Order Shear Deformation Theory Research Articles

Dynamic stability of different kinds of sandwich plates using third order shear deformation theory

In this paper dynamic buckling of different kinds of sandwich plates under harmonic axial load is analyzed by using combination of numerical finite strip analysis and Bolotin’s solution which offer a kind of method to find the regions of dynamic instability for dynamic stability of plates. The sandwich plates which are considered here consist of homogeneous core sandwich plates, functionally graded sandwich plates, trapezoidal corrugated core sandwich plates, Z and C form corrugated core sandwich plates and truss core sandwich plates. Third order shear deformation theory is used as numerical method to study stability regions of sandwich plates. The effect of damping is neglected and the effect of length to thickness ratio, different boundary conditions, number of face sheet layers and different geometry and material properties of core is studied. According to the results the dynamic buckling should also be investigated in sandwich plate. Also, according to the results, it is observed that the finite strip method has an acceptable accuracy in investigating the dynamic buckling of sandwich plates.

An investigation into vibrational behaviour of the functionally graded carbon nanotube reinforced shell structure

The requirement for space structures that can absorb stress caused by meteorite impacts and mitigate the thermal and inherent vibrations produced as rapidly as feasible motivates this research. Carbon nanotubes (CNT) produce better stiffness and elastic moduli when combined randomly or in a specific distribution in the matrix. As a result, they can be used as a better material than what is now available. Because most space projects have a cylindrical body, this study investigates the vibrational response of functionally graded carbon nanotube reinforced composites (FG-CNTRC) cylindrical panels. Furthermore, earlier research has shown that the problem of vibration control for FGCNTRCpanels has received little attention. In this study, straight and aligned carbon nanotubes of varied distributions are employed, and they are functionally graded in a specific orientation. For the FGCNTRC cylindrical panel, different configurations have been numerically modeled using the finite element technique. The kinematic and constitutive equations are obtained using first order shear deformation theory (FSDT). Furthermore, Hamilton's principle is applied in order to determine the equation of motion. The volume proportion and CNT efficiency parameters used in this study are as follows: for VCNT of 11%, 1 = 0.149, 2 = 3 = 0.934, for VCNT of 14%, 1 = 0.150, 2 = 3 = 0.941, and for VCNT of 17%, 1 = 0.149, 2 = 3 = 1.381. The boundary conditions are imposed along two straight edges of the FG-CNTRC cylindrical panel and are not applied along the curved edges. The free vibration response of the FGCNTRC cylindrical panel is illustrated for three distinct volume proportions, each of which corresponds to one of the configurations under examination, in order to have better insight of the mechanical behavior of the FGCNTRC cylindrical panel. The study lists the influence of varying volume fractions of CNTs on the first five natural frequencies. The influence of varied CNT volume fractions and configurations on the mode shapes is numerically illustrated. The effect of temperature is also included in numerical simulations because space structures are vulnerable to large temperature variations during space day and night. The piezoelectric patches are strategically placed to achieve excellent vibration reduction. For active vibration control, a nonconventional controller is developed and implemented.

Fundamental Frequencies of Elliptical Plates using Static Deflections

Fundamental frequencies of solid and annular elliptical plates were approximated using the static deflections by means of finite element method (FEM) without computing the eigenvalues. The problem was formulated within the framework of the first order shear deformation theory (FSDT). The effects of (i) the inner and outer boundary conditions, (ii) the size of the perforation, (iii) the aspect ratio, and (iv) the thickness of the plate on the performance of the method were examined via a large variety of numerical simulations. Convergence study was performed through h-refinement. Accuracy of the results was validated through comparison studies. The results reveal that the application of the Morley’s formula which does not require eigenvalue analysis approximates the fundamental frequency with finer mesh compared to the eigenvalue analysis. The method can be considered as a practical technique to approximate the fundamental frequency. However, the boundary conditions have dominant role on the accuracy of the solution particularly when the plate is perforated.

Flexural Analysis of Thick Isotropic Rectangular Plates Using Orthogonal Polynomial Displacement Functions

This paper analysed the flexural behaviour of SSSS thick isotropic rectangular plates under transverse load using the Ritz method. It is assumed that the line that is normal to the mid-surface of the plate before bending does not remain the same after bending and consequently a shear deformation function f (z) is introduced. A polynomial shear deformation function f (z) was derived for this research. The total potential energy which was established by combining the strain energy and external work was subjected to direct variation to determine the governing equations for the in – plane and out-plane displacement coefficients. Numerical results for the present study were obtained for the thick isotropic SSSS rectangular plates and comparison of the results of this research and previous work done in literature showed good convergence. However, It was also observed that the result obtained in this present study are significantly upper bound as compared with the results of other researchers who employed the higher order shear deformation theory (HSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT) theories for the in – plane and out of plane displacements at span – depth ratio of 4. Also, at a span - depth ratio of and above, there was approximately no difference in the values obtained for the out of plane displacements and in-plane displacements between the CPT and the theory used in this study.

Flexural Analysis of Thick Isotropic Rectangular Plates Using Orthogonal Polynomial Displacement Functions

This paper analysed the flexural behaviour of SSSS thick isotropic rectangular plates under transverse load using the Ritz method. It is assumed that the line that is normal to the mid-surface of the plate before bending does not remain the same after bending and consequently a shear deformation function f (z) is introduced. A polynomial shear deformation function f (z) was derived for this research. The total potential energy which was established by combining the strain energy and external work was subjected to direct variation to determine the governing equations for the in – plane and out-plane displacement coefficients. Numerical results for the present study were obtained for the thick isotropic SSSS rectangular plates and comparison of the results of this research and previous work done in literature showed good convergence. However, It was also observed that the result obtained in this present study are significantly upper bound as compared with the results of other researchers who employed the higher order shear deformation theory (HSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT) theories for the in – plane and out of plane displacements at span – depth ratio of 4. Also, at a span - depth ratio of and above, there was approximately no difference in the values obtained for the out of plane displacements and in-plane displacements between the CPT and the theory used in this study.

Dynamic thermal buckling of spherical porous shells

The aim of present work is to address nonlinear dynamic thermal buckling of shallow spherical functionally graded porous shells subjected to transient thermal loading using the first order shear deformation theory (FSDT). A power-law distribution as well as cosine-type porosity distribution are used to model the variation of constituents through the shell thickness. Thermomechanical properties are assumed to be temperature dependent. Using Crank–Nicolson time marching scheme, an iterative procedure is employed to solve nonlinear transient heat conduction equation. For thermal boundary conditions, the outer surface of shells is kept at a reference temperature, while the inner surface experiences a sudden temperature rise. Geometrical type of nonlinearity in the sense of von-Karman is taken into account. The highly coupled nonlinear governing equations of motion are extracted by constructing the appropriate weak form and also using multi-term Ritz–Chebyshev method. The resulting ODEs are then reduced to a system of nonlinear algebraic equations by employing the well-known Newmark family of time integration schemes. The latter equations are solved by means of Newton–Raphson iteration procedure. Budiansky criterion is used to recognize critical parameters of dynamic instability of shells due to applied thermal shocks. Some comparison studies are conducted in order to verify the accuracy of results of the present work. Moreover, various parametric studies are performed to assess the influence of involved parameters.

Nonlinear impulsive and vibration analysis of nonlocal FG-CNT reinforced sandwich plate by considering agglomerations

This paper is focused on impulsive and subsequent vibration response of a nano-scale composite sandwich plate considering both nonlocal effect and nanoparticle agglomerations. In particular, the core is homogeneous polymethyl methacrylate (PMMA) and the face sheet is PMMA matrix with functionally graded carbon nanotubes (short for FG-CNT). The material properties are given by Eshelby-Mori-Tanaka's approach taking into account of CNT agglomerations. The equations of motion for the FG-CNT reinforced nanoplate are derived based on nonlocal strain gradient theory, Reddy's higher order shear deformation theory and von-Karman nonlinear deformation theory. The impulsive and vibration problems for simply supported FG-CNT plates are solved using Galerkin method and Newton iteration method where the solutions have been verified with the literature. By considering impulsion of step load, exponential load and sinusoidal load and, it is found that weaker size effect, fewer CNT agglomerations, higher CNT content, smaller functional gradient, thinner core, bigger aspect ratio and larger length thickness ratio can enhance the stiffness of the nanoplate and thus will induce smaller center displacement during the load process and higher free vibration frequency after unloading, regardless of load type. However, the subsequent vibration amplitude is significantly influenced by the impulsive load type. Among the material parameters, minimizing size effect and agglomerations could greatly improve the structural stiffness whereas the improvement by varying functional gradient is very limited. It is the first time that a FG-CNT reinforced sandwich plate, especially considering nonlocal effect and nanoparticle agglomerations, is treated in the context of impulsive and subsequent vibration problems. The solutions presented in this paper could provide a benchmark for the design and manufacture of such a composite nanoplate.

Numerical instability investigation of composite pipes reinforced by carbon nanotubes based on higher-order shear deformation theory

Various types of pipelines and risers used in oil and gas industry undergo dynamic instabilities (buckling and flutter) due to the internal fluid flows. This paper aims to study the possibility of increasing pipeline stability by using pipes made of composite materials and reinforced by carbon nanotubes. Hamilton principle for open systems is used to derive the equations of motion of the pipe based on a higher order shear deformation theory. The rule of mixture is implemented to express the change of the material properties of the pipe through the pipe thickness. Finite element analysis with super convergent elements is used to solve the governing differential equations of the problem. Static equilibrium equations of the pipe without fluid are used to develop the shape functions of the elements. The results of the present study are compared with those available in the literature for dynamic behavior of pipes with Euler-Bernoulli and Timoshenko beam theories. The effects of different parameters on (i) vibration characteristics and (ii) stability limits of composite pipes conveying fluid are also examined.

Vibration and nonlinear dynamic response of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio

This paper reports a study on natural frequency, nonlinear dynamic response and frequency amplitude relation of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio. The nonlinear motion equations and the compatibility equation are retrieved from first order shear deformation theory including the effect of von Karman nonlinearity terms, elastic foundations and initial imperfection. The system of dynamic governing equations is obtained by utilizing the Airy stress function and Bubnov–Galerkin procedure. The nonlinear dynamic response is also obtained by applying the 4th-order Runge–Kutta method. The frequency–amplitude relations of the shell is determined by using the assumption of inertia caused by a rotation. Besides, the analytical method and the finite element method are proposed to calculate the vibrational frequencies. Five types of FG-CNTRC laminate shell, three types of the structures and different thermal environments with the shell’s material properties depending on the temperature are also considered in this study. The numerical results are presented, verified with available studies in the literature. Finally, the effects of elastic foundations, initial imperfection, and temperature-dependence, typing double curved shallow shell, angle of lamination against the shell x-axis on the natural frequency and nonlinear dynamic response of FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio are discussed.

FREE VIBRATION OF SKEW LAMINATES – A BRIEF REVIEW AND SOME BENCHMARK RESULTS

This study investigates and reviews prior research works on skew composite laminates. The equivalent single layer theories are explored and discussed. An exhaustive review on static and dynamic analysis of composite skew laminates is also presented. Subsequently, a nine node isoparametric plate bending element is used for free vibration analysis of laminated composite skew plate with central skew cut out. The effect of shear deformation is incorporated in the formulation considering first order shear deformation theory. Two types of mass lumping schemes are analysed to study the effect of rotary inertia. Certain numerical examples of plates having different skew angles, skew cut out sizes, boundary conditions, thickness ratios (h/a), aspect ratios (a/b), fiber orientations and number of layers are solved which will be useful for benchmarking of future studies.

STRESS DISTRIBUTION IN THE HOLLOW STIFFENED HYBRID LAMINATED COMPOSITE PANELS IN SHIP STRUCTURES UNDER SINUSOIDAL LOADING

A simplified hollow stiffened hybrid laminated plate model has been developed for the marine structures. The detailed stress analysis through the thickness of the stiffened plate based on the higher order shear deformation theory has been carried out under sinusoidal loading. The hybrid laminates are made by wrapping the GFRP laminates with CFRP at the outermost layers of the stiffened panel. This hybridization technique can be an optimum solution from the point of view of cost reduction as well as enhancement of strength properties. The layer-wise stresses for the stiffened plate have been calculated in the present paper. A 3D polynomial curve fitting technique has been used to obtain higher accuracy and consistency in the computation of stresses. The computer code has been developed using MATLAB considering the plates as eight noded isoparametric plate bending element and the stiffener has been modeled as three noded isoparametric beam element. The stiffened panel has also been analysed using the ANSYS14.0 software package considering 2D model. The results obtained from the present formulation have been compared with those available in the published literature to validate the present formulation. The stiffened panels made of GFRP, CFRP and GFRP-CFRP hybrid laminates have been studied here. An extensive parametric study has been carried out with varying fibre content in the laminates.

Investigation of the bending response of carbon nanotubes reinforced laminated tapered spherical composite panels with the influence of waviness, interphase and agglomeration

In the present study, the bending analysis of multi walled carbon nanotubes (MWCNTs) reinforced laminated tapered spherical composite panels subjected to transverse loading conditions is carried out using the finite element method (FEM) with displacement fields derived from high order shear deformation theory. In this regard, the elastic properties of the polymeric nanocomposite, considering the MWCNTs agglomeration, waviness and interphase thickness between the MWCNTs and polymer matrix were computed by employing various micromechanical approaches. Subsequently, glass-fiber were introduced as a reinforcement phase, and the overall elastic properties of a three-phase MWCNT/fiber/polymer hybrid nanocomposite material were obtained using Chamis model. The accuracy of the developed FE formulation were verified using the experimental and numerical data available in the open literature. After verification, the deflection and stresses of the panel are numerically determined for various taper configurations. Further, detailed parametric analysis is carried out to explore the influence of various parameters such as CNT content, agglomeration, waviness, interphase area, thickness ratio, curvature ratio and taper configurations on the bending results of laminated tapered spherical panels. It was observed that an increase in the weight fraction of CNTs, an equal number of agglomeration parameter and the thickness of the interphase have a positive effect in minimizing the deflection and stresses. It was also observed that the TC-2 and TC-3 taper configurations results the highest and lowest deflection. Finally, it can be concluded the current work can serve as a guide for the effective design and development of CNT reinforced composite structures.

Nonlinear response of doubly curved sandwich panels with CNT-reinforced composite core and elastically restrained edges subjected to external pressure in thermal environments

An analytical investigation on the nonlinear response of doubly curved panels constructed from homogeneous face sheets and carbon nanotube reinforced composite (CNTRC) core and subjected to external pressure in thermal environments is presented in this paper. Carbon nanotubes (CNTs) are reinforced into the core layer through uniform or functionally graded distributions. The properties of constituents are assumed to be temperature dependent and effective properties of CNTRC are determined using an extended rule of mixture. Governing equations are established within the framework of first order shear deformation theory taking into account geometrical imperfection, von Kármán–Donnell nonlinearity, panel-foundation interaction and elasticity of tangential edge restraints. These equations are solved using approximate analytical solutions and Galerkin method for simply supported panels. The results reveal that load carrying capacity of sandwich panels is stronger when boundary edges are more rigorously restrained and face sheets are thicker. Furthermore, elevated temperature has deteriorative and beneficial influences on the load bearing capability of sandwich panels with movable and restrained edges, respectively.

Agglomerated impact of CNT vs. GNP nanofillers on hybridization of polymer matrix for vibration of coupled hemispherical-conical-conical shells

This paper is dedicated to study the vibrational behavior of Coupled Hemispherical-Conical-Conical Shells (CHCCS) structures made composite materials reinforced with nanofillers. One of the most important issue in nanocomposites is the agglomeration effect of nanofillers in hybridization procedure. In this study, two well-known nanofillers including Carbon Nano Tubes (CNTs) and Graphene Nano Platelets (GNPs) are used for reinforcement of polymer matrix. In addition, the distribution of agglomerated nanofillers throughout the thickness of the structures is assumed to be uniform and functional. Therefore, three patterns of distribution including X, O and V are employed. To study these topics, first, the authors obtain the displacement fields of CHCCS using Donnell's shell theory. In this formulation, the First Order Shear Deformation Theory (FOSDT) is utilized to consider the shear deformation effect. Second, the Hamilton's principle is used for achieving the governing equations of motion associated with the CHCCS structure. Afterwards, an efficient and robust approximation numerical solution method, namely Generalized Differential Quadrature (GDQ) method, is employed to discretize the governing system of differential equations, apply boundary and connective conditions related to the CHCCS. Since there is no applicable benchmark for this analysis, the authors prove their proposed formulation by comparing the obtained results with the outputs of FEM program for some specific problems. Furthermore, several new and complex problems are designed and solved to investigate the effect of various factors such as geometrical characteristics, weight proportions, and distribution patterns of CNTs and GNPs on the vibration parameter of CHCCS. It is worth mentioning that the effect of agglomeration parameters associated with these nanofillers on the vibrational behavior of CHCCS structures is also studied numerically.

Buckling analysis of shear‐deformable orthotropic laminated plates with rotational restraints

AbstractBuckling of shear‐deformable laminated plates is investigated in the present study. A closed‐form method to determine the buckling load is developed within the framework of Reddy's Third Order Shear Deformation Theory (TSDT). The considered orthotropic plates are under uniaxial compression and simply supported at all edges with rotational restraints at the unloaded edges. This allows the modelling of a number of structural situations, e.g. local buckling of beams or the skin fields of stiffened composite panels, for which only the rotational stiffness of the surrounding structure is required. The analysis approach is based on the Ritz method and employs simple shape functions with few variables for the deflection and rotations, which finally leads to an explicit approximate solution for the critical buckling load. The comparison to the Lévy‐type solution shows a satisfactory agreement. The new approach is very simple and highly computationally efficient, which provides a great advantage for example in optimisations.

Design of laminated conical shells using spectral Chebyshev method and lamination parameters

This study presents a modeling approach to accurately and efficiently predict the dynamics of laminated conical shells. The governing equations are derived based on the first order shear deformation theory kinematic equations following the Hamilton’s principle. To express the strain energy of the shells, in-plane and bending lamination parameters are used. A two-dimensional spectral approach based on Chebyshev polynomials is implemented to solve the governing equations. The developed framework including the spectral-Chebyshev approach and lamination parameters results in an accurate and computationally efficient solution method. To demonstrate the performance of the presented solution approach, various case studies including straight panels, curved shells, and truncated conical shells are investigated. The benchmarks indicate that the calculated non-dimensional natural frequencies excellently match the results found using finite element method and the simulation duration can be decreased by 100 folds. To leverage the computational performance of the presented approach, a stacking sequence optimization is performed to maximize the fundamental frequency of a shell geometry, and the corresponding fiber angles are retrieved from the optimized lamination parameters. Furthermore, a parametric analysis is performed to investigate the effect of geometry on the optimized lamination parameters (and fiber angles) based on fundamental natural frequency maximization.

On linear and nonlinear bending of functionally graded graphene nanoplatelet reinforced composite beams using Gram-Schmidt-Ritz method

The current study examined the linear and nonlinear bending results of functionally graded graphene nanoplatelet reinforced composite beams. The equation system was constructed using Reddy's third order shear deformation theory and a von Kármán type nonlinear strain-displacement relationship. The nonlinear bending results of this system were solved using the Gram-Schmidt-Ritz method in conjunction with the direct iteration technique. By employing the Gram-Schmidt orthogonalization procedure for generating shape functions in the method, numerically stable functions for nonlinear bending analysis of beams were obtained. Our solutions were validated for accuracy by comparing them to some published results. The numerical results indicate that the geometrical nonlinearity of beams subjected to uniform distributed load is greater than that of beams subjected to sinusoidal, linear, or parabolic distributed loads. New results on several significant effects of reinforcement patterns, graphene nanoplatelet weight fraction, and different types of distributed loads are presented and discussed in this study, in which the findings can serve as a benchmark for future research.

Analytical solutions for nonlinear vibration of porous functionally graded sandwich plate subjected to blast loading

This paper presents the nonlinear vibration of porous functionally graded sandwich plate on elastic foundations subjected to blast loading by the analytical approach. The sandwich plate consists of two FGM face sheets and a homogeneous core which is made from metal or ceramic. Two types of porosity distribution, including evenly and unevenly distributed porosity have been considered for sandwich plate. The material properties of sandwich plate are assumed to vary in the thickness direction according to simple power law distribution with a volume fraction index and a porosity coefficient. The blast loading is assumed to be uniformly distributed on the surface of the sandwich plate and modeled by an exponential function. The Reddy’s higher order shear deformation theory with von Kármán type nonlinearity is used to establish governing equations for the vibration of sandwich plate. By applying Galerkin and fourth-order Runge–Kutta methods, the numerical results show the effects of volume fraction index, porosity coefficient, type of porosity distribution, geometrical parameters, elastic foundations and parameters of blast loading on the nonlinear vibration of the sandwich plate. Comparisons are conducted to evaluate the reliability of the obtained results.

Isogeometric free and forced vibration analyses of FG-CNTs plates based on a logarithmic higher order shear deformation theory

This paper develops the new logarithmic higher order shear deformation theory (LHSDT) incorporating isogeometric method for free and forced vibration analyses of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) plates. In this theory, a logarithmic function is employed to approximate the distribution of shear strains along the plate thickness which satisfies the condition of zero-tractions on the top and bottom surfaces of the plate. It is assumed that the plate is fabricated from a mixture of carbon nanotubes (CNTs) and a polymetric matrix. The CNTs are either uniformly distributed or functionally graded (FG) along the thickness direction of the plate. The modified rule of mixture scheme is applied to estimate effective mechanical properties of FG-CNTRC plates. The governing equations are derived from the Hamilton’s principle. Furthermore, the Newmark approach is utilized to predict the temporal response of FG-CNTRC plates under different transverse dynamical loadings. The applicability and efficiency of the present formulation in predicting vibrational characteristics of FG-CNTRC plates is investigated through an extensive set of numerical examples considering different configurations of the plate. It is revealed that the computed results are in excellent agreement with other solution methods extracted by 3D model as well as other plate theories. Eventually, a detailed parametric study is conducted to explore the influence of related parameters on the natural frequencies and temporal response of FG-CNTRC plates.

A new higher order shear and normal deformation theory for the free vibration analysis of sandwich curved beams

The free vibration analysis of functionally graded sandwich beams curved in elevation is presented in this paper using a fifth-order curved beam theory accounting for the effects of transverse shear and normal strains i.e. thickness stretching effects. The theory assumes higher-order variations of axial (tangential) as well as transverse (radial) displacements. Equations of motion are derived from the dynamic version of the principle of virtual work and solved analytically by using Navier’s technique. The present theory is applied for the free vibration analysis of sandwich curved beams with functionally graded face sheets and the homogenous core. The material properties of face sheets are graded in the thickness direction according to the power law. The non-dimensional fundamental frequencies are obtained for different parameters such as radius of curvature, the power-law index, and lamination scheme and compared with previously published results wherever possible and found in good agreement with those. It is observed that as the radius of curvature is decreased keeping length and thickness constant the value of non-dimensional fundamental frequency increases. It is also observed that as the power-law index increased the non-dimensional frequency decreases for all length to thickness ratios.