河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第十二章 薄板弯曲问题和经典解答（Bending of Thin Plates，Classical Solutions）

12.2 Differential Equation for Bending of Thin Plates Basic unknown function w(x,y Fifteen equations for spatial problems one equation in term of w(x y) for plate bending pi roblem 第二版2002 弹性力学第十二章 6

第二版2002 弹性力学第十二章 16 12.2 Differential Equation for Bending of Thin Plates • Basic unknown function w(x,y) • Fifteen equations for spatial problems------ one equation in term of w(x,y) for plate bending problem

A express u and v in terms of w rx=Ou/azt aw(x, y)/0x=0 u=-z Ow /x+f,(x,y) rx=av/az+ ow(x, y)/ay=0 V=-z Ow/ay+f(x,y) u)=0=v)20=0 f1=f2=0 UE-ZOW/OX V=-Z OW/Oy 第二版2002 弹性力学第十二章

第二版2002 弹性力学第十二章 17 A.express u and v in terms of w • rzx= u/z+ w(x,y)/x=0 u=-z w /x+f1 (x,y) rzy= v/z+ w(x,y)/y=0 v=-z w/y+f2 (x,y) • u)z=0= v)z=0=0 f1 = f2=0 • u=-z w /x v=-z w/y

B express strain components in terms of w U=-ZOW/OX V=-ZOWOy 8=0u/Ox=-z02w/x2 a=ov/ay=-z aZw/ay2 r=ou/ay+ov/ax=-2z0Z w/axa an are neglected 第二版2002 弹性力学第十二章 18

第二版2002 弹性力学第十二章 18 B.express strain components in terms of w • u=-z w /x v=-z w/y • x =u/x= -z  2w /x 2 y =v/y= -z  2w/y 2 rxy =u/y+v/x= -2z 2w/xy • z rzx and rzy are neglected

C. express stress componentsσG气小 in terms of w--use physical equation 0u/x=-02w/x2 E、=Ov0y=-202wy2 r=ou/ay+ov/ax=-2z0Zw/axa =E(1-2)(8+uE)=Ez(12)(02w/Ox2+u02w/Oy2) OVE/(1-H2)(E+u Ex=-E7/(1-H2)(a2w/ay2+u aZw/ax2) txy=E/((1+)l xy=-Ez/(1+u)aw/OxY 第二版2002 弹性力学第十二章

第二版2002 弹性力学第十二章 19 C.express stress components xy xy in terms of w--use physical equation • x =u/x= -z  2w/x 2 y =v/y= -z  2w/y 2 rxy =u/y+v/x= -2z 2w/xy x=E/(1- 2 )(x+ y )=-Ez/(1- 2 )( 2w/x 2+  2w/y 2 ) y=E/(1- 2 )(y+ x )=-Ez/(1- 2 )( 2w/y 2+  2w/x 2 ) xy =E/[2(1+)] rxy = -Ez/(1+)  2w/xy

Dexpress strain components tt n terms of w--use equations of equilibrium dox/ax+atv ay+oT/aztX-=0(8.1.1) atxy/ox+ do dyotz artY=0 (8.1.2) atx/az=-(0o/Ox+oty /ay) x/OX+ E7/(1-u 2)a(02w/0x2+a w/ay2)/Ox atzyaF-(Otx/ax+ doy/ay Ez(1-12)(02w/Ox2+02wy2)/Oy 第二版2002 弹性力学第十二章 20

第二版2002 弹性力学第十二章 20 D.express strain components zxzy in terms of w--use equations of equilibrium • x /x+yx/y+zx/z+X=0 (8.1.1) xy/x+ y /y+zy/z+Y=0 (8.1.2) • zx/z =-(x /x+yx/y) = Ez/(1- 2 ) ( 2w/x 2+ 2w/y 2 )/x zy/z=-(xy/x+ y /y) = Ez/(1- 2 ) ( 2w/x 2+ 2w/y 2 )/y