Abstract
This paper presents an approximate analytical solution for the vertical dynamic impedance and shaft resistance of rectangular closed diaphragm walls (RCDWs) embedded in a homogeneous soil medium and resting on rock base. The vertical continuous displacement model for the steady-state response of RCDWs is proposed to ensure proper RCDWs-soil contact and satisfy the boundary condition of soil at infinite horizontal distance. The soil horizontal displacement is neglected and the effect is indirectly taken into account by the modification of soil modulus. The functional of system potential and kinetic energies is established via the soil and RCDWs displacement functions. The governing equations of the system and the relevant conditions are obtained and the vertical shaft resistance of RCDWs is established by employing Hamilton’s variational principle for RCDWs-soil system and thin layer element, respectively. Some representative numerical examples are presented to portray the influences of soil core ratio, height–width ratio, diaphragm wall-soil stiffness ratio, and aspect ratio of RCDWs cross sections on the dynamic impedances and stiffness of the shaft resistance.
Funding source: National Natural Science Foundation
Award Identifier / Grant number: 51808112
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 51678145
Funding source: Natural Science Foundation of Jiangsu Province
Award Identifier / Grant number: BK20180155
Acknowledgments
The study presented herein was supported by the National Natural Science Foundation of China (Nos. 51808112; 51678145), the Natural Science Foundation of Jiangsu Province (BK20180155). The authors are grateful for their support.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This research was supported by the National Natural Science Foundation of China (Nos. 51808112; 51678145), the Natural Science Foundation of Jiangsu Province (BK20180155).
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Hamilton’s variational principle is employed for the thin layer element as follows
Substituting Eqs. (28)–(30) into Eq. (A.1), we get
Considering the stress-strain relationship Eqs. (2) and (3) and the modification of replacing (λ*+2μ*) with ημ*, integrate by parts with respect to time, we get
Collecting all the terms associated with δw, the follow equation can be got by equating their sum to zero
Based on Eq. (A.4), we get the expression Eq. (31a) of the dynamic shaft resistance of RCDWs. By contrast to Eq. (A.2), Eq. (A.3), and Eq. (A.4), the shearing stress τ xz and τ yz generate the linear term of the vertical displacement w and the stress increment ((∂σ z /∂x)dz) accounts for the item in proportion to the derivative of the second-order the vertical displacement.
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