 16.875 mm, 5wL^4/384EI = 16.875 mm





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

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


.. _sphx_glr_download_api_gallery_structural_engineering_energy_plot_02_mohr_unit_load.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/energy/plot_02_mohr_unit_load.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_mohr_unit_load.ipynb <plot_02_mohr_unit_load.ipynb>`

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

      :download:`Download Python source code: plot_02_mohr_unit_load.py <plot_02_mohr_unit_load.py>`

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

      :download:`Download zipped: plot_02_mohr_unit_load.zip <plot_02_mohr_unit_load.zip>`


.. only:: html

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

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