Pile up blocks until the tower collapses. But why does the tower always end up collapsing? Is it possible that it can be built ad infinitum? A study published in the International Journal of Solids and Structures explores the fascinating and complex dynamics of the stacking of blocks subjected to hazards. Carried out by Vincent Denoël, an engineer at the University of Liège, this research looks at the stochastic stability of stacks, providing crucial insights for engineering, construction and materials science.
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magine a tower of kaplas with each block slightly out of alignment. As the tower rises, the misalignment increases until an inevitable breaking point is reached, a situation familiar to all kapla fans. This simple phenomenon, which echoes the games of our childhood, raises a question of stability: how high can the blocks be stacked before the structure collapses? Vincent Denoël, an engineer in the SSD (Structural & Stochastic Dynamics) laboratory at the University of Liège, set out to gain a better understanding of these failures in order to develop a statistical model that could accurately predict the critical height and failure points by looking at a random stacking of blocks in a configuration where each block is slightly misaligned. So how do small positioning errors when stacking blocks influence the overall stability of the stack?
Modelled as Gaussian random variables*, these errors lead to progressive misalignments that inevitably lead to collapse," explains the researcher. This problem goes beyond the simple child's game of stacking objects; it is a scientific challenge with major implications. From the construction of dry masonry walls to the optimisation of automated storage systems, understanding the probabilistic nature of these collapses can improve safety and efficiency in a variety of fields."
Vincent Denoël has modelled this as a 'first-pass problem', a probabilistic approach to analysing the conditions leading to system failure. "As blocks are added, random misalignments gradually modify the stack's centre of gravity. When this exceeds a critical limit, the stack collapses." This approach revealed two main areas of vulnerability: the base of the stack, where cumulative errors become unsustainable, and an intermediate zone, where hidden instabilities accumulate insidiously.
The maximum height of a pile before it collapses is inversely proportional to the square of the amplitude of the positioning errors. Thus, small errors allow much greater heights to be reached, while larger errors lead to rapid collapse. Monte Carlo* simulations, used to validate the theoretical model, were used to visualise the behaviour of the piles. These simulations confirmed the bimodal nature of the failure points, for a given drop height, and highlighted the distribution of weak interfaces within the piles.
This research is not limited to abstract modelling. It has many practical applications. In construction, for example, the results may help to design more stable structures, even in the presence of small imperfections. In automated warehouses, where precise stacking of goods is essential, the probabilistic models resulting from this study could reduce the risk of collapse. Furthermore, in emerging fields such as nanotechnology, where precision is crucial, this study could inspire new strategies for optimising the arrangement of layers of material deposited on a microscopic scale.
Beyond its practical implications, this study illustrates how an apparently trivial question can lead to fundamental discoveries. By combining tools from mechanics, system dynamics and probability theory, the research opens up new perspectives on the interactions between randomness and stability. It also highlights a universal lesson: unpredictability, when properly understood, can be harnessed to produce more robust systems.
This study sheds new light on the stability of random structures. It not only provides tools to predict and prevent collapse, but also shows how imperfect systems can be optimised through a better understanding of their intrinsic dynamics. This work provides a bridge between scientific curiosity and practical application, demonstrating once again that even seemingly simple questions can lead to major breakthroughs.
A Gaussian variable, also known as a normal random variable, is a random variable whose values follow a normal or Gaussian distribution. This distribution is characterised by its bell-shaped curve, known as the Gauss curve.
Stochastic - comes from the ancient Greek stochastikos (στόχος), meaning "target" or "ability to guess". The term illustrates the idea of estimating or predicting something in an uncertain setting.
Monte Carlo simulations are a numerical method used to solve complex problems involving uncertainty, random systems or variables that are difficult to model analytically. They use random number generation techniques to simulate a large number of possible scenarios for a system or phenomenon, enabling statistical estimates of the results to be obtained. The basic principle is to repeat a random experiment a large number of times in order to observe variations in the results and deduce statistical properties such as the mean, variance or probability of occurrence of certain events. The greater the number of simulations, the more accurate and reliable the results.
Denoël, V. (2024). Stochastic stability of random stacking of blocks. International Journal of Solids and Structures, 305, 113094. DOI : 10.1016/j.ijsolstr.2024.113094
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