Rock Mechanics and Rock Engineering

, Volume 52, Issue 12, pp 5123–5135 | Cite as

The Elasto-Plastic Response of Deep Tunnels with Damaged Zone and Gravity Effects

  • Ahmadreza HedayatEmail author
  • Jacob Weems
Original Paper


Analysis of stresses and deformations around circular tunnels and shafts are is critical for evaluation of the interaction between the support system and the ground and, hence, the tunnel support design and stability. The damage induced in the material by the excavation method (i.e., blasting or mechanical excavation) can significantly influence the ground response as the excavation alters the rock mass properties over the damaged zone. When the provided support pressure inside the tunnel falls below a critical value, a zone of plastic (broken) material develops around the tunnel. The self-weight of the broken material is significant at the roof (crown) of the tunnel and may exert higher pressures to the support system at the roof than the sidewalls. This study presents a new analytical–numerical solution for the determination of stresses, strains, and deformations around a circular deep tunnel with the consideration of the gravity effect and the damaged zone. A modified equilibrium equation for the ground is used and the elastic and plastic zones of the tunnel are analyzed. The results indicate that gravity and the damaged zone have significant effects on the tunnel convergence and the distribution of stresses in the rock mass. The presented method in this paper is novel and allows tunnel designers to assess the combined effect of damage induced by excavation and the gravity on tunnel convergence.


Tunnel Analytical–numerical solution Damaged zone Gravity effect 

List of Symbols


Young’s modulus of rock mass


Shear modulus of rock mass


Geological Strength Index of Rock Masses


Coefficient of dilation


Initial value of Hoek–Brown constant

mb, s, a

Hoek–Brown constants for rock mass

mr, sr, ar

Residual Hoek–Brown constants

mbd, sd, mrd, srd

Hoek–Brown constants for damaged rock mass


Disturbance factor


Critical stress at the elastic–plastic boundary


Internal pressure


Far-field hydrostatic stress


Radius of plastic zone


Radial distance


Normalized radial distance


Damaged zone radius


Radial displacement


Poisson’s ratio


Rock mass unit weight

\(\varepsilon_{\theta }\)

Tangential strain

\(\varepsilon_{\theta }^{\text{p}}\)

Plastic component of tangential strain

\(\varepsilon_{\theta }^{\text{e}}\)

Elastic component of tangential strain


Radial strain


Plastic component of radial strain


Elastic component of radial strain


Major principal stress


Minor principal stress


Radial stress

\(\sigma_{\theta }\)

Tangential (circumferential) stress


Initial uniaxial compressive strength of rock


Residual value of uniaxial compressive strength of rock


Damaged zone uniaxial compressive strength of rock


Plastic potential function



The authors would like to thank the reviewers for their valuable comments and suggestions.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringColorado School of MinesGoldenUSA
  2. 2.Department of Geology and Geological EngineeringColorado School of MinesGoldenUSA

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