Cascading failures are common in real-life networks. To better explore the robustness against cascading failures on complex networks，by adopting the initial load of a node j to be Lj=kαj，where kj is the degree of the node j，a new cascading model is proposed based on the local preferential redistribution rule of the broken node's load. According to a new measure to the robustness of a network，we further investigate cascading failures on four typical networks. The numerical simulations show that four networks reach the strongest robustness level against cascading failures and that the robustness against cascading failures on four networks has a negative correlation with the average degree 〈k〉 when α=1. The simulation results are also verified by the theoretical analysis.