河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第三章 平面问题直角坐标解答 solution of plane problems in rectangular coordinates

Chapter 3 solution of plane problems in rectangular coordinates 第三章平面问题直角坐标解答 3.1 solution by polynomials 3.1多项式解答 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 1 Chapter 3 solution of plane problems in rectangular coordinates 第三章 平面问题直角坐标解答 3.1 solution by polynomials 3.1 多项式解答

Review: Inverse method逆解法 Select o satisfying the compatibility equation 设定φ,并满足相容方程V4φ=0(2.211) find the stress components by由下式求出应力分量 0x=a2o/Oy2-Xx 0v=02/0x2-Yx Txv=-84o/0 XOV (21210) find the surface force components b 由下式对给定坐标的物体求出面力分量 (ox+m tys=X(mo +Ixvs=Y (2.7.2) Identify the problem which the o selected can solve 确定所设定的φ能解决的问题 徐汉忠第一版20007 弹性力学第三章 2

徐汉忠第一版2000/7 弹性力学第三章 2 Review: Inverse method 逆解法 • Select  satisfying the compatibility equation 设定 ，并 满足相容方程 4  =0 (2.12.11) • find the stress components by 由下式求出应力分量 x = 2/y 2 -Xx y =  2/x 2 -Yy xy =- 2/xy (2.12.10) • find the surface force components by 由下式对给定坐标的物体求出面力分量 (lx+m yx)s = X (my+lxy)s =Y (2.7.2) • Identify the problem which the  selected can solve 确定所设定的  能解决的问题

A.φ=a+bx+cy,X=0,Y=0 It satisfies the compatibility equation V4=0 find the stress components by x=020y2=0y=020x2=0Ty=02d/0xoy=0 find the surface force components by (ox+m tyxs x=0(moy+It. XV/S Y=0 a linear stress function corresponds to the case of no surface forces and no stress The superposition of a linear function to the stress function for any problem does not affect the stresses 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 3 A. =a+bx+cy , X=0, Y=0 • It satisfies the compatibility equation 4  =0 • find the stress components by x = 2/y 2=0 y =  2/x 2=0 xy =- 2/xy=0 • find the surface force components by (lx+m yx)s=X=0 (my+lxy)s =Y=0 • a linear stress function corresponds to the case of no surface forces and no stress . • The superposition of a linear function to the stress function for any problem does not affect the stresses

A.φ=a+bx+cy,X=0Y=0 满足相容方程ⅴ4φ=0 由下式求出应力分量 o=020y2=0y=020x2=0x=020x0y=0 由下式对给定坐标的物体求出面力分量 (ox +m tsx=0(mo +Is=Y=0 确定所设定的φ能解决的问题为:任意物体无 体力,无面力,无应力。 应力函数加或减一个线性项不影响应力。 徐汉忠第一版20007 弹性力学第三章 4

徐汉忠第一版2000/7 弹性力学第三章 4 A. =a+bx+cy , X=0 Y=0 • 满足相容方程 4  =0 • 由下式求出应力分量 x = 2/y 2=0 y =  2/x 2=0 xy =- 2/xy=0 • 由下式对给定坐标的物体求出面力分量 (lx+m yx)s=X=0 (my+lxy)s =Y=0 • 确定所设定的  能解决的问题为：任意物体无 体力，无面力，无应力。 • 应力函数加或减一个线性项不影响应力

B.￠=ax2,X=0,Y=0 It satisfies the compatibility equation V49=0 find the stress components by x=02y2=0y=00x2=2a3y=02xoy=0 for a rectangular plate with its edges parallel to the coordinate axes find the surface force components by (ox+m tsX (mo+ltxys=Y the stress function =ax can solve the problem of uniform tension(a>0)or uniform compression(a<) of a rectangular plate in y direction. P37 Fig3.1.1(a) 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 5 B. =ax2 , X=0, Y=0 • It satisfies the compatibility equation 4  =0 • find the stress components by x = 2/y 2=0 y =  2/x 2=2a xy =- 2/xy=0 • for a rectangular plate with its edges parallel to the coordinate axes, find the surface force components by (lx+m yx)s=X (my+lxy)s =Y • the stress function =ax2 can solve the problem of uniform tension (a>0) or uniform compression (a<0) of a rectangular plate in y direction. P37 Fig.3.1.1(a)

B.￠ax2,X=0,Y=0 满足相容方程V4φ=0 由下式求出应力分量 o=020y2=0G=020x2=2atxy=020x0y=0 ·对和坐标轴平行的矩形板求出面力分量知 ￠ax2能解决矩形板y向受均匀拉力(a>0)或均 匀压力(a<0)的问题。P37Fg31.1(a) 徐汉忠第一版20007 弹性力学第三章 6

徐汉忠第一版2000/7 弹性力学第三章 6 B. =ax2 , X=0, Y=0 • 满足相容方程 4  =0 • 由下式求出应力分量 x = 2/y 2=0 y =  2/x 2=2a xy =- 2/xy=0 • 对和坐标轴平行的矩形板求出面力分量知 =ax2 能解决矩形板y向受均匀拉力(a>0)或 均 匀压力(a<0)的问题。P37 Fig.3.1.1(a)

x 2C 2C 2a↑t (a) (C Fig 3.1.1 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 7

C.φ=cy2,x=0,Y=0 It satisfies the compatibility equation V49=0 find the stress components by x=0240y2=2cay=020/0x2=0tx=020xoy=0 for a rectangular plate with its edges parallel to the coordinate axes find the surface force components by (ox+m tsX (mσ+lxy)s=Y the stress function =cy can solve the problem of uniform tension(c>0)or uniform compression (c<0)of a rectangular plate in x direction. P37 Fig3.1.1(c) 徐汉忠第一版20007 弹性力学第三章 8

徐汉忠第一版2000/7 弹性力学第三章 8 C. =cy2 , X=0, Y=0 • It satisfies the compatibility equation 4  =0 • find the stress components by x = 2/y 2=2c y =  2/x 2=0 xy =- 2/xy=0 for a rectangular plate with its edges parallel to the coordinate axes, find the surface force components by (lx+m yx)s=X (my+lxy)s =Y • the stress function =cy2 can solve the problem of uniform tension (c>0) or uniform compression (c<0) of a rectangular plate in x direction. P37 Fig.3.1.1(c)

C.φ=cy2,X=0,Y=0 满足相容方程V4φ=0 由下式求出应力分量 o=020y2=2coy=0200x2=0y=02Oxy=0 ·对和坐标轴平行的矩形板求出面力分量知 cy2能解决矩形板x向受均匀拉力(c>0)或均 匀压力(c<0)的问题。P37Fig311(c) 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 9 C. =cy2 , X=0, Y=0 • 满足相容方程 4  =0 • 由下式求出应力分量 x = 2/y 2= 2c y =  2/x 2=0 xy =- 2/xy=0 • 对和坐标轴平行的矩形板求出面力分量知 =cy2 能解决矩形板x向受均匀拉力(c>0)或 均 匀压力(c<0)的问题。P37 Fig.3.1.1(c)

x 2C 2C 2a↑t (a) (C Fig 3.1.1 徐汉忠第一版20007 弹性力学第三章

徐汉忠第一版2000/7 弹性力学第三章 10