. The action of materials under stress; . / un-Fij^.SO known quantities to admit of a solu- tion by the principles of statics alone. The required equa-tions involve expressions for the inclination or s/ope of thetangent to the curved neutral axis of the bent beam at anypoint, and its deflection, or perpendicular displacement, atany point from its original straight line, or from a given Formula for Curvature.—If, through the pointsA and B, on the neutral axis of a bent beam. Fig. 50, anddistant ds apart, normals C D and K G to the curve ot thisneutral axis are drawn, the distance from
. The action of materials under stress; . / un-Fij^.SO known quantities to admit of a solu- tion by the principles of statics alone. The required equa-tions involve expressions for the inclination or s/ope of thetangent to the curved neutral axis of the bent beam at anypoint, and its deflection, or perpendicular displacement, atany point from its original straight line, or from a given Formula for Curvature.—If, through the pointsA and B, on the neutral axis of a bent beam. Fig. 50, anddistant ds apart, normals C D and K G to the curve ot thisneutral axis are drawn, the distance from A B to their inter-section will be the radius of curvature p for that portion ofthe curve. If through A a plane F H is passed parallel to. DEFLECTION OF SIMPLE BEAMS. 97 K G, the distance F C will be the elongation, or H D will bethe shortening, from the unit stress f, of the extreme fibrewhich was ds long before flexure. Cross-sections planebefore flexure are plane after flexure, § 75. /A O = p, A C = _yi; C F = — ^i, § 10. From similar tri- f Ev angles A C F and O A B, /> : ds = y\ \ —-ds, or p = ~. As, by M ^ J 77, M = fl ^ y,, I _ M~J ~ El the reciprocal of the radius of curvature, called the curvatureor the amount of bending at any one point. 103. Slope and Deflection.—If the curve of the neutralaxis is referred to rectangular co-ordinates, x being parallel to^the original straight axis of the beam, and v being perpen-dicular to the same, the differential calculus gives for the radius of curvature, p = • For very slight curvature, such as is found in practical, safe beams, ds along the curvemay be assumed equal to dx along the axis of x. Then I ^^z; M p dx- E I ?As M is a function of x, as has been se
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Keywords: ., bookcentury1800, bookdecade1890, booksubjectstrengt, bookyear1897
