Shell Equations Research Articles

Vibration Control of a Smart Shell Reinforced by Graphene Nanoplatelets

Smart control and dynamic investigation of a graphene nanoplatelets reinforced composite (GPLRC) cylindrical shell surrounded by a piezoelectric layer as actuator and sensor based on a numerical solution method called generalized differential quadrature method (GDQM) are presented for the first time. The strains and stresses can be determined via the first-order shear deformable theory (FSDT). For accessing to various mass densities, thermal expansion as well as Poisson ratio, the rule of mixture is applied, although a modified Halpin–Tsai theory is used for obtaining the module of elasticity. The external voltage is applied to the sensor layer, while a proportional-derivative (PD) controller has been utilized for controlling the output of sensor. GPLRCs boundary conditions are derived through governing equations of the cylindrical shell using an energy method known as Hamilton’s principle. The outcomes show that the PD controller, viscoelastic foundation, slenderness factor ([Formula: see text]/[Formula: see text], external voltage and graphene nanoplatelets (GPLs) weight fraction have a considerable impact on the amplitude, and vibration behavior of a GPLRC cylindrical shell. As an applicable result in related industries, the parameter and consideration of the PD controller have a positive effect on the static and dynamic behaviors of the structure.

Dynamic analysis of FG stepped truncated conical shells surrounded by Pasternak elastic foundations

This research presents a continuous element model for solving vibration problems of FG stepped truncated conical shells having various material properties and surrounded by Pasternak foundations. Based on the First Order Shear Deformation Theory (FSDT) and the equations of the FGM conical shells, the dynamic stiffness matrix is obtained for each segment of the shell having constant thickness. The interesting assembly procedure of continuous element method (CEM) is employed for joining those segments in order to analyze the dynamic behavior of the FG stepped truncated conical shells an assembly procedure of continuous element method (CEM) is employed for joining those segments. Free vibrations of different configurations of FG stepped truncated conical shells on elastic foundations are examined. Effects of structural parameters, stepped thickness and elastic foundations on the free vibration of FG stepped truncated conical shells are also presented.

The Concept modeling method: An approach to optimize the structural dynamics of a vehicle body

Although the concept modeling method has already been proposed in the literature, there is still very limited knowledge about the validation and the application of this method for vehicle body design. This paper substantially increases this limited knowledge by developing a concept model for predicting and optimizing the structural dynamics of a vehicle body-in-white and validating this concept model against a detailed finite element model. The geometry and parameters of the concept model are extracted from its detailed finite element model. The major members and panels of the detailed finite element model are replaced by their equivalent beam and shell elements models. The joints of the concept model are represented by stiffness and mass matrices extracted from the detailed finite element model using the Guyan Reduction Method. The developed concept model is validated by comparing its structural dynamics, including the resonant frequencies and the vibration mode shapes, with the original detailed finite element model and the experimental results. The simplicity and small size of the concept model enable it to easily enhance the structural dynamics of the body-in-white by optimizing the cross-sections of the load-carrying members of the structure. The optimization in this case increased the resonant frequencies of the body-in-white while reducing the total mass by about 6 kg. The results prove that the concept modeling method can significantly enhance the body-in-white structural dynamics by reducing the complexity of the model and allowing the focus for the optimization to be on the main members of the structure at the development stage when the final design parameters are not well known and have not been fixed.

Nonlinear vibration of full-filled fluid corrugated sandwich functionally graded cylindrical shells

This article proposes a semianalytical approach for the nonlinear free and forced asymmetric vibration of corrugated sandwich functionally graded cylindrical shells containing fluid under harmonic radial load. The corrugation is considered with two types: trapezoidal corrugation and round corrugation. By using the Donnell shell theory, taking into account the geometrical nonlinearity of von Kármán and an equivalent model for corrugated structures, the governing equations of shell are established. The nonlinear motion equations are reduced to nonlinear differential equation in terms of time by applying the Galerkin procedure. The numerical investigations for the nonlinear dynamic response of cylindrical shells are obtained by using the fourth-order Runge–Kutta method. Numerical results show the very large effects of corrugation and fluid on the natural frequency and nonlinear vibration behavior of shells.

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

Magneto-electro-elastic semi-analytical models for free vibration and transient dynamic responses of composite cylindrical shell structures

Application of the scaled boundary finite element method (SBFEM) in free vibration and transient dynamic problems of the composite magneto-electro-elastic (MEE) cylindrical shell is considered in this paper for the first time. The mathematic model of the composite MEE cylindrical shell is governed by the three-dimensional (3D) theory of magneto-electro-elasticity in the cylindrical coordinate system, and the dynamic equilibrium equations are then derived into a second-order ordinary differential equation by using the SBFEM. In this method, discretizations are confined only to the surfaces or boundaries of the computational domain but no fundamental solutions and singular integrals are required, which reduces the spatial dimension of problem by one and making mesh generation much easier. At the same time, the transverse shear locking can be successfully avoided due to the analyticity of the SBFEM governing equation in the radial direction of the scaled boundary coordinate system. Based on the state space approach, the basic equations of the composite MEE cylindrical shell in the scaled boundary coordinate system are reduced to a system of first-order ordinary differential equation in terms of the independent variable involving the generalized displacement and internal nodal force fields, and then the free vibration and transient dynamic responses of the system are solved by using the generalized eigenvalue technique and Newmark's integration scheme, respectively. Numerical results illustrate that the convergence of the proposed formulations is satisfactory, and excellent agreement can be achieved for limiting cases with the solutions available in published work or that obtained using the commercial software ANSYS. In addition, the influences of the structural parameters (including thickness ratio, aspect ratio and stacking configuration), material properties and boundary conditions on the vibration and dynamic behaviors of the composite MEE cylindrical shells are discussed.

Studying propagation of wave in metal foam cylindrical shells with graded porosities resting on variable elastic substrate

This investigation deals with wave propagation analysis of porous metal foam cylindrical shells resting on the variable elastic substrate within the framework of the first-order shear deformation shell theory. Magnesium, nickel, titanium and tungsten foams are considered as constitutive materials of cylindrical shell. The pores are distributed through the thickness uniformly, symmetrically and asymmetrically. The principle of Hamilton is employed in order to reach motion equations of porous metal foam cylindrical shell. Next, governing equations of porous metal foam are derived for a first-order shear deformation shell and then solved analytically. The effects of various parameters including porosity coefficient, various types of porosity distribution, length-to-thickness ratio, radius-to-thickness ratio, circumferential wave number, Pasternak coefficient and variable Winkler coefficient on the variation of wave frequency and phase velocity of metal foam cylindrical shells are covered and presented within the framework of a group of figures which can be seen in detail.

Propagation of Flexural Waves in Anisotropic Fluid-Conveying Cylindrical Shells

In the present article, first-order shear deformation theory (FSDT) of the shell has been employed, for the first time, in order to analyze the propagation of the flexural waves in anisotropic fluid-conveying cylindrical shells. Four various anisotropic materials are utilized and their wave propagation behavior surveyed. Viscous fluid flow has been regarded to be laminar, fully developed, Newtonian, and axially symmetric. The Navier–Stokes equation can be utilized to explore the flow velocity effect. FSDT of the shell and Hamilton’s principle have been employed in order to achieve governing equations of anisotropic fluid-conveying cylindrical shells and finally, the obtained governing equations have been solved via an analytical method. In addition, the influences of different variables such as flow velocity, radius to thickness ratio, and longitudinal and circumferential wave numbers have been investigated and indicated within the framework of a detailed set of figures.

A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle.

Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

Development of a shell superelement for large deformation and free vibration analysis of composite spherical shells

Finite element analysis of huge and/or complicated structures often requires long times and large computational expenses. Superelements are huge elements that exploit the deformation theory of a specific problem to provide the capability of discretizing the problem with minimum number of elements. They are employed to reduce the computational cost while retaining the accuracy of results in FEM analysis of engineering problems. In this research, a new shell superelement is presented to study linear/nonlinear static and free vibration analysis of spherical structures with partial or full spherical geometries that exist in many industrial applications. Furthermore, this study investigates the effects of changing the superelement size and its number of nodes on solution accuracy. The governing equations of composite spherical shells are derived based on the first-order shear deformation theory and considering large deformations. For solving the nonlinear governing equations, the tangent stiffness matrix has been extracted and the Newton–Raphson method is employed. The capability of the presented shell superelement is investigated in several problems under linear/nonlinear static and free vibration analysis. The results acquired by the presented shell superelements are compared with available results in the literature and conventional shell elements in a commercial software. Results comparisons reveal high accuracy at a reduced computational cost in the superelement model.

Non-linear collapse behavior of externally pressurized resin toroidal and cylindrical shells: numerical and experimental studies

ABSTRACT The purpose of this paper is to elaborate the non-linear collapse of externally pressurized resin toroidal and cylindrical shells, along with a comparison between a toroid and cylinder. For this purpose, the collapse characteristics, critical loads, and imperfection sensitivities of volume equivalent shells were numerically evaluated for the first eigenmode imperfection. Furthermore, three nominally identical toroidal shells and three equivalent cylindrical shells were rapidly prototyped, optically scanned, ultrasonically measured, hydrostatically tested, and numerically examined. The numerical and experimental data were consistent with each other. The average pressure supporting capacity of the toroidal shell was approximately 29% higher than that of its volume equivalent cylindrical shell, thus demonstrating a superior pressure supporting capacity of the toroid. The knock down factors of the cylindrical shell were considerably lower than those of the toroidal shell, thus indicating that the toroidal shell is considerably less sensitive to initial geometrical imperfection than the cylindrical shell.

Method for Determining Orthotropic Elastic Constants of Equivalent Shell Elements for the Boiler Membrane Wall of Coal-Fired Power Plants

Method for Determining Orthotropic Elastic Constants of Equivalent Shell Elements for the Boiler Membrane Wall of Coal-Fired Power Plants

Electro-magnetic potential effects on free vibration of rotating circular cylindrical shells of functionally graded materials with laminated composite core and piezo electro-magnetic two face sheets

In this study, the effect of electric and magnetic potential on the free vibration of rotating circular cylindrical shell of functionally graded piezo electromagnetic composed with fiber reinforced polymers has been investigated. The laminated fiber reinforced polymer is in the core and is supported with the functionally graded material layer on the both sides, and two piezoelectric layers are mounted on the outside and inside of the cylindrical shells. Relationships between strain and displacement are expressed based on Love’s first approximation shell theory. Then according to Hamilton’s principle, the governing equations for the cylindrical shell are extracted. Also, the Navier’s solution is used to solve the equations of the cylindrical shell for the simply–simply supported boundary conditions. The numerical results of the analytical solution for the rotating functionally graded piezoelectric–fiber reinforced polymer cylindrical shells based on the effect of magnetic and electric potential, axial and circumferential wave number and speed rotation are obtained. These results are compared with other research and demonstrated a good agreement.

A Shell Description of Beams Incorporating Transverse Thickness Strain Energies for Receding Contact

Abstract The geometrically exact nonlinear deflection of a beamshell is considered here as an extension of the formulation derived by Libai and Simmonds (1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge, UK) to include deformation through the thickness of the beam, as might arise from transverse squeezing loads. In particular, this effect can lead to receding contact for a uniform beamshell resting on a smooth, flat, rigid surface; traditional shell theory cannot adequately such behavior. The formulation is developed from the weak form of the local equations for linear momentum balance, weighted by an appropriate tensor. Different choices for this tensor lead to both the traditional shell equations corresponding to linear and angular momentum balance, as well as the additional higher-order representation for the squeezing deformation. In addition, conjugate strains for the shell forces are derived from the deformation power, as presented by Libai and Simmonds. Finally, the predictions from this approach are compared against predictions from the finite element code abaqus for a uniform beam subject to transverse applied loads. The current geometrically exact shell model correctly predicts the transverse shell force through the thickness of the beamshell and is able to describe problems that admit receding contact.

Forced Vibrations of Multi-phase Crystalline Porous Shells Based on Strain Gradient Elasticity and Pulse Load Effects

Based on strain gradient theory (SGT), the present paper investigates pulse load effects on forced transient vibrations of porous crystalline shells. A crystalline material contains many voids inside it and also there are nano-size grains which define the material character. The formulation for crystalline cylindrical shell is provided by first-order shell theory and a numerical approach is used to solve shell equations. SGT gives a scale factor related to stiffness hardening provided by nano-grains. For more accurate description of size effects due to nano-grains or nano-void, their surface energy influences have been included. The size and radius of grains demonstrate notable impact on transient vibrations of crystalline shells. However, time responses of the crystalline shell are also dependent on strain gradient factor, void percentage, and shell radius.

Magnetoelastic Deformation of Isotropic Variable-Stiffness Shells of Revolution: Allowing for Joule Heat and Geometrical Nonlinearity

The theory and derivation of the nonlinear thermomagnetoelastic equations of flexible variable-stiffness shells of revolution with allowance for Joule heat in a nonstationary magnetic field are considered. A technique for design of flexible variable-thickness shells of revolution in a magnetic field is proposed. The thermomagnetoelasticity of a truncated flexible conical axisymmetric shell with Joule heating taken into account is analyzed.

Microscopic control actions of LaSMP actuators on hemispherical shells

Light activated shape memory polymer (LaSMP) is one of novel shape memory polymers with dynamic strain and Young’s modulus when exposed to UV lights. This study investigates distributed microscopic control actions of LaSMP patches on hemispherical shells with free boundary. Based on strain models established in the experiments, governing equations of the spherical shell are defined and control forces in the meridional, circumferential and transverse directions are derived respectively. In case studies of LaSMP patch laminated on the shell by covering specific wavelength at each mode in the circumferential direction, the meridional and circumferential forces turn to zero. Then, the dominant transverse control forces and its components resulting from the bending moments and membrane forces are compared at different modes. Parametric analyses are carried out to evaluate the microscopic actuation effects of actuator position, thickness and shell radius, LaSMP thickness, etc. LaSMP induced meridional and circumferential microscopic membrane/bending control actions on hemispherical shells are evaluated. The analyses indicate that the transverse control forces increase with increasing of LaSMP patch thickness and decreasing of hemispherical shell thickness and radius. Then, the normalized control forces are derived by dividing LaSMP patch size. The parametric analyses confirm that the normalized control forces caused by membrane forces dominate the control effect and the best control position is near the boundary of hemispherical shell. This work provides an analysis procedure on estimating LaSMP vibration control actions and gives theoretical predictions for idealized LaSMP vibration control of hemispherical shells.

On an improved 3D stress analysis for elastic composite shells

In this paper inhomogeneous shells subjected to static loading are considered. The theory is based on a multi–field functional, where the associated Euler-Lagrange equations include besides the global shell equations formulated in stress resultants, the local equilibrium in terms of stresses. Within representative volume elements the displacement field is enriched with warping displacements and relative thickness displacements. Elimination of a set of parameters by static condensation leads to a material matrix for the stress resultants. It is used in displacement based elements or in mixed elements with the usual 5 or 6 nodal degrees of freedom. A second model, where decoupling of the local and global boundary value problems is assumed, proves to be computationally very effective.

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

AbstractIn this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three‐dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two‐dimensional‐three‐components (2D‐3C) fluid problems. By introduce the S‐coordinate system in E3 and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.

Generalised assumed strain curved shell finite elements (CSFE-sh) with shifted-Lagrange and applications on N-T’s shells theory

AbstractWe present a simple methodology to design curved shell finite elements based on Nzengwa-Tagne’s shell equations. The element has three degrees of freedom at each node. The displacements field of the element satisfies the exact requirement of rigid body modes in a ‘shifted-Lagrange’ polynomial basis. The element is based on independent strain assumption insofar as it is allowed by the compatibility equations. The element developed herein is first validated on analysis of benchmark problems involving a standard shell with simply supported edges. Examples illustrating the accuracy improvement are included in the analysis. It showed that reasonably accurate results were obtained even when using fewer elements compared to other shell elements. The element is then used to analyse spherical roof structures. The distribution of the various components of deflection is obtained. Furthermore, the effect of introducing concentrated load on a cylindrical clamped ends structure is investigated. It is found that the CSFE3-sh element considered is a very good candidate for the analysis of general shell structures in engineering practice in which the ratio h/R ranges between 1/1000 and 2/5.