Pasternak Elastic Foundation Research Articles

Modal analysis of plates resting on elastic foundation based on the first-order shear deformation theory and finite element method

This paper investigates the free vibration characteristics of plate structures supported by a Pasternak elastic foundation, utilizing the first-order shear deformation theory (FSDT). FSDT simplifies the plate theory by considering only first-order shear deformation, enhancing formulation simplicity. Additionally, employing plate theory reduces computational complexity, as 2D models entail fewer degrees of freedom compared to their 3D counterparts. The finite element method (FEM) with 8-node quadrilateral element is employed for computational analysis, implemented using MATLAB. First, a comparison is made with some existing data to show the accuracy and reliability of the research. Numerical examples are then presented of the influence of the effects of thickness variation, foundation parameters and boundary conditions on frequency are investigated. The results show that the method converges very fast and reliability when compared to other research findings. The results of the research can be applied to many different engineering applications related to plates resting on elastic foundation.

Analysis of the underlying shield tunnel deformation caused by a newly built channel

As the development of underground space progresses, newly constructed waterways inevitably intersect with existing tunnels, posing significant challenges to the safety of underlying tunnels and making deformation control a critical issue in engineering design and construction. This study investigates the impact of waterway foundation pit excavation on the deformation of underlying tunnels. The research considers not only the vertical unloading at the bottom of the pit but also the horizontal unloading from the four side walls, which induces additional vertical stress fields in the underlying soil. These additional stresses are applied to the tunnel structure, modeled as a Pasternak elastic foundation beam, and the vertical tunnel deformation due to excavation unloading is calculated using the finite difference method. To validate the analytical results, a three-dimensional numerical model is employed. Furthermore, the study analyzes deformation trends under varying excavation sizes and the relative positions of the foundation pit and tunnel. The findings demonstrate that the proposed method provides a highly accurate, cost-effective, and efficient solution for predicting tunnel deformation during the excavation of small foundation pits. This research offers valuable insights for the design and assessment of underground structures in urban environments.

Effect of circular and elliptical cutouts on the free vibration analysis of sandwich FGM plate under the elastic foundations

The present article investigates the effect of circular and elliptical cutouts on vibrational characteristics of porous sandwich functionally graded material (SFGM) plates with FGM core resting on Pasternak elastic foundation. The structural kinematics of the plate are formulated based on nonpolynomial-based higher-order shear deformation theory (HSDT). Geometric discontinuities have been incorporated in terms of circular and elliptical cutouts in the SFGM plates. The sandwich structure is considered to be comprised of two homogeneous face sheets and a porous FGM core with various cutout shapes and sizes. The variation of material properties from the homogeneous face sheet to the FGM core has been carefully modeled using modified power law distribution. Further, a C0 continuous finite element formulation with a four-noded, isoparametric quadrilateral element with seven degrees of freedom (DOFs) per node has been employed to accomplish the results. The accuracy of the present results has been demonstrated through convergence and validation studies. A comprehensive study has been carried out to investigate the influence of volume fraction index, even and uneven porosity distribution, circular and elliptical cutouts, as well as H-elliptical and V-elliptical type cutout shapes on the frequency parameter of SFGM plates with the elastic foundation. The numerical results demonstrate the aforementioned parameters significantly impact the vibrational frequency of the SFGM plates.

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

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

Exact solution for free vibration analysis of non-uniform thickness functionally graded porous nanosheet with surface effect based on variable nonlocal and length-scale parameters

In this article, the analytical solution for free oscillation of non-uniform thickness functionally graded porous (FGP) rectangular nanosheets resting on a Pasternak foundation with various boundary conditions (BCs) based on the nonlocal strain gradient theory and surface effect are discovered. The nonlocal stress and strain effects are captured on the nonlocal strain gradient hypothesis, while the surface energy effects are based on the surface elasticity hypothesis. The FGP nanosheet has a nonlinear thickness change in both x and y directions and is placed on the Pasternak elastic foundation. The unique point of this study is to investigate the change of nonlocal and length-scale coefficients along the thickness as mechanical properties of the material. Applying classical shear deformation theory (ignore the effect of shear deformation) combined with Hamilton’s principle, the motion balance equation of non-uniform thickness nanosheets is established. The accuracy of the proposed method is verified through reliable publications. A set of results of natural vibration frequencies of variable thickness FGP nanosheets under the influence of factors are discovered and discussed. The results of this study can be used in design calculations of MEM and NEMs systems in mechanics.

Chebyshev–Ritz and Navier's methods for hygro‐magneto vibration of Euler–Bernoulli nanobeam resting on Winkler–Pasternak elastic foundation

AbstractThis research employs Chebyshev–Ritz method along with Navier's method to investigate the vibration characteristics of a nanobeam subject to a longitudinal magnetic field and linear hygroscopic environment. The nanobeam is characterized by a Winkler–Pasternak elastic foundation and follows the nonlocal Euler–Bernoulli beam theory. The governing equation of motion is derived using Hamilton's principle, and non‐dimensional frequency parameters are computed for Simply Supported‐Simply Supported (SS), Clamped‐Clamped (CC), and Clamped‐Free (CF) boundary conditions. The motivation behind this study is to provide a comprehensive and efficient analytical framework for understanding the dynamic behavior of nanobeams in complex environments. By investigating the influence of magnetic and hygroscopic factors on the vibration characteristics of nanobeams, this research aims to offer valuable insights for the design and optimization of nanoscale structures. Employing shifted Chebyshev polynomials as shape functions in Chebyshev–Ritz method offers several advantages in the proposed model. Firstly, these polynomials possess orthogonal properties, which can significantly enhance computational efficiency. The orthogonality of shifted Chebyshev polynomials allow for simpler and more streamlined numerical computations compared to non‐orthogonal basis functions. Additionally, the orthogonality ensures that the resulting system of equations is well‐conditioned, even for higher‐order polynomial approximations. A closed‐form solution for SS boundary condition is obtained through Navier's method. Convergence analysis is performed to validate the accuracy and effectiveness of the proposed model against existing models. The non‐dimensional frequency parameters obtained using both Navier's method and Chebyshev–Ritz method demonstrate strong agreement, further validating the proposed nanobeam model. Additionally, a comprehensive parametric study evaluates the impact of various characteristics, including the small‐scale parameter, Winkler modulus, shear modulus, magnetic parameter, and hygroscopic parameter. The findings contribute to a nuanced understanding of nanobeam vibrations under the influence of a magnetic field and hygroscopic environment, providing valuable insights for the design and optimization of nanoscale structures in practical applications.

NAVIER SOLUTION FOR DEFLECTION AND STRESSES ANALYSIS OF FGM SANDWICH PLATE RESTING ON PASTERNAK ELASTIC FOUNDATION

In this article, the first-order shear deformation theory (FSDT) and Navier’s solution are used to establish analytical solutions for analyzing the deflection and stresses of FGM sandwich plate structures resting on a Pasternak elastic foundation. The top and bottom layers consist of functionally graded materials (P-FGM) that vary according to the exponential law along the thickness direction, while the core layer is composed of a porous material. The boundary condition of the plate is simply supported on all the edges, and the plate is subjected to a sinusoidal distributed load perpendicular to the mid-surface. The reliability of the model is validated through comparison with findings published in reputable journals. The effects of material parameters, geometric dimensions, foundation coefficients, and face-core-face thickness ratios on deflection and stress components of rectangular FGM sandwich plates are investigated in detail in parametric studies.

EFG meshless-ANN approach for free vibration analysis of functionally graded material plates on elastic foundation in thermal environments

This study focuses on free vibration analysis of functionally graded material (FGM) plates supported by Winkler–Pasternak elastic foundation in thermal environment using element-free Galerkin (EFG) meshless method. Plate kinematics depend on first-order shear deformation theory. Uniform, linear, and nonlinear temperature variations through the thickness direction are considered, along with the temperature-dependent material properties. The numerical outcomes obtained from EFG method are compared with those available in the published literature to validate the proposed method’s accuracy. An artificial neural network (ANN) model that can easily predict the natural frequencies of the plate is constructed from the EFG method outcomes. Further, the effect of foundation parameters, power law index, thickness ratio, temperature variations, and different boundary conditions are investigated; results show that these significantly influence the vibration response of FGM plates supported by the elastic foundation. Increasing the temperature of FGM plates supported by the Winkler–Pasternack foundation causes a decrease in the dimensionless fundamental natural frequency, and the uniform temperature influence is greater than that of linear and nonlinear temperature variation. The proposed EFG-ANN prediction model saves approximately 98.80% computation time when predicting the natural frequency with an accuracy of approximately 98.76% compared to that by EFG meshless method alone.

Determination of Temperature Stresses during the Construction of Massive Monolithic Foundation Slabs, Taking into Account the Subgrade Compliance

Background The problem of early cracking caused by the heat of concrete hardening is relevant for massive reinforced concrete structures, including foundation slabs. The purpose of this work is to develop the methodology for determining temperature stresses during the construction of foundation slabs, taking into account the interaction with the subgrade. Methods The Pasternak elastic foundation model with two-bed coefficients is used for the soil. The temperature of the foundation slab is considered a function of only one coordinate z (temperature changes only along the thickness of the slab). As a result, to determine the stress-strain state of the slab, a fourth-order differential equation for deflection was obtained. A technique for numerically solving the resulting equation using the finite difference method is proposed. The calculation of the stress-strain state is preceded by the calculation of the temperature field, which is performed by the finite element method in a simplified one-dimensional formulation. Results The solution to the test problem is presented for a constant modulus of elasticity of concrete over time. The results were compared with finite element calculations in a three-dimensional formulation in the ASNYS software. The calculation was also performed taking into account the dependence of the mechanical characteristics of concrete on its degree of maturity. In this case, the picture of the stress-strain state changes significantly. The proposed method was also successfully tested on experimental data. Conclusion The proposed approach can significantly save calculation time compared to the finite element analysis in a three-dimensional setting.

Effect of connection and grouting level on mechanical behavior of pipe roof

ABSTRACT Pipe roof support suffers from engineering defects such as connection and insufficient grouting in tunnel engineering. To analyze the influence of connection and grouting level on the mechanical behavior of pipe roof support during tunnel construction, a series of pure bending tests were conducted to measure the mechanical property of pipe roof, and quantitatively analyze the influence of the connection and grouting level on the bearing capacity and bending stiffness of the pipe roof. In addition, the double shear mechanical analysis model of pipe roof based on the Timoshenko beam theory and the Pasternak elastic foundation theory was established by considering the shear deformation of pipe roof, the supporting effect of surrounding rock of tunnel face and the delayed effect of shotcrete for primary support. Then, the solution methods for pipe roof deflection and internal force were derived based on this model. Based on an engineering case, the deflection and internal force distribution, as well as the effect of connection and grouting levels on the pipe roof were discussed. Corresponding construction suggestions were proposed and had achieved good results in the case. This research can provide reference for the design and construction of pipe roof.

Frequency Analysis of Asymmetric Circular Organic Solar Cells Embedded in an Elastic Medium under Hygrothermal Conditions

This research represents the first theoretical investigation about the vibration behavior of circular organic solar cells. Therefore, the vibration response of asymmetric circular organic solar cells that represent a perfect renewable energy source is demonstrated. For this purpose, the differential quadrature method (DQM) is employed. The organic solar cell is modeled as a laminated plate consisting of five layers of Al, P3HT:PCBM, PEDOT:PSS, ITO, and Glass. This cell is rested on a Winkler–Pasternak elastic foundation and assumed to be exposed to various types of hygrothermal loadings. There are three different kinds of temperature and moisture variations that are taken into account: uniform, linear, and nonlinear distribution throughout the cell’s thickness. The displacement field is presented based on a new inverse hyperbolic shear deformation theory considering only two unknowns. The motion equations including hygrothermal effect and plate–foundation interaction are established within the framework of Hamilton’s principle. The DQM is utilized to solve these equations. In order to ensure the accuracy of the proposed theory, the present results are compared with those reported by other higher-order theories. A comprehensive parametric illustration is conducted on the impacts of different parameters involving the geometrical configuration, elastic foundation parameters, temperature, and moisture concentration on the deduced eigenfrequency of the circular organic solar cells.

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

Investigating the influence of excavating a tunnel undercrossing an existing tunnel at zero distance.

In urban areas with limited underground space, the new tunnel construction introduces additional loads and displacements to existing tunnels, raising serious safety concerns. These concerns become particularly pronounced in the case of closely undercrossing excavation at zero-distance. The conventional elastic foundation beam model, which assumes constant reaction coefficients for the subgrade, fails to account for foundation loss. In this study, the existing tunnel is modeled as an Euler-Bernoulli beam supported by the Pasternak elastic foundation, and the foundation loss caused by zero-distance undercrossing excavations is considered. Furthermore, an analytical solution is proposed to evaluate the mechanical response in segments, by establishing governing differential equations and boundary conditions for the excavation and neutral zones, and underpinning loads are also considered. The analytical solution is validated in two case studies. Finally, a parametric analysis is performed to explore the influence of various parameters on the mechanical response of the existing tunnel.

Static, buckling, and free vibration responses of functionally graded carbon nanotube-reinforced composite beams with elastic foundation in non-polynomial framework

In this work, the static, buckling, and free vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam resting on a Pasternak elastic foundation are studied. The secant function-based shear deformation theory (SFSDT) is used for this analysis. This theory fulfills the traction-free boundary conditions at the top and bottom surfaces of the beam, hence there is no need for a shear correction factor. Hamilton’s principle is used to determine the governing differential equations and boundary conditions whereas Navier’s solution technique is used for determining the closed-form solution. The analytical approach is used to examine the deflection, stresses, critical buckling load, and natural frequencies of the FG-CNTRC beam resting on the Pasternak elastic foundation including a shear layer and Winkler springs. To determine the material characteristics of FG-CNTRC beams, the Rule of the mixture is used. Uniform distribution (UD-beam), FG-X beam, FG-O beam, and FG-V beam are the different forms of CNT reinforcement distribution that are used in this study. Considering different span thickness ratios, the volume fraction and distribution of CNT, the Winkler spring, and the shear layer constant factors, all the structural responses are predicted. It is also observed that the present theory predicts the structural responses of the FG-CNTRC beam accurately when compared to other existing theories. A few new results are also included as the benchmark solutions for the new research.

Power flow analysis of the three-beam system on double parameter elastic foundations based on the wave propagation approach

This article investigates the power flow of the three-beam system on Winkler–Pasternak elastic foundations under elastic boundary conditions. The time-averaged power flow of the beam on elastic foundations with elastic boundary conditions is calculated based on the internal shear force and bending moment and the wave propagation approach. The reliability and accuracy of the calculation method are verified by comparing the vibration acceleration calculated by calculation examples and the finite element method. In addition, the effects of structural parameters and boundary conditions on the power flow of the three-beam system are investigated. This contributes to the analysis of the dynamic characteristics of the beam on double parameter elastic foundations under complex boundary conditions.

Nonlinear stability of ES-FG porous sandwich cylindrical shells subjected to axial compression in thermal environment

Stability analysis of cylindrical shells subjected to axial compression is a common problem in engineering applications. The phenomenon of instability can easily appear with thin cylindrical shells. Therefore, to increase the load-carrying capacity of this type of structure, the cylinders are increased in thickness by adding a porous core layer and reinforcing orthogonal stiffeners. In this study, the nonlinear buckling and postbuckling behavior of functionally graded (FG) porous sandwich cylindrical shells with orthogonal eccentrically stiffeners (ES) are investigated via analytical approach. The FG porous sandwich cylindrical shells are rested on Pasternak elastic foundations and subjected to axial compression load in a thermal environment. The reinforcement of the shells is achieved through closely spaced stringers and rings, with the material properties of both the shell and stiffeners graded continuously in the thickness direction. The governing equations are derived based on the Donnell’s shell theory with von Karman geometrical nonlinearity and the smeared stiffeners technique. The Galerkin procedure is employed to ascertain the critical load and post-buckling response, employing a three-term deflection solution. The study analyzed the effects of parameters such as porosity coefficient, material, temperature, dimensional parameters, stiffener, and foundation characteristics.

Linear finite element formulation for free vibration and buckling analyses of multi-directional FGP doubly curved shallow shells in thermal environment

ABSTRACT This paper presents the free vibration and buckling analyses of multi-directional functionally graded porous doubly curved shallow shells resting on Pasternak elastic foundations in a thermal environment. For the first time, a linear finite element formulation using the refined high-order shear deformation theory, combined with the Hermitian function, is developed to compute the vibration and buckling behaviour of doubly curved shallow shells. Mechanical properties of MFGP material change according to the length, width and thickness directions and the porosity distribution including even and uneven. The doubly curved shallow shell is operated in uniform, linear and non-linear temperature environments. The convergence and accuracy of the proposed method are verified by comparing numerical results with published works. In addition, a comprehensive numerical investigation was carried out to evaluate the influence of parameters on the natural frequency and critical buckling behaviour of MFGP doubly curved shallow shells.

Buckling analysis of nonuniformly compressed rectangular FG-CNT reinforced laminated composite plate resting on elastic foundation: An analytical solution

The present study is focused on the development of an analytical solution for the buckling analysis of carbon nanotube reinforced functionally graded (FG-CNTRC) laminated plates resting on Pasternak elastic foundation and subjected to different nonuniform edge loads. The analytical solution is formulated using a two-step procedure: First, an accurate in-plane stress solution is developed by superimposing three suitable Airy’s stress functions; second, the stress solution is utilized to compute the critical buckling loads using the Galerkin’s procedure. Results from parametric studies show that the buckling behavior of FG-CNTRC plates greatly depends on the type of nonuniform loads, plate configurations, and support parameters.