STATICS
1. BASIC CONCEPTS AND PRINCIPLES
1.1 The subject of statics. Statics is the branch of mechanics which studies the
laws of composition of forces and the conditions of equilibrium of material bodies
under the action of forces.
Conditions of equilibrium depend on whether a given body is solid, liquid or
gaseous. The equilibrium of liquids and gases is studied in the courses of
hydrostatics and aerostatics respectively. General mechanics deals essentially with
the equilibrium of solids.
All solid bodies change their shape to a certain extent when subject to external
forces. This is known as deformation. The amount of deformation depends on the
material, shape and dimensions of the body as well as on the acting forces. In order
to ensure the necessary strength of engineering structures and elements, the
material and dimensions of various parts are chosen in a way that the deformation
under specified loads would remain relatively small. This makes it possible, in
studying the general conditions of equilibrium, to treat solid bodies as
undeformable or perfectly rigid, ignoring the small deformations that actually
occur. A rigid body is said to be one in which the distances between any pair of
particles forming this body remains constant.
For a rigid body to be in equilibrium when subject to the action of a system
forces, the system must satisfy certain condition of equilibrium. The determination
of these conditions is one of the principal problems of statics. In order to find out
the equilibrium conditions for various force systems and to solve other problems of
mechanics one must know the principles of the composition, or addition, of forces
acting on a rigid body, the principles of reducing of a given system to as simple as
possible. According, statics of rigid bodies deals with two basic problems:
1) the composition of forces and reduction of force system acting on rigid
bodies to a form as simple as possible, and
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2) determination of the conditions for the equilibrium of force system acting on
rigid bodies.
1.2. Force. Since a force is a vector, all statements given in the chapter Elements
of Vector Algebra remain valid for a force. Now we shall further extend our
knowledge about the force accepting following definitions:
- a body not connected with other bodies and capable of displacement in any
direction is called a free body;
- a body whose displacement is restricted by other body is called a constrained
body; we shall call a constraint anything that restricts the displacement of a
given body;
- the force with which a constraint acts on a body, thereby restricting its
displacement, is called the force of reaction of the constraint, or simply the
reaction of the constraint;
- if a free rigid body subject to a force system is at rest, the system is said to be
balanced;
- if a force system acting on a free body can be replaced by another force
system without changing body’s condition of rest or motion, the two systems
are said to be equivalent;
- if a given force system is equivalent to a single force, that force is the
resultant of a system. Thus, a resultant is a single force capable of replacing the
action of a system of forces on a rigid body;
- a force equal in magnitude, collinear with, and opposite in direction to the
resultant force is called an equilibrant force;
- forces acting on rigid body can be divided into two groups: external and
internal forces. External forces represent the action of other material bodies on
the particles of a given body. Internal forces are those with which the particles
of a given body act on each other;
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- a force applied to one point of a body is called a concentrated force. Forces
acting on all the points of a given volume, given area or given line of a body are
called distributed forces.
1.3. Fundamental Principles. All theorems and equations in statics are deduced
from a few fundamental principles, which are accepted without mathematical
proof, and are known as the principles, or axioms, of statics. Some of these
principles are corollaries of the fundamental laws of mechanics, the other result
from a vast number of experiments and observations which, furthermore, have
been consistently confirmed by actual experience.
1st Principle. The free rigid body subjected to the action of two forces can be in
equilibrium if, and only if, the two forces are equal in magnitude, collinear, and
opposite in direction.
The 1st principle defines the simples balanced force system (Fig.1.1)
2nd Principle. The action of a given force system on rigid body remains unchanged
if another balanced force system is added to, or subtract from, the original system.
This principle establishes that two force systems differing from each other by a
balanced system are equivalent.
Corollary from the 1st and 2nd principles. Conditions of equilibrium or of motion
of a rigid body will remain unchanged if a force F acting at a given point of the
rigid body will be transmitted along its line of action.
The Fig.1.2 outlines a proof of this corollary which is also known as the principle
of transmissibility. Note that it doesn’t apply to deformable bodies.
3rd Principle. Two forces applied at one point of a body have as their resultant a
force applied at the same point and represented by the diagonal of a
parallelogram constructed with the two given forces as its sides.
This principle is known as the parallelogram law which has been explained
previously. Now we would like to stress the essential difference between the sum
of two forces and their resultant. To show this difference let us consider a body to
which two forces F1 and F2 are applied at points A and B (Fig.1.3). The sum of the
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two forces can be found by laying them off from any point in the body. In Fig.1.3,
force Q is the sum of F1 and F2. But Q is not the resultant of the two forces, for it
will be readily observed that Q alone cannot replace the action of F1 and F2 on the
body.
4th Principle. To any action of one material body on another there is always an
equal and opposite reaction, or, reaction is always equal to action.
It should be noted that though the reaction force F’ is equal in magnitude,
collinear with, and opposite in sense to the force of action F, both forces do not
form a balanced system as they are applied to different bodies.
5th Principle. If a freely deformable body subjected to the action of a force system
is in equilibrium, the state of equilibrium will not be disturbed if the body solidifies
(becomes rigid).
The idea expressed in this principle (known as the principle of solidification) is
self-evident, for, obviously, the equilibrium of chain will not be disturbed if its
links are welded together, and a flexible string will remain in equilibrium if it turns
into a bent rigid rod. Since the same force system acts on a reposing body before
and after solidification, the 5th principle can also be formulated as follows:
If a freely deformable body is in equilibrium, the forces acting on it satisfy the
conditions for the equilibrium of a rigid body; for a deformable body, however,
these conditions, though necessary, may not be sufficient.
For example, for a flexible string with two forces applied at its ends to be in
equilibrium, the same conditions are necessary as for rigid rod. These conditions,
though, are not sufficient. For the string to be in equilibrium the forces must be
tensile, i.e., they must stretch the body.
The principle of solidification is widely employed in engineering problems. It
enables a determination of equilibrium conditions of a flexible body (a belt, cable,
chain, etc.), or a collapsible structure by treating it, as a rigid body and to apply to
it the methods of rigid-body statics.
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6th Principle. Any constrained body can be treated as a free body detached from its
constraints, provided the latter are represented by their reactions.
The determination of the reactions of constraints is of practical importance
because, from the 4th principle, if we know them, we shall know the loads acting
on the constraints, i.e., the basic information necessary to calculate the strength of
structural elements. An illustrative example will be shown at the lecture.
Bibliography
Beer F.P, Johnston E.R., Jr., Vector Mechanics for Engineers, McGraw-Hill
Shames I.H., Engineering Mechanics – Statics, Prentice-Hall, 1959
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