<p>For Eurocode_8 as well as for other similar design standards, acceptance criteria are based on the concept of ductility capacity of structures or structural elements. However, in the process of Eurocode_8 generation, an issue was raised because the concept is not adequate for rigid-plastic behaviour, as exhibited by some ancillary elements. Consequently, a specific formula was introduced in the code to deal with this situation. The purpose of this communication is to present the scientific background of this formula. In a first step, the response of a sliding block, with Coulomb friction model, on a support animated with a harmonic motion is considered. Three regimes of response are identified, designated by stick regime, stick-slip regime (sliding phases separated by sicking phases), and slip-slip regime. Considering a non-dimensional input motion level, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda = 1\)</EquationSource> </InlineEquation> at the transition between stick and stick-slip regimes, it is analytically established that the slip-slip regime is attained for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">\({\lambda _1} = 1.862\)</EquationSource> </InlineEquation>. Concurrently, a non-dimensional sliding <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{\text{ }}\left( \lambda \right)\)</EquationSource> </InlineEquation> is introduced, an expression of it established for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda &lt; {\lambda _1}\)</EquationSource> </InlineEquation>, and turned into the Eurocode_8 formula. In a second step, the same approach is applied, considering a set of 100 natural accelerograms. For every of them, the friction coefficient is calibrated so that the stick-slip regimes is attained. Then, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1^0\)</EquationSource> </InlineEquation> is the amplification factor that should be applied on the input motion so that the slip-slip regime is attained. Concurrently, an expression of the induced non-dimensional sliding, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{\text{ }}^0\left( \lambda \right)\)</EquationSource> </InlineEquation>, is established. In a third step, the same approach applies with seismic input motions transferred to floors 2 and 5 of a 5-storey structure. Corresponding <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1^2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda _1^5\)</EquationSource> </InlineEquation> values are identified as well as non-dimensional sliding <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq10.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{\text{ }}^2\left( \lambda \right)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10518_2025_2240_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{\text{ }}^5\left( \lambda \right)\)</EquationSource> </InlineEquation>. Based on steps 2 and 3 outputs, it is eventually concluded that the Eurocode_8 formula need not being amended.</p>

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Harmonic response of sliding blocks and Eurocode_8 formula for rigid-plastic ancillary elements

  • P. Labbé,
  • P. Su

摘要

For Eurocode_8 as well as for other similar design standards, acceptance criteria are based on the concept of ductility capacity of structures or structural elements. However, in the process of Eurocode_8 generation, an issue was raised because the concept is not adequate for rigid-plastic behaviour, as exhibited by some ancillary elements. Consequently, a specific formula was introduced in the code to deal with this situation. The purpose of this communication is to present the scientific background of this formula. In a first step, the response of a sliding block, with Coulomb friction model, on a support animated with a harmonic motion is considered. Three regimes of response are identified, designated by stick regime, stick-slip regime (sliding phases separated by sicking phases), and slip-slip regime. Considering a non-dimensional input motion level, \(\lambda \) , with \(\lambda = 1\) at the transition between stick and stick-slip regimes, it is analytically established that the slip-slip regime is attained for \({\lambda _1} = 1.862\) . Concurrently, a non-dimensional sliding \(H_{\text{ }}\left( \lambda \right)\) is introduced, an expression of it established for \(\lambda < {\lambda _1}\) , and turned into the Eurocode_8 formula. In a second step, the same approach is applied, considering a set of 100 natural accelerograms. For every of them, the friction coefficient is calibrated so that the stick-slip regimes is attained. Then, \(\lambda _1^0\) is the amplification factor that should be applied on the input motion so that the slip-slip regime is attained. Concurrently, an expression of the induced non-dimensional sliding, \(H_{\text{ }}^0\left( \lambda \right)\) , is established. In a third step, the same approach applies with seismic input motions transferred to floors 2 and 5 of a 5-storey structure. Corresponding \(\lambda _1^2\) and \(\lambda _1^5\) values are identified as well as non-dimensional sliding \(H_{\text{ }}^2\left( \lambda \right)\) and \(H_{\text{ }}^5\left( \lambda \right)\) . Based on steps 2 and 3 outputs, it is eventually concluded that the Eurocode_8 formula need not being amended.