A text book on the mechanics of materials

The following pages contain an elementary course of study in the resistance of materials and the mechanics of beams, columns and shafts, designed for the use of classes in technical schools and colleges. It should be preceded by a good training in mathematics and theoretical mechanics, and be followed by a special study of the properties of different qualities of materials, and by detailed exercises in construction and design.

As the plan of the book is to deal mainly with the mechanics of the subject, extended tables of the results of tests on different kinds and qualities of materials are not given. The attempt, however, has been made to state average values of the quantities which express the strength and elasticity of what may be called the six principal materials. On account of the great variation of these values in different grades of the same material the wisdom of this attempt may perhaps be questioned, but the experience of the author in teaching the subject during the past eleven years has indicated that the best results are attained by forming at first a definite nucleus in the mind of the student, around which may be later grouped the multitude of facts necessary in his own particular department of study and work.

As the aim of all education should be to develop the powers of the mind rather than impart mere information, the author has endeavored not only to logically set forth the principles and theory of the subject, but to so arrange the matter that students will be encouraged and required to think for themselves. The problems which follow each article will be found useful for this purpose. Without the solution of many numerical problems it is indeed scarcely possible for the student to become well grounded in the theory. The attempt has been made to give examples, exercises, and problems of a practical nature, and also of such a character as to clearly illustrate the principles of the theory and the methods of investigation.

The principal materials used in engineering constructions are timber, brick, stone, cast iron, wrought iron, and steeJ. The following table gives their average unit-weights and average specific gravities.  

These weights, being mean or average values, should be carefully memorized by the student as a basis for more precise knowledge, but it must be noted that they are subject to more or less variation according to the quality of the material. Brick, for instance, may weigh as low as 100, or as high as 150 pounds per cubic foot, according as it is soft or hard pressed.

Unless otherwise stated the above average values will be used in the examples and problems of this book. In all engineering reference books are given tables showing the unit-weights far different qualities of the above six principal materials, and also for copper, lead, glass, cements, and other materials used in construction.

A stress is a force which acts in the interior of a body and resists the external forces which tend to change its shape. If a weight of 400 pounds be suspended by a rope, the stress in the rope is 400 pounds. This stress is accompanied by an elongation of the rope, which increases until the internal molecular stresses or resistances are in equilibrium with the exterior weight. Stresses are measured in pounds, tons, or kilograms.
A unit-stress is the amount of stress on a unit of area; this is expressed either in pounds per square inch, or in kilograms per square centimeter. Thus, if a rope of two square inches cross-section sustains a stress of 400 pounds, the unit-stress is 200 pounds per square inch, for the total stress must be regarded as distributed over the two square inches of cross-section.

A deformation is the amount of change of shape of a body caused by the external forces. If a load be put on a column its length is shortened, and the amount of shortening is a deformation. So in the case of the rope, the amount of elongation is a deformation. Deformations are generally measured in inches, or centimeters.

The word strain is often used in technical literature as synonymous with stress, and sometimes it is also used to designate the deformation, or change of shape. On account of this ambiguity the word will not be employed in this book.

Three kinds of simple stress are produced by forces which tend to change the shape of a body. They are,

-    Tensile, tending to pull apart, as in a rope.
-    Compressive, tending to push together, as in a column.
-    Shearing, tending to cut across, as in punching a plate.

The nouns corresponding to these three adjectives are Tension, Compression, and Shear. The stresses which occur in beams, columns, and shafts are of a complex character, but they may always be resolved into the three kinds of simple stress. The first effect of an applied force is to cause a deformation. This deformation receives a special name according to the kind
of stress which accompanies it. Thus,

-    Tension produces an elongation.
-    Compression produces a shortening.
-    Shear produces a detrusion.

This change of shape is resisted by the stresses between the molecules of the body, and as soon as these internal resistances balance the exterior forces the change of shape ceases and the body is in equilibrium. But if the external forces be increased far enough the molecular resistances are finally overcome and the body breaks or ruptures.

In any case of simple stress in a body in equilibrium the total internal stresses or resistances must equal the external applied force. Thus, in the above instance of a rope from which a weight of 400 pounds is suspended, let it be imagined to be cut at any section; then equilibrium can only be maintained by applying at that section an upward force of 400 pounds; hence the stresses in that section must also equal 400 pounds. In general, if a steady force P produce either tension, compression, or shear, the total stress produced is also P, for if not equilibrium does not obtain. In such cases, then, the word stress may be used to designate the external force as well as the internal resistances.

Tension and Compression are similar in character but differ in regard to direction. A tensile stress in a bar occurs when two forces of equal intensity act upon its ends, each in a direction away from the other. In compression the direction of the forces is reversed and each acts toward the bar.


ART. 3. EXPERIMENTAL LAWS.

Numerous tests or experiments have been made to ascertain the strength of materials and the laws that govern stresses and deformations. The resistance of a rope, for instance, may be investigated by suspending it from one end and applying weights to the other. As the weights are added the rope will be seen to stretch or elongate, and the amount of this deformation may be measured. When the load is made great enough the rope will break, and thus its ultimate tensile stress is known. For stone, iron, or steel, special machines, known as testing machines, have been constructed by which the effect of different stresses on different qualities and forms of materials may be accurately measured.

All experiments, and all experience, agree in establishing the five following laws for cases of simple tension and compression, which may be regarded as the fundamental principles of the science of the strength of materials.

(A) When a small stress is caused in a body a small deformation is produced, and on the removal of the stress the body springs back to its original form. For small stresses, then, materials may be regarded as perfectly elastic.

(B] Under small stresses the deformations are approximately proportional to the forces, or stresses, which produce them, and also approximately proportional to the length of the bar or body.

(C) When the stress is great enough a deformation is produced which is partly permanent, that is, the body does not spring back entirely to its original form on removal of the stress. This permanent part is termed a set. In such cases the deformations are not proportional to the stresses.

 (D] When the stress is greater still the deformation rapidly increases and the body finally ruptures.

(E) A sudden stress, or shock, is more injurious than a steady stress or than a stress gradually applied.

The words small and great, used in stating these laws, have, as will be seen later, very different values and limits for different kinds of materials and stresses.

The ultimate strength of a material under tension, compression, or shear, is the greatest unit-stress to which it can be subjected. This occurs at or shortly before rupture, and its value is very different for different materials.


ART. 4. ELASTIC LIMIT AND COEFFICIENT OF ELASTICITY.

The elastic limit is that unit-stress at which the permanent set is first visible and within which the stress is directly proportional to the deformation. For stresses less than the elastic limit bodies are perfectly elastic, resuming their original form on removal of the stress. Beyond the elastic limit a permanent alteration of shape occurs, or, in other words, the elasticity of the material has been impaired. It is a fundamental rule in all engineering constructions that materials cannot safely be strained beyond their elastic limit.

The coefficient of elasticity of a bar for tension, compression, or shearing, is the ratio of the unit-stress to the unit-deformation, provided the elastic limit of the material be not exceeded. Let 5 be the unit-stress, s the unit-deformation, and E the coefficient of elasticity.