Effect Of Transverse Shear Deformation Research Articles

Effect of thickness stretching on the static deformations, natural frequencies, and critical buckling loads of laminated composite and sandwich beams

The present study investigates the bending, buckling, and vibration responses of shear deformable laminated composite and sandwich beams using trigonometric shear and normal deformation theory. The most important feature of the present theory is that it includes the effects of transverse shear and normal deformations, i.e., the effect of thickness stretching. Therefore, the theory is also called as a quasi-2D theory. The axial displacement uses sine function in terms of the thickness coordinate to include the effect of transverse shear deformation, and the transverse displacement uses cosine function in terms of the thickness coordinate to include the effect of transverse normal deformation, i.e., the thickness stretching. The present theory satisfies the zero shear stress conditions at top and bottom surfaces of the beam without using shear correction factor. Governing differential equations and associated boundary conditions of the theory are derived by employing the dynamic version of principle of virtual work. Navier-type closed-form solutions are obtained for simply supported boundary conditions. The numerical results are obtained for deflections, stresses, natural frequencies, and critical buckling loads for isotropic, laminated composite, and sandwich beams. Since exact elasticity solutions for laminated composite and sandwich beams are not available in the literature, the results are compared with those obtained by using other higher-order shear deformation theories to demonstrate the accuracy of the proposed theory.

1D analysis of laminated composite and sandwich plates using a new fifth-order plate theory

Abstract In the present study, a new fifth-order shear and normal deformation theory (FOSNDT) is developed for the analysis of laminated composite and sandwich plates under cylindrical bending. The theory considered the effects of transverse shear and normal deformations. To account for the effect of transverse shear deformation, in-plane displacement uses polynomial shape function expanded up to fifth-order in-terms of the thickness coordinate. Transverse displacement uses derivative of shape function to account for the effect of transverse normal deformations. Therefore, the present theory involves six independent unknown variables. The theory satisfies traction free boundary conditions at top and bottom surfaces of the plate and does not require the shear correction factor. The principle of virtual work is used to obtain the variationally consistent governing differential equations and associated boundary conditions. Analytical solutions for simply supported boundary conditions are obtained using Navier’s solution technique. Non-dimensional displacements and stresses obtained using the present theory are compared with existing exact elasticity solutions and lower and higher-order theories to prove the efficacy of the present theory. The comparison shows that the displacements and stresses predicted by the present theory are in good agreement with those obtained by using the exact solution.

Parametric studies on buckling behavior of functionally graded graphene-reinforced composites laminated plates in thermal environment

This paper investigates buckling behavior of functionally graded graphene-reinforced composite (FG-GRC) laminated plates. The extended Halpin-Tsai model is used to estimate the effected material properties of the GRCs. The effects of rotary inertia and transverse shear deformation are incorporated by the first-order shear deformation theory (FSDT). Using the meshless kp-Ritz method, the buckling solutions are obtained. In order to derive the discretized eigenvalue equation, the kernel particle approximation is employed in the field variables and the energy functional is minimized through the Ritz procedure. The stability and accuracy of the meshless kp-Ritz method is verified through comparison and convergence studies by considering the effect of support size and number of nodes. The effects of boundary temperature, distribution of graphene, plate width-to-thickness ratio, plate aspect ratio and number of layers are investigated by detail parametric studies. Besides, the influences of boundary conditions, the types of buckling load and lamination angle are considered in this paper.

Bending analysis of laminated composite plates using finite element method

Laminated composite plate structures find numerous applications in aerospace, military and automotive industries. The role of transverse shear is very important in composites, as the material is weak in shear due to its low shear modulus compared to extensional rigidity. Hence, an accurate understanding of their structural behaviour is required, such as deflections and stresses. In this paper, a number of finite element analyses have been carried out for various side-to-thickness ratios, aspect ratios and modulus ratios to study the effect of transverse shear deformation on deflection and stresses of laminated composite plates subjected to uniformly distributed load. The numerical results showed, on the deflections and stresses, that the effect of coupling is to decrease the deflections with the increase in the aspect ratio and modulus ratio and increase the stresses with the increase in the side-to-thickness ratio and modulus ratio.Keywords: Laminated composite plate, bending analysis, uniformly distributed load, finite element method.

The effect of nonlinear temperature distribution on the vibrational behavior of a size‐dependent FG laminated rectangular plates undergoing prescribed overall motion

In this article, the effect of the nonlinear temperature distribution on the free vibration analysis of size‐dependent rotating laminated rectangular FG micro‐/nanoplates are developed based on modified couple stress theory and exponential shear deformation theory. The modified couple stress theory model contains one length scale parameter to consider the small size effect. In exponential shear deformation theory, exponential functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. The temperature is assumed to be constant in the plane of the plate and to vary in the thickness direction only. Material properties are assumed to be temperature dependent, and vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The Eigen frequency equation is obtained by the use of Rayleigh–Ritz method. Displacement components are expressed in simple algebraic polynomial forms which can handle any sets of boundary conditions. The effect of material compositions, rotation speed, hub ratio, length‐to‐thickness ratio, aspect ratio, material length scale parameter, number of layered, and temperature filed on the vibration characteristics is examined. The present theoretical results are verified by comparing with those in literature. POLYM. COMPOS., 40:766–778, 2019. © 2018 Society of Plastics Engineers

On vibrations of porous nanotubes

In this study, the vibration behaviors of porous nanotubes are investigated for the first time. The nonlocal strain gradient theory in conjunction with a refined beam model are employed to formulate the size-dependent model. It is presumed that the porous nanotubes are made from functionally graded materials, and the material parameters of nanotubes relate to temperature variation and vary continuously in the radial direction. Employing a refined beam theory which includes the effects of transverse shear deformation, the equations of motion are derived based on Hamilton's variation principle and solved by the Navier solution method. Some comparisons are presented to valid the correctness of present solution method. The effects of the nonlocal parameter, strain gradient parameter, temperature variations, porosity volume fraction and material variation on the vibration characteristic of the nanotubes are discussed in detail.

Vibration analysis of carbon nanotubes-based zeptogram masses sensors and taking into account their rotatory inertia

In this research work, the transverse vibration behaviour of single-walled carbon nanotubes (SCNT) based mass sensors is studied using the Timoshenko beam and nonlocal elasticity theories. The nonlocal constitutive equations are used in the formulations and the CNT with different lengths, attached mass (viruses and bacteria) and the general boundary conditions are considered. The dimensionless frequencies and associated modes are obtained for one and two attached masses and different boundary conditions. The effects of transverse shear deformation and rotatory inertia, nonlocal parameter, length of the carbon nanotubes, and attached mass and its location are investigated in detail for each considered problem. The relationship between the frequencies and mode shapes of the sensor and the attached zeptogramme masses are obtained. The sensing devices for biological objects including viruses and bacteria can be elaborated based on the developed sensitivity and frequency shift methodological approach.

A study on dynamic response of functionally graded sandwich beams under different dynamic loadings

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends.

Characterizing buckling behavior of matrix-cracked hybrid plates containing CNTR-FG layers

In this paper, the effect of matrix cracks on the buckling of a hybrid laminated plate is investigated. The plate is composed of carbon nanotube reinforced functionally graded (CNTR-FG) layers and conventional fiber reinforced composite (FRC) layers. Different distributions of single walled carbon nanotubes (SWCNTs) through the thickness of layers are considered. The cracks are modeled as aligned slit cracks across the ply thickness and transverse to the laminate plane, and the distribution of cracks is assumed statistically homogeneous corresponding to an average crack density. The first-order shear deformation theory (FSDT) is employed to incorporate the effects of rotary inertia and transverse shear deformation, and the meshless kp-Ritz method is used to obtain the buckling solutions. Detailed parametric studies are conducted to investigate the effects of matrix crack density, CNTs distributions, CNT volume fraction, plate aspect ratio and plate length-to-thickness ratio, boundary conditions and number of layers on buckling behaviors of hybrid laminated plates containing CNTR-FG layers.

Dynamic properties and damping predictions for laminated plates: High order theories – Timoshenko beam

The main aim of this study is to predict the elastic and damping properties of composite laminated plates. This problem has an exact elasticity solution for simple uniform bending and transverse loading conditions. This paper presents a new stress analysis method for the accurate determination of the detailed stress distributions in laminated plates subjected to cylindrical bending. Some approximate methods for the stress state predictions for laminated plates are presented here. The present method is adaptive and does not rely on strong assumptions about the model of the plate. The theoretical model described here incorporates deformations of each sheet of the lamina, which account for the effects of transverse shear deformation, transverse normal strain-stress and nonlinear variation of displacements with respect to the thickness coordinate. Predictions of the dynamic and damping values of laminated plates for various geometrical, mechanical and fastening properties are presented. Comparison with the Timoshenko beam theory is systematically made for analytical and approximation variants.

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.

Parametric Instability of Static Shafts-Disk System Using Finite Element Method

Parametric instability condition is an important consideration in design process as it can cause failure in machine elements. In this study, parametric instability behaviour was studied for a simple shaft and disk system that was subjected to axial load under pinned-pinned boundary condition. The shaft was modelled based on the Nelson’s beam model, which considered translational and rotary inertias, transverse shear deformation and torsional effect. The Floquet’s method was used to estimate the solution for Mathieu equation. Finite element codes were developed using MATLAB to establish the instability chart. The effect of additional disk mass on the stability chart was investigated for pinned-pinned boundary conditions. Numerical results and illustrative examples are given. It is found that the additional disk mass decreases the instability region during static condition. The location of the disk as well has significant effect on the instability region of the shaft.

Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading

Abstract The stochastic dynamic stability analysis of laminated composite curved panels under non-uniform partial edge loading is studied using finite element analysis. The system input parameters are randomized to ascertain the stochastic first buckling load and zone of resonance. Considering the effects of transverse shear deformation and rotary inertia, first order shear deformation theory is used to model the composite doubly curved shells. The stochasticity is introduced in Love's and Donnell's theory considering dynamic and shear deformable theory according to the Sander's first approximation by tracers for doubly curved laminated shells. The moving least square method is employed as a surrogate of the actual finite element model to reduce the computational cost. The results are compared with those available in the literature. Statistical results are presented to show the effects of radius of curvatures, material properties, fibre parameters, and non-uniform load parameters on the stability boundaries.

Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements

In the paper, the dynamic stiffness method based on the HSDT and FSDT plate theories is applied to study free vibration characteristics of composite stiffened and cracked plate assemblies. The dynamic stiffness matrices for transverse vibration of laminated composite plate based on the HSDT and FSDT previously derived by authors have been coupled with the dynamic stiffness matrix of laminated composite plate undergoing in-plane vibration. The numerical analysis has been carried out through several illustrative examples in order to check the applicability and accuracy of the proposed method in the free vibration analysis of stiffened plate assemblies. The effects of transverse shear deformation, boundary conditions, side-to-thickness ratios, crack length or reinforcement amount within the GFRP beam on the free vibration characteristics have been discussed, and the variety of new results has been provided as a benchmark for future investigations.

Nonlocal large-amplitude vibration of embedded higher-order shear deformable multiferroic composite rectangular nanoplates with different edge conditions

Based on the nonlocal elasticity theory, a unified nonlocal, nonlinear, higher-order shear deformable nanoplate model is developed to investigate the size-dependent, large-amplitude, nonlinear vibration of multiferroic composite rectangular nanoplates with different boundary conditions resting on an elastic foundation. By considering a unified displacement vector and using von Kármán’s strain tensor, the strain–displacement components are obtained. Using coupled nonlocal constitutive relations, the coupled ferroelastic, ferroelectric, ferromagnetic, and thermal properties of multiferroic composite materials and small-scale effect are taken into account. The electric and magnetic potential distributions in the nanoplate are calculated via Maxwell’s electromagnetic equations. Furthermore, Hamilton’s principle is utilized to obtain the mathematical formulation associated with the coupled governing equations of motions and boundary conditions. The developed model enables us to consider the effects of rotary inertia and transverse shear deformation without using any shear correction factor. Also, it can be degenerated to the models based on the Kirchhoff and existing shear deformation plate theories. To solve the large-amplitude vibration problem, an efficient multistep numerical solution approach is utilized. Effects of various important parameters such as the type of the plate theory, and parameters of nonlocality and coupled fields on the nonlinear frequency response are investigated.

Buckling of moderately thick arbitrarily shaped plates with intermediate point supports using a simple hp-cloud method

This paper presents the simple hp-cloud meshless method for the stability analysis of moderately thick plates with various shapes subjected to uniaxial and biaxial in-plane compressive and pure shear loads. To allow the effect of transverse shear deformation on the critical buckling load of the plate, the Mindlin's plate theory is employed. Simple hp-cloud method is applied for constructing the cloud shape functions and for discretization of the domain. Shepard functions are also used for the partition of unity and complete polynomials are utilized for the enrichment functions part. The simple hp-cloud method has Kronecker delta property, so contrary to most of meshless methods the essential boundary conditions can be imposed directly. Constructing the stiffness and geometry matrices leads to an eigenvalue problem that should be solved to determine the critical buckling load of the plate. Numerical results are verified against the results reported elsewhere to illustrate the accuracy and effectiveness of the present method. To show the applications of simple hp-cloud method in the buckling analysis of moderately thick plates, local buckling coefficients of various shapes of plates, rectangular, skew, trapezoidal, triangular, circular, semi-circular, hexagonal and general shape with different boundary conditions are determined. Also, local buckling coefficients of plates with point supports and intermediate support are calculated.

Analytical solution for buckling of Mindlin plates subjected to arbitrary boundary conditions

The study investigates the buckling behavior of isotropic plates subjected to axial, biaxial and pure shear loads. The effect of transverse shear deformation is taken into account by adopting the Mindlin first order shear theory. By applying the extended Kantorovich method, an exact solution is presented without any approximation on the boundary conditions. The procedure is proposed for thin, moderately thick and thick isotropic plates. The obtained results are in good agreement with those available in literature and they demonstrate the accuracy of the proposed procedure.

Size-dependent bending, buckling and vibration of higher-order shear deformable magneto-electro-thermo-elastic rectangular nanoplates

The aim of the present study is to propose a unified size-dependent higher-order shear deformable plate model for magneto-electro-thermo-elastic (METE) rectangular nanoplates by adopting the nonlocal elasticity theory to capture the size effect, and by utilizing a generalized shape function to consider the effects of transverse shear deformation and rotary inertia. By considering various shape functions, the proposed plate model can be reduced to the nonlocal plate model based upon the Kirchhoff, Mindlin and Reddy plate theories, as well as the parabolic, trigonometric, hyperbolic and exponential shear deformation plate theories. The governing equations of motion and corresponding boundary conditions of METE nanoplates subjected to external in-plane, transverse loads as well as magnetic, electric and thermal loadings, are obtained using Hamilton’s principle. Then, as in some case studies, the static bending, buckling, and free vibration characteristics of simply-supported METE rectangular nanoplates are investigated based upon the Navier solution approach. Numerical results are provided in order to investigate the influences of various parameters including the nondimensional nonlocal parameter, type of transverse loading, temperature change, applied voltage, and external magnetic potential on the mechanical behaviors of METE nanoplates. Furthermore, comparisons are made between the results predicted by different nonlocal plate models by utilizing the developed unified nonlocal plate model and selecting the associated shape functions. It is illustrated that by using the presented unified nonlocal plate model, the development of a nonlocal plate model based upon any existing higher-order shear deformable plate theory is a simple task.

Investigation of the Free Vibrations of Composite Anisogrid Lattice Conical Shells Formed by Geodesically Spiral and Circumferential Ribs

This paper focuses on the dynamic behavior of composite anisogrid lattice conical shells. Lattice composite conical shell consists of composite helical and circumferential ribs and thin outer skin. The free vibration analysis of anisogrid composite lattice conical shell is presented. A smeared method is employed to calculate the variable coefficients of stiffness of conical shell and more close to the realistic applications. The lattice part of conical shell is modeled as a beam, so in addition to the axial loads, ribs endure shear loads and bending moments. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations and rotary inertia. The current results are verified with 3D finite element model of conical shell by ANSYS Software and those reported in the literature. Some special cases as influences of geometric parameters of lattice part of shell, effects of boundary conditions and circumferential wave number on natural frequencies of the shell are discussed. It was concluded that employment of the smear method could be recommended for determining the coefficients of stiffness of the composite lattice conical shells with outer skin. Also increasing the vertex angle of cone increases the natural frequencies of conical shell.

Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model

In this paper, the critical buckling temperature of single-walled Boron Nitride nanotube (SWBNNT) is estimated using a new nonlocal first-order shear deformation beam theory. The present model is capable of capturing both small scale effect and transverse shear deformation effects of SWBNNT and is based on assumption that the inplane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. Results indicate the importance of the small scale effects in the thermal buckling analysis of Boron Nitride nanotube.