Abstract: It is aimed to develop a strength criterion complying with the principle of objectivity for sand with cross-anisotropy. First, how the peak strength of anisotropic granular material varies with the loading direction is observed through some typical tests. Then, the mechanism of the strength mobilizing is analyzed, and a novel dimensionless anisotropy parameter Λ(σ, F), which measures the relative direction of stress tensor and fabric tensor, is defined. Using this parameter, the spatial mobilized plane (SMP) criterion is extended to describe the strength of sand with cross-anisotropy. If the position of the depositional plane is known, there are only two parameters, k_f0 and k, to be determined in the criterion. They can be both easily determined by two experiments with different loading directions. The essential effectiveness of the new criterion is confirmed by the experimental facts and numerical modeling.