The k-$varepsilon$ turbulence models by Launder et al.(1977, LPS) and Leschziner and Rodi(1981, LR) are modified to account for the secondary straining effect with having a generality in the present paper. The modified models are obtained by replacing the gradient Richardson number used to account for the secondary straining effect in the original models by a new parameter with a tensor-invariant correction form. These two modified models are used to predict the turbulent flow over a backward-facing step. In contrast to both standard and modified LR models, the modified LPS model is found to predict the reattachment point fairy well, as well as mean velocity, wall static pressure, turbulent kinetic energy and Reynolds shear stress in the recirculating region.