This study introduces a new definition of the strain-concentration factor (SNCF) of thick walled internally pressurized cylinders. The stress state has been considered in this new definition; i.e. triaxial and biaxial stress states for closed and open ends, respectively. Primarily, the curvature effect of the strain concentration has been studied here. To this end, the inner radius of the employed cylinders has been changed from 0.5 to 50.8 mm. On the other hand, the thickness has been kept constant at 16.7 mm. Moreover, the thickness has been fragmented to 37 elements to study the thickness effect for each case. The results show that the tangential (hoop) strain regularly spread over the whole thickness. It has been revealed that the maximum value of the tangential strain occurs on the inner surface of the cylinder. In particular, it rapidly decreases from a maximum value on the inner surface to reach its minimum value on the outer surface, which is nearly equal to the average value of hoop strain through the thickness. The results also demonstrate that tangential strain values decrease with the increase of the inner radius for any thickness. It is clear that the rate of decrease of the hoop strain changes abruptly with decreasing the inner radius of the cylinder. This led to localization of the strain concentration on the inner surface of the cylinder due to curvature, making the values of the strain concentration factor very high on the inner surface of the cylinder. In addition, the strain concentration factor decreases through the thickness of the cylinder from the inner to outer surfaces, and the rate of the decrease is increasing with a decreasing inner radius of the cylinder. The current results introduce the serious effect of the curvature on the strain concentration even if there are no irregularities in the cylinder.