河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第四章 平面问题极坐标解答 solution of plane problems in polar coordinates

e=Ou( ra0)+due/ar-ue/r 人 A A Fig 4.2.1 徐汉忠第一版20007 弹性力学第四章 6

徐汉忠第一版2000/7 弹性力学第四章 16 rr =ur /(r) +u /r-u /r

physical equations in polar coordinates 极坐标中的物理方程 The physical equations in the two coordinate systems must have the same form, but with r and 0 in place of x and y respectively. 8=lo, -HOVE (26.4) E=o-Ho/E r=τ./G - y--0 E=|o-μoE (4214-16) 0=oo-HoVE re=te/G 徐汉忠第一版20007 弹性力学第四章

徐汉忠第一版2000/7 弹性力学第四章 17 physical equations in polar coordinates 极坐标中的物理方程 • The physical equations in the two coordinate systems must have the same form, but with r and  in place of x and y respectively. • x=[x - y ]/E (2.6.4) y=[y - x ]/E rxy =xy /G x--r y-- • r=[r -  ]/E (4.2.14-16) =[ - r ]/E rr=r /G

4.3 stress function and compatibility equation in polar coordinates 极坐标中的应力函数及相容方程 o=0(ror)+02(2002)o6=0 e=-(OOr)o/(ro)=-1ra2d/Oro)+1r2o60 σr+=0/(ror)+a2(02)+02d/r2 +σ=00y2+020x2σ+ σx+σ ∂φ/(ror)+∂2d/(r2002)+02dOr2=aOy2+a2d/0x2 V2=o/(ror)+02/(r2002)+a2r2=a210y2+02/6x2 V4=O/(ror)+02/(r262)+a2/r22=0(43.9) 徐汉忠第一版20007 弹性力学第四章 18

徐汉忠第一版2000/7 弹性力学第四章 18 4.3 stress function and compatibility equation in polar coordinates 极坐标中的应力函数及相容方程 • r =/(rr)+ 2/(r22 ) = 2/r 2 r=- (/r)[/(r)]= -1/r  2/(r)+1/r2 / • r+=/(rr)+ 2/(r22 )+ 2/r 2 • x+y =  2/y 2+ 2/x 2 r+= x+y • /(rr)+ 2/(r22 )+ 2/r 2=  2/y 2+ 2/x 2 • 2=/(rr)+ 2 /(r22 )+ 2 /r 2= 2 /y 2+ 2 /x 2 • 4 =[/(rr)+ 2 /(r22 )+ 2 /r 2 ] 2  =0 (4.3.9)

4.4 coordinate transformation of stress components应力分量的坐标变换式 °Egs.(44.1-(44.12) a B (b) 上ig.4.4.I 徐汉忠第一版20007 弹性力学第四章

徐汉忠第一版2000/7 弹性力学第四章 19 4.4 coordinate transformation of stress components应力分量的坐标变换式 • Egs. (4.4.1)---(4.4.12)

4.5 Axisymmetrial stresses and corresponding displacements轴对称应力和相应的位移 Axisymmetrical stresses:轴对称应力: I the normal stress components are independent of e 2. the shearing stress components vanish 3.hence the stress distribution is symmetrical with respect to any plane passing through the z axis. o=0(r e-co(r re 0 徐汉忠第一版20007 弹性力学第四章 20

徐汉忠第一版2000/7 弹性力学第四章 20 4.5 Axisymmetrial stresses and corresponding displacements轴对称应力和相应的位移 • Axisymmetrial stresses：轴对称应力： 1.the normal stress components are independent of  2.the shearing stress components vanish 3.hence the stress distribution is symmetrical with respect to any plane passing through the z axis. r= r (r) =  (r) r=0