By deforming and evolving the Weibull model formula

a gradation equation for quantitatively representing the continuous gradation distribution is obtained. The gradation equation can express three typical gradation distribution forms of hyperbola

approximate straight line and reversed S-shaped curve. The gradation distribution of silt

clay

sand

and coarse-grained soil is fitted

and the fitting results are all good. Considering the distribution range of the particle size

the value range of the gradation parameter is studied in detail with the method of theoretical deduction. By combining the actual fitting results with the theoretical derivation results

the final grading parameter range is obtained: c is 0.001-1.5

n is 0.065-4.8. Comparing the new gradation equation with Talbot fractal gradation equation and Wu Er-lu’s single-parameter gradation equation

it is found that the Talbot fractal gradation equation cannot reflect the reversed S-shaped gradation distribution

and the single-parameter gradation equation cannot reflect the uniform and narrow gradation distribution of particles

and the proposed gradation equation fits the above different gradation distributions well. The proposed gradation equation has wider applicability. In addition

the evolution of gradation parameters under the elimination method

similar gradation method

equivalent substitution method

and hybrid method is derived

and the quantitative expression of the gradation distribution before and after scaling is realized.