Foundation Parameters Research Articles

Nonlinear forced vibration of functionally graded hybrid three-phase nanocomposite toroidal shell segments reinforced by carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs)

This study proposes a unique three-phase functionally graded (FG) hybrid nanocomposite material reinforcing the toroidal shell segments to investigate the nonlinear forced vibration using Reddy's higher-order shear deformation theory, von Karman geometrical nonlinearity and Stein-McElman's assumption along with the Hamilton principle. The examined toroidal shell's material consists of polymeric resin, carbon nanotubes (CNTs), and graphene platelets (GPLs) as nanocomposites. The material properties have been derived based on a modified Halpin-Tsai micromechanical model. Four distribution patterns have been studied: uniform (UD), FG-X, FG-O, and FG-V. The continuous distribution of GPLs and CNTs leads to inhomogeneous position-dependent properties throughout the shell thickness. The obtained differential equations of motion have been reduced to ordinary equations using Galerkin's technique. A multi-scales method (MSM) is used to estimate a closed-form solution representing the frequency-amplitude relation, and the state space representation is used along with numerical fourth-order Runge Kutta method (RK4) to obtain the nonlinear dynamic response of the toroidal shell. The accuracy of the current results obtained has been verified by comparing them with the relevant literature and numerical results using (RK4). In addition, the influence of both GPLs and CNTs weight fractions, nanofillers’ distribution types, longitudinal and circumferential wave numbers, elastic foundation parameters, static axial compression load, transverse excitation load, damping ratio, geometrical characteristics of the toroidal shell on the dimensionless natural frequency, nonlinear primary resonance (frequency-amplitude curve), and nonlinear dynamic response are carefully studied. The results found that combining (GPLs & CNTs) into the shell's matrix improves performance, especially in concave shells compared to convex ones. The FG-X pattern reduces peak resonance amplitude and improves natural frequency. A reduction of 12.3 % is observed for convex shells, and 21.5 % for concave shells compared to FG-O.

Vibrational response of a sandwich microplate considering the impact of flexoelectricity and based on a novel porous-FGM formulation

Flexoelectricity indicates a unique characteristic of insulators and dielectric materials that results in electrical polarization induction in response to the non-uniform strain gradient. The impact of this property is bolder on the microscale and nanoscale rather than the macroscale. This article is conducted to derive the governing equations of motion for a flexoelectric-multilayer structure and evaluate its vibrational behavior in response to various variable variations. The model includes a functionally graded (FG) porous core which is confined between two flexoelectricity-based face sheets at both sides and a Vlasov type of foundation. To gain the governing equations, the new type of power law for porous material beside Hamilton’s principle (HP) and modified couple stress theory (MCST) is implemented. Capturing the effects of length scale parameter, different materials, aspect and thickness ratios, porosity index, foundation parameter and boundary conditions (B.C.s) on the normalized natural frequency (NNF) of such a multilayer model based on the novel FG porous material formulation serve as the main novelty of this article. According to the results, it is revealed that the lateral ratio enhancement causes NNF and stiffness reduction in the model. Furthermore, among all considered B.C.s, simply support (SSSS) and clamped support (CCCC) refer to the lowest and the highest NNF respectively. The content of the recent article serves as a good source to provide a better understanding of the vibrational response of high stiffness-to-weight ratio structures to different variable variations.

Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler-Pasternak elastic foundation

In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler-Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are estimated through Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization techniques. The Timoshenko microbeam model considering the length scale parameter is applied. The free vibration differential equations of FGM microbeams are established based on the Finite Element Method (FEM) and Kosmatka’s shape functions. The influences of the size-effect, foundation, material, and geometry parameters on the vibration frequency are then analyzed. It is shown that the study can be applied to other FGMs as well as more complex beam structures.

Analyzing the nonlinear torsional buckling and post-buckling of ES-FG porous cylindrical shells in thermal environment using FSDT in terms of the displacement components

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of eccentrically stiffened sandwich functionally graded porous (FGP) cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach in terms of the displacement components. The shells are reinforced by eccentric rings and stringers attached to the inside and material properties of face sheets and stiffeners are assumed to be continuously graded in the thickness direction. The sandwich cylindrical shell is composed of FG porous core and two FG layer coating. Based on the first order shear deformation theory (FSDT) with von Kármán geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method, the closed form to find critical torsional load and post-buckling load-deflection curves are obtained. The effects of porosity parameters, the thickness of the porous core, temperature, stiffener, foundation, material and dimensional parameters are analyzed.

Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets

The nonlinear vibration analysis of functionally graded reinforced with graphene platelet (FG-GRC) porous truncated conical shells surrounded by the Winkler-Pasternak elastic foundation is presented in this paper. An improved model for evaluating the material properties of porous composites is proposed. Three types of porous distribution and three patterns of graphene nanoplatelets (GPLs) dispersion are estimated. Coupled with the effect of the Winkler-Pasternak elastic foundation, the nonlinear governing equations are developed by using the Hamilton principle. The Galerkin integrated technique is employed to obtain the linear and nonlinear frequencies of the shells. After the present method is validated, the effects of the pores, GPLs, the Winkler-Pasternak foundation, and the semi-vertex are investigated in detail. The results show that the linear frequency can be raised by increasing the values of the mass volume of the GPL and foundation parameters. In contrast, the ratio of nonlinear to linear frequency declines as the mass volume of the GPLs and foundation parameters rises. Furthermore, it is found that the minimum ratio of nonlinear to linear frequency can be obtained as the semi-vertex angle is about 55º, and the effect of porosity distribution on the linear and linear frequencies might be neglected.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Thermal effects on nonlinear vibration of nonlocal nanobeam embedded in nonlinear elastic medium

Nanotechnology has an impact on our lives in a many ways, from better medical treatments and more efficient energy sources to stronger and lighter materials and advanced electronics and this article presents the implementation of a perturbation method for the vibration analysis of simply supported and clamped–clamped Euler–Bernoulli nanobeams resting on nonlinear elastic foundations in thermal environment using nonlocal elasticity theory. Hamilton's principle is used to construct the differential equation of motion of a nanobeam in conjunction with appropriate boundary conditions. The equations of motion of the Euler–Bernoulli nanobeam are determined using nonlocal elasticity theory. It is shown how thermal loadings affect the vibration of the Euler–Bernoulli nanobeam. The multiple scale method, which is one of the perturbation method, is used to get an approximated solution for the presented system. The effects of temperature, Winkler, Pasternak and nonlinear foundation parameters on the vibration analysis of simply supported and clamped–clamped nanobeams are determined and results are given in tables and graphs.

Vibration analysis of embedded porous nanobeams under thermal effects using boosting machine learning algorithms and semi-analytical approach

This study presents a thermal vibration analysis of functionally graded porous nanobeams using boosting machine learning models and a semi-analytical approach. Nonlocal strain gradient theory is employed to explore vibration behavior, accounting for thermal and size effects. A semi-analytical approach solution utilizing Fourier series and Stokes’ transform to establish an eigenvalue problem capable of examining vibrational frequencies of porous nanobeams in both rigid and deformable boundary conditions is presented. Four boosting models including gradient boosting (GBoost), light gradient boosting (LGBoost), extreme gradient boosting (LGBoost), and adaptive boosting (AdaBoost) are employed to study the impact of seven crucial parameters on natural frequencies of a nanobeam. Sobol quasi-random space-filling method is used to generate the samples by varying input feature combinations for different porous nanobeam distributions. The model performance is assessed using statistical metrics, visualization tools, 5-fold cross-validation, and SHAP analysis for feature importance. The results highlight the effectiveness of boosting ML models in predicting natural frequencies, particularly XGBoost, which achieved an exceptional R2 value of 0.999, accompanied by the lowest MAE, MAPE, and RMSE values among the models assessed. LGBoost and AdaBoost follow XGBoost in performance, while GBoost exhibits relatively lower effectiveness, as highlighted by radar plots. The SHAP analysis revealed the significant impact of the foundation parameter on frequency prediction, with porosity coefficients notably influencing higher vibration modes.

Crystal Structure and Optical Characteristics of K2Nd2O(BO3)2

A rare earth oxyborate crystal K2Nd2O(BO3)2 has been grown by means of high temperature solution method for the first time. It crystallizes in monoclinic P21/c (No. 14) space group, and the lattice parameters are a=11.318(6) Å, b=6.575(4) Å, c=10.689(6) Å, β=117.062(12)°, and Z=4. The absorption spectrum, emission spectrum, and fluorescence lifetime of K2Nd2O(BO3)2 have been evaluated, and the foundational spectral parameters have been analysed according to the Judd‐Ofelt theory. The high concentration of activated ions (1.13×1022 ions/cm3), large absorption coefficient (87.63 cm‐1), long fluorescence lifetime (91.2 μs), and large emission cross section (2.03×10‐20 cm2) of K2Nd2O(BO3)2 suggest its potential applications of microchip lasers.

Presentation of the Berry–Tabor conjecture in Lévy plates

Abstract The Berry–Tabor (BT) conjecture is a famous statistical inference in quantum chaos, which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena. In this paper, the BT conjecture has been extended to Lévy plates. As predicted by the BT conjecture, level clustering is present in the spectra of Lévy plates. The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P ( r ∼ ) , which is calculated through the analytical solution obtained by the Hamiltonian approach. Our work investigates the impact of varying foundation parameters, rotary inertia, and boundary conditions on the frequency spectra, and we find that P ( r ∼ ) conforms to a Poisson distribution in all cases. The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies, which can be understood through mode functions.

Механическое взаимодействие метаемого тела со стенками канала рельсового электромагнитного ускорителя. Численное решение

The vibration of the accelerator rails under the action of electromagnetic forces, the right boundary of which is moving along with the armature along the barrel bore, is considered. The rail accelerator is simply considered as a Bernoulli-Euler beam of finite length, lying on a viscoelastic foundation, with cantilever support from the side of the accelerator breech. The rail vibration is described by a differential equation in partial derivatives of the fourth order in coordinate and the second order in time. The equation is solved numerically by combined method: finite differences in time and finite elements in coordinate. A comparison of the results of numerical calculations with an analytical solution showed that the developed code evaluates the behavior of the rail well under the influence of an EM load moving along its length along the entire trajectory of the accelerated body. Using the developed code, the influence of the current profile in the accelerator circuit and the foundation parameters on the dynamics of bending waves arising and propagating in the rail, the position and magnitude of the maximum rail deflection was studied. It is shown that it is possible to select a current profile that minimizes the rail accelerator deflection during acceleration. It is also found that the choice of the optimal current profile has no less effect on the maximum rail deflection than improving the parameters of the accelerator foundation. The calculation results may be useful in the design of an electromagnetic rail accelerator.

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.

Inclined crack identification in bidirectional FG beams on an elastic foundation using the h-version of the finite element method

This article examines the impact of inclined transverse cracking on the natural frequency of Euler-Bernoulli-model imperfect FG beams on an elastic foundation, taking into account the pinned-pinned and clamped-clamped boundary conditions. The equations are diffused using the classic finite element method (h-version). Material properties are considered to vary in both directions of the beam; width and thickness, using the power-law formula. The approximate porosity model is adopted with a uniform distribution. Cracked element stiffness is determined by reducing the cross-section of a bi-directional FG beam. The numerical results are compared with those of previous convergence studies for dimensionless fundamental frequencies. Case studies were carried out to evaluate the effect of the porosity values, the crack depth, the crack location, slope angle, and the Winkler-elastic foundation parameters on the first three natural frequencies of a beam with different support conditions and material gradient. In addition, the results demonstrated that the two-way distribution function plays an important role in the convergence of frequencies.

Soil deformation and failure analysis of suction caisson in marine clay under combined loading

The suction caisson is a typical offshore wind turbine(OWT) foundation, which is widely used due to its low construction cost and simple structure. However, the soil deformation and failure mechanisms of the suction caisson under combined loading are very complex and variable, which are different from those of the embedded foundation. In this study, the finite element method is adopted to capture the soil deformation and failure mechanism of a typical suction caisson under combined loading. A series of finite element models of suction caisson were performed to investigate the effects of the soil and foundation parameters on the failure and deformation behavior of the foundation, and a framework for calculating the bearing performance of suction caisson was provided. The results show that the failure modes of a suction caisson can be depicted in four patterns, namely internal Hansen, scoop-slide, scoop and slide. The deformation and failure characteristics of the soil are closely related to the ratio of M/HD, while less influenced by the interface condition, length-dimeter ratio, vertical loading and soil strength profile. The results are of great benefit in understanding the bearing characteristics of the suction caisson under combined loading for the promotion of the foundation.

Vibration analysis of porous orthotropic cylindrical panels resting on elastic foundations based on shear deformation theory

Cylindrical panels are one of the most essential structural members of engineering structures, with mechanical, civil, aeronautical, and marine engineering applications. They are subjected to a wide range of vibrational loads. This article presents a novel higher-order porosity distribution and a free vibration analysis for porous orthotropic cylindrical panels resting on elastic foundations under higher-order shear deformation theory. It is assumed that cylindrical panels are composed of porous materials with uniformly and non-uniformly distributed pores. The porous panels' material properties are distributed in the thickness direction using specific functions. The equations of motion are derived using Hamilton's principle based on trigonometrical shear deformation theory and solved by performing the Galerkin solution procedure with simply supported edge conditions. The accuracy of the obtained natural frequency equation is confirmed by comparing the results to those of previously published in literature. Under comprehensive parametric studies, the influence of porosity coefficient, porosity distribution patterns, radius-to-curve length ratio, orthotropy, and stiffness of elastic foundation parameters on the free vibration response of porous orthotropic cylindrical panels are discussed in detail.

Thermal vibration analysis of FG-GPLRC doubly curved shells partially resting on Kerr foundation based on higher-order shear deformation theory

In this paper, we present a novel analytical approach for investigating the influence of non-homogeneous Kerr elastic foundations on the nonlinear dynamical characteristics of functionally graded graphene platelets reinforced composite (FG-GPLRC) doubly curved shells. A new aspect of this study is the investigation of various distributions of the Kerr foundation, either centrally within or along the edges of the nanocomposite shells, a model not previously explored. Three distinct distributions of graphene nanoplatelets reinforcement are examined, utilizing the rule of mixtures and the Halpin-Tsai micromechanical model. The theoretical framework is based on Reddy's third-order shear deformation theory. The equations of motion, formulated as partial differential equations, are efficiently resolved through the application of Galerkin's method. The effectiveness of this approach resides in its utilization of integration to tackle the distribution issue of the elastic foundation, all without the need for deploying intricate algorithms. The validity of our proposed method is established through meticulous comparisons with findings from prior literature. Furthermore, this paper delves into the effect of material properties, thermal conditions, boundary conditions, geometrical parameters, and four configurations of the shells on the vibrational responses. Of notable significance is our thorough investigation of the impact of variations in the distributions, stiffness, and geometrical parameters of the elastic foundation on natural frequencies and the nonlinear transient response of these shells, all of which hold great importance in structural design. We present the obtained results systematically in various formats, including tables, 2D and 3D graphs, and contour plots, providing a comprehensive overview of the research outcomes. These findings have profound implications for the field of engineering, contributing to advancements in novel structural designs and their diverse applications.

Dispersion of an inhomogeneous sandwich plate having imperfect interfaces and supported by the Pasternak foundation

The purpose of this investigation is to see the dispersion of an inhomogeneous sandwich plate with imperfect interfaces between the layers and supported by the two parameters Pasternak foundation under long-wave low-frequency conditions. The governing equation of motion has been considered from the perspective of an anti-plane shear propagation to achieve simplicity. The overall cut-off frequency and the exact dispersion relation (EDR) are computed. In the context of the structure under investigation, one material contrast setup has been considered. The shortened polynomial dispersion relation, which corresponds to the EDR under material contrast setup, has been reported and investigated further. Additionally, the variational effects of the Pasternak foundation parameters as well as the interface imperfect parameter on the lowest dispersion curve subject to the long-wave low-frequency domain have been investigated using numerical simulations and graphical representations. This study is noteworthy because it sheds light on the behavior of elastic waves in multilayered structures and may be utilized to enhance the layout of three-layered structures used in a variety of industrial fields. Furthermore, we have provided the optimum values of the involved parameters to control and confine the sandwich plate’s vibration within the long-wave low-frequency regime.

A new analytical approach for nonlinear thermo-mechanical postbuckling of FG-GPLRC circular plates and shallow spherical caps stiffened by spiderweb stiffeners

For the first time, the problem of nonlinear postbuckling of circular plates and shallow spherical caps reinforced by meridian, parallel stiffeners, and spiderweb stiffeners based on the Donnell shell theory (DST) and von Kármán geometric nonlinearities is presented. The caps/plates and stiffeners are made from functionally graded graphene platelet-reinforced composite (FG-GPLRC). These stiffened structures are subjected to uniformly distributed external pressure or/and uniformly distributed thermal loads and are rested on a nonlinear elastic foundation. By expanding Lekhnitskii's smeared stiffener technique and employing the Ritz method of energy, the formulas to determine the postbuckling curves of the external pressure–deflection and thermal load-deflection relations of stiffened plates/spherical caps are derived. Meaningful discussions of the various influences of FG-GPLRC stiffeners, material distributions of plate/cap and stiffeners, and geometrical, material, and foundation parameters are shown in the content of the numerical investigations.

Study on the Influence of Excavation of Superlarge and Ultra-Deep Foundation Pits on the Pile Foundation of Existing Viaducts

The rapid development of urban above- and underground transportation has brought an increasing number of deep foundation pit projects that close to existing urban viaducts. Excavation of the deep foundation pit projects leads to soil movement around the pile foundation, which adversely affects the adjacent existing pile foundation and seriously threatens the operation safety of the adjacent existing viaduct. In this paper, the three-dimensional numerical model of foundation pit excavation, viaduct, and pile foundation was established by using the MIDAS GTS against the background of the basement foundation pit project in the new Guangzhou Baiyun Station comprehensive transportation hub building, production, and life complex, where the interaction between foundation pit excavation and viaduct pile foundation was simulated. The influence of various excavation-related factors and different parameters of the pile foundation on lateral displacements of the pile body adjacent to the existing pile foundation was analyzed. The variables controlled in the numerical model included the different excavation depths of the foundation pit, the distance between the excavation surface and the adjacent pile foundation, the spatial effect of different excavation sequences, the diameter of the pile that adjacent to the existing pile foundation, and the elastic modulus of the bored occlusal pile support structure. The results indicated that the lateral displacement of the pile body of the bridge pile foundation was significantly affected by the excavation of the deep foundation pit. The excavation depth exerted the greatest influence on the lateral displacement of the pile foundation, followed by the distance between the pile foundation and the excavation surface, the excavation sequence of the foundation pit, and the elastic modulus and the diameter of the bored occlusive pile support structure. Among them, the pile foundation located at the corner of the foundation pit was negligibly affected by these influencing factors. The findings in this study can provide valuable experiences for similar projects.

Analysis of longitudinal stress and deformation characteristics of shield tunnel under faulting dislocation

Fault dislocation seriously threatens the structural security of cross-fault tunnels during operation. Regarding the mechanical response of the tunnel across faults, the tunnel was equated as the Timoshenko beam on the Vlasov foundation in the present study, and the mechanical analytical model of the cross-fault tunnel was established and solved by the finite difference method. Published cases were utilized to validate the reasonableness of the analytical model. The influence law of critical parameters of the model on the response of the tunnel was investigated by parameter sensitivity analysis. Finally, the mechanical properties of the tunnel under the non-homogeneous foundation conditions were discussed. The calculation results of the proposed analytical model were similar to the existing model test monitoring data, and the trend was the same, demonstrating the correctness of the analytical model. The tunnel-fault distance, foundation reaction coefficient, foundation shear stiffness, and effective factor of bending stiffness changed the maximum tunnel bending moment, while the fault inclination changed the position where the maximum bending moment occurred. For the non-homogeneous foundation, the difference of adjacent foundation parameters significantly affected the response law of tunnel longitudinal internal forces, and the maximum tunnel bending moment presented a particular nonlinearity with the change of foundation parameters.