Governing Equilibrium Equations Research Articles

Frictional Slippage of Annular Elastomeric Disks Compressed Between Rigid Platens

Abstract Compression of compressible, linearly elastic, annular disks by flat rigid platens is analyzed. Coulomb (Amonton) friction is assumed to act at the interfaces between the disk and the platens. Slip may occur in an outer annular region while the inner annular (bonded, stick) region of the disk does not slip. The critical radius (slip boundary) is of major interest. The governing equilibrium equations in terms of the deflections are satisfied exactly. Approximations are made in some of the boundary conditions and the transition (matching) conditions at the critical radius. Numerical results are presented for nearly incompressible disks. The effects of the radius ratio, aspect ratio, and Poisson's ratio of the disk, and of the coefficient of friction at the platens, on the critical radius, effective compression modulus, stresses, and radial deflection are investigated. Applications include structural (especially bridge) bearings, seismic-isolation devices, mounting blocks and bushings, gaskets, and sealing components.

Frictional slippage of elastomeric disks compressed between rigid platens and subjected to torsion

A new analysis is applied to compressible, linearly elastic disks that are compressed by flat rigid platens. The disks are not bonded to the platens and Coulomb (Amonton) friction is assumed to act at the interfaces between the disk and the platens. Slip may occur in an outer annular region, while the inner circular (stick) region of the disk does not slip. The critical radius (slip boundary) is of major interest. The governing equilibrium equations in terms of the deflections are satisfied exactly. Approximations are made in some of the boundary conditions and the transition (matching) conditions at the critical radius. Numerical results are presented for nearly incompressible disks, both for (a) compression only and (b) compression plus torsion. In the latter case, during twisting, either the compressed thickness of the disk is fixed while the compressive force varies, or the compressive force is fixed while the thickness varies. The effects of the aspect ratio and Poisson's ratio of the disk, the coefficient of friction at the platens, and the twist angle on the critical radius, stresses, compressive force, torque, and effective compression and shear moduli are investigated. Applications include structural (especially bridge) bearings, seismic-isolation devices, mounting blocks and bushings, and rheometry.

Analysis of doubly curved laminated composite shells using hybrid-Trefftz finite element model based on a high order shear deformation theory

Static analysis of doubly curved laminated composite shells with rectangular plan form has been carried out using a novel hybrid Trefftz finite element (HTFE) model. A typical higher order shear deformation theory (HSDT) has been used for describing the kinematics of deformations delineating both internal and auxiliary displacement fields. This HSDT allows to derive the system of governing partial differential equations of the single layer equivalent to the element domain. The characteristic functions forming the exact solutions of these homogenous governing equilibrium equations are used as the Trefftz functions. A key aspect of our HTFE strategy is its efficiency in bypassing the use of particular solution of the governing equations. Validation against the exact solutions reveals the precision of this novel HTFE model. The versatility of the Trefftz functions allows them to accommodate a spectrum of geometric configurations, ranging from standard to complex polygonal HTFE designs. The HTFE model for the doubly curved composite shells employing the HSDT marks a significant advancement in the analysis of doubly curved spherical, paraboloid and hyperboloid cross-ply and antisymmetric angle-ply shells.

Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

On the Static Instability of FG-GNP-Reinforced Composite Cylindrical Shells Under Thermo-Mechanical Loading

The nonlinear buckling response of laminated composite cylindrical shells reinforced with graphene nanoplatelets (GNPs) is studied in this paper. The functionally graded (FG) shell reinforced by GNPs is analytically studied under external pressure and uniform temperature rise loadings. It is also assumed that the GNP-reinforced laminated composite shell is in contact with an elastic foundation. Various types of profiles are employed for the GNP distribution patterns in the shell thickness including 10 nanocomposite layers. The nonlinear strain-displacement relations of the shallow cylindrical panel are established utilizing the third-order shear deformation shell theory. Governing equilibrium equations of the laminated GNP-reinforced composite shell are formulated employing the principle of virtual displacement. The coupled system of nonlinear differential equations is solved analytically for the hinged–hinged and fixed–fixed boundaries of the shell using a perturbation-based technique. Correctness of presented formulations and obtained solutions is proved by comparisons with results from previous studies for an isotropic cylindrical shell. Novel numerical results reveal that the material properties, geometrical characteristics and load parameters significantly affect on the buckling behavior of laminated composite cylindrical shells.

Nonlinear eigenstrain analysis for compressible Blatz–Ko solids

The objective of this research is to present eigenstrain analysis for a certain class of compressible, isotropic nonlinear elastic solids known as Blatz–Ko materials. The deformations and stress fields caused by eigenstrain distributions in (1) a cylindrical wedge with axisymmetric cylindrical eigenstrains, (2) a cylindrical bar with axisymmetric torsional and cylindrical eigenstrains, and (3) a spherical ball with spherically-symmetric eigenstrain distribution, which are supposed to be composed of the special Blatz–Ko material, are studied. In each problem, the form of deformation is first presumed to put the associated material manifold with Riemannian geometry in Euclidean space, and then the governing equilibrium equations are derived. The obtained nonlinear equations are solved and assessed when the structure has an inclusion with uniform eigenstrains. Moreover, this paper investigates the special solutions in which the stress becomes uniform or hydrostatic inside and outside the inclusion, or singular at the sphere center or cylinder axis. To have a better understanding of the analysis, an example is evaluated numerically in each section.

Effect of variable volume fraction distribution and geometrical parameters on the bending behavior of sandwich plates with FG isotropic face sheets

This article analyzes the bending behavior of functionally graded sandwich plate structures submitted to sinusoidal loads using the hyperbolic quasi-3D shear deformation plate theory. In this theory, the number of unknowns is reduced from six to only five unknowns using an undefined integral without needing any shear correction factor. Furthermore, the effective constituent material properties of the upper and lower layers of the functionally graded sandwich plate are assumed to vary smoothly and continuously only in the thickness direction. In contrast, the core layer is still homogeneous and isotropic. The governing equilibrium equations of simply supported functionally graded sandwich plates are derived using the principle of virtual displacement. Their analytical solution is obtained using Navier’s technique for various schemas of functionally graded sandwich plates. The effects of the power law index, side-to-thickness ratio, and aspect ratio on the bending behavior are investigated.

A C0-HSDT free vibration of magneto-electro-elastic functionally graded porous plates using a moving Kriging meshfree method

This research examines the natural vibration analysis of magneto-electro-elastic functionally graded porous (MEE-FGP) plates. The study employs a moving Kriging (MK) meshfree method to analyze the MEE-FGP plates composed of piezoelectric and piezomagnetic materials with both even and uneven distributions of porosity. The electric and magnetic potentials in the three-dimension are assumed to follow a combination of cosine and linear variations in the thickness direction to satisfy the Maxwell equations in the two-dimension. The displacement field, electric potential and magnetic potential are approximated by the MK interpolation shape function, which satisfies the Kronecker's delta property. This property allows to impose directly essential boundary conditions as those in the finite element method. The governing equilibrium equations are derived using a C0-type higher-order shear deformation plate theory (C0-HSDT) in conjunction with the virtual work principle and then solved using the MK meshfree method. Various geometries are performed to determine the natural frequency of MEE-FGP plates. The impact of the porosity distribution, porosity volume fraction, volume fraction index, initial external electric voltage, initial magnetic potential and geometry on the natural frequency of the MEE-FGP plates is fully investigated. Numerical results provide an effective approach to analyze and calculate the natural vibration of MEE-FGP plates.

On Bending of Piezoelectrically Layered Perforated Nanobeams Embedded in an Elastic Foundation with Flexoelectricity

Analysis of the electromechanical-size-dependent bending of piezoelectric composite structural components with flexoelectricity has been considered by many researchers because of the developments of nanotechnology and the applicability of piezoelectric composite nanobeam structures in Micro/Nano-Electro-Mechanical Systems (MEMS/NEMS). Therefore, the work investigates the size-dependent electromechanical bending of piezoelectrically layered perforated nanobeams resting on elastic foundations including the flexoelectric effect. Within the framework of the modified nonlocal strain gradient elasticity theory, both the microstructure and nonlocality effects are captured. The governing equilibrium equations including piezoelectric and flexoelectric effects are derived using Hamilton’s principle. Closed forms for the non-classical electromechanical bending profiles are derived. The accuracy of the proposed methodology is verified by comparing the obtained results with the available corresponding results in the literature within a 0.3% maximum deviation. Parametric studies are conducted to explore effects of perforation parameters, elastic foundation parameters, geometric dimensions, nonclassical parameters, flexoelectric parameters, as well as the piezoelectric parameters on the bending behavior of piezoelectrically layered perforated nanobeams. The obtained results demonstrate that incorporation of the nondimensional elastic foundation parameters, Kp = 2 and Kw = 20, results in a reduction in the relative percentage reduction in the maximum nondimensional mechanical transverse deflection due to increasing the perforation filling ratio from 0.2 to 1 from 199.86% to 91.83% for a point load and 89.39% for a uniformly distributed load. On the other hand, with Kp = 5 and Kw = 50, the relative percentage difference of the electromechanical bending deflection due to increasing the piezoelectric coefficient, e311, reaches about 8.7% for a point load and 8.5% for a uniformly distributed load at a beam aspect ratio of 50. Thus, the electromechanical as well as mechanical behaviors could be improved by controlling these parameters. The proposed methodology and the obtained results are supportive in many industrial and engineering applications, i.e., MEMS/NEMS.

Analytical model for the nonlinear buckling responses of the confined polyhedral FGP-GPLs lining subjected to crown point loading

This paper focuses on the nonlinear buckling responses of the confined functionally graded porous (FGP) lining with polyhedral shapes reinforced by graphene platelets (GPLs). The FGP-GPLs polyhedral lining is surrounded tightly and rigidly by a medium. It is assumed that the interface is smooth between the lining and pipeline. The deformation of the lining may be represented by an admissible displacement expression when a point load is applied at the crown position. Afterward, the nonlinear governing equilibrium equations are obtained explicitly by combining the thin-walled shell theory and the principle of minimum potential energy. Solving the nonlinear governing equilibrium equations yields the critical buckling load. A comparison study is strictly completed with the other closed-form solutions when the polyhedral lining reduces to a circular one. In addition, an improvement factor is quantified to understand the effect of polyhedral shapes on the bending stiffness. Finally, the effects of polyhedral shapes, porosity coefficient, weight fraction, and geometric parameters of the GPLs are explored in terms of buckling load, hoop force, and bending moment.

Stochastic Buckling of Geometrically Imperfect Beams on Elastic Foundation

Abstract Geometrical imperfections are ubiquitous in load-bearing structures, including beams, columns, and shells. Fabrication processes of structural members most often create geometrical imperfections of random size and shape, which lead to non-deterministic load-carrying capacity. This study investigates the statistics of the buckling load of a beam with a random initial imperfection profile that rests on a nonlinear elastic foundation. The geometrical imperfection is represented by a zero-mean Gaussian random field, generated using the Karhunen–Loève expansion. The spatial distribution of the random imperfection is characterized by the probability distribution of the local imperfection magnitude and a spatial autocorrelation function. A finite-difference scheme is used to solve the governing equilibrium equation for a given initial imperfection profile, from which the buckling load is determined. Through a set of Monte Carlo simulations, the mean and variance of the buckling load are determined. The simulations reveal the influence of different length scales on the statistics of the buckling load, including the beam length and the autocorrelation length of the geometrical imperfection. The size effects predicted with the simplified model have implications for reliability-based structural design.

Bending and Buckling of FG-GRNC Laminated Plates via Quasi-3D Nonlocal Strain Gradient Theory

To improve the structural stiffness, strength and reduce the weight of nanoplate structure, functionally graded (FG) graphene-reinforced nanocomposite (GRNC) laminated plates are exploited in this paper. The bending and buckling behaviors of FG-GRNC laminated nanoplates are investigated by using novel quasi-3D hyperbolic higher order shear deformation plate theory in conjunction with modified continuum nonlocal strain gradient theory, which considered both length and material scale parameters. The modified model of Halpin–Tsai is employed to calculate the effective Young’s modulus of the GRNC plate along the thickness direction, and Poisson’s ratio and mass density are computed by using the rule of mixture. An analytical approach of the Galerkin method is developed to solve governing equilibrium equations of the GRNC nanoplate and obtain closed-form solutions for bending deflection, stress distributions and critical buckling loads. A detailed parametric analysis is carried out to highlight influences of length scale parameter (nonlocal), material scale parameter (gradient), distribution pattern, the GPL weight fraction, thickness stretching, geometry and size of GPLs, geometry of the plate and the total number of layers on the stresses, deformation and critical buckling loads. Some details are studied exclusively for the first time, such as stresses and nonlocality effect.

A comprehensive damping study of variable stiffness composite rectangular/skew laminates reinforcement with curvilinear fibers by higher-order shear flexible model

In this work, a comprehensive investigation of curvilinear fiber reinforcement on damping of rectangular and skew composite plates is computationally estimated using higher-order shear flexible model. A set of governing equilibrium equations developed here in form of eigenvalue analysis is solved by adopting the Q-R algorithm. The damping factors associated with different vibrational modes are evaluated from the complex eigenvalues. The proposed model is validated against the available analytical and experimental results. The damping capability of laminated rectangular/skew composite plates is thoroughly analyzed by varying the curvilinear fiber path angles in the layers, lay-up orientations, structural boundary conditions, skew angle of the laminate, and nature of the material. Results reveal that the damping pretending to the curvilinear fibers plate is better than the conventional composite laminate; it varies significantly according to the variation in center and edge of the curvilinear fiber angles. It is also noted that the damping increases with the skew angle of the plate. The study conducted here shows the suitability of such variable stiffness composite structure for safe design under dynamic/impact loading situation.

Thermomechanical analysis of snap-buckling phenomenon in long FG-CNTRC cylindrical panels resting on nonlinear elastic foundation

Snap-buckling phenomenon in an infinitely long nanocomposite cylindrical panel subjected to uniformly distributed transverse pressure loading is analyzed in this research. It is assumed that the functionally graded (FG) carbon nanotube reinforced composite (CNTRC) shell is resting on nonlinear elastic foundation. The uniform thermal field is also included into the formulation. Thermomechanical properties of the shell are assumed to be temperature dependent. The CNTs distribution pattern can be uniform or functionally graded through the shell thickness. The shell is formulated using a general high-order shear deformation theory. Governing equilibrium equations of the shell which undergoes thermal–mechanical coupling loading are established via the virtual displacement principle. To capture the large deflections, the von Kármán type of kinematic assumptions is utilized. Two types of edge conditions are considered for the shell with infinite length which are immovable simply supported and clamped–clamped. To obtain the limit buckling load of the shell, the two-step perturbation technique and Galerkin procedure are used. Results of this study reveal that the snap-buckling phenomenon can take place in a shallow cylindrical panel subjected to lateral pressure loading. This attractive behavior is observed due to the immovability of the flat edges of the shell. It is shown that the snap-buckling phenomenon in a long cylindrical panel is highly dependent upon the elastic foundations, thermal environments, geometrical parameters and material properties.

Ultimate strength characteristics of unstiffened cylindrical shell in axial compression

Cylindrical shell is a common structural design solution in many engineering fields, such as the foundation supporting structures and buoyant column of the offshore wind turbine in the maritime sector. Operating in an ocean environment, these cylindrical shell structures need to withstand a combination of axial compression, bending moment, torsion and external pressure. This study contributes a parametric investigation for unstiffened cylindrical shells subjected to axial compression. Emphasis is placed on the ultimate strength characteristics and their relation with the initial geometric imperfection. The nonlinear finite element method is adopted by considering geometric and material nonlinearities in conjunction with an arc-length incrementation to solve the governing equilibrium equation. The numerical prediction is compared with the prevailing code-based approach, i.e. DNV, ABS, API and Eurocode. This study shows that the ultimate strength of unstiffened cylindrical shells in axial compression is highly sensitive to initial geometric imperfection. The code-based formulae appear to be overly conservative for predicting the ultimate compressive strength of cylindrical shells.

Thermal stress effects on size-dependent nonlinear axial vibrations of nanorods exposed to magnetic fields surrounded by nonlinear elastic medium

Taking von Kármán geometric nonlinearity into account, this research proposes a nonlinear model of nanorod, based on the nonlocal elasticity and axial beam theories. Imperfect nanorod is surrounded by nonlinear elastic medium and is subjected to the axial compression or various fields in terms of thermal and transvers magnetic loads. Pinned–pinned boundary conditions are immovable and free for the nanorod under axial compression considering thermal and magnetic loads. The governing equilibrium equations of the nanorod are derived by means of Hamilton's principle. The coupled nonlinear dimensionless differential equations are solved employing He's variational method. To evaluate the accuracy of the results, the results of this method are compared with the values obtained from the finite element method. Numerical results are provided to explore the influences of the low and high temperatures, nonlocal parameter, magnetic force of transverse field, amplitude of vibrations, and linear and nonlinear elastic materials for nanorods.

Simplified analytical method for lateral torsional buckling assessment of RHS beams with web openings

The present paper treats the lateral torsional buckling (LTB) of castellated rectangular hollow section (CRHS) beams. For the purpose, an analytical model was proposed to provide critical buckling loads in (LTB) resistance. In the present model two types of web openings are considered, referring to rectangular and hexagonal shape perforating. The sectional geometrical properties attributed to each type of opening are controlled by selecting the appropriate value of the reduction parameter. Ritz method is applied in order to found the governing equilibrium equations, following the critical buckling loads are determined from the eigenvalue problem imposed to the tangential stiffness matrix. The validity of the elastic lateral buckling resistances evaluated by the proposed method is checked with the finite element simulation using ABAQUS software. The numerical results for cantilevers and simply supported CRHS beams, with hexagonal and rectangular web openings, are presented for both short and long-span beams.

Buckling of partially-compressed laminated composite plates

The web buckling is one failure mode of web crippling for thin-walled fiber-reinforced plastic (FRP) composite beams subjected to transverse concentrated or distributed compression load from both the top and bottom bearing plates, and the web between the top and bottom flanges can be modeled as a plate with the rotationally-restrained (RR) and partially-compressed loaded edges and the simply-supported (SS) unloaded edges. Explicit solution and universal analytical charts/formulas for buckling analysis of such RRSS composite plates partially compressed along the transverse direction are obtained. The unique Heaviside function and trigonometric series expansion are used to express the partial edge load with effective load length in the governing equilibrium equation. The explicit solution is established for the orthotropic plates subjected to concentrated point compression load in the middle; while the generic buckling analytical curves are developed for infinitely long laminated composite plates transversely compressed partially at the middle of edges as well as near the corners. The accuracy of the present method is validated with available simple analytical solution, numerical finite element method and experimental data. The effect of rotational restraint stiffness, plate aspect ratio, the location and length of partial edge compression load on buckling behavior of laminated composite plates is investigated in the parametric study. A practical guideline for laminated composite plates with the elastic rotational restraints along the loaded edges and partially compressed along the transverse direction is proposed, and it can be further applied in discrete plate analysis to predict the web buckling strength (one failure mode in web crippling) of FRP composite structural shapes under transverse loading at the locations of load bearing or supports.

Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory

This manuscript presents the comprehensive study of thickness stretching effects on the free vibration, static stability and bending of multilayer functionally graded (FG) carbon nanotubes reinforced composite (CNTRC) nanoplates. The nanoscale and microstructure influences are considered through a modified nonlocal strain gradient continuum model. Based on power-law functions, four different patterns of CNTs distribution are considered in this analysis, a uniform distribution UD, FG-V CNTRC, FG-X CNTRC, and FG-O CNTRC. A 3D kinematic shear deformation theory is proposed to include the stretching influence, which is neglected in classical theories. Hamilton's principle is applied to derive the governing equations of motion and associated boundary conditions. Analytical solutions are developed based on Galerkin method to solve the governing equilibrium equations based on the generalized higher-order shear deformation theory and the nonlocal strain gradient theory and get the static bending, buckling loads, and natural frequencies of nanoplates. Verification with previous works is presented. A detailed parametric analysis is carried out to highlight the impact of thickness stretching, length scale parameter (nonlocal), material scale parameter (gradient), CNTs distribution pattern, geometry of the plate, various boundary conditions and the total number of layers on the stresses, deformation, critical buckling loads and vibration frequencies. Many new results are also reported in the current study, which will serve as a benchmark for future research.