Rigid Block Research Articles

Enhancing Detection of Small Earthquake Events from the 2017 Mw 6.5 Botswana Earthquake Using Cross-Correlation Back-Projection Analysis

On 3 April 2017, an Mw 6.5 earthquake occurred in central Botswana, situated inside the southern African plate. However, in a region with sparse station coverage like Botswana, the earthquake catalog is usually largely incomplete at smaller magnitudes. Here we propose a modified back-projection method using a correlation characteristic function, which represents energy similarity and mitigates interference due to waveform complexity in regional seismic networks, aiming to enhance the detection of small seismic events. The energy-related characteristic function exhibits robust detections for low signal-to-noise ratio data, a capability not achievable by traditional arrival time picking methods. Detection reliability is ensured through a dual search constraint, optimizing effective signal count, considering nearby epicentral station signals, and accounting for seismic events clustering, which involves discarding seismic events from regions with infrequent earthquake occurrences. A total of 365 events (magnitude completeness of ∼ M 2), about four times more than the catalog obtained by other methods, have been detected within seven days after the Botswana mainshock. Most are low signal-to-noise ratio events for that are not localized by traditional methods. The resulting earthquake sequence occurred in at least three NW-SE fault strands between the Kaapvaal Craton and the Limpopo Belt. Our results, when combined with other geophysical data, suggest that the Botswana earthquake mainly results from the difference of intraplate basal dragging forces at various tectonic regions driven by the clockwise plate motions, which would generate extensional strains along the weak edges of rigid blocks.

Sliding and rocking response of a flexible structure on a rigid block using floating frame of reference formulation

Sliding and rocking response of a flexible structure on a rigid block using floating frame of reference formulation

Non‐linear time history analyses of a rigid block isolated with unbonded fiber‐reinforced elastomeric isolators (UFREIs): A comparison between 3D finite element and phenomenological models

AbstractNumerical modeling represents a pivotal tool in the seismic analysis and design of structural systems, enabling the detailed prediction and examination of structural responses under seismic loading. This research conducts a comparative analysis of two numerical modeling approaches aimed at simulating the seismic response of unbonded fiber‐reinforced elastomeric isolators (UFREIs). The research focuses on a finite element (FE) model developed using Abaqus and a developed phenomenological model implemented in OpenSees, outlining the development and calibration processes for each. The FE model is developed based on simple rubber material testing data, while the phenomenological model is calibrated using experimental results from cyclic shear tests conducted on the UFREI device and the FE model. The primary objective of this study is to assess the effectiveness of these modeling approaches in predicting UFREI behavior under seismic conditions. This evaluation entails comparing model predictions with experimental data obtained from unidirectional shake table tests performed on a rigid block isolated by two UFREIs. This paper highlights the distinct advantages and limitations of each model in simulating UFREI dynamic responses during seismic events. Furthermore, it provides insights into the modeling techniques and discusses the computational demands and data requirements of each model, thereby aiding in their application to various aspects of seismic analysis and design.

Three-Dimensional Discontinuum Modeling and Rigid Block Stability Analysis of a Group of Three Caverns for Hydroproject in Northeast India

Abstract This paper presents various methods for calculating the factor of safety (FOS) and utilizes the Isoblock method to analyze the stability of underground powerhouse caverns located in the North-East state of India. A three-dimensional discontinuum analysis simulates the stability of the downstream sidewall, upstream sidewall, and crown of the caverns. The results indicate that the powerhouse cavern contains approximately 2.36% unstable blocks, while the Main Inlet Valve (MIV) and Transformer Caverns exhibit 0.14% and 0.84% unstable blocks, respectively. A comprehensive parametric analysis investigates the impact of joint stiffness, joint friction angle, in-situ stress, cover depth, and the orientation of the cavern's longer axis on stability. The findings reveal that the cavern's stability is significantly influenced by variations in lateral stress ratio and the direction of in-situ principal stress. Stability is optimized at stress ratios between 1.2 and 1.8, with maximum instability observed in the powerhouse cavern, correlating with its larger size. Notably, the stability of the crown is greater than that of the sidewalls, particularly when the cavern orientation is aligned between N68°E and N78°E. The study underscores the importance of these factors in the design and stability assessment of underground structures, providing valuable insights to enhance their safety and reliability. This analysis highlights the necessity for considering joint stiffness and in-situ stress orientations in future designs to mitigate risks associated with instability.

A comprehensive investigation into the effect of vertical component of ground motion on response of sliding unanchored blocks

This study aims to quantify the effect of the vertical component of ground motion on the seismic response of sliding rigid blocks, in terms of maximum and residual displacements. A multi-stripe analysis approach is employed to examine the response with and without the vertical component under various levels of ground motion intensity. First, a parametric study is conducted using a Coulomb friction model with a constant coefficient of friction that does not account for any dependence of the friction coefficient on velocity or pressure. It is shown that the effect of the vertical component on maximum and residual displacement is more pronounced for ground motions that can barely initiate sliding. Moreover, the influence of the vertical component on the residual sliding displacement is higher than that on the maximum displacement. Subsequently, to account for the friction coefficient dependence on pressure and sliding velocity, two interface cases (steel-steel, and concrete-concrete) are analyzed. It is found that even when a variable coefficient of friction is used, the overall trends observed for a constant coefficient of friction do not change.

Coupling of Spectrin Repeat Modules for the Assembly of Nanorods and Presentation of Protein Domains.

Modular protein engineering is a powerful approach for fabricating high-molecular-weight assemblies and biomaterials with nanoscale precision. Herein, we address the challenge of designing an extended nanoscale filamentous architecture inspired by the central rod domain of human dystrophin, which protects sarcolemma during muscle contraction and consists of spectrin repeats composed of three-helical bundles. A module of three tandem spectrin repeats was used as a rigid building block self-assembling via coiled-coil (CC) dimer-forming peptides. CC peptides were precisely integrated to maintain the spectrin α-helix continuity in an appropriate frame to form extended nanorods. An orthogonal set of customizable CC heterodimers was harnessed for modular rigid domain association, which could be additionally regulated by metal ions and chelators. We achieved a robust assembly of rigid rods several micrometers in length, determined by atomic force microscopy and negative stain transmission electron microscopy. Furthermore, these rigid rods can serve as a scaffold for the decoration of diverse proteins or biologically active peptides along their length with adjustable spacing up to tens of nanometers, as confirmed by the DNA-PAINT super-resolution microscopy. This demonstrates the potential of modular bottom-up protein engineering and tunable CCs for the fabrication of functionalized protein biomaterials.

Increasing measurement accuracy by estimating multiple dissipations for the impedance tube calibration

Impedance tube measurements, employing transfer function/matrix methods, are commonly used by acousticians to characterize the acoustic properties of materials in normal incident conditions. Sound dissipation in impedance tubes can impact the accuracy of measurement results. Various effects contributing to sound dissipation in tubes are studied and analyzed. This research investigates different dissipative effects and dissipation models. This paper uses model-based Bayesian inference to estimate critical parameters in multiple dissipations. The experiments involve using an empty impedance tube terminated with a rigid metal block. A hypothetical air layer in front of the metal block is assumed for establishing a prediction model in contrasting estimation results with theoretical values. The results incorporating dissipation models demonstrate a good match with the theoretical calculation of the air layer. Applying the estimated dissipations to actual resonator materials demonstrates a significantly improves measurement accuracy.

Hierarchical Self-Organization and Disorganization of Helical Supramolecular Columns Mediated by H-Bonding and Shape Complementarity.

H-bonding, shape complementarity, and quasi-equivalence are widely accepted as some of the most influential molecular recognition events mediating biological and synthetic self-organizations. H-bonds are weaker than ionic but stronger than van der Waals forces. However, the directionality of H-bonds makes them the most powerful among all nonbonding interactions. Here, we selected two taper-shaped self-assembling dendrons, one flexible and one rigid, and equipped them with -CO2CH3, -CH2OH, and -COOH at their apex. They demonstrated the hierarchical way in which shape-complementarity in the presence of -CO2CH3 mediated highly ordered helical self-organization for the case of the rigid building block and less ordered helical arrays for the flexible one. Weak H-bonding by -CH2OH unwound the helix from the rigid dendron, yielding a porous column. Due to its quasi-equivalence, the flexible dendron tolerated better the H-bonding by -CH2OH self-organizing a different helical column. The rigid and the flexible dendrons yielded only disorganized nonhelical columns in the presence of -COOH at the apex. This balance between rigidity, flexibility, and tolerance or lack of it to diverse H-bonding architectures indicates that mechanistic elucidation of the self-organization process helps endow it with the same building block, both helical organizations approaching biological precision, and disorganized nonhelical arrangements.

Rheology modification in a subduction channel due to eclogite facies metasomatism (Rocky Beach Metamorphic Mélange, Port Macquarie, Australia)

The rheological properties of the interface between the down-going and overriding plates in subduction zones provides insight into how plate convergence is accommodated and the controls on seismic and aseismic slip. This interface is known as the subduction channel and exhumed examples provide the only direct information on deformation mechanisms and the impact of metamorphism on rheology. The Rocky Beach Metamorphic Mélange in eastern Australia is one such exhumed subduction channel, composed of eclogite, blueschist and greenschist facies blocks within a mélange matrix. Previous phase equilibria modelling indicates that high pressure blocks were subducted to ca. 100 km depth and then retrogressed during return flow and exhumation. We found that the rheology of blocks is modified by metasomatism, consistent with studies on other subduction channels. However, through comparison of blocks from different metamorphic grades we found that the effect of metasomatism on rheology varied depending on the pressure and temperature conditions of metasomatism. While unmetasomatised eclogites behaved as rigid objects in the mélange matrix, rocks with mineral assemblages that equilibrated during eclogite facies metasomatism accumulated significant strain, forming isoclinal folds and refolded folds. Deformation of these blocks began at eclogite facies and continued during return flow and retrogression to blueschist facies. At blueschist facies, metasomatised blocks developed mm-scale isoclinal folds with shearing parallel to fold limbs forming rootless isoclinal folds. At the transition between blueschist and greenschist facies, pressure solution became important, preferentially focusing along layers of lawsonite, dissolving it from the rock. At greenschist facies, dissolution-precipitation processes caused significant mass loss, producing mm-spacing between pressure solution seams and cuspate folds, analogous to dewatering structures in sediments. In the Rocky Beach Metamorphic Mélange eclogite facies metasomatism reduces the competence of rigid blocks, reducing overall subduction channel heterogeneity during return flow. We suggest that subduction channels that experience widespread eclogite facies metasomatism may be less likely to generate seismic slip during return flow, since the proportion of rigid blocks and block strength are both reduced.

A variational-based non-smooth contact dynamics approach for the seismic analysis of historical masonry structures

A variational formulation of the non-smooth contact dynamics method is proposed to address the dynamic response of historical masonry structures modeled as systems of 3D rigid blocks and subjected to ground excitation. Upon assuming a unilateral-frictional contact law between the blocks, the equations of motions are formulated in a time-discrete impulse theorem format in the unknown block velocities and contact impulses. The variational structure of the problem to be solved at each time step is proven. On that basis, the numerical method requires at each time step to perform a collision detection that identifies antagonist contact points based on the given structural configuration, to solve a second-order conic programming problem that outputs block velocities and contact impulses, and to update the structural configuration for the solution to advance in time. As a merit of the formulation, large-scale problems can be robustly and efficiently addressed thanks to the convex setting of the time-step optimization problem. Numerical results are presented to test the computational performances of the proposed approach. Benchmark problems provide numerical evidence that the formulation is consistent with event-driven solutions based on the classical Housner impact model. The dynamic response, failure domains, and fragility functions of real-size masonry structures are then explored under ground impulse or earthquake excitation. The obtained results prove the reliability of the present computational method for the dynamic analysis and seismic assessment of historical masonry constructions of engineering interest.

Applying a 2D Variational Rigid Block Modeling Method to Rubble Stone Masonry Walls Considering Uncertainty in Material Properties

ABSTRACT The computational cost of detailed micro-modeling of masonry structures can be minimized with rigid block analysis methods, where a variational formulation is particularly interesting as it allows for nonlinear pushover analysis. This work investigates the accuracy of this modeling method in predicting the response of rubble stone masonry walls in shear compression tests. To this end, we extend the mathematical programming problem by including cohesion, compressive strength, and tensile strength in the formulation and implement an improved pushover procedure, which guards compatibility of the displaced shape and crack patterns at the intersection of elastic and rigid contact models. Additionally, we reduce model uncertainty, introduced through sampling material properties with distributions estimated based on results from tests on mortar masonry joints and on mortar specimens, by correlating the predicted crack patterns with the observed ones. The findings show that (i) model falsification based on crack pattern is an effective method for reducing the parameter uncertainty of stone masonry walls; and that (ii) the improved pushover procedure predicts the pre- and post-peak response of stone masonry walls very well with analysis times of less than 20 minutes for 2D models where stones and mortar are modeled explicitly.

A generalized Bouc–Wen model for simulating the quasi-static and dynamic shear responses of helical wire rope isolators

This research investigates the mechanical behavior of a helical wire rope isolator deforming along its shear direction. In particular, we present the results of an extensive experimental campaign including both quasi-static and dynamic tests. The former provide hysteresis loops characterizing the device quasi-static behavior; the latter, performed by using an electro-mechanical shaker, furnish frequency response curves describing the dynamic behavior of a rigid block supported by the tested device. To simulate such a complex behavior, we adopt a generalized Bouc–Wen model and identify its parameters on the basis of the quasi-static test results. Subsequently, such a model is employed to reproduce the frequency response curves of the isolated rigid block. Since the results of the dynamic tests suggest the presence of rate-dependent hysteresis phenomena in the isolated system, the generalized Bouc–Wen model is enhanced by introducing a linear viscous component. Finally, to substantiate the model validation, the experimental results obtained by applying a series of white noise signals are compared with those obtained numerically to demonstrate the model capability of reproducing the device behavior in non-stationary response conditions.

Nonlinear static analysis of masonry structures with mortar joints and cracking units by optimization‐based rigid block models

AbstractA rigid block model with elasto‐plastic softening interfaces is developed for nonlinear static analysis of masonry structures subject to monotonic loading. Cracking, crushing, and shear failures are taken into account at interfaces, following a simple micro‐modeling approach. An optimization‐based formulation is used for the solution of the equation systems governing the behavior of the rigid block assemblage. A simple incremental solution procedure is implemented to take into account the material softening behavior and the effects of large displacements on equilibrium conditions. The interface models are validated against tension and shear tests on bi‐block prisms from the literature. Applications to numerical and experimental out‐of‐plane loaded masonry walls as well as to circular arches with mortar joints are presented to evaluate the effects of tensile strength and the accuracy of the developed model against responses involving P‐Δ effects. Comparisons with experimental tests on shear walls also involving cracking of the units in the failure mechanisms are finally reported to discuss the potentialities and limitations of the proposed modeling approach.

Block Rotations in NW Iran in Response to the Arabia‐Eurasia Collision Constrained by Paleomagnetism

AbstractNorthwest Iran is a seismically active region dominated by NW‐SE trending strike‐slip faults, such as the North Tabriz and Qosha Dagh faults, and smaller NNE‐SSW striking faults. The Bozgush Mountains are shaped by these faults and divided into two domains that show a difference in strike. To quantify rotational tectonic deformation in NW Iran, we performed a paleomagnetic study along three transects of the Bozgush and Qosha Dagh Mountains with 127 sites. Our large new paleomagnetic data set shows that the Bozgush Mountains did not rotate as a single rigid block. In the western domain of the Bozgush Mountains, we find evidence for clockwise vertical axis rotations of ∼40°, while the eastern domain has rotated up to ∼80° clockwise. Declinations of the western Bozgush domain fit well with observed declinations in the Qosha Dagh Mountains. Fault patterns show that the eastern domain of the Bozgush Mountains is divided by a set of NNE‐SSW striking sinistral strike‐slip faults, which created domino‐style blocks that accommodated the additional 40° of rotation. We estimate that these extra rotations have resulted in around 4 km of N‐S shortening and more than 1.5 km of differential uplift.

Free Rocking of a Rigid Block on a Flexible Structure with Non-Smooth Contact Dynamics

Free Rocking of a Rigid Block on a Flexible Structure with Non-Smooth Contact Dynamics

Urban Livability and Vertical Inclusion: Exploring Mixed-Use Development Strategies in Khulna, Bangladesh

As urban populations expand and future needs evolve amidst limited land availability, rethinking traditional planning and construction approaches becomes imperative. Architect Frank Schlesinger advocates for urban environments that embrace mixed- use complexes, conceptualizing urban land as dynamic, three-dimensional blocks accommodating diverse functions above, at, and below ground levels. Shopping facilities, pivotal in urban landscapes, reflect regional and temporal variations in consumer behavior. Integrating public and private amenities is essential for cultivating a thriving urban ecosystem. However, rigid block structures in current city master plan to constrain diversity and livability, are exacerbated by inadequate setbacks that challenge urban identity. In Khulna, bustling daytime cores contrast with deserted evenings and holidays, reflecting segregated land uses that hinder societal inclusivity. Despite demand, office and commercial spaces often remain vacant. This study employs a mixed-method approach, combining quantitative surveys with qualitative methods such as interviews and observational studies. It aims to address critical local issues including land scarcity, traffic management, vertical development challenges, and the provision of sophisticated services to enhance urban comfort. By exploring innovative configurations of civic facilities, both horizontally and vertically, the study seeks to demonstrate how integrating functions can create cohesive and vibrant urban environments. Ultimately, the adoption of vertically integrated spaces holds promise for fostering inclusivity and sustainability in Khulna’s urban fabric.

Seismic Response Prediction of Rigid Rocking Structures Using Explainable LightGBM Models

This study emphasizes the explainability of machine learning (ML) models in predicting the seismic response of rigid rocking structures, specifically using the LightGBM algorithm. By employing SHapley Additive exPlanations (SHAP), partial dependence plots (PDP), and accumulated local effects (ALE), a comprehensive feature importance analysis has been performed. This revealed that ground motion parameters, particularly peak ground acceleration (PGA), are critical for predicting small rotations, while structural parameters like slenderness and frequency are more significant for larger rotations. Utilizing an extensive dataset generated from nonlinear time history analyses, the trained LightGBM model demonstrated high accuracy in estimating the maximum rotation angle of rigid blocks under natural ground motions. The study also examined the sensitivity of model performance to lower bound thresholds of the target variable, revealing that reduced feature sets can maintain predictive performance effectively. These findings advance ML-based modeling of seismic rocking responses, providing interpretable and accurate models that enhance our understanding of rocking structures’ dynamic behavior, which is crucial for designing resilient structures and improving seismic risk assessments. Future research will focus on incorporating additional parameters and exploring advanced ML techniques to further refine these models.

Comparison of Continuous and Discrete Modeling Strategies for the Structural Assessment of a Masonry Vault Under Dynamic Seismic Loading

ABSTRACT To begin, the paper introduces methodologies employing both homogeneous continuous Finite Element Method (FEM) and block-by-block Discrete Element Method (DEM) to model masonry behaviors. In the continuous approach, the masonry is modeled as an anisotropic damageable material (homogeneous or not). In contrast, the block-based approach represents rigid blocks interacting through contact joints governed by frictional cohesive behaviors. The DEM framework allows for significant displacements, rotations, and complete detachments of blocks, aspects often overlooked in traditional FEM models. The primary application of this research involves the seismic assessment of a masonry cross vault. Notably, the numerical results exhibit the approach’s capability to provide realistic predictions of failure mechanisms, crucial for retrofitting efforts. This includes an accurate representation of the actual cracking pattern, accounting for significant displacements. The numerical implementation is accessible through the open-source LMGC90 software. Comparative analyses are then carried out, focusing on the strengths and weaknesses of FEM macro-modeling (continuous homogeneous description) and DEM utilizing rigid blocks.

3D DEM investigation of shear behavior and interaction mechanism of woven geotextile-sand interfaces

This paper presents a numerical study on the investigation of microscopic mechanism governing the interaction of woven geotextile and angular sand employing the 3D discrete element method (DEM). The surface texture and tensile properties of the geotextile were simulated using overlapping spherical particles, and the angular sand was simulated using rigid blocks. The DEM models were fully calibrated based on previous experimental data. The shear and dilation zones of sand near the interface were quantitatively determined based on particle displacement gradients. Analysis of contact forces was conducted to explain the microscopic mechanism behind the macroscopic strength evolution. The influence of geotextile surface roughness on the shear strength of the geotextile-sand interface is also discussed. The results show that the failure mode of the woven geotextile-sand interface is a combination of particle sliding failure along the geotextile surface and shear failure of the sand within the shear zone above the interface. There is a rapid redistribution of contact forces prior to reaching peak shear resistance, and the average normal contact force within the shear zone remains relatively constant after the peak shear stress is achieved. A completely developed shear zone stabilizes soil deformation, typically after achieving the peak shear resistance.

Magnetic restoring forces on rocking blocks

AbstractThis study investigates the idea of adding an extra magnetic restoring force to a rocking block to improve its overall dynamic performance. The proposed concept ensues by introducing a pair of identical magnets to the rocking block. Both magnets are considered lumped on their respective volume centers and are embedded within the rocking block and the supporting base. When properly magnetized, this pair of magnets provides the rocking block with an extra magnetic restoring force which, although it takes on its maximum value when the two magnets are in contact, decreases as the distance between the two magnets increases. The proposed concept, subjected to pulse‐type base excitations, reveals the inherent problem of magnetic restoring forces. From the overturning spectra of the rocking block, it is found that there are cases where the block fails (overturning) in the presence of magnets, while the same free‐of‐magnets block rocks safely (no overturning) when its own weight acts as the only restoring force. This interesting finding appears to be counterintuitive. Is it possible that by providing additional restoring force the block is “driven” to overturn? This study shows that when the rocking block returns toward the vertical position, the angular velocity, in the presence of magnets, is higher than the angular velocity, in the absence of them. This increase in the angular velocity is a direct outcome of the nature of the magnetic restoring forces, and it is mainly the reason that causes the overturning of the rigid block during its free vibration regime. To mitigate the shortcomings of using magnetic restoring forces, the idea of a semi‐active control of the pair of magnets is introduced and explained in detail. This paper concludes with the advantages and potential disadvantages of the overall performance of rigid blocks in the presence of magnetic restoring forces.