Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . #, || to the tangent at Z, a leverarm, and consequent moment, about the gravity axes of allthe sections, whence for I ()=0 we have, (more ex-actly than from eq. (3) when x=T) M= P (Sin a)l+ PooaaSI^E (6) (We have supposed P replaced by its components || and~| to the fixed tangent at L, see Fig. 301). But even (6)will not give an exact value for p2 at L ; for the lever armof P cos «, viz. d, is >(P si
Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . #, || to the tangent at Z, a leverarm, and consequent moment, about the gravity axes of allthe sections, whence for I ()=0 we have, (more ex-actly than from eq. (3) when x=T) M= P (Sin a)l+ PooaaSI^E (6) (We have supposed P replaced by its components || and~| to the fixed tangent at L, see Fig. 301). But even (6)will not give an exact value for p2 at L ; for the lever armof P cos «, viz. d, is >(P sina)?3-j-3-ET, on account of thepresence and leverage of P cos a itself. The true value ofd in this case may be obtained by a method similar to thatindicated in the next paragraph. 297. Elastic Curve of Oblique Cantilever with Terminal Exact Solution. For variety place the cantilever as in Fig. 302, so that the deflectionOY=d tends to decrease themoment of P about the gravityaxis of any section, n. Wemay replace P by its X and Ycomponents, Fig. 303, || and~| respectively to the fixedtangent line at L. The origin,0, is taken at the free end of-the beam. Let «= angle bet-. Fig. 303. Fig. 302. ween P and X. For a free body On, n being any section,we have 2 ()=0 whence EI d*y dx* = P(cos a)y—P(sin a) x (i) [See eq. (1) § 295a]. In this equation the right handmember is evidently (see fig. 303) a negative quantity;this is as it should be, for EId2y-7-dx2 is negative, the curvebeing concave to the axis X in the first quadrant. (Itmust be noted that the axis X is always to be taken || tothe beam, for Eldty^-dx2 to represent the moment of thestress-couple.) FLEXURE. OBLIQUE FORCES. 357 Eq. (1) is not in proper form for taking the a>anti-deri-vative of both members, since one term contains the vari-able y, an unknown function of x. Its integration is in-cluded in a more general case given in some works on cal-culus, but a special solution by Prof. Kobinson, of Ohio
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectenginee, bookyear1888
