# Why is the normal stress assumed to be 0 in plane stress?

If a plate is elongated in the x and y direction, why is the stress sigma z considered negligible with respect to sigma x and sigma y?

• It is not "considered negligible," it is zero by definition - that is what "plane stress" means! If you have non-zero stress components in all three directions, then the stress isn't "all in one plane." May 29 '17 at 1:08
• @alephzero It is considered zero but it's not necessarily zero in reality, it's an approximation. May 29 '17 at 10:01
• Everything in real-world engineering structural analysis is an approximation. I don't understand what point you are trying to make here. May 29 '17 at 10:30
• I'm trying to know why the stress sigma z in the plane stress case is negligible with respect to sigma x and sigma y, I want to know the reason for this approximation. May 29 '17 at 11:44
• the title and text of your question ask entirely different things. please clarify May 29 '17 at 13:12

Because along z direction, body thickness is so small (ideally zero) that practically it consists of only the two surface boundaries (top and bottom, see figure) . Because no pressure is allowed in these boundaries (otherwise we violate plane stress conditions), equillibrium of internal stresses and external actions cant't be satisfied on them, unless $\sigma_z = 0$. Because no material exists (ideally) between these two boundaries, no $\sigma_z$ is possible "internally".

Typical applications of plane stress conditions include thin shells, loaded only in their plane.