Mr. Tredgold has computed the value of the con stant a, when the deflection is 1-40th of an inch to a foot, or 1-480th of the length, from the experiments reported in the following tables, consequently his con stant is obtained by the formula a = 40bd3J P W • When the length, depth, and weight are given, to find the breadth b= a 1 a W These are the only cases which arise in practice, and the numerical operations are sufficiently evident with out any numerical examples.
On the Resistance of Columns to a Vertical Pressure.
The application of the results contained in the pre ceding table is sufficiently obvious, at least while the deflection is intended to be of the length, and if in any case a greater deflection be admissible, or a less be requisite, it is only necessary to find a new value of a from the tabular one, by saying, as the proposed deflection, divided by the length, is to so is the tabular value of a to the value sought; with which proceed according to the following rules or formula: When the length, breadth, and weight are given, to find the depth then V a l a NV 1.
We have already given a detail of various experi ments on the force requisite to crush certain materials, but in the case here to be considered the material is not destroyed by the actual crush, but by a flexure which always takes place, and after which the me chanism of the operation resembles that already con sidered, under the denomination of the transverse strain. The results, however, from actual experiment, are, in this case, by no means so uniform as in the former; and theory, it must be acknowledged, can assist us but very little, because theory always supposes an uniformity of result in practical cases, where all the conditions are the same, but here with two trees, of the same wood, the same specific gravity, and the same dimensions, owing to some internal and hidden cause, will give results very wide of each other; the bending will take place in different parts and in different direc tions, and the fracture produced by different weights. It is therefore only within certain limits that practical rules can be laid down in these cases. In pieces of timber submitted to a transverse strain, the effect of the weight is greatest in a certain and determinate point, and if the material be sound at that place, a de fect iti'another part is seldom of any consequence; but when the whole piece is submitted to pressure end wise, every defective part has an influence in the ope ration; and to this cause may be principally attributed the irregular nature of results obtained in experiments of this description. Therefore, in stating the following
principles of computation, the reader will be aware that the same security cannot be placed in them as in those which have preceded, and that they are only applicable when the timber is uniformly sound through out. They may be stated as follows, viz.
The resistance to flexure varies directly as the breadth into the cube of the depth, or as the least di mension into the cube of the greater; And the strain varies as the square of the length into the weight; Consequently, if we have a series of experiments, showing the flexure which has been produced by certain weights in beams of given dimensions, we may hence compute the weight that would produce any other given flexure, and hence again the length and weight, in any other case, being given, we may com pute the requisite breadth and depth, these being either equal, or bearing any given proportion to each other.
The two following tables are of this description, the one extracted from " Construction des Pouts, &e." of M. Gauthey, and the second from the " Trait6 Ana lytique de la Resistance des Solides," by M. Girard, as they are given by Mr. Tredgold in his Elementary Principles of Carpentry. The elasticity c, however, as deduced from these experiments, the latter author considers too strong for general practice, and be re commends in practical cases to use the mean value e= for oak, and this being assumed, he has computed the mean value for the other woods stated in a preceding table for the transverse strain; this table we have also given. And since, acccording to the preceding / analo- v gies, it appears that = e or constant quantity, 21, we shall have at once for computing the depth or least thickness of beams (all the rest being given), v W or e W e when d is given, and b the greater side is sought.