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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="No"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Construction of the spatially varying ground motion field of a bedrock-overburden layer site and its influence on the seismic response of earth-rock dams | Arabian Journal of Geosciences</title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Construction of the spatially varying ground motion field of a bedrock-overburden layer site and its influence on the seismic response of earth-rock dams"/> <meta name="twitter:description" content="Arabian Journal of Geosciences - It is of great significance to consider the effects of the propagation characteristics of seismic waves and overburden layer, to reasonably describe the spatial..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig1_HTML.png"/> <meta name="journal_id" content="12517"/> <meta name="dc.title" content="Construction of the spatially varying ground motion field of a bedrock-overburden layer site and its influence on the seismic response of earth-rock dams"/> <meta name="dc.source" content="Arabian Journal of Geosciences 2021 14:18"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2021-08-28"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2021 Saudi Society for Geosciences"/> <meta name="dc.rights" content="2021 Saudi Society for Geosciences"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="It is of great significance to consider the effects of the propagation characteristics of seismic waves and overburden layer, to reasonably describe the spatial variations in the ground motion of overburden layer sites, and to study their influence on the seismic response of earth-rock dams. First, based on the stress equilibrium and displacement compatibility conditions, the reflection and transmission coefficients at the interface of the bedrock-overburden layer under the oblique incidence of P and SV waves are deduced, respectively. Second, based on the condition that the measured ground motion or design ground motion of the control point is the same as the free field motion, the time series of the incident waves is deduced by a Fourier transform and an inverse transform. Furthermore, the spatially varying ground motion field of the bedrock-overburden layer site under the combination of oblique incidence of P and SV waves is constructed and compared with two special cases that combine the vertical incidence of the wave to the bedrock-overburden layer site and combined oblique incidence of the waves to the bedrock site. The results show that the spatial variations in the ground motion are mainly caused by the oblique incidence of seismic waves, which results in changes in the wave front, wave type, first arrival time, and other propagation characteristics at different spatial positions, resulting in variations in particle motion. The existence of an overburden layer increases the spatial variations in the ground motion. Finally, the seismic response of the high earth-rockfill dam is analyzed. The numerical analysis shows that the spatial variations in the ground motion will cause the symmetry axis of the acceleration distribution on the foundation surface to deflect and shift, and the deflection angle is related to the incident angle of the seismic wave."/> <meta name="prism.issn" content="1866-7538"/> <meta name="prism.publicationName" content="Arabian Journal of Geosciences"/> <meta name="prism.publicationDate" content="2021-08-28"/> <meta name="prism.volume" content="14"/> <meta name="prism.number" content="18"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="20"/> <meta name="prism.copyright" content="2021 Saudi Society for Geosciences"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s12517-021-08295-6"/> <meta name="prism.doi" content="doi:10.1007/s12517-021-08295-6"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s12517-021-08295-6.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s12517-021-08295-6"/> <meta name="citation_journal_title" content="Arabian Journal of Geosciences"/> <meta name="citation_journal_abbrev" content="Arab J Geosci"/> <meta name="citation_publisher" content="Springer International Publishing"/> <meta name="citation_issn" content="1866-7538"/> <meta name="citation_title" content="Construction of the spatially varying ground motion field of a bedrock-overburden layer site and its influence on the seismic response of earth-rock dams"/> <meta name="citation_volume" content="14"/> <meta name="citation_issue" content="18"/> <meta name="citation_publication_date" content="2021/09"/> <meta name="citation_online_date" content="2021/08/28"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="20"/> <meta name="citation_article_type" content="Original Paper"/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s12517-021-08295-6"/> <meta name="DOI" content="10.1007/s12517-021-08295-6"/> <meta name="size" content="461976"/> <meta name="citation_doi" content="10.1007/s12517-021-08295-6"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s12517-021-08295-6&api_key="/> <meta name="description" content="It is of great significance to consider the effects of the propagation characteristics of seismic waves and overburden layer, to reasonably describe the sp"/> <meta name="dc.creator" content="Wang, Fei"/> <meta name="dc.creator" content="Song, Zhiqiang"/> <meta name="dc.creator" content="Liu, Yunhe"/> <meta name="dc.creator" content="Luo, Bohua"/> <meta name="dc.creator" content="Zhang, Wenbing"/> <meta name="dc.subject" content="Earth Sciences, general"/> <meta name="citation_reference" content="Alves SW (2004) Nonlinear analysis of Pacoima dam with spatially non-uniform ground motion. 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First, based on the stress equilibrium and displacement compatibility conditions, the reflection and transmission coefficients at the interface of the bedrock-overburden layer under the oblique incidence of P and SV waves are deduced, respectively. Second, based on the condition that the measured ground motion or design ground motion of the control point is the same as the free field motion, the time series of the incident waves is deduced by a Fourier transform and an inverse transform. Furthermore, the spatially varying ground motion field of the bedrock-overburden layer site under the combination of oblique incidence of P and SV waves is constructed and compared with two special cases that combine the vertical incidence of the wave to the bedrock-overburden layer site and combined oblique incidence of the waves to the bedrock site. The results show that the spatial variations in the ground motion are mainly caused by the oblique incidence of seismic waves, which results in changes in the wave front, wave type, first arrival time, and other propagation characteristics at different spatial positions, resulting in variations in particle motion. The existence of an overburden layer increases the spatial variations in the ground motion. Finally, the seismic response of the high earth-rockfill dam is analyzed. The numerical analysis shows that the spatial variations in the ground motion will cause the symmetry axis of the acceleration distribution on the foundation surface to deflect and shift, and the deflection angle is related to the incident angle of the seismic wave."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig1_HTML.png"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" 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First, based on the stress equilibrium and displacement compatibility conditions, the reflection and transmission coefficients at the interface of the bedrock-overburden layer under the oblique incidence of P and SV waves are deduced, respectively. Second, based on the condition that the measured ground motion or design ground motion of the control point is the same as the free field motion, the time series of the incident waves is deduced by a Fourier transform and an inverse transform. Furthermore, the spatially varying ground motion field of the bedrock-overburden layer site under the combination of oblique incidence of P and SV waves is constructed and compared with two special cases that combine the vertical incidence of the wave to the bedrock-overburden layer site and combined oblique incidence of the waves to the bedrock site. The results show that the spatial variations in the ground motion are mainly caused by the oblique incidence of seismic waves, which results in changes in the wave front, wave type, first arrival time, and other propagation characteristics at different spatial positions, resulting in variations in particle motion. The existence of an overburden layer increases the spatial variations in the ground motion. Finally, the seismic response of the high earth-rockfill dam is analyzed. The numerical analysis shows that the spatial variations in the ground motion will cause the symmetry axis of the acceleration distribution on the foundation surface to deflect and shift, and the deflection angle is related to the incident angle of the seismic wave.","datePublished":"2021-08-28T00:00:00Z","dateModified":"2021-08-28T00:00:00Z","pageStart":"1","pageEnd":"20","sameAs":"https://doi.org/10.1007/s12517-021-08295-6","keywords":["Earth-rock dam","Bedrock-overburden layer site","Spatial variations in ground motion","Seismic response","Earth 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Journal of Geosciences","issn":["1866-7538","1866-7511"],"volumeNumber":"14","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer International Publishing","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Fei Wang","affiliation":[{"name":"Xi’an University of Technology","address":{"name":"State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Zhiqiang Song","affiliation":[{"name":"Xi’an University of Technology","address":{"name":"State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"email":"zqsong@xaut.edu.cn","@type":"Person"},{"name":"Yunhe Liu","affiliation":[{"name":"Xi’an University of Technology","address":{"name":"State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Bohua Luo","affiliation":[{"name":"Xi’an University of Technology","address":{"name":"State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, China","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Wenbing Zhang","affiliation":[{"name":"Hohai University","address":{"name":"State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"}],"isAccessibleForFree":false,"hasPart":{"isAccessibleForFree":false,"cssSelector":".main-content","@type":"WebPageElement"},"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" 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href="/article/10.1007/s12517-021-08295-6/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>It is of great significance to consider the effects of the propagation characteristics of seismic waves and overburden layer, to reasonably describe the spatial variations in the ground motion of overburden layer sites, and to study their influence on the seismic response of earth-rock dams. First, based on the stress equilibrium and displacement compatibility conditions, the reflection and transmission coefficients at the interface of the bedrock-overburden layer under the oblique incidence of P and SV waves are deduced, respectively. Second, based on the condition that the measured ground motion or design ground motion of the control point is the same as the free field motion, the time series of the incident waves is deduced by a Fourier transform and an inverse transform. Furthermore, the spatially varying ground motion field of the bedrock-overburden layer site under the combination of oblique incidence of P and SV waves is constructed and compared with two special cases that combine the vertical incidence of the wave to the bedrock-overburden layer site and combined oblique incidence of the waves to the bedrock site. The results show that the spatial variations in the ground motion are mainly caused by the oblique incidence of seismic waves, which results in changes in the wave front, wave type, first arrival time, and other propagation characteristics at different spatial positions, resulting in variations in particle motion. The existence of an overburden layer increases the spatial variations in the ground motion. Finally, the seismic response of the high earth-rockfill dam is analyzed. The numerical analysis shows that the spatial variations in the ground motion will cause the symmetry axis of the acceleration distribution on the foundation surface to deflect and shift, and the deflection angle is related to the incident angle of the seismic wave.</p></div></div></section> <div class="c-notes"> <p class="c-notes__text c-status-message--info"> <svg width="24" height="24" focusable="false" role="img" aria-hidden="true" class="c-status-message__icon"> <use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-info-filled-medium"></use> </svg> This is a preview of subscription content, <a id="test-login-banner-link" href="//wayf.springernature.com?redirect_uri=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs12517-021-08295-6%3Ferror%3Dcookies_not_supported%26code%3D6f444b24-2721-4101-b42d-527f7a8342fa" data-track="click" data-track-action="login" data-track-label="link" class="c-preview-message__link">log in via an institution</a> <svg width="16" 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srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig8_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig8_HTML.png" alt="" loading="lazy" width="312" height="179"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-9"><figure><figcaption><b id="Fig9" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 9</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig9_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig9_HTML.png" alt="" loading="lazy" width="312" height="163"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-10"><figure><figcaption><b id="Fig10" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 10</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig10_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig10_HTML.png" alt="" loading="lazy" width="312" height="115"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-11"><figure><figcaption><b id="Fig11" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 11</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig11_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig11_HTML.png" alt="" loading="lazy" width="312" height="115"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-12"><figure><figcaption><b id="Fig12" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 12</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig12_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig12_HTML.png" alt="" loading="lazy" width="312" height="117"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-13"><figure><figcaption><b id="Fig13" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 13</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig13_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig13_HTML.png" alt="" loading="lazy" width="312" height="115"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-14"><figure><figcaption><b id="Fig14" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 14</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig14_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig14_HTML.png" alt="" loading="lazy" width="312" height="95"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-15"><figure><figcaption><b id="Fig15" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 15</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig15_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig15_HTML.png" alt="" loading="lazy" width="312" height="120"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-16"><figure><figcaption><b id="Fig16" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 16</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig16_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig16_HTML.png" alt="" loading="lazy" width="312" height="120"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-17"><figure><figcaption><b id="Fig17" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 17</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig17_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig17_HTML.png" alt="" loading="lazy" width="312" height="117"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-18"><figure><figcaption><b id="Fig18" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 18</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig18_HTML.png?as=webp"><img src="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig18_HTML.png" alt="" loading="lazy" width="312" height="116"></picture></div></div></figure></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-19"><figure><figcaption><b id="Fig19" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 19</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><picture><source type="image/webp" srcset="//media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs12517-021-08295-6/MediaObjects/12517_2021_8295_Fig19_HTML.png?as=webp"><img 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availability"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Code availability</h2><div class="c-article-section__content" id="Ack1-content"><p>Not applicable</p></div></div></section><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>This work was supported by the Key Program of National Natural Science Foundation of China [grant number 52039008] and the National Natural Science Foundation of China [grant numbers 51779208, 51479165].</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, 710048, China</p><p class="c-article-author-affiliation__authors-list">Fei Wang, Zhiqiang Song, Yunhe Liu & Bohua Luo</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, 210098, China</p><p class="c-article-author-affiliation__authors-list">Wenbing Zhang</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li 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href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Bohua%20Luo%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Wenbing-Zhang-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">Wenbing Zhang</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=Wenbing%20Zhang" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Wenbing%20Zhang" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Wenbing%20Zhang%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p><b>Fei Wang</b>: Conceptualization, methodology, software, data curation, writing—original draft, validation, writing—review and editing. <b>Zhiqiang Song</b>: Formal analysis, supervision, funding acquisition, project administration. <b>Yunhe Liu</b>: Formal analysis, funding acquisition. <b>Bohua Luo</b>: Software, visualization, investigation. <b>Wenbing Zhang:</b> writing—review and editing, visualization.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:zqsong@xaut.edu.cn">Zhiqiang Song</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar1">Conflict of interest</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p>Responsible Editor: Longjun Dong</p></div></div></section><section aria-labelledby="appendices"><div class="c-article-section" id="appendices-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="appendices">Appendices</h2><div class="c-article-section__content" id="appendices-content"><h3 class="c-article__sub-heading" id="App1">Appendix 1</h3> <div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ a=-\sin 2{\theta}_0\left(2\cos {\theta}_0\cos {\theta}_2{c}_{\mathrm{S}1}-\cos 2{\theta}_0{c}_{\mathrm{P}1}\right)/\left(\cos {\theta}_0\cos {\theta}_2-\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{S}1}+2\sin 2{\theta}_0, $$</span></div><div class="c-article-equation__number"> (16) </div></div> <div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}b=-\sin 2{\theta}_0\left(\cos {\theta}_0\cos {\theta}_2\frac{c_{\mathrm{P}2}}{c_{\mathrm{P}1}}+\cos {\theta}_2\cos {\theta}_3\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)+\cos 2{\theta}_0\left(\sin {\theta}_0\cos {\theta}_0+\cos {\theta}_0\cos {\theta}_3\right){c}_{\mathrm{P}1}\\ {}/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{S}1}+\sin 2{\theta}_0{c}_{\mathrm{P}2}/{c}_{\mathrm{P}1}+\sin 2{\theta}_3{\rho}_2{c}_{\mathrm{S}2}^2{c}_{\mathrm{P}1}/{\rho}_1{c}_{\mathrm{S}1}^2{c}_{\mathrm{P}2},\end{array}} $$</span></div><div class="c-article-equation__number"> (17) </div></div> <div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}c=-\cos {\theta}_2\left(2{\cos}^2{\theta}_0\cos {\theta}_4-\sin 2{\theta}_0\sin {\theta}_4\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)+\cos 2{\theta}_0\left(\cos {\theta}_0\cos {\theta}_4-\sin {\theta}_0\sin {\theta}_4\right){c}_{\mathrm{P}1}\\ {}/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{S}1}+2\cos {\theta}_0\cos {\theta}_4+\cos 2{\theta}_4{\rho}_2{c}_{\mathrm{S}2}{c}_{\mathrm{P}1}/{\rho}_1{c}_{\mathrm{S}1}^2,\end{array}} $$</span></div><div class="c-article-equation__number"> (18) </div></div> <div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ d=2\cos {\theta}_0\cos {\theta}_2\cos 2{\theta}_2/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)+\sin 2{\theta}_0\sin 2{\theta}_2{c}_{\mathrm{S}1}/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{P}1}, $$</span></div><div class="c-article-equation__number"> (19) </div></div> <div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}e=\cos 2{\theta}_2\left(\cos {\theta}_0\cos {\theta}_2\frac{c_{\mathrm{P}2}}{c_{\mathrm{P}1}}+\cos {\theta}_2\cos {\theta}_3\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)+\sin 2{\theta}_2\left(\cos {\theta}_0\sin {\theta}_3+\sin {\theta}_0\cos {\theta}_3\right){c}_{\mathrm{S}1}\\ {}/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{P}1}-\cos 2{\theta}_2{c}_{\mathrm{P}2}/{c}_{\mathrm{P}1}+\cos 2{\theta}_4{\rho}_2{c}_{\mathrm{P}2}/{\rho}_1{c}_{\mathrm{P}1},\end{array}} $$</span></div><div class="c-article-equation__number"> (20) </div></div> <div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}f=-\cos 2{\theta}_2\left(\cos {\theta}_2\sin {\theta}_4+\sin {\theta}_2\cos {\theta}_4\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)+\sin 2{\theta}_2\left(\cos {\theta}_0\cos {\theta}_4-\sin {\theta}_0\sin {\theta}_4\right){c}_{\mathrm{P}1}\\ {}/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right)/{c}_{\mathrm{S}1}-\sin 2{\theta}_4{\rho}_2{c}_{\mathrm{S}2}/{\rho}_1{c}_{\mathrm{P}1},\end{array}} $$</span></div><div class="c-article-equation__number"> (21) </div></div> <div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ g=\sin 2{\theta}_0/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right), $$</span></div><div class="c-article-equation__number"> (22) </div></div> <div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ h=\left(\cos {\theta}_0\sin {\theta}_3+\sin {\theta}_0\cos {\theta}_3\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right), $$</span></div><div class="c-article-equation__number"> (23) </div></div> <div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ i=\left(\cos {\theta}_0\cos {\theta}_4\hbox{-} \sin {\theta}_0\sin {\theta}_4\right)/\left(\cos {\theta}_0\cos {\theta}_2+\sin {\theta}_0\sin {\theta}_2\right). $$</span></div><div class="c-article-equation__number"> (24) </div></div> <h3 class="c-article__sub-heading" id="App2">Appendix 2</h3> <div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \left\{\begin{array}{c}{B}_1=1+\left(\sin {\theta}_2^{\hbox{'}}{B}_2+\cos {\theta}_3^{\hbox{'}}{B}_3+\sin {\theta}_4^{\hbox{'}}{B}_4\right)/\cos {\theta}_0^{\hbox{'}}\\ {}{B}_2={uB}_3+{vB}_4\\ {}{B}_3=-{jqB}_4/\left( kq- mn\right)\\ {}{B}_4= jn/\left( kq- mn\right)\end{array}\right., $$</span></div><div class="c-article-equation__number"> (25) </div></div><p>in which </p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ j=2\cos 2{\theta}_0^{\hbox{'}}, $$</span></div><div class="c-article-equation__number"> (25.1) </div></div><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}k=\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\sin {\theta}_3^{\hbox{'}}\right)\cos 2{\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/\cos {\theta}_0^{\hbox{'}}\\ {}+\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\sin {\theta}_3^{\hbox{'}}\right)\sin 2{\theta}_2^{\hbox{'}}{c}_{\mathrm{S}1}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/{c}_{\mathrm{P}1}\\ {}+\cos 2{\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}/\cos {\theta}_0^{\hbox{'}}+{\rho}_2{c}_{\mathrm{S}2}\cos 2{\theta}_3^{\hbox{'}}/{\rho}_1{c}_{\mathrm{S}1},\end{array}} $$</span></div><div class="c-article-equation__number"> (25.2) </div></div><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}m=-\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}-\cos {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}\right)\cos 2{\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/\cos {\theta}_0^{\hbox{'}}\\ {}+\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}-\cos {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}\right)\sin 2{\theta}_2^{\hbox{'}}{c}_{\mathrm{S}1}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/{c}_{\mathrm{P}1}\\ {}+\cos 2{\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}/\cos {\theta}_0^{\hbox{'}}+{\rho}_2{c}_{\mathrm{S}2}^2\sin 2{\theta}_4^{\hbox{'}}/{\rho}_1{c}_{\mathrm{S}1}{c}_{\mathrm{P}2},\end{array}} $$</span></div><div class="c-article-equation__number"> (25.3) </div></div><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}n=-\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\sin {\theta}_3^{\hbox{'}}\right)\sin 2{\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/\cos {\theta}_0^{\hbox{'}}\\ {}-\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\sin {\theta}_3^{\hbox{'}}\right)\cos 2{\theta}_2^{\hbox{'}}{c}_{\mathrm{P}1}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/{c}_{\mathrm{S}1}\\ {}+\sin 2{\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}/\cos {\theta}_0^{\hbox{'}}-{\rho}_2{c}_{\mathrm{S}2}^2\sin 2{\theta}_3^{\hbox{'}}/{\rho}_1{c}_{\mathrm{S}1},\end{array}} $$</span></div><div class="c-article-equation__number"> (25.4) </div></div><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ {\displaystyle \begin{array}{l}q=-\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}-\cos {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}\right)\sin 2{\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/\cos {\theta}_0^{\hbox{'}}\\ {}-\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}-\cos {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}\right)\cos 2{\theta}_0^{\hbox{'}}{c}_{\mathrm{P}1}/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right)/{c}_{\mathrm{S}1}\\ {}+\sin 2{\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}/\cos {\theta}_0^{\hbox{'}}+{\rho}_2{c}_{\mathrm{P}2}\cos 2{\theta}_3^{\hbox{'}}/{\rho}_1{c}_{\mathrm{S}1},\end{array}} $$</span></div><div class="c-article-equation__number"> (25.5) </div></div><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ u=-\left(\sin {\theta}_0^{\hbox{'}}\cos {\theta}_3^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\sin {\theta}_3^{\hbox{'}}\right)/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right), $$</span></div><div class="c-article-equation__number"> (25.6) </div></div><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ v=-\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_4^{\hbox{'}}-\cos {\theta}_0^{\hbox{'}}\cos {\theta}_4^{\hbox{'}}\right)/\left(\sin {\theta}_0^{\hbox{'}}\sin {\theta}_2^{\hbox{'}}+\cos {\theta}_0^{\hbox{'}}\cos {\theta}_2^{\hbox{'}}\right). $$</span></div><div class="c-article-equation__number"> (25.7) </div></div><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \left\{\begin{array}{c}{B}_5=\left[\left({\rho}_2{c}_{\mathrm{S}2}\right)/\left({\rho}_1{c}_{\mathrm{S}1}\right)-1\right]/\left[\left({\rho}_2{c}_{\mathrm{S}2}\right)/\left({\rho}_1{c}_{\mathrm{S}1}\right)+1\right]\\ {}{B}_6=0\\ {}{B}_7=2/\left[\left({\rho}_2{c}_{\mathrm{S}2}\right)/\left({\rho}_1{c}_{\mathrm{S}1}\right)+1\right]\\ {}{B}_8=0\end{array},\right. $$</span></div><div class="c-article-equation__number"> (26) </div></div><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \left\{\begin{array}{c}{B}_9=2{c}_{\mathrm{P}1}{c}_{\mathrm{S}1}\sin 2{\theta}_0^{\hbox{'}}\cos 2{\theta}_0^{\hbox{'}}/\left({c}_{\mathrm{S}1}^2\sin 2{\theta}_0^{\hbox{'}}\sin 2{\theta}_2^{\hbox{'}}+{c}_{\mathrm{P}1}^2{\cos}^22{\theta}_0^{\hbox{'}}\right)\\ {}{B}_{10}=-\left({c}_{\mathrm{S}1}^2\sin 2{\theta}_0^{\hbox{'}}\sin 2{\theta}_2^{\hbox{'}}-{c}_{\mathrm{P}1}^2{\cos}^22{\theta}_0^{\hbox{'}}\right)/\left({c}_{\mathrm{S}1}^2\sin 2{\theta}_0^{\hbox{'}}\sin 2{\theta}_2^{\hbox{'}}+{c}_{\mathrm{P}1}^2{\cos}^22{\theta}_0^{\hbox{'}}\right)\\ {}{B}_{11}=0\\ {}{B}_{12}=0\end{array}\right.. $$</span></div><div class="c-article-equation__number"> (27) </div></div></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"><p class="c-article-rights"><a data-track="click" data-track-action="view rights 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Wang, F., Song, Z., Liu, Y. <i>et al.</i> Construction of the spatially varying ground motion field of a bedrock-overburden layer site and its influence on the seismic response of earth-rock dams. <i>Arab J Geosci</i> <b>14</b>, 1914 (2021). https://doi.org/10.1007/s12517-021-08295-6</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s12517-021-08295-6?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2021-04-24">24 April 2021</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2021-08-19">19 August 2021</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2021-08-28">28 August 2021</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1007/s12517-021-08295-6</span></p></li></ul><div data-component="share-box"><div 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