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Geomechanics and Engineering Volume 19, Number 5, December10 2019 , pages 463-472 DOI: https://doi.org/10.12989/gae.2019.19.5.463 |
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Characteristic equation solution of nonuniform soil deposit: An energy-based mode perturbation method |
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Danguang Pan, Wenyan Lu, Qingjun Chen and Pan Lu
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| Abstract | ||
| The mode perturbation method (MPM) is suitable and efficient for solving the eigenvalue problem of a nonuniform soil deposit whose property varies with depth. However, results of the MPM do not always converge to the exact solution, when the variation of soil deposit property is discontinuous. This discontinuity is typical because soil is usually made up of sedimentary layers of different geologic materials. Based on the energy integral of the variational principle, a new mode perturbation method, the energy-based mode perturbation method (EMPM), is proposed to address the convergence of the perturbation solution on the natural frequencies and the corresponding mode shapes and is able to find solution whether the soil properties are continuous or not. First, the variational principle is used to transform the variable coefficient differential equation into an equivalent energy integral equation. Then, the natural mode shapes of the uniform shear beam with same height and boundary conditions are used as Ritz function. The EMPM transforms the energy integral equation into a set of nonlinear algebraic equations which significantly simplifies the eigenvalue solution of the soil layer with variable properties. Finally, the accuracy and convergence of this new method are illustrated with two case study examples. Numerical results show that the EMPM is more accurate and convergent than the MPM. As for the mode shapes of the uniform shear beam included in the EMPM, the additional 8 modes of vibration are sufficient in engineering applications. | ||
| Key Words | ||
| soil deposit; perturbation method; characteristic equation; variational principle; dynamic characteristics | ||
| Address | ||
| Danguang Pan: 1.) Department of Civil Engineering, University of Science and Technology Beijing, Beijing, China 2.) State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China Wenyan Lu: Department of Civil Engineering, University of Science and Technology Beijing, Beijing, China Qingjun Chen: State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China Pan Lu: Department of Transportation and Logistics/Upper Great Plains Transportation Institute, North Dakota State University, Fargo, North Dakota, U.S.A. | ||