Field Consistency Approach Research Articles

Flexural vibration of elastic composite beams using a third-order deformation kinematics

The present study introduces a one-dimensional finite element (FE) model that utilizes third-order deformation kinematics to analyze the flexural vibration of two-layered composite beams with partial shear slip. The proposed model incorporates a three-node straight beam element, utilizing the field-consistent approach and full numerical integration of the stiffness matrix. This approach is adopted to effectively address stress oscillation and shear locking issues, ensuring accurate and reliable results. The analysis focuses on the third-order deformation kinematics of longitudinal displacement within the beam depth, examining each of the two layers individually. The interface between the two material layers is joined using deformable shear connectors, which are represented as dispersed shear springs over the length of the beam. This study focuses on conducting a flexural vibration analysis of two-layered composite beams. The analysis includes the examination of both free vibration and forced vibration responses while considering or omitting the influence of damping under various loading conditions. The validity of the proposed FE model is assessed by the utilization of a two-dimensional model developed within the ABAQUS software. Additionally, the suggested model is evaluated through the process of verification by comparing it with previously published results.

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.

Isogeometric analysis for geometrically exact shell elements using Bézier extraction of NURBS with assumed natural strain method

In the present study, isogeometric analysis for geometrically exact shell considering the first-order shear deformation is studied. Since kinematics of the shell is formulated in the form of a general tensor in generalized curvilinear coordinates, isogeometric analysis of arbitrarily shaped shells is possible with a direct geometry link to NURBS. For the first-order shear deformable shell element, the assumed natural strain (ANS) method is normally used to alleviate membrane and transverse shear locking phenomenon. Due to a higher-order regularity between integration knot elements of NURBS basis function has, a locking phenomenon still appears even though the field consistent approach such as ANS is utilized. To solve this problem, we adopted Bézier extraction method which maps a piecewise C0 Bernstein polynomials basis onto NURBS basis to reduce element regularity for isogeometric shell analysis. For an arbitrarily shaped shell geometry generation using NURBS of the shell, the point data sets from the exact geometry shape can be used by the global interpolation. The test results of numerical examples using the developed method show the accuracy and robustness with a higher convergence rate of the proposed approach.

Adjusted approximation spaces for the treatment of transverse shear locking in isogeometric Reissner–Mindlin shell analysis

Transverse shear locking is an issue that occurs in Reissner–Mindlin plate and shell elements. It leads to an artificial stiffening of the system and to oscillations in the stress resultants for thin structures. The thinner the structure is, the more pronounced are the effects. Since transverse shear locking is caused by a mismatch in the approximation spaces of the displacements and the rotations, a field-consistent approach is proposed for an isogeometric degenerated Reissner–Mindlin shell formulation. The efficiency and accuracy of the method is investigated for benchmark plate and shell problems. A comparison to element formulations with locking alleviation methods from the literature is provided.

Nonlinear Dynamic Thermal Buckling of Sandwich Spherical and Conical Shells with CNT Reinforced Facesheets

Owing to their superior mechanical and thermal properties, carbon nanotube (CNT) reinforced composite materials have wide range of applications in various technical areas such as aerospace, automobile, chemical, structural and energy. In this paper, the nonlinear axisymmetric dynamic behavior of sandwich spherical and conical shells made up of CNT reinforced facesheets is studied. The shell is subjected to thermal loads and discretized with three-noded axisymmetric curved shell element based on field consistency approach. The in-plane and the rotary inertia effects are included within the transverse shear deformation theory in the element formulation. The present model is validated with the available analytical solutions from the literature. A detailed parametric study is carried out to showcase the effects of the shell geometry, the volume fraction of the CNT, the core-to-face sheet thickness and the environment temperature on the dynamic buckling thermal load of spherical caps.

Hygrothermal analysis of laminated composites using C0 FE model based on higher order zigzag theory

A C0 FE model developed based on an efficient higher order zigzag theory is used for hygrothermal analysis of laminated composite plates. The C0 FE model satisfies the inter-laminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory. In the present theory the above mentioned C0 continuity of the present element is compensated in the stiffness matrix formulation by using penalty parameter approach. In order to avoid stress oscillations observed in the displacement based finite element, the stress field derived from temperature/moisture fields (initial strains) must be consistent with total strain field. Special steps are introduced by field consistent approach (e.g., sampling at gauss points) to compensate this problem. A nine noded C0 continuous isoparametric element is used in the proposed FE model. Comparison of present numerical results with other existing solutions shows that the proposed FE model is efficient, accurate and free of locking.

A discrete layer finite element model for cylindrical shells

This paper presents a new discrete layer finite element to model thin as well as moderately thick orthotropic and laminated composite cylindrical shells. The element formulation is based on the first order shear deformation theory of shells. The twenty degrees of freedom plane stress element is modeled with inplane displacements defined at the interfaces of the element layers in addition to the, through the thickness, radial displacement. A field consistency approach is implemented to insure that the element is free from locking due to membrane tangential, membrane shear and transverse shear strains. The field consistency approach used eliminates the inconsistent terms from the original displacement shape functions that correspond to the targeted strains. The new element is validated through a series of benchmark problems and has shown accurate and fast converging results.

Static, Vibration and Buckling Analysis of Skew Composite and Sandwich Plates Under Thermo Mechanical Loading

Static, vibration and buckling behavior of laminated composite and sandwich skew plates is studied using an efficient C<sup>0</sup> FE model developed based on refined higher order zigzag theory. The C<sup>0</sup> FE model satisfies the interlaminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. In this model, the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C<sup>1</sup> continuity associated with the plate theory. The C<sup>0</sup> continuity of the present element is compensated in the stiffness matrix formulation by adding a suitable term. In order to avoid stress oscillations observed in the displacement based finite element, the stress field derived from temperature is made consistent with the total strain field by using field consistent approach. Numerical results are presented for different static, vibration and buckling problems by applying the FE model under thermo mechanical loading, where a nine noded C<sup>0</sup> continuous isoparametric element is used. It is observed that there are very few results available in the literature on laminated composite and sandwich skew plates based on refined theories. As such many new results are also generated for future reference

Analysis of composite plates by a unified formulation-cell based smoothed finite element method and field consistent elements

In this article, we combine Carrera’s Unified Formulation (CUF) [13,7] and cell based smoothed finite element method [28] for studying the static bending and the free vibration of thin and thick laminated plates. A 4-noded quadrilateral element based on the field consistency requirement is used for this study to suppress the shear locking phenomenon. The combination of cell based smoothed finite element method and field consistent approach with CUF allows a very accurate prediction of field variables. The accuracy and efficiency of the proposed approach are demonstrated through numerical experiments.

Vibrations and thermal stability of functionally graded spherical caps

Here, the axisymmetric free flexural vibrations and thermal stability behaviors 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 functionally graded material and base radius-to-thickness on the vibrations and buckling characteristics of spherical shells.

Dynamic Buckling of Functionally Graded Spherical Caps

The dynamic behavior of functionally graded spherical caps under suddenly applied loads is studied by using 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. Geometric nonlin- earity is introduced in the formulation using von Karman's strain-displacement relations. 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. The governing equations obtained are solved employing the Newmark's integration technique coupled with a modified Newton-Raphson iteration scheme. The load corresponding to a sudden jump in the maximum average displace- ment in the time history of the shell structure is taken as the dynamic buckling pressure. The present model is validated against the available isotropic cases. A detailed numerical study is carried out to bring out the effects of shell geometries, power-law index of functional graded material, boundary conditions, and finite pressure pulse with different duration on the axisymmetric dynamic buckling load of shallow spherical shells.

A C1 finite element for flexural and torsional analysis of rectangular piezoelectric laminated/sandwich composite beams

AbstractThis work deals with the development of a new C1 finite element for analysing the bending and torsional behaviour of rectangular piezoelectric laminated/sandwich composite beams. The formulation includes transverse shear, warping due to torsion, and elastic–electric coupling effects. It also accounts for the inter‐layer continuity condition at the interfaces between layers, and the boundary conditions at the upper and lower surfaces of the beam. The shear strain is represented by a cosine function of a higher order in nature and thus avoiding shear correction factors. The warping function obtained from a three‐dimensional elasticity solution is incorporated in the present model. An exact integration is employed in evaluating various energy terms due to the application of field consistency approach while interpolating the transverse shear and torsional strains. The variation of the electric potential through the thickness is taken care of in the formulation based on the observation of three‐dimensional solution. The performance of the laminated piezoelectric element is tested comparing with analytical results as well as with the reference solutions evaluated using three‐dimensional finite element procedure. A detailed study is conducted to highlight the influence of length‐to‐thickness ratio on the displacements, stresses and electric potential field of piezoelectric laminated beam structures subjected to flexural and torsional loadings. Copyright © 2004 John Wiley & Sons, Ltd.

Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory

Here, the free vibration characteristics of functionally graded elliptical cylindrical shells are analyzed using finite element formulated based on the theory with higher-order through the thickness approximations of both in-plane and transverse displacements. The power law variation of properties is assumed in the thickness direction. The finite element employed in the study is based on field-consistency approach and free from shear and membrane locking problems. The strain–displacement relations are accurately introduced in the formulation without making any approximation in the thickness co-ordinate to radius ratio terms. The detailed parametric studies are carried out to study the influences of non-circularity, radius-to-thickness ratio, material composition and material profile index on the free vibration frequencies and mode shape characteristics of functionally graded elliptical shells. The significance of thickness stretch/contraction terms is highlighted through the mode shape study.

Active vibration control of smart shells using distributed piezoelectricsensors and actuators

Smart/intelligent/adaptive structures with distributed, integratedsensors, actuators and control electronics are being investigated extensivelyfor possible use in high-performance, light-weight structural systems. Due tothe unique electromechanical coupling, these piezoelectric materials could beused as sensors and actuators in smart structures. This work deals with theactive vibration control of smart/intelligent shells with a distributedpiezoelectric sensor and actuator layer bonded onto the top and bottom surfacesof the structures. A shear-flexible nine-noded shell finite element derived fromthe field consistency approach has been used for the investigation. This elementhas been developed so as to have one electrical degree of freedom perpiezoelectric layer per element. The mass, stiffness and electromechanicalcoupling effects of the piezoelectric sensor and actuator layer are considered.The control performance with different types of loading is investigated. Alinear quadratic regulator optimal control scheme is used to obtain the controlgains.

Field-consistent higher-order free-interface component mode synthesis

A new free-interface component mode synthesizing technique based on the notion of higher-order field consistency is proposed. The present technique employs higher-order residual attachment modes in addition to kept normal modes while consistency in matching field variables at the substructure interface is maintained. The present field-consistency approach does not increase the size of the synthesized system even if higher-order residual attachment modes are included. Higher-order analyses greatly improves solution accuracy while the final system size always remains the same as the size of the kept normal modes regardless of the order of residual attachment modes. A general procedure of deriving higher-order residual attachment modes is first described and then a new field-consistent higher-order synthesis technique is presented. Copyright © 2001 John Wiley & Sons, Ltd.

AC1 finite element including transverse shear and torsion warping for rectangular sandwich beams

A new three-noded C1 beam finite element is derived for the analysis of sandwich beams. The formulation includes transverse shear and warping due to torsion. It also accounts for the interlaminar continuity conditions at the interfaces between the layers, and the boundary conditions at the upper and lower surfaces of the beam. The transverse shear deformation is represented by a cosine function of a higher order. This allows us to avoid using shear correction factors. A warping function obtained from a three-dimensional elasticity solution is used in the present model. Since the field consistency approach is accounted for interpolating the transverse strain and torsional strain, an exact integration scheme is employed in evaluating the strain energy terms. Performance of the element is tested by comparing the present results with exact three-dimensional solu-tions available for laminates under bending, and the elasticity three-dimensional solution deduced from the de Saint-Venant solution including both torsion with warping and bending. In addition, three-dimensional solid finite elements using 27 noded-brick elements have been used to bring out a reference solution not available for sandwich structures having high shear modular ratio between skins and core. A detailed parametric study is carried out to show the effects of various parameters such as length-to-thickness ratio, shear modular ratio, boundary conditions, free (de Saint-Venant) and constrained torsion. Copyright © 1999 John Wiley & Sons, Ltd.

A field-consistency approach to plate elements

The design of robust plate and shell elements has been a very challenging area for several decades. The main difficulty has been the shear locking phenomenon in plate elements and the shear and membrane locking phenomena together in the shell elements. Among the various artifices or devices which are used to develop elements free of these problems is the field-consistency approach. In this paper this approach is reviewed, It turns out that not only Mindlin type elements but also elements based on higher-order theories could be developed using the technique.

Dynamic Buckling of Laminated Anisotropic Spherical Caps

The dynamic axisymmetric behavior of clamped laminated composite spherical caps subjected to suddenly applied loads is investigated using an eight-noded quadrilateral doubly curved shear flexible shell element based on the field-consistency approach. Geometric nonlinearity is considered using von Karman’s strain-displacement relations. The solution is obtained using the Wilson-θ numerical integration scheme. The pressure corresponding to a sudden jump in the maximum average deflection in the time history of the shell structure is taken as dynamic buckling pressure. A detailed parametric study is carried out to bring out the effects of shell geometries and material properties, number of layers, lamination schemes, and type of loading on a dynamic buckling load.

Nonlinear free flexural vibrations of laminated circular cylindrical shells

This paper deals with large-amplitude free flexural vibrations of laminated composite circular cylindrical shells. The formulation is based on first-order shear deformation theory and Lagrange's equation of motion. The nonlinearity due to finite deformation of the shell is included in the formulation. Amplitude-frequency relationships are obtained through dynamic response history using the Wilson-θ numerical integration scheme. An element based on the field consistency approach is used to investigate the nonlinear behaviour of cylindrical shells. Detailed numerical results based on various parameters are presented for orthotropic, cross-ply and angle-ply laminated circular cylindrical shells with different boundary conditions.

Supersonic Flutter of Laminated Curved Panels

Supersonic flutter analysis of laminated composite curved panels is investigated using doubly-curved, quadrilateral, shear flexible, shell element based on field-consistency approach. The formulation includes transverse shear deformation, in-plane and rotary inertias. The aerodynamic force is evaluated using two-dimensional static aerodynamic approximation for high supersonic flow. Initially, the model developed here is verified for the flutter analysis of flat plates. Numerical results are presented for isotropic, orthotropic and laminated anisotropic curved panels. A detailed parametric study is carried out to observe the effects of aspect and thickness ratios, number of layers, lamination scheme, and boundary conditions on flutter boundary.