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

Chapter 8 Theory of Spatial Problems 第八章:空间问题的理论 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 1 Chapter 8 Theory of Spatial Problems 第八章：空间问题的理论

8.1 Differential equations of equilibrium 平衡微分方程 Plane problems平面问题的平衡方程 do ox+ovx/ay+X=0 Otxy/+ooyoy+Y=0(2.2.2) Spatial problems空间问题的平衡方程 do/ax+oT dy+oT/azt=0(8.1.1) Otxy/ox+ doy/oy+ouz/ozt Y=0(8.1.2) dtx/ox +otv/ay+ do /az+Z-0(8.1.3) 徐汉忠第一版20007 弹性力学第二章 2

徐汉忠第一版2000/7 弹性力学第二章 2 • Plane problems平面问题的平衡方程 x /x+yx/y+X=0 xy/x+y /y+Y=0 (2.2.2) • Spatial problems空间问题的平衡方程 x /x+yx/y+zx/z+X=0 (8.1.1) xy/x+ y /y+zy/z+Y=0 (8.1.2) xz/x +yz/y+ z /z+Z=0 (8.1.3) 8.1 Differential equations of equilibrium 平衡微分方程

z r,+=d rra atsdr Tur+=ds dy r,x+=4 -1 dox/ax+oT oy+ot/aztX0(8.1.1) OTxyox+ doy/oy+ouzozt Y=0(8.1.2) x/Ox+0vyOy+o20z+Z=0(8.1.3 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 3 x /x+yx/y+zx/z+X=0 (8.1.1) xy/x+ y /y+zy/z+Y=0 (8.1.2) xz/x +yz/y+ z /z+Z=0 (8.1.3

82 State of Stress at a point.一点的应力 Problem 1: When the six stress components o Oy O, txy tx ty at a certain point P are known, we want to find the stress acting on any inclined plane passing through the point. Let the outward normal to the inclined plane be n and the direction cosines of n be I=coS(N, x)m=cos(N, y) n=cos(N Z 问题1:已知1P点的σ、,TT2过P 点的斜面的法线方向余弦,m,n,求斜面上应力 徐汉忠第一版20007 弹性力学第二章 4

徐汉忠第一版2000/7 弹性力学第二章 4 8.2 State of Stress at a point. 一点的应力 • Problem 1: When the six stress components x y z xy xz yz at a certain point P are known, we want to find the stress acting on any inclined plane passing through the point. Let the outward normal to the inclined plane be N and the direction cosines of N be l=cos(N,x) m=cos(N,y) n=cos(N z) • 问题1：已知 1.P点的x y z xy xz yz 2.过 P 点的斜面的法线方向余弦l,m,n,求斜面上应力

ProblemI. 1: Stress components XNYN ZN acting on any inclined plane 斜面上应力 XYZ XN-lox+m tyx YN ITxy+ moy (23.3) XNIo+m tuxtnt YN-Itxy+ mo +tzv((8.2.1 I tx+munoz 徐汉忠第一版2000/7 弹性力学第二章 5

徐汉忠第一版2000/7 弹性力学第二章 5 XN=lx+m yx+n zx YN=lxy + my +nzy (8.2.1) ZN=l xz+myz+nz Problem1.1: Stress components XN YN ZN acting on any inclined plane 斜面上应力 XN YN ZN XN=lx+m yx YN= lxy+ my (2.3.3)

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