Rock Mechanics and Rock Engineering

, Volume 52, Issue 11, pp 4509–4526 | Cite as

An Analytical Solution for an Arbitrary Cavity in an Elastic Half-Plane

  • Gui-sen Zeng
  • Hui CaiEmail author
  • Ai-zhong Lu
Original Paper


The complex variable method is used for an analytical solution for an arbitrary cavity in a homogeneous elastic half-plane. Proposing a method for solving the coefficients of the function by conformally mapping the region that contains an arbitrary cavity onto a circular ring utilizing a mapping function, which can be used for arbitrary shape cavity. Considering the stress-free condition of the upper surface, the relationship between two analytic functions is derived; consequently, the problem of solving two analytic functions is attributed to solving one analytic function, and the solve process for the analytic function in general case is given in detail, finally, achieving the solution of an arbitrary shape cavity in an elastic half-plane. The effects of cavity shape and depth were analyzed to the required terms of analytic function under a given accuracy. Examples for a vertical-wall semicircle cavity and a horseshoe cavity are given through the present method, and verification example is given by ANSYS software.


Elastic half-plane Analytical solution Complex variable method Mapping function Arbitrary cavity 

List of Symbols


Pending constants of mapping function


A complex constant


The net height of cavity


Elastic modulus


Objective function

\(F_{x} ,F_{y}\)

Resultant force in x- and y-directions

\(g_{{q}} \left( X \right),g_{{l}} \left( X \right)\)

Constraint functions


The depth of cavity


The imaginary unit


The lateral pressure coefficient

L1, L2

The horizontal surface and cavity boundary on physical plane

\(P_{m} \left( {x_{m} ,y_{m} } \right)\)

Control point on the physical plane, m = 1,2,…, which is the number of control points, xm and ym are the corresponding horizontal and vertical coordinates, respectively


Control point on the image plane, m = 1,2,…, which is the number of control points

real, image

The real and imaginary parts of a complex

Xn, Yn

The force component of x- and y-directions


A point on the physical plane


The bulk density


A point on the image plane


Poisson’s ratio

σx, σy, τxy

The x- and y-directions and shear stress component of the surrounding rock mass after excavation

\(\sigma_{x}^{0} ,\sigma_{y}^{0} ,\tau_{xy}^{0}\)

The x- and y-directions and shear stress components of the surrounding rock mass before excavation

\(\sigma_{x}^{*} ,\sigma_{y}^{*} ,\tau_{xy}^{*}\)

The x- and y-directions and shear stress components of the surrounding rock mass produced by excavation

σθ, σρ, τρθ

The tangential, radial and shear stresses

\(\varphi \left( z \right),\varphi_{0} \left( z \right),\; \psi \left( z \right)\)

Analytic function on the z-plane

\(\varphi \left( \zeta \right),\varphi_{0} (z)\)

Analytic function on the ζ-plane

\(\omega (\zeta )\)

Mapping function



The study is supported by the Natural Science Foundation of China (Grant numbers 11572126 and 51704117).

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Hydroelectric and Geotechnical EngineeringNorth China Electric Power UniversityBeijingChina
  2. 2.Electric Power Research Institute of Guizhou Power Grid Co., Ltd.GuiyangChina

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