EQUILIBRIUM

 INTRODUCTION

3.1.1 The Concept Of Equilibrium

The concept of equilibrium is introduced to describe a body which is stationary or which is moving with a constant velocity.  A body under such a state is acted upon by balanced forces and balanced couples only.  There is no unbalanced force or unbalanced couple acting on it.  In statics, the concept of equilibrium is usually used in the analysis of a body which is stationary, or is said to be in the state of static equilibrium.

The concept of equilibrium is the most basic and most important concept in engineering analysis.  The concept must be really understood by every student.  The ability to understand mechanics and many other engineering disiciplines is dependent on mastering  the concept of equilibrium.

 Particles and Rigid Bodies

Particles.  A particle is a body whose size does not have any effect on the results of mechanical analyses on it and, therefore, its dimensions can be neglected.  The size of a particle is very small compared to the size of the system being analysed.

Rigid body.  A body is formed by a group of particles.  The size of a body affects the results of any mechanical analysis on it.  A body is said to be rigid when the relative positions of its particles are always fixed and do not change when the body is acted upon by any load (whether a force or a couple).   Most bodies encountered in engineering work can be considered rigid from the mechanical analysis point of view becase the deformations that take place within these bodies under the action of loads can be neglected when compared to other effects produced by the loads.  All bodies to be studied in this book are rigid, except for springs.  Springs undergo deformations that cannot be neglected when acted upon by forces or moments.  For the analyses ini this book, only the effects of the deformations of springs on a rigid body interacting with the springs are considered but the springs themselves will not be analysed as a body.

 Effects Of  The Action Of Forces on Particles and Rigid Bodies

In general, a force acting on a particle tends to cause the particle to translate.  Also, a force on on a body not only tends to cause the body to translate (as in the case of the particle) but also tends to cause the body to rotate about any axis which does not intersect with or is not parallel to the line of action of the force.  This tendency to rotate is called the “moment” and is measured quantitatively by the methods described in Chapter 2.

3.2 FREE BODY

Mechanical analysis of a structure, in general, is started by applying Newton’s law to the whole structure, or to part of the structure.  To see what actually happens to any particular part of a structure, that part has to be isolated from the other parts of the body.  The concept of isolating the part which is the target of analysis, is a very important concept in mechanical analysis.  The part which is isolated is called a free body.

In this section, we will see how a targeted part is isolated and how the free body formed through the isolation process is displayed graphically.

3.2.1 Mechanical System

A mechanical system is defined as a body system that can be isolated from other bodies.  The system can be formed by a single body, part of a body, or a group of connected bodies.  The bodies forming the system can either be rigid or non-rigid.  A mechanical system can be solid, fluid, or even a combination of solild and fluid.

3.2.2 Free-body Diagram (FBD)

The isolation of a mechanical system is achieved by cutting and isolating the system from its surroundings.  The isolation enables us to see the interactions between the isolated part and the other parts.   The part which has been cut (imaginarily), forms a free body.  A diagram which portrays the free body, complete with the system of external forces acting on it due to its interaction with the parts which have been removed, is called the free-body diagram (FBD) of the isolated part.

The FBD of of a body system shows all loads acting on the external boundary of the isolated body.  The loads include active forces applied on the free body by the parts removed from it.  Note that the internal forces acting on a structure becomes external forces when exposed by the imaginery “cutting” process carried out to form the free body.

As an example, consider the structure of the arm of a lift truck, Figure 3.1.  Assume that an analysis is to be carried-out on the whole structure of the arm when it is carrying a load as shown, where the weight of the component members of the arm can be nglected compared to the weight of the load. Assume also that all joints of the arm do not prevent rotation around the respective joints, i.e. every joint produces a reactive force only and does not produce any reactive moment.  Because the direction and the sense of every reactive force are not known, the direction nad sense shall be assumed.

The arm can be isolated from the body of the lift truck at point A where it is pinned to the body of the lorry and at point C where it is acted upon by the active forceof the hydraulic piston rod.  The isolated arm is shown in Figure 3.1(b).  On the diagram, FA is the force produced by the interaction between point A on the arm and that on lorry body, FC is the interactive force between point C on the lorry arm and the hydraulic piston, mg is the weight of the load, and  G is the centre of gravity of the load.  Figure 3.1(b) is the FBD for the entire arm for the conditions specified.

If what is to be analysed is the load container only, the FBD shown in Figure 3.1(c) is drawn.  If member BC is what one wants to analyse, the FBD is shown in Figure 3.1(d).

Please note that, in the FBDs shown, the direction and sense of all the reactions are drawn arbitarily because they are assumed to be unknown.  We will learn later how to determine the direction of some types of reactivce forces through observation.