[3] Components of the stress tensor ? at a point, with...

[3] Components of the stress tensor ? at a point, with respect to the (x, y, z) coordinate system, are given as ? = ? ? ? 18 24 0 24 32 0 0 0 20 ? ? ? (MPa). (a) Find the principal invariants of ? . (b) Find the principal stresses ?_i and the unit vectors ê_i along the principal directions. Arrange ?_i such that ?_1 ? ?_2 ? ?_3 . choose the senses of ê_i such that ê_1 ? ê_2 × ê_3 = +1. (c) Find the maximum shear stress |?_s|max and the normals to the planes on which |?_s|max acts. Express the normals in terms of the unit vectors (i', j', k').

Transcript

00:01 Hello everyone.

00:01 So this is the question that we have.

00:03 We have components of stress tensor sigma at a point with respect to x comma y comma z which is given by a sigma is equal to 18, 24, 0, 24, 32, 0 and 0 .20.

00:30 Part a of the question we need to find the principle invariance of sigma part p we need to find the principal stresses sigma 1 and the unit vectors p 1 cap pi cap for the principal directions such that sigma 1 is greater than or equal to sigma 2 is greater than or equal to sigma 3 and part c we need to find the maximum sheer stress so let's jump on to the solution of the problem so this is my sigma compare this with sigma on comparing with sigma this is going to be sigma xx is going to be 18 sigma y y is equal to 32 sigma z z is equal to 20 sigma x y is equal to 24 sigma zy is equal to 0 and sigma z x x is equal to 0 now the characteristic equation this is going to be in variations in variations of sigma i 1 is equal to sigma xx plus sigma y y plus sigma z z so this is going to be 18 plus 32 plus 20 so this implies my i 1 would be to seven now i1 is equal to 70 let's find the minor of all the signals so 32 0 020 plus 18 0 020 plus 18 24 24 32 so if we saw this we're going to get 640 plus 0 so my i 2 is equal to thousand now for i 3 i'll take a determinant so i 3 is going to be 18 24 0 24 32 32 0 and 0 0 0 20 so expand this so this is going to be 18 multiplied by 32 multiplied at 20 minus 0 minus 24 multiplied by 24 multiplied by 20 minus 0 so this is going to be equal to zero itself so i have the principal stresses so this is for part a i 1 is equal to 70 i 2 is equal to thousand and i 3 is equal to zero so this is shear in variance shear in various here now coming out to part b where we need to find the principal stresses so b principal stresses so you're going to have sigma 1 which is equal to 50 sigma 2 which is equal to 20 and sigma 3 which is equal to 0 so as it is given in the question that sigma 1 is greater than or equal to sigma 2 which is greater than it equal to sigma 3...

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