When some external forces (which may be concurrent or parallel) are acting on a stationary body, the body may start moving or may start rotating about any point. But if the body does not start moving and also does not start rotating about any point, then the body* is said to be in equilibrium. In this chapter, the conditions of equilibrium for concurrent forces (i.e., forces meeting at a point), for parallel forces, moment of a force about a point and about an axis, vectorial representation of moments and couples and Varignon’s theorem will be described. Also the concept of free body diagram, different types of support reactions and determination of reactions will be explained.
2.1.1. Equations of Equilibrium.
Equilibrium of rigid bodies has been described in chapter-1, Article 1.12. A stationary body which is subjected to coplanar forces (concurrent or parallel) will be in equilibrium if the algebraic sum of all the external forces is zero and also the algebraic sum of moments of all the external forces about any point in their plane is zero. Mathematically, it is expressed by the equations:
F = 0 … (2.1)
M = 0 … (2.2)
The sign is known as sigma which is a Greek letter. This sign represents the algebraic sum of forces or moments.
The equation (2.1) is also known as force law of equilibrium whereas the equation (2.2) is known as moment law of equilibrium.
The forces are generally resolved into horizontal and vertical components. Hence equation (2.1) is written as
Fx = 0
Fy = 0
where Fx = Algebraic sum of all horizontal components
and Fy = Algebraic sum of all vertical components.
2.1.2. Equations of Equilibrium for Non-concurrent Force Systems.
A non-concurrent force systems will be in equilibrium if the resultant of all forces and moment is zero.
Hence the equations of equilibrium are
Fx = 0, Fy = 0 and M = 0.
*A body will be in equilibrium if both the resultant of the forces and resultant moment are zero, whereas a particle will be in equilibrium if the resultant force acting on it is zero.
2.1.3. Equations of Equilibrium for Concurrent Force System.
For the concurrent forces, the lines of action of all forces meet at a point, and hence the moment of those fore: about that very point will be zero or M = 0 automatically.
Thus for concurrent force system, the condition M = 0 becomes redundant and only two conditions, i.e., Fx = 0 and Fy = 0 are required.
2.1.4. Action and Reaction.
From the Newton’s third law of motion, we know that to every action there is equal and opposite reaction. Hence reaction is always equal and opposite to the action.
Fig. 2.1 (a) shows a ball placed on a horizontal surface (or horizontal plane) such that is free to move along the plane but cannot move vertically downward. Hence the ball will exert it force vertically downwards at the support as shown in Fig. s.io». This force is known as action. The support will exert an equal force vertically upwards on the ball at the point of contact as shown in Fig. 2.l(c).