$7.9 Circular Plates and Diaphragms 207 For the outer portion of the plate r>Ri and eqn.(22.20)reduces to Fr2 .C1r2 y=-8品og,r-1+C￥+c1og.r+C (7.38) and from egn.(7.19) 0=0=%2照r-+号+月 (7.39) Equating these values with those obtained for the inner portions, CDog,R1 C log,:+C 4 8πD 4 and CiRL =-FRL [2l0g,R-1C 2 8D 2 R Similarly,from(7.16),equating the values of M,at the common radius RI yields pu+loe风+1-1+号+0-C”=9a+w F R Further,with M=0 at r=R,the outside edge,from eqn.(7.16) 52(1+1ogR+1-1+号(1+)- F C(-)=0 (7.40) There are thus four equations with four unknowns Ci,C,C2 and C3 and a solution using standard simultaneous equation procedures is possible.Such a solution yields the following values: F C二4rD 21ogR+1-R2-R3) (1+)R2 FR2 C2=- 8πD C3= FRi tog,R-1) 8πD The central deflection is found,as before,from the deflection of the edge,r =R,relative to the centre. Substituting in egn.(7.38)yields F ymax= +(R2-R)-R)-好1og (7.41) 8πD2(1+) The maximum radial bending moment and hence radial stress occurs at r =R1,giving 3F 4πt2 2+b+-e2 R2 (7.42) It can also be shown similarly that the maximum tangential stress is of equal value to the maximum radial stress.$7.9 Circular Plates and Diaphragms For the outer portion of the plate r > R1 and eqn. (22.20) reduces to Fr2 C’, r2 8nD 4 y= -- [log, r - 11 + - + C; log, r + C; and from eqn. (7.19) Fr C;r C; [2log,r - 11 + - + - 2r e=-=-- dY dr 8nD Equating these values with those obtained for the inner portions, and Similarly, from (7.16), equating the values of M, at the common radius R1 yields Further, with M, = 0 at r = R, the outside edge, from eqn. (7.16) 207 (7.38) (7.39) There are thus four equations with four unknowns Ci , C’, , C; and C; and a solution using standard simultaneous equation procedures is possible. Such a solution yields the following values: The central deflection is found, as before, from the deflection of the edge, r = R, relative Substituting in eqn. (7.38) yields to the centre. (R2 - R:) - R:) - R: log, - R1 “I (7.41 ) The maximum radial bending moment and hence radial stress occurs at r = RI , giving 1 R (R2 - Ri) RZ - It,, + U)lO& - + (1 - u) 4ntz R1 - (7.42) It can also be shown similarly that the maximum tangential stress is of equal value to the maximum radial stress