【正文】 l Equations in Polar Coordinates O xyAPdθdrθrP?A?BθuB?O xyABPdθdrθrP?A?B?ru? ?1124 ..,ru rr ???ε? ? ???? ur1ruε r? ? 1γ θθrr θ ??????彈性力學(xué) 第四章 20 167。 Geometrical Equations in Polar Coordinates O xyAPdθdrθrP?A?BθuB?The angle of rotation of PA will be ? ?824 ..,rudrudrruuPAPPAAα????????????????????θθθθand that of PB will be ? ?924 ..ruOP PPPPOβ θ???????????彈性力學(xué) 第四章 18 167。 Geometrical Equations in Polar Coordinates The displacements of points P,A,and B are Thus the normal strain of PA will be O xyAPdθdrθrP?A?BθuB?d θθuuBB θθ ??????θuPP ???drruuAA θθ ??????彈性力學(xué) 第四章 16 167。 Geometrical Equations in Polar Coordinates O xyABPdθdrθrP?A?B?ruHence, the shearing strain is, ? ? . rr θ ?????Next, we assume that only the circumferential displacement takes place. 彈性力學(xué) 第四章 14 167。 Geometrical Equations in Polar Coordinates O xyABPdθdrθrP?A?B?ruand that of the circumferential line element PB will be ? ?124 ..,rudrudrruuPAPPAAPAPAAPrrrrr??????????????????????ε? ? ? ?4. 2. 2.rurd θrd θd θurPBPBBPεrrθ????????彈性力學(xué) 第四章 12 167。 Geometrical Equations in Polar Coordinates O xyABPdθdrθrP?A?B?ruFirstly, we assume that only the radial displacement takes 彈性力學(xué) 第四章 10 167。 Geometrical and Physical Equations in Polar Coordinates極坐標(biāo)中的幾何物理方程 167。 Differential equations of equilibrium in polar coordinates Replacing by ,simplifying the equation, dividing it by and then neglecting the infinitesimal terms, we obtain rdθdθθrτ rθτ? ?114 .. σσθσr1rσ rθrrr ?????????Similarly, the equilibrium condition of the element in the tangential direction will yield 彈性力學(xué) 第四章 7 167。 Differential equations of equilibrium in polar coordinates 彈性力學(xué) 第四章 5 167。彈性力學(xué) 第四章 1 Chapter 4 solution of plane problems in polar coordinates 第四章 平面問(wèn)題極坐標(biāo)解答 彈性力學(xué) 第四章 2 Chapter 4. Solution of Plane Problems in Polar Coordinates 167。 Differential equations of equilibrium in polar coordinates In discussing stresses and displacements in circular disks and rings, solid and hollow circular cylinders, curved beams of rectangular section with a circular axis etc., it is advantageous to use polar coordinates instead of rectangular coordinates. 彈性力學(xué) 第四章 3 Polar coordinates 極坐標(biāo) ? The position of a point P in polar coordinates is defined by the radial coordinate r and the angular coordinate ?. 一點(diǎn) P的極坐標(biāo)用徑向坐標(biāo) r和角坐標(biāo) ?表示 P (r, ?) ? displacements:位移 ： ur u? ? strains:應(yīng)變 : ?r ?? rr? ? stresses: 應(yīng)力 ： ?r ?? ?r? ? body force:體力 : Kr K? 彈性力學(xué) 第四章 4 O xyrσrθτdrrσσ rr ???drrττ rθrθ ???θσθrτd θθσσ θθ ???d θθττ θrθr ???θK rKABCdθdrPθrThe position of a point P in polar coordinates is defined by the radial coordinate r and the angular coordinate ?. 167。 Differential equations of equilibrium in polar coordinates O xyrσrθτdrrσσ rr ???drrττ rθrθ ???θσθrτdθθσσ θθ ???dθθττ θrθr ???θK rKABCdθdrPθrSumming up all the forces acting on the element in the radial direction, we obtain the equilibrium equation ? ?.0????????????????????????????????rd θ d θKdrdrd θ2d θdrσ2d θdrd θσσrd θσd θdrrdrrσσrrrrrrrθθθθθθτθττθ彈性力學(xué) 第四章 6 167。 Differential equations of equilibrium in polar coordinates which reduces to ? ? d r d θK2d θdrτ2d θdrd θθττrd θ τd θdrrdrrττddrσddrd θθσσθθrθrθrr θr θr θθθθ??????????????????????????????????2c o s2c o s??? ? τθτθσr1 θr θr θθ ????????彈性力學(xué) 第四章 8 167。 Geometrical Equations in Polar Coordinates In considering the displacement in polar coordinates, let us denote by and the ponents of the displacement in the radial and circumferential directions,respectively. ru θuWe use for the strain in radial direction, for the strain in the circumferential direction, for shearing strain. rε θεrθγ彈性力學(xué) 第四章 9 167。 Geometrical Equations in Polar Coordinates The displacements of points P,A,and B are OxyABPdθdrθrP?A?B?rud θθuuBB rr ?????drruuAA rr ?????ruPP ??Thus the normal strain of the radial line element PA will be 彈性力學(xué) 第四章 11 167。 Geometrical Equations in Polar Coordinates O xyABPdθdrθrP?A?B?ruThe angle of rotation of PA will be ? ?4 . 2 .30 ，α ?and that of PB will be ? ?.θur1rd θud r )ru(uPBPPBBβrrrr????????????彈性力學(xué) 第四章 13 167。 Geometrical Equations in Polar Coordinates O xyAPdθdrθrP?A?BθuB?Next, we assume that only the circumferential displacement takes place. 彈性力學(xué) 第四章 15 167。 Geometrical Equations in Polar Coordinates O xyAPdθdrθrP?A?BθuB?? ?4 . 2 . 7θθθθ θθθθθ??????????????????????????ur1rduduu