Principle Of Virtual Displacements Research Articles

Thermal bending response of functionally graded magneto-electric–elastic shell employing non-polynomial model

The present mathematical model for complex shells is given in the framework of Carrera unified formulation. The mechanical, electrical, and magnetic equations are derived in terms of the principle of virtual displacement, Maxwell’s equations and Gauss equations. Fourier’s heat conduction equation is used. The governing equations are discretized by the Chebyshev–Gauss–Lobatto and solved with the differential quadrature method. The three-dimensional (3D) equilibrium for mechanical, electrical, and magnetic equations are employed for recovering the transverse stresses, electrical displacement and magnetic induction. Finally, quasi-3D solutions for cycloidal shell of revolution and a funnel panel are introduced in this paper.

Isogeometric buckling analysis of orthotropic thin-walled laminated beams using the Carrera unified formulation

The current work presents the combination of isogeometric approach (IGA) and Carrera unified formulation (CUF) to perform the buckling analysis of isotropic thin-walled and orthotropic beams. Toward this end, the layerwise (LW) theory based CUF and B-spline basis functions in IGA have been used. The LW theory uses Lagrangian expansion (LE) to estimate the displacement field in the cross-section. However, in the present study, B-spline basis functions in both longitudinal direction and cross-section have substituted the corresponding functions. The CUF expresses the 3 D displacement fields as a set of 1 D generalized unknown displacement. Combined model in the present study obliterates the costly use of 3 D finite element analysis with its high accuracy and loyalty to 3 D problems. The principle of virtual displacement (PVD) is used to obtain the fundamental nuclei. One of the features of CUF is that it’s higher-order expansion is free of the Poisson locking phenomenon. Also, there is no need to use the shear correction factor. Some examples of the isogeometric analysis of laminated composite beams with different geometries and boundary conditions are conducted and comparison of the critical buckling load results illustrate the proper validation and formulation of the present study.

Influence of initial geometrical imperfections on thermal stability of laminated composite plates using layerwise finite element

In this paper, the influence of initial geometrical imperfections on thermal stability of laminated composite plates using Layer wise plate model is presented. The mathematical model assumes layer wise variation of in-plane displacements, non-linear strain-displacement relations (in von Karman sense) and linear thermo mechanical material properties. The Koiter’s model for initial geometrical imperfection is adopted. Principle of virtual displacements (PVD) is used to derive Euler-Lagrange differential equations of linearized buckling problem. The weak form is discretized using nine-node Lagrangian isoparametric finite element. The original MATLAB program is coded for finite element solution. The effects of imperfection mode and amplitude, temperature distribution, side to thickness ratio b/h, aspect ratio b/a, boundary conditions and lamination scheme on critical buckling temperature are analysed. The accuracy of the numerical model is verified by comparison with the available results from the literature and new results are presented.

Local Buckling Development of H-Section Steel Core of Buckling-Restrained Brace

To enhance the theoretical basis for the half-wavelength evaluation of high-order local buckling of section steel used as inner core of buckling-restrained brace, this paper conducts theoretical and numerical studies on the local buckling development of an H-section steel core of buckling restrained brace. Firstly, the elastic buckling development of the flange under monotonic compression is theoretically discussed based on the elastic buckling theory of plate and the principle of virtual displacement. The numerical model for the buckling restrained brace with H-section steel core is then established based on Abaqus, and the elastic buckling development of the flange is validated. Finally, further numerical studies are conducted to reveal the elasto-plastic buckling development of the flange and web under cyclic loading. It is found that the local buckling development of the flange and web of the H-section steel core are different from that of the flat plate core of buckling restrained brace. Under cyclic loading, the shortest buckling wave of the flange and web are induced by the buckling of plates on the two sides of the contact point near the ends. It is confirmed that there is no need to consider the lateral support from the restraining members to evaluate the minimum half-wavelength of high-order local buckling for section steel core of buckling-restrained brace.

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.

A Quasi-3D Higher-Order Theory for Bending of FG Nanoplates Embedded in an Elastic Medium in a Thermal Environment

This paper presents the effects of temperature and the nonlocal coefficient on the bending response of functionally graded (FG) nanoplates embedded in an elastic foundation in a thermal environment. The effects of transverse normal strain, as well as transverse shear strains, are considered where the variation of the material properties of the FG nanoplate are considered only in thickness direction. Unlike other shear and deformations theories in which the number of unknown functions is five and more, the present work uses shear and deformations theory with only four unknown functions. The four-unknown normal and shear deformations model, associated with Eringen nonlocal elasticity theory, is used to derive the equations of equilibrium utilizing the principle of virtual displacements. The effects due to nonlocal coefficient, side-to-thickness ratio, aspect ratio, normal and shear deformations, thermal load and elastic foundation parameters, as well as the gradation in FG nanoplate bending, are investigated. In addition, for validation, the results obtained from the present work are compared to ones available in the literature.

SCHOUTEN'S FORCE STRESS TENSOR AND AFFINOR DENSITIES OF POSITIVE WEIGHT

The paper deals with the concept of the force stress pseudotensor and the derivation of equilibrium equations in terms of the Schouten's stress pseudotensor being an affinor density. The definition of Schouten's force stress pseudotensor is mainly based on the notion of a pseudoinvariant element of area. The requisute equations and notions from algebra and the analysis of pseudotensors is revisited. A fundamental orienting pseudoscalar is introduced and discussed. Conventional and non-conventional definitions of the force stress tensor are given. A unit normal vector to a level surface of a pseudoscalar field is introduced. The exceptional importance of using the theory of orientable manifolds in modeling micropolar continua in mechanics of solids is noted. The notion of M-cell and its orientation algorithm are recalled. Algorithms for constructing the tensor elements of the area of M-manifold immersed in N-dimensional space are discussed. The notions of vector, pseudovector, invariant and pseudoinvariant elements of surface area in three-dimensional space are revisited. The possibility of using pseudotensor volume elements of a given integer weight due to the formula for a pseudotensor field transformation to an absolute tensor field by a fundamental orienting pseudoscalar is discussed. Various realisations of covariant differentiation of pseudotensors are considered. Covariant derivatives are given for a pseudoscalar and a contravariant pseudotensor of the second rank of an arbitrary integer weight. The principle of virtual displacements is formulated in terms of pseudo-invariant volume and area elements. The hypothesis of the absolute invariance of the virtual work is assumed, i.e. insensitive to rotations, 3D inversion and mirror reflections. Equations of equilibrium and dynamics are derived in terms of the affinor density of Schouten's force stresses. Equilibrium equations are obtained for the case of using pseudo-invariant volume and area elements of an arbitrary integer weight.

FOURTH-ORDER STRAIN GRADIENT BAR-SUBSTRATE MODEL WITH NONLOCAL AND SURFACE EFFECTS FOR THE ANALYSIS OF NANOWIRES EMBEDDED IN SUBSTRATE MEDIA

This paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. A fourth-order strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the Gurtin-Murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). The proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. The analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. Two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order strain gradient model on nanowire responses, when compared to the classical model and the second-order strain gradient model. The first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model.

Design of curved composite panels for maximum buckling load using sequential permutation search algorithm

New buckling optimal solutions for curved composite panels are presented in terms of principle of virtual work and sequential permutation search (SPS) algorithm. By utilizing the principle of virtual displacements, the buckling load is calculated and the accuracy of the derived buckling solutions is verified by comparison with experimental data and numerical results. SPS utilizes the inherent mechanical properties of composite laminates in which the bending stiffness of the outer layers is more sensitive than those close to the mid-surface. Curved composite panels exhibit a variety of length-to-arc ratios and radius-to-length ratios, under axial compression, pure shear, and combined loadings, are optimized. Some of the results are compared with the global optima obtained by enumeration method, and SPS shows good robustness in locating the global optima. In addition, the buckling load carrying capacity of optimized curved composite panels can be improved by as much as twice over cross-ply laminates. Low computational costs of SPS algorithm imply its potential for solving high-dimensional discrete variable optimization problems of composite structures.

Hygrothermal environment effect on the critical buckling load of FGP microbeams with initial curvature integrated by CNT-reinforced skins considering the influence of thickness stretching

Abstract Due to the need for structures with refined properties to bear against different loading conditions, recently, carbon nanotubes (CNTs) have been used widely to reinforce them. The extremely high stiffness of CNTs makes them significant as one of the best reinforcements to improve the mechanical behaviors of structures. This work focuses on microbeam buckling response with an initial curvature that includes three layers. The mid-layer that is known as the core is constituted of functionally graded porous (FGP) materials and two CNT-reinforced composite skins are bonded to the core to integrate it. The whole structure is affected by the hygrothermal environment and springs and shear layers are put below it. For the first time, for such a structure, a refined shear deformation theory (RSDT) as a higher-order theory that considers thickness stretching effect in polar coordinates is used that presents more accurate results, especially for deeply curved beams. Modified couple stress theory (MCST) in combination with the virtual displacement principle is utilized to establish the governing equations. The obtained results demonstrate the significance of porosity percentage and CNTs’ addition to the skins on the critical nanotubes buckling load. Also, the different behaviors of the microstructure at various temperatures are analyzed and discussed in detail.

Large deflection analysis of FG-CNT reinforced composite pipes under thermal-mechanical coupling loading

This study is concerned with the analysis of large deflection in composite pipes reinforced with carbon nanotubes (CNTs) resting on nonlinear elastic foundation. It is assumed that the nanocomposite pipe is subjected to uniformly distributed transverse pressure loading and also uniform temperature rise. Distribution of CNTs is functionally graded (FG) through the radius of composite pipe and material properties are assumed as temperature-dependent. Two kinds of immovable boundary conditions are considered for the composite pipe which are simply-supported and clamped. The principle of virtual displacement is employed to establish the nonlinear equilibrium equations of the composite pipe under thermo-mechanical loading. Governing equations are extracted based on the higher-order shear deformation theory and the von Kármán geometric measure. The system of nonlinear differential equations is analytically solved utilizing the two-step perturbation technique and Galerkin method. Results of this paper are compared with the available data in the literature for the homogeneous isotropic pipes. The effects of boundary conditions, geometric factors, CNTs distribution profile, foundation stiffness, and the volume fraction coefficient on the bending behavior of FG-CNT reinforced composite pipes are discussed in detail. The novel results of this study demonstrate that the maximum deflection of the nanocomposite pipe occurs in the FG-Λ pattern and the minimum deflection belongs to the FG-V case.

High-fidelity analysis of multilayered shells with cut-outs via the discontinuous Galerkin method

A novel numerical method for the analysis of multilayered shells with cut-outs is presented. In the proposed approach, the shell geometry is represented via either analytical functions or NURBS parametrizations, while generally-shaped cut-outs are defined implicitly within the shell modelling domain via a level set function. The multilayered shell problem is addressed via the Equivalent-Single-Layer approach whereby high-order polynomial functions are employed to approximate the covariant components of the displacement field throughout the shell thickness. The shell governing equations are then derived from the Principle of Virtual Displacements of three-dimensional elasticity and solved via an Interior Penalty discontinuous Galerkin method over a discretization of the shell modelling domain that is obtained by intersecting a background structured grid with the level set function defining the cut-outs. To maintain high-order accuracy even in proximity of the embedded cut-outs, high-order accurate quadrature rules for implicitly-defined regions are employed to compute the integrals of the method while a cell-merging technique avoids the presence of overly small cut cells. The combined use of these features represents the novelty of the proposed method and provides a high-fidelity approach to the analysis of multilayered shells with cut-outs. Numerical tests are performed to model the static response of a cylindrical shell and a NURBS-based shell with a cut-out. The obtained results are compared with those obtained using the Finite Element method and show the accuracy and the computational efficiency of method.

Hygro-thermo-mechanical bending behavior of advanced functionally graded ceramic metal plate resting on a viscoelastic foundation

In this work, the bending behavior of an advanced functionally graded ceramic–metal plate subjected to a hygro-thermo-mechanical load and resting on a viscoelastic foundation is studied using a simple higher-order integral shear deformation theory. The power-law function in terms of volume fraction is used to vary the elastic material constituents through the plate's thickness. The in-plane displacement field uses a sine shape function which changes linearly through the plate thickness to calculate the out-of-plane shear deformation. Both the linear and nonlinear influence of temperature and moisture concentration on the bending response are investigated. For the first time, a three-parameter viscous foundation model is used to study the bending response utilizing the damping coefficient in addition to Winkler’s and Pasternak’s parameters. The governing equations are derived using the principle of virtual displacement, and the analytical solution is obtained by the Navier method. Non-dimensional numerical results is validated by existing results in the literature. A parametric investigation is established to discuss the effects of the power-law gradient index, temperature rise and moisture concentration, elastic foundation coefficients, and the viscoelastic damping coefficient on the FGM plate's bending response.

Thermo-mechanical buckling analysis of FG-GNPs reinforced composites sandwich microplates using a trigonometric four-variable shear deformation theory

The current study is aimed to investigate the thermo-mechanical buckling responses of the sandwich microplate with Graphene nanoplatelets (GNPs)-reinforced Epoxy core fully bonded to single-walled carbon nanotubes (SWCNTs)-reinforced piezoelectric patches. The face sheets were exposed to the electric field and the microplate was assumed to locate on the Pasternak elastic substrate. The GNPs and SWCNTs were dispersed throughout the core and face thickness according to some specific functions. To consider the shear deformation effect, tangential shear deformation theory (TSDT), as a trigonometric higher-order theory, was used. The size effects were also included through applying and the modified strain gradient theory (MSGT). The virtual displacement principle was utilized to derive the governing equations and then by employing an analytical method, they were solved. The validity of the results was confirmed by comparing the results for simpler state with previously published ones. A parametric study is provided to observe behavior of the microstructure in different conditions. It was observed that GNPs and CNTs dispersion patterns play an important role in the microplate buckling behavior, as well as temperature variations. The outcomes of this work may help to manufacture more efficient engineering smart structures, such as micro/nanoelectromechanical systems.

Stability Analysis of Composite Cylindrical Shell Containing Rotating Fluid

A semi-analytical finite element method is used to analyze the stability of composite cylindrical shells interacting with a rotating fluid inside them. A mathematical formulation of the problem of deformable structure dynamics is based on the variational principle of virtual displacements and classical shell theory. The behavior of an ideal compressible fluid is described within the framework of the potential theory. The validity of the obtained results is supported by comparing them with the known solutions. Numerical experiments were performed for two- and three-layer cross-ply shells made of boron-epoxy resin with different boundary conditions and geometrical dimensions. It is demonstrated that, for the examined configurations, an increase in the fibre angles leads to a significant increase in the critical rotation velocities of the fluid, regardless of the conditions for fixing the edges of a thin-walled structure.

A size dependent meshfree model for functionally graded plates based on the nonlocal strain gradient theory

A nonlocal strain gradient meshfree model based on the nonlocal strain gradient theory (NSGT) and higher-order shear deformation theory (HSDT) is presented and developed to examine the bending and free vibration behaviors of functionally graded (FG) nanoplates. Regarding two scale parameters, nonlocal and strain gradient effects in nanostructures are appeared in the current model. The power index and Mori-Tanaka schemes are utilized to compute material properties. A weak form of governing equations of motion for NSGT can be obtained by using the principle of virtual displacements. Later, the deflections and natural frequencies of the FG nanoplates are resolved by using the moving Kriging (MK) meshfree method. A detailed study to assess effects of exponential factor, length-to-thickness ratio, geometry, boundary condition, strain gradient parameter and nonlocal parameter on the deflections and natural frequencies of the FG nanoplates is revealed by numerical results. It is shown that large different results of deflections and natural frequencies between NSGT and classical theory are indicated through numerical examples.

Free vibration analysis of curved metallic and composite beam structures using a novel variable-kinematic DQ method

The present paper investigates the 3 D free vibration behavior of curved metallic and composite beams via a novel beam theory. The refined beam theory is constructed within the framework of the Carrera Unified Formulation (CUF), which expands 3 D displacement fields as 1 D generalized displacement unknowns over the cross-section. As a novelty, a set of improved hierarchical Legendre polynomials called the improved hierarchical Legendre expansion (IHLE) is used to describe the cross-sectional deformation. In this way, displacements at shared sides between piles can be interpolated by Lagrange polynomials, while displacements at the rest of the cross-section remain to be defined by hierarchical Legendre polynomials. Such determined cross-sectional kinematics not only retain the hierarchical properties of HLE in part but also facilitate the implementation of the Layer-Wise approach without paying much attention to the order that expansion terms appear over the cross-section. Due to the absence of the mesh generation and the convenience of collocation techniques, the differential quadrature based-meshless method is employed for the approximate solution of strong form governing equations derived by the principle of virtual displacements. Several numerical cases, including curved beams with various material properties and boundary conditions, are proposed to illustrate the optimized computational efficiency of this novel model over the 3 D finite element method and consistent convergence properties over the previous CUF-HLE model.

Peridynamics for geometrically nonlinear analysis of three-dimensional beam structures

This study presents a novel peridynamic model for geometrically nonlinear analysis of three-dimensional beam structures. The formulations of nonlinear strain energy densities for a beam is obtained by using Total Lagrange formulation. The peridynamic formulations for beam structures are obtained based on the principle of virtual displacements. The capability of the proposed peridynamic model is demonstrated by considering large deformations of straight beams and curved beams subjected to different loading conditions. To further demonstrate the capabilities of the proposed nonlinear model, damages on a dry spaghetti subjected to different loading conditions are simulated.

Numerical Simulation of Non-Darcy Flow Caused by Cross-Fracture Water Inrush, Considering Particle Loss

A flow erosion model of water inrush was developed that couples the Darcy, Forchheimer, and Navier–Stokes fields under the theory of continuum mechanics. Water flow and fluidized particles were regarded as single-phase mixed fluids based on the fundamentals of flow transition (aquifer laminar flow to turbulent flow) being the main cause of mine water inrush; the effects of rock disintegration and the coupled effects of flow and erosion were incorporated. The water source of the aquifer, water-inrush channel of the fracture network, and flow path of the slope roadway export were organically connected in a unified flow field. The weak integral forms of the Darcy, Forchheimer, and Navier–Stokes equations were constructed according to the principle of virtual displacement. The convective terms were discretized by the finite volume method and the other terms were discretized by the finite element method; the calculation source program was developed based on the finite element language and its compiler (FELAC). The source program system was used to numerically simulate water inrush in a fractured zone in the Jiangjiawan Mine. It reproduced the entire dynamic process of a water inrush and revealed the distribution and variation characteristics of the pressure field, velocity field, porosity, and concentration, as well as the mechanisms underlying the sudden change in water flow.

Static analysis of simply supported porous sandwich plates

In this study, it is aimed to analyze the bending of porous sandwich plates using the four-variable shear deformation theory. The core of the sandwich plate is assumed to be functionally graded, and face sheets are assumed to be isotropic. The pore distribution of the sandwich plate is considered even and uneven type of porosity distribution. Displacement fields are defined with four variable shear deformation theory. Equilibrium equations of porous sandwich plates are derived from virtual displacement principle. An analytical solution is obtained by Navier's approach. Results are presented for uniformly and sinusoidally distributed loaded porous sandwich plates. Face sheet -core thickness ratio, porosity distribution, amount of porosity is investigated.