Discrete Springs Research Articles

Modeling and parameter analysis of in-plane dynamics of a suspension bridge with transfer matrix method

In this paper, the model of a suspension bridge is composed of the main cable and beam with many discrete springs, which are used to model columns between the main cable and the deck. Based on Hamilton’s variational principle, the governing equations for the in-plane dynamics of the model are obtained. The static and dynamic configuration of the beam and cable is considered in the derivation of governing equations in this paper. The boundary and continuity conditions of the subsystems (cable and beam) and the transfer matrix method are utilized to determine the only six unknown non-dimensional constants of the system. Then, the in-plane eigenvalue problem can be solved easily, and the effects of the parameters on the in-plane natural frequencies and mode shapes are systematically investigated. Finally, the results obtained by the proposed model and method are compared with those obtained by the finite element method, which shows that the proposed method applied to analyze the suspension bridge is efficient.

Interação solo-estrutura para sistemas estruturais reticulados sobre fundações rasas

Neste artigo apresenta-se um programa para consideração da interação solo-estrutura na análise espacial de estruturas reticulares. O método empregado baseia-se na hipótese de Winkler, que admite molas discretas adjacentes as fundações rasas, simulando a influência dos recalques de apoio em estruturas tridimensionais. Embora o modelo seja 3D e, portanto, todos os seis graus de liberdade de cada apoio possam sofrer deslocamentos devidos a um recalque, neste artigo, a análise foi feita considerando apenas a influência da translação verticaldos apoios. O trabalho consiste em incorporar a metodologia para consideração de apoios flexíveis aos fluxogramas apresentados por Geree Weaver Jr. [1], utilizando o Método da Rigidez, o qual é amplamente empregado para análise de estruturas reticulares. Através da análiseintegrada, contemplando parâmetros da infraestrutura, supraestrutura e do terreno de fundação, comprovou-se que a deformabilidade do solotem influência significativa na redistribuição dos esforços e a vinculação entre solo e a estrutura que melhor descreve o comportamento físicode uma edificação é condição de apoios flexíveis.

Exact closed-form solutions for the static analysis of multi-cracked gradient-elastic beams in bending

Cracks and other forms of concentrated damage can significantly affect the performance of slender beams under static and dynamic loads. The computational model for such defects often consists of a localised reduction in the flexural stiffness, which is macroscopically equivalent to a beam where the undamaged parts are hinged at the position of the crack, with a rotational spring taking into account the residual stiffness (“discrete spring” model). It has been recently demonstrated that this model is equivalent to an inhomogeneous Euler–Bernoulli beam in which a Dirac’s delta is added to the bending flexibility at the position of each damage (“flexibility crack” model). Since these models concentrate the increased curvature at a single abscissa, a jump discontinuity appears in the field of rotations. This study presents an improved representation of cracked slender beams, based on a general class of gradient elasticity with both stress and strain gradient, which allows smoothing the singularities in the flexibility crack model. Exact closed-form solutions are derived for the static response of slender gradient-elastic beams in flexure with multiple cracks, and the numerical examples demonstrate the effects of the nonlocal mechanical parameters (i.e. length scales of the gradient elasticity) in this context.

Dynamic analysis of Mindlin plates on viscoelastic foundations under a moving vehicle by CS-MIN3 based on C0-type higher-order shear deformation theory

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed to improve the performance of the existing three-node Mindlin plate element (MIN3) for static and dynamic analyses of Mindlin plates. In this paper, the CS-MIN3 is extended to the C0-type higher-order shear deformation plate theory (C0-HSDT) and incorporated with damping-spring systems for dynamic analyses of Mindlin plates on the viscoelastic foundation subjected to a moving vehicle. The plate-foundation system is modeled as a discretization of triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. A two-step process for transforming the weight of a four-wheel vehicle into loads at nodes of elements is presented. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results. A parametric examination is also conducted to determine the effects of various parameters on the dynamic response of the plates on the viscoelastic foundation subjected to the moving vehicle.

Nonlinear analysis of damped semi-rigid frames considering moment–shear interaction of connections

Abstract Nonlinear vibration analysis of semi-rigid frames, considering moment and shear interaction (MVI) for the joint, is performed in this article. The numerical model of the semi-rigid frames is created using the nonlinear finite element (FE) formulation. In the numerical model, the beams and the columns are formulated based on the Euler–Bernoulli beam theory and the Partially Restrained (PR) connections are simulated as a discrete rotational springs and a parallel rotational damper. The previously formulated three-parameter model is modified and the nonlinear moment–rotation relation of the connection is tied to its shear force to assess the interaction effects of the moment and shear of the connections. An analysis algorithm is proposed to solve the eigenvalue problem of the semi-rigid frames subjected to pre-loading conditions, taking into account the MVI effects. The method is verified through comparison with analytical examples.

A force-based frame finite element formulation for analysis of two- and three-layered composite beams with material non-linearity

Presented in this paper is a formulation for an efficient frame finite element (FE) model for the non-linear analysis of timber–concrete and timber–timber composite structures. The element formulation is based on force interpolation concept and takes advantage of discrete non-linear springs to model the shear deformability between layers. The material non-linearity is taken into account, where the geometrical non-linearities are ignored. The developed formulation and analytical tool are employed to predict the ultimate loading capacity and load–deflection response of some experimental tests taken from the literature. The comparison between experimental results and numerical simulation shows the accuracy of the proposed model.

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamics analyses of Mindlin plates. In this paper, the CS-FEM-DSG3 is extended to the layerwise deformation theory for dynamic response of sandwich and laminated composite plates resting on viscoelastic foundation subjected to a moving mass. The plate-foundation system is modeled as a discretization of triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results.

A New Design of Vibration Absorber for Periodic Excitation

The authors designed a novel type of dynamic vibration absorber, called periodic vibration absorber (PVA), for mechanical systems subjected to periodic excitation. Since the periodic rather than single harmonic excitation is themost occurring case in mechanical systems, the design of PVA is hence of engineering significance. The PVA designed in this paper is composed of a dual-beam interconnected with a discrete spring in between. By appropriately choosing the design parameters, the PVA can be of resonance frequencies in integer multiples of the base frequency such that the PVA can absorb significant amount of higher harmonics in addition to the base harmonic. The designed PVA was first experimentally verified for its resonance frequencies. The PVA implemented onto a mechanical system was then tested for its absorption ability. From both tests, satisfying agreement between experiments and numerical calculations has been obtained. The sensitivities of the design variables, such as the discrete spring’s stiffness and location, were discussed as well. The parameters’ sensitivities provided us with the PVA’s adjustable room for excitation frequency’s mismatch. Numerical results showed that within 3% of frequency mismatch, the PVA still performed better than a single DVA via adjusting the spring’s constant and location. All the results proved that the novel type of PVA could be a very effective device for vibration reduction of mechanical systems subjected to periodic excitation.

Coupling up: How Interactions between Cell Stresses and Intracellular Biochemistry Affect Cell Spreading

We present a two-dimensional mathematical model and finite element simulations that allow us to better understand how local cellular deformations must be coupled to the evolution of an intracellular plaque protein that controls the formation of focal adhesions. Specifically, we explore effects of alternate formulations for coupling cellular response to substrate mechanics. Further, we also investigate the effect of initial cellular shape on cell spreading and intracellular stresses. Our aim is to determine whether the initial anisotropy of a cell predisposes it to remain anisotropic during spreading. In addition, we examine the role of focal adhesion strength in maintaining anisotropy. In the models for cell and substrate mechanics we assume that the cell is an active hypoelastic material and the substrate is linearly elastic. Focal adhesions are modeled as a collection of discrete springs that can be added and removed dynamically. This work aims to unearth some of the fundamental mechanisms in cell-substrate interactions.

A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation

A cell-based smoothed finite element method using three-node Mindlin plate element (CS-FEM-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS-FEM-MIN3 is extended and incorporated with damping-spring systems for dynamic responses of sandwich and laminated composite plates resting on viscoelastic foundation subjected to a moving mass. The plate-foundation system is modeled as a discretization of three-node triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results. A parametric examination is also conducted to determine the effects of various parameters on the dynamic response of the plates on the viscoelastic foundation subjected to a moving mass.

A Mechanistic 1D Finite Element Model for Nonlinear Analysis of FRP-Strengthened Reinforced Concrete Beams

An efficient frame finite element (FE) is developed and applied for nonlinear analysis of reinforced concrete beams flexurally strengthened with externally bonded fibre reinforced polymer (FRP) strips/plates. The developed nonlinear model is able to capture different failure modes associated with concrete crushing and cracking, yielding of steel, FRP rupture and intermediate crack-induced debonding of FRP sheets/plates. The element is formulated using force interpolation concept and a simple approach based on application of discrete springs and linear interpolation of bond forces along the element axis is adopted to take account of shear-slip. The developed formulation and analytical tool are employed to predict the loading capacity and the load-deflection response of reinforced concrete beams strengthened with FRP and the numerically simulated responses agree well with the corresponding experimental results. The major features of this frame element are its simplicity and efficiency compared with more complex FEs which makes it a suitable tool for practical use in design-oriented parametric studies.

Winkler Springs (p-y curves) for pile design from stress-strain of soils: FE assessment of scaling coefficients using the Mobilized Strength Design concept

In practice, analysis of laterally loaded piles is carried out using beams on non-linear Winkler springs model (often known as p-y method) due to its simplicity, low computational cost and the ability to model layered soils. In this approach, soil-pile interaction along the depth is characterized by a set of discrete non-linear springs represented by p-y curves where p is the pressure on the soil that causes a relative deformation of y. p-y curves are usually constructed based on semi-empirical correlations. In order to construct API/DNV proposed p-y curve for clay, one needs two values from the monotonic stress-strain test results i.e., undrained strength (su) and the strain at 50% yield stress (e50). This approach may ignore various features for a particular soil which may lead to un-conservative or over-conservative design as not all the data points in the stress-strain relation are used. However, with the increasing ability to simulate soil-structure interaction problems using highly developed computers, the trend has shifted towards a more theoretically sound basis. In this paper, principles of Mobilized Strength Design (MSD) concept is used to construct a continuous p-y curves from experimentally obtained stress-strain relationship of the soil. In the method, the stress-strain graph is scaled by two coefficient NC(for stress) and MC(for strain) to obtain the p-y curves. MC and NC are derived based on Semi-Analytical Finite Element approach exploiting the axial symmetry where a pile is modelled as a series of embedded discs. An example is considered to show the application of the methodology.

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Abstract We analyze the “fractional continuum limit” and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton’s (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian $ - ( - \Delta )^{\tfrac{\alpha } {2}} $ with 0 < α < 2. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions: We find a characteristic scaling law for the density of normal modes ∼ $\omega ^{\tfrac{{2n}} {\alpha } - 1} $ with a positive exponent $\tfrac{{2n}} {\alpha } - 1 > n - 1 $ being always greater than n−1 characterizing a regular lattice with local interparticle interactions. Furthermore, we study in this setting anomalous diffusion generated by this Laplacian which is the source of Lévi flights in n-dimensions. In the limit of “large scaled times” ∼ t/r α >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t -n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Lévi parameter α.

Buckling mode interaction in prestressed stayed columns

Prestressed stayed columns offer an innovative and aesthetically attractive solution where designs demand long and slender elements under compression. The addition of cross-arms and pretensioned cables to the columns provides significant lateral and rotational support that can result in a marked increase of the buckling capacity. The current work presents a simplified model, comprising discrete rigid links and springs with a mathematically exact formulation including the destabilising effect of cable slackening, which mimics the response of stayed columns and reveals the governing physical parameters that drive the behaviour. A recent finite-element study had indicated that these structural components are likely to exhibit interactive buckling phenomena, particularly if the critical buckling mode is antisymmetric, leading to highly unstable responses. However, by its very nature, that study could not reveal the bifurcational structure of the equilibrium response. The results from the present model reveal similar behaviour to that observed in the earlier finite-element study. A sequence of bifurcations is found that switch the post-buckling response from one mode to another by way of a path of interactive buckling.

Discussion of “Pull-out test and discrete spring model of fibre-reinforced polymer perfobond rib shear connector”

The discussed paper presented a series of pull-out tests for fibre-reinforced polymer (FRP) perfobond rib shear connector and proposed a discrete spring model to investigate the connector's shear behavior. The discussers appreciate the author's meaningful work. Some findings are interesting to the discussers and are worthy of further discussion. The statement that “the ultimate load caused by the failure of concrete dowel increases with larger areas of the rib hole and a larger number of the rib holes” and the conclusions that “the resistance produced by concrete dowel action was practically linearly proportional to the area of the rib hole” and “the resistance of concrete dowel action was not directly proportional to the number of the rib holes and experienced slight loss for a larger number of rib holes” are useful and can be introduced to near surface mounted (NSM) FRP strips for the strengthening of reinforced concrete (RC) beams (Bianco et al. 2009). According to this thought, if we make some holes on the FRP strips, the resistance may be improved greatly. It is worthy of further study. The predicted ultimate load by FRP failure is calculated by the following equation in the discussed paper.

Reply to the Discussion by Xingyan et al. of “Pull-out test and discrete spring model of fibre-reinforced polymer perfobond rib shear connector”

Reply to the Discussion by Xingyan et al. of “Pull-out test and discrete spring model of fibre-reinforced polymer perfobond rib shear connector”

Numerical Modeling of Buried HDPE Pipelines Subjected to Normal Faulting: A Case Study

A systematic study is presented herein on the seismic response of buried pipelines subjected to ground fault rupture in the form of normal faulting. In this study, advanced computational simulations are conducted in parallel with physical testing using a geotechnical centrifuge. For the numerical simulations, the pipeline was modeled using isotropic 3-D shell elements and the soil was modeled using either 1-D spring elements or 3-D solid (continuum) elements. The results from continuum finite-element analyses are compared with those from a Winkler-type model (in which the pipe is supported by a series of discrete springs) and with results from centrifuge tests. In addition, via appropriate modeling of the soil-pipe interaction, the q-z relation of the soil medium is elucidated for normal faulting events. The numerical analysis results demonstrate the potential for continuum modeling of events that induce pipe-soil interaction and results in improved understanding of pipe-soil interaction under normal faulting.

Parallelization of lattice modelling for estimation of fracture process zone extent in cementitious composites

This paper is focused on the verification and validation of the developing technique for estimation of the extent (the size and shape) of the fracture process zone (FPZ) in quasi-brittle silicate-based specimens/structures during failure process (termed the ReFraPro -Reconstruction of Fracture Process - technique). Most experimental data published in the literature are incomplete for its sound validation; therefore, numerical simulations by means of physical discretization of continuum are used for supplementing the verification of the technique. A discrete spring network/lattice particle-type model formulated as a nonlinear dynamical system is utilized. Parallelized implementation within the CUDA environment helps to decrease the computational cost of the simulations to an admissible level. The conducted analysis demonstrates satisfactory agreement of the size and shape of the FPZ reconstructed by the ReFraPro technique with both the data of the performed simulations and selected experimental data from literature.

The Role of Mechanical and Biochemical Interactions in Focal Adhesion and Cellular Shapes

We present findings from a two-dimensional mathematical model of a biological cell interacting with a soft substrate and focus on the role of the interaction of intracellular mechanical stresses and plaque protein concentrations on cell and focal adhesion shapes. The cell is treated as a hypoelastic actively-deforming continuum and the substrate is modeled as a linearly elastic continuum. The active deformation, captured by the addition of an active rate of deformation tensor, models local cytoskeletal reorganization. Focal adhesions connecting the cell and the substrate are modeled as a collection of discrete elastic springs. The model allows for the focal adhesion complexes to grow and shrink depending on mechanical forces that are acting on them and local concentrations of plaque protein. A model of stress-induced plaque protein dynamics, which is based on earlier work of Besser and Safran (2006), is coupled to the model of cell-substrate mechanics. Finite element simulations of the model allow us to explore the effects of original focal adhesion configuration, cytoskeletal dynamics, and focal adhesion strength on the shape evolution of individual focal adhesions and on overall cell shape.

Pull-out test and discrete spring model of fibre-reinforced polymer perfobond rib shear connector

In this study, a series of pull-out tests on fibre-reinforced polymer (FRP) perfobond rib shear connectors were conducted to investigate their shear behavior. The test specimens were designed to examine only the effects of the resistance by rib holes. The test variables were the size and number of rib holes. In the tests, the resistance by concrete dowel action was practically linearly proportional to the area of the rib hole. However, the resistance appeared to be nonlinearly proportional to the number of rib holes with a slight loss for a larger number of the rib holes, due to the sequential failure of the rib holes. The constitutive equation of the discrete spring model representing the concrete dowel including post-failure frictional effects, adopting the diameter of the rib hole as parameter, was derived through regression analysis of the load – relative displacement curve obtained for specimens with a single rib hole. The proposed discrete spring model resulted in good correlation with the experimental results for specimens with a larger number of rib holes. It is expected that the derived constitutive equation would be useful for the analysis and design of structures using FRP perfobond rib shear connectors.