# Chris Fox's Engineering Section

## Equilibrium

A body is in equilibrium if the resultant of all the forces - applied forces and reactions - acting on it is 0N, and if the sum of the moments exerted about any point is 0Nm.

These conditions can be used to determine the unknown values of forces acting on a body. To do this, one can use a free body diagram. Draw an outline of the body, separated from other bodies, then add all the applied forces and reactions. A force is denoted by a straight arrow and a moment by a curved arrow, accompanied in each case by either the magnitude or a symbol denoting a variable.

Having drawn the free body diagram, one can write equilibrium equations. For the purpose of taking moments, the mass of a body can be considered to be concentrated at a point, termed the centre of gravity. At the centre of gravity, the sum of the moments exerted by the parts of the body is zero. For a body of uniform cross-section and density, this point lies halfway along its length.

For a more complex body, ∑(mg(x-¯x))=0, where m is the mass of each part of the body, x is the distance (measured along one particular axis) to that part's centre of gravity from a datum point and ¯x is the distance from the same datum point to the whole body's centre of gravity. Multiplying out this equation gives ∑(mgx-mg¯x)=0. Because additions and subtractions can be carried out in any order, this is the same as ∑(mgx)-∑(mg¯x)=0. Factorising gives g∑(mx)=g¯x∑m. Finally, dividing both sides by ∑m and cancelling gives ¯x=∑(mx)/∑m.

In two-dimensional problems, there are four categories of equilibrium. In the first, forces are collinear. There will be one equilibrium equation, of the form ∑Fx=0, which can only be solved if there is only one unknown quantity.

In the second category, forces are concurrent. There will be two equations, of the form ∑Fx=0 and ∑Fy=0. These can be solved if there are up to two unknowns.

If the forces are parallel, there will be two equations, of the form ∑Fx=0 and ∑Mo=0. These too can be solved if there are up to two unknowns.

For a general system of forces and moments, there will be three equations, of the form ∑Fx=0, ∑Fy=0 and ∑Mo=0. These can be solved if there are up to three unknowns.

For a two-dimensional problem, one can write a maximum of three independent equilibrium equations. Three is therefore the maximum number of unknown quantities that can be determined using one free body diagram.

In each of the above cases, the force equations can be replaced with moment equations, provided that in any problem each moment equation is based on a different pivot. Where there is more than one unknown, it is an advantage if the pivot used for each moment equation lies on the line of action of one of the unknowns; that quantity is then removed from the equation.

One common case of equilibrium is a two force system in which a component is supported by one having pin joints at its ends; the forces in the pin joints must be equal in magnitude, opposite in direction and collinear. Another case is a three force system in which the lines of action of the three forces act through a single point.

A free body diagram can be used to determine internal forces in components and systems; these in turn can be used to calculate stresses and displacements. To find the internal forces acting at a section (as in cross-section), start by splitting the component in two at that section. Then, treating each part as a separate body, add at the break the forces and moments needed to keep each part in equilibrium. The internal forces will be tension, which acts along the length of a component, and shear, which acts perpendicular to the length.