Abstract This paper presents closed-form analytical solutions for stresses and strains in a long rectangular elastic plate subjected to gravity loading and periodically distributed vertical surface loads, with direct application to roof stability assessment in room-and-pillar (RP) mining layouts. The solutions are derived using Fourier series expansions within linear isotropic elasticity and are formulated for three distinct boundary-value problems of practical relevance. The first two problems, which address long plate strips, pertain to: (i) a single-level RP excavation at depth and (ii) a two-level RP system separated by a protective sill pillar, including the effect of uniformly distributed roof support loads induced by systematic rockbolting. Based on these solutions, a safety factor against roof tensile failure is defined as the ratio of the critical extensional strain of the rock mass to the maximum induced extension strain. The strain-based criterion is adopted because roof failure initiates in bending and because extensional strains are routinely monitored in situ using multi-point borehole extensometers. The analytical predictions of the first problem are verified against field stress and strain measurements from an operating underground marble quarry, demonstrating the capability of the solution to support back-analysis of rock mass properties and roof safety assessment. The third problem provides an approximate analytical solution for short beams under three-point bending, obtained as a by-product of the plate formulations. This solution is also verified against laboratory bending tests on marble specimens, confirming its applicability to experimental configurations. Overall, the proposed analytical framework offers a transparent and fast method for both in-situ monitoring interpretation and laboratory data analysis in rock engineering applications.
Stavropoulou et al. (Wed,) studied this question.