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

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.