We study the scaling properties of randomly folded aluminum sheets of different thicknesses h and widths L . We found that the fractal dimension D=2.30+/-0.01 and the force scaling exponent delta=0.21+/-0.02 are independent of the sheet thickness and close to those obtained in numerical simulations with a coarse-grained model of triangulated self-avoiding surfaces with bending and stretching rigidity. So our findings suggest that finite bending rigidity and self-avoidance play the predominant roles in the scaling behavior of randomly folded plastic sheets.