# Solid statics

Solid statics. Section of Theoretical Mechanics that studies the general properties of forces and equilibrium conditions of solid bodies under the action of the forces applied.

## Summary

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• 1 Definition
• 2 main notions of statics
• 1 Material point and rigid body
• 2 Strength
• 3 Axioms of Statics
• 4 Sources

## Definition

Statics is the section of Theoretical Mechanics that studies the general properties of forces and the equilibrium conditions of solid bodies under the action of applied forces. (Nikitín, 1980, p.23).

By balance of a solid body in the Statics is understood its state of rest with respect to a coordinate system that is considered immobile.

Statics as a science that studies the balance of solid bodies has great importance in the construction of engineering works . To ensure the stability of these works, the material and the dimensions of its elements are chosen so that the deformations under the loads acting on them are very small.

## Main notions of statics

### Material point and rigid body

In order to study any phenomenon of nature more deeply , in science we resort to abstractions, concentrating our attention on the most important parts of this phenomenon and rejecting the secondary ones. In Theoretical Mechanics, abstractions of this type are notions about the material point and the rigid body.

The material particle, whose dimensions in the conditions of the studied problem can be neglected, is called a material point. This differs from the geometric point in that the material point presupposes the concentration of a certain amount of matter . Thanks to this the material point has the property of inertia and the ability to interact with other material points.

Every physical body is represented in Mechanics as a system of material points. A system of material points is understood as a certain set of material particles that act against each other in accordance with the law of equality of action and reaction. A body is called rigid in which the distance between each pair of points remains unchanged in all conditions. In other words, the rigid body invariably retains its geometric shape, as well as each of its parts, that is, this body does not deform.

### Force

In mechanics force is called the quantitative measure of the mechanical interaction of material bodies. As a result of this interaction a change in the kinematic state of material bodies can occur , that is, they can change not only their position in space , but also the velocities of the points of the body. Taking into account the problems of statics, the action of one body on another expressed as pressure, attraction or repulsion is considered as force.

The simplest example of a force is the force of gravity . This is the force with which every body is drawn to Earth . As a result, a non-free body exerts pressure on its support (static action of the force) and, when free, falls to the Earth with an acceleration equal to g (dynamic action of the force). In the international system the unit of force is the newton (N). A newton is the force that must be applied to a mass of one kilogram to give it an acceleration of 1 m / s2.

The action of force on a body is characterized by its application point, its direction and numerical value. The line by which said force is directed is called the line of action of the force. The numerical value, modulus of force, is found by comparing it with the unit of force.

Force is a vector quantity, so it is represented graphically by means of a vector. The length of the vector represents, determined scale, the modulus (numerical value) of the force; the line, in which the vector is located, and its direction indicate the line of action and the direction of the force.

## Static Axioms

Aside from the first ( law of inertia ) and third ( law of equality of action and reaction ) of classical mechanics , Statics is based on some more theses confirmed by practice, which are called axioms of Statics. Based on them, logically, the other concepts of Statics are established. Previously agree to the following definitions:

1. The set of forces acting on a given body is called the force system. The forces that are part of a given system are called components of this system.
2. If the system of forces is such that under its action a free body does not modify its movement or, as a particular case, continues to remain at rest, this system of forces is called a balanced system.
3. The force which, when added to a system of forces acting on the body, puts the system in equilibrium, is called the compensating force of the given system of forces.
4. Two systems of forces are called equivalents if they exert equal mechanical action on the same free solid body.
5. A force, which is equivalent to a given system of forces, is called the resultant of this system.
6. The forces acting on a given body by other bodies are called external forces. The interacting forces of the particles of the given body are called internal forces.

All the theorems and equations of Statics are deduced from some initial statements, which are accepted without mathematical proofs, called axioms or principles of Statics. The axioms of Statics are the result of generalizations of numerous experiences and observations of the balance and movement of bodies, which have been repeatedly confirmed by practice

• Axiom 1. A rigid body remains in equilibrium under the action of two forces, if and only if these forces have the same modulus and are directed along the same line in opposite directions.
• Axiom 2. The action of a system of forces on a rigid body will not be modified if an equilibrium system of forces is added or taken away from it.

This axiom establishes that two systems of forces that are differentiated by a balanced system are equivalent. From axioms 1 and 2 we obtain (the principle of transmissibility). The action of a force on a body will not be changed if the point of application of the force is moved along the line of action to any other point on the body.

• Axiom 3. (Principle of the parallelogram of forces). The resultant of two forces applied at one point is applied to the same point and is represented by the diagonal of the parallelogram constructed with these forces taken as sides (Fig-1).

Principle of the parallelogram of forces

The parallelogram built on the given forces is called the parallelogram of forces, while the procedure of finding the resultant by constructing the parallelogram is called the parallelogram principle. The module of the resultant is found with the help of the expression

Axiom 4. (Principle of rigidity). The balance of a deformable body that is under the action of a system of forces, is preserved if this body is considered solidified (rigid).

The stiffness principle is widely applied in engineering calculations . This principle allows, when determining the equilibrium conditions, to consider any body (strap, cable, chain, etc.) or any deformable construction, as rigid, and to apply, when calculating, the procedures of the Static of the rigid solid.