A Boundary Virtual Load Method for Prestress-Field Prediction in Corner-Tensioned Membranes

Abstract

Flexible membrane structures, such as solar sails and membrane reflectors, rely on accurate characterization of in-plane prestress for structural reliability and functional performance. This study proposes a Boundary Virtual Load Method (BVLM) for the semi-analytical prediction of prestress fields in corner-tensioned rectangular membranes. An initial corner-dominated radial stress field is constructed by superposing the stress solutions generated by four corner loads. Residual normal stresses along the nominally free edges are then evaluated, and boundary virtual loads of equal magnitude and opposite direction are introduced to approximate the zero-normal-traction condition. Polynomial representations of these virtual-load distributions are incorporated into the Airy stress-function framework, yielding a boundary-corrected closed-form prestress solution. Comparisons with finite element results for three membrane configurations show that BVLM accurately captures the principal spatial characteristics and aspect-ratio-dependent evolution of the prestress field while requiring substantially lower computational cost for the cases examined. The predicted prestress fields are further incorporated into a free-vibration model that accounts for the added mass of the surrounding air. The resulting natural frequencies generally reproduce the experimentally measured modal-frequency trends, although mode-dependent discrepancies are observed for several modes. The 12 frequency comparisons yield a mean absolute relative error of 8.46%, indicating reasonable overall agreement for the approximate analytical formulation. These comparisons provide an indirect dynamic consistency assessment rather than a direct validation of the spatial prestress field. BVLM therefore offers an efficient framework for prestress analysis, preliminary vibration prediction, and parametric design of corner-tensioned membranes.