1: Fundamental Concepts

1.2: Equilibrium

Equilibrium is a fundamental concept in structural analysis. Briefly, it is the requirement that the resultant of each and every set of forces that act on a structure, or on any part of the structure, be zero. This requirement is a mathematical tool that we can use to setup and solve equations that will allow us to solve for unknown forces.

1.2.1: Two Dimensional (2D) Equilibrium

In this course, we will deal almost exclusively with structures that can be modeled as 2D, or planar, structures. All forces, then are assumed to lie in the plane of the structure. In such a system, there are three independent scalar equations of equilibrium that are necessary and sufficient to establish equilibrium.

The particular equations used can vary from problem to problem, but the idea that there are never more than three available on any one force system is crucial. Some special force systems have fewer than three independent equations available.

Couple/Moment Equivalency

Before proceeding, we note the equivalency regarding a couple. A couple is a pair of equal, opposite and parallel forces, $F$, separated by a non-zero distance, $d$. A couple is equivalent to a pure moment of magnitude $M = F \times d$; wherever one appears in a force system, it can be replaced by the other with no effect on the resulting system.

The three equations normally used for a general force system are the sums of forces in two different directions (usually orthogonal directions) and sum of the moments about some point in the plane, often expressed as:

\[\begin{eqnarray} \sum F_x &=& 0\\ \sum F_y &=& 0\\ \sum M_o &=& 0\\ \end{eqnarray}\]

These equations are commonly used because they are convenient and there are few restrictions as to their applicability.

Other sets of equations are possible, for example moments about three different points as long as they are not co-linear:

\[\begin{eqnarray} \sum M_a &=& 0 \\ \sum M_b &=& 0 \\ \sum M_c &=& 0 \\ \end{eqnarray}\]

as long as all points are different and point c is not on the line through points a and b.

Even other equations are possible. For example, for summing forces in two different directions, the directions need not be orthogonal (though they obviously cannot be parallel). Or, you can use a force and two moment equations, as long as a line through the two moment points is not parallel to the force direction.

1.2.2: Special Force Systems

In some force systems, there are even fewer than three equations available.

If forces are concurrent, $\sum M_c = 0$ happens automatically when point c is the point of concurrency, so only two equations remain. You may still use a moment equation to solve for unknowns, but you can only use $\sum M_o = 0$ when point o is not the point of concurrency.

If forces are parallel, $\sum F_y = 0$, where y is perpendicular to the forces, happens automatically no matter what the magnitude of the forces are, and so only two equations remain. You could still use two force equations such as $\sum F_x = 0$ and $\sum F_z = 0$, where direction z is neither parallel to nor perpendicular to x.