( E=200GPa, v=0.29 ) is subjected to a state of plane strain ( e zz = e xz = e y

Question

(E=200GPa, v=0.29)

is subjected to a state of plane strain (ezz=

exz =eyz=0) when

the design loads are applied. At the critical point in the member,

three of the stress componentsare

sxx=60 Mpa, syy = 240 MPa, and sxy=-80 MPa.

The material has a yield stresss0=490

MPa. Based on the

maximum shear-stress criterion, determine the factor of safety used

in the design.

I understand how to get two principle stress values of 270.42MPa and 29.58MPa. My lecturer used

v(sigma_x + sigma_y)

to find the third principle stress. Where did that formula come from?

Explanation / Answer

We know that longitudinal strain

e1 = (1/2*E) (sigma1 - v(sigma2+sigma3)

We have been given the plane stress condition,,, so eZ = 0

0 = (1/2*E) (sigmaZ - v(sigmaX+sigmaY)

therefore

sigmaZ = v(sigmaX+sigmaY)

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