K Xia Y Guiyun 2022 Generalized foundation Timoshenko beam and its calculating methods Arch. J Fariborz RC Batra 2019 Free vibration of bi-directional functionally graded material circular beams using shear deformation theory employing logarithmic function of radius Compos. Y Tang X Lv T Yang 2019 Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration Compos. K Xie Y Wang X Fan T Fu 2020 Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories Appl. M Soltani B Asgarian 2019 New hybrid approach for free vibration and stability analyses of axially functionally graded Euler–Bernoulli beams with variable cross-section resting on uniform Winkler–Pasternak foundation Lat. ĪE Alshorbagy MA Eltaher F Mahmoud 2011 Free vibration characteristics of a functionally graded beam by finite element method Appl. M Song L Chen J Yang W Zhu S Kitipornchai 2019 Thermal buckling and postbuckling of edge-cracked functionally graded multilayer graphene nanocomposite beams on an elastic foundation Int. Y Ma X Du J Wu G Chen F Yang 2020 Natural vibration of a non-uniform beam with multiple transverse cracks J. ![]() The tunnel’s overall deflection adopts an “M” shape, and the computation using the variable-section FGB portrays the existing tunnel’s deformation trend with heightened accuracy, diminishing the calculation error from 35 to 8.3% compared to the conventional normal section beam. A comparative analysis of theoretical calculations, monitoring data, and numerical simulation results exhibits substantial concurrence across the three methods. The proposed method is then integrated with the Mindlin stress solution to evaluate construction impact on an existing tunnel within a foundation pit project in Shenzhen. Furthermore, when the beam stiffness adheres to a Gaussian distribution along the axial direction of the beam, a significant increase in displacement at the boundary position of the beam is observed. The results indicate that an asymmetrical distribution of stiffness on either side of the midpoint could increase the displacement at the middle section. Additional analysis is conducted on displacement and internal force variation when the beam stiffness follows different distribution along the axial direction. The P–T model degenerates to the Winkler–Timoshenko model (W–T model) when the foundation’s shear layer stiffness is set to zero. The semi-analytical solution is subsequently compared with finite difference solution results from prior studies, affirming the accuracy and precision of the proposed computational theory. ![]() Employing the variational principle and the transfer-matrix method, and considering the shear stiffness of the axially FGB structure itself, along with the continuity and shear strength of the soil, a semi-analytical solution for displacement and internal force of axially FGBs on Pasternak foundation (termed as P–T model) is derived in this paper. Therefore, the incorporation of variable-section FGBs theory into the analysis of structural and soil interactions is crucial for advancing engineering applications. They enhance the performance of functionally graded beams (FGBs) under various loading conditions. Functionally graded materials (FGMs) are commonly utilized in construction projects.
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