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: 193
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 - [’ 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
194 Mechanics of Materials 2 For a circular plate,radius R.freely supported at its edge and subjected to a load F distributed around a circle radius R F R ymax [3v)(R2)-Ri log.R] 8πD2(1+y) 3F 2(+)logR -)R + (R2-R) and 三0zm Table 7.1.Summary of maximum deflections and stresses. Loading Maximum Maximum stresses condition deflection (ymax) Urmas Ozmux Uniformly loaded, 3gR4 1661-2) 3qR2 3gR2 edges clamped 4r2 82I+v) Uniformly loaded. edges freely 3qR 3qR2 3g supported 16E735+1-) 823+ 82(3+) Central load F, 3FR2 3F 3vF edges clamped 4πE1-2) 2πt2 2n2 Central load F, From 3FR2 From edges freely 4玩E3+(1- 3 R 3F 2721+)log (I+v)log +(1-v) supported 2πt2 For an annular ring,freely supported at its outside edge,with total load F applied around the inside radius Ri,the maximum stress is tangential at the inside radius, 3F(1+v) R2 R i.e. πt2 (R-R) loge Ri If the outside edge is clamped the maximum stress becomes 3F (R2-R) Omax= 2πt2 R2 For thin membranes subjected to uniform pressure g the maximum deflection is given by ymax =0.662 R 9R7'3 Et For rectangular plates subjected to uniform loads the maximum deflection and bending moments are given by equations of the form 9b4 ymax a- Et3 M=Bgb2
194 Mechanics of Materials 2 .. Central load F, edges clamped Central load F, edges freely support e d For a circular plate, radius R, freely supported at its edge and subjected to a load F distributed around a circle radius RI 1 3FR2 3F 3uF 2nr2 2nr2 From From 3FR2 3F R 3F r 2nr2 4?rEt" (I - 2) ~ - 1 and Table 7.1. Summary of maximum deflections and stresses. I Loading I ~ condition I ~aflxri~~~ Maximum stresses I Uniformly loaded, edges freely supported 16Et" For an annular ring, freely supported at its outside edge, with total load F applied around the inside radius RI , the maximum stress is tangential at the inside radius, i.e. If the outside edge is clamped the maximum stress becomes 3F (R' - R:) = [ R2 ] For thin membranes subjected to uniform pressure q the maximum deflection is given by For rectangular plates subjected to uniform loads the maximum deflection and bending moments are given by equations of the form 9b4 ymax = aEt3 M = &h2
$7.1 Circular Plates and Diaphragms 195 the constants a and B depending on the method of support and plate dimensions.Typical values are listed later in Tables 7.3 and 7.4. A.CIRCULAR PLATES 7.1.Stresses Consider the portion of a thin plate or diaphragm shown in Fig.7.1 bent to a radius Rxy in the XY plane and Ryz in the YZ plane.The relationship between stresses and strains in a three-dimensional strain system is given by eqn.(7.2), 6=glo,-w 1 1 lo:-vox-voyl Fig.7.1. Now for thin plates,provided deflections are restricted to no greater than half the plate thickness,the direct stress in the y direction may be assumed to be zero and the above equations give E 0x≠ 0-2e:+e] (7.1) E 0-2吗8，+e (7.2) EJ.Hearn,Mechanics of Materials /Butterworth-Heinemann,1997. S.Timoshenko,Theory of Plates and Shells,2nd edn.McGraw-Hill.1959
$7.1 Circular Plates and Diaphragms 195 the constants a and ,~9 depending on the method of support and plate dimensions. Typical values are listed later in Tables 7.3 and 7.4. A. CIRCULAR PLATES 7.1. Stresses Consider the portion of a thin plate or diaphragm shown in Fig. 7.1 bent to a radius RxU in the XY plane and RYZ in the YZ plane. The relationship between stresses and strains in a three-dimensional strain system is given by eqn. (7.2): 1 E - -[a, - uay - ua,] ,-E 1 E E? = -[a, - ua, - !Jay] Y Z / Fig. 7.1. Now for thin plates, provided deflections are restricted to no greater than half the plate thickness3 the direct stress in the Y direction may be assumed to be zero and the above equations give E a, = [E, + UE,] ~ (1 - u2) E.J. Hearn, Mechanics of Materials I, Butterworth-Heinemann, 1997. S. Timoshenko, Theory of Plates and Shells, 2nd edn., McGraw-Hill, 1959
196 Mechanics of Materials 2 $7.1 If u is the distance of any fibre from the neutral axis,then,for pure bending in the XY and YZ planes, Mo E 7=y=R and E-R =8 8x- RxY and 8:= Ryz 1 d2y u Now R dx2 and,for small deflections, tan =0(radians). dx 1 d2y de RxY dxdx de and x=4-(=radial strain) (7.3) dx Consider now the diagram Fig.7.2 in which the radii of the concentric circles through C and D on the unloaded plate increase to [(x+dx)+(0+d)u]and [x +ue],respectively, when the plate is loaded. dx Unlogded plate E N.A. /de e Looded plate N.A.(Unstrained) u(8+d8) u8 Fig.7.2. Circumferential strain at D2 2π(x+u0)-2πx =6:= 2πx 0 -(circumferential strain) (7.4)
196 Mechanics of Materials 2 $7.1 If u is the distance of any fibre from the neutral axis, then, for pure bending in the XY and YZ planes, MaE (Tu - ---- and -E- =E IYR ER -- U U and E? = - Rxr RYZ E, = ~ 1 d2y du R dx2 dx Now - = ~ and, for small deflections, - = tan0 = 0 (radians). d0 dx and E, = u- (= radial strain) (7.3) Consider now the diagram Fig. 7.2 in which the radii of the concentric circles through CI and D1 on the unloaded plate increase to [(x + dx) + (6' + dO)u] and [x + uO], respectively, when the plate is loaded. Circumferential strain at 02 2n(x + ue) - 2rx 2TcX - - E; = Ue = - (= circumferential strain) X (7.4)
§7.2 Circular Plates and Diaphragms 197 Substituting egns.(7.3)and (7.4)in egns.(7.1)and (7.2)yields E de ox=(1)"dx Eu [de ie. 0x= 1-吗a+vg (7.5) Eu 「d81 Similarly, =0-x (7.6) dx Thus we have equations for the stresses in terms of the slope and rate of change of slope de/dx.We shall now proceed to evaluate the bending moments in the two planes in similar form and hence to the procedure for determination of 0 and de/dx from a knowledge of the applied loading. 7.2.Bending moments Consider the small section of plate shown in Fig.7.3,which is of unit length.Defining the moments M as moments per unit length and applying the simple bending theory, Unit length Fig.7.3. Substituting eqns.(7.5)and (7.6). Et3 8 MxY= 121-2)dc (7.7) Et3 Now 12(1-y2) =D is a constant and termed the flexural stiffness 「d81 so that Mxy =D (7.8) dx x 「6 de and,similarly, Myz =D +V (7.9) d
$7.2 Circular Plates and Diaphragms 197 Substituting eqns. (7.3) and (7.4) in eqns. (7.1) and (7.2) yields E d6 u6 a, = ~ i.e. Similarly, Eu Eu de u, = ~ + v- (1-9) [: dx] (7.5) (7.6) Thus we have equations for the stresses in terms of the slope 6 and rate of change of slope d6/dx. We shall now proceed to evaluate the bending moments in the two planes in similar form and hence to the procedure for determination of 6 and d@/dx from a knowledge of the applied loading. 7.2. Bending moments Consider the small section of plate shown in Fig. 7.3, which is of unit length. Defining the moments M as moments per unit length and applying the simple bending theory, az c 1 x t3 ct3 M=-=- - -- y u [ 12 1 - 12u m Fig. 7.3. Substituting eqns. (7.5) and (7.6), Et3 [ 9 + 21 12(1- 9) dx Mxr = Et3 Now = D is a constant and termed the j-lexural stiffness 12(1 - I?) so that and, similarly, (7.7)