Linear Elastic Spring Research Articles

Plane straight element with semi-rigid connections: formulation from an energetic approach

Discrete connectors, such as screws, nails, and others, are commonly used to fasten various structural elements. The established hypothesis regarding the behavior of joints conceived with discrete connectors directly impacts the distribution of forces. For example, in the design of truss structures, the resisting moments that arise due to the arrangement of the connectors are often disregarded, as considering the semi-rigid behavior of the joints introduces greater complexities in the modeling required to determine displacements and internal forces. This work presents the stiffness matrix of a plane straight element with semi-rigid connections and its support reactions for three loading cases from the application of the Principle of Virtual Work (PVW). Semi-rigid connections are represented by translational and rotational linear elastic springs at both ends of the element, giving it six degrees of freedom. One example shows the application of this formulation using the Classical Displacement Method and the results are also obtained numerically via the Finite Element Method (FEM)This work presents the stiffness matrix of a plane straight element with semi-rigid connections and its support reactions for three loading cases from the application of the Principle of Virtual Work (PVW). Semi-rigid connections are represented by translational and rotational linear elastic springs at both ends of the element, giving it six degrees of freedom. One example shows the application of this formulation using the Classical Displacement Method and the results are also obtained numerically via the Finite Element Method (FEM).

Development and experimental study of disc spring-based negative-positive-uncoupled stiffness devices for structural multi-level seismic fortification

To achieve the structural multi-level seismic fortification objectives, this paper proposes a novel negative-positive-uncoupled stiffness device (NPUSD) based on combined disc springs with multi-linear elasticity characteristics. This device allows for independent variations in the negative stiffness range under small deformations and the positive stiffness range under large deformations. It primarily utilizes the negative stiffness mechanism for seismic mitigation during medium and small seismic events referring to existing research on traditional negative stiffness devices (NSDs) while providing positive stiffness to prevent collapse during major seismic events. The configuration and working principle of the NPUSD are discussed. By replacing the preloaded linear elastic spring of traditional NSDs with multi-linear elastic spring, the NPUSD can effectively and conveniently achieve stiffnesses decoupling. A design algorithm based on the successive area equivalence principle for the multi-linear elastic spring is proposed. Finally, based on a 56-story steel core-outrigger structure, a comparison between NSDs and NPUSDs is made under the four seismic intensity levels. Results show that the novel NPUSDs contribute to achieving structural multi-level seismic fortification objectives.

Effective boundary conditions for second-order homogenization

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter ɛ≪1, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Asymptotic, second-order homogenization of linear elastic beam networks

We propose a general approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The network may be nearly (rather than exactly) periodic: its elastic and geometric properties are allowed to vary slowly in space, in addition to being periodic at the scale of the unit cell. The reference configuration may be prestressed. A homogenized strain energy depending on both the macroscopic strain ɛ and its gradient ∇ɛ is obtained by means of a two-scale expansion. The homogenized energy is asymptotically exact two orders beyond that obtained by classical homogenization. The homogenization method is implemented in a symbolic calculation language and applied to various types of networks, such as a 2D honeycomb, a 2D Kagome lattice, a 3D truss and a 1D pantograph. It is validated by comparing the predictions of the microscopic displacement to that obtained by full, discrete simulations. This second-order method remains highly accurate even when the strain gradient effects are significant, such as near the lips of a crack tip or in regions where a gradient of pre-strain is imposed.

Analytical formulations of tunnel–soil–pile interaction under seismic excitations

AbstractTunnel–soil–pile interaction (TSPI) is a vital issue during seismic excitation because of the evaluation of the interactive structural behaviors of the tunnel. This study derives analytical formulae for the proposed TSPI model under transverse horizontal shear, vertical shear, and body waves, considering a series of piles along the tunnel's longitudinal axis. For this reason, the soil medium is considered to be an isotropic, homogeneous, elastic, and infinite. Also, the tunnel is assumed to be a beam element to connect to the series of piles by the linear elastic springs. Parametric studies of proposed formulae show the parabolic and exponential variations of tunnel forces and soil pressures when increases the number of piles. Also, recalculated tunnel moments of previous studies by using proposed formulae vary closely, which may indicate the accuracy of the present formulations. Similar variations are obtained for the previous field study verification by the present formulations. Therefore, the design charts and graphs are addressed in the present research which may be used as a standard to enhance this research in the future.

Analytical solution of Tunnel-Soil-Tunnel (TST) interaction under seismic excitation

ABSTRACT This research proposes the analytical formulations for the vertically aligned twin tunnels under the sinusoidal form of seismic excitation in the case of the damped free vibration. The analytical formulae are derived to calculate the interactive tunnel moment, tunnel shear force and soil pressure for the transverse horizontal shear (SH), vertical shear (SV) and body (P) waves. For the derivation of the analytical formulations, the twin tunnels are assumed as beam elements connected by the linear elastic springs. The tunnel-soil-tunnel (TST) interaction model is verified by the previous analytical solution and numerical analysis of the case studies. The ranges of the difference in results between the present research and the previous analytical solution are found to be (3.8 ~ 8.1) %, (6.73 ~ 10.95) % and (2.3 ~ 11.79) % in the case of the SH, SV and P waves, respectively. These differences may represent good agreement about verifications.

Hybrid Rayleigh wave along a nonlocal nonlinear metasurface with two-degree-of-freedom spring-mass resonators

Metasurfaces are devices employing the notion of resonance to transform seismic surface waves into body waves, thereby protecting the subwavelength structures. Using metasurfaces to control surface waves is a rapidly expanding field with intrinsic theoretical value and potential applications everywhere. In this paper, we have combined the concepts of nonlinearity, double mass system, and nonlocal elasticity to unveil the dispersive properties of hybrid Rayleigh waves. To enable a more compact construction for multi-frequency attenuation, two spring-mass resonators coupled with a nonlinear spring are used instead of single-mass resonators. A novel metasurface consisting of an array of nonlinear two-degree-of-freedom (two-mass) spring-mass systems is explored. This metasurface is connected to a nonlocal elastic substrate through a linear elastic spring. The paper provides a simple but effective analytical approach to study the dispersive properties of seismic metasurfaces. The complete explicit solutions are derived for the displacement of spring-mass systems and the Rayleigh waves in the nonlocal host substrate. Two frequency bandgaps are formed due to the presence of a dual spring-mass system in the metasurface. An attempt is made to explain the existence of these multi-frequency bandgaps as a result of the presence of multi-resonators. The effects of nonlinearities (softening and hardening) of the springs, nonlocal elasticity and relative amplitude inputs of the spring-mass system on the frequency (spectral) bandgaps are analyzed in detail via numerous plots.

General boundary conditions implementation for the static analysis of anisotropic doubly-curved shells resting on a Winkler foundation

In the present work an Equivalent Single Layer (ESL) formulation is proposed for the static analysis of doubly-curved anisotropic structures of arbitrary geometry and variable stiffness resting on a Winkler elastic foundation. In-plane and out-of-plane general distributions of linear elastic springs are provided for the implementation of general external constraints along the edges of the structure. The structure is geometrically described accounting for the principal curvatures of the shell object of analysis. A generalized set of blending functions based on Non-Uniform Rational Basis Spline (NURBS) curves is adopted so that arbitrary shaped structures can be modelled with the same approach. The fundamental governing equations are obtained in terms of displacement field unknowns, which has been effectively described accounting for a unified formulation based on the minimum potential energy principle. General anisotropic lamination schemes are considered, setting a general orientation of each lamina, as well as all possible material symmetries. The numerical implementation is performed by means of the Generalized Differential Quadrature (GDQ) method, thus allowing a strong formulation of the structural problem. A series of validation examples is performed on shells with zero, single and double curvatures in which the static structural response provided with the proposed formulation has been compared to that obtained from a refined three-dimensional finite element model, showing a great accordance between these different approaches. The research shows that the employment of higher order theories, together with the GDQ method, allows to obtain very accurate results with a reduced computational cost, compared to finite element simulations.

Micromechanical modeling of the visco-hyperelastic–viscoplastic​ behavior and fracture of aged semicrystalline polymers

This work presents a micromechanical-based visco-hyperelastic–viscoplastic constitutive model for low-density polyethylene (LDPE) films subjected to accelerated ultraviolet (UV) aging. The model captures the evolution of the mechanical and fracture properties of the material. Our proposed framework assumes that the semicrystalline polymer’s macromechanical behavior is the sum of the intermolecular and macromolecular network contributions. The intermolecular part of the model follows a nonlinear composite mixture rule, combining the contribution of the crystalline and amorphous phases, each represented rheologically by a linear elastic spring in series with a nonlinear viscous dashpot. The rheological representation of the macromolecular network consists of a series of two Langevin springs capturing deformation-induced conformation change and interatomic bond deformation between Kuhn segments, as well as a viscous dashpot for chain relaxation. We have developed a novel framework that reformulates the eight chain model to include the energy densities associated with both conformation change and interatomic bond deformation, providing a more accurate description of macromolecular network mechanical and fracture behavior. The aging of semicrystalline polymers results from two competing mechanisms: oxidation-induced strengthening resulting from chemicrystallization and crosslinking, and oxidation-induced softening resulting from chain scissions and chemicavitation. Our proposed model accurately captures the effect of UV aging on the mechanical behavior of the semicrystalline material. Additionally, we introduce the novel concept of a ”healthy network” to predict the evolution of the strain to fracture. The model accurately predicts the evolution of the mechanical behavior and elongation at break over a wide range of UV irradiation doses.

Free vibration analysis of laminated doubly-curved shells with arbitrary material orientation distribution employing higher order theories and differential quadrature method

In the present work, the dynamic behaviour of laminated anisotropic doubly-curved shells characterized by a generalized distribution of the material orientation angle is investigated employing higher order theories. The structural problem is developed following the Equivalent Single Layer (ESL) methodology, setting up a unified approach for the assessment of the displacement field variable with higher order theories. Accordingly, a generalized three-dimensional distribution of the material orientation angle is associated to each layer of the stacking sequence, accounting for an in-plane bivariate power distribution and out-of-plane symmetric and unsymmetric profiles described with both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian Principle, and they are numerically tackled employing the Generalized Differential Quadrature (GDQ) method directly in the strong form. Moreover, a generalized three-dimensional set of linear elastic springs is implemented for the assessment of con-conventional boundary conditions. Furthermore, a generalized isogeometric mapping of the physical domain accounts for arbitrarily-shaped structures. The model is validated successfully with respect to refined three-dimensional classical models, and it is, then, applied systematically to check for the sensitivity of the mechanical response to the structural curvature, external constraints, and material orientation angle distributions.

On the proofs of orthogonality of eigenfunctions for heat conduction, wave propagation, and advection-diffusion problems

The orthogonality of eigenfunctions in problems of unsteady heat conduction in an infinite slab with symmetric and nonsymmetric convective boundary conditions are demonstrated by performing actual integration of the products of the derived forms of eigenfunctions and by implementing the corresponding eigenvalue conditions. The analysis also applies to longitudinal vibrations of an elastic rod attached at its ends to linear elastic springs, and to advection-diffusion problems under appropriate boundary conditions. The same form of eigenfunctions and the same type of eigenvalue condition apply in all three considered cases. The proofs are compared with the well-known general proof of orthogonality of eigenfunctions, which circumvents the actual integration. The presented analysis is pedagogically appealing for use in engineering and applied physics education.

Centrifuge investigation of soil–foundation–superstructure interaction under static loading

Extensive efforts have been made to investigate soil–structure interaction (SSI) through numerical simulations whereas little experimental research is reported that tests the applicability of existing numerical models to SSI problems. To provide such experimental data, a unique series of experiments with 3D-printed aluminium framed structures supported by isolated spread footings were conducted in a geotechnical centrifuge. The centrifuge models included equipment and instrumentation that allowed the application of a uniformly distributed static load to the structures as well as the measurement of column loads and foundation movements. The effect of the stiffness of the structure relative to that of the soil was evaluated by varying the number of storeys in the structures and the type/relative density of foundation soils. The variability of soil properties was also studied by modifying the support conditions at single columns at various plan locations of the structure. It is shown that there is appreciable load redistribution in the superstructure, the amount of which depends primarily on the relative structure–soil stiffness and the location of weak and/or rigid foundations. The comparison between the centrifuge test results and the predictions from a simple numerical model with the footings simulated by appropriate linear elastic springs indicates that such a simple numerical model can provide a reasonable prediction of SSI.

On the Regulation of Oscillations of a Galloping-Based Wind Power Harvesting System

Currently, various possibilities for obtaining energy from renewable sources, in particular, flows of water or wind, are intensively investigated. The most widely used wind power harvesters are those where the working element rotates (a propeller or a vertical axis turbine, such as a Darrieus or Savonius rotor). However, the possibility of using the flow-induced oscillations of elastic structures in order to generate energy is now actively considered. One of the types of such oscillations is galloping, i.e. vibrations of bluff bodies in the direction perpendicular to the incident flow. The occurrence of galloping is due to the fact that aerodynamic forces acting on a bluff body, under certain conditions, create a negative damping. In this paper, we consider a mechanical system consisting of three bodies that can move in a direction perpendicular to the flow. One of these bodies is a square prism, and the other two are material points. The bodies are connected in series with each other and with a fixed support by linear elastic springs. A permanent magnet is rigidly connected to the prism. This magnet moves inside an induction coil. As a result, an electric current is generated in the electrical circuit connected to the coil. For such installations, on the one hand, it is required that galloping occurs at the lowest possible flow speed. On the other hand, at high flow speeds, it is necessary to reduce the amplitude of oscillations so that the device would not be damaged. The influence of the system parameters (in particular, the spring stiffness coefficients) on the stability of the equilibrium and on the characteristics of periodic solutions is studied. It is shown that by changing the stiffness of the springs, it is possible to significantly expand the range of flow speeds where the galloping occurs. The amplitudes of oscillations of bodies increase as the flow speed grows. In order to increase the limit flow speed, at which the amplitudes of oscillations start exceeding the maximum permissible value, a regulating algorithm is proposed. Within the framework of this algorithm, the displacement of one of mass points with respect to the prism is locked/unlocked depending on the current flow speed.

Solvability for a dynamical beam problem on a frictionally damped foundation under a moving load

This paper deals with a dynamic Euler-Bernoulli beam of infinite length subjected to a moving concentrated Dirac mass. The beam relies on a foundation composed of a continuous distribution of linear elastic springs associated in parallel with a uniform distribution of Coulomb friction elements and viscous dampers. The problem is stated in distributional form, and the existence and uniqueness results are established by means of a combination of $ {\rm{ L}} ^\infty$–$ {\rm{L}}^2 $–$ {\rm{L }}^1$ estimates together with a monotonicity argument. Traveling wave solutions are studied in detail in the case without Coulomb friction, and they are shown to be globally exponentially stable under positive viscous damping.

Dynamic analysis on axially functionally graded plates resting on elastic foundation

In this paper, the dynamics of functionally graded plates resting on elastic foundation is investigated using finite element method. The properties of the plate material like modulus of elasticity and material density are considered to be varying exponentially along the axial direction. The elastic foundation is of Winkler type which is modelled as a combination of evenly spaced linear elastic springs. The formulation of the plate is based on the Mindlin plate theory where the effect of rotary inertia and shear deformation are incorporated. The governing equations for the free vibration problem are derived from Hamilton's principle. Two discrete boundary conditions viz. all sides clamped and all sides simply supported are considered. The results of the present methodology are compared with already established results in the literature and good agreement is observed. The effect of different parameters like foundation stiffness and material variation parameter is reported. It is found that the natural frequency of the plate increases with increasing foundation stiffness which suggests that the elastic foundation adds to the stiffness of the system.

A dynamic Euler–Bernoulli beam equation frictionally damped on an elastic foundation

This paper deals with a dynamic Euler–Bernoulli beam equation. The beam relies on a foundation composed of a continuous distribution of linear elastic springs. In addition to this time dependent uniformly distributed force, the model includes a continuous distribution of Coulomb frictional dampers, formalized by a partial differential inclusion. Under appropriate regularity assumptions on the initial data, the existence of a weak solution is obtained as a limit of a sequence of solutions associated with some physically relevant regularized problems.

Viscoelastic mechanical behavior of periodontal ligament: Creep and relaxation hyper-viscoelastic constitutive models

In this contribution, creep and relaxation hyper-viscoelastic constitutive models of periodontal ligament (PDL) was proposed to define PDL as a steady state rheology (linear)-nonlinear viscoelastic fluid biomaterial. The physical elements of the hyperelastic spring, non-Newtonian fluid damper and strain gap were proposed. The hyperelastic spring, linear elastic spring, linear viscous damper, stress lock, and strain gap constituted the creep and relaxation models. The qualitative energy analysis was made in the viscoelastic zone between the instantaneous elastic element and the infinite elastic element in the creep and relaxation models. The mechanical behavior of viscoelastic materials is qualitatively unified. The universal static, creep, and relaxation mechanical behaviors of materials were explained from the energy point of view (graphische viscoelastic). After defining a virtual “infinite elastic curve of viscoelastic solid” in the viscoelastic fluid model, the universal creep recovery mechanical behavior was analyzed. The unity of the opposites between viscoelastic solid and viscoelastic fluid was clarified. In addition, A physical thinking of “anchored” was proposed to qualitatively modify the Boltzmann superposition principle. Instantaneous/infinite elastic-plastic-fracture curve, yield curve and fracture curve were defined to distinguish viscoelastic zone and viscoplastic zone. Finally, the unsteady-rheological (nonlinear) behavior of nonlinear viscoelastic fluid materials and the hyper-viscoelastic-plastic behavior of nonlinear viscoelastic solid materials were predicted.

A simplified and versatile element model for elastomeric seismic isolation bearings

A novel approach for two-dimensional modeling of elastomeric bearings using three springs in parallel is presented. This simplified element model considers as follows: (1) an elastoplastic spring with a smooth transition between branches; (2) a linear elastic spring; and (3) a non-linear elastic spring, and is fully defined by only six parameters. The main advantages of the simplified model are twofold: (1) versatility, as a single model is capable of accurately reproducing the main characteristics of the hysteretic behavior of different types of rubber-based seismic isolators, including low damping rubber bearings (LDRBs), high damping rubber bearings (HDRBs), and lead-core rubber bearings (LRBs) and (2) simplicity, as it requires fewer parameters and it is easier to calibrate from experimental cyclic test results than most currently available models. Model parameters’ identification is illustrated using quasi-static cyclic and earthquake simulator tests of HDRBs and LRBs, demonstrating that the model shows a good agreement between the test-measured and model-predicted hysteretic behavior. Different objective functions are evaluated in the optimization procedure, and their effect on the identified parameters is studied and discussed. This practitioner-oriented model is particularly amenable for implementation in general-purpose structural analysis software. Its usage is strongly recommended as an initial-stage design tool to select the optimal isolation system for a specific project.

A superconvergent elastically coupled double beam element for analysis of adhesively bonded lap joints

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan (“Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element,” Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers.