This paper investigates the strength and critical state (CS) behaviours of sand under axisymmetric stress paths with different shearing modes using discrete element method (DEM). The stress paths include axial compression (AC) and axial extension (AE), and the shearing modes include conventional triaxial (CT) mode and constant mean pressure (CP) mode. A series of dense and loose sand samples are generated for this purpose with confining pressures ranging from 100 kPa to 900 kPa. The CS is achieved for all samples after an axial strain (absolute value) of about 45%. The CS value of deviator stress is unique and independent of the initial packing densities for the samples with a given confining pressure, but the unique deviator stress under AC is generally larger than that under AE. The CS values of the stress ratio are independent of the shearing modes and the confining pressures, but are dependent on the stress paths. The CS friction angle for a given confining pressure is found to be