Shear Correction Factor Research Articles

Research on bending vibration characteristics of phononic crystal plates based on Mindlin’s piezoelectric plate theory

Based on Mindlin’s theory and the plane wave expansion method, the formulas are proposed for the governing equations and dispersion relations of bending waves in piezoelectric phononic crystal plates. The shear correction factors can be obtained through transcendental equations based on forced vibrations of the plate. The plates are made of inclusions of different shapes and lattice types, finding that the inclusion shape dramatically affects the mid-to-high frequency band gaps. Piezoelectric materials exhibit distinct eigenfrequencies at the high-symmetry point Γ at low frequencies. Thickness affects the band gap width differently than in two-dimensional models, and cuts influence band gap width significantly.

Transient responses of double-curved sandwich two-layer shells resting on Kerr’s foundations with laminated three-phase polymer/GNP/fiber surface and auxetic honeycomb core subjected to the blast load

This work uses refined first-order shear theory to analyze the free vibration and transient responses of double-curved sandwich two-layer shells made of auxetic honeycomb core and laminated three-phase polymer/GNP/fiber surface subjected to the blast load. Each of the two layers that make up the double-curved shell structure is made up of an auxetic honeycomb core and two laminated sheets of three-phase polymer/GNP/fiber. The exterior is supported by a Kerr elastic foundation with three characteristics. The key innovation of the proposed theory is that the transverse shear stresses are zero at two free surfaces of each layer. In contrast to previous first-order shear deformation theories, no shear correction factor is required. Navier’s exact solution was used to treat the double-curved shell problem with a single title boundary, while the finite element technique and an eight-node quadrilateral were used to address the other boundary requirements. To ensure the accuracy of these results, a thorough comparison technique is employed in conjunction with credible statements. The problem model’s edge cases allow for this kind of analysis. The study’s findings may be used in the post-construction evaluation of military and civil works structures for their ability to sustain explosive loads. In addition, this is also an important basis for the calculation and design of shell structures made of smart materials when subjected to shock waves or explosive loads.

Bending and free vibration analyses of CNTRC shell structures considering agglomeration effects with through-the-thickness stretch

The present work investigates an enhanced finite element model and numerical analysis to illustrate the agglomeration effect of Carbon Nanotubes (CNTs) on static and free vibration responses of reinforced composite plate and cylindrical panel shell. The developed model is based on the kinematics of the improved first-order-shear deformation theory (FSDT), assuming a parabolic variation of the transverse shear strains and fulfilling the zero circumstance of the transverse shear stresses at the extreme structure’ surfaces omitting the use of shear correction factors. The material properties of the nanocomposite are determined using a developed micromechanical model based on the Mori–Tanaka approach. The developed finite element formulation is based on three-dimensional constitutive equations without reduction involving the addition of one parameter of thickness change combined with the Enhanced Assumed Strain (EAS) method. Four enhancement models are established and discussed. In a first attempt, the improvement of the third bending strain using one or four parameters are presented. In a second attempt, the membrane strain is enhanced with four parameters. Due to lack of investigations on structural behavior of composite structures reinforced with agglomerated CNTs, new static and free vibration results for shell structures with three level of CNT agglomeration are examined. Parametric studies are here in developed to present the influences of the characteristic parameters on the structural behavior of the CNTRC shell in a new graphic way; such as the volume fraction and agglomeration of nanofibers, geometrical parameters, and boundary conditions. Based on numerical results, it is found that the agglomeration and distributions of the reinforcing fibers are really crucial and play a significant role on the static and vibrational performances of the structure.

On the Buckling of Simply Supported Anisotropic Sandwich Plates

Here, the elastic stability behavior of simply supported anisotropic sandwich flat panels subjected to mechanical in-plane loads is investigated using analytical approach. The formulation is based on first-order shear deformation theory and the shear correction factors employed are based on energy consideration that depends on the lay-up as well as material properties. The governing equations are obtained using the Raleigh-Ritz method assuming a combination of sine and cosine functions in the form of double Fourier series for the displacement fields. The effectiveness of the integrated formulation is tested for global characteristics considering examples related to multi-layered laminates and sandwich panels for which solutions are available.

A comprehensive analysis of in-plane functionally graded plates using improved first-order mixed finite element model

In this study, the mechanical behaviors of the in-plane functionally graded plate (IFG) are comprehensively explored via a finite element (FE) model. The current FE model is developed base on improved first-order shear deformation theory (FSDT) in conjunction with the mixed FE formulation. The volume fraction as well as the material properties change smoothly along the width and length of the plates for six different patterns. A simple improved transverse shear stress is applied to modify its distribution across the thickness of the IFG plates. Hence, the transverse shear stress equals zeros on the lower and upper surfaces of the IFG plates. Besides, no shear correction factor is needed for calculation of shear strain energy. A four-node mixed plate element, IMQ4, is established via mixed FE formulation for mechanical analysis of the IFG plates. The accuracy and fast convergence rate are validated via some examples. A comprehensive parametric analysis is provided to demonstrate the roles of several parameters including the material gradient index, the side-to-thickness ratio, aspect ratio, and boundary conditions on the mechanical behaviors of the IFG plates.

A new higher-order beam theory for buckling and free vibration responses of laminated composite and functionally graded porous beams

The present study introduces a novel higher-order shear deformation theory for assessing buckling and free vibration characteristics in laminated composite and functionally graded porous beams. The proposed theoretical framework effectively considers three variables and eliminates the need for a shear correction factor. The governing equations are derived from the Lagrange principle, while Legendre-Ritz functions are utilised to solve the resulting problem. Various types of laminated composite beams with arbitrary lay-ups and functionally graded porous beams with symmetric or unsymmetric configurations are analysed. To validate the accuracy and efficiency of the proposed theory, several numerical examples are conducted and compared against the results of existing research endeavours.

Geometrically nonlinear shape sensing of anisotropic composite beam structure using iFEM algorithm and third-order shear deformation theory

The inverse finite element method (iFEM) has been used to achieve the shape sensing of small displacements based on linear elastic theory. However, with the development of smart structure technology, the formulation is not suitable for the anisotropic composite structures with large deformation in practical engineering. Therefore, a nonlinear iFEM algorithm is proposed to monitor the linear and nonlinear deformation of the anisotropic composite beams. The formulation not only involves the effect of orientation of composite fibers on strain distribution into the shape sensing model, but also accounts for the effect of shear deformation without any requirement of shear correction factor. Initially, the third-order shear deformation theory (TSDT) is reviewed along with deriving the nonlinear strain field based on von-Karman strain theory. Considering the problem that the couple term of the shear strain and bending strain is unmeasurable, the relationship between shear and bending displacements is established according to the derived constitutive equations. Then, the proposed nonlinear iFEM method reconstructs the deformed structural shape, where isogeometric analysis (IGA) approach is used to construct the displacement functions and the experimental section strains are calculated from the discretized surface strains. Finally, several examples are solved to verify the proposed methodology. Numerical results demonstrate that the nonlinear iFEM algorithm can improve the reconstruction accuracy by 4% with respect to the linear iFEM method for beam structures. Hence, the proposed approach can be used as a viable tool to predict nonlinear deformation of composite structure.

Static and free vibration analysis of sandwich shells with double curvature considering the effects of transverse normal strain

Higher-order closed-form solutions for the static and free vibration problems of sandwich shells with double curvature are presented in the current study based on a new hyperbolic shell theory considering the effects of transverse normal strain. A theory involves six unknowns and satisfies traction-free boundary conditions at the top and the bottom surfaces of the shell. The theory does not require a shear correction factor to account for the strain energy due to the shear deformation effect. A shell consists of three layers, wherein the top and the bottom layers, i.e. face sheets are made up of hard material and the middle layer, i.e. core is made up of soft material. The governing equations and associated boundary conditions of the theory are produced by employing Hamilton’s principle. Semi-analytical closed-form solutions for the static and free vibration problems are produced by the Navier technique for simply supported boundary conditions of the shell. The present results are compared with results that have already been published to confirm the accuracy and efficacy of the current higher-order hyperbolic shell theory.

Isogeometric Analysis for Free Vibration of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

This paper presents a novel quasi-three-dimensional shear deformation theory called the spectral displacement formulation (SDF) for analyzing the free vibration of functionally graded plates. The SDF expresses the unknown displacement field as a unique form of the Chebyshev series in the thickness direction. By increasing the truncation number in the Chebyshev series, the bending analysis results can approach the three-dimensional elasticity solution and satisfy the traction-free boundary conditions without requiring a shear correction factor. The SDF is an extension of the classical plate theory, thereby naturally avoiding the shear-locking phenomenon. These characteristics enable the SDF to apply to plates of arbitrary thickness while maintaining accuracy. The nonuniform rational B-spline-based isogeometric approach is employed to enhance the applicability of this theory to free vibration analysis of functionally graded plates with complex geometries and different boundary conditions. Numerical examples are presented to demonstrate the accuracy and reliability of the proposed method in analyzing the free vibration of functionally graded plates.

Buckling analysis of plates using an efficient sinusoidal shear deformation theory

Mechanical buckling response of isotropic and orthotropic plates using the two variable refined plate theory is presented in this paper. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate; hence it is unnecessary to use shear correction factors. Governing equations are derived from the principle of virtual displacements. The nonlinear strain-displacement of Von Karman relations are also taken into consideration. The closed-form solution of a simply supported rectangular plate subjected to in-plane loading has been obtained by using the Navier method. Numerical results are presented for the present efficient sinusoidal shear deformation theory, demonstrating its importance and accuracy in comparison to other theories.

Flutter of carbon-based nanohybrid composite panels

Recently, carbon-based composite aerospace and mechanical structures have considerable attention due to their great material and mechanical properties. Carbon nanotubes (CNTs) or graphene platelets (GPLs) reinforcements for different matrices are common for the design of modern multilayer structures. In the present paper, the supersonic flutter of nanohybrid composite multilayer panels is examined and discussed. The nanohybrid multilayer panel is made of different technologically justified configurations of polymer layers reinforced by CNTs and GPLs The effective material properties of each CNTs and GPLs layer are determined according to two well-established homogenization techniques: the Mori–Tanaka scheme and the modified Halpin–Tsai model, respectively. Reddy’s third-order shear deformation theory is applied to avoid determining shear correction factors for different distributions of material properties. Hamilton’s principle and the first-order piston theory are applied to formulate aeroelastic coupled equations of motion of the panel. The numerical meshless method is employed to discretize dynamic equations and find generalized multi-parametric characteristic equations functionally dependent on the parameters of the structure with different boundary conditions. The accuracy of the obtained solution and results is examined by comparing the determined natural frequency and critical aerodynamic pressure with those obtained in previous studies. Then, a comprehensive parametric study is carried out to show the effects of hybridization of different distribution patterns of CNTs and GPLs, weight fractions, total layer number, and aspect ratio of hybrid composite panels on the flutter bounds. Results show that the present method provides a highly effective general numerical model for flutter analysis of hybrid nanocomposite panels with different boundary conditions for all thickness categories including thick plates. Moreover, an optimization nanoparticles distribution for the flutter of the nanohybrid composite panel has been obtained.

Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate.

A nonlinear static analysis of a circular/annular nanoplate on the Winkler-Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler-Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.

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.

On the pressure–deflection relations and instability of carbon-based composite nonlinear pipes

In this paper, for the first time, the nonlinear buckling of curved pipes is investigated for different distributions of single-walled carbon nanotubes (SWCNTs) in the matrix. The curved pipe is subjected to uniformly distributed transverse pressure and experiences uniform temperature rise. Additionally, the pipe is in one-sided contact with an elastic nonlinear medium. Defined thickness-dependent and temperature-dependent material properties of SWCNTs are taken from the results of molecular dynamics (MD) simulations, well-established in the literature on the subject. Five technologically justified types of CNT distributions are considered. Shen’s rule of the mixture is applied to determine the effective properties of CNT-based composites. To avoid determining shear correction factor, the higher-order shear deformation theory is used. The nonlinear components of the Green–Lagrange strain tensor and the principle of virtual displacements are applied to derive the nonlinear system of partial differential equations of motion of the curved pipe. The solution to formulated boundary value problem is obtained by the perturbation technique for both simply-supported and clamped-clamped boundary conditions. The pressure–deflection curves of the curved pipe are obtained and discussed. The investigation of snap-buckling intensity and upper/lower limit load of the curved pipe is presented and new insights for its nonlinear stability are presented. For instance, it is shown that the V-type of CNT distribution profile affects on lower deflection of the curved pipe while the higher deflection exists for the Λ-type profile.

Free vibration analysis of the structural integrity on the porous functionally graded plates using a novel Quasi-3D hyperbolic high order shear deformation theory

In this study, a novel quasi-three dimensional hyperbolic high-order shear deformation theory (quasi-3D HHSDT) is developed for free vibration analysis of porous functionally graded plates (FGPs). There are six unknowns in the current displacement field, and no shear correction factor is required. The mechanical properties are varied continuously through the thickness of porous FG plates using a modified power law function while considering the effect of porosities on the plate’s structural integrity. Two distinct porosity distribution models are considered, including even and uneven porosity distributions. The Navier technique is employed to obtain the closed-form solutions of motion's equations. An exhaustive parametric study is presented to show the influence of the different parameters on the fundamental frequencies.

Effects of Transverse Normal Stress on Hygrothermomechanical Analysis of Laminated Shallow Shells

Hygrothermomechanical analysis of doubly curved laminated composite shallow shells is presented in this paper using a two-dimensional higher-order shell theory considering the effects of transverse normal stress. The present theory accounts for the fifth-order expansion of the thickness coordinates in the in-plane displacements and the fourth-order expansion in the transverse displacement. The main focus of this work is to study the effects of both transverse normal stress and the fifth-order expansion of thickness coordinates on the hygrothermomechanical responses of doubly curved laminated shallow shells. The governing equations of the theory are derived based on the principle of virtual work and solved by using Navier’s method. The theory ensures traction-free boundary conditions using constitutive relations and does not require a shear correction factor. Interlaminar transverse shear stresses are recovered using the three-dimensional equilibrium equations of the theory of elasticity to ascertain the stress continuity at the layer interface. Three-dimensional elasticity solutions are taken as a basis for comparing the present results in thermal problems, whereas this study contributes some useful results to the hygrothermomechanical analysis of doubly curved laminated composite shallow shells.

Evaluation of static responses for layered composite arches

Abstract Layered composite materials are widely used across a variety of sectors, including the automotive industry, aerospace engineering, offshore, and various mechanical domains, because of their strong yet lightweight structures. Therefore, various emergent theories are available on the deformation of layered beams. The previous research studies are insufficient as they are based on deformation of layered composite and sandwich arches with simply supported (SS) end conditions. Therefore, it is a good opportunity for researchers to investigate the arches using exponential shear deformation and normal deformation theory. The leading hypothesis mainly adds to the research of bending for sandwich and layered composite arches adopting the exponential theory. The present theory does not require any shear correction factor to satisfy zero transverse shear stress condition at the bottom and top fibers of arches. Governing equations and associated end conditions are derived through principle of virtual work. Navier’s techniques used for sandwich and layered composite arches are SS boundary conditions subjected to uniformly distributed load. The results of the current study showed that the exponential normal and shear deformation theories may be used to evaluate static responses for layered composite and sandwich arches. The obtained results from the present theory are validated through the results available in published literature.

The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates

This is the first study of the static bending and free vibration response of organic nanoplates based on the combination of nonlocal theory with various shear strain theories, where the theory of shear deformation of the plate has the advantage of requiring no shear correction factor. The equilibrium equation of the plate is derived using Hamilton’s principle, the analytic solution is derived using the Navier solution form, and the finite element technique is implemented using a quadrilateral element with four nodes and six degrees of freedom for each node. Moreover, this is the first work to calculate for nanoplates with a nonlocal parameter whose value varies with plate thickness. This study’s credibility has been established by comparing it to previously published findings, which are utilized to validate the results of analytical and numerical calculations, respectively. In addition, the study investigates how a range of components impact the displacement and stress response of organic nanoplates. The findings of the study indicate that there are some circumstances in which it is not feasible to disregard the nonlocal parameter while doing calculations for organic nanostructures.

A closed-form solution for accurate stress analysis of functionally graded Reddy beams

In the present paper, a closed-form solution of the Reddy beam theory is developed and applied, for the first time, to investigate the bending behavior of straight and curved functionally graded (FG) beams, where the material properties change continuously from one surface to another in the thickness (or height) direction. The obtained closed-form solution, sum of polynomial and exponential terms, enriches the polynomial displacement field usually proposed in a finite element (FE) approach, with effects also on the derived strain and stress quantities, particularly relevant in FG beams. The adopted beam model is exploited to satisfy parabolic variation of the shear stress distribution along the thickness direction and does not require the use of shear correction factors, particularly difficult to obtain when the beam is inhomogeneous in the thickness direction. Comparative studies are carried-out to establish the robustness and the performance of the present model, and numerical results are presented and discussed in detail to investigate the effects of volume fraction index, radius of curvature, length-to-height ratio, and boundary conditions on the stress response of FG beams. The obtained results can serve as benchmarks for future research.

An analysis of vibration and buckling behaviors of nano-composite beams reinforced with agglomerated carbon nanotubes via differential quadrature finite element method

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.