(330.) Experimental enquiries into the laws which regulate the strength of solid bodies, or their power to resist forces variously applied to tear or break them, are obstructed by practical difficulties, the nature and extent of which are so discouraging that few have ventured to encounter them at all, and still fewer have had the steadiness to persevere until any result showing a general law has been obtained. These difficulties arise, partly from the great forces which must be applied, but more from the peculiar nature of the objects of those experiments. The end to which such an enquiry must be directed is the development of a general law; that is, such a rule as would be rigidly observed if the materials, the strength of which is the object of enquiry, were perfectly uniform in their texture, and subject to no casual inequalities. In proportion as these inequalities are frequent, experiments must be multiplied, that a long average may embrace cases varying in both extremes, so as to eliminate each other’s effects in the final result.
The materials of which structures and works of art are composed are liable to so many and so considerable inequalities of texture, that any rule which can be deduced, even by the most extensive series of experiments, must be regarded as a mean result, from which individual examples will be found to vary in so great a degree, that more than usual caution must be observed in its practical application. The details of this subject belong to engineering, more properly than to the elements of mechanics. Nevertheless, a general view of the most important principles which have been established respecting the strength of materials will not be misplaced in this treatise.
A piece of solid matter may be submitted to the action of a force tending to separate its parts in several ways; the principal of which are,—
1. To a direct pull,—as when a rope or wire is stretched by a weight. When a tie-beam resists the separation of the sides of a structure, &c.
2. To a direct pressure or thrust,—as when a weight rests upon a pillar.
3. To a transverse strain,—as when weights on the ends of a lever press it on the fulcrum.
(331.) If a solid be submitted to a force which draws it in the direction of its length, having a tendency to pull its ends in opposite directions, its strength or power to resist such a force is proportional to the magnitude of its transverse section. Thus, suppose a square rod of metal A B, fig. 185., of the breadth and thickness of one inch, be pulled by a force in the direction A B, and that a certain force is found sufficient to tear it; a rod of the same metal of twice the breadth and the same thickness will require double the force to break it; one of treble the breadth and the same thickness will require treble the force to break it, and so on.
The reason of this is evident. A rod of double or treble the thickness, in this case, is equivalent to two or three equal and similar rods which equally and separately resist the drawing force, and therefore possess a degree of strength proportionate to their number.
It will easily be perceived, that whatever be the section, the same reasoning will be applicable, and the power of resistance will, in general, be proportional to its magnitude or area.
If the material were perfectly uniform throughout its dimensions, the resistance to a direct pull would not be affected by the length of the rod. In practice, however, the increase of length is found to lessen the strength. This is to be attributed to the increased chance of inequality.
(332.) No satisfactory results have been obtained either by theory or experiment respecting the laws by which solids resist compression. The power of a perpendicular pillar to support a weight placed upon it evidently depends on its thickness, or the magnitude of its base, and on its height. It is certain that when the height is the same, the strength increases with every increase of the base, but it seems doubtful whether the strength be exactly proportional to the base. That is, if two columns of the same material have equal heights, and the base of one be double the base of the other, the strength of one will be greater, but it is not certain whether it will exactly double that of the other. According to the theory of Euler, which is in a certain degree verified by the experiments of Musschenbrock, the strength will be increased in a greater proportion than the base, so that, if the base be doubled, the strength will be more than doubled.
When the base is the same, the strength is diminished by increasing the height, and this decrease of strength is proportionally greater than the increase of height. According to Euler’s theory, the decrease of strength is proportional to the square of the height; that is, when the height is increased in a two-fold proportion, the strength is diminished in a four-fold proportion.
(333.) The strain to which solids forming the parts of structures of every kind are most commonly exposed is the lateral or transverse strain, or that which acts at right angles to their lengths. If any strain act obliquely to the direction of their length it may be resolved into two forces (76.), one in the direction of the length, and the other at right angles to the length. That part which acts in the direction of the length will produce either compression or a direct pull, and its effect must be investigated accordingly.
Although the results of theory, as well as those of experimental investigations, present great discordances respecting the transverse strength of solids, yet there are some particulars, in which they, for the most part, agree; to this it is our object here to confine our observations, declining all details relating to disputed points.
Let A B C D, fig. 186., be a beam, supported at its ends A and B. Its strength to support a weight at E pressing downwards at right angles to its length is evidently proportional to its breadth, the other things being the same. For a beam of double or treble breadth, and of the same thickness, is equivalent to two or three equal and similar beams placed side by side. Since each of these would possess the same strength, the whole taken together would possess double or treble the strength of any one of them.
When the breadth and length are the same the strength obviously increases with the depth, but not in the same proportion. The increase of strength is found to be much greater in proportion than the increase of depth. By the theory of Galileo, a double or treble thickness ought to increase the strength in a four-fold or nine-fold proportion, and experiments in most cases do not materially vary from this rule.
If while the breadth and depth remain the same, the length of the beam, or rather, the distance between the points of support, vary, the strength will vary accordingly, decreasing in the same proportion as the length increases.
From these observations it appears, that the transverse strength of a beam depends more on its thickness than its breadth. Hence we find that a broad thin board is much stronger when its edge is presented upwards. On this principle the joists or rafters of floors and roofs are constructed.
If two beams be in all respects similar, their strengths will be in the proportion of the squares of their lengths. Let the length, breadth, and depth of the one be respectively double the length, breadth, and depth of the other. By the double breadth the beam doubles its strength, but by doubling the length half this strength is lost. Thus the increase of length and breadth counteract each other’s effects, and as far as they are concerned the strength of the beam is not changed. But by doubling the thickness the strength is increased in a four-fold proportion, that is, as the square of the length. In the same manner it may be shown, that when all the dimensions are trebled, the strength is increased in a nine-fold proportion, and so on.
(334.) In all structures the materials have to support their own weight, and therefore their available strength is to be estimated by the excess of their absolute strength above that degree of strength which is just sufficient to support their own weight. This consideration leads to some conclusions, of which numerous and striking illustrations are presented in the works of nature and art.
We have seen that the absolute strength with which a lateral strain is resisted is in the proportion of the square of the linear dimensions of similar parts of a structure, and therefore the amount of this strength increases rapidly with every increase of the dimensions of a body. But at the same time the weight of the body increases in a still more rapid proportion. Thus, if the several dimensions be doubled, the strength will be increased in a four-fold but the weight in an eight-fold proportion. If the dimensions be trebled, the strength will be multiplied nine times, but the weight twenty-seven times. Again, if the dimensions be multiplied four times, the strength will be multiplied sixteen times, and the weight sixty-four times, and so on.
Hence it is obvious, that although the strength of a body of small dimensions may greatly exceed its weight, and, therefore, it may be able to support a load many times its own weight; yet by a great increase in the dimensions the weight increasing in a much greater degree the available strength may be much diminished, and such a magnitude may be assigned, that the weight of the body must exceed its strength, and it not only would be unable to support any load, but would actually fall to pieces by its own weight.
The strength of a structure of any kind is not, therefore, to be determined by that of its model, which will always be much stronger in proportion to its size. All works natural and artificial have limits of magnitude which, while their materials remain the same, they cannot surpass.
In conformity with what has just been explained, it has been observed, that small animals are stronger in proportion than large ones; that the young plant has more available strength in proportion than the large forest tree; that children are less liable to injury from accident than men, &c. But although to a certain extent these observations are just, yet it ought not to be forgotten, that the mechanical conclusions which they are brought to illustrate are founded on the supposition, that the smaller and greater bodies which are compared are composed of precisely similar materials. This is not the case in any of the examples here adduced.