Abstract
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.
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Abbreviations
- E :
-
Young’s modulus
- υ :
-
Poisson’s ratio
- c :
-
Cohesion
- φ :
-
Internal friction angle
- r :
-
Rad
- r 0 :
-
Tunnel excavation radius
- r p :
-
Plastic zone radius
- P :
-
Country rock stress
- P 0 :
-
Internal water pressure
- σ r :
-
Radial stress
- σ θ :
-
Tangential stress
- σ z :
-
The stress along the axis of the hole
- I 1 :
-
First stress invariant
- J 2 :
-
Second partial stress invariant
- α :
-
Constant related to the surrounding rock strength, \(\alpha = \frac{\tan \varphi }{{\sqrt {9 + 12\tan^{2} \varphi } }}\)
- k :
-
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
- dλ :
-
Instantaneous stress deviator
- S r S θ S z :
-
Transient deviation stress
- C 1 :
-
Integral constant
- C 2 :
-
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
- ɛ r :
-
Radial strain
- ɛ θ :
-
Tangential strain
- u :
-
Displacement
- K 1 :
-
Integral constant
- K 2 :
-
Integral constant
- G :
-
Shear modulus
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Acknowledgements
This research was sponsored by the Natural Science Foundation Project of Chongqing (cstc2013jcyjA30005).
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Wang, S., Zhong, Z. & Ren, Y. Elastic–plastic solutions for a circular hydraulic pressure tunnel based on the D–P criterion considering the fluid field. Innov. Infrastruct. Solut. 3, 31 (2018). https://doi.org/10.1007/s41062-018-0135-6
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DOI: https://doi.org/10.1007/s41062-018-0135-6