When the beam A B (fig. 6), supports a weight W, at E, it is equally strong between the points A and B, if the upper sides, AE, BE, be two parabolas whose vertex is A and B respectively. Let the weight W have a bearing E F (fig. 7), equal on both sides of the centre G, and also let the weight be equally distributed on the bearing E F. The units of strain at G are equal to W.AB W.EF Now, if the weight W were a sphere, and were laid on the middle of the beam at G, the units of strain at G would be equal to W. AB 4 If the same weight be formed into a cube, whose side is EF, the units of strain at the centre G will be less than in the case of the W.EF sphere by 8 Let A B be any beam suspended vertically from the point A (fig. 8): and let the sectional area be constant from A to B, where a weight W lbs. is acting to extend the beam. Put a area of the section of the beam in square inches. 7 = length of the beam in feet before the e= the elongation produced by the weight W. = e equal to l. Fig. 8. A B W of elasticity of the material of which the beam is composed. In the case of the beam being compressed by the weight W acting in the opposite direction, Put c = compression produced by the weight W. C = force which is necessary to make c equal to half of (l). The quantity C is called the modulus of elasticity of the material, when it is subject to compression. Units of work done to elongate the beam e feet Units of work done to compress the beam c feet = W c 2 Mean results of experiments on four different kinds of Cast-iron bars, 10 feet long and Weight laid on bar square inch in section. Extension of bar in Set of bar in inches. The value of 12 W inches 12 e. e Hence, the breaking weight per square inch of section is 14793 lbs. 6.6 tons nearly; and the ultimate extension is .1859 inches, or of the whole length, 10 feet. 5 If we deduct the set 0209 from 1859, we shall have 165 inches for the elongation produced by the weight 14793 lbs. .. Breaking weight = 6.6 tons × area of section in square inches. If the weight 5269 be taken, the modulus of elasticity will be considerably increased. Deduct .00175 the set from .05, leaving .04825 inches for the elongation due to the weight 5269 lbs.. E = modulus of elasticity = 5269 × 10 × 12 .04825 = 13104249. This difference in the modulus of elasticity arises from the circumstance of the law of elasticity not being proportional to the weight. The bar broke with a weight of 24 tons per square inch of section. Hence the tensile force of wrought iron is nearly four times as great as the tensile force of cast iron. TABLE Of the Compressive Strength of Wrought Iron. The Bar was 10 feet long and 1 square inch section. The crushing force of wrought iron is 12 tons per square inch. It is a curious fact, that cast iron is decreased in length nearly double what wrought iron is, by the same weight; but the wrought iron bar will sink to any degree with little more than 12 tons per square inch, whilst cast iron will bear 43 56 tons to produce the same effect. A wrought bar will bear a compression of 3 of its length, without its utility being destroyed. Compression of Cast Iron. Mean results of experiments on four different kinds of Cast Iron, 10 feet long, and 1 square inch in section. The crushing or compressive force of cast iron per square inch is 43.56 tons, which has been obtained from eleven kinds of cast iron. But the tensile force of cast iron is 6.6 tons; therefore the compressive force is equal to the square of the tensile force, or (6·6)2. ff' tensile and compressive forces of the material, in a square inch of section, as exerted at a distance (a) on opposite sides of the neutral line. For the determination of the neutral line ƒ { ba2— (b—3) (a—c)2 } = ƒ' { b' d'2— (b'—b) (a'—c')2 } And a + a' = D, where D is the whole depth of the beam. · ·. ba2—(b—ß) (a—c)2='b' (D—a)3—(b'—ß) (D—a—c')2 {{ f ba3— (6—6) (a—c)' } - Moments of compression = { b' d°—(V'—B) (a'—c')' } 3 a If W be the weight laid on the middle, and equal length between supports, Therefore, for the neutral line ß a2 = b' (D — a)2 — (b' — ß) (D — a — c)2 N O |