**Materials resistance** . It is the discipline that studies internal stresses and deformations that occur in the body subjected to external loads which can cause its failure. The difference between Theoretical __Mechanics__ and the Strength of Materials lies in the fact that for this the properties of deformable bodies are essential, while in general they are not important for the former. Russian author VI Feodosiev has said that the Strength of Materials can be considered as the Mechanics of Deformable Solids.

Failure of a body or of certain parts of it is understood: the breakdown, or without reaching it, the existence of an inadequate state. The latter can occur for several reasons: too large deformations, lack of stability of the materials, cracks, loss of static balance due to buckling, dents or overturning, etc. In this course we will limit the study to failure due to breakage, excessive deformation or buckling.

Summary

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- 1 Objective
- 2 Real System and Analysis Scheme
- 3 Fundamental Hypotheses
- 4 Basic concepts
- 5 External and internal forces
- 6 Deformations
- 7 Sources

objective

The purpose of the Strength of Materials is to develop simple calculation methods, acceptable from the practical point of view, of the most frequent typical elements of structures, elements of machines and electromechanical equipment, using various approximate procedures. The need to obtain concrete results by solving practical problems forces us to resort to simplifying hypotheses, which can be justified by comparing calculation results with tests, or those obtained by applying more accurate theories, which are more complicated and therefore usually little expeditious.

The problems to be solved using this science are of two types:

- Dimensioning: it is about finding the most suitable material, shapes and dimensions of a part, so that it can fulfill its mission: Safely, in perfect technical condition and with adequate expenses.
- Verification: when the material, shape and dimensions have already been predetermined and it is necessary to know if they are adequate to resist the state of acting stresses

Real System and Analysis Scheme

In the resistance of materials as in any branch of the natural sciences, the study of the real object or system begins by choosing the analysis scheme. When starting the calculation of the structure, it is necessary first of all to separate the important from the unimportant, that is, the structure should be schematized, disregarding all those factors of little importance, that is, those that do not significantly influence the behavior of the system as such. The real system, free from all that is unimportant, is called the analysis scheme. This simplification is absolutely essential, since an analysis that takes fully into account all the properties of the real system is impossible in principle, given the inexhaustible variety of these.

Fundamental Hypotheses

The constructions that the engineer finds in his practice have, in most cases, quite complex configurations. As part of the process of conceiving analysis schemes from real systems, simplifications are made in terms of the shape of the elements. The typical forms of the various elements are reduced to the following simple types:

**Bar**: It is a body that has two small dimensions compared to the third, as a particular case, they can be of constant cross section and rectilinear axis. The line connecting the centers of gravity of its cross sections is called the axis of the bar.**Plate**: It is a body limited by two planes, at a small distance compared to the other dimensions.**Vault**: It is a body limited by two curvilinear surfaces, at a small distance compared to the other dimensions.**Block**: It is a body whose three dimensions are of the same order.

With a view to establishing analysis schemes from real systems, simplifications are also made related to the nature of the bodies, their properties, the acting loads and the nature of their interaction with the pieces; these are:

The material is considered solid (continuous). The actual behavior of materials complies with this hypothesis even when the presence of pores can be detected or the discontinuity of the structure of matter is considered, made up of __atoms__ that are not in rigid contact with each other, since there are spaces between them and forces that keep them linked, forming an orderly network. It is this hypothesis that allows material to be considered within the field of continuous functions.

The material of the piece is homogeneous (identical properties at all points). Steel is a highly homogeneous material; on the other hand, wood, concrete and stone are quite heterogeneous. However, experiments show that the calculations based on this hypothesis are satisfactory. The material of the part is isotropic. This means that we admit that the material maintains identical properties in all directions.

The original internal forces that precede the loads are nil. The internal forces between the particles of the material, whose distances vary, oppose the change in the shape and dimensions of the body under load. When speaking of internal forces we do not consider the molecular forces that exist in uncharged solid. This hypothesis is practically not fulfilled in any of the materials. In steel pieces these forces originate due to cooling, in wood due to drying and in __concrete__ during setting. If these effects are important, a special study should be done.

The principle of superposition of effects is valid. As they are deformable solids, this principle is valid when: The displacements of the application points of the forces are small compared to the dimensions of the solid. The displacements that accompany the deformations of the solid depend linearly on the loads. These solids are called “linearly deformable solids”. On the other hand, since the deformations are small, the equilibrium equations corresponding to a loaded body can be posed on its initial configuration, that is, without deformations.

The latter is valid in most cases, however, when analyzing the problem of the buckling of an elastic bar it will be seen that this criterion cannot be applied. The __Saint – Venant Principle__ applies . This principle establishes that the value of the internal forces at the points of a solid, located sufficiently far from the places of application of the loads, depends very little on the specific way of applying them. Thanks to this principle, in many cases we will be able to replace one system of forces with another statically equivalent, which can lead to simplification of the calculation.

Loads are static or quasi-static. Loads are said to be static when they take an infinite time to apply, while they are called quasi-static when the application time is long enough. The loads that are applied in a very short time are called dynamic, and as we will see later, the internal stresses they produce are significantly greater than if they were static or quasi-static.

Basic concepts

When choosing the analysis scheme, simplifications are obligatorily introduced in:

- The geometry of the object
- The links.
- The systems of applied forces.
- The properties of the materials.

The next step to the elaboration of the analysis scheme corresponds to the numerical resolution of the problem, for which, the fundamental bases of the Resistance of Materials are based on the Statics, which is extremely important in the determination of the internal solicitations and of deformations.

Even though from the channeling of the study through mathematical operations it seems that the work has been completed, we must make it clear that the calculation does not consist solely of the use of formulas. Indeed, we must bear in mind that what has been solved is not the real system but a mathematical model. This means that the results must be properly interpreted, and eventually corrected to get as close as possible to the real solution.

The method of the Strength of Materials, which is only that of __Applied Mechanics__ can be stated as follows:

- Choice of an analysis scheme (elaboration of a mathematical model).
- Mathematical resolution of the problem.
- Interpretation of results based on the actual physical system.

External and internal forces

For a body subjected to a general system of external forces, if we apply a section to the body and separate these portions, they will remain in balance due to the emergence of internal forces (equal and opposite).

These internal forces will be the result of distributed forces in the cross section (stresses) and are introduced to characterize the law of distribution of internal forces in the cross section, as a measure of the intensity of the internal forces.

Deformations

Real bodies can be deformed, that is, change their shape and dimensions. The deformations of the bodies happen because of their load with external forces or temperature change. During it the points, lines or sections mentally drawn on the bodies, move in the plane or in space with respect to their initial position. When loading a solid body, internal forces of interaction arise between it between the particles that oppose the external forces and tend to return the particles of the body to the position they occupied before deformation.

The deformations can be elastic, which disappear after the action of forces has been canceled, and plastic or permanent deformations that do not disappear when the loads are removed. The following main types of deformations are studied in the resistance of materials:

- Traction – axial compression: arises, for example, in the case that forces directed in the opposite direction are applied to a bar along its axis, causing an advance displacement of the sections along the axis of the bar that during traction it lengthens, and during compression it shortens. The change in the initial length l, designated l, is called the absolute elongation (during tension) or absolute reduction (during compression). Many elements of structures work in tension or compression, for example: reinforcement bars, columns, piston machines, clamp bolts, etc.
- Slippage or shear arises when external forces tend to move two parallel flat sections of the bar with respect to each other, the distance between them being constant (see figure). The magnitude is called the absolute slip. The ratio of the absolute slip to the distance between two slid planes (the tangent of the angle) is called the relative slip. As the angle is small, it can be considered that: The relative slip is an angular deformation that characterizes the obliqueness of the element. A sliding or shearing works, for example, rivets and bolts that join the elements, which external forces tend to displace each other.
- Torsion arises when external forces act on a bar, forming a moment with respect to its axis (see figure). Torsional deformation is accompanied by rotation of the cross sections of the bar relative to one another about its axis. The angle of rotation of one section of the bar with respect to another located at a distance l is called the length distortion angle l. the ratio of the angle of distortion to the length l is called the relative angle of distortion: The shafts, spindles of lathes and drills as well as other parts work in torsion.
- Bending consists of the deviation of the axis of a straight bar or the change in the curvature of a curved bar. During this deformation the linear displacement arises (arrow f displacement of a point directed perpendicular to the initial position of the axis) and the angular deformation (angle of rotation α the rotation of the sections with respect to their initial positions).

Bending work intermediate deck girders, bridge girders, railroad car axles, leaf springs, axles, gear teeth, wheel spokes, levers and many other parts.

The previously described simple deformations of the bar give an idea about the paths of its shape and dimensions in general, but say nothing about the degree and character of the deformed state of the material. Research shows that the deformed state of a body, generally speaking, is heterogeneous and changes from one point to another. The deformed state at a point in the body is perfectly determined by six components of the deformation: three unit linear deformations and three unit angular deformations.