Elastic Foundation Coefficients Research Articles

Subsidence Prediction Method Based on Elastic Foundation Beam and Equivalent Mining Height Theory and Its Application

Grouting technology in overburden separation is recognized as an effective method to prevent surface subsidence and reuse solid waste. This study used mechanical analysis to explore deflection characteristics of key strata and accurately predict and control surface subsidence. Conceptualizing the coal–rock mass beneath the key strata as an elastic foundation, we developed a method to calculate the elastic foundation coefficients for various regions and established an equation for key strata deflection, validated through discrete element numerical simulations. This simulation also examined subsidence behavior under different grout injection–extraction ratios. Additionally, combining the equivalent mining height theory with the probability integral method, we formulated a predictive model for surface subsidence during grouting. Applied to the 8006 working face of the Wuyang Coal Mine, this model was supported by numerical simulations and field data, which showed a maximum surface subsidence of 546 mm at a 33% injection–extraction ratio, closely matching the theoretical value of 557 mm and demonstrating a nominal error of 2%. Post-grouting, the surface tilt was reduced to below 3 mm/m, meeting regulatory standards and eliminating the need for ongoing surface structure maintenance. These results confirm the model’s effectiveness in forecasting and controlling surface subsidence with grouting. The study can provide a basis for determining the grouting injection–extraction ratios and evaluating the effectiveness of surface subsidence control in grouting into overburden separation projects.

Nonlinear Forced Vibration of Functionally Graded Graphene-Reinforced Composite (FG-GRC) Laminated Cylindrical Shells under Different Boundary Conditions with Thermal Repercussions

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

Thermal buckling and postbuckling of functionally graded multilayer GPL-reinforced composite beams on nonlinear elastic foundations

Under the influence of axial forces and uniform temperature variations, the thermal buckling and postbuckling of composite beams reinforced of functionally graded multilayer graphene platelets (GPLs) resting on nonlinear elastic foundations are examined. The Halpin-Tsai model is used to calculate the elastic modulus of each layer of GPL-reinforced composite (GPLRC). According to the virtual work principle, the nonlinear governing equations for the beam are obtained from the first-order shear deformation beam theory. The impact of axial force and nonlinear elastic foundation on thermal buckling and postbuckling is discussed using the differential quadrature method (DQM), and the analytical expression is given by the two-step perturbation method (TSPM). The effects of axial force, boundary conditions, slenderness ratio, GPL geometry, GPL weight fraction, GPL distribution pattern, and elastic foundation coefficient on thermal buckling and postbuckling are examined through parameter analysis.

Investigation of Free Vibrations of Stepped Nanobeam Embedded In Elastic Foundation

The free vibration of stepped nanobeams embedded in the elastic foundation was investigated using Eringen's nonlocal elasticity theory. It is fixed at the system ends with a simple-simple support. The stepped nanbeam’s equations of motion are obtained by using Hamilton's principle. Multi-time scale, which is the perturbation methods, was used for the analytical solution of the equations. To observe the effects of nano size effect, elastic basis coefficient and step location, natural frequencies of the first three modes of the system were obtained for different non-local parameter values, elastic foundation coefficients, step rates and step positions. In the results, it was seen that the non-local parameter had a negative effect on the natural frequency. The elastic foundation coefficient has been shown to reduce vibration amplitudes.

Porosity-dependent wave propagation in multi-directional functionally graded nanoplate with nonlinear temperature-dependent characteristics on Kerr-type substrate

The present study investigates the wave propagation characteristics of porous, multi-directional, functionally graded (FG) nanoplates embedded in a Kerr-elastic substrate. A first-order shear deformation model (FSDM) and the nonlocal strain gradient approach (NSGA) are utilized to address this issue. The NSGA is enhanced by considering softening and hardening material effects, leading to improved accuracy in the obtained results. In accordance with the law of mixing, the material properties of the nanoplates exhibit nonlinear temperature-dependent variations along the structural axes. Both even and uneven porosity profiles are considered in this study. The nonclassical governing equations for the proposed structure are derived using Hamilton's approach. The phase velocity and wave frequency are determined through the analytical solution of an eigenvalue problem, which depends on the wave number. The dispersion properties of waves in porous FG nanoplates are examined with respect to various factors, including porosity distributions, wave numbers, nonlocal and strain gradient effects, elastic foundation coefficients, and volume fraction index. The main findings of the study indicate that increasing the volume fraction index in the thickness direction leads to a decrease in the phase velocity and frequency of wave propagation in a porous FG nanoplate for a given wave number. Additionally, regardless of the porosity pattern, an increase in the porosity coefficient results in a decrease in wave frequency and phase velocity.

Wave propagation phenomenon of functionally graded graphene oxide powder-strengthened nanocomposite curved beam

In this study, it is tried to probe the wave propagation phenomenon in an embedded functionally graded graphene oxide powder-strengthened nanocomposite curved beam exposed to different thermal loadings. In order to achieve different thermal environments, three different temperature rises including uniform temperature rise, linear temperature rise and sinusoidal temperature rise are considered. The graphene oxide powders are distributed in a polymer matrix along the curved beam thickness by regarding various patterns (i.e. X, Λ, V, O and uniform patterns). The Halpin–Tsai homogenization model is implemented to compute the effective properties of the nanocomposite structure. The kinetic relations of nanocomposite curved beam are derived by utilizing Euler-Bernoulli beam theory incorporating Hamilton's principle. The derived governing equations of nanocomposite curved beams are analytically solved to achieve the wave frequency and phase velocity values. The precision of the calculated results is verified by the published outcomes in the literature. Thereupon, a parametric investigation is performed to study the influences of different remarkable parameters like different distribution patterns of graphene oxide powders, the weight fraction of graphene oxide powders, wave number, thermal environment, elastic foundation coefficients and opening angle.

Wave dispersion analysis of natural fiber-strengthened composite beam lying on variable elastic foundation

For the first time, the dispersion of elastic waves in natural fiber-strengthened composite high-order beam rested on an elastic substrate is probed in the current article. Furthermore, a theoretical damage model is used to describe the mechanical degradation of natural fiber-strengthened composite beam during hygrothermal aging. Different three types of Winkler foundations are considered as a linear layer of the elastic foundation. To derive the equations of motion of natural fiber-strengthened composite beam, principle of Hamilton is implemented. In addition, the analytical solution is utilized to solve the derived governing equation of natural fiber-strengthened composite beam rested on Winkler–Pasternak substrate. The accuracy of using methodology is verified by those reported in the literature. The impacts of various parameters, i.e., wave number, aging time, fiber content, different types of Winkler foundation, Winkler and Pasternak parameters, α parameter and slenderness ratio on the variation of wave propagation of natural fiber strengthened composite beam are explored. Some tables and figures are provided to illustrate the obtained outcomes. The obtained results indicate that elastic foundation coefficient and wave number have an increasing effect and slenderness ratio and α parameter have decreasing effect. Furthermore, the effect of fiber content on change in wave propagation behavior depends on aging time.

Dynamic stability, and absorbed energy measurements in a sandwich structure resting on the novel frictional viscoelastic torsional substrate and horizontal friction force

Like metallic materials, fiber metal laminate (FML) lets the material has a simple metal structure behavior but with such advantages as specialized strength properties, impact, fire resistance, weight savings, corrosion resistance, and metal fatigue. The main focus of this article is to indicate the dynamic response of circular size-dependent circular plates made of FML material via 3D-elasticity theories. The size-dependent structure is simulated by coupling the modified nonlocal couple stress and 3D-elasticity theories (MNCS-3DET). For accurate modeling of the elastic substrate, torsional and linear spring and Pasternak foundation are modeled using the equations of horizontal friction force and torsional interaction. Harmonic differential quadrature method and Chebyshev–Gauss–Lobatto grid distribution points are coupled for solving the partial differential equations of the current micro/nano-structure. Novelties of this work consider the effects of MCS-3DET, torsional interaction, horizontal friction force, and FML material in the construction of the circular micro/nanoplate. In the results section, the outputs are verified with other published articles, and the outcomes show that the MCS-3DET parameter, elastic foundation coefficient, horizontal friction force coefficient, and geometry have an essential role in the dynamic stability of the FML micro/nano-structure.

A Theoretical Model of Roof Self-Stability in Solid Backfilling Mining and Its Engineering Verification

Roof self-stability in backfilling mining was proposed to explore its connotation and characteristics after a comparative analysis of roof structures under long-wall caving and backfilling mining. The mechanical analysis models of roof self-stability along strike and dip were established. After that, the mechanical equations for cooperative roof control were constructed to analyze the elastic foundation coefficients of the backfill, support peak load, unsupported-roof distance, and drilling effect of the working face along strike, the size of the working face, and the section pillar effect along dip. Research showed that the roof self-stability was greatly impacted by the elastic foundation coefficient of backfill, and it was less impacted by the support peak load along strike. The unsupported-roof distance had no obvious effect on roof self-stability. Roof self-stability was significantly affected by the working face and coal-pillar length along the dip. Therefore, the engineering control method of roof self-stability was proposed. The backfilling engineering practice in Xinjulong Coal Mine showed that the maximum roof subsidence was 438 mm, and the backfill ratio was 86.3% when the supporting intensity of backfilling hydraulic support was 0.94 MPa; the advanced distance of the working face was greater than 638 m; the foundation coefficient of backfilling material was 4.16 × 108 Nm−3. The roof formed the self-stability structure, which satisfied safe coal mining under buildings, water bodies, and railways.

Interlayer shearing and bending performances of ballastless track plates based on high-order shear deformation theory (HSDT) for laminated structures

Based on higher-order shear deformation theory (HSDT), the bending of China Railway Track System type II (CRTS-II) track laminated plates on elastic foundations and interlayer shearing performances are analyzed by meshfree radial point interpolation method (RPIM), which has the properties of the Kronecker function and discretizes the governing equations derived from HSDT. To verify the applicability of the model, the geometric configuration and key parameters of the CRTS-II slab ballastless track structure are analyzed, and the interlaminar slip model of the track structure is derived as a benchmark solution when calculating the track structure using HSDT-RPIM. The accuracy of the derivation is verified by comparing the HSDT-RPIM model with a double-layer slab model considering bidirectional interlayer slip. Then, the effect of resistant slip coefficient on the bending deformation of the track structure was analyzed using the bidirectional interlaminar slip model. The effect of elastic foundation coefficient on the bending of the track structure and the interlayer slip of the track structure (track slab – base slab) was analyzed. Finally, the effects of boundary conditions and elastic foundation coefficients on the interlaminar stresses of the track structure are investigated using HSDT-RPIM, and some interlaminar damages of the track structure are explained.

Modified size-dependent theory for investigation of dynamic stability and critical voltage of piezoelectric curved system

Nowadays, using piezoelectric materials for electrical purposes has attracted much attention among researchers. So, in the current report, dynamic stability and critical voltage analysis of an nanoplate resting on an elastic foundation. The current nanostructure has been made of piezoelectric material. Modified couple stress theory (MCST) and Nonlocal theory (NT) are mixed (modified nonlocal couple stress theory) with one nonlocal and one length scale parameter for modeling the current nanostructure has been presented. Higher-order shear deformation plate theory (HSDPT), and Hamilton’s principle have been used for obtaining the boundary and governing equations of the piezoelectric nanoplate. A two-dimensional generalized differential quadrature element method (2D-GDQEM) has been presented for solving the displacement domains of the current nanostructure. By comparing the outputs of the current report and those of published articles, there is good agreement between the results. Finally, the results show that elastic foundation coefficients (Winkler and Pasternak type), boundary conditions, applied voltage, size-dependent parameters (length scale and nonlocal ones), and geometry properties have important role in the critical voltage and dynamic stability performance of the piezoelectric nanostructure. The outputs of the current report can be used for future researchers of NEMS by incorporating modified nonlocal couple stress theory (MNCST).

Pseudo-dynamic analysis of reinforced slope with anchor cables

The anchor cable is an excellent reinforcement device for enhancing slope performance subjected to earthquakes. Thus, this work combines the minimum potential energy principle with the pseudo-dynamic method and develops a novel approach to determining a reinforced slope safety factor (SF). Specifically, the mobilized shear stress on the slip surface is solved utilizing the moment equilibrium equation of the sliding mass. Moreover, the proposed scheme introduces the elastic foundation coefficient characterizing the soil’s compressibility. Subsequently, the potential energy equation is derived. Finally, the SF is calculated as the ratio of the anti-sliding moment to the sliding moment for evaluating the slope stability. The developed method is evaluated on four examples, with the results highlighting that our method is efficient and accurate. Meanwhile, we examine the influence of the variation of the axis forces on the SF when the foundation coefficient is smaller, revealing that the depth of the sliding mass is negatively correlated with the SF, while the landslide area and length of the sliding mass are positively correlated with the SF. Furthermore, this work also studies the influences of amplification factor, earthquake acceleration factor, anchored force, cohesion, and internal friction angle on the SF. Overall, this work provides suggestions for the design and construction management of reinforced slopes with anchor cables.

Nonlocal and surface effects on nonlinear vibration response of a graded Timoshenko nanobeam

The free and forced vibration of a graded geometrically nonlinear Timoshenko nanobeam supported by on a nonlinear foundation is considered in this paper. The main contribution of this study is to propose a new formulation for the dynamic response of this beam by combining nonlocal and surface elasticity in addition to employing the physical neutral axis method which eliminates the quadratic nonlinearity from the equation of motion. Using the principle of virtual work, a fourth-order nonlinear partial differential equation is formulated and Galerkin technique is employed to yield a fourth-order ordinary differential equation with cubic nonlinearity in the temporal domain. The method of multiple scales is employed to obtain the analytical expression of the nonlinear frequency of the beam and its frequency response curve from a primary resonance analysis. To assess the accuracy of this analytical solution, it is compared with a numerical solution obtained using the differential quadrature method. The obtained analytical results are successfully validated for particular cases of the considered problem with results published by other authors. The effects of surface elasticity, nonlocality, the physical neutral axis, the beam aspect ratio, the power-law index and the elastic foundation coefficients on the free and forced vibration response of the graded Timoshenko nanobeam are thoroughly investigated for different types of boundary conditions .

Exact analytical solution for static deflection of Timoshenko composite beams on two-parameter elastic foundations

New exact analytical solutions are presented for the static deflection of coupled Timoshenko composite beams resting on two-parameter elastic foundations subject to arbitrary boundary and loading conditions. Governing differential equations are obtained from the principle of virtual work in which four degrees of freedom, namely axial elongation, twist, bending and transverse shear are coupled. In order to obtain the analytical solutions describing the static deflection of coupled Timoshenko composite beams resting on elastic foundations, the governing equations are first rewritten in matrix form to enable decoupling of axial displacement and twist from bending and transverse shear. Applying a separation of variables technique, axial elongation and twist are eliminated and the matrix differential equation for bending and transverse shear is transformed into a system of first-order equations which is solved using a fundamental matrix approach. The results obtained from the analytical solutions are firstly verified by comparison with solutions from the literature for isotropic homogeneous beams on elastic foundations and then with the numerical results from the Chebyshev collocation method obtained for composite beams on elastic foundations, and found to be in excellent agreement. The influence of elastic foundation coefficients, coupling terms and length-to-thickness ratio on the static deflection of Timoshenko composite beams resting on elastic foundations are investigated and discussed.

Nonlinear Free Vibration Analysis of In-plane Bi-directional Functionally Graded Plate with Porosities Resting on Elastic Foundations

This paper deals with the nonlinear free vibration analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with porosities which are resting on Winkler–Pasternak elastic foundations. The material properties of the IBFG plate are assumed to be graded along the length and width of the plate according to the power-law distribution, as well as, even and uneven types are taken into account for porosity distributions. Equations of motion are developed by means of Hamilton’s principle and von Karman nonlinearity strain–displacement relations based on classical plate theory (CPT). Afterward, the time-dependent nonlinear equations are derived by applying the Galerkin procedure. The nonlinear frequency is determined by using modified Poincare–Lindstedt method (MPLM). Numerical results are obtained in tabular and graphical form to examine the effects of some system key parameters such as porosity coefficients, distribution patterns, gradient indices, elastic foundation coefficients, aspect ratio and vibration amplitude on the nonlinear frequency of the porous IBFG plate. To validate the analysis, the results of this paper have been compared to the published data and good agreements have been found.

Nonlinear dynamic stability analysis of clamped and simply supported organic solar cells via the third-order shear deformation plate theory

This study presents a semi-analytical framework to explore the nonlinear dynamic buckling characteristics of the imperfect organic solar cell (OSC) with clamped and simply supported restrains grounded on the third-order shear deformation plate theory (TSDT). The exerting axial displacement loading is divided into infinite and finite durations. Moreover, the spatially dependent Winker-Pasternak elastic foundation and damping effect are incorporated in the current analysis. Combining the von Kármán nonlinearity, the governing equations are derived with the aid of the Airy stress function. By applying the Galerkin method and fourth-order Runge–Kutta procedure, the dynamic buckling critical condition of the OSC subjected to these two kinds of loadings is determined by Budiansky–Hutchinson (B-H) criterion. Based on the numerical results, the influence of some critical parameters, including the geometrical dimension, boundary condition, loading configuration, initial imperfection, damping ratio, and elastic foundation coefficient are investigated. Not limited to the solar cell, this method is also suitable to the dynamic buckling behaviours of other laminated plates.

Aerodynamic Analysis of Temperature-Dependent FG-WCNTRC Nanoplates under a Moving Nanoparticle using Meshfree Finite Volume Method

Transporting of nanomachines using the nanoplates has been acknowledged as an emerging frontier of nanotechnology over recent years. This paper investigates the aerodynamic response of a temperature-dependent functionally-graded composite nanoplate reinforced by wavy carbon nanotubes (FG-WCNTRC nanoplate) under a nanoparticle moving in a straight path exposed by air drag. The meshfree finite volume (MFV) approach is implemented to develop the governing equations based on the Mindlin's plate and Eringen's non-local theories. Therefore, one uniform distribution and three linear distributions of wavy CNTs are considered through the thickness of the FG composite nanoplate. Moreover, a generalized rule of mixtures is employed to predict the mechanical properties of the nanoplate. In this sense, the material properties of the matrix and wavy CNTs are presumed to depend on temperature. A comprehensive investigation is then carried out for the first time to assess the effects of nanoplate elastic foundation coefficients, nanoparticle velocity, temperature dependency of material, volume fraction, waviness, distribution pattern, and orientation of the wavy CNTs on the amount of air drag effect. It is revealed that in the aerodynamic response of temperature-dependent FG-WCNTRC nanoplates under a moving nanoparticle, the air drag amount can be influenced considerably by the material properties of the nanoplate.

A size-dependent nonlinear finite element free vibration analysis of multilayer FG-GPLRC toroidal micropanels in thermal environment

A nonlinear finite element model based on the modified strain gradient theory (MSGT) in combination with the first-order shear deformation theory (FSDT) of shells for the free vibration of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) toroidal micropanels is developed. The nine-noded shell elements with five degrees of freedom per node are used to accurately simulate thin as well as moderately thick toroidal micropanels without shear locking phenomena. In addition to initial thermal stresses, the influences of nonlinear elastic foundation and the elastic rotational springs at the micropanel edges are considered. The formulation and corresponding computer codes are validated by performing convergence study and doing comparison studies with some existing results in the literature. Then, the impacts of amplitude ratio, geometric parameters, ratio of thickness-to-material length scale parameter (MLSP), temperature rise, coefficients of elastic foundation and rotational springs, and different GPLs distribution patterns on the nonlinear vibrational behaviors of the multilayer FG-GPLRC toroidal micropanels are investigated. The results show that the size dependence of material properties increases the impact of geometric nonlinearity on the vibrational behavior of the micropanels. In addition, it affects the nonlinear vibrational behavior more than the linear one in a widespread range of thickness-to-MLSP ratio.

Exact solution for nonlinear static behaviors of functionally graded beams with porosities resting on elastic foundation using neutral surface concept

In this paper, an exact solution for nonlinear static behaviors of functionally graded (FG) beams with porosities resting on the elastic foundation is presented. The FG material properties with porosities are assumed to vary along the thickness of the beam, and two types of porosity distributions are considered. Actually, the geometrical middle surface of the FG beam selected in computations is very popular in the literature. By contrast, in this study, the physical neutral surface of the beam is utilized. Based on the Timoshenko beam theory, von Kármán nonlinear assumption, together with neutral surface concept, the nonlinear governing equations of the FG beam resting on the elastic foundation are derived. By using the physical neutral surface, the nonlinear governing equations have simple forms and can be solved directly. The exact solution for the problem with all immovable and moveable boundary conditions is conducted in detail. Some numerical investigations to show the effects of boundary conditions, material properties, length-to-thickness ratio, elastic foundation coefficients and several types of applied load on nonlinear static bending behaviors of the beam are given.