Calculations of structural stabilities for thin iron layers as grown on Cu(001) are reported. It is found that a free-standing fcc(001) monolayer is unstable with respect to 3x1 and 4x1 distortions. The former instability is directly related to the Pitsch transformation from fcc(001) to bcc(110), and calculations along the path find bcc(110) as the stable structure. Inclusion of the Cu substrate in the calculation tends to stabilize fcc(001). The calculated instabilities are of similar form to the superstructures observed for Fe grown on Cu(001). The relation to the experimental situation is discussed.