C0 Finite Element Formulation Research Articles

Post-buckling analysis of CNT-reinforced hybrid FG plates using MTSDT

The post-buckling behavior of a hybrid functionally graded (FG) rectangular plate reinforced with carbon nanotubes (CNT) is presented. The mathematical model developed based on transverse displacement variation of second-order in third-order shear deformation theory, hence referred to as modified third-order shear deformation theory (MTSDT). Transverse shear stress is imposed zero on top and bottom surface of plate. 2D C0 finite element (FE) formulation model developed based on MTSDT. Inhouse MATLAB code developed of present FE formulation. The rectangular plates consist of matrix material, fiber, and CNT as reinforcement material. The grading of fiber material in the thickness direction ensured using a power–law distribution. The material characteristics of the hybrid FG plates predicted using the Halpin-Tsai equation and the rule of mixing technique. No shear correction factor is required due to realistic parabolic behavior considered in the present work. The comparison study results of the present model are in good agreement with the published work. After confirming the accuracy of the present model a parametric study was carried outd to check the influence of different parameters.

Free vibration response of cnt-reinforced multiscale functionally graded plates using the modified shear deformation theory

ABSTRACT This article represents the free vibration analysis of multiscale functionally graded plates, reinforced with a carbon nanotube. The modified third-order shear deformation theory with a variation in transverse displacement is employed in the present work. MATLAB code is developed for C0 finite element (FE) formulation. A rectangular nine-noded element containing 117 nodal unknowns at each element was considered. The top and bottom transverse shear stress of the plate is considered zero. No shear correction factor is required in the present model. In this study, the FGM plates are assumed to be reinforced with MWCNT and SWCNT and also graded with fibre material throughout the thickness. The effective elastic properties of the multiscale composite plate are estimated by the combination of the Halpin–Tsai equation and the homogenisation scheme. The effects of various parameters such as the volume fraction of fibre material, a/h ratio, aspect ratio and weight percentage of both types of CNTs on frequency are studied.

Transient dynamic response of doubly curved laminated composite shells under pulse loading

Transient response behavior of laminated doubly curved shells with varying geometry and boundary conditions subjected to various time-dependent pulse loads is investigated using Higher order Shear Deformation Theory (HSDT) in the present study. The ratio of thickness co-ordinate to radius of shell (z/R) is incorporated in the mathematical formulation based on HSDT. The condition of zero transverse shear stresses at free surfaces of laminated shell is also included in the displacement function. A C0 finite element formulation using eight noded isoparametric shell element with seven degrees of freedom per node is applied to evaluate the dynamic response of shells. Three types of pulse loading such as sine, triangular and rectangular pulses are considered for the present investigation. Newmark's β method is applied to solve the dynamic equilibrium equation of the shells. The accuracy of the present analysis is examined by comparing the results obtained with those available in the published literature. Several numerical examples are illustrated to show the effects of pulse loading and boundary condition on the central displacement and stresses of laminated composite shells.

Dynamic response with mass variation of laminated composite twisted plates

In this study, the dynamic response of free vibration of laminated composite twisted (skewed) plate with mass variation via cutout and additional mass is investigated. The mathematical model employs cubic variation of the thickness coordinates in the displacement field. The mathematical formulation also includes the effects of transverse shear and rotatory inertia. A realistic parabolic distribution of transverse shear strains through the plate thickness is assumed and the use of the shear correction factor is avoided. A C0 finite element formulation of the mathematical model is developed to analyze the dynamic behavior of laminated composite twisted plate with cutout and additional mass. Given that no results on the present problem of free vibration of laminated composite twisted plate with cutout and additional mass are available in literature, novel results for different twist angles of laminated composite plates along with mode shapes are presented by varying the geometry, boundary conditions, and ply orientations.

Evaluation of free vibration and bending analysis of laminated composite and sandwich plates using non-polynomial zigzag models: C0 Finite element formulation

The present work deals with prediction of bending and free vibration analysis of laminated composite and sandwich plates using non-polynomial zigzag theories inline with C0 finite element formulation. The assumed mathematical model is the combination of zigzag parameters and non-polynomial shear strain functions with constant variation of transverse displacement. It represents continuous form of out-plane stresses and parabolic variation across the plate thickness. Moreover, improved form of in-plane responses also achieved. The mathematical model is layer independent and consist of less number of unknowns like in the case of first order theory. Hence, the present theory required less computational efforts. Higher mode shapes of frequencies are obtained for the multilayered sandwich plates and composite plates. By ranging the geometrical parameters, loading conditions, material properties and boundary conditions the numerical examples are carried out to establish the superiority of the current zigzag models than the available results.

Failure Analysis of Laminated Composite Skew Laminates

Failure analysis of laminated composite skew laminates subjected to transverse loads is presented by developing a C0 finite element (FE) model based on higher order shear deformation theory (HSDT). In this theory the transverse shear stresses are taken as zero at the laminate top and bottom. A realistic parabolic variation of transverse shear strains through the laminate thickness is assumed and the use of shear correction factor is avoided. The C0 finite element formulation has been done quite efficiently to overcome the problem of C1 continuity associated with the HSDT. The isoparametric FE used in the present model consists of nine nodes with seven nodal unknowns per node. Five different failure criterions have been used for each problem. Since there is no result available in the literature on the problem of skew laminates subjected to transverse loads based on HSDT, present results are validated with few results available on laminates subjected to transverse loads without skew angle. Many new results are presented on the failure load for skew laminates subjected to transverse loads considering different geometry, boundary conditions, ply orientation and skew angles.

Behaviour of sandwich laminates subjected to thermal loading using higher-order zig-zag theory

In the present work, a higher-order zig-zag laminate theory is utilized to investigate the behaviour of sandwich laminates having low-density core materials subjected to thermal loading. The theory satisfies continuity condition of transverse shear stress at the layer interfaces and zero transverse shear stress at the top and bottom surfaces of the laminated plate. The assumed in-plane displacements are having cubic variations through the entire thickness of the plate. On the other hand, the variation of the transverse displacement is taken to be quadratic within the core and constant through the facesheets. In order to reduce the computation effort, an efficient C0 finite element formulation for the above sandwich plate model is adopted. Some examples of laminated composite and sandwich plate with different material properties, core thickness ratios, aspect ratios, boundary conditions, number of layers and ply orientations are considered for the analysis. The efficacy of the present model in predicting various responses is verified by comparing with three-dimensional elasticity solutions along with the available published results.

The (3,2)-Mixed Refined Zigzag Theory for generally laminated beams: Theoretical development and C0 finite element formulation

A third-order Refined Zigzag Theory for multilayered composite and sandwich beams is developed based on the Reissner Mixed Variational Theorem. The assumed kinematics enriches the Timoshenko’s beam Theory: a second- and a third-order smeared polynomial terms along with a piece-wise linear contributions are added to the axial displacement, whereas the transverse displacement is approximated with a power-series expansion up to the second-order term. Transverse shear and normal stress, continuous and able to satisfy the boundary stress conditions on the outer beam surfaces, are assumed by the model: the transverse shear stress is derived with the aid of the Elasticity equations, whereas a third-order power-series expansion is adopted for the transverse normal stress. Based on the proposed model, an efficient C0-continuous beam element is formulated by adopting the anisoparametric interpolation strategy to avoid the shear locking phenomenon. The accuracy of the proposed model and the finite element implementation is assessed by solving problems concerning the elasto-static behavior of generally laminated beams, both simply supported and clamped at the ends. Comparison with reference solution (Elasticity or high-fidelity FE model) demonstrates the remarkable accuracy of the proposed model, both in terms of displacements and stresses distributions, as well as the finite element implementation efficiency.

Efficient C0 Finite Element Formulation for Thermal Stress Analysis of Laminated Composite and Sandwich Plates

AbstractBy introducing thermal expansion coefficients in the transverse displacement field, this paper proposes an approach to improve the performance of the global-local higher-order theory for thermal stress analysis of laminated composite plates. Transverse normal deformation has been considered in the proposed model, whereas the additional displacement variables have not increased because thermal loads could be included in the generalized force vector. The proposed model a priori satisfies the continuity conditions of transverse shear stresses at interfaces, and the number of displacement variables involved in the present model is independent of the number of layers. The derivatives of transverse displacement have been eliminated from the displacement field so that the C0 interpolation functions are only required for the finite-element implementation. Based on the proposed model, a six-node triangular element is proposed for thermal expansion problems of laminated composite and sandwich plates. Numeri...

Efficient Failure Analysis of Laminated Composites and Sandwich Cylindrical Shells Based on Higher-Order Zigzag Theory

AbstractAn efficient failure analysis of laminated composite and sandwich cylindrical shell is done for the first time using an efficient C0 finite-element (FE) model based on higher-order zigzag theory (HOZT). The FE implementation based on HOZT to study the failure of composite and sandwich shells incorporating all three radii of curvature is presented. The proposed two-dimensional (2D) FE model satisfies the interlaminar shear stress continuity at the layer interfaces and also ensures zero transverse shear stress conditions at the shell top and bottom. The problem of C1 continuity associated with the HOZT is circumvented by using an appropriate C0 FE formulation. The piecewise parabolic shear stress variation across thickness of each layer is considered and hence, shear deformation is accurately modeled for composites and sandwich shells. The failure load results for laminated composite and sandwich shells obtained by using the present 2D FE model are quite close to the 3D elasticity results. Many new ...

Buckling of soft-core sandwich plates with angle-ply face sheets by means of a C0 finite element formulation

In order to avoid using C1 interpolation functions in finite element implementation of the previous zig–zag theories, artificial constraints, in which the first derivatives of transverse displacement will be replaced by the assumed variables, are usually employed. However, such assumption will violate continuity conditions of transverse shear stresses at interfaces. Differing from previous work, this paper will propose a C0-type zig–zag theory for buckling analysis of laminated composite and sandwich plates with general configurations. The first derivatives of transverse displacement have been taken out from a displacement field of the proposed zig–zag theory. Thus, the C0 interpolation functions are only required in finite element implementations of the proposed model. Without use of any artificial constraints, an eight-node quadrilateral element based on the proposed model is presented by incorporating the terms associated with the geometric stiffness matrix. In order to verify performance of the proposed model, several buckling problems of sandwich plates with soft core have been analyzed. Numerical results show that the proposed model is able to predict accurately buckling loads of the soft-core sandwich plates with varying fiber orientations of face sheets.

Bending analysis of functionally graded skew sandwich plates with through-the thickness displacement variations

This paper analyzes the bending response of functionally graded skew sandwich plates with graded layer as face sheet/core using higher order shear deformation theory in the frame work of finite element method considering through-the-thickness variation of displacements. The assumed kinematics field incorporates the quadratic variation of thickness co-ordinate in defining the transverse displacement. The C0 finite element formulation has been done efficiently to overcome the problem of C1 continuity associated with the higher order shear deformation theory. For this, a rectangular isoparametic Lagrangian element with nine nodes and 13 nodal unknowns at each node is developed. Two types of plates are modeled: functionally graded skew plates with graded face sheets and graded layer as core part. By considering the symmetry of the plate, different combinations of bottom-core-top thickness has been considered for the numerical study. Wide variety of numerical problems are presented for functionally graded sandwich plates with various skew angles, thickness schemes, aspect ratio, thickness ratio and boundary conditions. It is noticed that isotropic plates exhibit diverge response in comparison with graded plates. Also, power law exponent and skew angle are vital parameters that dictate the response of the plate under bending.

Dynamic response of functionally graded skew shell panel

The dynamic response of functionally graded skew shell is investigated using a C0 finite element formulation. Reddy's higher order theory has been employed to perform the analysis and the volume fractions of the ceramic and metallic components are assumed to follow simple linear distribution law. The present study attempts to focus mainly on the influence of skew angle on frequency parameter and displacement of shell panel with various geometries. Comprehensive numerical results are demonstrated for cylindrical, spherical and hypar shells for different boundary conditions and skew angles.The findings obtained for functionally graded skew shell panels are new and can be used as bench mark for researchers in this field.

Accurate dynamic response of laminated composites and sandwich shells using higher order zigzag theory

Forced vibration response of laminated composite and sandwich shell is studied by using a 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the forced vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders' approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at each layer interface in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C0 finite element formulation has been done to overcome the problem of C1 continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells besides normal curvature shells like cylindrical, spherical shells etc. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.

Analysis of laminated composite skew shells using higher order shear deformation theory

Static analysis of skew composite shells is presented by developing a C0 finite element (FE) model based on higher order shear deformation theory (HSDT). In this theory the transverse shear stresses are taken as zero at the shell top and bottom. A realistic parabolic variation of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. Sander's approximations are considered to include the effect of three curvature terms in the strain components of composite shells. The C0 finite element formulation has been done quite efficiently to overcome the problem of C1 continuity associated with the HSDT. The isoparametric FE used in the present model consists of nine nodes with seven nodal unknowns per node. Since there is no result available in the literature on the problem of skew composite shell based on HSDT, present results are validated with few results available on composite plates/shells. Many new results are presented on the static response of laminated composite skew shells considering different geometry, boundary conditions, ply orientation, loadings and skew angles. Shell forms considered in this study include spherical, conical, cylindrical and hypar shells.

Vibration of laminated composites and sandwich shells based on higher order zigzag theory

Free vibration response of laminated composite and sandwich shell is studied by using an efficient 2D FE (finite element) model based on higher order zigzag theory (HOZT). This is the first finite element implementation of the HOZT to solve the vibration problem of shells incorporating all three radii of curvatures including the effect of cross curvature in the formulation using Sanders’ approximations. The proposed finite element model satisfies the inter-laminar shear stress continuity at the interfaces in addition to higher order theory features, hence most suitable to model sandwich shells along with composite shells. The C0 finite element formulation has been done efficiently to overcome the problem of C1 continuity associated with the HOZT. The present model can also analyze shells with cross curvature like hypar shells, etc., besides normal curvature shells like cylindrical, and spherical shells. The numerical studies show that the present 2D FE model is more accurate than existing FE models based on first and higher order theories for predicting results close to those obtained by 3D elasticity solutions for laminated composite and sandwich shallow shells. Many new results are presented by varying different parameters which should be useful for future research.

Vibration analysis of laminated composite skew cylindrical shells using higher order shear deformation theory

A C0 finite element (FE) formulation based on higher order shear deformation theory (HSDT) is developed for free vibration analysis of composite skew cylindrical shells. The problem of C1 continuity associated with the HSDT has been overcome quite efficiently in the present FE model. The curved isoparametric element used in the model consists of nine nodes with seven nodal unknowns per node. Use of shear correction factor is avoided by assuming a realistic parabolic variation of transverse shear strain through the shell thickness. The transverse shear stresses are taken as zero at the shell top and bottom. Sander’s approximations are considered in the FE formulation to include the effect of three curvature terms in the strain components of composite shells. Since there is no result available in the literature on the problem of skew composite cylindrical shell based on HSDT, the present results are validated with some results available on composite plates/shells. Many new results are presented on free vibration response of laminated composite skew cylindrical shells considering different geometry, boundary conditions, ply orientation and skew angles.

Finite element analysis of laminated composite and sandwich shells using higher order zigzag theory

Static analysis of laminated composite and sandwich shell is presented by developing a C0 finite element (FE) formulation based on higher order zigzag theory (HOZT) using Sander’s approximations. The proposed model satisfies the inter-laminar shear stress continuity at the interfaces; and zero transverse shear stress conditions at shell top and bottom. This is the first finite element implementation of the HOZT to solve the problem of shells incorporating cross curvature effects in shells. The present HOZT predicts transverse shear stresses more accurately than FE results previously published in literature. Numerical results show that the present 2D model is very efficient in predicting the static response of laminated composite and sandwich shallow shell very close to 3D elasticity solutions.

Vibration of laminated composite skew hypar shells using higher order theory

Free vibration analysis of laminated composite skew hypar shells are presented using a C0 finite element (FE) formulation based on higher order shear deformation theory (HSDT). Effect of cross curvature is included in the formulation. The C0 finite element formulation has been done quite efficiently to overcome the problem of C1 continuity associated with the HSDT. The isoparametric FE used in the present model consists of nine nodes with seven nodal unknowns per node. Since there are no results available in the literature based on HSDT on the problem of free vibration of laminated composite skew hypar shells, present results are validated with the results available for laminated composite hypar shells without skew angle and with the results available for laminated composite skew plates. New results are presented by varying geometry, boundary conditions, ply orientations and skew angles.

Natural Frequencies and Mode Shapes of Laminated Composite Skew Hypar Shells with Complicated Boundary Conditions Using Finite Element Method

In the present investigation, free vibration behaviour is studied for the laminated composite skew hypar shells having twist radius of curvature. A higher-order shear deformation theory is employed in the C0 finite element formulation. Higher-order terms in the Taylor’s series expansion are used to represent the higher-order transverse cross sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effect of transverse shear. The structural system is considered to be undamped. The correctness of the formulation is established by comparing the present results of problems with those available in the published literature. The effects of different parameters are studied on the free vibration aspects of laminated composite skew hypar shells. Effect of cross curvature is included in the formulation. The C0 finite element formulation has been done quite efficiently to overcome the problem of C1 continuity associated with the HSDT. The isoparametric FE used in the present model consists of nine nodes with seven nodal unknowns per node. Since there is no result available in the literature based on HSDT on the problem of free vibration of laminated composite skew hypar shells, new results are presented by varying geometry, boundary conditions, ply orientations and skew angles which will serve as benchmark for future researchers.