.. _background:

Background knowledge
--------------------

In this page we will go through introductory bending theory, as typically covered in an introductory statics course.
We will start by a quick overview of the required knowledge, and then derive our analysis from basic principles.

Our starting point will be the fact that for a body to be at rest, the **vector sum** of all external forces :math:`\mathbf{F_i}` and moments :math:`\mathbf{M_i}` acting on it must be zero.

.. math::
   :label: static

   \sum \mathbf{F_i} = 0; \ \ \ \ \ \sum \mathbf{M_i} = 0

.. .. figure:: /_static/placeholder_01.png

.. figure:: /../../examples/foundation01.png
   :scale: 50 %
   :align: center
   :alt: rigid body with forces acting on it (resultant zero)

   A rigid body in equilibrium (i.e. whose sum of both forces and moments equal zero)

Furthermore, for a system to be at rest, each of its components need to be at rest.
This means that Eq. :eq:`static` must be satisfied **for each** component in our system.

.. .. figure:: /_static/placeholder_02.png

.. figure:: /../../examples/foundation02.png
   :scale: 50 %
   :align: center
   :alt: system where the resultant force acting on each one of the rigid bodies is zero

   For a system to be in equilibrium, each of its subsystems **must** be in equilibrium.

Note that :math:`\mathbf{F_{1C}} = \mathbf{-F_{2C}}`, according to the *action and reaction principle*.
If you are still not 100% comfortable with the action and reaction principle, you should review that before proceeding.
As a self-test, the concept presented in this `educational video
<https://www.edumedia-sciences.com/en/media/80-action-reaction-principle>`_ should be completely obvious to you.

So far, we have only thought about reaction forces as applied *externally* to one or more rigid bodies.
In other words, each rigid body has been considered to be fully contained in a single subsystem.
However, Eq. :eq:`static` can tell us much more than that if we lift that restriction.
This equation applies to every arbitrary subset of a body, and not only to full bodies.
We are going to exploit this to a larger extent in the next section.