Shear Deformable Model Research Articles

Higher-order shear deformation model for free vibration analysis of imperfect functionally graded plates

Higher-order shear deformation model for free vibration analysis of imperfect functionally graded plates

Smart analysis of sandwich foldable cylinders as gymnastic accessories

A novel smart analysis of sandwich composite cylindrical shell is presented in this article. It is assumed that the sandwich shell is manufactured from the graphene origami reinforced copper core between two intelligent layers. The proposed structure in this article can be used as an idealized model for cylinder shapes in gymnastic sport and physical therapy units. The virtual work principle in the piezo elasticity framework and the cylindrical coordinate system is extended to derive governing equations. To arrive at more accurate results, a third order shear deformable model is extended for kinematic relations. The electro elastic responses are explored using the analytical method to seek impact of main parameters of the composite structure such as foldability parameter, graphene content percent, and applied electric potential on the electro elastic strain, deformation and stress results.

Bending responses of graphene nanoplatelets reinforced sandwich cylindrical micro panel with piezoelectric layers

Bending responses of a nanocomposite-reinforced cylindrical panel are studied in this article. A cylindrical shell panel is assumed in micro scale and sandwiched by piezoelectric layers. In this article, a micro-size dependent theory named as the modified couple stress theory (MCST) is analytically employed and the kinematic relations are extended through employing shear deformable model in order to investigate the electroelastic bending responses of a three-layered micro-shell bonded between smart layers subjected to an applied voltage, external and internal pressures. The micro-shell is assumed rested on a two parametrically elastic foundation. In order to develop constitutive relations, the mixture’s rule as well as Halpin-Tsai model are utilized to compute governing equations. Electroelastostatic responses are analytically obtained through trigonometric functions. A large extended parametric analysis is presented to explore deflection of the micro-shell with a change in applied voltage, thickness of the piezoelectric layer to radius, length to radius ratio, different characteristics of nanoplatelet reinforcement for the both external and internal pressure. The proposed composite electromechanical structure may be used in the smart structures and systems. A controllable system can be suggested through the usage of graphene nanoplatelets because of flexibility and affecting parameters.

A Novel Calculate Model of Shear Deformation and Relative Displacement of Pile–Soil Interface in Warm Frozen Soil Foundation

In permafrost regions with warm frozen soil, the pile foundation is commonly used, but most currently available models for the WFS foundation pile–soil system are either highly empirical or overcomplicated, without a simplified theoretical manner in engineering. This study derives a novel and simplified calculated model of the WFS pile–soil system. The model is formulated in terms of the shear deformation theory and load transfer method based on the rigorous deformation mechanism of the WFS foundation soil around the pile. Considering the different depth soil features and the equilibrium state of the pile–soil system, dividing warm frozen soil foundation into three regions (TPPR, ER, and BPPR) to calculate the Dp and Ds can simply obtain the total displacement of pile under different loads. The results demonstrate that the present theoretical model can well predict the WFS foundation load–displacement response of the pile. The present model provides a simple, practical, and effective approach for the estimation of the load–displacement behavior of piles installed in the WFS foundation.

Free vibration analysis of graphene origami-reinforced nano cylindrical shell

Vibrational behavior of a shear deformable cylindrical shell is studied in this article based on Hamilton’s principle. The shell is manufactured by a Copper (Cu) core reinforced with graphene origami auxetic metamaterial and subjected to mechanical and thermal loads. The constitutive relations are developed based on the shear deformable model where the effective modulus of elasticity, Poisson’s ratio, and density are evaluated using micromechanical models including volume fraction and folding degree of graphene origami auxetic metamaterial, geometric characteristics of graphene, material properties of matrix, and reinforcement and local temperature. The natural frequencies are obtained based on Navier’s and Galerkin’s approaches for simply supported, clamped-simply, and clamped-clamped boundary conditions. The formulation, solution procedure, and numerical results are confirmed through comparison with the available results of the literature. The complete numerical results are presented in terms of volume fraction and folding degree of graphene origami auxetic metamaterial, thermal loading, and various types of boundary conditions.

Static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix

A higher-order shear deformable model is extended in this paper for static analysis of a sandwich cylindrical panel composed of porous core sandwiched by graphene-reinforced copper matrix. Principle of virtual work is used to derive governing equations of static bending through the computation of strain energy and external work. A higher-order shear deformable model is used to develop the kinematic relations. The sandwich cylindrical panel is subjected to mechanical and thermal loads. The governing equations are solved using Eigenvalue–Eigenvector method for clamped–clamped boundary conditions. The eigenvalues of the characteristic equation are obtained, and the solution is obtained for real and complex roots. The boundary conditions are applied to obtain integration constants. The axial and radial variation of the displacement, strain, and stress components are presented in terms of significant input parameters such as volume fraction, and folding angle of the graphene origami, various distribution patterns, porosity coefficient, thermal and mechanical loads, and geometric parameters. The results indicate that the stress, strain, and deformation components are increased with an increase in porosity coefficient and folding angle of the graphene. Furthermore, an enhancement in the volume fraction of the graphene leads to a significant decrease in various stress, strain, and deformation components. The results of this work can be used in aerospace science and technology specially in airplanes.

Modified third order shear-deformation model for anisotropic glulam beams via three-field mixed finite-element formulation

ABSTRACT Glulam timber structural linear elements often show low span-to-depth ratios. Moreover, timber is intrinsically orthotropic with very low shear moduli. These two circumstances justify the substantial contribution of shear strain to the overall deformation of linear elements. Since design specifications for timber beams commonly adopt 1D models, two such elastic models are proposed in this paper for static plane behavior, where, instead of fully orthotropic, glulam is assumed to be transversely isotropic. These models provide a good compromise between accuracy and simplicity for design proposes. A 3-field mixed formulation is generalized in order to produce two new 1D finite elements suitable for modeling transversely isotropic elastic glulam beams. To assess the accuracy of these new finite elements the results they produce are compared with those determined with a plane stress transversely isotropic 2D finite element. The included examples show that the proposed combination of transverse isotropy with a third order shear deformation model provides a suitable background to the 1D modeling of glulam beams with improved accuracy when compared to the models conventionally used in design. The paper is concluded with a parametric analysis which identifies the cases where the improved models should be used instead of simpler ones.

Stress analysis of laminated and sandwich beams subjected to concentrated load by using quasi-two-dimensional theory

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

Buckling of constant/variable stiffness curved arbitrary layup composite beams by renewed constitutive relations-based sine finite element with improved in-plane and transverse responses

The elastic stability performance of constant- and variable-stiffness based composite curved beams with general layups is evaluated using an enriched finite element that accounts for three-dimensional structural response features. The variable stiffness is spatially created over the beam by layers with curvilinear fibers. The structural model in the present work is represented by sinusoidal variation based higher-order shear deformable model, augmented with improved in-plane response by zig-zag function and stretching in thickness direction by higher polynomial order. The appropriate constitutive relationships for laminated composite curved beam with arbitrary layup producing stiffness coupling effects are deduced from three-dimensional elasticity relations. The beam governing equations are formed applying the total potential energy minimum principle and the buckling parameters are predicted employing the eigenvalue analysis. The curved laminated beam element efficacy is examined for the known analytical solutions in the literature. Extensive numerical analysis by opting for different structural parameters like curved beam angle, thickness, ply-angle, curvilinear fiber angle variation from center-to-edge, layup, and beam end condition are made to visualize the buckling characteristics of general layup composite curved beams.

Variable-stiffness curved laminated-beams by curvilinear fibers with arbitrarily layup – Vibrational features by sine-based higher-order beam model with renewed-constitutive relations and improved-kinematics

The vibration characteristics of variable-stiffness based composite curved beams with general layups are examined using a higher-order beam theory that accounts for three-dimensional structural response features. The variable stiffness is spatially considered in the laminated beam by introducing the curvilinear fibres-based layers. The structural model is here represented by a sine-based shear deformable model considering zig-zag function for improving in-plane response and higher order polynomial for capturing through-thickness stretching effect. The appropriate constitutive relationships for laminated composite curved beam with arbitrary layup producing stiffness coupling effects are deduced from three-dimensional elasticity equations. The beam governing equations are then formed applying Hamilton’s principle and the free vibration features are predicted employing the eigenvalue analysis. The effectiveness of the curved laminated beam element developed here is tested against the available analytical solutions in the literature. A detailed numerical analysis by opting for different structural parameters like curved beam angle, thickness, ply-angle, layup, and beam end condition are made to investigate the vibration characteristics of general layup constant stiffness laminated curved beams and by including additional parameter, curvilinear fibre angle variation from centre-to-edge, for variable stiffness laminates.

Nonlinear Damping and Vibration Assessment of Sandwich Beam with Composite Faces and Viscoelastic Core Layer

This work deals with the damping and vibration characteristics evaluation of sandwich beams based on higher-order shear deformation theories (HSDT’s). The studied sandwich beams are composed of composite materials at faces and homogeneous, isotropic viscoelastic material in the core layer. This latter (viscoelastic material) is described by complex Young’s modulus in the frequency domain. Indeed, the real part represents the stiffness, and the imaginary one describes the damping property. Nonlinear damping parameters and frequency curves are obtained using the nonlinear equation of frequency amplitude governed by multiple complex coefficients. In conjunction with a one-mode Galerkin’s procedure, the harmonic balance method is employed to obtain the nonlinear loss factor and the response frequency curves. The nonlinear frequency value and, in turn, the system loss factor ratio are obtained for various amplitudes to examine the impacts of the ply angle of fiber-reinforced composite faces, the viscoelastic shear modulus, the shear deformation model, for considering different geometric and material parameters. In the framework of this study, one attempt is to describe a better way to achieve satisfactory passive vibration control of the laminated beams.

Buckling responses and instability mode landscapes of composite sandwich panels with extreme auxeticity based on higher-order shear and normal deformation theory

Sandwich structures have been a subject of intense scrutiny and utilization in engineering applications. However, there has been a limited understanding of the mechanical behaviors of novel sandwich structures composed of auxetic metamaterials. This study aims to contribute to the body of knowledge by examining the buckling responses of sandwich beams or wide plates with an orthotropic anti-tetra chiral lattice (ATCL) aluminum cores and symmetric angle-ply carbon-fiber/epoxy face sheets, both of which can exhibit highly negative Poisson’s ratios or extreme auxeticity. The linear stability analysis of the sandwich structures under axial compression is conducted by applying the minimum potential energy principle and solving the eigenvalue problem using a numerical method. To achieve an accurate representation of arbitrary two-dimensional buckling morphologies, the first- and fifth-order shear and normal deformation models are employed to simulate the transverse kinematics of the face sheets and cores, while the finite-element model is utilized to discretize the longitudinal displacement of all constituents. The investigation reveals new insights into the interrelationship between global buckling, facial wrinkling, and shear crimping in conventional sandwich structures. The effects of material auxeticity on buckling responses are analyzed across an extensive range of geometrical configurations for the first time, unveiling five unprecedented buckling mode shapes, sophisticated buckling mode landscapes, and Poisson’s ratio-sensitive buckling criteria. These findings lay a crucial foundation for designing and optimizing auxetic sandwich structures, bringing diverse possibilities in lightweight constructions, acoustic insulation, energy absorption, and dynamic and shock mitigation.

The shear resistance mechanism of prestressed anchor cables in slope reinforcement

At present, for the reinforcement effect of prestressed anchor cables, only the shear resistance produced by prestress is considered in slope reinforcement design, which greatly underestimates the shear resistance of anchor cables. To investigate the shear resistance effect of prestressed anchor cables, a large-scale shear test system of anchored joint with sample dimensions of 70 × 70 × 140 cm was developed in this study. The test results revealed the interaction mechanism of extrusion and disengaging between the anchor cable and local surrounding rock in the process of shear deformation, the reversed ‘S’-shaped deformation characteristics of the anchor cable, and the combined tensile-shear failure mechanism. Based on the test results, the shear deformation and shear resistance characteristics of the anchored joint in different phases were thoroughly analysed, and a three-stage shear deformation model for anchor cables was proposed. In addition, the corresponding mechanical analysis model for shear resistance of anchor cables was developed. The comparative verification revealed that the error between the model calculated results and the test results was within 6%.

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.

A general electroelastic analysis of piezoelectric shells based on levy-type solution and eigenvalue-eigenvector method

Eigenvalue-Eigenvector approach as well as Levy type solution are used for electroelastic analysis of a doubly curved shell made of piezoelectric material based on a shear deformable model and piezoelasticity relations. The electroelastic governing equations are derived using virtual work principle. The solution is proposed for a Levy type boundary conditions with two simply-supported boundary conditions and two clamped ones. After derivation of the governing equations, a solution satisfying two simply supported boundary conditions is assumed to arrive a system of ordinary differential equations. The latest governing equations are solved using Eigenvalue-Eigenvector method to satisfy clamped-clamped boundary conditions. The distribution of displacements, rotations, electric potential, strain and stress is presented along the planar coordinate. Accuracy of the proposed solution is justified through comparison with results of previous papers.

Thermo-Electro-Mechanical Analysis of Micropolar FGP Cylindrical Shell Covered with Piezoelectric Actuator Layers

An investigation is performed in this paper to analyze the nonlinear thermo-electro-mechanical response of long sandwich cylindrical shells with functionally graded porous (FGP) core and thin piezoelectric actuator layers. The FGP core of the sandwich shell is assumed to be temperature- and microstructure-dependent. It is assumed that the sandwich shell is subjected to uniform lateral pressure loading in a thermo-electrical environment. The equilibrium equations of the shell with infinite length are established with the aid of the virtual displacement principle and von Kármán kinematic assumptions. The governing equations are obtained in terms of displacement components based on the first-order shear deformation model of shallow cylindrical shells and the modified couple stress theory. These nonlinear differential equations are analytically solved for a sandwich shell having both the simply-supported and clamped–clamped edge conditions by employing a two-step perturbation technique. Analytical closed-form solutions are determined as a relationship including the load parameter and the mid-span deflection. The comparison examples are made with the existing results in the literature for a simple functionally graded shell, where good agreement is obtained. It is shown that the nonlinear response of the sandwich shell is highly dependent upon the temperature variation, power law index, porosity coefficient, couple stress parameter, piezoelectric layers, and geometrical parameters of the shell.

Dynamic analysis of axially loaded cantilever shear‐beam under large deflections with small rotations

AbstractThe modal analysis and seismic response of a vibrating cantilever, with or without tip mass and rotary inertia, are investigated in this study using a shear deformable beam model and including the effect of vertical load. Based on existing approaches, an original method is proposed that does not use fourth‐order uncoupled equations to determine modal deflection and rotation. In fact, the approach presented herein transforms the second‐order coupled system into a first‐order system which can be solved more easily using matrix algebra and Laplace transform. Furthermore, the proposed form allows a straightforward demonstration of orthogonality conditions, that is, the problem is self‐adjoint, and the solution in the case of forced response using modal superposition. In addition, even if the solution presented herein is applicable only to the cantilever with a tip mass and rotary inertia, the scope is general, and the approach can be applied to shear deformable beams with other boundary conditions. Finally, the seismic response by modal superposition is shown, and some examples are proposed and discussed for the case of uniform or continuously varying cross‐sectional properties.

A refined first-order piecewise shear deformation model for vibration analysis of composite sandwich plates

In this paper, for reducing the number of independent generalized displacements of the traditional piecewise model to decrease the number of vibration equations, a refined piecewise shear deformation model considering the effect of rotary inertias and shear deformation for vibration analysis of laminated composite and sandwich plates is presented. Unlike the conventional piecewise model, the significant novelty of the proposed theory is that the new proposed piecewise theory reduces the number of independent generalized displacements from 9 to 8. Some relevant assumptions, dividing the transverse displacement into bending and shear parts, are introduced into the traditional piecewise model, and the displacement fields of the refined piecewise shear deformation theory are obtained. Next, the governing equations are derived by employing Hamilton’s principle and solved by the closed-form Navier method. Then, the present solutions are compared with those available in the previous literature for confirming their validity. Finally, the variations of frequencies and loss factors with system parameters are evaluated and presented graphically.

Thermo-mechanical analysis of porous functionally graded graphene reinforced cylindrical panels using an improved third order shear deformable model

In this paper, a novel shear deformable model is developed to investigate the temperature-dependent thermo-mechanical responses of porous functionally graded graphene reinforced composite (FG-GPLRC) cylindrical panels subjected to an initial blast pressure. An improved third order shear deformation theory, which has more rigorous kinematic of displacement, is implemented in the construction of the proposed model. Then, the current model is used to predict the thermo-mechanical responses of the nanocomposite panels made of metal foams reinforced by graphene platelets (GPLs). Lagrange equations are used to derive the governing equations corresponding to thermal buckling as well as the free and forced vibrations of the porous FG-GPLRC cylindrical panels. A Navier-type solution, which is free from computational errors, is implemented. Validation works demonstrate that the present model provides accurate predictions of mechanical responses for composite structures. A comprehensive parameter study is carried out to explore the influences of several key parameters regarding the thermo-mechanical behaviors of the porous FG-GPLRC cylindrical panels.

Vibration characteristics of arbitrary thick sandwich beam with metal foam core resting on elastic medium

An analysis is performed in this research to obtain the natural frequencies and mode shapes of a thick sandwich beam with metal foam core. Different types of foams are considered for the sandwich beam where distribution of pores is a functionally graded patterns. The governing equations of the sandwich beam are established by means of a shear and normal deformable thick beam model. This theory considers the thickness stretching and admits the nonuniform through-the-thickness shear strain. The governing equations of the sandwich beam are established which are four in number by means of the Hamilton principle. The established equations are solved for the case of thick sandwich beams resting on elastic foundation via the Navier solution method. Results of this study are first compared with the available data in the open literature and after that novel results are given to explore the effects of different parameters on the vibration characteristics of the sandwich beam. It is shown that porosity of the core is an important factor on vibration characteristics of sandwich beam with metal foam core.