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Static and Seismic Response of Deep Spherical Cavities

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Abstract

An analytical solution is derived for a supported deep spherical cavity, and for the following scenarios: static loading with far-field vertical and horizontal stress, with the ground either dry or saturated with flow or no flow towards the cavity, and seismic loading for drained and undrained loading conditions. The solutions are obtained with the assumption that the ground and the support are elastic and remain elastic during loading, for tied ground-liner interface, excavation and liner support occur at the same time, the liner is thin and its response can be approximated as that of a membrane and the seismic demand can be represented as a far-field static shear stress. The correctness of the analytical solution is verified by comparing its results with those of the finite element code ABAQUS. Inspection of the closed-form solutions shows that the static response of the cavity strongly depends on the relative stiffness of the support relative to the ground and on the coefficient of earth pressure at rest. In addition, the seismic response also depends on the type of loading, either drained or undrained. Results from the seismic analysis show that undrained loading results in larger stresses in stiffer liners, while drained loading imposes larger displacements in flexible liners.

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Abbreviations

E, ν, G:

Young’s modulus, Poisson’s ratio, shear modulus of the ground

Es, νs, Gs :

Young’s modulus, Poisson’s ratio, shear modulus of the liner

K, Ks :

Permeability of the ground and liner

Ko :

Coefficient of earth pressure at rest

\({\mathrm{N}}_{\theta }^{\mathrm{s}}, {\mathrm{N}}_{\phi }^{\mathrm{s}}\) :

Axial forces in the liner, in spherical coordinates

ro :

Radius of the spherical cavity

r, θ, ϕ:

Spherical coordinates

t:

Thickness of the liner/support

u:

Pore pressures

ui :

Pore pressures at the liner-ground interface

uo :

Pore pressures inside the cavity

uw :

Far-field pore pressures

x, y, z:

Cartesian coordinates

Q:

Flow through the cavity

R:

Radial distance where far-field pore pressures are recovered

T:

Rotation matrix

Ur, Uθ , Uϕ :

Displacements in spherical coordinates

\({\mathrm{U}}_{\mathrm{r}}^{\mathrm{net}}\), \({\mathrm{U}}_{\theta }^{\mathrm{net}}\) :

Net displacements of the ground in spherical coordinates

\({\mathrm{U}}_{\mathrm{r}}^{\mathrm{s}}\), \({\mathrm{U}}_{\theta }^{\mathrm{s}}\) :

Displacements of the liner, in spherical coordinates

α, β:

Polar and azimuthal angles

γw :

Unit weight of water

εrr, εθθ, εϕϕ, εr θ, εr ϕ, εθϕ :

Strains in spherical coordinates

\({\sigma }_{\mathrm{r}}^{\mathrm{s}}\), \({\tau }^{\mathrm{s}}\) :

Normal and shear stresses acting on the liner

σv, σH, σh :

Far-field total vertical and horizontal stresses

σ’v, σ’H, σ’h :

Far-field effective vertical and horizontal stresses

σrr, σθθ, σϕϕ, σr θ, σr ϕ, σθϕ :

Total normal and shear stresses in spherical coordinates

σ’rr, σ’θθ, σ’ϕϕ :

Effective normal stresses in spherical coordinates

χ:

Non-dimensional factor equal to: ro/t G/Gs

\({\Sigma }_{\mathrm{r},\theta ,\phi }, {\Sigma }_{\mathrm{r},\mathrm{\alpha },\beta }\) :

Stress tensors in spherical coordinates r, θ, ϕ and r, α, β

\({\mathrm{U}}_{\mathrm{r},\theta ,\phi } , {\mathrm{U}}_{\mathrm{r},\mathrm{\alpha },\beta }\) :

Displacement vectors in spherical coordinates r, θ, ϕ and r, α, β

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  1. Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, USA

    Antonio Bobet

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Bobet, A. Static and Seismic Response of Deep Spherical Cavities. Rock Mech Rock Eng 56, 9135–9148 (2023). https://doi.org/10.1007/s00603-023-03540-8

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