$7.7 Circular Plates and Diaphragms 203 The maximum deflection is at the centre and again equal to the deflection of the supports relative to the centre. Substituting for the constants with r=R in eqn.(7.22), maximum deflection =)R 64D+ 8D(1+)4 qR4「(5+)7 64DL(1+)」 i.e.substituting for D. 3qR4 mx=i6E5+1-0 (7.26) With v=0.3 this value is approximately four times that for the clamped edge condition. As before,the stresses are obtained from eqns.(7.14)and(7.15)by substituting for de/dr and 0/r from eqn.(7.21), Eu 0= (1-2) 8++e+ This gives a maximum stress at the centre where r =0 E 1qR2 3+) (1-v2)216D 3qR2 82(3+) Similarly, 0m= 3qR2 82(3)also at the centre i.e.for a uniformly loaded circular plate with edges freely supported, 3qR2 Ormax 2max 823+) (7.27) 7.7.Circular plate with central concentrated load F and edges clamped For a central concentrated load, Q×2πr=F F Q=2Tr The fundamental equation for slope and deflection is,therefore, 品品（器 2nrD I d F Integrating, loger+Ci 2xD57.7 Circular Plates and Diaphragms 203 The maximum deflection is at the centre and again equal to the deflection of the supports relative to the centre. Substituting for the constants with r = R in eqn. (7.22), 4R4 640 qR2 (3 + u) R2 maximum deflection = - - + - ___ - 80 (1 + u) 4 i.e. substituting for 0, 3qR 16Et3 Ymax = - (5 + v)(l - v) (7.26) With u = 0.3 this value is approximately four times that for the clamped edge condition. and O/r from eqn. (7.21), As before, the stresses are obtained from eqns. (7.14) and (7.15) by substituting for dO/dr [ 4r2 (3 + ”) + -(3 + u) qR2 160 1 Eu or=- -__ (1 - u2) 160 This gives a maximum stress at the centre where r = 0 E t qR2 (1 - u2) 2 160 urmax = ___-- (3 + u) 3qR’ 89 Similarly, CJZm,, - -(3 + u) also at the centre i.e. for a uniformly loaded circular plate with edges freely supported, 7.7. Circular plate with central concentrated load F and edges clamped For a central concentrated load, Q x 2nr = F F 27rr Q=- The fundamental equation for slope and deflection is, therefore, F (7.27) F Integrating, Id (rs) = _- log, r + CI r dr 2nD