Rigid Spring Research Articles

Annular Sector Element for Applied Element Method

Applied element method (AEM) is a numerical procedure that allows simplified modelling of structures. In AEM, the structure is modelled using rigid elements and springs. Any shape can be modelled by properly placing the springs, without increasing the computational effort. So far, AEM has not been utilised for the analysis of curved structures such as arches and rings using curved element. For the broad use of AEM, curved element is required. Arches are widely used as structural member. The analysis of circular ring is of interest today, as they are used as frames and parts of machines. In this paper, a new element is developed for the linear elastic analysis of curved structures. For this, a generalised two-dimensional annular sector element with infinite number of springs is considered. The element is used for both static and modal analyses of arches and rings. The results are compared with available analytical results, and good agreement is observed.

The effect of isolation in controlling the variability of point connected structures with uncertain structural attachments

As the demand for the transmission of electric power and communication signals in automotive and aerospace vehicles increases, so does the number of structures comprising simplified one-dimensional attachments, such as electrical cabling, affixed to a host plate like primary structure. These attachments are typically uncertain in their geometric or material properties, potentially affecting the response of the built-up structure. Difficulties then arise in the prediction of the response of the assembly. This study shows how the variability, due to the uncertain attachments, might be reduced by considering flexible connections. A mobility analysis compares systems connected with either rigid links or elastic springs. A frequency is identified at which the assembly dynamically uncouples; the effect on the host response variability due to the uncertain attachment decreases above this frequency with a reduction of the order of 60 dB in the coefficient of variation. This uncoupling or effective isolation frequency can be simply estimated from the mobility of the elastic connection and the properties of the nominal structural attachment. For design purposes, this frequency can be adjusted to achieve a more predictable response above a given frequency.

Topology optimization of connections in mechanical systems

One of the issues for the automotive industry is weight reduction. For this purpose, topology optimization is used for mechanical parts and usually involves a single part. Its connections to other parts are assumed to be fixed. This paper deals with a coupled topology optimization of both the structure of a part and its connections (location and number) to other parts. The present work focuses on two models of connections, namely rigid support and spring that prepares work for bolt connection. Rigid supports are modeled by Dirichlet boundary conditions while bolt-like connections are idealized and simplified as a non-local interaction to be representative enough at a low computational cost. On the other hand, the structure is modeled by the linearized elasticity system and its topology is represented by a level set function. A coupled optimization of the structure and the location of rigid supports is performed to minimize the volume of an engine accessories bracket under a compliance constraint. This coupled topology optimization (shape and connections) provides more satisfactory performance of a part than the one given by classical shape optimization alone. The approach presented in this work is therefore one step closer to the optimization of assembled mechanical systems. Thereafter, the concept of topological derivative is adapted to create an idealized bolt. The main idea is to add a small idealized bolt at the best location and to test the optimality of the solution with this new connection. The topological derivative is tested with a 3d academic test case for a problem of compliance minimization.

Effect of pivot stiffness on nonlinear dynamic characteristics of tilting pad journal bearings

The role of pivot stiffness on dynamic response is explored in a large stiffness range for tilting pad journal bearings in vertical and horizontal rotors. Two designs are compared: rigid pivot and disc spring set pivot. The latter is adjustable to vary from completely soft to almost rigid. Effect of journal unbalance is also discussed. It is shown that two threshold points of 0.1 and 10 are discovered for pivot stiffness ratio. Specific pivot stiffness at different intervals has totally different effects on orbits. In addition, another point exists as turning point for pivot stiffness ratio and results in the significant orbit peak. Further, pivot stiffness effect is sensitive to journal unbalance, particularly at the peak point.

Presentation and validation of a specific RBSM approach for the meso-scale modelling of in-plane masonry-infills in RC frames

In the last few years, the scientific community has been strongly involved in the development of approaches able to incorporate the contribution of infill walls in seismic vulnerability analysis of infilled RC framed buildings. The detailed meso-scale modelling of panels within global models as an alternative to equivalent strut models is still a challenge because of the difficulties in controlling constitutive parameters and the high computational effort. The potential of rigid body and spring models (RBSMs) in this field is investigated by exploiting a specific code for the nonlinear analysis of in-plane masonry with an efficient computational management. The reference benchmark is an infilled one-bay one-storey frame that was subjected to cyclic tests within an experimental campaign and has been here modelled by RBSM, performing a set of numerical parametrical analyses under lateral loads. A good general matching of results and effective computational performance have been found.

Presentation and validation of a specific RBSM approach for the meso-scale modelling of in-plane masonry-infills in RC frames

In the last few years, the scientific community has been strongly involved in the development of approaches able to incorporate the contribution of infill walls in seismic vulnerability analysis of infilled RC framed buildings. The detailed meso-scale modelling of panels within global models as an alternative to equivalent strut models is still a challenge because of the difficulties in controlling constitutive parameters and the high computational effort. The potential of rigid body and spring models (RBSMs) in this field is investigated by exploiting a specific code for the nonlinear analysis of in-plane masonry with an efficient computational management. The reference benchmark is an infilled one-bay one-storey frame that was subjected to cyclic tests within an experimental campaign and has been here modelled by RBSM, performing a set of numerical parametrical analyses under lateral loads. A good general matching of results and effective computational performance have been found.

Implementation of smeared crack approach in rigid block and spring modeling of reinforced concrete

The main purpose of this study is to investigate the structural behavior and cracking response of reinforced concrete (RC) members on the basis of Rigid Block and Spring Method (RBSM). The main problem of this method i.e. the independent evaluation of springs’ stiffness is discussed and modified based on the concept of smeared crack approach. The response of RC membrane element is evaluated by considering the stress transfer mechanisms involving axial behavior of the cracked/un-cracked concrete, axial behavior of the reinforcement and the stress transfer across cracks due to aggregate interlock. A non-iterative approach for estimating the reinforcing bars’ strains based on the shear strain of cracked RC domain is proposed using approximate relationships. Accordingly, the application of smeared crack concept in the RBSM, leads to a more precise simulation of the shear behavior of cracked RC members.

A new discrete method for modeling hydraulic fracturing in cohesive porous materials

A new 2D hydromechanical rigid block spring method is developed for studying hydraulic fracturing in cohesive porous materials. The continuum medium is discretized into an assembly of discrete rigid blocks. Macroscopic deformation and failure are related to local properties of interfaces between blocks. Fluid flow in porous continuum is described by an equivalent discrete fracture network model. Biot's theory is extended to discrete porous interfaces for studying interactions between fluid flow and mechanical behavior. Further, the hydraulic conductivity evolution due to fracture propagation is taken into account through the change of aperture of interfaces. The proposed discrete hydromechanical model is first verified through the response of an elastic isotropic thick-walled cylinder subjected to an inner pressure under the steady state flow condition. The obtained numerical results agree well with the corresponding analytical solutions. Further, the numerical model is applied to studying the hydraulic fracturing of the thick-walled cylinder. The effect of transient flow on fracturing pressure is highlighted. The effect of materials strength anisotropy on fracturing process is also studied. It is shown that the proposed discrete model provides an efficient tool for modeling hydraulic fracturing in cohesive rock-like materials.

Stiffness modelling of 2D welded joints using metamodels based on mode shapes

Connections are a key component of steel structures. Current methods of analysis of steel frames mostly rely on approximate joint models based on zero-length rotational springs or mechanical models of rigid bars and springs. A method is presented in this paper to accurately model the stiffness of 2D welded steel joints that is based on surrogate modelling. The joint modelling process starts by developing an accurate 4-node (12 degree of freedom) cruciform finite element from a complete finite element model using substructuring techniques. Then a modal decomposition of these matrices yields the eigenvalues and mode shapes that are used to develop the design points of the training and validation sets of the surrogate model. The modal decomposition leads to a reduced set of variables that accurately models the design space. A uniform Rectangular Grid of design sites is proposed. The Kriging method and Support Vector Machine Regression are used to generate the metamodels and predict the stiffness at untried sites. Both methods provide accurate results. Simulations of steel frames are performed to show the accuracy of the proposed surrogate modelling.

Validation of a two-step simplified compatible homogenisation approach extended to out-plane loaded masonries

A two-step homogenisation model, formulated by the authors for the in-plane case, is herein extended for the nonlinear out-of-plane analysis of masonry structures. A rectangular running bond elementary cell is discretised by 24 elastic CST elements and inelastic zero-thickness interfaces. The mechanical meso-scale problem is briefly recalled, whereas the out-of-plane homogenised behaviour is evaluated by means of a simple on-thickness integration of the in-plane homogenised curves. At a macro-scale, the rigid body and spring model is slightly modified to allow both flexural and torsional failure mechanisms. The validation of the numerical approach is achieved comparing with some full-scale masonry panels tested in two-way bending up to failure. A series of nonlinear structural analyses are conducted considering different parameters, which have been varied during the experimental campaign. The numerical results are promising and demonstrate the capability to deal with different failure mechanisms as result of a combination of various experimental aspects.

Validation of a two-step simplified compatible homogenisation approach extended to out-plane loaded masonries

A two-step homogenisation model, formulated by the authors for the in-plane case, is herein extended for the nonlinear out-of-plane analysis of masonry structures. A rectangular running bond elementary cell is discretised by 24 elastic CST elements and inelastic zero-thickness interfaces. The mechanical meso-scale problem is briefly recalled, whereas the out-of-plane homogenised behaviour is evaluated by means of a simple on-thickness integration of the in-plane homogenised curves. At a macro-scale, the rigid body and spring model is slightly modified to allow both flexural and torsional failure mechanisms. The validation of the numerical approach is achieved comparing with some full-scale masonry panels tested in two-way bending up to failure. A series of nonlinear structural analyses are conducted considering different parameters, which have been varied during the experimental campaign. The numerical results are promising and demonstrate the capability to deal with different failure mechanisms as result of a combination of various experimental aspects.

Failure Process Simulation of Interlayered Rocks under Compression

Anisotropy in strength and deformation of rock mass induced by bedding planes and interlayered structures is a vital problem in rock mechanics and rock engineering. The modified rigid block spring method (RBSM), initially proposed for modeling of isotropic rock, is extended to study the failure process of interlayered rocks under compression with different confining pressures. The modified rigid block spring method is used to simulate the initiation and propagation of microcracks. The Mohr–Coulomb criterion is employed to determine shear failure events and the tensile strength criterion for tensile failure events. Rock materials are replaced by an assembly of Voronoi-based polygonal blocks. To explicitly simulate structural planes and for automatic mesh generation, a multistep point insertion procedure is proposed. A typical experiment on interlayered rocks in literature is simulated using the proposed model. Effects of the orientation of bedding planes with regard to the loading direction on the failure mechanism and strength anisotropy are emphasized. Results indicate that the modified RBSM model succeeds in capturing main failure mechanisms and strength anisotropy induced by interlayered structures and different confining pressures.

Extension of Hencky bar-net model for vibration analysis of rectangular plates with rectangular cutouts

Comprising rigid bars and spring systems at the joints and in the cells, the Hencky bar-net model (HBM) has been shown to be a physical structural representation of the finite difference plate model (FDM). In this paper, the HBM is extended for the vibration analysis of rectangular plates with rectangular cutouts. This extension addresses the rotationally elastic and transverse spring stiffnesses for the HBM at the cutout corners. After verifying the HBM model by comparing the vibration results with existing solutions for some plate problems, the model is used to obtain some new free vibration solutions for plates having various boundary conditions including cracked corners modelled by free small rectangular cutouts.

Investigation of Shear Wave Velocity by Using Multi-Channel Analysis of Surface Wave Method for Microzonation Map Development and Its Application to Industrial Frame Structures

The present study focuses on determining soil dynamic property, specifically the shear wave velocity at the top 30 m, to establish seismic microzonation maps and to evaluate the effect of soil flexibility on the frame structures of industrial building design in terms of drift. The model of industrial building steel frame was built on the basis of a 3D model using the SAP2000 software. The applications of spring stiffness for rigid footing spring constraints include average shear wave velocity to define the shallow foundation of the structure. The flexible- and fixed-base models were developed to evaluate the structural performance.

Waggawagga-CLI: A command-line tool for predicting stable single α-helices (SAH-domains), and the SAH-domain distribution across eukaryotes.

Stable single-alpha helices (SAH-domains) function as rigid connectors and constant force springs between structural domains, and can provide contact surfaces for protein-protein and protein-RNA interactions. SAH-domains mainly consist of charged amino acids and are monomeric and stable in polar solutions, characteristics which distinguish them from coiled-coil domains and intrinsically disordered regions. Although the number of reported SAH-domains is steadily increasing, genome-wide analyses of SAH-domains in eukaryotic genomes are still missing. Here, we present Waggawagga-CLI, a command-line tool for predicting and analysing SAH-domains in protein sequence datasets. Using Waggawagga-CLI we predicted SAH-domains in 24 datasets from eukaryotes across the tree of life. SAH-domains were predicted in 0.5 to 3.5% of the protein-coding content per species. SAH-domains are particularly present in longer proteins supporting their function as structural building block in multi-domain proteins. In human, SAH-domains are mainly used as alternative building blocks not being present in all transcripts of a gene. Gene ontology analysis showed that yeast proteins with SAH-domains are particular enriched in macromolecular complex subunit organization, cellular component biogenesis and RNA metabolic processes, and that they have a strong nuclear and ribonucleoprotein complex localization and function in ribosome and nucleic acid binding. Human proteins with SAH-domains have roles in all types of RNA processing and cytoskeleton organization, and are predicted to function in RNA binding, protein binding involved in cell and cell-cell adhesion, and cytoskeletal protein binding. Waggawagga-CLI allows the user to adjust the stabilizing and destabilizing contribution of amino acid interactions in i,i+3 and i,i+4 spacings, and provides extensive flexibility for user-designed analyses.

Numerical simulation of macro-meso mechanical behaviours of sandstone containing a single open fissure under uniaxial compression

By employing the improved rigid block spring method, the failure process of a sandstone example containing a single open fissure under uniaxial compression is modelled. In this method, the intact rock is considered as an assemblage of rigid polygonal blocks. Macro mechanical behaviours are governed by mechanical properties of interfaces between two neighbouring blocks. In the local scale on interfaces, both tensile splitting failure and shearing sliding failure are considered. Micro properties are obtained by a calibration procedure against the stress–strain curve of intact rock under uniaxial compression in the lab. Influences of fissure inclination and length on the uniaxial compression strength (UCS) and failure modes are discussed. Comparisons between numerical simulations and laboratory tests are presented. The growth process of micro-cracks is studied and the mechanism of micro-crack propagation is studied through analysis on the displacement evolution. Some conclusions are drawn: numerical results generally agree well with those from lab tests; the UCS of fissured rock is lower than the intact one; the UCS first goes down then goes up as inclination increases, but decreases gradually as fissure length increases; the micro failure modes are governed mainly by tensile splitting failures.

The mechanical effects of cup inclination and anteversion angle on the bearing surface.

Regarding the cup setting in total hip arthroplasty, range of motion and prevention of dislocation are important. From past reports, the wear of the bearing surface may affect long-term results. This study evaluated the stress applied to the bearing surface by the combined use of the three-dimensional rigid spring model and the finite-element analysis. For contact pressure distribution of the bearing surface, interference analysis was performed using a three-dimensional rigid body spring model. Furthermore, stress was applied to the inner surface of the cup installed in the bone so that the same stress distribution obtained from the interference analysis was achieved. The finite-element analysis was then performed at each condition, which changed the inclination and anteversion angles of the cup to examine the relationship of maximum equivalent stress. The maximum equivalent stresses on the bearing surface under the condition with fixation of an anteversion angle of 0° were 0.78, 0.85, and 1.15MPa at inclination angles of 25°, 40°, and 55°, respectively. The stress value at 55° was approximately 1.5 times greater than that at 25°. The maximum equivalent stresses on the bearing surface under the condition with fixation of an inclination angle of 40° were 0.85, 0.9, and 1.02MPa at anteversion angles of 0°, 15°, and 30°, respectively. The stress value at 30° was approximately 1.2 times greater than that at 0°. This study suggests that large inclination and anteversion angles may enhance the stress on the bearing surface and affect long-term results.

Sensitivity characterisation of a parametric transducer for gravitational wave detection through optical spring effect

We present the characterisation of the most recent parametric transducers designed to enhance the Mario Schenberg gravitational wave detector sensitivity. The transducer is composed of a microwave re-entrant cavity that attaches to the gravitational wave antenna via a rigid spring. It functions as a three-mode mass-spring system; motion of the spherical antenna couples to a 50 μm thick membrane, which converts its mechanical motion into a frequency shift of the cavity resonance. Through the optical spring effect, the microwave transducer frequency-displacement sensitivity was measured to be at 4 K. The spherical antenna detection sensitivity is determined analytically using the transducer amplification gain and equivalent displacement noise in the test setup, which are and , respectively.

A numerical study on the cumulative out‐of‐plane damage to church masonry façades due to a sequence of strong ground motions

SummarySeismic shocks occur sometimes as a sequence, close in space and time, of destructive events of comparable intensity. In these cases, a significant portion of the damage to historical buildings can be related with the cumulated damage on structures that become progressively more vulnerable. This research investigates the specific increase of damage determined by a sequence of strong ground motions, focusing the interest on the out‐of‐plane response of 2 church masonry façades. The dynamic analyses were performed by a specific rigid body and spring model RBSM, which only accounts for out‐of‐plane damage mechanisms. Two idealized models of façade, each made of 2 different masonry bonds, have been studied by applying various sequences of recorded accelerograms. The results highlighted a complex relationship between the spectral content of the seismic shocks and the characteristics of the structures that change in the course of the loading sequence due to the development of damage. The Housner spectral intensity proved to be a reliable scalar measure of the ground motion destructiveness for these façades. Moreover, when considering a design‐consistent accelerogram that causes a relevant damage pattern, ie, with a significant elongation of the effective first period of vibration, the numerical results indicated a possible spectral intensity threshold below which the occurrence of repeated seismic shocks, both before and after the reference design shock, can be considered as irrelevant. On the other hand, a catastrophic increase of damage should be expected when this threshold is overcome.

Rigid block and spring homogenized model (HRBSM) for masonry subjected to impact and blast loading

In the present study, a simple and reliable Homogenization approach coupled with a Rigid Body and Spring Model (HRBSM) accounting for high strain rate effects is utilized to analyse masonry panels subjected to impact and blast loads.The homogenization approach adopted relies into a coarse FE discretization where bricks are meshed with a few elastic constant stress triangular elements and joints are reduced to interfaces with elasto-plastic softening behaviour including friction, a tension cut-off and a cap in compression. Flexural behaviour is deduced from membrane homogenized stress-strain relationships by on-thickness integration (Kirchhoff– Love plate). Strain rate effects are accounted for assuming the most meaningful mechanical properties in the unit cell variable through the so-called Dynamic Increase Factors (DIFs), with values from literature data. The procedure is robust and allows obtaining homogenized bending moment/torque curvature relationships (also in presence of membrane pre-compression) to be used at a structural level within the HRBS model, which has been implemented in a commercial software. At structural level, the approach resorts to a discretization into rigid quadrilateral elements with homogenized bending/torque non-linear springs on adjoining edges.The model is tested on a masonry parapet subjected to a standardized impact and on a rectangular masonry slab subjected to a blast load. In both cases, a number of previous results obtained by literature models are available for comparison, as well as experimental data. Satisfactory agreement is found between the present results and the existing literature in the field, both experimental and numerical.