s9.2 Thin Cylinders and Shells 201 New circumference nd+nde# =d(1+eH) But this is the circumference of a circle of diameter d(1+8#) New diameter =d(1+) Change in diameter dem d起班一H Diametral strainod i.e. the diametral strain equals the hoop or circumferential strain (94) d。 Thus change in diameterden-[v E2-] (9.5) (c)Change in internal volume Change in volume volumetric strain x original volume From the work of§14.5，page364. volumetric strain sum of three mutually perpendicular direct strains 8L+26D 1 2 =ELoL-vom]+ECon-voL] 1 =E[oL+20-v(on+201)] pd1+4-v2+2] 5-4幻 Therefore with original internal volume V change in internal volumep 4E5-4n]y (9.6) 9.2.Thin rotating ring or cylinder Consider a thin ring or cylinder as shown in Fig.9.3 subjected to a radial pressure p caused by the centrifugal effect of its own mass when rotating.The centrifugal effect on a unit length$9.2 Thin Cylinders and Sheh 201 New circumference = xd + 7cd~ H = d(1 +EH) But this is the circumference of a circle of diameter d (1 +E,,) .. .. New diameter = d (1 + E~) Change in diameter = dEH d&H Diametral strain E,, = - = eH the diametral strain equals the hoop or circumferential strain d i.e. (9.4) d E Thus change in diameter = deH = - [aH - voL] Pd’ = ---[2-v] 4tE (c) Change in internal volume Change in volume = volumetric strain x original volume From the work of $14.5, page 364. volumetric strain = sum of three mutually perpendicular direct strains = EL+ 2ED 1 E = -[UL+2aH-v(aH+2aL)J = -[ Pd 1 +4-v(2+2) J 4tE Pd = -[5-4v] 4t E Therefore with original internal volume V Pd cbange in internal volume = - [5 - 4v] Y 4tE 9.2. Thin rotating ring or cylinder (9.5) Consider a thin ring or cylinder as shown in Fig. 9.3 subjected to a radial pressure p caused by the centrifugal effect of its own mass when rotating. The centrifugal effect on a unit length