Treatise on natural philosophy . ical harmonics of x, y,with z = 0, of degrees i, 2i, 3i, etc., and - i, - 2i, - 3i, etc. These are completeharmonics when i is unity or any integer. 17—2 260 ABSTRACT DYNAMICS. L711. Problem offlexure. 711. To prove the law of Hexure (§§ 591, 592), and toinvestigate the flexural rigidity (§ 596) of a bar or wire ofisotropic substance, we shall first conceive the bar to be bentinto a circular arc, and investigate the application of forcenecessary to do so, subject to the following conditions:— (1) All lines of it parallel to its length become circular arcsin or
Treatise on natural philosophy . ical harmonics of x, y,with z = 0, of degrees i, 2i, 3i, etc., and - i, - 2i, - 3i, etc. These are completeharmonics when i is unity or any integer. 17—2 260 ABSTRACT DYNAMICS. L711. Problem offlexure. 711. To prove the law of Hexure (§§ 591, 592), and toinvestigate the flexural rigidity (§ 596) of a bar or wire ofisotropic substance, we shall first conceive the bar to be bentinto a circular arc, and investigate the application of forcenecessary to do so, subject to the following conditions:— (1) All lines of it parallel to its length become circular arcsin or parallel to the plane ZOX, with their centres in one lineperpendicular to this plane; OZ and all lines parallel to itthrough OY being bent without change of length. (2) All normal sections remain plane, and perpendicularto those longitudinal lines, so that their planes come to passthrough that line of centres. (3) No part of any normal section experiences deformation. Forced con-dition of nodistortionin normalBcctions. IX. A section DOEof the beam be-ing chosen forplane of i-efer-ence, XO V, let?P, {x,i/,z)he huypoint of the un-bent, and P,{x,y\z) the samepoint of the bent,beam ; each seenin jiiojection, ontluplane i^(^>X, inthe diagnim: andlet p be the radiusof the arc ON,into which the line ON of the strait;ht beam is bent. AW- have « = a: + (p-a:) (1-cos-), y = y, 2= (p-x) sin *.\ P/ P But, according to the fundamental limitation (§ 588), x is atmost infinitely small in comparison with p : and through anylength of the bar not exceeding its greatest transverse dimen- 711.] STATICS. 261 sion, z is so also. Hence we neglect higher powers of x\p andz\p than the second in the preceding expressions; and putting a; - X = a, ?/ - 2/ = /3, « - « = r. xz 1-, /3 = 0, y^-P P we have These, substituted in § 693 (5) and § 697 (2), give .{!). P=-{r\i-n)-, Q = -{Tii-n)-, R = -{7)1 + 11)- \ P }? ??{^)- P P ^=0, T=0, U = 0, J •(3). Surfacetracnon(-P, Q), re-quired topreventd
Size: 1967px × 1270px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjecttides, bookyear1912
