Damping considerations for rocking block dynamics using the discrete element method

Abstract

It is well established that unreinforced masonry (URM) buildings develop damage-forming collapse mechanisms during high-intensity earthquakes, with these mechanisms exhibiting large rocking displacements before collapsing. The Discrete Element Method (DEM) of analysis can realistically capture phenomena that involve large movements of elements, resulting in the technique being ideal for simulating the collapse of URM building elements. Consequently, extensive research using DEM to analyse the seismic response of URM buildings and building components has recently been published. However, the variety of reported damping approaches that have apparently led to DEM results that successfully replicate physical observations underscores the need for consistent guidance related to the assignment of damping factors. The Rayleigh damping distribution model implemented in the DEM software 3DEC was used to study the differences between mass proportional (MP) and stiffness proportional (SP) damping configurations. After reviewing phenomena that need to be damped and previous works where damping was implemented, the capabilities and drawbacks of the time-efficient MP damping configuration were studied and the results compared to simulations with SP damping. When considering numerical simulations that incorporated MP damping and led to results that were seemingly well-matched to experimental tests, it was found that the apparent robustness of decisions pertaining to the adopted input parameters was deceptive in most cases. Consequently, SP damping was recommended for all DEM rocking simulations, even though MP damping could be used with satisfactory accuracy in certain situations discussed herein. A pragmatic relationship between both damping strategies was proposed.

Seismic rocking simulation of unreinforced masonry parapets and façades using the discrete element method

“…As E reduces and b increases the energy damping needed to perform a realistic 1SR simulation increases. All the trends studied herein were experimentally validated in Galvez et al 38 . and their consistency was tested in Galvez et al 72 …”

“…A lower value of β can be obtained by adapting the formulation given by DeJong 39 for vertically stacked blocks ( ω Vcrit = ω crit ·$$\sqrt 2 $$) by considering the 1SR problem as a series of horizontally stacked blocks. When substituting in the ω crit formulation, height would be thickness and vice versa, resulting in $${\omega _{Hcrit}} = \sqrt {\frac{{2 \cdot E \cdot {h^2}}}{{m \cdot b}}} $$ (further information can be found in Galvez et al 38 . As seen in Figure 6C, some overlap between the façade and the return walls was only observed for β < 0.006.…”

“…DeJong 39 derived the impact frequency ($${\omega _{crit}} = \sqrt {\frac{{E \cdot {b^2}}}{{m \cdot h}}} $$, where E is the Young's modulus and m is the mass of the block) for impact of the blocks against the ground, which should be critically damped as happens in nature. Once that ω crit is calculated, Equation () is used to obtain the corresponding β , keeping α as zero for the SP damping approach (further information can be found in Galvez, et al 38 …”

“…The trend intercepting the horizontal axis suggests that when all external damping is turned to zero, the model is capable of simulating rocking behavior with damped motion equal to the performance of the Housner model with e H applied. However, this fortunate constraint for the regression is mainly theoretical because when zero viscous damping is selected, high‐frequency undamped numerical noise and impact bouncing trigger unrealistic behavior 38,52 . — 55 If x is plotted against the h/b ratio of the blocks, the power trends observed in Figure 2A can be expressed as: $$\begin{equation}x = y \cdot {\left( {\frac{h}{b}} \right)^{ - 0.37}}\end{equation}$$where y is, as x , a fitting factor.…”

“…Galvez et al 38 studied the possibilities and challenges of an MP damping approach to simulate 3D DEM rocking blocks versus an SP damping approach, reaching the conclusion that SP damping provides greater benefits in terms of accuracy and parametric stability at the cost of higher computation time. For this reason, SP damping was further implemented and studied in the study reported here.…”