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Study on Seismic Ground Motion in the Layered Elastic Ground of S-Wave Incident Under the Thermal Effect

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Abstract

Purpose

Previous studies on the seismic response of the site were carried out under isothermal conditions and homogeneous foundations, but there were few studies on the seismic response of the layered site under thermal effect. However, since the coupling effect between temperature and stress–strain affects the wave propagation, and natural soil usually exists in the form of stratification, which has strong stratification characteristics. Therefore, the paper studies the seismic response problem of S-wave incident on the layered elastic ground under thermal effect.

Methods

Based on the wave propagation theory in thermoelastic media, the conversion matrix of amplitude coefficients in layered foundations is derived according to the Helmholtz vector decomposition principle and the transfer matrix method. The analytical solution of the seismic response of S-wave incident on layered elastic foundation under thermal effect is obtained.

Results

The results show that there is a pretty wide gap between the displacement magnification factors obtained under the two theoretical models considering the thermal effect and not considering the thermal effect; the effect of solid relative heat on displacement magnification factors mainly depends on the thermal stability of soil, and the implications of thermal expansion coefficient on displacement magnification factors is closely related to the size of solid relative heat of soil; in addition, the effects of frequency, medium temperature and the arrangement sequence of the soft and hard soil layers on the displacement magnification factors of the ground surface are significant.

Conclusions

This conclusion promotes the application of fluctuation theory in geotechnical engineering seismic resistance, seismic exploration, and other engineering practices and has significant practical value and guiding significance for actual engineering.

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Data availability

The datasets generated during and analyzed during the current study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors express their gratitude for the financial assistance provided by the Natural Science Foundation of China (No.52168053) and the Qinghai Province Science and Technology Department Project (No. 2024-ZJ-922). The authors also sincerely thank the editors and reviewers for their constructive comments and comments on this article.

Funding

National Natural Science Foundation of China, 52168053, qiang Ma, Qinghai Provincial Department of Science and Technology, 2024-ZJ-922, qiang Ma.

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Authors and Affiliations

  1. School of Civil Engineering and Hydraulic, Qinghai University, **ning, 810016, China

    Yiqi Yang & Qiang Ma

  2. Qinghai Provincial Key Laboratory of Energy-Saving Building Materials and Engineering Safety, **ning, 810016, China

    Qiang Ma

Authors
  1. Yiqi Yang
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  2. Qiang Ma
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Correspondence to Qiang Ma.

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Appendices

Appendix 1

$$T_{si11} = ( - \lambda^{{\text{i}}} k_{{{\text{itp1}}}}^{2} - 2\mu^{{\text{i}}} k_{{{\text{itp1z}}}}^{2} - 3K_{{{\text{bi}}}} \beta_{{{\text{Ti}}}} \delta_{{{\text{iTp1}}}} )\exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )$$
$$T_{si12} = ( - \lambda^{{\text{i}}} k_{{{\text{itp}}2}}^{2} - 2\mu^{{\text{i}}} k_{{{\text{itp2z}}}}^{2} - 3K_{{{\text{bi}}}} \beta_{{{\text{Ti}}}} \delta_{{{\text{iTp2}}}} )\exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )\;T_{si13} = 2\mu^{{\text{i}}} k_{{{\text{itsx}}}}^{{}} k_{{{\text{itsz}}}}^{{}} \exp (ik_{{{\text{itsz}}}} H_{{\text{i}}} )$$
$$T_{{{\text{si14}}}} = ( - \lambda^{{\text{i}}} k_{{{\text{irp1}}}}^{2} - 2\mu^{{\text{i}}} k_{{{\text{irp1z}}}}^{2} - 3K_{{{\text{bi}}}} \beta_{{{\text{Ti}}}} \delta_{{{\text{iTp1}}}} )\exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )\,T_{{{\text{si14}}}} = ( - \lambda^{{\text{i}}} k_{{{\text{irp1}}}}^{2} - 2\mu^{{\text{i}}} k_{{{\text{irp1z}}}}^{2} - 3K_{{{\text{bi}}}} \beta_{{{\text{Ti}}}} \delta_{{{\text{iTp1}}}} )\exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )$$
$$T_{si15} = ( - \lambda^{{\text{i}}} k_{{{\text{irp2}}}}^{2} - 2\mu^{{\text{i}}} k_{{{\text{irp2z}}}}^{2} - 3K_{{{\text{bi}}}} \beta_{{{\text{Ti}}}} \delta_{{{\text{iTp2}}}} )\exp ( - ik_{{{\text{irp2z}}}} H_{{\text{i}}} )\;T_{si16} = - 2\mu^{{\text{i}}} k_{{{\text{irsx}}}}^{{}} k_{irsz}^{{}} \exp ( - ik_{{{\text{irsz}}}} H_{{\text{i}}} )$$
$$T_{si21} = 2\mu^{{\text{i}}} k_{{{\text{itp1x}}}}^{{}} k_{{{\text{itp1z}}}}^{{}} \exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )\;T_{si22} = 2\mu^{{\text{i}}} k_{{{\text{itp2x}}}}^{{}} k_{{{\text{itp2z}}}}^{{}} \exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )\;T_{si23} = \mu^{{\text{i}}} (k_{{{\text{itsz}}}}^{2} - k_{{{\text{itsx}}}}^{2} )\exp (ik_{{{\text{itsz}}}} H_{{\text{i}}} )$$
$$T_{si24} = - 2\mu^{{\text{i}}} k_{{{\text{irp1x}}}}^{{}} k_{{{\text{irp1z}}}}^{{}} \exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )\;T_{si25} = - 2\mu^{{\text{i}}} k_{{{\text{irp2x}}}}^{{}} k_{{{\text{irp2z}}}}^{{}} \exp ( - ik_{{{\text{irp2z}}}} H_{{\text{i}}} )\;T_{si26} = \mu^{{\text{i}}} (k_{{{\text{irsz}}}}^{2} - k_{{{\text{irsx}}}}^{2} )\exp ( - ik_{{{\text{irsz}}}} H_{{\text{i}}} )$$
$$T_{si31} = - ik_{{{\text{itp1x}}}}^{{}} \exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )\;T_{si32} = - ik_{{{\text{itp2x}}}}^{{}} \exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )\;T_{si33} = - ik_{{{\text{itsz}}}}^{{}} \exp (ik_{{{\text{itsz}}}} H_{{\text{i}}} )\;T_{si34} = - ik_{{{\text{irp1x}}}}^{{}} \exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )$$
$$T_{si35} = - ik_{{{\text{irp2x}}}}^{{}} \exp ( - ik_{{{\text{irp2z}}}} H_{i} )\;T_{si36} = ik_{{{\text{irsz}}}}^{{}} \exp ( - ik_{{{\text{irsz}}}} H_{{\text{i}}} )\;T_{si41} = ik_{{{\text{itp1z}}}}^{{}} \exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )\;T_{si42} = ik_{{{\text{itp2z}}}}^{{}} \exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )$$
$$T_{si43} = - ik_{{{\text{itsx}}}}^{{}} \exp (ik_{{{\text{itsz}}}} H_{{\text{i}}} )\;T_{si44} = - ik_{{{\text{irp1z}}}}^{{}} \exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )\;T_{si45} = - ik_{{{\text{irp2z}}}}^{{}} \exp ( - ik_{{{\text{irp2z}}}} H_{{\text{i}}} )\;T_{si46} = - ik_{{{\text{irsx}}}}^{{}} \exp ( - ik_{{{\text{irsz}}}} H_{{\text{i}}} )$$
$$T_{si51} = \delta_{{{\text{iTp1}}}} \exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )\;T_{si52} = \delta_{{{\text{iTp2}}}} \exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )\;T_{si53} = 0\;T_{si54} = \delta_{{{\text{iTp1}}}} \exp ( - ik_{{{\text{irp1z}}}} H_{{\text{i}}} )$$
$$T_{si55} = \delta_{{{\text{iTp2}}}} \exp ( - ik_{{{\text{irp2z}}}} H_{{\text{i}}} )\;T_{si56} = 0\;T_{si61} = K_{{\text{i}}} ik_{{{\text{itp1z}}}} \delta_{{{\text{iTp1}}}} \exp (ik_{{{\text{itp1z}}}} H_{{\text{i}}} )\;T_{si62} = K_{{\text{i}}} ik_{{{\text{itp2z}}}} \delta_{{{\text{iTp2}}}} \exp (ik_{{{\text{itp2z}}}} H_{{\text{i}}} )$$
$$T_{si63} = 0\;T_{si64} = - K_{i} ik_{irp1z} \delta_{iTp1} \exp ( - ik_{irp1z} H_{i} )\;T_{si65} = - K_{i} ik_{irp2z} \delta_{iTp2} \exp ( - ik_{irp2z} H_{i} )\;T_{si66} = 0$$

Appendix 2

$$T_{se11} = - k_{{{\text{isx}}}} k_{{{\text{isz}}}} \exp (ik_{{{\text{isz}}}} H)\;T_{{{\text{se12}}}} = ( - \lambda^{e} k_{rp1}^{2} - 2\mu^{e} k_{{{\text{rp1z}}}}^{{2}} - 3K_{{\text{b}}} \beta_{{\text{T}}} \delta_{{{\text{Tp1}}}}^{e} )\exp ( - ik_{{{\text{rp1z}}}} H)$$
$$T_{se13} = ( - \lambda^{e} k_{{{\text{rp2}}}}^{{2}} - 2\mu^{e} k_{{{\text{rp2z}}}}^{2} - 3K_{{\text{b}}} \beta_{{\text{T}}} \delta_{{{\text{Tp1}}}}^{e} )\exp ( - ik_{{{\text{rp2z}}}} H)\;T_{se14} = - 2\mu^{e} k_{{{\text{rsx}}}}^{{}} k_{{{\text{rsz}}}}^{{}} \exp ( - ik_{{{\text{rsz}}}} H)$$
$$T_{se21} = \mu^{e} (k_{{{\text{isz}}}}^{2} - k_{{{\text{isx}}}}^{2} )\exp (ik_{{{\text{isz}}}} H)\;T_{se22} = - 2\mu^{e} k_{{{\text{rp1x}}}} k_{{{\text{rp1z}}}}^{{}} \exp ( - ik_{{{\text{rp1z}}}} H)\;T_{se23} = - 2\mu^{e} k_{{{\text{rp2x}}}}^{{}} k_{{{\text{rp2z}}}}^{{}} \exp ( - ik_{{{\text{rp2z}}}} H)$$
$$T_{se24} = \mu^{e} (k_{{{\text{rsz}}}}^{2} - k_{{{\text{rsx}}}}^{2} )\exp ( - ik_{{{\text{rsz}}}} H)\;T_{se31} = - ik_{{{\text{isz}}}}^{{}} \exp (ik_{{{\text{isz}}}} H)\;T_{se32} = - ik_{{{\text{rp1x}}}}^{{}} \exp ( - ik_{{{\text{rp1z}}}} H)$$
$$T_{se33} = - ik_{{{\text{rp2x}}}}^{{}} \exp ( - ik_{{{\text{rp2z}}}} H)\;T_{se34} = ik_{{{\text{rsz}}}}^{{}} \exp ( - ik_{{{\text{rsz}}}} H)\;T_{se41} = - ik_{{{\text{isx}}}}^{{}} \exp (ik_{{{\text{isz}}}} H)\;T_{se42} = - ik_{{{\text{rp1z}}}}^{{}} \exp ( - ik_{{{\text{rp1z}}}} H)$$
$$T_{se43} = - ik_{{{\text{rp2z}}}}^{{}} \exp ( - ik_{{{\text{rp2z}}}} H)\;T_{se44} = - ik_{{{\text{rsx}}}}^{{}} \exp ( - ik_{{{\text{rsz}}}} H)\;T_{se51} = 0\;T_{se52} = \delta_{{{\text{Tp1}}}}^{e} \exp ( - ik_{{{\text{rp1z}}}} H)$$
$$T_{se53} = \delta_{{{\text{Tp2}}}}^{e} \exp ( - ik_{{{\text{rp2z}}}} H)\;T_{se54} = 0\;T_{se61} = 0\;T_{se62} = - K^{e} ik_{{{\text{rp1z}}}} \delta_{{{\text{Tp1}}}}^{e} \exp ( - ik_{{{\text{rp1z}}}} H)$$
$$T_{se63} = - K^{e} ik_{{{\text{rp2z}}}} \delta_{{{\text{Tp2}}}}^{e} \exp ( - ik_{{{\text{rp2z}}}} H)\;T_{se64} = 0$$

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Yang, Y., Ma, Q. Study on Seismic Ground Motion in the Layered Elastic Ground of S-Wave Incident Under the Thermal Effect. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01306-2

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