Features of the calculation of a flat sheet as a membrane element of a combined retaining wall

Abstract

The construction of underground structures is one of the key trends in modern construction, with additional consideration given to the threat posed by an aggressor state. This inevitably involves the excavation of deep pits, which requires the installation of costly retaining wall structures, particularly in water-saturated soils. An effective design for a combined flexible retaining wall is proposed, incorporating a flat sheet pile that functions within the system under tension as a membrane element, also serving as an anti-seepage curtain.

Classical elasticity theories are used to model the behaviour of flat sheet piles: the Kirchhoff–Love theory for thin plates, the Mindlin–Reissner theory for elements of intermediate thickness, and the Föppl–von Kármán theory for cases involving significant deflections and geometric non-linearity. The choice of theory depends on the ratio of the membrane thickness to its characteristic dimensions and on the type of loading. Combining analytical solutions with numerical methods, in particular the finite element method (FEM), ensures the accuracy and reliability of the results.

The effect of membrane thickness on deflections, axial forces and the convergence of numerical algorithms was investigated. It was found that as the thickness increases from 2 to 20 mm, deflections decrease from 147,3 to 65,6 mm, whilst the maximum axial force Nx increases from 491,29 to 904,5 kN/m. Thicker membranes are characterised by greater stiffness, lower deformation and faster convergence of calculations. For thin membranes (2–6 mm), the application of the Föppl–von Kármán theory is necessary, whilst for medium thicknesses (8–12 mm), a balance between permissible deflections and load-bearing capacity is advisable. For thick membranes (14–20 mm), the use of the Mindlin–Reissner theory, taking into account shear effects, is recommended. The optimal thickness range of 8–12 mm ensures acceptable deflections, sufficient load-bearing capacity and material savings.

Modelling in the LIRA-FEM software package is based on the displacement-based finite element method. The membrane is treated as a shell with minimal bending stiffness, where membrane forces dominate. The use of the Kirchhoff–Love and Föppl–von Kármán theories allows the effect of geometric stiffening to be taken into account, whilst the application of physically-based geometric non-linearity ensures realistic results. The study involved modelling a steel membrane 6 m high, 5 m wide and 10 mm thick under trapezoidal soil pressure. Three modelling variants were considered: geometric non-linearity only, a shell with full stiffness, and physical-geometric non-linearity. The comparison showed that taking physical non-linearity into account allows the bearing capacity to be correctly estimated and the safety factor to be accurately assessed.

Thus, an integrated approach combining analytical models and numerical methods makes it possible to establish a multi-level methodological framework for the design and calculation of composite retaining wall structures incorporating flat sheet piles (membrane elements). This ensures the accuracy of calculations and the reliability of results for modern engineering practice.