Statics of loose triangular embankment under Nadai's sand hill analogy

TL;DR

This paper revises Nadai's sand hill analogy by analyzing stress distribution in a wedge-shaped sand heap, correcting previous inaccuracies, and confirming the analogy's validity through theoretical and experimental agreement.

Contribution

It introduces a corrected stress analysis model for loose triangular embankments based on Nadai's analogy, aligning theory with experimental data.

Findings

Corrected stress distribution aligns with experimental measurements

Nadai's analogy is validated with revised theoretical analysis

Stress analysis accounts for variable friction angle and principal stress orientation

Abstract

In structural mechanics, Nadai's sand hill analogy is the interpretation of an ultimate torque applied to a given structural member with a magnitude that is analogously twice the volume of stable sand heap which can be accommodated on a transverse cross-section basis. Nadai's analogy is accompanied by his observation of a loose triangular embankment, based on the fact that gravitating loose earth is stable if inclined just under the angle of repose. However, Nadai's analysis of stress distribution in a planar sand heap was found to be inaccurate because the total pressure obtained from Nadai's solution is greater than the self-weight calculated from the heap geometry. This raises a question about the validity of his observation in relation to the analogy. To confirm his criterion, this article presents and corrects the error found in Nadai's solution by analyzing a radially symmetric stress field for a wedge-shaped sand heap with the purpose of satisfying both force balance and Nadai's closure. The fundamental equation was obtained by letting the friction state vary as a function of angular position and deduce it under the constraint that the principal stress orientation obeys Nadai's closure. The theoretical solution sufficiently agreed with the past experimental measurements.