Discussion
Determine the correctness or otherwise of the following statements, [a] and [r]:
[a]: In a plane stress problem, the shear strains along the thickness direction of a body are zero but the normal strain along the thickness is not zero.
[r]: In a plane stress problem, Poisson effect induces the normal strain along the thickness direction of the body.
Answer: A
Both [a] and [r] are true, and [r] is the correct reason for [a].
Plane stress means the stresses on the faces normal to the thickness are zero: \(\sigma_{z} = \tau_{xz} = \tau_{yz} = 0\). With no out-of-plane shear stress, the shear strains \(\gamma_{xz}\) and \(\gamma_{yz}\) are zero. But the normal strain in the thickness direction is
\[ \varepsilon_{z} = -\frac{\nu}{E}(\sigma_{x} + \sigma_{y}) \]
which is not zero, so [a] is true. That strain arises because the in-plane stresses make the body contract or expand sideways, the Poisson effect, which is what [r] says, so [r] explains [a]. 'Both false' and '[r] false' are therefore wrong, and so is 'not the correct reason'.
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