Adopting the initial load of a node j to be L_j = [k_j(∑_m∈Γ_jk_m)]^α with k_j and Γ_j being the degree of the node j and the set of its neighbouring nodes respectively, we propose a cascading model based on a local preferential redistribution rule of the load after removing a node. Assuming that a failed node leads only to a redistribution of the load passing through it to its neighbouring nodes, we explore the response of scale-free networks subject to two different attack strategies on nodes and find some interesting and counterintuitive results in our cascading model. On the one hand, unexpectedly, the attack on the nodes with the lowest degree is more harmful than the attack on the highest degree nodes when α<1/2. On the other hand, when α=1/2, the effects of two attacks for the robustness against cascading failures are almost identical. In addition, the numerical simulations are also verified by the theoretical analysis. These results may be very helpful for real-life networks to protect the key nodes selected effectively and to avoid cascading-failure-induced disasters.