First-order Shear Deformation Theory Research Articles

Behaviour of Shear Deformable Plate Bending Finite Elements for Composites

A thorough study of the behaviour of shear deformable plate bending finite elements is carried out to investigate their performance when applied to static and free vibration analyses of Iaminated composite plates. 4-noded, 8-noded and 9-noded quadrilateral isoparametricfinite elements based on first-order shear deformation theory and a 4-noded element based on higher-order shear deformation theory are considered. The numerical results indicating the effect of order of integration on the accuracy of the results and convergence of transverse displacements and stresses are presented. The resuLts indicate that a 4-noded element with seven degrees offreedom per node based on higher-order shear deformation theory is required to predict the deflection as well as the stresses accurately and a I 6x I 6 mesh division for full plate is necessary for obtaining converged resuLts. For the case offree vibration analysis, the element based onfirst-order shear deformation theory is found to be sufficient.

The Effect of Cracks on the Free Vibration of a Plate with Parabolic Thickness

This paper uses first-order shear deformation theory and the finite element method to analyze the vibrations of rectangular plates with one or more cracks. The study investigated the influence of cracks (length, angle of inclination), the number of cracks, and the ratio of plate thickness to the natural vibration frequency of the plate, using phase field simulation. Plate thickness varies nonlinearly with the parabolic function. The results of the proposed method were compared with reputable studies to verify its reliability. In addition, some pictures of the characteristic vibration patterns of the plate with varying thickness are presented when cracks appear.

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.

Static Behaviour of Laminated Composite Shells Containing Piezoelectric Actuators and Sensors

The behavior of laminated composite shells with piezoelectric actuators and sensors subjected to external load and electric potential is investigated. The analysis is carried out using the finite element method based on first-order shear deformation theory. An eight nodded degenerated isoparametric shell element with five degrees of freedom at each node is considered. Results are presented for the variation of deflection and stresses in the simply supported laminated composite cylindrical shell panel subjected uniform normal pressure and electric potential.

Thermal Postbuckling Characteristics Of Angle-ply Laminated Truncated Circular Conical Shells

Here, the nonlinear thermoelastic pre- and post-buckling characteristics of angle-ply laminated composite conical shells subjected to uniform temperature rise are studied using semi-analytical finite element approach. The finite element formulation is based on first-order shear deformation theory and field consistency principle. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton-Raphson iteration procedure coupled with displace- ment control method to trace the prebuckling followed by postbuckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, ply-angles, number of layers and number of circumferential waves on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the deformation shape analysis.

On the Vibrations and Thermal Buckling of Functionally Graded Spherical Caps

Here, the axisymmetric free flexural vibrations and thermal buckling characteristics of functionally graded spherical caps are investigated employing a three-noded axisymmetric curved shell element based on field consistency approach. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents of the material. The effective material properties are evaluated using homogenization method. A detailed numerical study is carried out to bring out the effects of shell geometries, power law index of functional graded material and base radius-to-thickness on the vibrations and buckling characteristics of spherical shells.

Finite Element-Based Free Vibration and Deflection Analysis of Bio-Composites Hybrid Laminated Skew Plates Using Experimental Elastic Properties

In this study, the frequency and deflection response analysis of basalt, soy, flax and hybrid pineapple and flax hybrid laminated skew plates are done. The conventional hand-layup method is used to fabricate the different bio-composites-based laminated structures. The uniaxial tension test is conducted in the INSTRON 5967 UTM to determine the elastic properties of the bio-composite laminated structure as per ISO 527-5 standards. The hybrid laminate skew plate is modeled using the first-order shear deformation theory (FSDT) and their elastic properties are obtained from the uniaxial tensile test from which the frequency and deflection of the skew laminated plates are calculated. The eight-noded shell elements (S8R5) with each node having five degrees-of-freedom are deputed in the commercial finite element method (FEM) package ABAQUS to beget the numerical solutions. The damping effect is neglected in the analysis. The accuracy of the proposed model is verified with available literature and found to be in good agreement. Later, the effect of different parameters such as boundary conditions, aspect ratio and skew angles on the natural frequency and deflection of the bio-composites hybrid laminated skew plates are investigated. The obtained results inferred the natural frequencies are decreasing with an increasing aspect ratio of skew plates. The skew angles help to increase the natural frequencies of skew plates. The central deflections are lower for high-skew angled laminated plates.

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.

Vibration Control of Composite Thick Shells Using Magnetostrictive Material and Higher Order Shear Deformation Theory

Analytical solutions of laminated composite shells with embedded actuating layers are presented in this study. The actuating layers are used to control natural vibration of laminated composite shell panels. The first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) for shells are used to represent the shell kinematics and equations of motion. The exact solution for symmetric laminated shells with embedded actuating layers under simply supported boundary conditions is obtained using the Navier solution procedure. Negative velocity feedback control is used. The parametric effect of the position of the magnetostrictive layers, material properties and control parameters on the vibration suppression are investigated in detail. It is found that (i) the shortest vibration suppression time is achieved by placing the actuating layers farthest from the neutral axis (ii) using thinner smart material layers leads to better vibration attenuation characteristics and, (iii) the vibration suppression time is longer for a lower value of the feedback control coefficient. For thicker shells HSDT predicts larger amplitude of vibration and longer vibration suppression time as compared to FSDT predictions.

Effects of Shear Deformation on Flexural Response of Laminated Composite I-Beam

In this paper, a computational approach based on a kinematic model for laminated composite I-section beam is presented. The flexural behavior of laminated composite I-beams is analyzed by using Classical Lamination Plate Theory (CLPT) and the First Order Shear Deformation Theory (FSDT). Based on these theories, a computer programme for analysis of laminated I-beam has been developed in MATLAB. The comparison of results obtained using both theories has been made. It is shown that the shear deformation has significant effect on the deflection of the beam. Furthermore, the laminated I-beam is also modeled using ABAQUS software package for numerical results using different value of load applied at mid-span of the beam. Results obtained using ABAQUS are compared with results obtained with self developed computer program in MATLAB. It is observed that there is good correlation between results obtained from ABAQUS and the computer program before local failure such as buckling of flanges and web occur. Finally, the effect of various parameters such as width-tothickness ratio of flanges, depth-to-thickness of web, and span length-to-depth ratio of I-beam has been investigated. It is concluded that the effect of web depth-to-thickness ratio has little effect on the overall flexural response of I-beam while the width-to-thickness ratio of flanges has significant effect on the flexural response of composite I-beams.

Transient response of delaminated composite curved beams subjected to a moving force

In this study, transient response of the delaminated composite curved beam (DCCB) under radial moving force, traveling with the constant velocity along the beam, has been investigated. First order shear deformation theory (FSDT) is used, and general boundary conditions have been considered by employing artificial springs. The finite element method (FEM) is employed to solve these equations. In order to simulate the mechanical behavior of the delaminated region, dynamic contact between nodes has been applied on each pair of adjacent nodes, examining two critical issues: relative displacement between adjacent nodes, and the type of reaction force between them. Results for the free and forced vibration of the beam have been verified against those available in another literature and 3D model in ANSYS. The effects of different parameters such as the delamination's length and location, fibers orientation, and material properties on the transient response of the DCCBs have been investigated. It has been concluded that the critical point of the beam, which has the greatest radial deformation, does not necessarily exist at the mid-point of the beam, and has different locations for different parameters. Also, the role of the dynamic contact is essential, particularly when delamination is in the lower layers.

Nonlinear transient responses of lattice core sandwich plates under a sudden heat flux

In this study, we investigated the nonlinear thermally induced transient response of lattice core sandwich plates under the sudden application of heat flux for the first time. First, the equivalent thermal conductivity of the lattice core sandwich plates is obtained by considering the thermal conduction within the trusses and heat radiation in the cells. The one-dimensional transient heat transfer equation was implemented and solved using the central finite difference method to calculate the transient temperature field along the direction of the thickness of the plate. The displacement fields of the facesheets and lattice core layer were modeled using the Kirchhoff plate theory and the first-order shear deformation theory, respectively. Based on the Lagrange method and the assumed modal method, the governing equations, with mechanical-thermal coupling, for the simply supported lattice core sandwich plates under sudden heat flux are derived. Through a series of examples, the influences of the different lattice core cell types, thermal boundary conditions, truss radii, and cavity heat radiation on the nonlinear thermally induced transient responses of the sandwich plates are explored.

Dynamic analysis of cracked plates based on a first-order shear deformation theory formulation by using an extended meshfree method

An extended meshfree method is employed in this paper for investigating the dynamic behaviour of cracked plates based on the first-order shear deformation theory (FSDT). The FSDT is a straightforward formulation with the assumption of first-order shear deformation as its name implies, which is appropriate for relatively thick plates. In this study, the meshfree method is chosen as an alternative to the conventional mesh-based methods to model plate structures. Among various meshfree formulations, Moving Kriging (MK) is a method that satisfies the Kronecker delta property, allowing for the easy imposition of essential boundary conditions. An extended MK formulation is proposed in this paper to model cracked plates without explicitly pre-defining the crack in the geometry domain. In the extended concept, the extrinsic enriched functions are employed to model the discontinuity due to the crack. Particularly, the Heaviside step function is employed to describe the discontinuity of the displacement fields on two sides of the crack surface. And the asymptotic enriched functions are used for stress singularity around the crack tip. In the dynamic analysis of cracked plates, one of the important factors that must be evaluated is the dynamic stress resultant intensity factor (DSRIF). In this paper, the DSRIFs are shown through many numerical examples and compared with analytical solutions and other numerical methods, showing the accuracy and efficiency of the present extended MK approach.

Analysis of Moderately Thick Functionally Graded Plates Subjected to Transverse Loads

In the present work, flexural analysis of a thin to moderately thick FGM plate subjected to transverse loads have been studied using finite element method. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. FGM’s are typically heterogeneous in nature and a homogenization scheme is generally adopted for the analysis. To ascertain the effect of homogenization schemes on the material properties, the variation of volume fraction through the thickness have been computed using two different material homogenization techniques; namely the Rule of Mixtures and Mori-Tanaka scheme. In calculating the effective material properties through the thickness numerical integration have been used to evaluate the integrals which is easier and faster when compared to symbolic integration. A detailed discussion comparing the results from both the homogenization schemes have been presented. In addition to that a detailed parametric study bringing out the effect of boundary conditions, loading intensities and volume fraction index have been presented. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present formulation.

On the poroelastic vibrations of lightweight FGSP doubly-curved shells integrated with GNPs-reinforced composite coatings in thermal atmospheres

Porous and nanocomposite materials are widely used in various engineering structures due to their exceptional thermos-mechanical properties. In this work, the poro-thermoelastic frequency analysis of three-layer functionally graded saturated porous (FGSP) lightweight shells covered with two graphene nanoplatelets (GNPs)-reinforced composite surfaces is explored. Both the volume fraction and the diffusion types of GNPs reinforcement in the patches are investigated and moreover, the effect of pressure exerted by the fluids in the openings is considered and the different pore arrangements are compared. Based on the first-order shear deformation theory (FSDT) combined with Hamilton's principle, the equations of motion are derived and then solved analytically. After confirming the accuracy of the results, the frequencies are presented for different types of shells, including cylindrical, spherical, and doubly-curved shells. These types of shells are widely used in various industries, so the results of this work can be useful for these industries. In addition, this is the first comprehensive frequency analysis for different types of shells, and the results can serve as a benchmark for future studies.

Dynamic response of the hybrid honeycomb sandwich panel with zero Poisson’s ratio under low-speed impact

The dynamic response of the hybrid honeycomb sandwich panel under low-speed impact is studied. According to Hamilton’s principle and first-order shear deformation theory, the motion equation of the hybrid honeycomb sandwich panel is deduced. Then the Navier method and Duhamel’s integral are used to solve the vibration displacement of panels. Besides, the mass-spring (MS) model is used to calculate the contact force between the honeycomb sandwich panel and the impactor. The results of this model are compared with those of Abaqus and published studies, which verifies the feasibility of the theoretical model in this paper. Based on the developed theoretical model, the influences of the unit cell angle, cell wall length, cell wall thickness, core layer height and impact speed on the dynamic response of sandwich panels have been studied. The results show that under the same low-speed impact, the maximum lateral displacement of hybrid sandwich panels was 11.65% smaller than that of conventional sandwich panels when honeycomb parameter θ = 60°, and was 15.45% when honeycomb parameter The energy absorption character of the hybrid honeycomb sandwich panel are better than those of the traditional hexagonal honeycomb sandwich panel and the concave hexagonal honeycomb sandwich panel.

Multi-field coupling and free vibration of a sandwiched functionally-graded piezoelectric semiconductor plate

Sandwiched functionally-graded piezoelectric semiconductor (FGPS) plates possess high strength and excellent piezoelectric and semiconductor properties, and have significant potential applications in micro-electro-mechanical systems. The multi-field coupling and free vibration of a sandwiched FGPS plate are studied, and the governing equation and natural frequency are derived with the consideration of electron movement. The material properties in the functionally-graded layers are assumed to vary smoothly, and the first-order shear deformation theory is introduced to derive the multi-field coupling in the plate. The total strain energy of the plate is obtained, and the governing equations are presented by using Hamilton’s principle. By introducing the boundary conditions, the coupling physical fields are solved. In numerical examples, the natural frequencies of sandwiched FGPS plates under different geometrical and physical parameters are discussed. It is found that the initial electron density can be used to modulate the natural frequencies and vibrational displacement of sandwiched FGPS plates in the case of nano-size. The effects of the material properties of FGPS layers on the natural frequencies are also examined in detail.

Random vibration analysis of FGM plates subjected to moving load using a refined stochastic finite element method

The article introduces a finite element procedure using the bilinear quadrilateral element or four-node rectangular element (namely Q4 element) based on a refined first-order shear deformation theory (r-FSDT) and Monte Carlo simulation (MCS), so-called refined stochastic finite element method to investigate the random vibration of functionally graded material (FGM) plates subjected to the moving load. The advantage of the proposed method is to use r-FSDT to improve the accuracy of classical FSDT, satisfy the stress-free condition at the plate boundaries, and combine with MCS to analyze the vibration of the FGM plate when the parameter inputs are random quantities following a normal distribution. The obtained results show that the distribution characteristics of the vibration response of the FGM plate depend on the standard deviation of the input parameters and the velocity of the moving load. Furthermore, the numerical results in this study are expected to contribute to improving the understanding of FGM plates subjected to moving loads with uncertain input parameters.

Electro-mechanical vibro-acoustic characteristics of submerged functionally graded piezoelectric plates with general boundary conditions

The present paper focuses on studying the underwater electro-mechanical vibro-acoustic characteristics of functionally graded piezoelectric material (FGPM) plates with general boundary conditions. For this purpose, the collocation points method is firstly developed to handle vibro-acoustic coupling, which can directly calculate the virtual work of secondary sound pressure via an easy double integral, without pre-simplifying the complex quadruple integral. The proposed FGPM plate model is established in the framework of first-order shear deformation theory (FSDT) and second type constitutive equations of piezoelectric materials, and the macroscopically inhomogeneous electro-mechanical properties are considered by means of Voigt model. The multi-physical fields including electric, mechanical and acoustic fields are uniformly extended in Chebyshev polynomials. Meanwhile, penalty function method is introduced to satisfy the arbitrary mechanical and electric constraints. All energy formulations are processed by Rayleigh-Ritz procedure and a standard governing equation is finally derived. Based on the proposed unified model, the convergence and accuracy for solving vibration and acoustic radiation of FGPM plates are validated firstly. Additionally, the influence of some key parameters on underwater vibro-acoustic characteristics of FGPM plates is fully examined. The results of this paper may provide a reference for future application of FGPM-based smart structures.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.