. Differential and integral calculus. Fig. 69. M= W(l- x). 388 Integral Calculus d2y WHence, § 256 (2), — = ^(/- *) When x = o, — = o, since the tangent at O is coincident with X\ hence C — o. dx Integrating, W1EI W (2 Ix — x2). ^ = 6J/^-4 («) since when :r = o, y — o and therefore C = o. Equa. («;) isthe equation of the curve OA, , the equation of the curvewhich the mean fiber takes under the action of the load W. If in (a) we make x = I and let 8 = value of y when x = /,we have Wlz for the maximum deflection of the beam. 8 = 258. Shape and deflection of a beam fixed at one end anduniform
. Differential and integral calculus. Fig. 69. M= W(l- x). 388 Integral Calculus d2y WHence, § 256 (2), — = ^(/- *) When x = o, — = o, since the tangent at O is coincident with X\ hence C — o. dx Integrating, W1EI W (2 Ix — x2). ^ = 6J/^-4 («) since when :r = o, y — o and therefore C = o. Equa. («;) isthe equation of the curve OA, , the equation of the curvewhich the mean fiber takes under the action of the load W. If in (a) we make x = I and let 8 = value of y when x = /,we have Wlz for the maximum deflection of the beam. 8 = 258. Shape and deflection of a beam fixed at one end anduniformly Let w = load per unit of length of beam; then at anysection S M=w(l-x) —- = ™ {l-xf. Mechanical Applications 389 dV UL) Hence ^ = Yei^ ~ 2 lx + ^; (px -lx>+ -\. dySince —- = o when x = o ; .*. C = o. Integrating again we have a/ //2*2 /x3 x4\ Since jr = o when x = o; .*. C = o. Equa. («) gives theshape the beam assumes under the action of the load. Repre-senting the maximum deflection by 8 which obviously occurswhen x = I we have Cor. Let W — wl = load on beam ; then y = Wl* 8J5Z Comparing 8 with 8 of § 257 we find 8 = f 8 = 3 8, nearly. That is, the deflection is nearly three times as great when theload is concentrated at the end as it would be if uniformly dis-tributed over the beam. 259. Shape and deflection of a beam supported at both ends a?idloaded in ce?iter. WIn this case M — — x\ 2 dy _ jv_ • dx* ~ 2 EI*dx 4 £/ 390 Integral Calculus If x = -, —- = o, since at the middle of the beam the tan-2 ax gent is || to X; ..
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
