Note 3.-If the load is to be applied at any other point than the middle, then the strength will be as the product of the two distances is to the square of half the length of the beam between the supports; - or, twice the distance from one end, multiplied by twice from the other, and divided by the whole length, equal the effective length of the beam. Ex. In a building 18 feet in width, an engine boiler of 5 tons is to be fixed, the centre of which to be 7 feet from the wall; and having two pieces of red pine, 10 inches by 6, which I can lay across the two walls for the purpose of slinging it at each end, may I with sufficient confidence apply them, so as to effect this object? 2240 X 5.5 =6160 lbs. to carry at each end. And 18 feet-7=11, double each, or 14 and 22, then 14 × 22 17 feet, or 204 inches, effective length of beam. 18 1341 x 4 x 10 x 60 Tabular value of S, red pine, = 204 = 15776 lbs. the absolute strength of each piece of timber at that point. To determine the dimensions of a rectangular beam capable of supporting a required weight, with a given degree of deflection, when fixed at one end. Rule. -Divide the weight to be supported, in lbs., by the tabular value of E, multiplied by the breadth and deflection, both in inches; and the cube root of the quotient, multiplied by the length in feet, equal the depth required in inches. Ex. A beam of ash is intended to bear a load of 700 lbs. at its extremity; its length being 5 feet, its breadth 4 inches, and the deflection not to exceed of an inch. Tabular value of E = 119 x 4× 5: =238 the divisor; then 700-238 = 3√/2.94 x 5 = 7.25 inches, depth of the beam. To find the absolute strength of a rectangular beam, when fixed at one end, and loaded at the other. Rule. Multiply the value of S by the depth of the beam, and by the area of its section, both in inches; divide the product by the leverage in inches, and the quotient equal the absolute strength of the beam in lbs. Ex. A beam of Riga fir, 12 inches by 42, and projecting 6 feet from the wall; what is the greatest weight it will support at the extremity of its length? When fracture of a beam is produced by vertical pressure, the fibres of the lower section of fracture are separated by extension, whilst at the same time those of the upper portion are destroyed by compression; hence exists a point in section where neither the one nor the other takes place, and which is distinguished as the point of neutral axis. Therefore, by the law of fracture thus established, and proper data of tenacity and compression given, as in the table, (p. 52) we are enabled to form metal beams of strongest section with the least possible material. Thus, in cast iron, the resistance to compression is nearly as 6 to 1 of tenacity; consequently a beam of cast iron, to be of strongest section, must be of the following form, and a parabola in the direction of its length, the quantity of material in the bottom flange being about 63 times that of the upper. But such is not the case with beams of timber; for although the tenacity of timber be on an average twice that of its resistance to compression, its flexibility is so great, that any considerable length of beam, where columns cannot be situated to its support, requires to be strengthened or trussed by iron rods, as in the following manner. And these applications of principle not only tend to diminish deflection, but the required purpose is also more effectively attained, and that by lighter pieces of timber. To ascertain the absolute strength of a cast iron beam of the preceding form, or that of strongest section. Rule. Multiply the sectional area of the bottom flange in inches by the depth of the beam in inches, and divide the product by the distance between the supports, also in inches; and 514 times the quotient equal the absolute strength of the beam in cwts. The strongest form in which any given quantity of matter can be disposed is that of a hollow cylinder; and it has been demonstrated that the maximum of strength is obtained in cast iron, when the thickness of the annulus, or ring, amounts to th of the cylinder's external diameter; the relative strength of a solid to that of a hollow cylinder being as the diameters of their sections. A Table showing the Weight or Pressure a Beam of Cast Iron, 1 inch in breadth, will sustain, without destroying its elastic force, when it is supported at each end, and loaded in the middle of its length, and also the deflection in the middle which that weight will produce By Mr. Hodgkinson, Manchester. Length. 6 feet. 7 feet. 8 feet. 9 feet. 10 feet. lbs. in in. Depth Wt. in Def. Wt. in Def. Wt. in Def. Wt. Defl. in in. 954 426 855 ⚫54 1298 365 1164 46 1700 32 1520 405 Wt. in Defl. 765 66 1041 57 1936 245 1360 5 1721 443 2125 4 3060 33 4165 29 5440 25 6885 22 11971 22 10260 31 Note.-This Table shows the greatest weight that ever ought to be laid upon a beam for permanent load; and, if there be any liability to jerks, &c., ample allowance must be made; also, the weight of the beam itself must be included. 8978 39 To find the weight of a cast iron beam of given dimen sions. Rule. Multiply the sectional area in inches by the length in feet, and by 3-2, the product equal the weight in lbs. Er. Required the weight of a uniform rectangular beam of cast iron, 16 feet in length, 11 inches in breadth, and 1 inch in thickness. 11 x 1.5 x 16 x 3.2844-8 lbs. Resistance of Bodies to Flexure by vertical Pressure. When a piece of timber is employed as a column or support, its tendency to yielding by compression is dif ferent according to the proportion between its length and area of its cross section; and supposing the form that of a cylinder whose length is less than seven or eight times its diameter, it is impossible to bend it by any force applied longitudinally, as it will be destroyed by splitting before that bending can take place; but when the length exceeds this, the column will bend under a certain load, and be ultimately destroyed by a similar kind of action to that which has place in the transverse strain. Columns of cast iron and of other bodies are also similarly circumstanced, this law having recently been fully developed by the experiments of Mr. Hodgkinson on columns of different diameters, and of different lengths. When the length of a cast iron column with flat ends equals about thirty times its diameter, fracture will be produced wholly by bending of the material. When of less length, fracture takes place partly by crushing and partly by bending. But, when the column is enlarged in the middle of its length from one and a half to twice its diameter at the ends, by being cast hollow, the strength is greater by 4th than in a solid column containing the same quantity of material. |