If the weight be removed, the reaction of the beam will cause it to regain its original figure if not resisted by a pressure at the ends. The force of this reaction will be proportional to the degree to which the fibres are strained, and as the strain upon the fibres is nothing at the ends A and B, and increases uniformly to the middle point, the force of reaction will be in the same proportion, and the point of application of the resultant of the whole of the reacting forces will correspond to the centre of gravity of a triangle whose base is Bf; it will consequently be at a distance from 2 3 The effect of this resultant acting at a distance. Bf, must be the same as the weight (22) acting at a distance Bƒ, and must consequently be in proportion to as 3: 2. The value of the resultant is therefore 2 3 w 4 The line of direction of the pressure at B being the tangent B C, the force of reaction at h may be considered as applied at the point k of its line of direction, and as kh B and CfB are similar triangles, Cf: ƒB:: 3 f Ᏼ 4 ƒC = ·wx. 3 w: horizontal pressure at B 4 senting this force by P we have 31 3 w 1 3 Repre As the deflexion of a beam within the elastic limits is always in proportion to the weight, if (w') = = the weight that will produce a deflection equal to unity, the deflexion (d) will require a weight = (d w'), and by substituting this value in the equation, we find In this expression (d), which represents the deflexion, has disappeared, and as (w) is a constant quantity for the same beam, representing the weight that produces a deflexion equal to the unit of measure, it follows that P is the same with every weight and every degree of deflexion within the elastic limits. This result seems at first view to be contrary to fact; it would appear that if the weight is increased, the horizontal strain should be increased in the same proportion; but when it is remembered that the deflexion increases with the weight, and that the former diminishes the value of P in precisely the same proportion that the latter increases it, the difficulty vanishes, and the reason why P should be constant for the same beam becomes obvious. The practical importance of this result is very great, as it furnishes the means of obtaining a formula which will give at once the extreme limit of the resistance to flexure, or the weight which, applied to a post, will cause it to yield by bending. As the formulæ used by Tredgold are calculated for a deflexion of th But from the ordinary formula for the stiffness of a beam supported at the ends we have The expression P90 w shows that the extreme limit of the strength of any post whatever, of any length, breadth, or depth, or of any kind of material, is ninety times the weight which causes a deflexion of ʊ of the length. 90b d3 will give the value of P directly, without first knowing the weight required to cause a given deflexion in a horizontally supported beam. In this expression, b = breadth in inches, d depth or least dimension in inches, l length in feet, and c = a constant to be determined by experiment for each species of material. The value of c for white pine is 01. By substituting this value, we 9000 b d3 find P = 12 -, a remarkably simple formula, which gives the ex treme limit of the resistance to flexure of a white pine post. The same expression may be used to determine the constants used in the ordinary formula for the stiffness of beams. For this purpose let the equation P = 90 b d3 Find 90 b d3 be transposed, which will give c = P 2 P by applying a string to a flexible strip of the material to be experimented upon, in the manner of a cord to an arc, and ascertain the tension on the cord with an accurate spring balance. It will be found that, whether the strip be bent much or little, the tension on the cord, as shown by the spring balance, will be constant, and this tension, in pounds substituted for P, will give the value of c without requiring, as is necessary with other formulæ, an observation of the deflexion. Experiments made upon these principles with strips of white pine, yellow pine, and white oak, 5 feet long, 14 inches wide, and inch deep, give the following results: The observed tensions were As the stiffness is inversely as their constants, it follows that white pine is stiffer than yellow pine or oak. The experiments of Tredgold give similar results. MECHANICS, PHYSICS, AND CHEMISTRY. Report on the Gases evolved from Iron Furnaces, with reference to the Theory of the Smelting of Iron. By Prof. BUNSEN, of Marburg, Hesse Cassel, and Dr. LYON PLAYFAIR, of the Museum of Economic Geology, Department of Her Majesty's Woods and Forests. From the Report of the British Association for the Advancement of Science, for 1845. (Continued from Vol. xvII, page 393.) The heat lost in a furnace may be easily compared with that actually realized. The following numbers exhibit the quantities of heat (expressed by the unities which we formerly described) generated during the combustion of the gases, and they show at the same time the part played by each constituent in the development of the heat: The numbers (II.) representing the units of heat are calculated from the data on the heat of combustion found in the posthumous papers of Dulong. Carbon burning to CO, heats 15444 grains of water to 1499°C. 1 kilogramme, or 15,444 grains of 73710 The quantity of heat actually generated in the furnace during the escape of the unused 110,570 units of heat may be determined by the amount of nitrogen in the gases, which corresponds to the quantity of the air consumed during their escape. The amount of nitrogen found, viz. 64.126, corresponds to 83-29 of atmospheric air, which is able to effect the conversion of 14.367 carbon into carbonic oxide gas. Proceeding on the • The ammonia and sulphuretted hydrogen are calculated from their constituents. experiments of Dulong, the quantity of heat thus liberated will be 21,536°. Thus it follows that a furnace filled with Gasforth coal could realize in the most favorable conditions only 16.30 per cent. of its combustible material. The remainder, 83-70 per cent., escapes as unused but useful combustible matter. The practical use of these gases does not depend merely upon the quantity of heat generated during their combustion, but involves another equally important consideration, viz. the temperature capable of being attained by their use as fuel. This may be determined without any new experiments, by founding the calculation on the composition of the gas, its combustible value, and the capacity for heat of the products generated during combustion. 1 kil. of the gas gives, by its combustion, as we have already seen, 1105.7 units of heat. The products of combustion weigh 2.1385 kil.; and if this last quantity* consisted of water, the heat liberated would raise 1105.7 it to a temperature Now as the capacity of the heat of water is 2-1385 to that of the products of combustion as 1 : 0.2665, and the elevation of temperature produced in different bodies of equal weight, by equal quantities of heat, is in inverse proportion to their capacity for heat, we obtain, as the expression for the temperature of the mixture of gas burning with 1105.7 = 1940° C., or 3522° Fahr. air, 2.1385 x 0.2665 In these calculations we have neglected the influence exerted on the composition of the products of combustion by the gases escaping from the iron ore and limestone. Of course this must differ according to the quantities of materials used in the furnaces, and we therefore select as a basis for the calculation the iron furnaces of Alfreton, belonging to Mr. Oakes, the dimensions of which are given in fig. 6. We proceed on the supposition that the carbonic acid of the limestone and the oxygen of the ore are separated as carbonic acid. The coal used in the furnace was subjected to distillation with the precautions already described, and the composition thus obtained gives us the limits to which the combustible constituent of the gases from the furnace might be deteriorated under the most unfavorable conditions. 0.1321 10. Weight of the condensed hydrocarbons * In this, and also in other similar calculations, the small quantity of sulphuretted hydrogen has been left out of the calculation. VOL. XVIII.-THIRD SERIES.-No. 1.—JULY, 1849. 3 The eudiometric analysis of the uncondensed gases gave the following results: If we suppose that the very inconsiderable quantity of nitrogen found in the calculation (0.4) was an unavoidable impurity, we obtain, by the use of the formula 1, 2, 3, the following composition for the gas examined: According to volume. According to weight. Hydrogen Light carburetted hydrogen The 2.2142 carbonic oxide, carburetted hydrogen, and hydrogen, found in the analysis consist therefore of— Light carburetted hydrogen Carbonic oxide Hydrogen 1.7067 0.4122 0.0953 2.2142 Thus 100 parts of the coal were broken up into the following products: The 67-228 carbon in the above analysis must escape altogether as carbonic oxide, if a part of it were not converted into carbonic acid at the expense of the oxygen of the iron ore. In order to determine the quantity of carbonic acid thus produced, we must refer to the details of the Alfreton The ammonia escaping with the gases out of the condensed alcoholic water is neglected in this calculation. |