Elastic–plastic solutions for a circular hydraulic pressure tunnel based on the D–P criterion considering the fluid field

  • Sui Wang
  • Zuliang ZhongEmail author
  • Yuqi Ren
Technical Paper


This work provides new findings on the theory of the Drucker–Prager (D–P) failure criterion applied to study the analytical solutions of the radius and stress of a plastic zone and the displacement at the periphery of a hydraulic pressure tunnel under a uniform ground stress field considering fluid. This work focuses on the elastic–plastic rock mass during the stages of construction and operation. The obtained results are compared to other calculation techniques and show that the elastic–plastic solution calculated based on the D–P criterion is more conservative than that based on the Mohr–Coulomb (M–C) model. The first principal stress varies with the internal water pressure, which first decreases to zero and then gradually increases in the plastic zone. As the value of the water head difference is large enough, there will be a threshold radius in the elastic zone, at which the maximum principal stress changes from the radial stress to the tangential stress with increasing distance from the tunnel, ultimately reaching the in situ stress, and the threshold radius is related to the magnitude of the water head difference. Considering the effect of seepage, the radial stress and tangential stress are not symmetrical about the axis of in situ stress within the elastic zone. In addition, the degree of deviation from the in situ stress increases with the increasing value of water head difference. Under the low internal water pressure condition, an increase in the osmotic pressure will further develop the plastic zone; however, under the high internal water pressure condition, an increase in the osmotic pressure will slow the development of the plastic zone.


Deeply buried tunnel Seepage force D–P criterion Elastic–plastic solution 



Young’s modulus


Poisson’s ratio




Internal friction angle




Tunnel excavation radius


Plastic zone radius


Country rock stress


Internal water pressure


Radial stress


Tangential stress


The stress along the axis of the hole


First stress invariant


Second partial stress invariant


Constant related to the surrounding rock strength, \(\alpha = \frac{\tan \varphi }{{\sqrt {9 + 12\tan^{2} \varphi } }}\)


Constant related to the surrounding rock strength, \(k = \frac{3c}{{\sqrt {9 + 12\tan^{2} \varphi } }}\)

\({\text{d}}\varepsilon_{r}^{\text{p}} ,\;{\text{d}}\varepsilon_{\theta }^{\text{p}} ,\;{\text{d}}\varepsilon_{z}^{\text{p}}\)

Plastic strain increment


Instantaneous stress deviator


Transient deviation stress


Integral constant


Integral constant

\(\sigma_{r}^{\text{p}} ,\sigma_{\theta }^{\text{p}} ,\;\sigma_{z}^{\text{p}}\)

Stresses in the plastic zone

\(\sigma_{r}^{e} ,\sigma_{\theta }^{e}\)

Rummy solutions in the elastic zone


Radial strain


Tangential strain




Integral constant


Integral constant


Shear modulus



This research was sponsored by the Natural Science Foundation Project of Chongqing (cstc2013jcyjA30005).


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Civil EngineeringChongqing UniversityChongqingChina
  2. 2.Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University)Ministry of EducationChongqingChina

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