Why plane stress condition is taken for thin plates Why plane stress is taken for thin plates? It says in the books that the stress variation is small for thin components and is close to zero. Why is that so?
Also why stress at free surface is zero? (talking with respect to a solid under simple loading e.g. a bar under uniaxial loading) and free surface is the boundary of a specimen right?
 A: The boundary conditions of free surfaces are $\sigma_{ij}n_j=0$, where $\vec n$ is the normal to the surface. If a body is thin in the $z$ dimension, then on the top surface (say, $z=h$) and on the bottom one (say, $z=0$) you have $\vec n=\vec z$ and therefore $\sigma_{iz}=0$.
Since $\sigma_{iz}$ vanishes of $z=0$ and for $z=h$, and since $0\le z\le h$ and $h$ is small (i.e. the body is thin), it is reasonable to assume that $\sigma_{iz}$ is identically zero for all values of $z$.
This is the standard argument. Note, however, that in contrast to plane-strain which is a rigorous reduction of the full 3D equations, plane-stress is an approximation, which can not be satisfied identically.
A: Plane stress and plane strain are both approximations to reduce 3-D problems to 2-D problems, which are easier to solve.
Plane stress is used to model structures that are very small in one direction compared to the other two, such as thin plates.   The normal stress at a free surface is zero because there's nothing (other than, usually, air) to react against it.   In a thin plate, the normal stress is zero on both faces of the plate and there's not much room for it to build up in between the faces.   So we just assume that it's zero all the way through.   The in-plane stresses aren't subject to this restriction and can be any size you want (within the limits of the material).
Plane strain is used to model structures that are very large in one dimension compared to the other two, such as columns.   Each thin slice of material is constrained by material on either side and so the strain in the 'long' direction is taken to be zero all the way through.   The crosswise strains aren't subject to this restriction and can be any size you like (within the limits of the material).
