tensile strain on the convex side, and the beam will endeavour to regain its normal form by the repulsion of its molecules, m, on each other, and the attractions of those, m', tending to restore each vertical layer, m m', to its original vertical position, by revolving it about a point in the layer I K. IK is termed the neutral layer, or neutral axis.

This illustrates the internal resistances of a beam to bending as a number of pushing and pulling efforts acting along the two arms of a lever, those of opposite kinds being on opposite sides of the fulcrum.

The intensities of these elastic resistances and their effects will be exactly ascertained when treating of the behaviour of materials subjected to bending or transverse

stress.

When material is placed between two edges so that on their being pressed towards each other a tendency to sever it arises, the strain is known as shearing strain; this occurs on the rivets joining plates on which longitudinal forces act: a diagram will clearly illustrate the manner in which this stress acts.

E

Α

F

In Fig. 2 let A and B be two edges acting upon the opposite surfaces of the body E F in such a manner as to shear it along the line c d. The action tends to force each molecule downwards, and away from that opposite it, and the result will be different according to the nature of the material. It may happen that as a layer of molecules is forced down from the opposite layer, in coming opposite the layer next below, it will be attracted by that with sufficient force to maintain the continuity of the mass, if the shearing force be now stopped. Such a case is ex

Fig. 2.

B

hibited by lead; but if this does not occur, the mass will be divided at cd, and the molecules having been forced asunder, it is evidently tensile stress that has been called into action; therefore, we should expect the shearing resistance of a body to be for equal area equal to its tensile resistance, and this is found to be the general rule in non-crystalline matter. The failure of bodies under compressive strain occurs in several different ways. If the member acted upon is not absolutely straight and of equal resistance per square inch in every part of any given section, it will give way by bending or crippling, without being actually crushed; and when it so happens the convex side will be in tension and the concave in compression, the same as in a beam under transverse load. As it is commercially impossible to secure homogeneity or uniformity of substance throughout the materials we use, it follows that failures under compression must, in elements of any length compared with their breadth or thickness, occur in the manner just described.

In dealing with this question we are at the outset met by the difficulty that we do not quite know how the work will break, for it may be by transverse bending, or it may be by a process of crushing or splitting, and this will depend not only on the nature of the material, but also on the circumstances in which it is placed. It is very seldom that an actual flattening out will occur practically.

A

1B

In Fig. 3 the methods of crushing are shown at A; the upper part has wedged asunder the lower, and here there evidently are in action both shearing force along the faces of division, and tensile stress on the parts forced open. In fracture, as shown at B, shearing stress alone seems to come into play, but the angle will depend upon the qualities of the material experimented on. The calcula

Fig. 3.

tions used in determining the proportions of columns and elements subject to compressive stress are of an empirical nature, being derived from extensive series of experiments.

It is impossible to examine the internal action of the external forces without being struck by the manner in which every kind of force seems to incline towards conversion into tensile stress on the molecules, and it must always happen that rupture ultimately occurs by these molecules being pulled or driven beyond the limits of their spheres of mutual attraction, for so long as they remain within those spheres they cannot be separated and fall asunder. When bodies are distorted, and so kept for a length of time, there is observed a tendency of the molecules to rearrangement by equalising the internal strains, and the less perfect the elasticity of the material the more extensive will be the adjustment thus occurring.

Passing on beyond the limit of elasticity of a material, we may yet stop short of actual rupture, but the integrity of the substance will have been invaded, some permanent distortion will have been caused, and consequently we may assume that a proportionate amount of damage has been done.

I would here warn my readers against the abuse of the term "permanent set," as it is applied commonly to effects which have no more similarity to that which it really means than has the lightest stress to absolute fracture of material.

Permanent set, in reference to structural details, means, actual molecular alteration in the internal arrangement of the substances acted upon, and it is incorrect to apply it to any other effect. The most common misapplication is as applied to the permanent subsidence of works due to the joints falling into their bearings, and defects in the mode of erection; but some folks have even gone to the extreme of applying the term to the deflection due to a permanent load, which really should be called the permanent deflec

tion. I have thought it desirable to allude to this matter, as nothing tends more to confusion than the slovenly misapplication of technical terms.

It is evident that the limit of elasticity cannot be exceeded with impunity, and practically a sufficiently wide margin of strength must be left. It seems only reasonable that the working strength of material should be taken in some proportion to the elastic limit of resistance and not to the ultimate resistance of the material, but up to the present time this latter ratio has been the guide almost universally. In some substances, however, it must be remarked the range of elasticity is so extremely small as not to be observable, as in some kinds of stone, brick, &c., and the material appears to give way without previous alteration of shape, although we know that some such alteration must take place before the normal arrangement of the internal forces can be altered.

As to the factors of working strength to be used in practice, I shall give those under the special headings of the various descriptions of constructive details upon which they bear.

The resistances offered by structures to disturbing forces may be put in two classes:-1st, the resistance due to the strength of the material—that is, to the cohesion of its constituent molecules; this is properly called the strength of the work.

2nd. The resistance offered by the dead weight of material, which may operate in opposition to an overturning effort, or to a force tending to slide its mass bodily on the surface on which it stands, or to both combined; this resistance is the "stability" of the work.

In the first form the force is directed to overcome gravity by causing a body to lift and revolve about one of its edges until the centre of gravity falls without such edge, when the mass will altogether upset; in the second, the

resistance to be overcome is that of the friction of the mass on the surface upon which it rests.

I will here point out the nature of the resisting force of friction between surfaces. The surfaces are taken to be physically smooth and free from viscosity or any special property of attraction for, or repulsion of, each other.

On account of the elasticity of matter, it follows that if a body rest upon another of larger surface than its surface of contact, it will to some extent sink into it, compressing the parts immediately beneath; hence, if the upper body be pushed along, it must as it were be pushed into a higher part, which, in its turn, sinks under the weight imposed upon it, so that virtually, in moving the upper body, we are constantly pushing it uphill. As time is required for a substance to be compressed, it is evident that if the upper body be moved rapidly upon the lower it will not at any time sink as much as if it were allowed to rest on one spot; hence the friction at starting, which is the friction of rest, is greater than the friction of motion; and the friction of slow motion will be greater than that of rapid motion, so far as the surfaces themselves are concerned.

The friction of surfaces, when the pressures upon them are slight, compared to what would be required to injure them, are taken as certain fractions of the insistent pressures, regardless of the areas in contact; but it is evident that regarding the matter strictly, there should be some variation with area of surface exposed to a varying pressure this, however, does not seem to be of sufficient magnitude to demand practical consideration.

Having determined the properties and capabilities of the materials at our disposal, it follows to consider the best modes of applying them for economic purposes: to arrive at these, the nature of the forces presenting themselves as acting upon structures and machinery must be carefully