Variation Of In-plane Displacements Research Articles

Thermomechanical loading and cut-out effect on static and dynamic responses of multilayered structure with TD properties

The influences of cut-out on the static and dynamic deflection of the multilayered composites under thermomechanical are predicted numerically. In this regard, a mathematical model is prepared using the third-order kinematics associated with the isoparametric finite element steps. Moreover, the solution accuracy of the derived model has been tested frequently by solving a series of examples (obtained from MATLAB code) similar to the published one. In addition, the model verification is extended further by comparing the numerical results with the in-house experimental data recorded under the static and dynamic loading conditions. The equivalent single-layer theoretical model (prepared for the cut-out abided laminated structure) exactness is maintained by considering the variation of in-plane displacement as cubic. In contrast, the state space variable along with the thickness is constant. Additionally, the experimental comparisons are made considering the influences of different cut-out geometry, temperature increment, and loading intensities. The effects of cut-out and the laminated structural geometrical parameters that affect the final design consideration due to the thermomechanical loading as well as the temperature-dependent properties are computed using the derived model, and the outcomes are discussed in detail.

Influence of initial geometrical imperfections on thermal stability of laminated composite plates using layerwise finite element

In this paper, the influence of initial geometrical imperfections on thermal stability of laminated composite plates using Layer wise plate model is presented. The mathematical model assumes layer wise variation of in-plane displacements, non-linear strain-displacement relations (in von Karman sense) and linear thermo mechanical material properties. The Koiter’s model for initial geometrical imperfection is adopted. Principle of virtual displacements (PVD) is used to derive Euler-Lagrange differential equations of linearized buckling problem. The weak form is discretized using nine-node Lagrangian isoparametric finite element. The original MATLAB program is coded for finite element solution. The effects of imperfection mode and amplitude, temperature distribution, side to thickness ratio b/h, aspect ratio b/a, boundary conditions and lamination scheme on critical buckling temperature are analysed. The accuracy of the numerical model is verified by comparison with the available results from the literature and new results are presented.

Dynamic response of functionally graded plates with a porous middle layer under time-dependent load

A dynamic analytical solution for a simply supported, rectangular functionally graded plate with a porous middle layer under time-dependent load based on a refined third-order shear deformation theory with a cubic variation of in-plane displacements according to the thickness and linear/quadratic transverse displacement is presented. The solution achieved in the trigonometric series form and rests on the Green's function method. Two porosity types and their influence on material properties, and mechanical behavior are considered. The network of pores is assumed to be empty or filled with low-pressure air, and the material properties are calculated using the power-law distribution idealization. Numerical calculations have been carried out to demonstrate the accuracy of the kinematic model for the dynamic problem, the effect of porosity, thickness of porous layers, power-law index, and type of loading on the dynamic response of an imperfect functionally graded material plate.

Buckling of laminated composite skew plate using FEM and machine learning methods

PurposeThe purpose of this paper is to attempt the buckling analysis of a laminated composite skew plate using the C0 finite element (FE) model based on higher-order shear deformation theory (HSDT) in conjunction with minimax probability machine regression (MPMR) and multivariate adaptive regression spline (MARS).Design/methodology/approachHSDT considers the third-order variation of in-plane displacements which eliminates the use of shear correction factor owing to realistic parabolic transverse shear stresses across the thickness coordinate. At the top and bottom of the plate, zero transverse shear stress condition is imposed. C0 FE model based on HSDT is developed and coded in formula translation (FORTRAN). FE model is validated and found efficient to create new results. MPMR and MARS models are coded in MATLAB. Using skew angle (α), stacking sequence (Ai) and buckling strength (Y) as input parameters, a regression problem is formulated using MPMR and MARS to predict the buckling strength of laminated composite skew plates.FindingsThe results of the MPMR and MARS models are in good agreement with the FE model result. MPMR is a better tool than MARS to analyze the buckling problem.Research limitations/implicationsThe present work considers the linear behavior of the laminated composite skew plate.Originality/valueTo the authors’ best of knowledge, there is no work in the literature on the buckling analysis of a laminated composite skew plate using C0 FE formulation based on third-order shear deformation theory in conjunction with MPMR and MARS. These machine-learning techniques increase efficiency, reduce the computational time and reduce the cost of analysis. Further, an equation is generated with the MARS model via which the buckling strength of the laminated composite skew plate can be predicted with ease and simplicity.

Isogeometric static analysis of laminated plates with curvilinear fibers based on Refined Zigzag Theory

In this study, we propose a new IsoGeometric formulation based on Refined Zigzag Theory (RZT), abbreviated as IG-RZT, for static analysis of laminated plates and sandwich panels with curvilinear fiber paths for the first time in literature. The original RZT formulation defines constant, linear, and zigzag deformation contributions of thickness coordinate to the in-plane displacements. This kinematic relation can accurately predict in-plane displacement of composite with straight fibers. However, estimating a realistic variation of in-plane displacements in a variable angle tow (VAT) composite is a more challenging problem as compared to the straight fiber composites. This difficulty has been addressed herein by including a quartic (fourth-order) polynomial thickness expansion that includes the transverse normal strain effects to the kinematic displacement fields of RZT. Moreover, the modelling of VAT composites results in the RZT zigzag functions to depend not only on the thickness coordinate but also on the in-plane positions. The present IG-RZT methodology is free of shear-locking and shear correction factors due to the integration of RZT with Non-Uniform Rational B-Splines (NURBS) functions of isogeometric analysis. The use of NURBS functions also enable the exact geometry data to be taken directly from a computer aided design (CAD) software, e.g., Rhinoceros, into an in-house Mathematica code. The accuracy and efficiency of the present IG-RZT formulation is assessed and validated by solving various examples of curvilinear fiber laminated plates with different aspect and span-to-thickness ratios. Comparison of IG-RZT results with reference solutions available in literature and generated by a commercial software (ANSYS) using 3D finite elements have demonstrated the remarkable benefits of the proposed IG-RZT method for predicting highly accurate both displacement and stress distributions of VAT composite structures.

Static analysis of functionally graded plates with a porous middle layer based on higher order shear deformation theory with linear/quadratic transverse displacement

In this article, we present an analytical solution for an imperfect functionally graded plate based on higher order shear deformation theory with cubic variation of in-plane displacements according to thickness and linear/quadratic transverse displacement. The developed solution is used to analyze the static responses of a plate with a porous layer to mechanical loading. Two porosity types and their influences on material properties, displacement, and stress behaviors are considered. The network of pores is assumed to be empty or filled with low-pressure air, and the material properties are calculated using power-law distribution idealization in terms of the volume fractions of constituents. The computed results are presented to illustrate the accuracy of the proposed solutions based on comparisons to previously reported analytical and numerical solutions in the literature. We also analyzed the effects of different volume fractions and thicknesses of porous layers on the mechanical loading and mechanical behavior of the imperfect functionally graded plate.

Stress analysis of composite plate with cutout of various shape

Composite plate found numerous applications in various structures due to its superior properties like high stiffness to weight and strength to weight ratio. Generally, cutouts in these structural elements are unavoidable for many purposes like access to internal parts, ventilation and sometimes to prevent resonance. These geometrical discontinuities in the form of cutouts may be of different shapes and arise nonuniform stress in the plate. Therefore, it becomes necessary to analyse stress concentration factor near cutouts and effect of shape of cutouts on axial deformation and stress distribution for better safety and stability of structure. In the present investigation a Carbon/epoxy composite plate with various shape of cutout subjected to in-plane tensile loading is considered. Finite element based plane stress analysis is used for mathematical modeling. A program for the present analysis is developed in MATLAB. Variation of in-plane displacement and stress concentration factor (SCF) with various shape of cutout is determined. The effect of the geometry of hole on the stress distribution around the hole is also studied. The results of various analyses have been presented with the help of tables, graphs, and nodal plots.

Cell- and Node-Based Smoothing MITC3-Finite Elements for Static and Free Vibration Analysis of Laminated Composite and Functionally Graded Plates

Cell- and node-based smoothing MITC3-finite elements for static and free vibration analysis of laminated composite and functionally graded plates are proposed in this paper. It is based on a higher-order shear deformation theory which accounts for a higher-order variation of in-plane displacements and traction-free boundary conditions on top and bottom surfaces of transverse shear stresses. Novel CS-MITC3 and NS-MITC3 finite elements are proposed in which an MITC3 three-node triangle element with seven degrees-of-freedom per nodes that only requires the [Formula: see text]-type continuity is developed, and then cell- and node-based smoothing techniques are applied. Numerical results for the laminated composite and functionally graded plates are proposed to validate the accuracy of developed theory and to investigate the effects of material distribution and side-to-thickness ratio on the deflection, stresses and natural frequencies of the plates.

Dynamic response of functionally graded plates resting on two-parameter-based elastic foundation model using a quasi-3D theory

In this article, an improved quasi 3-D theory, which considers transverse normal strain effect along with inverse tangential variation of in-plane displacements across plate thickness, is employed to encapsulate the response of thick functionally graded plates. The present model satisfies both non-linear variation and zero transverse shear stresses across plate thickness. The governing equations of plate are obtained using Lagrange equation and solved by applying finite element procedure. The adaptability of the proposed theory is handled by considering the impact of various parameters on dynamic response of plate through several illustrations. The present model gives incredible concurrence with accessible literature.

Bending Analysis of Thick Isotropic Plates by Using 5th Order Shear Deformation Theory

A 5th order shear deformation theory considering transverse shear deformation effect as well as transverse normal strain deformation effect is presented for static flexure analysis of simply supported isotropic plate. The assumed displacement field accounts for non-linear variation of in-plane displacements as well as transverse displacement through the plate thickness. The condition of zero transverse shear stresses on the upper and lower surface of plate is satisfied. Hence the present formulation does not require the shear correction factor generally associated with the first order shear deformable theory. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Closed-form analytical solutions for simply supported square isotropic thick plates subjected to single sinusoidal distributed loads are obtained. Numerical results for static flexure analysis include the effects of side to thickness ratio and plate aspect ratio for simply supported isotropic plates. Numerical results are obtained using MATLAB programming. The results of present theory are in close agreement with those of higher order shear deformation theories and exact 3D elasticity solutions.

Delamination of laminated composite plates by p-convergent partial discrete-layer elements with VCCT

The simulation of the delamination process in laminated composite plates is quite complex and requires advanced finite element modeling techniques. Failure analysis tools must be able to predict initiation, size and propagation of delamination process. This paper presents the p-convergent partial discrete-layer elements with the virtual crack closure technique (VCCT) for the delamination analysis of laminated composite plates. The proposed element can be formulated by the suitable dimensional reduction from three-dimensional solid to two-dimensional plate. It is assumed that the piecewise linear variation of in-plane displacements and the constant value of out-of-plane displacements across the thickness. The higher-order approximation based on integrals of Legendre polynomials is used to define displacement fields. The three-dimensional VCCT is also slightly modified to incorporate with the proposed elements to estimate the energy release rate. The initiation of delamination occurs when the energy release rate for a displacement increment is same as the critical energy release rate corresponding to fracture toughness. The approach is to use a fracture mechanics criterion, but to avoid the complex moving mesh technique. At first, the validation and characteristic of the proposed elements are investigated on isotropic plates and orthotropic laminated plates, compared with referenced values. Then for fracture analysis, the efficiency of proposed approach is demonstrated with the help of additionally two problems such as the double-cantilever-beam test and the orthotropic laminated square plate with interior delamination.

Layerwise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations

Layerwise plate theory of Reddy, extended for the analysis of delaminations, has served as a basis for development of enriched finite elements. The proposed model assumes layerwise linear variation of in-plane displacements and constant transverse displacement through the plate thickness. Jump discontinuities in displacement field in three orthogonal directions are incorporated using Heaviside step functions, depending on delamination position through the plate thickness. Equations of motion are derived using Hamilton’s principle. Using the proposed model laminated composite and sandwich plates were analyzed. All numerical solutions are obtained using originally coded MATLAB programs. Proposed model is verified using existing results from the literature. Results for natural frequencies, mode shapes and critical buckling loads for intact and damaged plates are compared. Effects of plate geometry, lamination scheme, degree of orthotropy and delamination size or position on dynamic characteristics of the plate are presented. Excellent agreement is obtained and a family of new results is presented as a benchmark for future investigations.

Free vibration response of functionally graded thick plates with shear and normal deformations effects

Free vibration response of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates is presented based on higher order shear/shear-normal deformations theories (HOSTs/HOSNTs). The theoretical models are based on Taylor’s series expansion of in-plane and transverse displacements in thickness coordinate defining the plate deformations. Some of these advanced models account for the effects of transverse shear deformations, transverse normal deformation and non-linear variation of in-plane displacements through plate’s thickness. Functionally graded materials (FGMs) are idealized as continua with mechanical properties changing smoothly with respect to spatial coordinates. The material properties of FG plates are assumed here to be varying through thickness of plate in a continuous manner. Poisson’s ratios of FG plates are assumed constant, but their Young’s modulii and material densities vary continuously in thickness direction according to the volume fraction of constituents which is modeled as exponential and power law functions. The equations of motion are derived using Hamilton’s principle on the basis of HOSTs/HOSNTs. Numerical solutions are obtained using Navier solution method. The accuracy of numerical solutions is first established through comparison with the exact three dimensional (3D) elasticity solutions and then compared with available other models’ solutions. New solutions are then provided for future use.

Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells

The improved third order zigzag theory and its smeared counterpart (without the zigzag effect), recently developed by the authors for static analysis of piezoelectric laminated cylindrical shells, are extended to dynamics. The piezoelectric layers are considered as radially polarized to make use of the extension actuation mechanism that is best suited for effective actuation and sensing. The zigzag theory accounts for the layerwise variation of inplane displacements and includes the transverse normal extensibility under electric field, and also satisfies the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces of the shell. Yet, the number of primary displacement variables is only five, same as its smeared counterpart. The two theories are critically assessed for their accuracy by direct comparison with the three dimensional piezoelasticity solutions for free and forced vibration response of simply supported smart angle-ply infinite-length and cross-ply finite-length shells, with a variety of heterogeneous composite and sandwich laminates. It is shown that the zigzag theory, in spite of being computationally efficient, is very accurate even for shells with highly inhomogeneous laminates. In contrast, the smeared third order theory is grossly inadequate for smart shells made of inhomogeneous composite and sandwich substrates.

Analysis of Cantilever Plates with Stepped Section Using p-Convergent Transition Element for Solid-to-Shell Connections

This paper presents efficient modeling technique using the p-convergent transition element for analysis of cantilever plates with stepped section. For the efficient modeling, firstly, the high-precision element is formulated by combination of one- and two-dimensional shape functions. Secondly, for another three-dimensional modeling, the reduced-precision element is proposed by assuming a piecewise linear variation of in-plane displacements and a constant value of the lateral displacement across the thickness. Thirdly, two-dimensional shell element with higher-order shape functions is formulated using first-order shear deformation theory. Lastly, the p-convergent transition element is proposed in order to connect three element types formulated. All the elements considered are on the basis of subparametric element concept. The elements are applied with cantilever plates with stepped section subjected to in-plane and transversal loadings. The present models using the p-convergent transition element are demonstrated to be more practical and economical than previously p-version FEM using only single element type.

Assessment of improved zigzag and smeared theories for smart cross-ply composite cylindrical shells including transverse normal extensibility under thermoelectric loading

The improved zigzag theory recently developed by the authors for smart, piezoelectric, and laminated cylindrical shells is assessed for the response of finite-length cross-ply shells and shell panels under mechanical, potential, and thermal loading, in direct comparison with the exact three-dimensional (3D) piezothermoelasticity solution. This theory has the unique features of including the transverse normal strain due to thermoelectric loading without introducing additional deflection variables, capturing the nonlinear potential field and actual temperature profile across laminate thickness, accounting for the layerwise (zigzag) variation of inplane displacements, and satisfying the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces. For the assessment, new results are obtained for the 3D exact solution for smart cylindrical shells having a test laminate with widely different material properties across layers, a piezo-composite laminate and a piezo-sandwich laminate. To ascertain the contribution of the layerwise terms in the inplane displacements, the theory is compared with its smeared counterpart with the same number of primary variables. The effect of inclusion of transverse normal extensibility in these theories is established by comparing with their conventional counterparts that assume constant deflection across the thickness. The effect of span angle (for shell panels), length, and thickness parameters on the error of the 2D theories is illustrated.

Computational Models for the Static and Dynamic Analyses of Laminated Composite Plates

The present study is to assess the accuracy of the few computational models based on various shear deformation theories in predicting the static and dynamic behaviour of antisymmetric angle-ply laminated composite plates. In the present investigation two higher order refined computational models incorporate the laminate deformations which account for the effects of both transverse shear and normal deformation and a non-linear variation of in-plane displacements are used. In addition to above, few higher order models and the first order model developed by other investigators and available in the literature are also considered for the evaluation. The equations of equilibrium are obtained using Principle of Minimum Potential Energy and the equations of motion using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s solution technique. The solutions of the static analysis are obtained by solving the boundary value problem and the natural frequencies are obtained by solving the eigenvalue problem. The accuracy of the solutions obtained from the different models are established by comparing the results with the solutions wherever available in the literature. Results generated independently using the various models are presented for antisymmetric angle ply composite plates which will serve as benchmark solutions for future investigations.

Hierarchic layer models for anisotropic laminated plates

Hierarchic p-refined formulations based on several macroscopic layer models are presented to analyze anisotropic laminated plates. Analysis of composite laminated plates is implemented with full discrete-layer, partial discrete-layer, and equivalent single-layer models, respectively. In the first approach all three displacement components are expressed as the product of one- and two-dimensional interpolation functions for applying the three-dimensional elasticity theory to each layer. Second approach considers thickness-wise variation of in-plane displacement in individual layers and a constant value of out-of-plane displacement across the plate thickness. The third approach assumes that a heterogeneous laminated plate stacked with several laminae is treated as a shell element using hierarchic interpolation functions. The integrals of Legendre polynomials and Gauss-Lobatto technique are adopted to interpolate displacement fields and to implement numerical quadrature, respectively. The validity and characteristics of the proposed numerical layer models are tested on anisotropic multilayered plates and sandwich plates, and compared with the values available in the published literature based on analytical methods and h-refined layer models.

Influence of fixation conditions and material anisotropy on the frequency spectrum of laminated beams

To accurately determine the influence of fixation conditions and material anisotropy of laminated beams on their dynamic properties, a new method for analyzing their stress is used. The theoretical model takes into account the transverse shear and normal deformations of layers and the nonlinear variation of in-plane displacements across the thickness of each layer. The stress-strain state in the region of fixation of a beam and its influence on the bending stiffness of the beam is examined. The frequency spectrum of laminated beams is also analyzed in relation to the anisotropic mechanical properties of layers and the configuration of the fixation unit.

Stress and failure analysis of laminated composites based on layerwise B-spline finite strip method

Abstract A layerwise B k , k −1 -spline finite strip method is developed for stress and failure analysis in fibre reinforced composite laminates. The composite laminates are divided into a number of numerical layers in the thickness direction. A linear variation of in-plane displacements are assumed across the thickness of each numerical layer to represent the sectional warping which can have significant effects on through thickness shear stresses. The development of B k , k −1 -spline finite strip method is presented. Material failure criteria and material degradation models for progressive failure are discussed and integrated in a progressive failure analysis for composite laminates. Numerical applications include a progressive failure analysis example are presented to verify the layerwise B-spline finite strip method developed for stress and failure analysis in composite laminates. Whenever possible the present predictions are compared with the existing analytical and numerical results.