In engineering geological formations, it is common to encounter discontinuous grading soils. However, existing grading equations have limitations in representing these discontinuous gradings, which are mainly manifested in a single morphological limitation and quantitative expression. This study proposes a unified equation for describing continuous and discontinuous grading of coarse-grained soil. The equation's basic properties were investigated to check its ability to reflect different forms of grading curves. Additionally, the practicality of using this equation for designing grading curves of coarse-grained soil was discussed. The main conclusions of this study are as follows: By adjusting the grading segmentation parameter n in the grading equation, the equation can be made applicable to both continuous and discontinuous grading curves. When n = 1, adjusting the grading curve anchor point parameter gt,i, the anchor point control parameter dcr,i,, and the curve shape control parameter ki can achieve the precise representation of three typical curves: hyperbolic, reversed S-shaped, and approximate straight line. When n = 2, inflection points may occur along a continuous grading curve, at which structural transitions in particle composition and the dominant particle size take place. When n > 3, adjusting the parameters n,gt,i, dcr,i, and ki can allow control over the number of inflection points and the particle content in different intervals of discontinuous grading. The unified grading equation applies to both continuous and discontinuous grading curves of coarse-grained soil in various regions. Its key parameters effectively link the granular skeleton state to macro-mechanical performance, providing a predictive mathematical model for permeability, shear strength, and other engineering properties, and offering a reference for assessing how grading continuity affects internal stability. The findings are thus valuable for designing both continuous and gap-graded coarse-grained soils.
Yiqian Xia — School of Civil Engineering, Central South University, Changsha, Hunan 410075, China Jinyang F: School of Civil Engineering, Central South University, Changsha, Hunan 410075, China; National Engineering Research Center of High-speed Railway Construction Technology, Changsha, Hunan 410075, China
Zhou Yang — Guangzhou Metro Construction Management Co., Ltd., Guangzhou, Guangdong 510330, China
Qianhui Sun — Power China Hubei Electric Engineering Co., Ltd., Wuhan, Hubei 430040, China
Jiawei Xie — School of Infrastructure Engineering, Nanchang University, Nanchang, Jiangxi 330031, China
Niu Zhang — National Engineering Research Center of High-speed Railway Construction Technology, Changsha, Hunan 410075, China; China Railway No. 10 Engineering Group Co., Ltd. Jinan, Shandong 250101, China
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