【正文】 PBPPBBPBPBBPεand that of the circumferential line element PB will be ? ?624 ..0ε r ?彈性力學 第四章 17 167。 Geometrical Equations in Polar Coordinates O xyAPdθdrθrP?A?BθuB?Hence, the shearing strain is, ? ?..ruruβαγ r θ θθ ??????In the general case,when both the radial and circumferential displacements take place,we can obtain the total strains by superposition. 彈性力學 第四章 19 167。 Physical Equations in Polar Coordinates Since the polar coordinates r and ? are orthogonal, just as the rectangular coordinates x and y are, the physical equations in the two coordinate systems must have the same form, but with r and ? in place of x and y respectively. ? ? ? ?1424 ..,μσσE1ε θrr ??? ? ? ?.Eμ12G1γr θr θr θ ττ???? ? ? ?1524 ..,μσσE1ε r?? θθ彈性力學 第四章 21 167。 Stress Function in Polar Coordinates 彈性力學 第四章 22 167。a r c t a n,222???ryrxxyyxr?????O xyPθrfrom which we have .c o s,s i n。 Stress Function in Polar Coordinates Noting that is a function of x and y and also a function of r and ,we have ??? ?.c oss i n,s i nc os???????????????????????????????????????????????????????????????rryyrryrrxxrrxRepetition of the above operation yields 彈性力學 第四章 24 167。 Stress Function in Polar Coordinates )(c o sc o ss i n2c o sc o ss i n2s i n)c o s) ( s i nc o s( s i n222222222222?????????????????????????????????????????????????????rrrrrrrrrrry彈性力學 第四章 26 167。 Stress Function in Polar Coordinates Now,noting that as ,we obtain 0s i n,1c o s,0 ???? ????? rx? ?)( 2220220 ???????? ???????????????????? rrryxrAlso, noting that as ,we find 0s i n,1c o s,0 ???? ????? ?y? ?)( 220220 rxy ????????????????????????彈性力學 第四章 28 167。 Compatibility Equation in Polar Coordinates On the other hand, since the addition of Eqs.() and () yields ? ?,11 222222222 ? ????? ?????????????? rrrryxthe patibility equation in rectangular coordinates 0)( 22222?????? ?yxbees that in polar coordinates as )(.0)11( 222222????????? ??rrrr彈性力學 第四章 30 coordinate transformation of stress ponents應力分量的 坐標變換式 ? Egs. ()() 彈性力學 第四章 31 Axisymmetrial stresses and corresponding displacements軸對稱應力和相應的位移 ? Axisymmetrial stresses： 軸對稱應力： normal stress ponents are independent of ? shearing stress ponents vanish the stress distribution is symmetrical with respect to any plane passing through the z axis. ?r= ?r(r) ??= ?? (r) ?r?=0 彈性力學 第四章 32 ?r= ?r(r) ??= ?? (r) ?r?=0 ?r(r) ?? (r) ?r(r) ?? (r) 彈性力學 第四章 33 Axisymmetrial stresses 軸對稱應力 221 , , 0r r rddr d r d r? ? ???? ? ? ?? ? ? ?22211r r r r??????????22r??? ???1r rr?????????? ??????()r???2222 2 211( ) 0 .r r r r ??? ? ?? ? ????2221( ) 0 .ddd r r d r ???Compatibility equation 彈性力學 第四章 34 2221( ) 0 .ddd r r d r ???2211( ) ( ) 0 .d d d dd r r d r d r r d r??? ? ?4 2 34 2 31 1 1 1( ) ( ) 0 .d d d d d dd r d r r d r r d r r d r r d r? ? ? ?? ? ? ?2 2 32 2 2 3 31 1 2 1 2( ) [ ( ) ] .d d d d d d d dd r r d r d r d r r d r r d r r d r r d r? ? ? ? ?? ? ? ? ?23 2 21 1 1 1( ) .d d d dr d r r d r r d r r d r? ? ?? ? ?彈性力學 第四章 35 2 4 3 222 4 3 2 2 31 2 1 1( ) 0 .d d d d d dd r r d r d r r d r r d r r d r? ? ? ??? ? ? ? ? ?This is an ordinary differential equation, which can be reduced to a linear differential equation with constant coefficients by introducing a new variable t such that .tre?td d d t ded r d t d r d t? ? ????222 ()td d ded r d t d t? ? ????3 3 233 3 2( 3 2 )td d d ded r d t d t d t? ? ? ??? ? ?4 4 3 244 4 3 2( 6 1 1 6 )td d d d ded r d t d t d t d t? ? ? ? ??? ? ? ?彈性力學 第四章 36 Substituting them into the patibility equations ,we can get 4 3 24 3 24 4 0 .d d dd t d t d t? ? ?? ? ?22122 ()td e c c tdt? ??solve Integration twice, we can get the general solution 22l n l nA r Br r C r D? ? ? ? ?彈性力學 第四章 37 21 ( 1 2 l n ) 2rdA B r Cr d r r?? ? ? ? ? ?1 0rddd r r d??????? ? ?????222 ( 3 2 l n ) 2dA B r Cd r r??? ? ? ? ? ? ?If there is no hole at the origin of coordinates, constants A and B vanish, since otherwise the stress ponents bee infinite when r=0. hence for a plate without a hole at the origin and with no body forces, only one case of stress distribution symmetrical with respect to the axis may exist, namely, that when ?r=??=constant and that the plate is in a condition of uniform tension or uniform pression in all directions in its plane. 彈性力學 第四章 38 將應變代入幾何方程，對應第一、二式分別積分， (3) 應變通解： Subst