A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses
Publication: International Journal of Geomechanics
Volume 22, Issue 4
Abstract
This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.
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Acknowledgments
The authors acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 52108297); the Fellowship of China Postdoctoral Science Foundation (Grant No. 2021M692742); the Open Research Project from Key Laboratory of Geotechnical and Underground Engineering, Tongji University (Grant No. KLE-TJGE-B2101); and the National Natural Science Foundation of China (Grant No. 41772290).
Notation
The following symbols are used in this paper:
- C0, D0, F0, Ck, Dk, Lk, Mk, Nk Pk
- undetermined coefficients of the stress function;
- Hk
- coefficient of Fourier series;
- K0
- earth pressure coefficient at rest;
- Rh, Rv
- semimajor axis and semiminor axis of the opening;
- R(θ),
- radius and reference radius of the opening;
- radial and tangential displacements;
- UT
- total displacement;
- ɛr, , ɛz, γrz, γθz, γrθ
- radial, circumferential, axial, and three shear strains;
- σa, τa
- normal and shear stress at the surface of the opening;
- σr, , σz, τrz, τrθ, τθz
- radial, circumferential, axial, and three shear stresses;
- σs, τs
- normal and tangential supporting pressures;
- σv0, σh0
- in situ vertical and horizontal stresses;
- σx, σy, τxy
- horizontal, vertical, and shear stresses in the Cartesian coordinate;
- Φ(r, θ)
- stress function;
- α
- angle between the tangential direction of the tunnel face and the perpendicular direction of the tunnel radius; and
- μ
- Poisson’s ratio.
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Published In
International Journal of Geomechanics
Volume 22 • Issue 4 • April 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Jun 27, 2021
Accepted: Nov 30, 2021
Published online: Feb 2, 2022
Published in print: Apr 1, 2022
Discussion open until: Jul 2, 2022
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