Two-parameter Foundation Research Articles

Refined model for rectangular thin plates on two-parameter foundations using an iterative procedure

The main purpose of this study is to investigate the influence of soil heterogeneity on the mechanical responses of rectangular plates resting on elastic foundations and to determine reasonable physical model parameters. The bending problem of rectangular thin plates lying on Gibson elastic foundations is solved using an iterative procedure. First, plates and foundations are discussed as a whole system, based on the principle of minimum potential energy. Second, the governing equations and boundary conditions for plates are established. And the equation for the attenuation parameter in the mathematical model is also derived. Then an iterative procedure is used to iterate the two parameters simultaneously, which solves a key issue for the two-parameter foundation model. Third, a double Fourier series with supplementary functions is adopted, and the solution to the bending problem is obtained. The agreements between the numerical results and the literature results prove that the refined model is practical and achievable. The proposed method exhibits a certain universality in analyzing the interactions between rectangular thin plates and elastic foundations. Furthermore, Gibson soil has an influence on the structural responses of rectangular plates on elastic foundations.

A Refined Model for Analysis of Beams on Two-Parameter Foundations by Iterative Method

It is of great significance to study the interactions between structures and supporting soils for both structural engineering and geotechnical engineering. In this paper, based on the refined two-parameter elastic foundation model, the bending problem for a finite-length beam on Gibson elastic soil is solved. The effects of axial force and soil heterogeneity on the bending behaviours and stress states of beams on elastic foundations are discussed, and the parameters of the physical model are determined reasonably. The beam and elastic foundation are treated as a single system, and the complete potential energy is obtained. Based on the principle of minimum potential energy, the governing differential equations for the beam bearing axial force on the Gibson foundation are derived, and the equations for attenuation parameters are also defined. The problem of the unknown parameters in foundation models being difficult to determine is solved by an iterative method. The results demonstrate that this calculation method is feasible and accurate, and that the applied theory is universal for the analysis of interactions between beams and elastic foundations. Both axial force and soil heterogeneity have a certain effect on the deformation and internal force of beams on elastic foundations, and the vertical elastic coefficient of foundations is mainly determined by the stiffness of the surface soil. Additionally, attenuation parameters can be obtained relatively accurately by an iterative method, and then the vertical elastic coefficient and shear coefficient can be further obtained. This research lays a foundation for the popularisation and application of the two-parameter elastic foundation model.

Dynamic Loading of the Rod at a Sudden Change of Elastic Foundation Structure

The mathematical model of dynamic process in the loaded constant evenly distributed load of structurally nonlinear system “beam - two-parameter foundation" is constructed, arising as a result of sudden change of physical and mechanical properties of the foundation, leading to zeroing of its shear rigidity. The solving of the static bending problem of a hinged beam fixed at the ends of the beam, supported by (если физически на него опирается, если нет, тогда operated) the Pasternak foundation, serves as an initial condition of the problem about the forced vibrations of the beam on the Winkler foundation, which occurred after a sudden formation of the defect. The solutions of static and dynamic problems are built using the method of initial parameters with the involvement of vectors of beam cross-sections states and matrixes of influence of initial parameters on the state of arbitrary cross-sections. To analyze forced vibrations, the decomposition of load and deflection of the initial static state into rows according to the forms of natural vibrations of the new state is applied.

Natural frequency calculation of reinforced earth retaining wall considering wall–soil interaction

Natural frequency is one of the important indicators that reflect the dynamic performance of the structure. On the basis of comprehensive consideration of the continuity of the deformation of the filling behind the reinforced retaining wall, if the filling behind the wall is regarded as the Pasternak two-parameter foundation model, then the retaining wall panels can be considered as Bernoulli–Euler beams. Considering the force characteristics of the panel, the calculation of the shear modulus of the panel and the foundation reaction force of the filling soil behind the wall is reasonably considered. On this basis, the natural frequency calculation model of the reinforced retaining wall interacting with the concrete faceplate and the backfill of the wall is established, and the natural frequency calculation equation of the reinforced retaining wall is derived through the wall–soil dynamic balance equation. The complex modal analysis method is used to obtain the analytical solution of the natural frequency of the reinforced retaining wall, and the comparative analysis with the existing method proves the validity of the calculation result; then, the method is used to analyze the frequency influence factors of the reinforced retaining wall.

ДИНАМИЧЕСКОЕ ДОГРУЖЕНИЕ СТЕРЖНЯ ПРИ ВНЕЗАПНОМ ИЗМЕНЕНИИ СТРУКТУРЫ УПРУГОГО ОСНОВАНИЯ

A mathematical model of the dynamic process in the structurally nonlinear system «beam – two-parameter foundation», which arises as a result of a sudden change in the physical and mechanical properties of the foundation, leading to zeroing of its shear stiffness, is constructed in a structurally-nonlinear system «beam – two-parameter foundation» The solution of the static problem of bending of a beam supported on a Pasternak base serves as the initial condition for the problem of forced vibrations of a beam on a Winkler base that arose after the sudden formation of a defect. Solutions to static and dynamic problems are constructed by the method of initial parameters with the use of vectors of states of beam sections and matrices of the influence of initial parameters on the state of arbitrary sections. In the analysis of forced vibrations, the decomposition of the load and deflections of the initial static state into rows according to the forms of natural vibrations of the new state is used.

An analytical method for shallow spherical shell free vibration on two-parameter foundation

The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of R-function, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method.

Finite element analysis for functionally graded porous nano-platesresting on elastic foundation

This paper proposes an improved triangular element based on the strain approach and the Reissner-Mindlin theory to investigate the static, free vibration, and buckling response of functionally graded porous (FGP) nano-plates resting on the Parternak's two-parameter elastic medium foundation. The internal pores of nano-plates are described by two distribution laws, including uneven porosity distribution and logarithmic-uneven porosity distribution. Using Hamilton's principle, equilibrium equations of FGP nano-plates lying on a two-parameter foundation are obtained. The most remarkable feature of the improved triangular element is the degrees of freedom of elements approximated by Lagrange functions for the membrane strain and by the high-degree polynomial functions for the bending strain. The numerical results of the present work are compared with the available results in the literature to evaluate the performance of the proposed approach. Effects of geometrical and material properties such as the power-law index n, the porosity coefficient S, the nonlocal coefficient u, and the parameters of the elastic foundation on the static, free vibration, and buckling behavior of the FGP nano-plates are examined in detail.

Analytical formulation and finite-element implementation technique of a rigorous two-parameter foundation model to beams on elastic foundations

ABSTRACT A simple way of computer implementation of an effective two-parameter subgrade model for the routine analysis of flexible foundations on an elastic subgrade is presented. It is based on a rigorous analytical subgrade model recently proposed by one of the authors that takes full account of the soil–structure interaction. The details of both finite element (FE) and analytical implementation of the model to beams are presented. Fully established closed-form relationships are provided for the subgrade parameters. The essence in the method is that the soil shear resistance within the loaded region is transferred to the foundation element as an additional rigidity in form of a geometric stiffness, thereby leaving the Winkler spring bed to take care of the vertical compression as accustomed. Additional edge springs are introduced to account for the edge forces due to the shear continuity extending beyond the edges. By so doing, the attractiveness of the long-enduring Winkler’s model is maintained, whereas the missing shear interaction is taken care of through the introduction of the geometric stiffness and the edge springs. The FE discretisation is thus limited to the foundation unit alone as initially intended. Subject classification codes: soil–foundation interaction

Thermoelastic Vibrations of Nonlocal Nanobeams Resting on a Pasternak Foundation via DPL Model

The present work introduces the thermoelastic vibrations of nonlocal nanobeams resting on a two-parameter foundation. The governing equations are formulated for linear Winkler–Pasternak foundation type based on the generalized dual-phase-lag heat conduction and nonlocal beams theories. The nanobeam is subjected to a temperature ramping function. The coupled equations of the problem are formulated and solved by Laplace transform technique. The effects of the nonlocal parameter and different foundation parameters on the field variables are illustrated graphically and discussed. The results obtained are consistent with previous analytical and numerical results.

Bending Analysis of Circular Thin Plates Resting on Elastic Foundations Using Two Modified Vlasov Models

The influence of soil heterogeneity is studied on the bending of circular thin plates using two modified Vlasov foundation models. The model parameters are determined reasonably using an iterative technique. According to the principle of minimum potential energy and considering transversely isotropic soils and Gibson soils, the governing differential equations and boundary conditions for circular thin plates on two modified Vlasov foundations are derived using a variational approach, respectively. The determination of attenuation parameters is a difficult problem, which has hindered the further application of the Vlasov foundation model. The equation that must be satisfied by the attenuation parameter is determined, and an iterative method is used to solve the problem. A comparative analysis is conducted between two modified Vlasov models and the traditional Vlasov model. The results show that the governing equations and boundary conditions for circular thin plates resting on two modified foundations are consistent with those for a circular thin plate on traditional two-parameter foundation after degradation. The accuracy and reliability of the proposed solutions are demonstrated by comparing the obtained results with those reported in the literature. The heterogeneity of soils, including the transversely isotropic soils and Gibson soils, has a certain effect on characteristic parameters of the foundation models as well as the deformations and internal forces of circular thin plates. The present study could be employed as a reference for future engineering designs.

Assessment of Hybrid Method on Investigation of Dynamic Behaviour of Isotropic Rectangular Plates Resting on Two-Parameters Foundation

Dynamic behaviour of isotropic rectangular plate resting on two-parameter foundation is investigated. The governing partial differential equation is transformed to ordinary differential equation due to Galerkin method of separation. The hybrid method of Laplace transform and variation parameters method is used to analyze the ordinary differential equation. Introduction of exact method helps in fast convergence of the results. Obtained analytical solutions are compared with existing literature and confirmed as accurate. They are used to examine the effect of controlling parameters on the plate natural frequencies. Due to obtained results it is obvious that, the increase of both elastic foundation parameter and aspect ratio results in increasing the natural frequency. The solution is found immediately by means of a few iterations

Strata Movement Model of Filling Coal Mining Based on Two-Parameter Elastic Foundation

To avoid the discontinuity defect of the Winkler elastic foundation beam model, a two-parameter elastic foundation model of Pasternak reflecting the continuity of the waste foundation was applied in the regular analysis of strata movement in waste filling coal mining. A two-parameter elastic foundation beam strata movement mechanical model was established. The formulae to calculate the shear force, bending moment, deflection and foundation reaction force of roof strata were obtained using elastic foundation beam theory. The two-parameter elastic foundation beam model and the Winkler elastic foundation beam model were applied to an engineering example to analyze the influence of different waste foundation coefficient k and shear stiffness Gp on the force and deformation of roof strata respectively, and the results were compared accordingly. The results show that the shear stiffness has a certain influence on the force state and subsidence of the roof strata, when waste foundation coefficient is smaller, the difference between the two elastic foundation models is greater. Considering the continuity of the waste filling material, the subsidence of the roof beam obtained by the two-parameter foundation model is smaller than that by the Winkler model. The prediction of the development height of water flowing fractured zone and surface subsidence by the new model is more in line with the engineering practice.

A transient dynamic process in a structural nonlinear system “beam – two-parameter foundation”

A method for analytical assessment of dynamic added stress in elastic loaded beam resting on elastic two-parameter Pasternak’s foundation due to sudden destruction a part of foundation is proposed. Equations of static bending, natural and forced oscillations are written in a matrix form using state vectors including deflection, rotational angles, bending moments, and shear forces at arbitrary cross section of a beam and also using the matrices of the initial parameters influence on the stress-strain state in arbitrary cross section. The influence of foundation failure on beam’s stress-strain state, taking into account a relation between the stiffness parameters of foundation, is analyzed. The condition of smallness for the shear stiffness parameter (Pasternak’s parameter) in comparison with the stretching-compressing stiffness parameter (Vinkler’s parameter) is accepted. It is shown that the accounting of Pasternak’s parameter reduces the level of dynamic added stress in a beam when sudden destructing of a foundation. The factor of sudden defect occurrence in the system “beam – foundation” increases considerably the internal forces in a beam in comparison with quasistatic formation of the same defect.

Combination resonances of imperfect SSFG cylindrical shells rested on viscoelastic foundations

The present paper investigates the combination resonance behavior of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal and external functionally graded stiffeners under two-term large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. With regard to classical plate theory of shells, von-Karman equation and Hook law, the relations of stress-strain are derived for shell and stiffeners. The spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. According to the Galerkin method, the discretized motion equation is obtained. The combination resonance is obtained by using the multiple scales method. Finally, the influences of the stiffeners angles, foundation type, the nonlinear elastic foundation coefficients, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

Study on size-dependent vibration and stability of DWCNTs subjected to moving nanoparticles and embedded on two-parameter foundations

Parametric resonance is an important phenomenon that may be evinced in applying carbon nanotubes for the delivery of nanoparticles. This paper aims to investigate dynamics instability of double-walled carbon nanotubes (DWCNTs) surrounded by elastic medium and excited by a sequence of moving nanoparticles. The DWCNT is modeled as two Euler-Bernoulli beams interacting between them through van der Waals (vdW) forces. Based on Eringen's nonlocal elastic theory to consider the small-scale effects, the governing equations are derived by using Hamilton's principle. All inertial terms of the moving nanoparticles are taken into account. In addition, the van der Waals force between the constitutive atoms of the moving nanoparticle and those of the nanotube is considered. By utilization of the Galerkin method, the partial differential equations (PDEs) of motion are reduced to couple ordinary differential equations with time-varying coefficients describing a parametrically excited nanosystem. Then, an incremental harmonic balance (IHB) method is implemented to calculate the instability regions of the DWCNT. The results show that considering the vdW effects, increasing the amplitude of the static axial tensile force, reducing the amplitude of axial oscillating force, and increasing the stiffness of the elastic medium improve stability of the system. A comparison between the results with those reported in the literature is performed to verify the precision of the presented analyses.

Dynamic Analysis of Circular Plates in Contact with Fluid and Resting on Two-Parameter Foundations

The dynamic behaviour of a circular plate in contact with fluid and resting on two-parameter elastic foundations is of interest in the field of geotechnics, structure, highway, railway, oil and gas and mechanical engineering. In this work, the dynamic behaviour of circular plate in contact with fluid and resting on Winkler and Pasternak foundations is investigated. The coupled differential equation of the system is analysed using differential transformation method. Good agreements are established when the results of the analytical solutions are compared to the results of the experimental investigation as reported in literature. The analytical solutions obtained are used to investigate the effects of elastic foundation parameters on natural frequency, combine foundation parameters on natural frequency, plate in contact with fluid and that of radial and circumferential stress on mode shapes. From the results, it is observed that, increases in elastic foundation parameter increases the natural frequency in all cases. Presence of fluid reduces the natural frequency of the plate. Also, it is established that mode shapes are not altered by the presence of fluid. However, mode shape displacement occurs due to presence of radial and circumferential stresses. Since the study provides a physical insight into the vibration mode of the structure, it is expected that the study will enhance better understanding on the dynamic behaviour of a circular plate in contact with fluid and resting on two-parameters elastic foundation.

Nonlinear vibrations of single- and double-walled carbon nanotubes resting on two-parameter foundation in a magneto-thermal environment

The excellent mechanical, electrical, structural and thermal properties coupled with high strength to weight ratio of carbon nanotubes have tremendously expanded their applications in various industrial, engineering, physical and natural sciences processes. In this work, nonlocal elasticity theory is used to analyze nonlinear vibrations of single and double-walled carbon nanotubes resting on two-parameter foundation in a thermal and magnetic environment. With the aid of Galerkin decomposition method, the systems of nonlinear partial differential equations are transformed into systems of nonlinear ordinary differential equations which are solved using homotopy perturbation method. The developed analytical solutions are used to investigate the influences of elastic foundations, magnetic field, temperature rise, interlayer forces, small scale parameter and boundary conditions on the frequency ratio. From the results, it is observed that the frequency ratio for all boundary conditions decreases as the number of walls increases from single to double. Also, it is established that the frequency ratio is highest for clamped–simple supported and lowest for clamped–clamped supported. Additionally, the results revealed that the frequency ratio decreases with increase in the value of spring constant (k1) temperature and magnetic field strength. This work will enhance the applications of carbon nanotubes in structural, electrical, mechanical and biological applications especially in a thermal and magnetic environment.

Analysis of Rectangular Plates on Resting Winkler and Two-Parameter Elastic Foundation Models by Finite Difference Method

Analysis of rectangular plates on a rising elastic foundation is common when designing the foundation of civil, traffic, and irrigation works. The current research presents the results of the analysis of rectangular plates on rising Winkler elastic and two-parameter foundation using the finite difference method. The computational programming was done on Matlab. The results of the research verify the accuracy of the FEM and are in agreement with findings in the literature. In addition, this research also compares the calculation results of two foundation models and investigates the change of plate height ratio to plate size.

Rotary Inertia and Higher Modes Effect on the Dynamic Response of Timoshenko Beams on Two-Parameter Elastic Foundation

This study investigates the dynamic response of the axially loaded Timoshenko beams on two-parameter foundation with generalized elastic end conditions. A simplified modal analysis procedure is presented for the vibration analysis by using the conventional separation of variables method. The effects of higher modes and rotary inertia on the dynamic response of the forced Timoshenko beam are highlighted. A numerical example is presented to compare the modal responses of bending moment and shear force of the beams on Winkler and Vlasov type foundations with and without the rotary inertia effect.

Decoupled technique for dynamic response of vehicle-pavement systems

This paper presents an iterative decoupled technique for the dynamic response of vehicle-pavement systems. The method involves the decoupling of the vehicle-pavement coupled system into two subsystems and solving them independently until the solution has converged using an iterative scheme. The technique uses a combination of the Galerkin method, finite difference method, and Newmark’s method for discretizing the differential equation and computing the dynamic response. The proposed method is illustrated for two pavement models. The first model idealizes the pavement as a Timoshenko beam resting on a nonlinear foundation. The second model considers the pavement as a double-layer rectangular thin plate on a two-parameter foundation. The vehicle is simplified as a spring-mass-damper oscillator. The effect of the two pavement models on the dynamic response is investigated. Random roughness profiles for two different quality of roads is considered, and Monte Carlo simulation is used to compute the stochastic system responses. Additionally, the effect of coupling action on pavement displacements and vehicle body vertical displacement are examined. Numerical study on the proposed technique reveals that the method has a linear rate of convergence. The competence of the technique in computing the dynamic response of a pavement traversed by multiple vehicles is demonstrated. Numerical investigations prove that the decoupled technique is not only computationally efficient but also has the potential to tackle complicated problems in vehicle-pavement system dynamics.