. The strength of materials; a text-book for engineers and architects. ^ we shallhave Force tending to cause bursting of ring = F, + {p + Sp).2{x + S x)Force resisting bursting of ring ^2fSx + Fig. 260. These must be equal • • -^ + iv + ^P) (^ + S x) ^ j Sx + px neglecting products of small quantities F (V + p X + p S X + X ^ p = f S X + p X F .•. if — p)Sx=xSp+ 2 = ^ + g -2 x^^ (by (1)) ROTATING DRUMS, DISKS, AND SHAFTS 555 . •. In the limit f = p + ^/- + ^ co^ .t^ (3) Next consider the strains. If the radius x increases to{x + u), the circumference increases from 2 tt a; to 2 tt {x + u
. The strength of materials; a text-book for engineers and architects. ^ we shallhave Force tending to cause bursting of ring = F, + {p + Sp).2{x + S x)Force resisting bursting of ring ^2fSx + Fig. 260. These must be equal • • -^ + iv + ^P) (^ + S x) ^ j Sx + px neglecting products of small quantities F (V + p X + p S X + X ^ p = f S X + p X F .•. if — p)Sx=xSp+ 2 = ^ + g -2 x^^ (by (1)) ROTATING DRUMS, DISKS, AND SHAFTS 555 . •. In the limit f = p + ^/- + ^ co^ .t^ (3) Next consider the strains. If the radius x increases to{x + u), the circumference increases from 2 tt a; to 2 tt {x + u).. increase in circumference = 2 tt u . •. Unital circumferential strain = ^ = - Z TT X X Also the thickness of the ring increases from 8a:;toSa: + Su o IJ u U .. Unital radial strain = ^— = -,— in the limit. ox dx Now the principal stresses acting in an element of this ring are / and p . •. (as shown on p. 25) Unital circumferential strain ^ ^ [f — r] p)Unital radial strain = ^ {p — v f)i-e. -^ == ^ (/ -V2?) (4) §-:=e(p-/) (5) . •. solving these two simultaneous equations we have E fu . 7] du ^-,-r^-)a + fS <^) Putting these results in (3) we haveE /u 7] du (1 -
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