Rigid Block Model Research Articles

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.

An investigation of force and pressure analysis during the process of equal channel angular pressing to M1 copper using finite element and upper-bound methods

This article presents two methods to determine pressing forces to the M1 copper plate. The first approach employs the upper-bound method, and the second method combines numerical simulation with a central composite design. The upper-bound method uses physical analysis to determine the deformation zone and establish the division of the rigid block model. The findings were identical to the slip-line solution method. The results obtained from the three-dimensional simulation performed using Qform software combined with a central composite design have provided regression equations that help determine the force and punching pressure. The impact of each parameter on the pressing force is similar in both approaches, and the difference between the two methods is insignificant and falls within a dependable range.

Spatio-temporal evolution laws of storage coefficient of coal mine underground reservoir and contact network of crushed rock.

Spatio-temporal evolution laws of storage coefficient and contact network of crushed rock are of great significance for the construction and utilization of underground reservoirs in coal mines. Based on discrete element method and irregular rigid block model, spatio-temporal evolution laws of storage coefficient and contact network of crushed rock under different overburden stresses are investigated and the following main conclusions are obtained: (1) The average storage coefficient and the storage coefficient at different vertical heights of the crushed rock packing system decrease exponentially as the overburden stress increases. When the overburden stress ranges from 0 to 20 MPa, the average storage coefficient decreases by 48.947%. (2) The average void radius and throat radius of water storage space decrease exponentially as the overburden stress increases. The increase in overburden stress leads to the transformation of large voids into smaller voids, causing a gradual decrease in void connectivity and a tendency towards irregular void shapes. (3) With the increase of overburden stress, the number of strong contacts in the packing system of crushed rock increases and gradually expands from the top to the bottom. The average contact force of crushed rock increases exponentially, while the coordination number increases linearly. (4) As the overburden stress increases, the majority of contact directions are concentrated within the ±30° range in the loading direction. This increase results in an enhancement of the anisotropy of the packing system structure of crushed rock.

Spatial Structure Characteristics of Underground Reservoir Water Storage Space in Coal Mines Considering Shape Characteristics of Crushed Rock

In order to investigate the impact of a crushed rock shape on the storage coefficient of underground reservoirs in coal mines, statistical analysis of the shape characteristics of crushed rocks was conducted, which was followed by numerical packing tests using the rigid block model. These tests aimed to investigate the spatial structure characteristics of underground reservoir water storage space in coal mines under the influence of different shapes of crushed rock. The results demonstrated the following: (1) Crushed rock exhibits a lognormal distribution in its shape characteristic parameters at different scales with a predominant discoid shape. The shape coefficient M can be utilized as a comprehensive indicator to characterize the shape characteristics of crushed rock. (2) The average storage coefficient of crushed rock increases exponentially as the shape coefficient M increases. There is a 50.1% increase in the storage coefficient from M = 1 to 3.5. (3) The spatial structure of the water storage space exhibits self-similarity, and both the void fractal dimension and the void boundary fractal dimension increase with an increase in the shape coefficient M. (4) When comparing the non-spherical particle system with the spherical particle system, it is observed that the spherical particle system has smaller water storage space, lower connectivity among voids, and more irregular void space. In the non-spherical particle system, the water storage space becomes larger as the shape of crushed rock becomes more irregular, resulting in more irregular void space. However, there is no significant effect on void connectivity.

A tectonic reconstruction model for Aotearoa-New Zealand from the mid-Late Cretaceous to the present day

ABSTRACT We present a mid-Late Cretaceous to present day tectonic reconstruction model for Aotearoa-New Zealand. Our GPlates model comprises 50 rigid crustal blocks grouped into regions with common deformation histories set within a well-defined Australia-Pacific-Antarctica plate circuit tied to a published global paleomagnetic absolute reference frame. Within the model, four distinct periods of deformation are recognised from both near- and far-field observations. A key model assumption is the continuity of basement terranes between North and South Zealandia prior to Middle Eocene rifting and Late Oligocene initiation of transform motion. To complement the rigid crustal block model, continuously closing polygons show a ∼25% decrease in plate boundary area since the Middle Eocene that has been compensated for by corresponding increases in crustal thickness. A kinematic fault-propagation fold model demonstrates the plausibility of post-Late Oligocene asymmetric oroclinal folding on both sides of the evolving transform boundary. ‘Missing’ areas of map section common to previous tectonic reconstructions can be reconciled through contraction, elongation and vertical-axis rotation of continental crust within the deformation zone ahead of northward propagation of the Alpine Fault. This tectonic reconstruction model provides an open, accessible, and testable foundation for current and future paleogeographic and tectonic studies across Zealandia.

Seismic performance assessment of medical equipment using experimentally validated rolling and toppling nonlinear models

During the last decades, several hospitals stopped service due to severe seismic damage to nonstructural components and medical equipment. This article investigates the dynamic behavior of medical equipment deployed in a full-scale, five-story building with two different support conditions, base isolated (BI) and fixed to the base (FB). Two nonlinear mathematical models, namely, rolling and toppling, are used to simulate the observed experimental responses using recorded data and the camera projection technique (CPT). CPT generated the possibility of measuring equipment horizontal displacements, slips, rotations, rocking, and toppling responses. The Euler–Lagrange formulation, along with the Stribeck friction model, was used to numerically model the rolling and in-plane rotation behavior of a four-wheel mobile cart (4WMC), which is considered to represent the trajectory of equipment supported on caster wheels. Besides, a simple two-dimensional (2D) rigid block model was used to describe the rocking and toppling behavior of locked and free-standing equipment. It was found that the 4WMC model was sensitive to the platform and wheels’ initial orientation angles, as well as the static and kinetic friction coefficients. As it should be, the toppling model was sensitive to the block dimensions and the intensity of the shaking. It was concluded that both numerical models successfully predicted the equipment rolling, in-plane rotation, rocking, and toppling behavior, as long as neither severe lateral impacts nor significant changes in the equipment mass occur during the motion.

Hosgri Fault Transpressional Slip Rates Reproduce Observed Central California Coast Uplift Rates

Abstract Coastal uplift is a common feature of Pacific Rim tectonic deformation. The 1989 Mw 7.0 Loma Prieta earthquake illustrated that reverse-oblique slip on a transpressive segment of the predominantly horizontal slip San Andreas transform fault produces coastal uplift. In west-central California the 400 km long San Gregorio-Hosgri fault coastal fault system of near-vertical to steeply northeast-dipping, reverse-oblique, dextral strike-slip faults also contribute to coastal uplift. Published southern Hosgri fault dextral slip rates are ∼4× reverse rates. Viscoelastic deformation modeling using published transpressional slip rates along the southern Hosgri fault reproduces observed emergent marine terrace uplift rates, offshore Hosgri fault deformation morphology, and crustal structure from San Luis Obispo Bay to Estero Bay. Our coastal uplift rate modeling is constrained by a unique combination of measurements: (1) offshore reverse- and strike-slip Hosgri fault zone (HFZ) slip rates, and (2) ≤125 ka onshore emergent terrace uplift rates adjacent to the HFZ. The previously proposed conceptual rigid block models of the southern central California coast (CCC) invoke local onshore thrust or reverse faults to explain coastal uplift. Observed transpressional offshore Hosgri fault deformation rates alone rigorously reproduce observed coastal uplift rates and crustal structure along the southern CCC.

Seismic displacement prediction methods, procedure, comparison and application

The methodology in the seismic design of slopes has evolved with technological advancement and the development of geotechnical engineering. Since methods with great limitations such as pseudo-static analysis to rigorous methods that require a lot of information for a better understanding of the behavior of the soil such as the dynamic analysis.
 This article develops the procedure, comparison, and application of methods for estimating seismic displacements, which are the Newmark (1965), Sarma (1975), Makdisi & Seed (1978) and Bray, Macedo & Travasarou (2018). Each of these methods considering the criteria of each one to obtain permanent displacements, such as yield acceleration, degraded period, spectral acceleration, magnitude, etc.
 As an applicative case, a rockfill dam and core central 50 m high placed in sandstone strata and moraine deposits interspersed with shale intrusions is shown, as seismic scenarios the Lima 1974 earthquake, the Attic 2001 earthquake and the earthquake were used. of Pisco 2007, considering both NS and EW directions. Likewise, the response spectrum on the base of the failure surface, resulting from a nonlinear seismic response analysis from 3 columns in the dam, is considered as information. The methods of Newmark (1965), Sarma (1975), Makdisi & Seed (1978), and Bray, Macedo & Travasarou (2018) were applied, obtaining deformation results between 1 to 30 cm that according to their order of magnitude represent a stable condition for the dam.

Seismic analysis of failure mechanisms in adjacent interacting stone masonry buildings via rigid block modeling

Groups of contiguous unreinforced stone masonry buildings are a common type of housing seen in old European downtowns. However, assessing their response to earthquakes poses several challenges to the analysts, especially when the housing units are laid out in compact configurations. In fact, in those circumstances a modeling technique that allows for the dynamic interaction of the units is required. The numerical study carried out in this paper makes use of a rigid block modeling approach implemented into in-house software tools to simulate the static behavior and dynamic response of an aggregate stone masonry building. Said approach is used to reproduce the results of bi-axial shake-table tests that were performed on a building prototype as part of the activities organized within the Adjacent Interacting Masonry Structures project, sponsored by the Seismology and Earthquake Engineering Research Infrastructure Alliance for Europe. The experimental mock-up consisted of two adjacent interacting units with matching layout but different height. Two rigid block models are used to investigate the seismic response of the mock-up: a 3D model allowing for the limit analysis of the building on one hand, and a 2D model allowing for the non-linear static pushover and time-history analysis on the other. The 3D model was built for the blind prediction of the test results, as part of a competition organized to test different modeling approaches that are nowadays available to the analysts. The 2D model was implemented once the experimental data were made available, to deepen the investigation by non-linear static pushover and time-history analysis. In both models, the stonework is idealized into an assemblage of rigid blocks interacting via no-tension frictional interfaces, and mathematical programming is utilized to solve the optimization problems associated to the different types of analysis. Differences between numerical and experimental failure mechanisms, base shears, peak ground accelerations, and displacement histories are discussed. Potentialities and limitations of the adopted rigid block models for limit, pushover and time-history analyses are pointed out on the basis of their comparisons with the experimental results.

Effects of non-plastic fines on liquefaction properties of saturated silt using discrete element modeling

This paper presents a numerical investigation using the three-dimensional discrete element method (DEM) to evaluate the macro-scale properties of saturated silt in a cyclic triaxial test. The three-dimensional particle model generated based on scanning electron microscopy (SEM) images of silt was imported into the discrete element program PFC3D and a rigid block model with different fines content was established. The effect of non-plastic fines content on the liquefaction properties of silty rigid block model was studied. The DEM results show that: (1) In the initial stage of dynamic stress loading, the axial strain of the sample develops slowly, and the stress-strain hysteresis loops almost overlap. With the increase of the number of cycles, the axial strain fluctuates violently, and the hysteresis loop becomes larger and develops toward the transverse axis. After the initial liquefaction, the dynamic elastic modulus and strength of the sample decrease significantly. (2) When the fines content increases from 0 to 30%, the liquefaction resistance of silt decrease with the increase of fines content. When the fines content increases from 30% to 100%, the liquefaction resistance of silt increases with the increase of fines content. (3) Under the same cyclic stress ratio, the number of cycles required for silt to liquefy decreases initially and then increases with the increase of fines content, and the critical fines content is 30%. The DEM results were compared with experimental results from the laboratory cyclic triaxial tests to provide insights into the underlying mechanism that governs the effects of fines content value on the liquefaction properties of silt.

Propagation Mechanism of Deep-Water Impulse Waves Generated by Landslides in V-Shaped River Channels of Mountain Valleys: Physical Model of Regular Rigid Block

Landslide-induced impulse waves in alpine valleys are a significant risk to large-scale dam and reservoir engineering projects in the surrounding area. In this study, a 1 : 200-scale physical model of landslide-induced impulse waves in a V-shaped river channel was established, and 18 groups of tests were conducted to evaluate the influence of different parameters, such as the volume and shape of the landslide body, water entry velocity, and water depth of the reservoir. Based on the test results, a dimensionless formula was established for the first wave height of impulse waves caused by a deep-water landslide in a V-shaped channel. An energy conversion law was determined for the impact of landslide-induced impulse waves on the reservoir bank. Finally, a distribution law was obtained for the initial maximum pressure caused by landslide-induced impulse waves along the water depth on the opposite bank. The theoretical predictions of the dimensionless formula showed good agreement with the experimental results, and the energy conversion rate of the landslide-induced impulse waves initially increased and then decreased with an increasing Froude number. The maximum dynamic water pressure showed a triangular distribution with increasing water depth below the surface of the still water body. The impact pressure of the impulse waves on the slope on the opposite bank increased with the water entry velocity. This study provides a scientific basis for the risk prevention and control of landslide-induced impulse waves in river channels feeding into reservoirs.

The effect of concentrated loads on open-well masonry spiral stairs

Concentrated loads applied on masonry open-well spiral stairs affect the compressive stress regime and, more importantly, may change the stress distribution drastically, producing undesirable torsional effects on the steps. Depending on geometry, materials, and construction technology, their application may compromise the stair stability. The current literature on the argument usually considers only uniformly load distribution, while the case of localised loads has not received any attention. The present paper aims to fill this gap by adopting a simplified method as an analytical tool, namely the Linear Arch Static Analysis (LASA), recently introduced and applied to analyse the structural behaviour of open-well spiral stairs under constant loads. An extensive parametric analysis of stairs with circular and elliptical planforms is carried out to define meaningful ranges of geometric stair parameters and concentrated load positions, ensuring safety against collapse.Lastly, to demonstrate the efficiency of the proposed approach to concentrated loads, the LASA method is applied and compared with a rigid block model on a real case study, represented by the stair in the Convent of San Domingo de Bonaval. Here the steps touch each other at their inner end, forming a virtual rib that transmits the external loads to the ground. Pros and cons of adopting the proposed procedures are highlighted, showing how LASA represents a simple but effective method to evaluate compressive equilibrium solutions for any stair geometry and vertical load distributions.

Palaeogeographic evolution of Zealandia: mid-Cretaceous to present

ABSTRACT We present a suite of 15 palaeogeographic maps illustrating the geological evolution of the entirety of Zealandia, from mid-Cretaceous to present, highlighting major tectonic phases, from initial Gondwana rifting through to development of the Neogene plate boundary. They illustrate palaeobathymetric and palaeofacies interpretations along with supporting geological datasets and a synthesis of regional tectonics. The maps are underpinned by a geologically-constrained and structurally-based rigid retro-deformation block model. This model, tied to the global plate circuit, is relatively simple for the main regions of Northern and Southern Zealandia, but breaks central Zealandia into numerous fault-bounded blocks, reflecting complex Neogene deformation associated with the modern plate boundary. Production of maps using GPlates and GIS allows for simple alteration or refinement of the block model and reconstruction of any geological dataset at any time. Reconstructions are within a palaeomagnetic reference frame, allowing assessment of palaeo-latitude, critical for palaeo-climatic and palaeo-biogeographic studies.

Digital Survey and Structural Investigation on the Triumphal Arch of Caracalla in the Archaeological Site of Volubilis in Morocco: Retracing the Timeline of Collapses Occurred during the 18th Century Earthquake

ABSTRACT The Lisbon earthquake occurred in 1755, which affected Morocco seriously, provoked the collapse of the Triumphal arch of Caracalla in Volubilis. The currently visible monument is the result of the rebuilding carried out in 1930 by French restores. In this paper, the timeline of collapses caused by the subsequent seismic quakes is retraced aimed at estimating the lower bound of the intensity of the 18th century earthquake, based on structural investigations carried out on rigid-block and continuous FE models. The historical investigations and the digital survey carried out by the authors in 2017 served as the base of knowledge to drive the analyses. Results of the analyses of the two models are very comparable and provide that a low acceleration of 0.168 g was sufficient to provoke the collapse of the arch and explain the reason why the expected earthquake acceleration in the zone of the archaeological site of Volubilis provided by the current Moroccan seismic code is exactly 0.16 g. Therefore, with the purpose of assessing also the current seismic vulnerability of the arch, this result leads to state that the capacity is equal to the seismic demand, and, therefore, too low to ensure the safety in the case of un unexpected event.

A thermo-hydro-mechanical approach to soil slope stability under climate change

Landslide initiation and dynamics are approached with different methods according to the scale of investigation. Individual landslides are typically described mechanistically, relying on the known coupling between the hydrological input – via atmosphere-soil interaction – and the mechanical response. As the scale increases, fully-coupled hydro-mechanical approaches gradually give way to simpler sequential couplings of hydrological and infinite slope or rigid block models, and finally to geostatistical models. At any scale, while the role of temperature in controlling evapotranspiration and thus the hydrological balance is well recognised, direct thermo-mechanical couplings are systematically neglected unless changes in water phase are involved. This contrasts with abundant experimental evidence of a fully-coupled thermo-hydro-mechanical behaviour of most geomaterials, even in ranges of temperature naturally experienced at the ground surface or in the near subsurface. Here, we review temperature-dependent processes that are potentially relevant to slope stability, focusing in particular on clayey slopes in temperate and warm regions. Our thought-provoking hypothesis is that temperature fluctuations and trends induced by climate change may exert, in short to long terms, a hydro-mechanical forcing on slopes (by altering permeability, water retention capacity, compressibility, shear strength). Together with other known effects (such as altered precipitation patterns and changes in land use), they could affect landslide activity and the distribution and frequency of slope failures. To verify this hypothesis across the scales, systematic field monitoring of temperature-related variables is necessary, together with geostatistical analyses entailing thermal remote sensing products. At the same time, fully-coupled approaches need to be upscaled to permit physically-based catchment- or regional-scale studies accounting for appropriate temperature-related variables and the inherent heterogeneity in materials and boundary conditions.

On blocks detection in unilateral masonry-like structures: a rigid-elastic displacement approach

In the present paper, a rigid block model for detection and identification of the cracks' pattern in a masonry-like structure subjected to dead loads and finite displacements is presented. The equilibrium problem is formulated by means of a suitable variational formulation for the boundary value problem of normal elastic no-tension material. Specifically, static equilibrium states of masonry-like structures are obtained through the linear conic optimisation of total potential energy. The independent variables of the optimisation problem are elastic strains and displacements, both assumed to be continuous. Indeed, the displacement and the strain fields are described in terms of nodal values through a FEM-like formulation and both of them are treated as independent variables of the optimisation problem. The study provides the basis for future developments in the field of mechanics and mechanisms of fracture. Two simple case studies are also discussed in order to tap the promising potential of the proposed approach.

A Dynamic Method to Predict the Earthquake-Triggered Sliding Displacement of Slopes

The earthquake-induced permanent displacement is an important index of the potential damage to a slope during an earthquake. The Newmark method assumes that a slope is a rigid-plastic body, and the seismic responses of sliding masses or seismic forces along the slide plane are ignored. The decoupled method considers no relative displacement across the sliding plane, so it overpredicts the seismic response of the sliding mass. Both dynamic and sliding analyses are performed in the coupled method, but when Ts/Tm is large, the results are unconservative. In this paper, a method is proposed to predict the earthquake-triggered sliding displacement of slopes. The proposed method is based on the Newmark rigid method, coupled method, and decoupled method considering both the forces at the sliding interface and the system dynamics under critical conditions. For the flexible system, the displacements are calculated with different stiffness values, and the results show that as the stiffness increases and tends to infinity, the critical acceleration and displacements of the proposed method are close to those of the Newmark method. The proposed method is also compared with the Newmark method with the period ratio Ts/Tm. At small values of Ts/Tm, the flexible system analysis results of the displacement are more conservative than those of the rigid block model; at larger values of Ts/Tm, the rigid block model is more conservative than the flexible system.

A variational rigid‐block modeling approach to nonlinear elastic and kinematic analysis of failure mechanisms in historic masonry structures subjected to lateral loads

AbstractDisplacement‐based methods contained in recent standards for seismic safety assessment require the determination of the full nonlinear pushover curve for local failure mechanisms in historic masonry structures. This curve should reflect both the initial elastic behavior and the rigid body behavior after the activation of rocking. In this work, a rigid block model is proposed for the displacement‐based seismic assessment of local collapse mechanisms of these structures. Masonry is modeled as an assemblage of two‐dimensional rigid blocks in contact through frictional interfaces. Two types of contact models are formulated to capture, respectively, the pre and postpeak branches of the pushover curve: a unilateral elastic contact model, capturing the initial nonlinear behavior up to the force capacity of the structure, corresponding to the activation of the collapse mechanism, and a rigid contact model with finite friction and compressive strength, which describes the rigid‐body rocking behavior up to the attainment of the displacement capacity of the structure. Tension‐only elements are also implemented to model strengthening interventions with tie‐rods. The contact problems associated with the elastic and rigid contact models are formulated using mathematical programming. For both models, a sequential solution procedure is implemented to capture the variation of the load multiplier with the increasing deformation of the structure (P–Δ effect). The accuracy of the modeling approach in reproducing the pushover curve of masonry panels subjected to horizontal seismic loads is evaluated on selected case studies. The solution is first tested against hand calculations, existing analytical models, and distinct element simulations. Then, comparisons against experimental tests follow. As a final application, the failure mechanism and pushover curve of a triumphal masonry arch are predicted by the model and its seismic assessment is performed according to codified force‐ and displacement‐based methods, demonstrating the adequacy of the proposed tool for practice.

Seismic vulnerability assessment of masonry arch bridges

A seismic assessment study of two historical masonry arch bridges, with different geometries and spans, is presented. The seismicity of the two locations was analyzed in order to define the seismic action to be applied. Artificial ground motion records were generated accordingly, and real ground motions were also used. The two numerical models were created with a 3D discrete element code. The accuracy of the deformability of the block models was verified by comparison with a finite element model. In situ measurements of the natural frequencies were available for one of the bridges, providing further means for model calibration. Time domain dynamic analyses were performed with each set of ground motions, allowing an evaluation of damage and failure modes under seismic action. The shorter span bridge was found to withstand the regulatory records with minimum damage. In contrast, the longer span bridge displayed a significant vulnerability to earthquake action. The study shows that rigid block models have the ability to represent effectively the dynamic elastic response of the bridges and can be calibrated by ambient vibration measurements. Furthermore, the discrete element models provide a powerful tool to investigate the nonlinear dynamic response and failure of these structures, which has to be based on a comprehensive set of dynamic records.

Free‐rocking tests of a freestanding object with variation of center of gravity

AbstractFreestanding nonstructural objects are very common in various infrastructure systems, but their dynamic properties with variations in the center of gravity (CG) are not yet fully understood. To investigate the dynamic properties of a freestanding object as its CG is varied, a steel frame model was fabricated using movable steel plates. Two scenarios, with varying and fixed CG, were considered in the free rocking tests. The experimental rotation responses generally matched the analytical responses in the initial cycles, and the discrepancy increased as the height of the CG (hcg) increased. The magnitude of the quarter period exhibited a similar trend. When the CG was fixed, the resulting magnitude of the quarter period varied slightly with hcg, indicating that the CG is one of the primary parameters affecting the quarter period. When the CG was varied, the experimental energy reduction factor was generally larger than the analytical results computed using the methods suggested in the literature, and it was scattered around the fixed CG value. The magnitude of the damping ratio was also scattered as the quarter cycle number increased, and it slightly decreased as hcg increased. An equivalent rectangular rigid block model with an equivalent height (heq) was employed to compute the geometric and dynamic parameters of the steel frame. The accuracy of this measure was investigated in terms of the quarter period and energy reduction factor. This work is beneficial for understanding the rocking behavior of freestanding objects.