《弹性力学》课程教学课件（英文讲稿）Chapter 7 The Basic Theory of the Plane Problem

The Basic Theory of the Space ProblemExample 1: The stress state at a certain9.=a,=a,一a,7point is known as the right equation0,:0,TXPlease solve the principal stress and the-aTxymaximumshearstress.Solution:3 - I? + I20 - I3 = 0Ii=ox+,+o,=a-TTy-tI2=axO,+a,,+,x-127-=-a? -a? +a? -a? =-2a?22I3=0x0,0, +2txyTyzTzx -0,1zr14x-α+α=0Q1 = 2a, 02 = 0,0=-a3a2-2a =03a1-03o(α - 2a)(α +a) = 0Tmax1622

16 Example 1: The stress state at a certain point is known as the right equation. Please solve the principal stress and the maximum shear stress. a a a a xy yz zx x y z = = = − = = − =       0, 0, , , , Solution: 2 3 0 2 1 3  − I  + I  − I = I1 =  x +  y +  z = a 2 2 2 2 x y y z z x x y yz zx I =   +   +   − − − 2 2 2 3 x y z 2 x y yz zx x yz y zx z x y I =    +    −  −  −  2 2 2 2 2 = −a − a + a − a = −2a 0 3 3 = −a + a = 2 0 3 2 2  − a − a  =  ( − 2a)( + a) = 0  1 = 2a,  2 = 0,  3 = −a 2 3 2 1 3 max a = − =   

The Basic Theory of the Space ProblemExample 2: The known stress state at，,=10,αx=0, α,=20,a certain point is:T =20Txy =10, T = 0,Unit: MPaSolve: Please solve the normal stress and shear stress of the point, and thedirection is the exterior normal of the plane 3x + /3y + 2z =1 .Solution: I: m: n=3: /3 : 2/331[?+m?+n?=1nm244Px =lox+mtyx +ntzPx =14.33MPaP, = 16.16MPaP, =ltxy +mo, +ntyP, = 20.00MPaP,=ltx +mty+noz17

17 Example 2: The known stress state at a certain point is: Unit: MPa Solve: Please solve the normal stress and shear stress of the point, and the direction is the exterior normal of the plane . 10, 0, 20 0, 20, 10, = = = = = = xy yz zx x y z       Solution: l :m : n = 3 : 3 : 2 1 2 2 2 l + m + n = 3x + 3 y + 2z = 1 4 3 l = 4 3 m = 2 1 n = x x m yx n zx p = l +  +  y xy m y n zy p = l +  +  z xz m yz n z p = l +  +  14.33MPa 16.16MPa 20.00MPa x y z p p p = = =