It appears also to have been determined that a column or cylinder has a less power of resistance than a rectangular beam (in which b = D being the diameter of a circle) in the ratio 1: 1.7, therefore fora cylinder, the formula for the diameter is D x 1.7 c)also a rectangular beam is said to be weakest in the direction of its diagonal, and when square to require the multiplier 4, so that in a square beam = \V X 4 e, where d' is the diagonal.
We have, in compliance with general practice, given these rules, the investigation of some of which may be seen in Tralgold's work above quoted, and in Dr. Young's Lectures on Natural Philosophy; but we have already stated our opinion of their inadequacy, that is, they would be true provided we could assume an uniformity of texture throughout, but this is seldom to be expected in practice. The tables, however, are computed on the lowest possible resisting powers, so that the dimensions determinable by the rules will never fail in ooint of strength, unless the timber be very de flective, and it is much better to have our dimensions in excess than our strength in defect. For the reasons stated in the preceding section, the numerical appli cation of these tables and formula are omitted.
On the Resistance to Torsion.
Mr. Tredgold, in his Practical Essay on the Strength of Cast Iron, has the following illustration of the na ture of this strain. If a rectangular plate be support ed at the corner A and B, Fig. 5, and a weight be sus pended from each of the other corners C, D, then the strains produced by loading it in that manner will be similar to the twisting strains which occur in shafts. In a cast iron plate, the fracture would take place in the directions AB, and CD, at the same time; but before the fracture, the ore of the strains will serve as a fulcrum for the other; and the resistance to the forces at C and D will be sensibly the same as if the plate were supported upon a continued fulcrum in the di rection AB.
Hence, the strains may be considered as a transverse strain of the same kind as that already treated of with the leverage a D, or c C, acting at D or C, the breadth of the strained section being AB.
To find the breadth of the section of fracture, and the leverage in terms of the length and breadth of the The following experiments were made by Mr. George Rennie, and published in the Philosophical Transactions for 1818. The apparatus consisted of a
wrought iron lever, two feet long, having an arched head of about and four feet diameter, of which the lever represented the radius: the centre round which it moved had a square hole made to receive the end of the bar to be twisted. The lever was balanced, and a scale hung on the arched head; the other end of the bar being fixed in a square hole, in a piece of iron, and that again in a vice. The following are the results of these experiments:— With respect to internal pressure, such as that sustained by water pipes, hydraulic cylinders, Stc. we know of no actual experiments, but the amount of re sistance is so directly dependent on the resistance to tension, that no experiments are in this case neces sary. If we examine the force which tends to pro duce the rupture in this case, it will be seen immedi ately that it varies as the longitudinal section of the cylinder; or which is the same, the circumferential strain on any given point of the interior of the cylinder, is equal to the pressure on a square inch multiplied by the number of inches in the radius. That is, the force tending to rend the cylinder along any line pa rallel to its axis, is equal to a pressure on a section between the circumference and axis.
Hence it would appear at first sight that the deter mination of the thickness to resist this pressure would be simply to determine the sectional area of the metal requisite for this purpose, on the supposition of every part bearing an equal torsion: this, however, is not sufficient, and practice has pointed out that in presses and pipes, it is always necessary to increase the thickness in a higher ratio than the pressure. This subject has been investigated by Mr. Barlow, and the following is the result of his inquiries. If we imagine, as we ought to do, that the metal, in conse quence of the internal pressure, suffers a certain de gree of extension, it will be found that the external circumference participates less in this extension than the internal, and as the resistance is proportional to the extension divided by the length, it follows that the interior circumference, and every successive circular lamina from the exterior to the interior surface, offers a less and less resistance to the interior strain. The law of which it is our object to investigate.