A shift approach for the dynamic response of rigid‐plastic systems

Abstract

A shift approach is presented for evaluating and interpreting the response of rigid-perfectly plastic singledegree-of-freedom systems to dynamic loading. Scaling laws for such systems are, as the term suggests, multiplicative in nature, relating peak dynamic response to products of key problem parameters such as linear spectral coordinates, force reduction coefficient and peak values of the excitation and its time derivatives. Contrary to classical laws, the proposed approach is additive, imposing a shift in the ordinates and the abscissa of the excitation function by means of a set of parameters uniquely related to the yielding resistance of the system. The dynamic response is then obtained by integrating the modified excitation function in a linear-like manner within a particular yielding branch, for the nonlinearity is incorporated into the forcing term. The mathematical validity of the approach is demonstrated analytically and its importance is highlighted for systems with symmetric yielding resistance subjected to near-fault earthquake motions. The modified excitation function may be discontinuous between different yielding branches and relates uniquely to the development of plastic deformation. It is thereby referred to as Plastic Input Motion (PIM). It is shown that the ordinates and the duration of this function may be significantly (yet not necessarily) smaller than those of the original ground motion depending on yield strength. The relationship of the proposed approach to the existing methods and parameters of earthquake engineering such as Newmark's sliding block and relative ground acceleration, is discussed. A SHIFT APPROACH FOR THE DYNAMIC RESPONSE 853 Figure 4. Definition of integration limits and Plastic Input Acceleration (PIA) according to the proposed approach.and the complexity of the associated time histories, all available methods encompass some degree of ambiguity. By virtue of the physics of a rigid-perfectly plastic system, a simple unambiguous procedure is employed in this study, as depicted in Figure 4: First, acceleration peaks associated with large velocity content are identified. The initiation of the pulse is set at the time wherë u g = |a y |, which coincides with the initiation of yielding and which can be readily established on the acceleration record (point 1, Figure 4(a)). Second, the ordinates and the abscissa of the time history are shifted by a y and t y , respectively, to set the origin of the new axes at the beginning of the pulse (point 1, Figure 4(b)). Third, integration of the modified ground acceleration is carried out considering zero initial conditions, up to the point where the modified ground velocity function becomes zero. This time instant (point 2, Figure 4(b)) corresponds to a possible end of yielding and manifests itself in the form of equal areas of opposite signs in the modified record. Fourth, if the acceleration level in the original record at the zero-velocity time instant (point 2) is smaller than a y (this case does not correspond to the one shown in F...

Rocking amplification and strong‐motion duration

“…With reference to Figure 2 (right column), the summation of the time intervals of the shaded areas provides the uniform duration t uni , whereas the summation of the pertinent areas yields the cumulative absolute velocity of exceedance CAV exc . Interestingly, the t uni conceptually aligns with the plastic input motion approach proposed by Voyagaki et al 55 for yielding oscillators, whereas the CAV exc is similar to the cumulative impact pulse velocity of Gelagoti et al 37 introduced for elastic frames rocking on deforming soil.…”

Dynamic response analysis of nonlinear secondary oscillators to idealised seismic pulses

“…Hong and Liu refined the frictional oscillator model formulation, deriving exact solutions to simple harmonic loading 11 . More recently, Voyagaki et al 12 proposed a new response evaluation method and interpretation of rigid‐plastic block dynamics by imposing a shift in the ordinates and the abscissa of the excitation function. The authors also presented analytical 13 and numerical 14 solutions for the response to idealised near‐fault acceleration pulses.…”

Stochastic Response Quantification of Fixed-Base and Base-Isolated Rigid-Plastic Blocks

“…Of interest is the case of the idealised sliding block, exhibiting rigid-plastic behaviour, a widely accepted model representing a broad range of structural and geotechnical systems, including buildings on moving foundation, equipment, retaining walls, slopes and masonry. Several studies have been devoted to the deterministic seismic analysis of such blocks, including those dealing with idealised ground acceleration pulses [5][6][7] and recorded earthquake ground motions [8]. The stochastic response of such systems has been examined in presence of white noise [9] and filtered white noise, characterised by the Kanai-Tajimi [10,11] power spectrum [12][13][14][15][16][17], mostly for applications dealing with rigid structures resting on a frictional foundation.…”