Calculating local geomembrane strains from a single gravel particle with thin plate theory

A new method is presented to calculate geomembrane strains induced by a single gravel particle (or grain) under vertical load from the deformed shape of the geomembrane for axisymmetric conditions. Past equations consider only vertical displacements of the geomembrane and neglect the contribution of radial displacements on strain and, consequently, underestimate the maximum strain. Axisymmetric large-strain-displacement relationships are used to relate radial strain to vertical and radial displacements. Vertical displacements are obtained from measurements of the deformed geomembrane from a physical experiment. Radial displacements do not need to be measured, but are related to tangential strain in the strain-displacement formulation. A linear elastic constitutive relationship is invoked and radial strains at the midsurface of the geomembrane from membrane elongation are solved for using Airy's stress function. Bending strains are obtained from the curvature of the deformed shape. Extreme fibre strains are the sum of the membrane and bending strains. Results from the new method match the maximum strain and pattern of strain when compared to large-displacement finite-element analysis. The new method is used to show that neglecting radial displacements underestimates the maximum strain by 25%, while neglecting radial displacements and bending strains underestimates the maximum strain by 60% (for a 2.5-mm-deep, Gaussian-shape indentation) and hence could affect selection of an appropriate geomembrane protection layer.