A Refined Vlasov Foundation Model and Finite-Difference Method for Computing Slope Anchor Frame Resting on Nonsingle Bearing Strata

The Vlasov foundation model and its variations belong to two-parameter elastic foundation beam models that can rationally reflect the displacement decrease law with increasing depth within the compressible elasticity layer. They are, however, only applicable for foundation beams resting on a single bearing stratum or horizontally layered soils. To meet the need to calculate slope anchor frames resting on nonsingle bearing strata or nonhorizontally layered soils that may often be encountered in practical engineering, a refined Vlasov model was presented in this paper. In this model, a general expression of the weighted thickness of the elasticity layer was established, and the corresponding expressions of the two parameters of soil resistance, as well as coefficient γ, were deduced. Meanwhile, it was found that by dividing the relative stiffness of beam–soil into two variables expressed with a logarithm, the calibration factor χP can be formulated by fitting its relationship with the two variables so that the problem of determining the thickness of the elasticity layer of the Vlasov model can be resolved. Based on the aforementioned, the finite-difference method was established considering the conditions of static equilibrium and compatible deformation of slope anchor frames. A corresponding computer program that can perform all operations was developed using Matlab (version 2019a). The three-dimensional finite-element method can truthfully simulate the slope anchor frame resting on nonsingle strata and obtain a high-precision numerical solution to this elastic mechanics problem. The example shows that the results by the finite-difference method based on the refined Vlasov model agreed well with those by the finite-element method, demonstrating the reliability of the finite-difference method. Furthermore, comparisons show that if a slope anchor frame resting on nonsingle bearing strata is simplified as a frame resting on a single bearing stratum like people used to do, it will lead to an overconservative or unsafe design result. The significance of this study lies in providing a valid and reliable approach for calculating the displacement and internal forces of slope anchor frames resting on nonsingle bearing strata.