河海大学：《弹性力学及有限元》课程PPT教学课件（双语版）第八章 空间问题的理论 Theory of Spatial Problems

X B 图 徐汉忠第一版20007 弹性力学第二章 6

徐汉忠第一版2000/7 弹性力学第二章 6

Problem1. 2: Stress components oN tN acting on any inclined plane斜面上应力aNN Plane problems: projection of XN YN on the normaN will give oN, projection ofXNYN perpendicular to the normal n will give tN XNYN(XN=ox+ m y YN=mo+3)投影到法线方 向为σN,投影到和法线垂直的方向为c ON=IXN+m YN=120x+mo\+2lmwy 40 4) IN-IYN-m XN=Im(oy- 0x)+(2-m)xy(2.3.5) 徐汉忠第一版20007 弹性力学第二章

徐汉忠第一版2000/7 弹性力学第二章 7 Problem1.2: Stress components N N acting on any inclined plane 斜面上应力 N N • Plane problems: 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)

Problem1. 2: Stress components oN IN acting on any inclined plane斜面上应力oNTN Spatial problems: XN-lox+m tyx+n t YN-Itxy+ moy +ntzy (8.2.1) zNτxz+myz+noz 1. Substitution of Eqs.(8.2.1)into σN=xN+mYN+ n ZN yields ON=120x+m2oy+n2 0, +2Imtx+ 2Intx +2mnt 2. TN=SQRT(XN+YN+ZN2-ON2 Note: The six stress components completely define the state of stress at a point in the body concerned 徐汉忠第一版20007 弹性力学第二章

徐汉忠第一版2000/7 弹性力学第二章 8 Problem1.2: Stress components N N acting on any inclined plane 斜面上应力 N N Spatial problems: XN=lx+m yx+n zx YN=lxy + my +nzy (8.2.1) ZN=l xz+myz+nz 1. Substitution of Eqs. (8.2.1) into N=l XN+mYN+n ZN yields N =l2 x +m2y+n2 z +2lmxy+ 2lnxz +2mnyz 2. N =SQRT(XN 2+YN 2+ZN 2 -N 2 ) Note: The six stress components completely define the state of stress at a point in the body concerned

Stress boundary condition应力边界条件 X=loxtm ty invo Y=IT +mo +nt (8.2.4) Z=1 lτx+myz+noz 徐汉忠第一版2000/7 弹性力学第二章

徐汉忠第一版2000/7 弹性力学第二章 9 Stress boundary condition应力边界条件 X=lx+m yx+n zx Y=lxy + my +nzy (8.2.4) Z=l xz+myz+nz

83 Principal stress主应力 应力主面- a principal plane of stress 主应力- a principal stress 应力主轴- a principal axis of stress Plane problems平面问题 0=(o+ov/2+ VI(ox-Oy)/212tx 2 2=(x+a3)2-V(ox-y)2 2 tan(o, x)=(o-ox)txy= txy/o-oy) 0 soro 0,to,=o +o Invariants of the state of stress 徐汉忠第一版20007 弹性力学第二章

徐汉忠第一版2000/7 弹性力学第二章 10 8.3 Principal stress 主应力 Plane problems 平面问题 • 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 Invariants of the state of stress • 应力主面-a principal plane of stress • 主应力-a principal stress • 应力主轴-a principal axis of stress