§7.8 Circular Plates and Diaphragms 205 7.8.Circular plate with central concentrated load F and edges freely supported The fundamental equation and hence the slope and deflection expressions will be as for the previous section ($7.7), dy F r ie. 0= d= 25g-+ (7.33) y- Fr2 C12 Dllog,-11+ (7.34) constants C2 and C3 being zero as before. As for the uniformly loaded plate with freely supported edges,the constant Ci is deter- mined from the knowledge that the bending moment Mr is zero at the free support, ie.at r=R, M,=0 Therefore from eqn.(7.16), 0 =0 and r dr and,substituting from eqn.(7.33)with r =R, 2g,R-+号=2oR--号 F vF 21+)= 8rD2(1+)log。R-(1-明 F C1= 4πD 2log.R+ 1-) 1+) As before,the maximum deflection is at the centre and equivalent to that obtained with r =R. Substituting in eqn.(7.34), FR2 FR2 maximum deflection 8 Dllog.R-刂+16rD 2log.R+ (1-) (1+) FR2(3+) 16πD(1+v) Substituting for D 3FR2 ymax =4Eg33+v)(1-v) (7.35) For v=0.3 this is approximately 2.5 times that for the clamped edge condition. From egn.(7.14), Eu 0,= -)a+ Substituting for de/dr and e/r as above, Eu「F R ,=-4D1+1oe87.8 Circular Plates and Diaphragms 205 7.8. Circular plate with central concentrated load F and edges freely supported The fundamental equation and hence the slope and deflection expressions will be as for the previous section ($7.7), i.e. %=-=-- Fr2 CI r2 y = -- [log,r - 11 + - 8x0 4 (7.33) (7.34) constants C2 and C3 being zero as before. mined from the knowledge that the bending moment M, is zero at the free support, i.e. at r = R, Therefore from eqn . (7.16), As for the uniformly loaded plate with freely supported edges, the constant CI is deter￾M,=O and, substituting from eqn. (7.33) with r = R, .. As before, the maximum deflection is at the centre and equivalent to that obtained with r = R. Substituting in eqn. (7.34), maximum deflection = -[log, FR~ R - 1 I + ~ FR2 [Zlog,R + 8nD 167rD -~ FR= (3 + U) - - 16nD (1 + u) Substituting for D 3FR2 43rEt3 Ymax = - (3 + v)(l - v) For v = 0.3 this is approximately 2.5 times that for the clamped edge condition. From eqn. (7.14), Eu Substituting for dO/dr and e/r as above, (7.35)