Strength of materials - Manual for students of engineering

The greater part of the matter contained in the following pages is based on the notes of lectures given to the day and evening students at the Manchester Municipal School of Technology during the last five sessions.

The book is intended for the use of those students of engineering who are desirous of obtaining a working knowledge of the fundamental principles involved in problems of machine and structural design. It should be found useful to candidates for the Third and Honours stages of the Examinations of the Board of Education, the examination for the admission of Associate Members to the Institution of Civil Engineers, as well as the examinations in the Engineering Schools of the Universities.

It will be seen that special attention has been paid to the unequal distribution of stress, and to the limits of elasticity in iron and steel. Many of the examples quoted are taken from experimental results obtained by the writer or his students. It is also to be noted that the majority of the proofs given are similar to those used in most of the text-books.

The author desires to acknowledge his indebtedness to the many writers of books and scientific papers to which he has referred in collecting these notes. W. C. P.

All material used by the engineer, whether it forms part of some piece of mechanism or of a fixed structure, has to withstand the application of force. The structural part must be so proportioned by the engineer who initiates its design as to be able to carry the loads which come upon it without injury to itself, and at the same time without the employment of more material than is necessary. It is for the carrying out of this safe and economical design that a knowledge of the "Strength of Materials" is required.

The causes producing the loads which come upon the various parts of a structure, and their magnitudes as depending on outside effects and upon the form of the structure as a whole, will not be discussed in any detail in what follows. That part of the subject is generally dealt with under the title of "Theory of Structures." Here the reader is more particularly concerned with a knowledge of the relations between the ascertained loads and the dimensions and forms as affecting the stresses in structural members.

The subject of "Strength of Materials" naturally divides itself into two parts. The former of these deals with the nature and intensity of the forces which come upon a part, as depending on its form and the loads which it has to carry. The second portion of the subject relates more especially to the effect which these forces have upon the internal structure of the material itself. One is analytical and depends upon mathematical proofs, the other is descriptive and experimental.

This order will be observed in the following chapters, the general laws bearing upon the relations of stress and strain, which apply equally to all materials, being taken first; to be followed by a more detailed discussion of the effect of stress upon particular materials.

Stress. Stress is the force exerted by a portion of material upon that part adjacent to it.

A stress may act normally to a surface, when it will be either a compressive or a tensile stress. In the former of these the tendency is for the two portions of material on opposite sides of the section to be pressed against one another; in the latter case the tendency is towards a separation of the parts. Or, the stress may be tangential to the surface in question, with a tendency to cause the portion on one side of the section to slide upon the other: this is called a shear stress. Again, the stress may be partly normal and partly tangential.

Load. This term is applied to the total force which acts upon a structural part. A body acted upon by a load is said to be in a state of stress. Thus, in the case of a metal rod which is with-standing a pair of loads or forces pulling away from one another at the opposite ends, the whole of the material between the two ends is said to be in a state of tensile stress, and it is only the mutual adhesion between the individual particles which prevents the metal from being torn asunder at any point.

Intensity of Stress. Stress is generally defined as being so many units of force acting upon a unit of the area referred to.

The load, on the other hand, is defined as so many units of force, irrespective of the extent of the surface upon which it acts. Thus in the case of the above tension bar, the load might be given as so many tons, and if the extent of the area of the cross-section were so many square inches, the stress would be given as so many tons acting on each square inch, or as so many tons per square inch.

The principal units employed for the measurement of stress
are as follows:

In Great Britain. For the metals and timber the stresses are usually given in tons per square inch, and, less frequently, in pounds per square inch.

For brickwork, masonry, and concrete, stresses are given in tons per square foot, and sometimes in pounds per square foot.

In the United States of America the units employed are respectively pounds per square inch and pounds per square foot.

In the countries using the metric system, kilogrammes or grammes per square centimetre or millimetre are the units.

Strain. Strain is the deformation brought about by stress. Stress never occurs without changing the shape of the piece of material on which it acts.

For example, in the case of the steel bar before referred to, a tension load or pull is put upon the bar, causing tensile stress throughout its length, with the result that the bar is stretched: this stretch is the strain.

The strain may be temporary or permanent, or part of it may be temporary and part permanent. Under the ordinary working loads which are put upon engineering materials the strains accompanying the stresses are relatively of very small magnitude, and are almost always temporary, disappearing on the removal of the loads. When, however, the loading is carried beyond the working limits the strains become larger, and the material only partially recovers itself on the removal of the load, leaving the remainder of the deformation as permanent strain.


Different Kinds of Stress and Strain.

Simple Stresses. There are three kinds of simple stress, namely: Tension or pulling; compression or thrusting; and simple shear.

In Tensile Stress the loads act outwards along the axis of the piece of material, giving rise to a tensile stress on any section normal to the axis. The strain in this case consists of the amount the bar is stretched ; this is shown by the dotted portion on Fig. I (a).

In Compressive Stress the line of action of the loads is the same, but they tend towards instead of away from one another, giving rise to a stress which compels the particles into closer union and at the same time causes a shortening, Fig. 1 (b).

The above are called direct stresses.

In Simple Shear the loads act parallel to one another in opposite directions, tending to cause the two portions of the material acted upon to slide one upon the other.

The strain caused by shear stress is one of distortion. This is shown by the dotted portion, Fig. 1 (c).

Torsion is the particular case of shear stress which occurs when a shaft is twisted, Fig. 1 (d). The effect is to cause any two adjacent normal sections of the shaft to revolve relatively to one another. The strain in this case is measured by the angle of rotation of one end of the shaft with respect to the other.

Bending or Cross-breaking is the kind of stress met with where the load is applied in such a way that it causes a bar to bend, as shown on Fig. 1 (l). Here the bending action of the load results in a curving of the beam, the material on the convex surface being lengthened and put in tension, while that on the concave side is in compression. Thus there are two kinds of simple stress occurring at the same time. The strain in this case is the amount of deflection of the centre from its original position, measured in a direction at right angles to the axis.


Distribution of Stress Uniform Stress.

It has been said that stress is expressed as the amount of the force (compression, tension, or shear) acting upon each unit of area exposed to it. When the load is caused to act in such a way that the intensity of the stress is the same at all points of the sectional area considered, it is said to be uniform. In such a case the stress is

F = P/a

where P is the total force or load and A the area of the section considered. This applies to simple tension, compression, and shear.

But it is possible to apply the load in such a way that the intensity of the stress is not the same at all points of the area. For instance, if the pull in a tie bar is applied along a line which lies outside the geometrical axis of the bar, the intensity of the stress on a section at right angles to the axis is not the same at all points. In such a case the above equation only serves to give the average stress.

Elasticity Plasticity. It has been said that the strain in a piece of material under stress is sometimes temporary and sometimes permanent, or partly temporary and partly permanent. If, after the removal of the load, the strain wholly disappears by reason of the material recovering its original form and dimensions, the strain is said to have been temporary and the state of the material to be elastic.

When, however, the material fails to recover its original dimensions and some of the strain remains after the removal of the stress, the material is said to have been strained beyond its elastic limit, and to have acquired permanent set.

Almost all the materials of engineering exhibit this property of taking permanent set after a certain stress has been reached. In some, permanent set is found after the application of very small stresses.

Materials like wrought iron and mild steel are found to arrive at a point in the loading when the strain goes on increasing with little or no increase in load. When this point has been reached the material is said to have become plastic.

In materials where this plastic state is reached, the period extending from the end of the elastic to the beginning of the plastic stage is spoken of as the semi-plastic stage. The part of this subject which relates to the stresses and strains in the plastic and semi-plastic stages will be left for later consideration.

For the present, all the material dealt with will be assumed to be perfectly elastic, homogeneous and isotropic that is, to have the same properties at all points and in every direction.