Geometrical assessment of internal instability potential of granular soils based on grading entropy

Abstract

Internal instability of a soil is closely related to its particle size distribution (PSD) that occurs when its coarser particles cannot protect the finer particles from erosion, thereby inducing permanent changes in its original PSD. This study proposes a new criterion based on grading entropy theory for prompt assessment of internal stability. PSD is discretized into several fractions to extract particle grading information through statistical analysis. Two normalized variables: base entropy ($h_{0}$) and entropy increment ($\Delta h$) are determined directly from the PSD curve, and the principle of maximum entropy is used to obtain a semi-ellipse within plane formed by $h_{0}$ and $\Delta h$, wherein a PSD can be simply expressed as a point. A clear boundary between stable and unstable soils is visualized at maximum $\Delta h$ line, which is used to correctly evaluate a large published experimental dataset and its performance is compared with the existing criteria.

Appendix 1: Principal of Maximum Entropy

According to the principle of maximum entropy, the maximum H can be achieved using Lagrange multipliers as follows [22]:

$$L_{\max \,H} = - \mathop \sum \limits_{i = 1}^{N} C_{i} \frac{{p_{i} }}{{C_{i} }}\log_{2} \frac{{p_{i} }}{{C_{i} }} + \lambda \mathop \sum \limits_{i = 1}^{N} \left( {p_{i} - 1} \right)$$

(18)

where $L_{{{\text{Max}} H}}$ is the Lagrange function of maximum H, while $\lambda$ is the Lagrange multiplier of the corresponding constraint ($\mathop \sum \nolimits_{i = 1}^{N} \left( {p_{i} - 1} \right)$). By differentiating the Lagrange function with respect to the relative frequencies ${ }p_{i}$, and equating the derivative to zero, Eq. (18) takes the following form:

$$\frac{{\partial L_{\max H} }}{{\partial p_{i} }} = - \log_{2} \frac{{p_{i} }}{{C_{i} }} - \frac{1}{\ln 2} + \lambda = 0$$

(19)

Simplifying Eq. (19) further to obtain,

$$\frac{{p_{i} }}{{C_{i} }} = 2^{{\lambda - \frac{1}{\ln 2}}} = {\text{constant}}$$

(20)

Following the same procedure as maximum H, the maximum $\Delta$ H can be obtained from the following steps:

$$L_{\max \,\Delta H} = - \mathop \sum \limits_{i = 1}^{N} p_{i} \log_{2} p_{i} + \lambda \mathop \sum \limits_{i = 1}^{N} \left( {p_{i} - 1} \right)$$

(21)

$$p_{1} = p_{2} = p_{3} = \cdots = p_{N} = 1/N$$

(22)

where $L_{{{\text{Max}}\,\Delta H}}$ is the Lagrange function of maximum $\Delta H$. The maximum $\Delta h$ can be obtained from the same procedure as follows:

$$L_{\Delta h} = - \frac{1}{\ln N} \mathop \sum \limits_{i = 1}^{N} p_{i} \log_{2} p_{i} + \lambda_{1} \mathop \sum \limits_{i = 1}^{N} \left( {p_{i} - 1} \right) + \lambda_{2} \left[ {\mathop \sum \limits_{i = 1}^{N} p_{i} \left( {i - 1} \right) - h_{0} \left( {N - 1} \right)} \right]$$

(23)

where $L_{\Delta h}$ is the Lagrange function of maximum $\Delta h$.