Pasternak Elastic Foundation Research Articles

Vibration of Non-Uniform Bernoulli-Euler Beam Under Moving Distributed Masses Resting on Pasternak Elastic Foundation Subjected to Variable Magnitude

This paper investigates the dynamic response of a clamped-clamped non-uniform Bernoulli-Euler beam resting on a Pasternak elastic foundation to variable magnitude moving distributed masses. The predicament is dictated by a partial differential equation of fourth order, which features coefficients that are both variable and singular. The primary aim is to derive an analytical solution for this category of a dynamic problem. To achieve this, we employ the method of Galerkin with a series representation of the Heaviside function to reduce the equation to second-order ordinary differential equations with variable coefficients. We simplify these transformed equations using (i) the Laplace transformation technique in conjunction with convolution theory for solving moving force problems, and (ii) finite element analysis in conjunction with the Newmark method for solving analytically unsolvable moving mass problems due to their harmonic nature. We first solve the moving force problem using the finite element method and compare it against analytical solutions as validation for its accuracy in solving analytically unsolvable moving mass problems. The numerical solution obtained from the finite element method is shown to be comparable favorably against analytical solutions of our moving force problem. Lastly, we calculate displacement response curves for both moving distributed force and mass models at various time t for our dynamical problem presentation purposes.

Vibration Characteristics of FML Cylindrical Shell Bonded by Thin Piezoelectric Actuator and Sensor Layer with and Without Fluid–Structure Interaction Resting on Pasternak Elastic Foundation

The present study investigates the vibration analysis of cylindrical shells composed of fiber metal laminate (FML) with embedded piezoelectric layers, undergoing fluid-structure interaction (FSI) and resting on a Pasternak elastic foundation based on the principles of three-dimensional elasticity theory. Using the state space approach, the equations of motion were derived under simply supported boundary conditions. The natural frequencies of the FML cylindrical shell, accounting for the presence of a moving fluid, were computed by solving the eigenfrequency equations. The study examined the influence of various parameters, including boundary conditions, length-to-radius ratio, fluid type, fluid velocity, circumferential wave number, and radius-to-thickness ratio, on glass-reinforced aluminum laminate (GLARE), aramid-reinforced aluminum laminate (ARALL), and carbon-reinforced aluminum laminate (CARALL). A constant composite/metal volume ratio was assumed. The results obtained were validated by comparing with natural frequency values from the existing literature, confirming the agreement and convergence with previous studies. The results confirm that the highest natural frequency values are assigned to the CARALL, ARALL and GRALE structures in descending order. Furthermore, an increase in fluid flow velocity through the cylindrical shell correlates with a reduction in natural frequency.

Bending analysis of FGSP nanoplate resting on elastic foundation by using nonlocal quasi-3D theory

In this paper, the bending response of a functionally graded saturated porous (FGSP) nanoplate resting on the Pasternak elastic foundation is analyzed within the framework of quasi-3D higher-order shear deformation theory (quasi-3D HSDT) for the first time. The material properties are presumed to change gradually along the thickness direction following three patterns of porosity distribution: uniform, non-uniform symmetric, and asymmetric. The theory of poroelasticity developed by Biot is utilized in modeling the stress-strain relationships for the saturated condition. Moreover, the nanoscale effects of the structures are considered by Eringen's nonlocal elasticity theory. The governing equations are derived by using the principle of minimum potential energy according to quasi-3D HSDT, which ensures transverse shear stress-free on the upper and lower surfaces of the nanoplate. Based on the obtained closed-form solution, the impacts of the porosity distribution patterns, porosity coefficient, Skempton coefficient, geometrical parameters, elastic foundation, and nonlocal parameters on the bending behavior have been explored. According to the findings, when the pores are saturated by the fluid, the plate stiffness increases. Additionally, increasing the values of the nonlocal parameter for FGSP nanoplates leads to an increase in deflection and stresses. Finally, the present study quantitatively reveals the size-dependent effects of a saturated porous medium.

Nonlinear vibration and stability of sandwich functionally graded porous plates reinforced with graphene platelets in subsonic flow on elastic foundation

This paper investigates the nonlinear vibration and stability of a functionally graded porous sandwich plate reinforced with graphene platelets (GPLR-SFGP) interacting with subsonic airflow on elastic foundation. The plate comprises of a functionally graded porous core with graphene platelet reinforcement and two metal face layers. Utilizing Hamilton's principle, the nonlinear equation of the plate is exported and discretized into ordinary equations using the assumed modes method. The influence of porosity, GPL weight fraction, surface thickness ratio and Winkler Pasternak elastic foundation arguments on the critical divergence velocity of the plate under subsonic flow is revealed by calculating the system characteristic values. The Matcont toolbox is occupied to generate nonlinear amplitude frequency resonance curves, allowing for a comprehensive examination of the influence of these parameters on the nonlinear resonance behavior of the system. The GPLR-SFGP plate exhibits outstanding characteristics, including superior stiffness and a reduced mass, rendering it a suitable choice for exterior applications in airplanes, automobiles, and high-speed railways. The findings in this study can provide valuable insight into the key design parameters that significantly affect the performance of GPLR-SFGP plates, enabling future design efforts aimed at enhancing their efficacy and robustness in real-world applications.

Dynamic Response to Moving Distributed Masses of Prestressed Orthotropic Cantilever Plate Resting on Variable Pasternak Elastic Foundation

Dynamic Response to Moving Distributed Masses of Prestressed Orthotropic Cantilever Plate Resting on Variable Pasternak Elastic Foundation

Free Vibration and Dynamic Analysis on Free-Constrained Layer of Graphene Based on Composite Conical Shell via Jacobi–Ritz Method

Excessive vibration has always been a serious problem for conical shell structures, while the application of the graphene-based free-constrained layer (GFCL) based on carbon fiber-reinforced composite (CFRC) structure is a novel way to improve structural performance. An analytical model for vibration and dynamic characteristics of the GFCL-CFRC conical shell resting on the Winkler–Pasternak elastic foundation with arbitrary boundary conditions is constructed, and four types of GFCL porosity distribution and GFCL dispersion pattern are considered in this model. The multi-segment technique and virtual spring technique are utilized to simulate arbitrary boundary conditions. Then, the first-order shear deformation theory (FSDT) and Hamilton’s principle are employed to obtain the motion equation of the GFCL-CFRC conical shell, and the motion equation of the GFCL-CFRC conical shell is solved by the Ritz method. In conclusion, the dispersion mode of GFCL, thickness ratio of GFCL, and fiber angle have influence on the dynamic performance. With a reasonable design, the dynamic performance of the GFCL-CFRC conical shell can be further improved.

On Vibration Responses of Advanced Functionally Graded Carbon Nanotubes Reinforced Composite Nanobeams

This article presents an analytical approach to explore the free vibration behaviour of new functionally graded carbon nanotube-reinforced composite beams (FG-CNTRC) based on a two-variable higher-order shear deformation theory and nonlocal strain gradient theory. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The kinematic relations of the shaft are proposed according to novel trigonometric functions. The vibrated nanobeam’s motion equations are obtained via the classical Hamilton’s principle and solved using Navier’s steps. A comparative evaluation of results against predictions from literature demonstrates the accuracy of the proposed analytical model. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanobeams to nonlocal length scale, strain gradient microstructure scale, material distribution, constant spring factors, and geometry. The current work presents the free vibration problem of supported (FG-CNTRC) beams reinforced by different patterns of carbon nanotube (CNT) distributions in the polymeric matrix.

Buckling and Free Vibration Analyses of Various Nanoparticle Reinforced Concrete Beams Resting on Multi-Parameter Elastic Foundations.

Given their considerable specific surface area and amorphous characteristics, nanoparticles exhibit excellent pozzolanic activity, and when undergoing a reaction with calcium hydroxide, this leads to the generation of a denser matrix by promoting the formation of a greater amount of C-S-H gel, thereby enhancing the strength and durability of the concrete and fortifying the overall structure. Indeed, the present study investigates a comparative study of the buckling and free vibration analyses of concrete beams reinforced with various types of nanoparticles. For its simplicity and accuracy, a higher-order shear deformation theory will be used to analytically model the reinforced concrete beam. Furthermore, the powerful Eshelby's model is used to derive the equivalent nanocomposite properties. The soil medium is simulated with Pasternak elastic foundation, including a shear layer, and Winkler's spring, interlinked with a Kerr foundation. The motion equations are derived using Hamilton's principle. Moreover, based on Navier's analytical methods, the closed-form solutions of simply supported beams have been obtained. Different parameters, such as type and volume percent of nanoparticles, geometrical parameters, choice of theory and soil medium, on the buckling and dynamic behavior of the beam, are exercised and shown. The major findings of this work indicate that the use of nanoparticles in concretes increases better mechanical resistance and amplifies the natural frequencies. In addition, the elastic foundation has a significant impact on the buckling and vibration performances of concrete beams.

Analytical solutions for double-plate systems with pasternak-type foundation in between

This paper deals with analytical solutions of double-plate system under static loads, where the connected system is composed by two parallel rectangular plates with an elastic layer in between. The plates are assumed to be represented by Kirchhoff plate model and the elastic layer is assumed to be a Pasternak elastic foundation. Analytical solutions of double-plate system based on Navier's method and Levy's approach are derived for specific conditions. Furthermore, a change of variables is used to decouple ordinary differential equations resulting from the Levy method as one of the steps to derive its corresponding homogeneous solutions. Examples for different loads and boundary conditions are presented.

Free Vibration Characteristics Analysis of Metal-Rubber Cylindrical Shells Based on Viscoelastic Theory

The cylindrical shell made of metal rubber has a strong ability to reduce and absorb vibration, which widens its application in the industrial field. Therefore, it is of great significance to study the vibration characteristics of metal-rubber cylindrical shells (MRCSs). However, there is relatively little research on this aspect. Based on this, the dynamic properties of MRCS are investigated in this paper based on viscoelastic theory, the Rayleigh–Ritz method, and the Gram–Schmidt orthogonal polynomials. The correctness of the proposed model was verified by comparison with the literature and experimental verification. The results show that the preloading state and boundary conditions have significant effects on the natural frequency and modal loss factor of MRCS. The effect of the Pasternak elastic foundation on the natural frequency and modal loss factor of MRCS varies with the change of the axial half wave number m. The effect of the Pasternak elastic foundation on higher-order vibrations is similar to that of the artificial spring technique.

Stability of multi-cracked FG plate on elastic foundations

Recently, the stability calculation of the functionally graded (FG) plate has attracted many scientists, especially when considering the FG plate with many cracks. In this work, the plate is made from a new generation composite material consisting of two components, ceramic and metal, with the law of continuous exponential material distribution. The plate is placed on a Winkler - Pasternak elastic foundation with two background parameters. Then, we used the third-order shear deformation plate theory to establish the dynamical equations. After applying Phase-Field theory to simulate the crack state, we applied the finite element method to solve the equations to find the critical force causing instability of the plate. Next, we investigated the influence of material index, number of cracks, crack length, crack shape as well as elastic foundation parameters on the plate stability. The results show that the crack length and elastic foundation parameter have the great influence on the stability of the FG plate. Especially, the elastic foundation with large shear coefficient, creating high stability for the plate. That is very meaningful in exploiting and using plate structure when the cracks appear.

Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate.

A nonlinear static analysis of a circular/annular nanoplate on the Winkler-Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler-Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.

Re-examination of nonlinear vibration and nonlinear bending of porous sandwich cylindrical panels reinforced by graphene platelets

Abstract This article re-examines the nonlinear vibration and nonlinear bending responses of porous sandwich cylindrical panels reinforced by graphene platelets resting on elastic foundations in thermal environments. The graphene platelet-reinforced composite (GPLRC) core is assumed to be of multilayers, and each layer may have different porosity coefficient values to achieve a piece-wise functionally graded pattern. By introducing an inhomogeneous model instead of the equivalent isotropic model (EIM), the Young’s moduli along with the shear modulus of the porous GPLRC core are predicted through a generic Halpin–Tsai model in which the porosity is included. The thermomechanical properties of metal face sheets and the porous GPLRC core are assumed to be temperature-dependent. Governing equations of motion for sandwich cylindrical panels with porous GPLRC core are formulated based on Reddy’s third-order shear deformation theory coupled with von Kármán nonlinear strain–displacement relationships. In the modeling, the panel–foundation interaction and the thermal effects are also considered. The analytical solutions for the nonlinear vibration and nonlinear bending problems are obtained by applying a two-step perturbation approach. Numerical studies are performed to compare the results obtained from the present model and the EIM. The results confirm that the EIM is not suitable for linear free vibration analysis of sandwich cylindrical panels with the porous GPLRC core, but the EIM may be valid for the cases of nonlinear vibration and nonlinear bending analyses of the same panel resting on Pasternak elastic foundations.

Porosity-dependent impact study of a plate with Winkler-Pasternak elastic foundations reinforced with agglomerated CNTs

PurposeThe purpose of this article is to investigate the porosity-dependent impact study of a plate with Winkler–Pasternak elastic foundations reinforced with agglomerated carbon nanotubes (CNTs).Design/methodology/approachBased on the first-order shear deformation plate theory, the strain energy related to elastic foundations is added to system strain energy. Using separation of variables and Lagrangian generalized equations, the nonlinear and time-dependent motion equations are extracted.FindingsVerification examples are fulfilled to prove the precision and effectiveness of the presented model. The impact outputs illustrate the effects of various distribution of CNTs porosity functions along the plate thickness direction, Winkler–Pasternak elastic foundations and different boundary conditions on the Hertz contact law, the plate center displacement, impactor displacement and impactor velocity.Originality/valueThis paper investigates the effect of Winkler–Pasternak elastic foundations on the functionally graded porous plate reinforced with agglomerated CNTs under impact loading.

Bending wave at the edge of a thermally affected functionally graded poroelastic plate

This study examines the propagation characteristics of flexural edge waves in a functionally graded poroelastic plate within a thermal environment. A transversely isotropic, functionally graded porous material plate supported by a Pasternak elastic foundation is introduced to analyze the dynamics of bending edge wave. Kirchhoff plate theory and Moore–Gibson–Thomson (MGT) thermoelasticity theory are introduced to study the displacement field and the temperature distributions on the plate respectively. Seven different porosity models are tested in this study to compare edge wave behavior in different porous structures. It is found that the decay’s constant for edge wave propagation depends implicitly on frequency and wave number for each of the seven cases. The dispersion relation is derived using the plane harmonic wave propagating along the free edge of the plate. On analyzing the model, it is evident that waves exhibit a cut-off frequency due to an elastic foundation, dependent only on the Winkler modulus. The impact of various parameters, such as porosity and temperature, on the traveling edge wave is illustrated numerically and discussed in terms of their physical behavior.

A Novel Analytical Approach for Nonlinear Thermo-Mechanical Buckling of Higher-Order Shear Deformable Porous Circular Plates and Spherical Caps with FGM Face Sheets

A novel analytical approach for nonlinear thermo-mechanical buckling of higher-order shear deformable porous circular plates and spherical caps with functionally graded material (FGM) face sheets resting on Pasternak elastic foundation is presented in this paper. The circular plates and spherical caps are assumed to be subjected to uniformly distributed external pressure and/or uniformly distributed thermal loads, and the nonlinear higher-order shear deformation theory (HSDT) is used for largely thick plates and caps with the shell-foundation interaction modeled by Pasternak elastic foundation. The caps are assumed to be shallow with clamped boundary conditions. The total potential energy expression of structures is established and the Ritz energy method is used to solve the problem directly from the total potential energy expression. The expressions between external pressure–deflection, thermal load–deflection, and thermo-mechanical combined load–defection can be obtained using the iterative algorithms. The critical buckling loads and postbuckling behavior of plates/caps are investigated numerically. Significant effects of foundation, porosity, structure parameters on the nonlinear thermo-mechanical responses of circular plates and spherical caps are numerically investigated and discussed, and the complex tendencies of postbuckling strength of plates and caps are obtained.

Thermoacoustic response of polymethyl methacrylate composite shells reinforced by carbon nanotube resting on Winkler–Pasternak elastic foundation

This paper presented an analytical method for investigating the vibroacoustic response of polymethyl methacrylate composite shells reinforced by CNTs resting on an elastic foundation under thermal loading. This structure is submerged in a moving fluid and excited by an acoustics plane wave. The CNTs are assumed to have a graded distribution in the polymethyl methacrylate matrix. The properties of the CNTs are assumed to be temperature-dependent. Different types of CNTs distributions such as UD, X, V, and O are considered. The Winkler–Pasternak model is applied to describe the vibrational behavior of the elastic foundation. The equations of motion of the structure are based on Hamilton’s principle using the TSDT, considering the effects of shear and rotation. To validate the present analytical method, the obtained results are compared with those of other researchers. Next, the effects of various significant parameters such as elastic foundation, temperature gradient, CNTs distributions, and Mach number on the TL are studied. The results show increasing the temperature reduces the TL because the stiffness bending of the structure decreases. It can be seen that the addition of CNTs to the matrix, with an appropriate volume fraction, improves the sound transmission loss due to the enhancement of stiffness.

Static analysis of functionally graded saturated porous plate rested on pasternak elastic foundation by using a new quasi-3D higher-order shear deformation theory

Static analysis of functionally graded saturated porous plate rested on pasternak elastic foundation by using a new quasi-3D higher-order shear deformation theory

VIBRATIONAL ANALYSIS OF SUBMERGED CYLINDRICAL SHELLS BASED ON ELASTIC FOUNDATIONS

In this study a vibration analysis was performed of an isotropic cylindrical shell submerged in fluid, resting on Winkler and Pasternak elastic foundations for simply supported boundary condition. Love’s thin shell theory was exploited for strain- and curvature- displacement relationship. Shell problem was solved by using wave propagation approach. Influence of fluid and Winkler as well as Pasternak elastic foundations were studied on the natural frequencies of submerged isotropic cylindrical shells. Results were validated by comparing with the existing results in literature.

Study on dynamic stability of magneto-electro-thermo-elastic cylindrical nanoshells resting on Winkler–Pasternak elastic foundations using nonlocal strain gradient theory

Study on dynamic stability of magneto-electro-thermo-elastic cylindrical nanoshells resting on Winkler–Pasternak elastic foundations using nonlocal strain gradient theory