CHAPTER 7 CIRCULAR PLATES AND DIAPHRAGMS Summary The slope and deflection of circular plates under various loading and support conditions are given by the fundamental deflection equation 品品()】=号 ar rar where y is the deflection at radius r:dy/dr is the slope 6 at radius r;is the applied load or shear force per unit length,usually given as a function of r;D is a constant termed the “flexural stiffness'”or“flexural rigidity”=Et/12(1-v2】and t is the plate thickness. For applied uniformly distributed load (i.e.pressure g)the equation becomes d「1d.dy1 ar rar'dr For central concentrated load F F 0= and the right-hand-side becomes- 2πr 2πrD For axisymmetric non-uniform pressure (e.g.impacting gas or water jet) g K/r and the right-hand-side becomes -K/2D The bending moments per unit length at any point in the plate are: de Mr Mxy D 十- d. M:=My= D dr Similarly,the radial and tangential stresses at any radius r are given by: Eu de radial stress o,= (1-v2)dr Eu 「d tangential stress 124 124 Alternatively, 0二 M,and d:M: 193CHAPTER 7 CIRCULAR PLATES AND DIAPHRAGMS Summary The slope and deflection of circular plates under various loading and support conditions are given by the fundamental deflection equation - [’ Q __ (.$)I - [’ _- (rz)] =-% dr rdr where y is the deflection at radius r; dyldr is the slope e at radius r; Q is the applied load or shear force per unit length, usually given as a function of r; D is a constant termed the “flexural stiffness” or “flexural rigidity” = Et3/[12(l - u2)] and t is the plate thickness. For applied uniformly distributed load (i.e. pressure q) the equation becomes qr dr rdr For central concentrated load F F F Q = - and the right-hand-side becomes - - 21rr 2nrD For axisymmetric non-uniform pressure (e.g. impacting gas or water jet) q = K/r and the right-hand-side becomes - K/2D The bending moments per unit length at any point in the plate are: Similarly, the radial and tangential stresses at any radius r are given by: radial stress a, = EU tangential stress a, = ___ Alternatively, 12u 12u t3 13 a, = -Mr and a: = -MZ 193