Higher-order Shear Deformation Theory Research Articles

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Static and free vibration response of FGM plates using higher order shear deformation theory and strain-based finite element formulation

The primary objective and novelty of this work lie in the development of a new four-node quadrilateral finite element based on strain-based high-order shear deformation theory (HSDT). This is the first study to apply this innovative approach to analyze both the static and free vibration behaviors of functionally graded (FG) plates. Another key novelty is the reduction in the number of unknowns to five, unlike other high-order shear deformation theories that employ a larger number of unknowns. This reduction is achieved by applying the condition of zero transverse shear stress at the top and bottom free surfaces of the FG plate and by assuming that the transverse shear strains are sinusoidally distributed through the thickness. The material properties of FG plates are modeled to vary according to a simple power law based on the volume fractions of their constituents. The developed finite element possesses five degrees of freedom (DOFs) per node, resulting from the combination of two strain-based elements: the first is a membrane with two DOFs, and the second is a bending plate with three DOFs. The displacement fields of the proposed element are expressed using high-order terms based on the strain approach, which satisfy compatibility equations and rigid body modes. Furthermore, the concept of the neutral surface is introduced to eliminate membrane-bending coupling. The elementary stiffness and mass matrices are derived using both the total potential energy principle and Hamilton’s principle. The performance and convergence of the proposed element are validated through examples from the literature.

Assessment of a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells

Abstract Static response of simply supported functionally graded (FG) shallow shells using a new higher-order shear and normal deformation theory is focused in this article. The effects of transverse strains and stresses on the bending response of FG shell are considered by the present theory. The current theory considers the impacts of transverse normal and shear deformations that meet the requirements for traction-free boundary conditions. The virtual work principle is applied to the mathematical formulation of the present theory. The simply supported doubly curved shallow shell problems under the static transverse load are analyzed using Navier’s solution technique. To verify the existing theory, the current results are, whenever possible, compared with those that have already been published. Additionally, a few benchmark results are presented in this article that will be helpful to researchers in the future.

Effect of Porosity on Stability Analysis of Bidirectional FGM Skew Plate via Higher Order Shear Deformation Theory and RBF Approach

Effect of Porosity on Stability Analysis of Bidirectional FGM Skew Plate via Higher Order Shear Deformation Theory and RBF Approach

Free vibration and buckling behavior of porous orthotropic doubly-curved shallow shells subjected to non-uniform edge compression using higher-order shear deformation theory

Due to the continuous enhancement of manufacturing technology for porous materials, it is expected to be increasingly used in aerospace, aviation, and other engineering applications, where the necessity for lightweight structures is paramount. Also, the non-uniform edge loads caused by service conditions, complex adjacent structures, and external loads lead to localized stress concentrations within the shells. Localized stress concentrations result in a loss of load-carrying capacity in specific regions, causing step-by-step failure of the shell. This study investigates the buckling and free vibration behavior of porous orthotropic doubly-curved shallow shells with four different porosity distribution patterns subjected to non-uniform edge compressive loads. The equations of motion of doubly-curved shallow shells are established by using higher-order shear deformation theory and Hamilton’s principle. Then, the pre-buckling in-plane stress distributions of doubly-curved shallow shells are determined and appended to the equations of motion. These fundamental partial differential equations are solved by Galerkin’s method, and the critical buckling load and natural frequency formulations are achieved. By comparing with the results in published literature, the feasibility and accuracy of present formulations are validated. Finally, systematic parametric studies are carried out to discuss the effects of different non-uniform edge compression patterns, porosity distribution patterns, porosity coefficients, aspect ratios, arc length-to-thickness ratios, radius-to-arc length ratios, orthotropy ratios, and shell types on the buckling and free vibration responses of porous orthotropic doubly-curved shallow shells. Parametric studies indicate that the reduction or increment in critical buckling loads (CBLs) and fundamental natural frequencies (FNFs) of spherical shells (SSs) are more sensitive than those of hyperbolic paraboloidal shells (HPSs). The CBLs and FNFs of SSs are larger than those of HPSs. The maximum and minimum CBLs are achieved for the triangular and uniform edge compression, respectively. The porosity effect is independent of edge compression pattern types.

High-frequency vibration analysis of laminated composite plates using energy flow and shear deformation theories

Energy Flow Analysis (EFA) is an efficient approach for characterizing high-frequency vibrations through time-averaged energy density distributions. This study employs EFA to investigate vibroacoustic behavior in laminated composite plates subjected to high-frequency excitation. Governing equations of motion are derived using Classical Plate Theory (CPT), First Order Shear Deformation Theory (FSDT), and Higher Order Shear Deformation Theories (HSDT). Wave propagation parameters like wave number and group velocity are obtained from each theory and compared to assess accuracy. Energy density and intensity formulations are then developed based on classical solutions of the governing equations. EFA analysis indicates that HSDT provides more accurate predictions of wave parameters, especially at very high frequencies where it accounts better for shear deformation effects. Validation against classical energy density solutions shows acceptable accuracy with less than 0.5dB difference in the far-field. Comparisons further demonstrate the superiority of HSDT over FSDT for thicker plates in both classical and EFA analyses. This research establishes the effectiveness of EFA for high-frequency vibration analysis of composite laminates. The main novelty of this research lies in the integration of HSDT into the application of EFA for composite laminates, providing a more precise consideration of shear deformation effects at high frequencies. Specifically, HSDT enhances modeling of critical shear deformation effects at elevated frequencies. EFA delivers computationally efficient solutions while maintaining acceptable accuracy, improving characterization and design of composite structures under high-frequency loading.

Influence of Transverse Normal Strain on Buckling and Vibration of Sandwich Plates

This study presents the buckling and free vibration behavior of simply supported square and rectangular sandwich plates. A third-order shear deformation theory considering both the transverse shear and normal deformations is used. The condition of zero transverse shear stresses on the top and bottom surfaces of the plate is satisfied. Hence, the present theory does not require the shear correction factor generally associated with the first-order shear deformation theory. Governing equations and boundary conditions are obtained using the principle of minimum potential energy. The closed-form solutions of simply supported sandwich plates are obtained. The effect of the side-to-thickness ratio and plate aspect ratio on buckling and free vibration responses of simply supported square and rectangular sandwich plates is examined. Parametric effects of face sheet thickness ratios and load factor upon the critical buckling load for uniaxial and biaxial buckling analysis of square and rectangular sandwich plates for different staking sequences are studied. All numerical solutions are obtained using MATLAB® programs. The results obtained from the present theory are compared with those from classical plate theory, first-order shear deformation theory, higher-order shear deformation theories, and the exact three-dimensional elasticity solutions as applicable.

RFOR-DQHFEM: Hybrid relaxed first-Order reliability and differential quadrature hierarchical finite element method for multi-physics reliability analysis of conical shells

In this current work, a hybrid reliability analysis and theoretical frequency technique are suggested for the reliability response of conical shells. Two levels of analyses are proposed as the main loop of the reliability method for finding the failure probability and the second level applied in the main loop for giving the performance function of frequency applied in conical shell structures with multi-physics vibration analysis. A dynamical adjusting procedure is proposed for computing the relaxed factor using the enough descent condition inside the reliability method. The superior convergence rate is considered for selecting the relaxed factor of the proposed first-order reliability method named RFORM. An elastic-electro-mechanical model based on the Higher-Order Shear Deformation Theory (HSDT) is extended for frequency analysis of conical shells. The innovative numerical procedure named Differential Quadrature Hierarchical Finite Element Method (DQHFEM) as a robust framework for giving the vibration behavior of studied mechanical structures is applied for solving motion equations. The developing DQHFEM and RFORM are applied for the laminated, nanocomposite, and piezoelectric conical shell structures with multi-source uncertainties. Increasing the volume percentage of nanoparticles from 0% to 10% significantly enhances the reliability index, with carbon nanoparticles showing a 132% increase, silica nanoparticles showing a 97% increase, and other nanoparticles showing an approximate 40% increase. Also, as moisture content increases from 0% to 30%, the reliability index for a thickness-to-large-radius ratio of 0.2 drops by about five times. Excessive moisture levels (above 20%) result in a negative reliability index, indicating a hazardous condition.

Effect of circular and elliptical cutouts on the free vibration analysis of sandwich FGM plate under the elastic foundations

The present article investigates the effect of circular and elliptical cutouts on vibrational characteristics of porous sandwich functionally graded material (SFGM) plates with FGM core resting on Pasternak elastic foundation. The structural kinematics of the plate are formulated based on nonpolynomial-based higher-order shear deformation theory (HSDT). Geometric discontinuities have been incorporated in terms of circular and elliptical cutouts in the SFGM plates. The sandwich structure is considered to be comprised of two homogeneous face sheets and a porous FGM core with various cutout shapes and sizes. The variation of material properties from the homogeneous face sheet to the FGM core has been carefully modeled using modified power law distribution. Further, a C0 continuous finite element formulation with a four-noded, isoparametric quadrilateral element with seven degrees of freedom (DOFs) per node has been employed to accomplish the results. The accuracy of the present results has been demonstrated through convergence and validation studies. A comprehensive study has been carried out to investigate the influence of volume fraction index, even and uneven porosity distribution, circular and elliptical cutouts, as well as H-elliptical and V-elliptical type cutout shapes on the frequency parameter of SFGM plates with the elastic foundation. The numerical results demonstrate the aforementioned parameters significantly impact the vibrational frequency of the SFGM plates.

A novel hybrid function-based Ritz method for static and free vibration of axially loaded bi-directional functionally graded porous beams

This paper introduces a novel hybrid function, synthesizing trigonometric and Hermitian polynomials, to analyze the structural responses of bi-directional functionally graded porous (2D-FGP) beams. The displacement field of the beam is based on the higher-order shear deformation theory. A power-law relation governs the material variation along the x- and z-directions, with the porosity manifesting in three distinct distributions. The governing equations are deduced through the application of Lagrange’s equation. The proposed hybrid function is developed to delineate the displacement field. The deflections, stresses, buckling loads, and natural frequencies of 2D-FGP beams are computed for diverse gradation exponents in the x- and z-directions, boundary conditions, porosity type, porosity coefficient, and slenderness ratio. A comparative assessment with finite element method results indicates that the findings agree with available data. It substantiates the accuracy and efficiency of the proposed methodology.

Thermal buckling of organic nanobeams resting on viscous elastic foundation

This study presents an analytical solution for organic solar nanobeams by including the new high-order shear deformation theory and considering the impact of size effects using non-local elasticity theory. The beam is supported by a viscoelastic base and experiences thermal loads that are distributed in accordance with uniform and nonlinear principles. The calculation formulae are derived from the consideration of significant deformations in the beam, resulting in increased complexity of the calculations. The equilibrium equation is derived from the fundamental principle of maximum work potential. The explicit equation for the critical thermal load is derived, providing a convenient means for the calculation and analysis of the impacts of relevant factors. The critical thermal buckling load encompasses both real and imaginary components, with the real component corresponding to the thermal load responsible for inducing beam instability, and the imaginary component associated with the dissipation of this load. This consideration is particularly pertinent when accounting for the foundation resistance. This finding enhances the level of interest in the research outcomes. This study also examined the impact of certain characteristic factors of the foundation and organic beams on the thermal buckling behavior of the beam, establishing a scientific basis for practical design applications.

Dynamic analysis of laminated plate and ballastless track slab on Pasternak foundation under moving load based on different shear deformation theory

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Thermoelastic analysis of FG-CNTRC cylindrical shells with various boundary conditions and temperature-dependent characteristics using quasi-3D higher-order shear deformation theory

Abstract. This study focuses on performing static analysis of FG-CNTRC cylinder shells with various boundary restrictions, including thermomechanical responses. The governing equations are developed by taking into account the temperature-dependent material features, the quasi-3D high-order shear deformation hypothesis, and the normal transverse stress effect. The temperature gradient inside the thickness is expected to fluctuate, and the distribution pattern is derived by using the heat transfer equation and considering the temperature boundary limitations. A singular trigonometric series and the Laplace transform are used in an analytics solution to address basic equations. This study primarily examines the stress levels at the border region. The findings indicate that it is crucial to take into account the abrupt rise in stress at the boundary area, particularly when the shell’s relative length is small. The reciprocal impact of pressure and temperature load is also emphasized. Significant findings indicate that thermal load may either augment or diminish stress levels, contingent upon the orientation of the pressure and thermal load effect. The results of the study of this issue serve as the foundation for the calculation and design of relevant structures in practical applications. Furthermore, this serves as a foundation for the creation of more intricate issues in the forthcoming.

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

Free and forced vibration mitigation of nonlinear functionally graded beams based on higher order shear deformation theories using NES

Free and forced vibration mitigation of nonlinear functionally graded beams based on higher order shear deformation theories using NES

Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow

The current work examines the thermal postbuckling, aero-elastic flutter and thermally induced postbuckled flutter about a static equilibrium state of tri-directional functionally graded (TFGM) rectangular and tapered plates in yawed supersonic flow. Based on the von Kármán nonlinear strains, the general higher-order shear deformation theory (GHSDT) and the first-order piston theory, the governing equations of motion for TFGM plates in yawed supersonic flow are established via the Hamilton's principle. The isogemetric analysis (IGA), the load continue strategy associated with the Newton Raphson iterative technique are exploited synthetically to capture the postbuckling paths, and then the postbuckled flutter behaviors are acquired with the static equilibrium state being obtained. Comparative and convergence studies are provided to improve reliabilities of the present material model, formulae and code implementations. Subsequently, detailed parametric investigations are carried out to evaluate the influences of material volume fractions, boundary conditions and airflow angles on the thermal postbuckling, aero-elastic flutter and postbuckled flutter behaviors of TFGM plates. The results show that it is the in-plane volume fractions, rather than the thickness volume fraction, that exert a greater influence on the thermal postbuckling and flutter behaviors of TFGM plates. Furthermore, TFGM plates exhibit more diverse postbuckling paths and more complex deformations due to the inhomogeneity of the in-plane material properties compared with z-directional FGM plates.

Nonlinear dynamic response of a sandwich plate with negative Poisson’s ratio honeycomb-core layer under low-velocity collision impact

In this paper, a nonlinear analytical model is presented for the low-velocity collision impact of a sandwich plate with auxetic honeycomb-core layer. The auxetic feature of the honeycomb material is realized by mathematically expressing the effective material coefficients in terms of material property and cellular geometric parameters through a homogenization method. The higher order shear deformation theory, the von Kármán nonlinearity theory, and a modified Hertz contact law which accounts for the contact pressure distribution and indentation effect, are employed to establish the kinematic relations. The Newmark time integration scheme in conjunction with the direct iterative method is utilized to establish a solution procedure for the nonlinear dynamic governing equation. The verification of the presented model with the data in published literatures is carried out, followed by a series of numerical analyses for influences of cellular geometric features and impactor’s initial conditions (such as impactor’s initial velocity, mass, and nose curvature radius) on the nonlinear dynamic response. The results of numerical analyses show that the geometric features of the unit cell in the honeycomb-core and impactor’s initial conditions can cause significant influences on the nonlinear collision impact behaviors of the system.

Nonlinear dynamic study of FG-GFRC-NPR structures

The functionally graded glass fibre-reinforced polymer composite(FG-GFRC) is one of the contemporary sophisticated materials that may be employed for a variety of technical purposes. The shock absorption capacity of FG-GFRP laminates can be improved by incorporating auxetic property (i.e., negative Poisson's ratio NPR) in the structures, which can be attained by a particular stacking sequence of glass fibres (GF) and a different percentage of GF distribution along the thickness direction within the FG-GFRP structure. The nonlinear dynamic behaviour of such structures must be investigated. The clamped-clamped FG-GFRP-NPR structure is stimulated under a uniform transverse load distribution in the current work. Based on the higher-order shear deformation theory (SDT), the displacement field and nonlinear stress–strain relationship are derived. The nonlinear differential equation with cubic non-linear components were derived using Hamilton’s principle and transformed using Galerkin’s technique and obtained governing equation of motion. MATLAB has been used to conduct all numerical simulations. A time series, a phase picture and a Poincare diagram (PCM) were generated to characterize the vibration behavior of such auxetic structures.

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

On the dynamics of improved perovskite solar cells: Introducing SVM-DNN-GA algorithm to predict dynamical information

This study presents an innovative approach to enhancing the performance of perovskite solar cells through the integration of a functionally graded triply periodic minimal surface (FG-TPMS) layer. The research focuses on the mechanical and vibrational characteristics of doubly curved panels embedded with three distinct iterations of the FG-TPMS model: the primitive, gyroid, and wrapped package graph (IWP). By employing higher-order shear deformation theory (HSDT), the analysis accounts for the complex geometrical and material gradations within the FG-TPMS structures. An advanced analytical method utilizing trigonometric functions is developed to accurately predict the natural frequencies and mode shapes of these novel composite structures. In order to assess the vibrations of TPMS-reinforced perovskite solar cells surrounded by an elastic foundation, this work proposes the implementation of a novel Support Vector Machine (SVM)-deep neural network (DNN)-Genetic Algorithm (GA) employing mathematical modeling datasets. Using the SVM-DNN-GA algorithm, predicted accuracy is improved. In order to simulate and forecast the vibrational behavior of the reinforced solar cells, the integrated methodology makes use of the advantages of each technique. The results indicate that the integration of FG-TPMS layers significantly enhances the mechanical stability of the perovskite solar cells. The application of HSDT reveals detailed insights into the dynamic responses of the doubly curved panels, highlighting the potential for fine-tuning their vibrational characteristics to further improve solar cell performance. This research underscores the potential of FG-TPMS structures in advancing solar cell technology, providing a foundation for future studies to explore the integration of complex geometries and material gradations in photovoltaic applications.