Shallow Shell Element Research Articles

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

Vibration analysis of scale-dependent thin shallow microshells with arbitrary planform and boundary conditions

The current article presents a thorough investigation into the natural frequencies and mode shapes of thin doubly curved shallow microshells by taking into account the influence of material length scale parameter. The mathematical model of system is obtained by applying the Hamilton's principle along with the modified couple stress theory and the Kirchhoff-Love's shell theory. In order to extract the free vibration characteristics of system, the finite element method is employed. Several comparative studies are then conducted to verify the quadrilateral thin shallow shell element. The numerical results are presented for the thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with different planforms including rectangle, triangle, circle, and ellipse. Thin microplate as a special case of thin shallow microshell is also analyzed. A profound examination on the numerical results is also performed to divulge the impacts of scale-dependency accompanied by the geometrical parameters and boundary conditions on the free vibration response of system.

Experimental and Numerical Free Vibration Analysis of Laminated Composite Plates with Arbitrary Cut-Outs

Cut-outs are the integral parts of laminated plate components in a variety of civil, aerospace and marine engineering applications. Cut-outs alter the dynamic behaviour of the structures by reducing mass and stiffness simultaneously. The glass epoxy laminates are fabricated by resin infusion method through vacuum bagging system in the laboratory. The experimental studies are carried out on composite plates with cut-out positioned at different locations in the laminate. A finite element formulation based on the first-order shear deformation theory is developed in the present study using nine noded isoparametric shallow shell elements. The effects of rotary inertial contribution of mass on the natural frequencies are studied in the present paper. The present finite element formulation is validated by comparing the solutions obtained in terms of natural frequencies with the relevant published results and also experimental data. The correlation between experimental and numerical mode shapes is established by using modal assurance criteria.

A quadrilateral shallow shell element based on the third-order theory for functionally graded plates and shells and the inaccuracy of rule of mixtures

Abstract A four-node quadrilateral element is developed for the dynamic analysis of doubly curved functionally graded material (FGM) shallow shells, using the refined third order theory. Two micromechanics models, the Voigt's rule of mixtures (ROM) and the Mori–Tanaka model, are considered for computing the effective material properties at a point. The accuracy of the element is examined by comparing with various three dimensional elasticity and two dimensional (2D) analytical and finite element solutions available in the literature for static and free vibration responses of FGM plates and shells. It is shown that the present element, with the least number of degrees of freedom, achieves similar or better accuracy compared to other available 2D finite elements some of which are even based on higher order theories. Using this element, we also make a systematic assessment of the accuracy of the widely used ROM in predicting the behavior of FGM structures, for different values of the inhomogeneity parameter, and different geometrical parameters, boundary conditions, and material combinations. It is revealed that there can be very significant error in the deflection, stresses and natural frequencies predicted by the ROM, depending primarily on the inhomogeneity parameter and the difference in the material properties of the constituents.

Coupled efficient layerwise finite element modeling for active vibration control of smart composite and sandwich shallow shells

The active vibration suppression of smart composite and sandwich shallow shells equipped with distributed monolithic piezoelectric and piezo-fiber reinforced composite (PFRC) sensors and actuators is studied using a four-node quadrilateral shallow shell element based on a fully coupled, accurate and efficient layerwise (zigzag) theory. The shell element uses the concept of electric nodes to satisfy the equipotential condition of electroded sensor surfaces without making any approximations or averaging. The effective electromechanical properties of the PFRC laminas are computed using a coupled three-dimensional iso-field micromechanical model. Both classical (constant gain velocity feedback (CGVF)) and optimal (linear quadratic Gaussian (LQG)) control strategies are studied. A truly collocated actuator–sensor arrangement is proposed and shown to remove the instability in CGVF control of shallow shells with conventionally collocated actuators and sensors. The effects of piezoelectric fiber orientation and volume fraction ratio of PFRC, and the radius of curvature and span to thickness ratio of the shell on the control performance, are studied. It is shown that the LQG control not only suppresses the transient vibration under step/impulse excitations, but also eliminates the beating phenomena under harmonic excitation when the forcing frequency is close to the natural frequency of the system.

Topology Optimization of Fluid-Loaded Shells by Minimizing the Acoustic Coupling to the Fluid Domain

Topology optimization of fluid-loaded shells is presented. A finite element model with shallow shell elements is developed using strain-based formulation. The considered elements are quadrilateral, where each node has five degrees of freedom to account for in-plane and out-of-plane deflections as well as rotations. The developed model was integrated in the optimization algorithm aiming at minimizing the coupling between the flexible shell and the adjacent fluid domain. The model is exercised to generate closed-form expressions of the sensitivities in order to enhance the topology optimization process. The Method of Moving Asymptotes is used along with the sensitivity expressions to predict the successive iterations. The fluid-structure coupling was manifested in the coupling matrix in the finite element model. The results of the finite element–based topology optimization were validated using ANSYS® commercial package. Radial loads with fixed-fixed as well as axial loads with fixed-free boundary conditions were implemented. Different values of minimum permissible shell wall thickness were developed. Numerical examples show that considerable attenuations are attained for the selected structural modes in the order of 9–12 dB for the radial as well as the axial excitation cases. Reducing the minimum permissible thickness from 35% of the original thickness to 15% has shown tremendous effect on the attenuation at the targeted modes for the different load cases in the order of 12 dB. The presented model and design methodology can be invaluable in the design of various critical cylindrical structures that must satisfy strict vibration and noise radiation require- ments.

Free vibration of composite skewed cylindrical shell panel by finite element method

In this paper a composite triangular shallow shell element has been used for free vibration analysis of laminated composite skewed cylindrical shell panels. In the present element first-order shear deformation theory has been incorporated by taking transverse displacement and bending rotations as independent field variables. The interpolation function used to approximate transverse displacement is one order higher than for bending rotations. This has made the element free from locking in shear. Two types of mass lumping schemes have been recommended. In one of the mass lumping scheme the effect of rotary inertia has been incorporated in the element formulations. Free vibration of skewed composite cylindrical shell panels having different thickness to radius ratios ( h/R=0.01–0.2), length to radius ratios ( L/R), number of layers and fiber orientation angles have been analyzed following the shallow shell method. The results for few examples obtained in the present analysis have compared with the published results. Some new results of composite skewed cylindrical shell panels have been presented which are expected to be useful to future research in this direction.

Non-linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large-amplitude multi-mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non-linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees-of-freedom (DOF) are transformed into modal co-ordinates, and the non-linear stiffness matrices are transformed into non-linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap-through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap-through between the two BEPs, and (iii) non-linear large-amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.

Free Vibration Analysis of Laminated Stiffened Shells

The free vibration of the laminated composite anticlastic doubly curved stiffened shells is investigated using the finite element method. The stiffened shell element is obtained by appropriate combination of the nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell forms include the hyperbolic paraboloid, hypar, and conoidal shells. The accuracy of the formulation is validated by comparing the authors' results of specific problems with those available in the literature. The additional problems are taken up for parametric studies to include the effects of fiber orientation and lamina stacking sequence of shells and stiffeners. Moreover, the effects of number, types, and orientations of stiffeners, and stiffener depth to shell thickness ratio on the fundamental frequency are also included in the present study. Further, mode shapes corresponding to the fundamental frequency for typical cases are obtained to verify the parametric trend of the results of the fundamental frequency.

Advanced Shell Element Formulations for Coupled Electromechanical Systems

AbstractWith the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A High Precision Shear Flexible Triangular Element for Vibration of Composite Shells

A high precision triangular shallow shell element is proposed and it is applied to free vibration analysis of composite and isotropic shells. The Mindlin's hypothesis is followed to include the effect of shear deformation. The formulation is made in an efficient manner to make the element free from shear locking problem. The element has some internal nodes, which are eliminated through static condensation technique to improve the computational elegance of the element. In the present vibration problem, the implementation of the static condensation became possible with the help of an efficient mass lumping scheme. It is quite interesting that the effect of rotary inertia can be included in the recommended scheme for lumped mass matrix. Numerical examples covering a wide range of problems are solved and the results obtained are compared with the published results in many cases, which show the precision and range of applicability of the proposed element. The performance of the proposed technique for rotary inertia is found to be excellent. Some new results are produced, which may be useful in future research.

Multi-field variational formulations and related finite elements for piezoelectricshells

Smart structures technology characterized by structurally integrated sensors and actuatorshas recently expanded significantly especially as regards lightweight constructions inaeronautics and robotics, e.g. to allow vibration suppression and noise attenuation. In orderto be capable of solving these complex issues the finite element method as a wellestablished design tool has to be extended. This paper focuses on shallow sandwichcomposite shell structures with thin piezoelectric patches bonded to the surfaces. For theproper design of plate and shell structures with integrated piezoelectric materials, variousvariational formulations and corresponding finite elements are presented. The starting pointis the well known two-field variational formulation where the linear piezoelectric effect istaken into account so that the displacements and the electric potential serve asindependent variables. Here, the mostly assumed linear variation of the electricpotential through the thickness is assumed. Next, it is shown that a quadraticvariation of the electric potential through the thickness can be deduced directlyfrom the charge conservation condition. This quadratic variation of the electricpotential in the thickness direction is compared with the linear gradient of the firsttwo-field variational formulation. Moreover, in order to allow the implementation ofalternative formulations of the constitutive equations by switching of the independentvariables and nonlinear material behaviour, a three-field variational formulation ispresented in analogy to the Hu–Washizu principle. Adopting this variationalprinciple a hybrid finite element is derived where the dielectric displacement isformulated as an additional degree of freedom. This independent variable canbe condensed on the element level and does not enter the system of equations.For the first time all these different variational formulations are developed for aReissner–Mindlin shallow shell element formulation and compared with each other.

A HIGH PRECISION SHELL ELEMENT WITH SHEAR DEFORMATION FOR THE ANALYSIS OF ISOTROPIC AND COMPOSITE SHELLS

A high precision triangular shallow shell element with shear deformation is proposed and it is applied to the analysis of isotropic and fiber reinforced laminated composite shells. To make the element free from locking in shear, a different order of interpolation function is taken for the transverse displacement and rotations of the normal. This idea is taken from a plate element developed by the last author of this paper. Initially, the proposed element has twelve external and four internal nodes containing fifty-five degrees of freedom which is reduced to forty-eight by elimination of the degrees of freedom at the internal nodes through static condensation. Numerical examples of isotropic and composite shells having different geometry, curvature ratio, thickness ratio, stacking sequence, boundary and loading are solved by this element. The results obtained in the form of displacements, stress and stresses resultants are compared with the published results, which show the accuracy and range of applicability of the proposed element. Some new results of composite shell are presented, which are expected to be useful to future research in this direction.

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/ y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

AbstractThe static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.

GEOMETRICALLY NONLINEAR STABILITY ANALYSIS OF SHELLS USING GENERALIZED CONFORMING SHALLOW SHELL ELEMENT

A generalized conforming finite element theory was presented to satisfy the C1 continuity condition for plate and shell element. The effectiveness of the theory in the linear analysis has been proved. This paper discusses the membrane locking phenomenon of shallow shell element based on the satisfaction of the requirement of rigid body motion, and a technique is developed to eliminate the membrane locking phenomenon. Accordingly a geometrically nonlinear generalized conforming rectangular shallow shell element with tangential degrees of freedom of midpoints of sides is formulated. Nonlinear numerical analysis of shell stability shows that the element exhibits high precision and fast convergence characteristics.

Finite Element Modeling of MFC/AFC Actuators and Performance of MFC

The anisoparametric three-node MIN6 shallow shell element is extended for modeling Macro-Fiber Composite/Active Fiber Composites (MFCTM/AFC) actuators for active vibration and acoustic control of curved and flat panels. The recently developed MFCTM/AFC actuators exhibit enhanced performance, they are anisotropic and highly conformable as compared to the traditional monolithic isotropic piezoceramic actuators. The extended MIN6 shell element includes embedded or surface bonded MFCTM/AFC laminae. The fully coupled electrical-structural formulation is general and it is able to handle arbitrary doubly curved laminated composite and isotropic shell structures. A square and a triangular cantilever isotropic plates are modeled using the MIN6 elements to demonstrate the anisotropic actuation of a surface bonded MFCTM actuator for coupled bending and twisting plate motions. Steady state modal bending and twisting amplitudes of the cantilever square and triangular plates with MFCTM actuator are compared with the plate’s steady state modal amplitudes with traditional PZT 5A actuator for different angle orientations. Frequency Response Functions (FRF) for the square plate with MFCTM and PZT 5A actuators are also obtained and their actuation performance is compared. The actuation performance of the MFCTM at different locations is also investigated.

A new stiffened shell element for geometrically nonlinear analysis of composite laminates

A new stiffened shell element combining shallow beam and shallow shell elements is developed for geometrically nonlinear analysis of stiffened composite laminates under mechanical loading. The formulation of this element is based on the principle of virtual displacements in conjunction with the co-rotational form of the total Lagrangian description of motion. In the finite element formulation, both the shell and the beam (stiffener) elements account for transverse shear deformations and material anisotropy. The cross section of the stiffener (beam) can be arbitrary in geometry and lamination. In order to combine the stiffener with the shell element, constraint conditions are applied to the displacement and rotation fields of the stiffener. These constraint conditions ensure that the cross section of the stiffener remains co-planar with the shell section after deformation. The resulting expressions for the displacement and rotation fields of the stiffener involve only the nodal unknowns of the shell element, thus reducing the total number of degrees-of-freedom. Also, the discretization of the entire stiffened shell structure becomes more flexible. The robustness of the stiffened shell element has been proven by comparison against other shell elements considered previously.

Nonlinear thermoelastic analysis of composite panels under non-uniform temperature distribution

The response of moderately thick laminated panels experiencing large displacements and rotations under non-uniform thermal loading is investigated through a nonlinear finite element analysis. The present nonlinear thermoelastic analysis incorporates an anisoparametric, doubly curved, shallow shell element that is free of the `locking' phenomenon. The effects of large displacements and rotations, transverse shear deformations, the coupling between stretching and bending due to shallow geometry, and Duhamel–Neumann-type thermoelastic material anisotropy are included in the element formulation. The equations of equilibrium are derived from the virtual work principle, along with the co-rotational form of the total Lagrangian formulation. A non-uniform temperature field across the shell surface is approximated by piecewise-uniform temperature distributions over individual elements. In the thickness direction, the temperature distribution is approximated linearly. Accuracy of the present analysis is established by comparison with benchmark solutions. The numerical results are presented for various configurations, including cutouts under uniform and non-uniform temperatures. The numerical results demonstrate that the present finite element analysis is computationally robust and efficient.

Nonlinear analysis of laminates through a mindlin-type shear deformable shallow shell element

This study presents a nonlinear analysis with application to a doubly curved shallow shell element free of ‘locking’. The ‘locking’ phenomenon is eliminated by explicitly determining the shear and membrane correction factors. The element formulation utilizes the Reissner-Mindlin and Marguerre theories. The analysis of thin and moderately thick composite shells undergoing large displacements and rotations is achieved by using the corotational form of an updated Lagrangian formulation. The validity of the analysis is established by correlating present results with various benchmark cases that involve large displacements and rotations, as well as elastic stability.