河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第二章 平面问题的理论 Theory of Plane Problems

O Ey S 引 Fig.23.1 徐汉忠第二版2001/10 弹性力学第二章

徐汉忠第二版2001/10 弹性力学第二章 26

Problem1. 1 stresses acting x on any plane 斜面上应力XN9 Y Fig. 23.1 AB=ds Pa=mds pB=lds >F =0: XNds-o,I ds-T m ds+XO. lds mds=0 2F =0: YNds-o,m ds-tyl ds+y0.5 lds mds=0 XN=lox+m yx YN=mov+lxy(2.3.3) 徐汉忠第二版2001/10 弹性力学第二章 27

徐汉忠第二版2001/10 弹性力学第二章 27 • AB=ds PA=mds PB=lds • Fx=0: XNds-x l ds-yxm ds+X0.5lds mds=0 • Fy=0: YNds-ym ds-xyl ds+Y0.5 lds mds=0 • XN=lx+m yx YN=my+lxy (2.3.3) Problem1.1: stresses acting on any plane 斜面上应力 XN YN

Problem1. 2: stresses A acting on any plane 斜面上应力N飞N 上ig.23.I projection of XN Yn on the normal n will give oN projection of XN yn perpendicular to the normal n will giveτ XNYN(XN=ox+ m x YN=mo+)投影到法线方 向为σN,投影到和法线垂直的方向为N ON-IXN+M YNox+moy+2 mtxy (2.3.4) IN=IYN- m XN=Im(o -ox)(2- m2)Tx(2.3.5) 徐汉忠第二版2001/10 弹性力学第二章 28

徐汉忠第二版2001/10 弹性力学第二章 28 Problem1.2: stresses acting on any plane 斜面上应力 N N • projection of XN YN on the normal N will give N , projection of XN YN perpendicular to the normal N will give N XN YN(XN=lx+m yx YN=my+lxy)投影到法线方 向为 N ，投影到和法线垂直的方向为 N N=lXN+m YN=l2 x +m2y+2lmxy (2.3.4) N=lYN - m XN=lm (y - x )+(l2 - m2 )xy (2.3.5) N

Problem2: Principal stress主应力 G(3+a)2+(σ-σ,)22+x2 (ax+a)2-V(ax-o)22+x2 tan(o, x=(o-ox) tx= tx /o-Ov) (o-o,oro) 1+σ,=σ、+σ, invariant of the state of stress 应力的不变量 徐汉忠第二版2001/10 弹性力学第二章

徐汉忠第二版2001/10 弹性力学第二章 29 Problem 2: Principal stress 主应力 • 1= (x + y )/2 + [(x - y ) /2]2 +xy 2 2= (x + y )/2 - [(x - y ) /2]2 +xy 2 • tan(,x)=(- x )/ xy= xy /(- y ) (= 1 or 2 ) • 1 + 2= x + y invariant of the state of stress 应力的不变量

Problem3: Maximum and minimum shearing stress 最大最小剪应力 Tmax=(01-o2)/2 tmin=-(01-02/2 They are acting on planes inclined at 45 with o oro 最大与最小剪应力发生在与应力主向成45度的斜面上 °N=(01+o2)2 The normal stresses on the planes of maximum and minimum shearing stresses at a point are equal and take the mean value of the two principal stresses at the point. 在发生最大与最小剪应力的面上,正应力的数值都 等于两个主应力的平均值。 徐汉忠第二版2001/10 弹性力学第二章

徐汉忠第二版2001/10 弹性力学第二章 30 Problem3:Maximum and minimum shearing stress 最大最小剪应力 • max= (1 - 2 )/2 min= - (1 - 2 )/2 They are acting on planes inclined at 45 with 1 or2 最大与最小剪应力发生在与应力主向成45度的斜面上 • N= (1 + 2 )/2 The normal stresses on the planes of maximum and minimum shearing stresses at a point are equal and take the mean value of the two principal stresses at the point. 在发生最大与最小剪应力的面上，正应力的数值都 等于两个主应力的平均值