# Strength of Materials

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## materials, strength of:

see strength of materialsstrength of materials,
measurement in engineering of the capacity of metal, wood, concrete, and other materials to withstand stress and strain. Stress is the internal force exerted by one part of an elastic body upon the adjoining part, and strain is the deformation or change in
.

## strength of materials,

measurement in engineering of the capacity of metal, wood, concrete, and other materials to withstand stress and strain. Stress is the internal force exerted by one part of an elastic body upon the adjoining part, and strain is the deformation or change in dimension occasioned by stress. When a body is subjected to pull, it is said to be under tension, or tensional stress, and when it is being pushed, i.e., is supporting a weight, it is under compression, or compressive stress. Shear, or shearing stress, results when a force tends to make part of the body or one side of a plane slide past the other. Torsion, or torsional stress, occurs when external forces tend to twist a body around an axis. Materials are considered to be elastic in relation to an applied stress if the strain disappears after the force is removed. The elastic limit is the maximum stress a material can sustain and still return to its original form. According to Hooke's law, the stress created in an elastic material is proportional to strain, within the elastic limit (see elasticityelasticity,
the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. All solids are elastic for small enough deformations or strains, but if the stress exceeds a certain amount known as the elastic
). In calculating the dimensions of materials required for specific application, the engineer uses working stresses that are ultimate strengths, or elastic limits, divided by a quantity called factor of safety. In laboratories materials are frequently "tested to destruction." They are deliberately overloaded with the particular force that acts against the property or strength to be measured. Changes in form are measured to the millionth of an inch. Static tests are conducted to determine a material's elastic limit, ductility, hardness, reaction to temperature change, and other qualities. Dynamic tests are those in which the material is exposed to a combination of expected operating circumstances including impact (e.g., a shell against a steel tank), vibration, cyclic stress, fluctuating loads, and fatigue. Polarized light, X rays, ultrasonic waves, and microscopic examination are some of the means of testing materials.

### Bibliography

See H. E. Parker, Simplified Mechanics and Strength of Materials (rev. ed. 1961); S. Timoshenko and D. H. Young, Elements of Strength of Materials (5th ed. 1968); M. G. Bassin, Statics and Strength of Materials (4th ed. 1988).

## Strength of Materials

the branch of science dealing with the strength of structural and machine elements (parts) and with the ability of these elements to undergo deformation. The principal objects of study in the strength of materials are bars and plates, for which methods of calculating the strength, stiffness, and stability under the influence of static and dynamic loads have been established.

Strength of materials is based on the laws and conclusions of theoretical mechanics, but in addition to this, it takes into account the ability of materials to undergo deformation under the influence of external forces. The physicomechanical properties (yield point, ultimate strength, elastic modulus) needed to evaluate the strength and the ability to undergo deformation of materials are determined by testing machines and special measuring instruments. such as extensometers. Through testing it is possible to arrive at the necessary load conditions and at highly precise measurements of the deformations in test specimens. The most typical test is the calibrated pull test, which is carried out on specimens that are rods having a circular cross section or strips having a narrow rectangular cross section. The results of these tests can be plotted as stress-strain diagrams. Test diagrams and the analytical methods of strength of materials permit a prediction of the behavior of actual structures made of the same materials.

Content and methods of strength of materials. When a solid body is deformed by a load, there is a change in the relative position of the body’s microscopic particles, which in turn gives rise to internal stresses. Strength of materials seeks to determine the maximum stresses in structural elements and machine parts. These stresses are compared with allowable stresses, that is, stresses that can be tolerated without damage to or destruction of the elements or parts. The deformation of the body and the displacements of the body’s individual points are also examined. In addition to required strength, a structure must also have stability; that is, it must deviate only insignificantly from its initial state when acted upon by small, random, short-lived influences. Whether a structure will have this property depends on the external forces, the geometry of the elements or parts, and the physical constants of the materials involved.

A number of approximate engineering methods have been developed in strength of materials for analyzing structural elements. These methods utilize kinematic and static hypotheses that prove to be sufficiently close to reality in most cases. When deriving design formulas for the determination of stresses and displacements, a schematic diagram is drawn up showing the structural element being analyzed, the element’s supports, and the loads involved; in other words, a structural model of the object is created.

Ideal bodies are used in constructing general design theories in strength of materials. These bodies have properties that only approximate the behavior of real objects. Ideal bodies are regarded as uniform (with identical properties at all points), solid (without voids), elastic (capable of assuming their original shape when the load is removed), and isotropic (with identical elastic properties in all directions). From the study of the simplest deformations—those caused by tension, compression, torsion, and bending—formulas are derived that permit a determination for each of these types of deformation of the stresses, displacements, and deformations experienced at separate points of a body. When two or more of the simplest deformations occur simultaneously and the stresses do not exceed the proportional limit (linear relationship between stress and deformation), the stresses and deformations are determined separately for each type of deformation and then added together.

Many materials, for example, concrete, possess a property known as creep, which manifests itself through an increase in the deformation over time with no change in the load. The laws governing the development of this gradual deformation and the time necessary for creep to become appreciable are established in strength of materials. The effect of impact loading on bars, which results in dynamic stresses, is also investigated; these stresses are determined using approximation formulas that are based on a number of assumptions. When designing elements having a complicated shape for which analytical formulas cannot be derived, experimental methods—for example, the optical and moiré method and the method using lacquer coatings—are used that make it possible to obtain a graphic picture of the distribution of deformations over the surface of the element and to calculate the stresses at separate points of the element. The most difficult determinations are those of residual stresses, which may occur in structural elements not subjected to loads, for example, during welding or during the rolling of steel shapes.

One of the important tasks of strength of materials is the formulation of theories of strength that can be used for evaluating the strength of elements in complex states of stress starting from strength characteristics obtained experimentally for simple tension and compression. There are a number of theories of strength; for each individual case, the theory that best conforms to the nature of the load and the failure of the material is used.

Historical survey. The history of strength of materials, as with many other sciences, is closely linked to the history of technology. Strength of materials originated as a science in the 17th century, and Galileo, who was the first to demonstrate the need to use analytical design methods instead of empirical rules, is regarded as the founder. The experimental studies of R. Hooke (1660–70), which established the linear relationship between the force applied to a stretched rod and the rod’s elongation (Hooke’s law), did much to advance the science. Great contributions to the development of analytical methods in strength of materials were made in the 18th century by D. Bernoulli, L. Euler, and C. A. de Coulomb, scientists who formulated important hypotheses and laid the foundation for theories used in analyzing bars subjected to bending and torsion. Euler’s studies on buckling served as the basis for theories dealing with the stability of bars and systems of bars. In 1807, T. Young introduced the concept of the elastic modulus for tension and proposed a method for the modulus’s determination.

An important step in the development of strength of materials was taken with the publication in 1826 by L. Navier of the first course on this subject. The course contained a systematic presentation of the theories used in analyzing structures and structural elements. The research of A. de Saint-Venant in the second half of the 19th century was of fundamental importance. Saint-Venant was the first to derive precise formulas for analyzing a curved beam upon bending and to formulate the principle according to which the stress distribution in sections at some distance from the point of application of a load is not related to the method of the load’s application but depends only on the resultant of the load.

Significant contributions to the development of strength of materials were made by the Russian scientists M. V. Ostrogradskii, whose studies in strength of materials, structural mechanics, mathematics, and the theory of elasticity achieved world renown, and D. I. Zhuravskii, who was the first to establish the existence of shearing stresses (1855) in the longitudinal sections of a beam and to derive a formula, still used in design calculations, for determining the stresses. Wide recognition was accorded to the research of F. S. Iasinskii, who developed (1893) a theory of buckling for stresses both below and beyond the elastic limit. (Iasinskii’s recommendations are reflected in normative documents used today in the USSR and abroad.)

In the early 20th century, the wider use of reinforced-concrete and steel structures and the advent of complex machines and mechanisms gave impetus to the development of strength of materials. The classic textbooks of S. P. Timoshenko on strength of materials and structural mechanics were published, as were the works of A. N. Dinnik on buckling and the stability of bars in compression.

Further improvements in the methods used in strength of materials were made possible through the establishment in the USSR of a number of scientific-research institutions to conduct investigations in structural design. These investigations have given rise to new areas of specialization. The development of strength of materials has been greatly influenced by the research of N. M. Beliaev on plastic deformation, A. A. Il’iushin on the theory of plasticity, and Iu. N. Rabotnov and A. R. Rzhanitsyn on the theory of creep. V. Z. Vlasov’s design theory for thin-walled bars and shells was a significant contribution to the science. In addition, basic research has been carried out by such Soviet scientists as N. I. Bezukhov, V. V. Bolotin, A. F. Smirnov, and V. I. Feodos’ev.

Modern trends. One of the most important tasks of strength of materials is to establish the causes and nature of failure in materials. This task requires a comprehensive theoretical and experimental study of the processes occurring within microscopic volumes of a body, especially the nature of the origin and development of cracks. It has been found that there are certain (ultimate) stresses above which cracks that have already formed grow until the body finally fails. If the stress is less than this threshold, a body having cracks is said to be in a crack-resistant state. Under the effect of a load, failure in a body’s microscopic elements in some cases is propagated throughout the volume of a body, especially at high temperatures. The study of these questions necessitates the creation of a new and important specialty area in the mechanics of bodies able to undergo deformation—the mechanics of failure. There are a number of questions concerning the fatigue strength of materials that have not yet been adequately studied, in particular, the strength of machine elements (parts) under prolonged cyclic loading.

With the advent of such new construction materials as plastics and light alloys, it has become necessary to develop theories of strength that reflect the particular properties of these materials. Modern technological processes, such as processes carried out under high pressures, make it possible to obtain materials with extremely high strength. Research must be done on these materials, whose behavior under loading has not yet been adequately studied.

### REFERENCES

Timoshenko, S. P. Istoriia nauki o soprotivlenii materialov s kratkimi svedeniiami iz istorii teorii uprugosti i teorii sooruzhenii. Moscow, 1957.
Rabotnov, Iu. N. Soprotivlenie materialov. Moscow, 1962.
Feodos’ev, V. I. Soprotivlenie materialov. Moscow, 1974.
Soprotivlenie materialov. Moscow, 1975.

Edited by A. F. SMIRNOV

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