quadrantal arc of which is 5 feet radius; required the cubic contents of the room. Ans. 30779-4595. 85. PROBLEM VI.-To find the curve surface of a saloon. 'Multiply the length of the arch by the mean perimeter." Let the length of the arc, and p = the mean perimeter measured along the middle of the arch, then spl. EXAMPLE. The breadth of the curve surface of a saloon is 10 feet, and the mean perimeter is 150 feet; what is its curve surface? spl 150 x 10 = 1500. EXERCISE. Find the curve surface of a saloon, whose breadth is 8 feet, and mean perimeter 164 feet. Ans. 1394. GROINS. 86. PROBLEM VII.-To find the cubic contents of the vacuity of a groin. 'Multiply the area of the base by the height, and this product by 904.' v = ·904 bh. EXAMPLE. Find the vacuity of a square circular groin, the side of its base being 24 feet. v=904 bh·904 × 242 × 12-6248. EXERCISE. Find the content of the vacuity of an elliptical groin with a square base, whose side is 20 feet, and the height of the groin 6 feet. Ans. 2169.6. 87. PROBLEM VIII.-To find the surface of a groin. 'Multiply the area of the base by 1.1416.' This rule will give very nearly the surface for circular and elliptical groins of small eccentricity. =1.14166. EXAMPLE. Find the surface of a circular groin with a square base, whose side is 12 feet. s=1.14166=1·1416 × 122-164.39. EXERCISE. What is the surface of a circular groin having a square base, whose side is 9 feet? Ans. 92-4696. STRENGTH OF MATERIALS. 88. In investigating the strength of beams of various materials, as timber and metals, they are conceived to be composed of equal elastic fibres disposed in the direction of their length. The distension is considered, within certain narrow limits, to be proportional to the distending force. When a beam is subjected to a strain exceeding its elastic force, it takes a set, that is, a permanent alteration of form, and a force just not sufficient to produce this effect is considered to measure the strength of the beam. The rules deduced by the aid of theory and experiment afford, in many important cases, a tolerably near approximation to fact, considering the variety of strength of materials of the same kind. 89. The measure of the absolute strength or direct cohesion of any material, is the greatest weight that a prism one inch square is capable of supporting, acting in the direction of its length. The weight thus supported by any body will evidently be proportional to its transverse section. 90. When a beam, as BN, with one end fixed, is strained by a weight, W, suspended from any part of it, the moment of the weight (that is, the product of the greatest weight supported into the length of the lever, BM', with which it acts) is the measure of its transverse or relative strength. on The effect of the straining weight is to distend the fibres the upper part of the beam, and to compress those in the lower part; and the length of the fibres in some intermediate position, as AX, will not be altered either by distension or compression, so that AX will be what is called the neutral axis; and the line passing through the neutral point A, about which the motion takes place, is called the neutral axis of rotation. When the beam is suffering the greatest strain it can bear, the fibres at B are exerting their absolute strength, and the intermediate fibres between B and A suffer distensions proportional to their distances from A. The fibres on the lower side are compressed, the quantity of compression being proportional to their distances from A. If BB' represent the distension of the fibre BM, PP' will be the distension of a fibre at the distance AP from A; DD' will be the compression of the fibre DN, and CC' that of a fibre at the distance AC from A. The distension and compression are by some considered, when not great, to be equal; although, in the case of unseasoned timber, the resistance to compression exceeds that to distension. The weight, W, supported will be inversely as the length of the beam AX; for it is the arm of the lever at which W acts. It is also evident that the weight will be proportional to the breadth; for if another beam, equal to BN, were joined to its side, the double beam would support a weight 2 W. The weight will also be proportional to the square of the depth. This is proved in treatises of theoretical mechanics, and also in practical treatises.* Hence, Wbd2 = the transverse strength is ÷ ι 91. The quantity by which a beam is bent from its position of rest by a transversely straining force, is called the deflexion, as MM'. 92. When a pressure acts against a beam in the direction of its length, as in the case of pillars or posts, it is called a crushing or compressive force. 93. A force acting in such a manner as to twist or wrench a beam, as when it acts at the free extremity of an arm of a lever, one end of which is fixed in the beam, as in the case of the axles of wheels, or the screw of a press, it is called a force of torsion; and the angle through which it is twisted is called the angle of flexure. 94. The strains to which a bar of timber, metal, or other * See Barlow's Essay on the Strength of Timber, and Tredgold's Essay on the Strength of Cast-Iron. hard materials, may be subjected, are reducible to the four already explained, namely, a direct strain, a transverse strain, a force of compression, and a force of torsion. 95. The modulus of elasticity is that length of a prismatic beam, whose weight would be capable of distending any portion of it to double its length, were it capable of such distension supposed to be uniform. 'Hence, any small distension of a beam is to its length, as the distending weight to the weight of the modulus of elasticity.' 96. The following table contains the data for determining the strength and flexibility of materials. The column G contains the specific gravity; C the absolute strength or direct cohesion; S the constant for transverse strain; E the constant for deflexion; U that for ultimate deflexion; and M the modulus of elasticity. 760 17000 2030 1900 750 540 5780 1030 7760 36000 U 6580000 395 4988000 5417000 615 4457000 6570000 5406000 3388000 5700000 2803000 509 3007000 3400000 588 2797000 5314000 588 4080000 5967000 757 6249000 4200000 411 4480000 490000 598 2872000 7000000 4702000 3880000 610 2257000 8950000 588 5674000 4760000 724 3607000 7360000 605 6423000 5000000 588 4364000 6760000 596 6488000 5830000 648 5789000 9660000 818 7417000 5530000 6770000 60000 9000 91440000 M 97. PROBLEM I.-To find the absolute strength of any piece of any material of given dimensions. 'Find the area of the transverse section in square inches, and multiply it by the value of C for the given material in the preceding table, and the product will be the required strength.' Let a = the area of the transverse section in inches, then When s is given, and a is required, a = and when a is a square, whose side is b, then b=√a. When a is a rectangle, whose breadth is b, and depth d, and if b is given, then d=7; and if d is given, 6 = a a b If ď a a is a circle, whose diameter is d, then d = √↓· 7854 The constant, C, could also be calculated, if a and s were found by experiment, for C = Ꭶ a When the specific gravity differs from that in the table, then the tabular specific gravity is to that given, as the strength found by the above rule to the required strength. Let g the given specific gravity, agC then, in this case, s = G' Ꭶ C' EXAMPLE.-What weight will be necessary to tear asunder a rectangular piece of beech, whose breadth (b) and depth (d) are respectively 6 and 3 inches? Here abd = 6 x 3 18 inches; hence, s=aC=18 × 11500=207000 lbs. But if the specific gravity were different from that in the agC 705 table, as 705, then s = = × 207000208478. G 700 EXERCISES. 1. What weight will be sufficient to tear asunder a square piece of ash, the side of which is 3 inches? Ans. 153000 lbs. |