# Plane Strain examples

Trying to find examples of structures, components etc. in plane strain (under normal loading conditions).

Is a thin-walled-pressure-vessel in plane strain?

(I thought it was plane stress but could be wrong)

regards

A section of a long straight pipeline can be considered as plane strain. When the internal pressure creates radial strain in the pipe, the Poisson's ratio effect can not reduce the length of the pipe so the strain in the axial direction is always zero. (Think about a pipeline that is say 100mm in diameter and 100km long - it doesn't make sense to say that if the diameter increased by 1 mm because of internal pressure or thermal expansion, Poisson's ratio would make the length of the pipeline decrease by a few hundred meters!)

A thin walled pressure vessel is approximately a plane stress situation, if you ignore the radial stress through the thickness of the vessel. To be consistent with the boundary conditions, the radial stress must be equal to the internal pressure on the inside of the wall and zero on the outside, but that is a small stress compared with the circumferential and axial stresses, which are both of the order of $$R/t$$ times bigger, where $$R$$ is the pipe radius and $$t$$ the wall thickness. (The previous sentence ignored a few factors of 2, but they don't affect the point I'm making).

• thanks for your comments, but the question was "Is a thin-walled-pressure-vessel in plane strain?" – William White Jun 13 '19 at 20:23
• You asked three things, of which that was the second one. And a one-word answer "no" to the second question would probably have been deleted by the moderators, even though it was correct! – alephzero Jun 13 '19 at 22:34
• without getting too argumentative, I asked one question: the one with the question mark. But thanks for your help anyway, much appreciated. – William White Jun 13 '19 at 23:58

If you confine a column laterally by rigid walls so it can not laterally expand under the load it is under plane strain.

In structural members under plane strains the principal plane strains can be found by rotating the coordinate system very similar to Mohr circle to find the principal max and min strain and the rotation angle is. $$tan2\theta=\frac{2 \epsilon_{xy}}{\epsilon_{xx}-\epsilon_{yy}}$$

Thin wall vessels are not in plain strain. The lower the strength the thicker a vessel must be to reach plain strain. A reference you maybe able to find is "Fracture of Structural Materials " by Tetelman and Mcevily , Pub. Wiley and sons 1967.A note on pg 21 :" In general , plain -strain conditions exist in thick bodies and plain-stress conditions exist in thin ones". For petrochemical vessels we generally would not do a fracture mechanics analysis /brittle fracture evaluation unless wall thickness was over 1 in. : that is, there was a small chance of plain strain conditions. ( Low temperature brittle fracture aside). For line pipe and casing; toughness tests ( eg. Charpy) are often required not because of plain strain ,but because the high strengths and high stresses; casing couplings can easily reach yield stress.