Elastic Foundation Research Articles

Dynamic analysis of laminated plate and ballastless track slab on Pasternak foundation under moving load based on different shear deformation theory

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Analytical Solution for Longitudinal Seismic Responses of Circular Tunnel Crossing Fault Zone

ABSTRACTThis paper proposes a simplified analytical solution for longitudinal seismic responses of a circular tunnel crossing a fault zone under longitudinally propagating shear waves. The transmissions and reflections of shear waves at two geological interfaces between the fault zone and intact rock are considered when calculating the free‐field displacement. An improved elastic foundation beam model considering different tangential contact conditions at the tunnel‒rock interface is also adopted. According to the continuous conditions at the two geological interfaces, explicit expressions for the tunnel displacement, bending moment, and shearing force are given. The effectiveness of the proposed analytical solution is validated via numerical simulations, and the importance of accounting for tangential contact conditions at the tunnel‒rock interface is emphasized. Moreover, parametric studies are performed to investigate the effects of the fault zone width, rock conditions, tunnel lining stiffness, tangential contact conditions, and earthquake frequency on the deformation and internal forces of tunnels subjected to seismic waves. This novel analytical solution can be utilized to quickly estimate the longitudinal seismic responses of circular tunnels crossing fault zones subjected to longitudinally propagating shear waves, particularly in the preliminary engineering design, and can be extended to geological conditions with multiple interfaces.

Shear behavior of multi-row perforated GFRP rib shear connectors in GFRP-concrete hybrid beams/desks

Perforated GFRP ribs (PFRs) and epoxy bonding are commonly used shear connections ensuring a full composite action in FRP-concrete hybrid beam/deck. This study demonstrates test results of 27 push-out specimens to compare the failure modes, shear load-slip (P-S) curves, and shear performance of PFRs and epoxy bonding. The variables included the sanding depth on GFRP profiles (0.5 mm/1.0 mm) and the rows of PFRs (1/2/3/5). The push-out tests highlighted that the PFR specimens with 0.5 mm or 1.0 mm sanding depth experienced interface debonding failure and shearing failure of PFR, respectively. The epoxy bonding specimens with 0.5 mm or 1.0 mm sanding depth suffered adhesive failure and cohesive failure, respectively. Increasing the sanding depth from 0.5 mm to 1.0 mm is an effective approach to prevent the debonding failure. The typical shear load-slip (P-S) curves for PFRs exhibit the micro-slipping stage (slip≤0.2 mm) and the slipping stage (slip>0.2 mm). The P-S curves of epoxy bonding showed only the elastic stage. The shear capacity of the double-row PFR connector was close to epoxy bonding. The shear capacity and stiffness of the PFRs increased 37.2∼47.3 % and 99.7∼119.7 % with the sanding depth improved from 0.5 mm to 1.0 mm, respectively. Increasing the sanding depth would improve the shear stiffness of epoxy bonding. The shear capacity and stiffness of PFRs decreased with the rows of PFRs increased. Calculation equations for the shear capacity of epoxy bonding and PFRs were evaluated by comparing the test results collected from the literature and calculated results. Based on the elastic foundation beam theory, a calculation equation for the shear stiffness of PFRs was proposed and a theoretical bi-linear model was suggested to predict the load-slip curves.

Comprehensive Analysis of Static, Buckling, and Free Vibration Behavior of Carbon Nanotube Reinforced Composite Plates on Pasternak’s Elastic Foundation

Comprehensive Analysis of Static, Buckling, and Free Vibration Behavior of Carbon Nanotube Reinforced Composite Plates on Pasternak’s Elastic Foundation

Flexural Rigidity Influence on Dynamic Response of Orthoropic Rectangular Plate Resting on Constant Elastic Foundation

This research article considers the flexural rigidity influence along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions. The orthotropic rectangular plate model is a coupled fourth order partial differential equation having variables and singular coefficients. The solutions to this model are arrived at by reconstructing the fourth order partial differential. This partial differential equation model is converted to a set of coupled second order ordinary differential equations by using a special technique adopted by Shadnam et al., [11]. This set of second order ordinary differential equations is then reduced using modified asymptotic method of Struble. The closed form solution is evaluated, resonance conditions are obtained and the results are showed in plotted curves to depict the influence of flexural rigidities along both x and y axes on the dynamic response of orthotropic rectangular plate resting on constant elastic foundation with elastic end conditions for both cases of moving distributed mass and moving distributed force.

Analysis of the transverse vibration of a multistepped FGM beam resting on a Winkler foundation in a thermal environment and carrying concentrated masses

This paper examines a novel case study involving the free vibration of a stepped FGM beam-mass system resting on elastic foundations and subjected to a nonlinear or uniform thermal load. The present study examines the complex interaction of thermal effects on dynamic behavior by varying the cross-section and length ratios, material parameters and mass magnitudes. After adopting Euler-Bernoulli's theory, the dynamic behavior of non-uniform FGM beams was investigated using the generalized finite element method (GFEM) and the Rayleigh quotient finite element method (RQFEM). In line with the objective set, relevant results derived from parametric studies, showcasing accentuated variation in dimensional frequency curves and bending moments, have been highlighted and discussed.

Analysis of bi-directional functionally graded plate on an elastic foundation subjected to low velocity impact using the refined plate theory

This study investigates the mechanical response of a bi-directional functionally graded plate consisting of three distinct materials when subjected to low-velocity impact. Utilizing the New Refined Plate Theory, the research explores the mechanical behavior of the plate resting on a Pasternak elastic foundation, providing insights useful for engineering applications. The investigation examines various parameters, including initial impact velocity, projectile radius, volume fraction indices, and elastic foundation stiffness, and their effects on critical response characteristics such as contact force, indentation, lateral deflection, and projectile velocity. The theoretical predictions are validated through detailed comparisons with existing literature and numerical simulations, ensuring the reliability and applicability of the proposed methodology. This study introduces a novel approach using refined plate theory to analyze low-velocity impact on 2D-FGMs, providing practical insights for structural analysis. The findings deepen understanding of functionally graded materials and enhance the evaluation of impact-resistant structures. The research highlights the importance of diverse parameters in improving structural performance and reliability, relevant across aerospace, civil, and mechanical engineering. Detailed exploration shows that initial impact velocities and projectile radius enhance impact force and deflection, while volume fraction indices and foundation stiffness influence the dynamic response.

Finite element algorithm for rigid pavement resting on Pasternak elastic foundation model under aircraft loading

Finite element algorithm for rigid pavement resting on Pasternak elastic foundation model under aircraft loading

Free Vibration of the Magneto-Electro-Elastic Plates Resting on Elastic Foundation Using the Refined Plate Theory with Two Variables

The objective of this article is to investigate the free vibration analyses of the functionally graded magneto-electro-elastic (FG MEE) plates supported by an elastic foundation using the refined plate theory (RPT) with two variables. The elastic foundation is modeled utilizing the Winkler-Pasternak theory. The power-law model is employed to characterize the graded material properties of the FG MEE plates. According to the RPT and Hamilton's principle, the governing equations of the FG MEE plate are derived. The displacement fields and electric and magnetic potentials are approximated using the Non-Uniform Rational B-Splines (NURBS) basic functions of the isogeometric approach (IGA). The proposed model’s advantages and accuracy are demonstrated by comparing the obtained results with those reported in the existing literature. The study comprehensively examines and discusses the impact of several parameters, including the power index, initial external magnetic potential and electric voltage, and the geometry, on the vibration frequency of the FG MEE plates. The numerical findings indicate that an increase in the power index leads to a decrease in the frequency of the FG MEE plates. Besides, the stiffness of FG MEE plates decreases with an increase in the initial external electric voltage, whereas it increases with an increase in the initial external magnetic potential. This article presents valuable perspectives on examining vibration analysis of the FG MEE plates, which can inform the design of innovative materials and structures with customized properties.

Dynamics of a compressed Euler–Bernoulli beam on an elastic foundation with a partly prescribed discrete spectrum

Dynamics of a compressed Euler–Bernoulli beam on an elastic foundation with a partly prescribed discrete spectrum

Comparative study on dynamic responses of ballasted and ballastless tracks at critical velocity

High-speed railways are becoming ubiquitous worldwide, demanding increased operational speeds. Yet, as trains near critical speeds, dynamic stress and embankment displacement escalate, jeopardizing safety. While significant strides have been made in studying these critical speeds, capturing the dynamic characteristics of all track responses remains elusive. This study presents a coupled train-track-ground model employing a 2.5D finite element approach integrated with a nonlinear soil model to investigate the influence of embankment and foundation properties on the critical speeds for both ballasted and ballastless tracks. The research resulted in the development of unified Dynamic Amplification Factor curves that consistently represent the dynamic behavior of various tracks as observed in multiple studies. Additionally, the applicability of simplified theoretical models for calculating critical speeds, such as the elastic half-space foundation and track-foundation models, was assessed. The findings suggest that the simplified models are suitable only under specific conditions. This research provides valuable perspectives on optimizing train speeds and ensuring the safety of diverse track types.

Study on the seismic wave input method for earth-rock dam on overburden foundation with dynamic nonlinearity based on nonlinear artificial boundary

Accurately obtaining the seismic response of earth-rock dams on overburden foundations is crucial for seismic safety assessment and design, and a reasonable seismic input method is necessary. The seismic wave input method based on artificial boundaries is widely used in elastic foundations. However, the overburden with obvious dynamic nonlinearity is an insurmountable obstacle for the traditional method. In this research, relying on a specific earth-rock dam project, the essence of the seismic wave input method for nonlinear foundations is analyzed in detail from equivalent loads and viscous boundaries, respectively. The mechanism of inapplicability of the traditional seismic wave input method is illustrated, and an effective modified method is validated and presented. The research indicated that equivalent loads for the nonlinear dynamic response of the overburden foundation can be accurately acquired with the shear box model. The dynamic parameters of soil can be captured dynamically by the nonlinear viscous coupling boundary. The states of the lateral boundary of the overburden foundation are consistent with the free field precisely with the modified seismic wave input method by combining the shear box model and the nonlinear viscous coupling boundary. A large error will be caused by neglecting the dynamic nonlinear behavior of the overburden foundation, where the maximum deviation of peak acceleration may be over 55 % or more. A numerical model with a relatively small lateral range of overburden foundation will be obtained when the modified seismic wave input method is engaged. Moreover, the computational accuracy is enhanced remarkably, with a maximum deviation of no more than 5 %. This study provides effective theoretical support for seismic analysis and safety assessment of earth-rock dams on deep overburden foundations.

Vibration suppression of a platform by a fractional type electromagnetic damper and inerter-based nonlinear energy sink

The theory of linear and nonlinear dynamic vibration absorbers is a well-established topic for many years. However, many recent contributions paid attention to the nonlinear vibration absorbers and different practical realizations of corresponding devices. Here, we propose a mechanical system constituted of the inerter-based nonlinear energy sink attached to the main body that is resting on an elastic foundation and is grounded through the fractional type electromagnetic damper. The two-degree-of-freedom system is described via two coupled differential equations with one of them having a fractional-order derivative term and the other one containing cubic stiffness nonlinearity. The incremental harmonic balance (IHB) method is employed to solve the equations and studies the strongly nonlinear periodic responses of the system. Applied approximated solution methodology is validated by the numerical Newmark method adapted to deal with the system of nonlinear fractional-order differential equations. The appropriate and necessary number of harmonics used in the IHB solution is commented and validated. This study can be a first step in understanding the dynamics and giving directions for the future design of vibration-isolating platforms based on inerter-based nonlinear vibration absorbers and electromagnetic dampers.

A Comprehensive Model to Compute Roll Gap Profile for a Rolling Mill Including Bearing and Mill Housing Deflections

Abstract Obtaining the desired strip profile is of paramount importance in cold rolling. The final strip profile depends on several parameters of the plants including crown and taper profiles and segmented backup roll positions. This paper aims to present a newly developed combined finite element–analytical model of the rolling mill able to model not only the roll cluster but also the mill housing and backup bearings deformations. This makes the proposed model especially suitable to simulate cluster-type rolling mills. The model exploits a modified elastic foundation formulation for a precise computation of the contact forces between rolls, taking into account rolls’ crown and taper. A reduced stiffness matrix approach is implemented for accurate mill housing modeling. A full 3D finite element model of the rolling mill and experimental data are used for the validation of the presented model. By exploiting the validated model, the paper shows that the deformation of supporting bearings and mill housing plays a crucial role in the strip profile calculation and should be included in any cluster-type roll mill model.

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

Analytical method for free vibration of steel pipe piles partially submerged in water and penetrated in layered soil

This paper presents a novel analytical method for dynamic characteristics of steel pipe piles submerged in water based on the thin shell theory by using the state space method. The state equation with respect to the length direction is first derived by idealizing the steel pipe pile in water as a thin cylindrical shell, where the soil around the pile is modelled as an elastic foundation with stiffnesses in three directions and the surrounding seawater is considered through added equivalent mass. The governing equations for the free vibration of steel pipe piles are derived after the expansion of Fourier series for the state vectors along the circumferential direction. The frequency equation for free vibration and its corresponding mode shapes are then obtained, which is used to analyze the dynamic characteristics of steel pipe piles in water. Due to the advantage of the state space method, arbitrary boundary conditions of the steel pipe pile are conveniently achieved as well as the effects of seawater, soil, the piling hammer and guiding frame. Finally, the present analytical method is verified and demonstrated in detail by several numerical examples, and results show its simplicity and efficiency.

Recent developments in computational modelling of the knee

ObjectiveModel calculations of knee joint loading range from an assumption of perfectly rigid articular surfaces to more realistic simulations of cartilage and meniscal deformation. Rigid-body musculoskeletal models simulate knee contact mechanics using the ‘bed of springs’ method from elastic foundation theory whereas finite-element models discretise each structure into a series of interconnected elements and ascribe material properties to each element. This mini-review describes some of the most recent developments in computational modelling of knee contact mechanics and suggests possible avenues for future improvements. DesignNarrative mini-review. ResultsMuscle and joint contact forces can be calculated synchronously at a reasonable computational cost (typically a few hours of CPU time) using rigid-body models and elastic foundation theory whereas similar calculations using fully deformable finite-element models can take several days and even weeks. The main computational expense incurred in finite-element musculoskeletal modelling is the solution of a muscle-force optimization problem. ConclusionCalculation of muscle and joint contact forces within the framework of a finite-element musculoskeletal model remains challenging. Integrating biomechanical data from human motion experiments with fully deformable finite-element models to simulate knee contact mechanics during dynamic activity is an evolving science. Future work should explore the use of efficient methods such as direct collocation to perform muscle-driven dynamic optimization simulations of movement using finite-element musculoskeletal models. Dynamic optimization may be combined with finite-element modelling to enable predictive simulations of movement so that the effects of changes in musculoskeletal anatomy on knee contact mechanics can be studied more systematically.

Analysis of thermo-elastoplastic bending behavior of FG skew sandwich plates on elastic foundation using an enhanced meshless radial basis reproducing kernel particle approach

Analysis of thermo-elastoplastic bending behavior of FG skew sandwich plates on elastic foundation using an enhanced meshless radial basis reproducing kernel particle approach

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

Analytical solutions for out-of-plane response of curved beams resting on an elastic foundation under a random moving load

In this study, the vibration behaviour of a horizontally curved beam resting on an elastic foundation and subjected to a random moving force is studied. The governing coupled differential equations for the out-of-plane vibration of curved beams are derived based on the mode superposition method, Duhamel's integral, and the Laplace transformation. Note that the out-of-dynamic responses of curved beams induced by a random moving load are composed of the centred random and mean value. Analytical solutions are proposed for the beam’s vertical and torsional deflections and the bending moment, where the rotary inertia, warping constant, and foundation stiffness are incorporated in these formulations. Because analytical studies on the random vibration of a curved beam resting on an elastic foundation are very limited, the aim of this study is to investigate the effect of relative parameters, namely, high-mode vibrations, the radius of curvature, the moving velocity, and foundation vertical and torsional stiffnesses, on the centred random and mean values of the beam dynamic responses. The results show that high-mode vibrations have a greater impact on the beam responses at higher velocities. To obtain precise results, the first 30 modes are considered in the calculation for the dynamic responses of the beams. The critical velocity is an inherent property of bridges, and it increases as the radius of curvature and foundation stiffness increase. In addition, an increase in the velocity and foundation stiffness results in a reduction in the centred random values for responses.