空间问题的理论课件.pptxVIP

1空间问题的理论

2Planeproblems平面问题的平衡方程??x/?x+??yx/?y+X=0??xy/?x+??y/?y+Y=0(2.2.2)Spatialproblems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)8.1Differentialequationsofequilibrium

平衡微分方程

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

48.2StateofStressatapoint.一点的应力Problem1:Whenthesixstresscomponents?x?y?z?xy?xz?yzatacertainpointPareknown,wewanttofindthestressactingonanyinclinedplanepassingthroughthepoint.LettheoutwardnormaltotheinclinedplanebeNandthedirectioncosinesofNbel=cos(N,x)m=cos(N,y)n=cos(Nz)问题1：已知1.P点的?x?y?z?xy?xz?yz2.过P点的斜面的法线方向余弦l,m,n,求斜面上应力

5XN=l?x+m?yx+n?zxYN=l?xy+m?y+n?zy(8.2.1)ZN=l?xz+m?yz+n?zProblem1.1:StresscomponentsXNYNZNactingonanyinclinedplane斜面上应力XNYNZNXN=l?x+m?yxYN=l?xy+m?y(2.3.3)

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Problem1.2:Stresscomponents?N?Nactingonanyinclinedplane斜面上应力?N?N

Planeproblems:projectionofXNYNonthenormalNwillgive?N,projectionofXNYNperpendiculartothenormalNwillgive?NXNYN(XN=l?x+m?yxYN=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+mYN=l2?x+m2?y+2lm?xy(2.3.4)?N=lYN-mXN=lm(?y-?x)+(l2-m2)?xy(2.3.5)

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Problem1.2:Stresscomponents?N?Nactingonanyinclinedplane斜面上应力?N?N

Spatialproblems:

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.SubstitutionofEqs.(8.2.1)into

?N=lXN+mYN+nZNyields

?N=l2?x+m2?y+n2?z+2lm?xy+2ln?xz+2mn?yz

2.?N=SQRT(XN2+YN2+ZN2-?N2)

Note:Thesixstresscomponentscompletelydefinethestateofstressatapointinthebodyconcerned.

9Stressboundarycondition应力边界条件X=l?x+m?yx+n?zx