Discrete Springs Research Articles

A unified dynamic stiffness modeling of multi-plate coupled systems with discrete spring connections

This work presents a unified dynamic stiffness modeling for the vibration analysis of multi-plate coupled systems with discrete spring connections (MPCS-DSC). First, based on the governing differential equation of the plate, the dynamic stiffness matrix (DS) of transverse and in-plane vibration for a completely free rectangular plate is separately derived by combining the generalized superposition method and the projection method. Then, according to the continuity of displacements at the connection point between the spring and the plate, the projected DS matrix of the discrete spring is developed. Then, using an element assembly concept similar to that in the finite element method (FEM), global DS matrices of various coupled systems are determined by assembling the spring's DS matrice and the transverse or in-plane DS matrices of the plate. In order to verify the accuracy and applicability of the proposed method, the free and forced vibration analysis of four types of coupling systems is carried out. The reliability and applicability of the proposed method are confirmed by comparing the present results with those from open literature and the finite element solutions. This study not only expands the application range of DS modeling theory but also provides a powerful tool to investigate the vibration characteristics of the MPCS-DSC.

Deformation features of high-pressure rotary pile reinforced strata by discrete lattice spring modeling (DLSM)

Deformation features of high-pressure rotary pile reinforced strata by discrete lattice spring modeling (DLSM)

Discrete element models for elasto-plastic wave propagation in nonlinear architected materials

Nonlinear architected materials are known to exhibit a plethora of dynamic phenomena that enhance our capacity to manipulate elastic waves. Since these properties stem from complex, subwavelength geometry, dynamic simulations at high resolutions are often intractable at scales of interest. Therefore, prior studies have turned to effective medium models that capture essential properties in the long-wavelength limit. One class of such models is a periodic structure composed of discrete mass, spring, and damper elements, whose constitutive relationships are computed to match behaviors observed in experiments or fine-scale simulations of representative volume elements. While models of this type have been implemented successfully in many cases, the majority of literature has been focused on recoverable deformation, possibly including linear damping. However, history-dependent effects, such as friction and plasticity, have been much less explored. In this work, we develop a discrete element modeling framework for nonlinear architected materials undergoing plastic deformation. Due to the history- and rate-dependent characteristics of plasticity, the framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than an elastic system of comparable size. We demonstrate the method using several examples from the literature and explore means to obtain phenomenological plasticity models for general geometries.

Efficient numerical implementation for nonlinear analysis of pile-supported structures during soil liquefaction through adaptive pile element formulations

The structural behavior of pile-supported structures during soil liquefaction is sometimes insufficiently investigated properly due to the inefficiency in the existing line finite element method (LFEM), which is widely used in practice. The traditional LFEM to simulate pile response requires dense element mesh for accurate results, as it simplifies the continuous surrounding soil as discrete spring elements, resulting in high computational costs, especially for pile-supported structures with a significant number of piles. Additionally, analyzing pile-supported structures in different stages of soil liquefaction requires repetitive updating of p-y curves and tedious starting-over iteration processes. To address these issues, an innovative line element formulation named the pile element formulation is employed and improved to adaptively simulate the surrounding soil conditions at different stages during liquefaction within the element formulation, eliminating the need to model discrete soil springs and significantly improving computational efficiency. The adaptive pile element formulation is then implemented within a refined Newton-Raphson iteration procedure that is proposed for a more effective nonlinear analysis process to consider soil liquefaction without repetitive iteration processes from starting over. Through several examples, the computational efficiency of the proposed method is validated, which requires only 20% of the computational cost compared to traditional LFEM while still providing accurate results, highlighting the application potential of the present study.

Centrifuge and numerical modeling of brittle damage of buried pipelines subjected to tunneling induced ground settlements

Tunnel excavations can seriously threaten the structural integrity of nearby pipelines. In the present study, centrifuge tests and numerical simulations are conducted to systematically investigate the brittle damage of buried pipelines subjected to tunneling induced ground settlements. In the numerical model, the pipeline is characterized as peridynamic (PD) shell structures, and the surrounding soil is modeled as discrete independent Winkler springs, and the modified Gaussian distribution profile is incorporated as displacement-controlled boundary conditions on springs. The effectiveness of the proposed numerical model is demonstrated by comparing against the centrifuge test results. A pipeline is subjected to bending due to tunneling until the initiation of cracks, after which the cracks propagate inwardly quickly and penetrate through the cross-section. Eventually, a ring crack occurs at the cross-section above the tunnel centerline. From parametric analyses, the following factors delay the pipeline damage, i.e., thicker pipe wall, shallower burial depth, loosely soils, smaller soil friction angle, and smaller tunnel-pipeline crossing angle. A pipeline with a thicker wall and a higher critical energy release rate can have a greater initial pipe fracture angle. In any case, the fracture angle always increases steadily if the amount of ground settlement continues to grow.

Dynamic analysis of buried pipeline with and without barrier system subjected to underground detonation

Failure of pipe networks due to blast loads resulting from terrorist attacks or construction facilities, may cause economic loss, environmental pollution, source of firing or even it may lead to a disaster. The present work develops a closed-form solution of buried pipe with barrier system subjected to subsurface detonation. The solution is derived based on the concept of double-beam system. Euler Bernoulli’s beams are used to simulate the buried pipe and the barrier system. Soil is idealized as viscoelastic foundation along with shear interaction between discrete Winkler springs (advanced soil model). The finite Sine-Fourier transform is employed to solve the coupled partial differential equations. The solution is validated with past studies. A parametric study is conducted to investigate the influence of TNT charge weight, pipe material, damping ratio and TNT offset on the response of buried pipe with and without barrier system. Further a statistical analysis is carried out to get the significant soil and pipe input parameters. It is perceived that peak pipe displacements for both the cases (with and without barrier) are increases with increasing the weight of TNT charge and decreases with increasing the damping ratio and TNT offset. The deformation of pipe also varies with pipe material. Pipe safety against blast loads can be ensured by providing suitable barrier layer. The present study can be utilized in preliminary design stage as an alternative to expensive numerical analysis or field study.

A New Technique for Calculating the Effective Length Factor of the Tower in Cable-Stayed Bridge

The stability issues of the tower column (hereinafter referred to as ‘tower’) in the cable-stayed bridge are studied in this paper. Two types of cable-stayed bridges are considered, namely inclined tower cable-stayed bridges without backstays and floating single-tower cable-stayed bridges. Based on the structure feature of the inclined tower cable-stayed bridge without backstays, a novel double-beam model with discrete springs is established to facilitate the linear elastic buckling analysis of the tower. To solve the model, the transfer matrix method is applied, and the stability characteristic equation of the system is derived by considering the boundary and continuity conditions. In this way, the effective length factors for the longitudinal stability of the tower are obtained. At the same time, the finite element method is utilized to verify the results obtained by the proposed model and method. Then, the parametric analysis is conducted to explore the effects of some structure parameters on the effective length factor of the tower. In addition, the influence on the effective length factor of the tower is investigated in detail when the steel cables are replaced with CFRP cables for an inclined tower cable-stayed bridge without backstays. Finally, the same processing method is extended to derive the effective length factor of the tower in the floating single-tower cable-stayed bridge. The comparison with the results of the finite element method confirms the feasibility of the present method again.

Three-dimensional vibration investigation of the thin web gear pair based on Timoshenko beam

In the aerospace transmission system, thin web gears are appreciated for reducing weight. Due to the force excitation from the helical gear, except the in-plane vibrations, gears are also vulnerable to out-of-plane vibrations which may cause system instability. However, few papers investigated how the out-of-plane vibrations influence the system in current years. This study analyzes both the in-plane and out-of-plane vibration modes of thin web gear based on the Timoshenko theory. The governing equations of three-dimensional displacements are derived from Hamilton’s principles. The results show that instability will likely happen in out-of-plane vibration and the existence of discrete springs will deteriorate the instability. Additionally, frequency veering is a common phenomenon in a non-axisymmetry gear system. A comparison between the Timoshenko and Euler–Bernoulli models indicates little differences are founded at zero speed, but a significate gap comes up at high rotation speed. Lastly, a Timoshenko gear pair model with a coupling mesh stiffness matrix is presented. The natural frequencies divers from those with independent discrete springs fixed to the ground. The results indicate the importance of the coupling mesh stiffness matrix in system vibration analysis. This work is introductive in the thin web gear transmission system design.

Experimental and Numerical Analysis of Pile–Rock Interaction Characteristics of Steel Pipe Piles Penetrating into Coral Reef Limestone

In order to study the characteristics of pile–rock action of steel pipe driven pile in coral reef limestone stratum, coral reef limestone at the China–Maldives Friendship Bridge site was selected to carry out indoor physical and model tests with red sandstone as the control group. The test outcomes indicate the following: (1) when substantial deformation is permitted, the coral reef limestone has a considerable strength dispersion, a low post-peak stress decrease rate, and a high residual strength, roughly 30% of the peak strength; (2) when the steel pipe pile penetrates the coral reef limestone, the pile top load shows an obvious sawtooth shape, and with the increase in penetration depth, the pile end load of the high-porosity rock sample gradually decreases, and the pile end load of the low-porosity rock sample gradually increases; (3) when the steel pipe pile is penetrated, the strain value of the red sandstone is about twice that of the coral reef limestone at the same position from the steel pipe pile. These findings indicate that the high porosity and heterogeneity cementation characteristics of the coral reef limestone make the extrusion effect during piling significantly less than that of the red sandstone. In addition, the steel pipe pile penetration process is numerically simulated using a four-dimensional discrete spring model method based on the multi-body damage criterion. The numerical simulation results further demonstrate that the pile-side rock fragmentation during steel pipe pile penetration is the primary reason for the lower bearing capacity of steel pipe piles in coral reef limestone stratums. This method provides a novel approach for studying the mechanical properties of reef limestone. The findings can serve as a guide for the design and construction of steel pipe piles in the reef limestone stratum.

A dynamic fracture finite element model of the buried gas transmission pipeline combining soil constraints and gas decompression

A long-distance natural gas transmission pipeline is the primary solution for natural gas resource distribution. In fact, the longitudinal crack propagation of gas transmission pipelines has always been a concern of the natural gas industry. Because it will cause disastrous accidents once the ductile crack in the pipeline propagates rapidly. Most natural gas transmission pipelines are buried underground to improve the safety of pipeline operation as the backfill soil can absorb energy during the fracture process and restrain the displacements of the flaps. Fracture control is a fundamental requirement for the safe operation of underground pipelines. In this study, the dynamic fracture finite element model of the buried gas pipeline is developed to simulate the actual fracture process instead of the costly full-scale burst experiments. The finite element model incorporates different external soil constraints and internal gas decompression behaviors during pipe rupture in which soil constraints are represented by discrete soil springs, and a user-defined subroutine realizes the gas decompression behavior. The explicit dynamic analysis is employed to simulate the high-speed dynamic fracture process of the pipeline, and the crack propagation is simulated by using the cohesive zone model. Finally, the crack propagation velocity is calculated by this model, and the influence of different internal pressures on the fracture velocity is studied; the crack tip opening angle (CTOA) is predicted, and the effect of soil spring stiffness on CTOA is evaluated.

Innovative Timoshenko soil-pile integrated element for lager diameter laterally-loaded piles considering soil-pile interactions

The large diameter piles (LDP) are designed to provide strong bearing capacity, which should be carefully analysed to ensure the superstructures’ reliability. The conventional discrete spring element (DSE) method with the Euler-Bernoulli beam element is inappropriate for LDPs as the pile’s shear deformations may be considerable. Besides, the nonlinear soil-pile interactions (SPI) cannot be conveniently simulated with the DSE method. To accurately and effectively evaluate the lateral performance of LDP under extreme load, this paper develops an innovative soil-pile integrated element based on the Timoshenko beam theory. The Green-Lagrange strain is used to formulate the potential energy function with the SPI considered. The Gauss integration method is adopted to simplify the element formulation and evaluate the SPI within the element end nodes. The secant and tangent matrixes are derived for calculating the element deformations and internal forces. The transformation matrix describing the nodal parameters’ relation between element local and global coordinate is presented for practical numerical analysis. The soil-pile integrated element avoids using discrete soil spring elements, significantly reducing the modelling and computing efforts. The verification examples demonstrate that the proposed soil-pile integrated element has high accuracy and efficiency in geometrically nonlinear analysis.

Experimental analysis, discrete modeling and parameter optimization of SLS-printed bi-pantographic structures

Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder–Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.

Acoustic Radiation Study of a Box Girder Viaduct Considering the Frequency-Dependent Viscoelasticity of the Rail Pad

In order to investigate the mechanism of the frequency-dependent viscoelasticity of the rail pad on the acoustic radiation characteristics of a box girder viaduct, this study establishes a high-order model of its dynamic parameters to reveal the frequency-varying viscoelasticity of the rail pad, and establishes a vehicle–track–viaduct vertical coupling model. Finally, the acoustic radiation characteristics of a box girder viaduct are analyzed by combining the finite element method and the boundary element theory. The results show that the S-stiffness and D-stiffness of the rail pad increase with the increase in frequency, and the frequency sensitivity of the S-stiffness is greater than that of the D-stiffness. The high-order characterization model of the dynamic parameters of the rail pad has a good fitting effect. The main influence frequency band of the frequency variable viscoelasticity of the rail pad on the wheel–rail force and the equivalent discrete spring force of the sliding layer is 30–90 Hz, resulting in the shift of the dominant frequency to a high frequency by 4 Hz. We consider that the frequency-varying viscoelasticity of the rail pad will cause the dominant frequency of the acoustic pressure level of the field point to shift to a high frequency of 4–6 Hz, which has the greatest influence on the sound pressure level of each field point at the Peak Frequency Point of Insertion Loss (PFPIL), and the influence degree is consistent, resulting in the maximum value of the total sound pressure level of the surface field increasing by 4.1 dB. Without considering the frequency-varying viscoelasticity of the rail pad, the sound pressure level of each field point at 20–53 Hz will be overestimated and the sound pressure level of each field point in the 53–100 Hz frequency band will be underestimated. The panel sound power level contribution coefficient of the box girder is obviously different at different frequency points.

Transfer matrix method for analyzing dynamic response of multi-span elastically supported SFT under moving load

This paper proposed a new theoretical model for analyzing the submerged floating tunnel(SFT) under moving load, whose boundary at two ends was regarded as elastic constraint, and the anchor cable was simplified as the discrete spring support. Besides, the tunnel joint was simulated as a flexural spring to connect two adjacent tube segments. The modes and natural frequencies of the model were solved by transfer matrix method, and the dynamic response of the SFT under moving load was investigated by the modal superposition method, which was verified by comparing with the results obtained by the general FEM software. The influence of boundary conditions, dissimilarity of cable stiffness and joint flexural stiffness on the dynamic response of the SFT was analyzed. The difference of the response between the beam on elastic supports (BOES) and the beam on elastic foundation (BOEF) was compared. The results showed that the boundary conditions had little influence on the vibration response of SFT's intermediate spans, but had great influence on the displacement and bending moment of the side span. Besides, even in the case when the frequency differentials of the BOES and the BOEF were small (1.84%), the displacements of the two models also had significant difference. The disparity of the cable stiffness and joint flexural stiffness had great influence on the distribution of displacement and bending moment of SFT.

Dynamic amplification in a periodic structure with a transition zone subject to a moving load: Three different phenomena

The study of periodic systems under the action of moving loads is of high practical importance in railway, road, and bridge engineering, among others. Even though plenty of studies focus on periodic systems, few of them are dedicated to the influence of a local inhomogeneous region, a so-called transition zone, on the dynamic response. In railway engineering, these transition zones are prone to significant degradation, leading to more maintenance requirements than the rest of the structure. This study aims to identify and investigate phenomena that arise due to the combination of periodicity and local inhomogeneity in a system acted upon by a moving load. To study such phenomena in their purest form, a one-dimensional model is formulated consisting of a constant moving load acting on an infinite string periodically supported by discrete springs and dashpots, with a finite domain in which the stiffness and damping of the supports is larger than for the rest of the infinite domain; this model is representative of a catenary system (overhead wires in railway tracks). The identified phenomena can be considered as additional constraints for the design parameters at transition zones such that dynamic amplifications are avoided.

Topological acoustic sensing of ground stiffness: Presenting a potential means of sensing warming permafrost in a forest

The exploitation of sound as a remote sensing platform for environmental monitoring is a novel avenue for detecting changes in subsurface temperatures and properties in permafrost. We present a workflow for using the geometric phase of ground-supported acoustic waves (i.e., seismic waves) due to scattering by trees in a forest as a means of detecting global changes in permafrost stiffness. We work with a simulated environment composed of discrete masses and springs as a testbed for the vibrational response of a forest. For two different spatial distributions of trees, the geometric phase of ground-supported acoustic waves is numerically calculated using MATLAB to show how changes in ground stiffness, due e.g., to changes in subsurface temperature, affect changes in the phase. For acoustic waves with frequencies near the resonant frequencies of the trees the phase is shown to vary more drastically with ground stiffness. This topological acoustic workflow may present a new approach to characterizing changes in permafrost and provide early warning of thawing permafrost, which is vital to maintaining infrastructure and access to natural resources.

Analytical and finite element analyses of flexural and torsional buckling of I-columns with discrete braces

The present study investigates the buckling strength of braced thin-walled columns under bending and torsional modes. Unlike the well-known bending modes, the torsion buckling modes of braced columns are not adequately assessed in design and are often overlooked. In order to control the buckling behaviour, the effects of elastic discrete springs in bending and torsion are taken into account. Analytical solutions are derived for bending and torsion buckling modes in the case of simply supported struts with doubly symmetric cross sections. Further, the threshold bracing stiffness to achieve the higher buckling modes is found in both bending and torsion modes. For more general cases, the finite element approach of the model is implemented and it is able to carry out the buckling modes of columns under compression in the presence of 3D elastic springs and for any arbitrary cross sections. The analytical and the numerical results of the present models are compared to some available solutions in the literature and to finite element simulations of the commercial code Abaqus. The efficiency of the closed-form solutions and the numerical approach is successfully verified. The post-buckling of braced columns is considered at the end in the presence of elastic bending and torsion springs. The effect of braces to enhance the buckling capacity concludes the study.

A Review on stability and passivity analysis of haptic devices

Haptic devices are robots that are used for simulating virtual objects for their operators. Since most real objects can be modelled by spring and dampers, the virtual object is also a (discrete) spring and damper. The discretization is done with a constant sampling time, while a time delay may exist in the control loop, due to do different sources. Simulating stiff objects is a challenging problem since severe vibrations occur in the body of the haptic device and may damage itself or even its operator. The stable operation of a haptic device is a key problem in all haptic applications. It is shown that using smaller values of sampling time and time delay, can make an increase in the stability domain; while increasing effective mass and friction has the same result. In this paper, a review of the stability and passivity of haptic devices is presented and helps the readers to know better recent activities in this field.

Experimental and numerical investigations on the mode I delamination growth behavior of laminated composites with different z-pin fiber reinforcements

Z-pinning can effectively improve delamination growth resistance of fiber reinforcing laminated composites and attracts much attention. Material and geometrical characteristics of z-pins have effects on the delamination behavior. Numerous studies have been conducted to investigate the effect of volume fraction, diameter, shape, surface treatment and angle of z-pin, insertion length and areal density. However, the effect of elastic modulus of z-pins on the delamination behavior is still lack of investigation. In this work, the delamination behavior of two kinds of z-pinned laminates is experimentally and numerically studied. The laminates are reinforced by z-pins with different elastic moduli, including carbon fiber pin and polyimide fiber pin. It is found the elastic modulus of z-pins has effects on the steady-state fracture toughness, the bridging zone length and the micro failure mechanism. A bilinear cohesive zone model is applied for the simulation of delamination in composite laminates and discrete nonlinear spring elements are used for the characterization of the mechanical behavior of z-pins. The FE model is validated by comparing the predictions with the experimental results. Validated FE model is further applied to numerically investigate the density, the distribution distance and the distribution form of z-pins on the mode I delamination behavior.

Sedimentary and geomorphic evidence of Saharan megalakes: A synthesis

It has long been recognised that the Sahara Desert contains sediment, landform and palaeoecological evidence for phases of increased humidity during the Quaternary period. Many authors have also suggested that during some of these humid periods very large lakes, termed megalakes, developed in several basins within the Sahara. Recent work has questioned their existence. In particular it has been argued that the lack of well-developed and spatially extensive shorelines in these basins suggests that discrete groundwater and spring deposits have been misinterpreted as evidence for megalakes. In this paper we re-evaluate the evidence used to identify megalakes. Firstly, we apply a comprehensive remote sensing and GIS analyses to the megalake shorelines, their catchments and the wider Sahara. This not only supports the previously proposed existence of numerous megalakes, but also indicates a previously unrecognised megalake in the Niger Inland Delta region, here named Megalake Timbuktu. Secondly, we review the geomorphic and sedimentary evidence for the megalakes, highlighting the importance of the sedimentary record in identifying lake highstands, particularly through the example of the Chotts Megalake in southern Tunisia where we provide new sedimentary information on lake shorelines. This analysis demonstrates that in much of the Sahara the dynamic aeolian systems preclude the preservation of well-developed shorelines, but the distribution of fragmented geomorphic features and localised lake deposits provide robust evidence for Quaternary megalake formation. The paper concludes by highlighting that although extensive evidence for Saharan megalake formation exists, the current chronology of lake highstands indicates that the vast majority date to Marine Isotope Stage (MIS) 5 or earlier. Only megalakes Chad and Timbuktu, which derive much of their water from outside the desert, show evidence for Holocene (African Humid Period or AHP) shorelines. The AHP record of the other megalakes indicate the existence of much smaller water bodies than those that developed earlier in the Pleistocene indicating that it was significantly drier than these earlier humid phases.