Having thus detailed various points connected with the strains or pressures which weights exercise on beams, under various circumstances, we have now briefly to explain how these influence the form of beams subjected to them. In a special diagram in the present chapter we have shown that a beam supported at both ends, and loaded in the centre, can resist only half the pressure or sustain half the weight which a beam similarly placed can sustain or support with the weight distributed equally over its whole surface. We have also shown how that weight exercises its greatest pressure on a beam when placed at its centre; it follows from this, that if the weight be moved from this point towards either one or other of its ends, where it is supported, the pressure exercised by the weight on the beam is less and less, in proportion as the weight recedes from the centre. This may be illustrated in a simple diagram, fig. 470, in which a b c represents a beam, supported at its ends by the walls d e, the arrow b f representing the weight at the centre, and therefore exercising its greatest pressure, while other weights represented by arrows, as g h i j k l m and n, may be taken as illustrating the gradual decrease of the various pressures at different points receding from the centre; or, in other words, the shorter arrows may represent the different positions to which the central weight f b is moved along the beam. As the pressure at the point h or g is less than that at the point b f, and j or i less than that at g or h, it follows that it is simply a waste of material to make the beam of the same depth at the points g or i, as at b, and as the pressures gradually decrease towards the points of support a and c, so that it may decrease on either side of the point b, towards a and c; this is illustrated by the curved dotted line m n. This curve gives of course apparently the decrease on the upper part of the beam, but for obvious reasons, in convenience of building, etc., the flat side, as o q, is placed uppermost, the curve p q being at the lower edge of the beam; again, as the narrow points of the curves - a parabolic one being the best, as o q - present obvious inconveniences, the ends are finished off with a flat surface, as at the points r and s in elevation and plan, these being built into the wall t, or the end may be finished with a curved part u secured to the wall by a bolt and nut v.
Again, in the case of beams projecting from a wall built into or supported by this, as the beam a, fig. 471, at point b; the pressure sustained by the beam is greatest at the point b, in the case where the weight, as c, is supported at the end d of the beam, inasmuch as the leverage which the weight c exercises on the beam decreases in its fracturing capacity towards the end d ; this is illustrated in fig. 472, after the manner used or employed in fig. 470, showing also that as the tendency of the weight c, fig. 471, to fracture the beam diminishes towards the end d, the beam will be as strong as when in the form of a b c, fig. 472, as when of the form a b d c. To find the strain at b, fig. 471, when so far as the beam itself is concerned, not taking into account any weight to which it may be subjected, multiply the weight of the beam by half its length. As in the case of beams supported at both ends, so in that of beams supported at one end; a beam, as b a d, fig. 471, loaded at its end by a weight c will support only half the weight of a beam similarly placed, as e f, over the surface of which the weight represented by the lines g h is uniformly distributed; and as in the case ab d c, fig. 472, the beam, as e f, fig. 471, will be as strong if formed as a triangle e f g, as if formed as a rectangle e f h g.
These points as yet named, with relation to the form of beams, have had reference to their outline considered as elevations, or the appearance they present when viewed from either side; we have now to take up the form as presented when looked at at right angles to the side or front view; in other words, their cross or transverse section. We have just seen that the side elevation of a beam in the improved form, as detailed by results of scientific observation and repeated experiments, varies according to circumstances, and indeed the ideas or taste of the designer; but so far as regards the cross or transverse section is concerned, a form and proportion have been decided upon and adopted by the best practitioners, as that which gives the maximum of strength with the minimum of material. That form and these proportions were only arrived at by long and patient observation, and by a series of experiments of the most elaborate and painstaking character.
Of those who devoted themselves to this important work - while many could be named who have done good service in connection with it - the name of one stands prominently and pre-eminently forward and first, Mr. Hodgkinson of Manchester and London. This gentlemen brought to the task the accomplishment of mathematical knowledge of a profoundity rarely met with, and which has perhaps never been excelled, at least in modern times; together with a remarkable skill in applying this to circumstances which were altogether novel, many of which might be said to be created by him to meet the peculiar wants and necessities of the investigations he entered upon, and all of which were characterised by difficulties of no common order; and further by the exercise of a wonderful patience in observing and recording, and by the application of new methods of investigation and experimenting, he succeeded in so placing the whole subject of strength of material in such a thoroughly practical position, and deducing formulae and rules so easily applied and so applicable to almost every variety or kind of practice, that it may be said he left little or no work for others to do who might follow after him. In connection with the labours of this distinguished scientist, it is impossible to omit naming one, who partly in the same field of mathematical investigation, but more immediately in that which may be called the practical, namely, William Fairbairn, whose reputation is world-wide as a mechanical engineer. The value of such labours cannot be over-estimated, for apart from the importance of being able to design parts of structures which have to support heavy weights or to sustain great pressures, with the least expenditure of material, and thus effect in most cases a considerable, but in some extensive works a very large saving in money, the fact is too often overlooked that the mere weight of a beam, for example, so badly designed and its proportions so carelessly calculated - if calculated at all - that its material is greatly in excess of what is really required, endangers the stability of the building, nay, has a tendency to weaken the part itself by throwing, by this extra weight, a pressure upon it which it should not be required to bear; so that the " rule of thumb " principle, which is said to be the safest, because it errs generally in making the parts too strong, in order to be " sure of them," as the phrase goes, introduces the very element of danger which it is supposed to avoid. Such remarks are not out of place here nor devoid of practical value, as they touch points often overlooked, which exercise an important influence in sound and economical construction.