§4.2 Rings,Discs and Cylinders Subjected to Rotation and Thermal Gradients 119 The cylinder wall is assumed to be so thin that the centrifugal effect can be assumed constant across the wall thickness. F=mass×acceleration=mw2r2×r This tension is transmitted through the complete circumference and therefore is resisted by the complete cross-sectional area. F mo2r2 hoop stress = A A where A is the cross-sectional area of the ring. Now with unit length assumed,m/A is the mass of the material per unit volume,i.e.the density p. hoop stress=pa2r2 4.2.Rotating solid disc (a)General equations (g,+8c,(r+8r)88 C.F.pr2uw28r8 CHx8rxI Fig.4.2.Forces acting on a general element in a rotating solid disc. Consider an element of a disc at radius r as shown in Fig.4.2.Assuming unit thickness: volume of element =r80 x 8r x I =r808r mass of element pr 806r Therefore centrifugal force acting on the element mo'r pr808ra2r pr2a280 8r$4.2 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients 119 The cylinder wall is assumed to be so thin that the centrifugal effect can be assumed constant across the wall thickness. .. F = mass x acceleration = mw2r2 x r This tension is transmitted through the complete circumference and therefore is resisted by the complete cross-sectional area. .. F mw2r2 hoop stress = - = - A A where A is the cross-sectional area of the ring. density p. .. hoop stress = po2r2 Now with unit length assumed, m/A is the mass of the material per unit volume, i.e. the 4.2. Rotating solid disc (a) General equations ( u, t 8 u~( r + 8 r 8 6 4 XI Fig. 4.2. Forces acting on a general element in a rotating solid disc Consider an element of a disc at radius r as shown in Fig. 4.2. Assuming unit thickness: volume of element = r SO x Sr x 1 = r SO& mass of element = pr SO&- Therefore centrifugal force acting on the element = mw2r = pr~~rw'r = pr2w260Sr