Analytical Solution for Rayleigh Wave-Induced Dynamic Response of Shallow Grouted Tunnels in Saturated Soil
Abstract
1. Introduction
2. Theoretical Modeling
2.1. Biot Theory of Poroelastic Medium
2.2. Potential Function of Governing Equation
2.3. Incident Wave Field
2.4. Scattering Wave Field in Soil
2.5. Total Wave Field Potential Function in Saturated Soil
2.6. Scattered Field Wave Potential Function in the Grouting Reinforcement Zone
2.7. Scattered Field Wave Potential Function in the Tunnel Lining
3. Boundary Conditions and the Solutions
3.1. Boundary Conditions
- Stress boundary condition at the surface of saturated soil;
- Displacement boundary condition at the interface between saturated soil and grouting reinforcement zone;
- Stress and displacement boundary conditions at the surface of the grouting reinforcement zone;
- Displacement boundary condition at the interface between grouting reinforcement zone and tunnel lining;
- Stress boundary condition at the interface between grouting reinforcement zone and tunnel lining;
- Stress boundary condition at the free surface of the lining;
3.2. Displacement and Stress Expressions
3.3. Processing of the Coefficients of Iincident R-Wave Potential Functions
3.4. Coefficients Solving
4. Numerical Results and Discussion
4.1. Model Verification
4.2. The Material Properties for Numerical Examples
4.3. Results Analysis
5. Conclusions
- (1)
- Incident frequency governs both the spatial distribution and magnitude of DSCFs and PPCFs. Low-to-medium frequencies (η = 0.25–0.5) amplify stress concentrations at the lining crown (DSCF = 7.2–12.1) and sidewalls (DSCF = 14.2–15.9), while high frequencies (η ≥ 1) shift stress peaks to the invert and foot arch regions (DSCF = 9.6–10.9). PPCF magnitudes diminish by 30% to 50% as frequency η increases from 0.25 to 2, reflecting attenuated pore pressure coupling at shorter wavelengths.
- (2)
- Enhancing the stiffness ratio between the lining and grouting zones can markedly reduce the DSCF and PPCF of the grouting area, while substantially escalating the DSCF of the lining. Furthermore, beyond a specific amplitude (EL/EP > 4), the lining DSCF amplifies by 80% in low-to-medium frequency and the seismic mitigation impact on the pore pressure is constrained in high frequency, while EL/EP < 2 inadequately meets the anti-liquefaction demand for pore pressure may increase several times. Thus, a stiffness ratio of two to four balances stress mitigation and structural practicality.
- (3)
- Augmenting the thickness ratio between the lining and grouting zones can effectively limit both the DSCF and PPCF of the soil in the grouting diffusion zone, while also partially reducing the DSCF of the lining. Nonetheless, above a specific amplitude (δL/δP > 2), the influence on the seismic mitigation efficacy of the lining stress is constrained. The thickness ratio is advised to be within the range of one to two.
- (4)
- Shallow tunnels (h/R1 < 5) exhibit pronounced R-wave-induced stress amplification (peak DSCF = 6.2–11.9 in low-to-mid frequency), with diminishing effects as depth increases (h/R1 > 10). Medium-to-high frequencies (η ≥ 1) induce oscillatory DSCF and PPCF attenuation with depth, underscoring the vulnerability of shallow tunnels to surface wave dominance.
6. Limitations and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
DSCF | Dynamic stress concentration factor |
PPCF | Pore Pressure Concentration Factor |
Appendix A
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Huang, H.; Chang, M.; Zhou, P.; Luo, Y.; Wang, C.; Shen, Y.; Fan, K.; Gao, B. Analytical Solution for Rayleigh Wave-Induced Dynamic Response of Shallow Grouted Tunnels in Saturated Soil. Buildings 2025, 15, 1589. https://doi.org/10.3390/buildings15101589
Huang H, Chang M, Zhou P, Luo Y, Wang C, Shen Y, Fan K, Gao B. Analytical Solution for Rayleigh Wave-Induced Dynamic Response of Shallow Grouted Tunnels in Saturated Soil. Buildings. 2025; 15(10):1589. https://doi.org/10.3390/buildings15101589
Chicago/Turabian StyleHuang, Haifeng, Mingyu Chang, Pengfa Zhou, Yang Luo, Chao Wang, Yusheng Shen, Kaixiang Fan, and Bo Gao. 2025. "Analytical Solution for Rayleigh Wave-Induced Dynamic Response of Shallow Grouted Tunnels in Saturated Soil" Buildings 15, no. 10: 1589. https://doi.org/10.3390/buildings15101589
APA StyleHuang, H., Chang, M., Zhou, P., Luo, Y., Wang, C., Shen, Y., Fan, K., & Gao, B. (2025). Analytical Solution for Rayleigh Wave-Induced Dynamic Response of Shallow Grouted Tunnels in Saturated Soil. Buildings, 15(10), 1589. https://doi.org/10.3390/buildings15101589