Strength of material - An elementary study

This book has been prepared for the use of the Midshipmen at the U. S. Naval Academy, and is designed to cover a short course in the subject taken up in the Department of Mathematics and Mechanics preliminary to the work in the Departments of Ordnance and Gunnery and of Steam Engineering at the Academy.

In arranging the subject matter many of the methods introduced by officers previously on duty in the Department of Mathematics have been employed, and the endeavor has been to lead the student to the opening point for the professional work carried on by the other Departments.


INTRODUCTION

Before beginning the study of Strength of Material, let us see what has been discovered by experimenting with test pieces of material and note some of the conclusions arrived at from the results obtained in this way.

Experiment shows us that whenever a force acts on a body formed of any substance the dimensions of that body are changed. In mechanics all bodies were assumed rigid and the results obtained under this assumption were true, for mechanics taught us to find the action of one body on another or the force transmitted by one body to another, while strength of material will teach us the effect in the body itself of a force acting upon it. Mechanics showed us that by means of a piece of material force could be moved from one point to another, and strength of material will show us that in transmitting the force the substance forming the conveyance suffers some slight temporary deformation if the force is within certain limits; that beyond these same limits the substance suffers permanent deformation and if the force be great enough will be completely ruptured.

The study of strength of material will include finding the safe limits of a force to be transmitted by any particular piece of material, finding the deformation caused by transmitting any force, and finding the dimensions of a piece of material in order that it may safely transmit any particular force.

With regard to deformation materials differ greatly, for example, the force which will double the length of a piece of rubber will not apparently change the length of a piece of steel of the same size, though both of these substances are elastic, and each if stretched within limits will return to its original length when the stretching force is removed. The force which makes an indentation in a piece of putty will scarcely affect a piece of lead, but in this case the indentation made will remain in both these substances as they are plastic.

Obviously, then, we must experiment with the different materials and find out some of their physical properties before proceeding with a mathematical investigation.

The materials used in building are elastic and we determine their physical properties by experimenting with small pieces of them in machines made for the purpose.

Take steel, for example; small test pieces of different shapes are tried under different forces. A pull is applied and we find the test pieces stretch; if we apply pressure the test pieces are compressed. Having applied all sorts of forces we compare our results and find that the stretching or compression is always of the same amount if the same value of force on unit cube of the steel is used. We also find that up to a certain limiting value of the force the material will always return to its original shape when the force is removed, but if we go beyond this value we find the piece will not return to its original shape; in other words it is not perfectly elastic for forces beyond this value.

If we go through the whole list of materials used in building and test each kind in the same way we will find that they will all behave in a similar manner, the difference will lie in the amount of deformation for any force and the value of the limiting force beyond which they are not perfectly elastic.

Experimenting further we find the value of this limiting force for the different materials and having it we can compare the various substances as to their usefulness under different circumstances.

If we continue to experiment, again and again with a single substance we find that we may apply as often as we like a force less than the limiting one we found, and that the piece will always return to its original shape when the force is removed; but when the force used exceeds the limiting one found, if by ever so little, and the force is applied and removed often enough the piece will break j though the deformation caused by the first application of the force was too small to be measured or even noticed. From this fact we see that we must never use a piece of material which will have to sustain a force which is in the least greater than the limiting value found by experimenting.

Materials differ in other ways. A piecer of glass is easily broken by a light blow and is therefore called brittle or fragile; wrought iron can be twisted and bent into almost any shape without rupture and therefore is called malleable or ductile.

Material which has been melted and cast into desired shapes cools quicker at some parts that it does at others, thereby setting up within it internal stresses which are irregularly distributed. Such castings can be broken by a comparatively light blow though they can usually withstand a large pressure. These internal stresses can be removed by a process called annealing f which consists in heating the body to a red heat and allowing it to cool slowly ^ thus allowing the particles an opportunity to rearrange themselves.

Metals have the peculiarity that if overstrained they harden in the vicinity of the overstrain and this hardening goes on with time. Thus a bar which is sheared off while cold will finally become extremely hard and brittle near the sheared end, as will a plate in the neighborhood of cold-punched rivet holes. To avoid this, bore the rivet holes and saw the bar, or, if feasible, anneal the cold-sheared bar and the plate near the cold-punched rivet holes.

Now in all practical cases we must of course use material that will not break, but in addition we must have material which will not change its dimensions to any considerable extent under the applied load.

The experiments by which the constants used in this study have been determined were very carefully conducted and are the results of many independent efforts on the parts of many different scientists. In this work the mathematical investigation only will be touched upon, the experimental part being beyond its scope.
 

In the study of strength of material we must consider two ways of arranging the different pieces used: first, when there is to be motion between the parts, and second, when the parts are to be relatively at rest. In the first case, force is transmitted from one piece to another and the combination of pieces is called a machine, the study of which involves the principles of dynamics; the second arrangement is called a framed structure, or simply structure, and we must employ the principles of statics in its investigation. In either arrangement, any two parts in contact have a mutual action between the touching surfaces, and the effects produced by this action depend in great measure upon the way in which it is applied. In any case it tends to change the shape or dimensions of the parts involved and, if the force is great enough, to crush or break them. So for permanence the machine or structure must be strong enough in each part to withstand any force to which it may be subjected. If then we can find the greatest force that any particular piece of material can endure without breaking or suffering a permanent change in shape we can be sure of its remaining intact for all force within that limit. In the first four chapters we will apply, separately, all the different forces to which a piece of material can be subjected and as any piece in our machine or structure may have more than one of these forces to sustain at any given instant we will, in the fifth chapter, show the effect of the combined action of two or more of them.

Let us first investigate the effect on a piece of material of an external force applied to it, and we will choose for our investigation a straight rod of uniform cross-section and will not consider the force of gravity as acting. We will apply to the end of our rod a pull, F (not sufficient to break or permanently change its shape), which, in order that the rod remain stationary, will require an equal pull in the opposite direction at the other end. These forces tend to tear apart the particles of the material, and as the bar remains intact the particles must be in a different condition, relative to each other, from that in which they were before we applied the pull. If instead of a pull we exert a pressure on one end, an equal and opposite pressure must be applied to the other end to keep the bar in equilibrium, and the particles of the material will now tend to crowd together and crush each other. In both of these cases we have arranged our forces so that the bar does not move and they have been taken small enough so as not permanently to change its dimensions. Now as the length of the bar separates the points of application of our forces, there must be some action set up among the particles of the bar itself which transmits the force from one end to the other, or to a common point of action. Let us now imagine a plane passed through the bar perpendicular to its axis: in the first case (the pull, Fig. 1) our forces would tend to pull apart the two pieces of the bar; in the second (not shown) they would press them together, each piece would tend to move in the direction of the external force acting on it and the amount of this tendency would be equal to that force so that the total action in the bar between the particles on either side of any imaginary section would be equal but opposite to the external forces applied. The action of all the particles on the right side of any section would be equal to but opposite to the force on the right end, and of those on the left side equal to but opposite to the force on the left end. This must be true in order that the bar remain intact, i. e. in equilibrium.

Stress. When any such action as the above is set up among the particles of a piece of material that piece is said to be under stress, and the external force which causes the stress is called the load. A rod under the action of a pull is said to be in tension, and the pull is the tensile load. A rod to which pressure is applied is said to be under compression, and the pressure is called the compressive load. Hereafter in using the word stress we will mean the amount of action between the particles in unit cross-sectional area, and will use "total stress" for the action over the area of the whole cross-section. The stress per unit cross-sectional area is sometimes called "intensity of stress" and is equal to the external force, F, divided by the cross-sectional area, A.