198 Mechanics of Materials 2 $7.3 It is now possible to write the stress equations in terms of the applied moments, 124 ie. Os =Mxy (7.10) 12w 0:Myz (7.11) 7.3.General equation for slope and deflection Consider now Fig.7.4 which shows the forces and moments per unit length acting on a small element of the plate subtending an angle 8 at the centre.Thus Mxy and Myz are the moments per unit length in the two planes as described above and o is the shearing force per unit length in the direction Oy. 8中/2 于☑M,sin Q+dQ Fig.7.4.Small element of circular plate showing applied moments and forces per unit length. For equilibrium of moments in the radial XY plane,taking moments about the outside edge, (MxY +8Mxy)(x +8x)8-Mxyx80-2Myz8x sin 38+Qx808x =0 which,neglecting squares of small quantities,reduces to MxYδr+δMxYx-Myz8r+Qx8x=0 In the limit,therefore, MxY+x dMxY-Myz =-Ox dx Substituting eqns.(7.8)and (7.9),and simplifying d281d00 dxi+xdx=- D This may be re-written in the form d「1dx dxx dx D (7.12) This is then the general equation for slopes and deflections of circular plates or diaphragms. Provided that the applied loading is known as a function of x the expression can be treated198 Mechanics qf Materials 2 57.3 It is now possible to write the stress equations in terms of the applied moments, i.e. (7.10) (7.11) 12u U, =Mxy￾t3 12u U, = MyZ -- t3 73. General equation for slope and deflection Consider now Fig. 7.4 which shows the forces and moments per unit length acting on a small element of the plate subtending an angle 84 at the centre. Thus Mxy and MYZ are the moments per unit length in the two planes as described above and Q is the shearing force per unit length in the direction OY. Fig. 7.4. Small element of circular plate showing applied moments and forces per unit length. For equilibrium of moments in the radial XY plane, taking moments about the outside edge, (Mxy + GMxy)(x + Sx)S$ - MxyxG$ - 2My~Sx sin 484 + QxS4Sx = 0 which, neglecting squares of small quantities, reduces to MxyGx + GMxyx - MyzGx + QXSX = 0 Substituting eqns. (7.8) and (7.9), and simplifying d20 1 dO 0 Q -+ dx2 xdx x2 D This may be re-written in the form Q D (7.12) This is then the general equation for slopes and deflections of circular plates or diaphragms. Provided that the applied loading Q is known as a function of x the expression can be treated