. The action of materials under stress; . Head.—If a thin hollow sphere of radius r has a uniform normal unitpressure / applied to it within, the total interior pressure on ameridian plane will be ^/^ /, and the tension per linear inchof shell will be Ttr^ p -^ 2-r = \p^- If / is applied externally, the stress in the material will becompression. It may be noted that the double curvature ofthe sphere is associated with half \\\q^ stress which is found inthe cylinder of single curvature having the same radius. If a segment of a sphere is used to close or cap the endof a cylinder or boiler, the s
. The action of materials under stress; . Head.—If a thin hollow sphere of radius r has a uniform normal unitpressure / applied to it within, the total interior pressure on ameridian plane will be ^/^ /, and the tension per linear inchof shell will be Ttr^ p -^ 2-r = \p^- If / is applied externally, the stress in the material will becompression. It may be noted that the double curvature ofthe sphere is associated with half \\\q^ stress which is found inthe cylinder of single curvature having the same radius. If a segment of a sphere is used to close or cap the endof a cylinder or boiler, the same value will hold good. In thiscase the radius / is greater than r for the cylinder. If the segmental end is fastened to the cylinder by abolted flange, the combined tension on the bolts will be r^p,as this is the total force on a right section of the cylinder. The flange itself will be in compression. The pressure/ from below, in Fig. 86, causes a pull per circumferentialunit, in the direction of a tangent at B, which pull has just. 2 20 STRUCTURAL MECHANICS. been shown to be equal to ^pr: It may be resolved intovertical and horizontal components. The vertical componentB C is, by § 227, \pi. The horizontal component h must beproportioned to the vertical component as A O to A B, the sides of the right angled triangle towhich they are respectively perpen-dicular. As A O = i/(r — r),h : ^pr = ?,/(r^ — r^) : r,or/^ = ^p-^{r —r). As h is a uniform normal pressureapplied from without (or tensionapplied from within) in the plane of the flange, the com-pression on the cross-section of the latter will be hr ori/r-j/(r^ — ^^^)» to be divided by that cross-section for find-ing the unit compression. Segmental bottoms of cylinders are sometimes turnedinward. The principles are the same. Example.—A segmental spherical top to a cylinder of 24 , under, 100 lbs. steam pressure, has a radius of 15 in. witha versed sine of 6 in. The tension in top = \ . 100 . 15 = 750lbs.
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectstrengt, bookyear1897
