Classical Shell Theory Research Articles

Elastic local buckling of periodic sinusoidal corrugated composite panels subjected to in-plane shear

The critical local buckling instability of simply supported sinusoidal panels subjected to in-plane shear loading is investigated semi-analytically using the Rayleigh-Ritz method. Due to significant weight-strength saving and increased out-of-plane rigidity, these thin corrugated structural elements have gained wide recognition as an alternative to flat plate structural elements. To utilize these structures effectively in applications, such as civil infrastructure, aircraft wings, among many others, it is important to attain a comprehensive understanding of the failure mechanisms of these structures. Such corrugated panels are easily analyzed by modeling as an equivalent flat plate; however, this approximate method cannot capture the local buckling effect due to the complicated geometries of sinusoidal panels. Hence, a more precise solution is developed to accurately predict the local buckling behavior based on classical shell theory using a representative periodic structural element which encapsulates the periodic buckling nature of the panel in shear. Excellent correlation is observed with the results based on the numerical finite element analysis. Parametric studies are conducted to explore the effects of the thickness, aspect ratio, corrugated amplitude and material properties of the panel on buckling. The derived semi-analytical solution can accurately capture the local buckling behavior at any thickness, any aspect ratios, and high corrugated amplitudes within the range of thin-walled shells. The proposed semi-analytical solution can be confidently used to aid in efficient and accurate design analysis and optimization of corrugated panels.

Nonlinear forced vibrations analysis of imperfect stiffened FG doubly curved shallow shell in thermal environment using multiple scales method

This study investigates the non-linear vibrations of stiffened imperfect functionally graded double-curved shallow shells, as rested on nonlinear elastic foundations. The shells are exposed to external harmonic excitation and are placed in the thermal situations. The modeling of shells is derived according to the classical shell theory and the non-linear geometric von Kármán relationships. It is considered that the distribution of material properties changes along the thickness direction based on a power law index. The smeared stiffener technique is considered to model the stiffened shells. An approximation, according to Galerkin’s approach, is utilized to reduction of the shell governing equations into the non-linear coupled ordinary differential relations. The ODE equations are analytically solved and analyzed through the perturbation methodology for investigating the resonance behavior of shells. Simulation results are reported to examine the influences of stiffeners, initial imperfection, foundation coefficients, thermal environment, and geometrical characteristics on the non-linear primary resonance response of doubly curved shallow shells. Also, the nonlinear dynamic behaviors are analyzed by numerical methods through the bifurcation diagrams, and the nonlinear dynamical behaviors of the shell for different value of parameters are examined.

The Effect of Porosity on Elastic Stability of Toroidal Shell Segments Made of Saturated Porous Functionally Graded Materials.

This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

Non-Linear Response of Torsional Buckling Piezoelectric Cylindrical Shell Reinforced with DWBNNTs Under Combination of Electro-Thermo-Mechanical Loadings in Elastic Foundation

Nanocomposites provide new properties and exploit unique synergism between materials. Polyvinylidene fluoride (PVDF) is an ideal piezoelectric matrix applicable in nanocomposites in a broad range of industries from oil and gas to electronics and automotive. And boron nitride nanotubes (BNNTs) show high mechanical, electrical and chemical properties. In this paper, the critical torsional load of a composite tube made of PVDF reinforced with double-walled BNNTs is investigated, under a combination of electro-thermo-mechanical loading. First, a nanocomposite smart tube is modeled as an isotropic cylindrical shell in an elastic foundation. Next, employing the classical shell theory, strain-displacement equations are derived so loads and moments are obtained. Then, the total energy equation is determined, consisting of strain energy of shell, energy due to external work, and energy due to elastic foundation. Additionally, equilibrium equations are derived in cylindrical coordinates as triply orthogonal, utilizing Euler equations; subsequently, stability equations are developed through the equivalent method in adjacent points. The developed equations are solved using the wave technique to achieve critical torsional torque. Results indicated that critical torsional buckling load occurred in axial half-wave number m = 24 and circumferential wave number n = 1, for the investigated cylindrical shell. The results also showed that with the increase in the length-to-radius ratio and in the radius-to-shell thickness ratio, the critical torsional buckling load increased and decreased, respectively. Lastly, results are compared in various states through a numerical method. Moreover, stability equations are validated via comparison with the shell and sheet equations in the literature.

Nonlinear hygrothermal vibration and buckling analysis of imperfect FG-CNTRC cylindrical panels embedded in viscoelastic foundations

This paper investigates the nonlinear free vibration and nonlinear dynamic hygrothermal buckling behavior of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical panels under hygrothermal environment. The nonlinear temperature distribution is assumed along the thickness direction. The structure is resting on a generalized nonlinear viscoelastic foundation which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness and damping considerations. The nonlinear von Kármán strain-displacement relations and the classical shell theory are considered to derive the governing equations of the cylindrical panels. The discretized equations of motion are obtained using the Galerkin method. The fourth order Runge-Kutta method is utilized to analyze the nonlinear dynamic hygrothermal buckling. The influences of the four CNTs distribution types, consist of the FG-O, FG-Λ, FG-X, and UD, are considered for the cylindrical panel. The effects of temperature, moisture, nonlinear viscoelastic foundations, initial imperfection, and material parameters on the nonlinear free vibration and nonlinear dynamic hygrothermal buckling behavior of the system are investigated.

Response investigation of viscoelastic cylindrical shells with geometrical nonlinearity effect under moving pressure: An analytical approach

In this research, an analytical method is presented to investigate the nonlinear response of viscoelastic cylindrical shells with the finite length under moving internal pressure by moderately large deflection assumption. The viscoelastic behavior is considered as elastic in the bulk and standard linear solid in the shear. The strain-displacement relations are defined using nonlinear von Karman relations. By employing Hamilton’s principle, the equations of motion are extracted based on the classical shell theory. These equations which are a system of nonlinear coupled partial differential equations, are solved by the method of multiple scale of the perturbation technique and the response due to moving pressure and the critical velocity are determined. Moreover, the effects of different geometrical and viscoelastic parameters on the results are investigated. The results are compared with the finite element method and the reported results in some literatures.

Vibrational behavior of composite cylindrical shells with auxetic honeycombs core layer subjected to a moving pressure

A mathematical procedure has been presented to study the vibrational behavior of a composite cylindrical shell under moving internal pressure. The shell contains three layers in which the internal and external layers have elastic and isotropic properties and the middle layer consists of an auxetic honeycomb structure which has negative Poisson’s ratio. The equations of motion have been extracted based on the classical shell theory and Hamilton’s principle. The governing equations which are a system of coupled partial differential equations, are solved using an analytical solution, and the natural frequencies, the critical velocities and dynamic response have been determined. To investigate the sensitivity of the geometrical parameters and material properties on the natural frequency, critical velocity, and dynamic response, a parametric study has been conducted. To verify the analytical method, the results have been compared with the finite element results and some available literature results.

Wave propagation analysis of laminated composite shell panels using a frequency domain spectral finite element model

This paper presents a frequency domain spectral finite element model (SFEM) for studying wave propagation in laminated composite shell panels. The SFEM formulation is derived using two different theories: (i) the first order shear deformation theory (FSDT) that takes shear deformation into consideration, and (ii) the classical shell theory (CST) that neglects shear deformation. The spectral approach here is based on the use of fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT). The spectral element uses exact shape functions and a frequency dependent exact dynamic stiffness matrix is obtained relating the nodal displacements to the nodal forces of an element. The formulation is validated by comparing the present results for wavenumber dispersion and natural frequencies with the published results. The present SFEM is then used to perform wavenumber dispersion, phase and group velocities, and wave propagation analyses of laminated composite shell panels. The numerical studies show that the wavenumbers obtained from the SFEM based on the CST are significantly different from those of the FSDT at high frequencies. The CST results for the group velocity and wave propagation response to tone burst excitation show significant deviation from the FSDT results for out-of-plane A0 flexural mode even at relatively low frequencies. The orthotropy ratio and lamination arrangements in a composite shell panel have significant effect on the group velocity as well as the flexural mode wave propagation response.

Boundary layer modeling of surface residual tension in postbuckling behavior of axially loaded silicon panels at nanoscale embedded in elastic foundations

Here, nonlinear buckling and postbuckling properties of cylindrical silicon nanopanels of finite length resting on elastic foundation exposed to axial compression have been studied by taking into account surface free energy (SFE) effect. Size-dependent governing equations were constructed by integrating classical shell theory and Gurtin-Murdoch elasticity theory. Surrounding elastic media were considered as Pasternak foundations. Using shell buckling boundary layer theory, the influences of large deflections and SFE were extended to nonlinear instability analysis of nanopanels under axial loads. Finally, a perturbation-based solution procedure was applied to extract explicit equations for nanopanel postbuckling equilibrium paths at different surface elastic constants and geometric parameter values. It was revealed that surface effect related to positive surface elastic constant values shifted the minimum postbuckling domain point to lower maximum deflection while materials with negative surface values of elastic constant, SFE effect shifted the minimum point of postbuckling domain to higher maximum deflections.

Nonlinear buckling and post-buckling behavior of shear deformable sandwich toroidal shell segments with functionally graded core subjected to axial compression and thermal loads

This paper presents an analytical investigation on buckling and post-buckling of shear deformable sandwich toroidal shell segments with functionally graded core and homogeneous face sheets. The effective material properties of core layer are graded in the thickness direction according to a simple power law distribution in terms of volume fraction index. The shells are surrounded by an elastic foundation and subjected to axial compressive or thermal loads. Reddy's third-order shear deformation shell theory (TSDT) is used to established formulations. A new solution using Galerkin method for solving a governing system of four-partial differential equations is carried out to obtain closed-form expressions of buckling stresses and post-buckling curves. Two cases of bowed-out and bowed-in sandwich toroidal shells are considered. Effects of material, geometric parameters and elastic foundation on stability behavior of the shells is analyzed in detail. The difference between bucking stresses basing on TSDT and those basing on classical shell theory (CST) is also clearly shown in this study.

A three‐dimensional static analysis of embedded single‐walled carbon nanotubes using the perturbation method

A three‐dimensional (3D) asymptotic local elasticity theory is reformulated for the static analysis of a simply supported, single‐walled carbon nanotube (SWCNT) embedded in an elastic medium under a transverse normal load at its inner and outer surfaces. Eringen's nonlocal constitutive relations are used to account for the small length scale effects in the formulation examining the static behavior of the SWCNT. The interaction between the SWCNT and its surrounding foundation is modeled as either a Winkler‐type or a Pasternak‐type model. The two‐dimensional (2D) nonlocal classical shell theory (CST) is derived as a first‐order approximation of the 3D nonlocal elasticity theory. The 2D nonlocal CST solutions can be modified order‐by‐order to asymptotically approach the exact 3D nonlocal elasticity solutions in a hierarchic and consistent manner. A parametric study related to some effects on the static behavior of the embedded SWCNT is undertaken, including the aspect ratio, the stiffness and shear modulus of the surrounding medium of the SWCNT, and the nonlocal parameter. The current 3D asymptotic solutions can provide a standard to assess the accuracy and convergence rate of various 2D nonlocal shell theories.

An analytical approach for the vibration behavior of viscoelastic cylindrical shells under internal moving pressure

In the current paper, the vibrational behavior of viscoelastic cylindrical shells under moving internal pressure is studied, analytically. The viscoelastic behavior is considered as viscoelastic in shear and elastic in the bulk. The equations of motion are extracted based on the classical shell theory by applying Hamilton’s principle. These equations which are a system of coupled partial differential equations are solved by employing a closed form mathematic method, and the natural frequencies, the critical velocity, and response due to moving pressure are determined. Moreover, the effects of different geometric and viscoelastic parameters on the results are studied. The results are compared with the finite element analysis and the results available in the literature.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Effect of agglomeration of the nanotubes on the vibration frequency of the multi-scale hybrid nanocomposite conical shells: a GDQ-based study

In the present manuscript, the generalized differential quadrature method (GDQM) will be employed in order to solve the free vibration problem of three-phase carbon fiber/nanotube/polymer hybrid nanocomposite conical shells, rested on a two-parameter elastic medium, regarding for the influence of the agglomeration of the carbon nanotubes (CNTs) on the dynamic responses of the shell for the first time. The equivalent material properties of the multi-scale hybrid nanocomposite will be derived according to a two-step homogenization procedure including the Eshelby-Mori-Tanaka method incorporated with the modified form of the rule of the mixture. Afterward, the strain-displacement relationships of the continuous system will be obtained using the classical shell theory for thin-walled conical shells. Extending the Hamilton’s principle for the conical shell, the motion equation of the problem will be enriched. At the end, the associated boundary condition will be used in associated with the GDQ discretization for the purpose of extracting the natural frequency of the continuous system. The verification tests reveal that the presented mathematical framework is powerful enough to estimate the dynamic response of the system. The results of this study show that existence of agglomeration phenomenon can dramatically affect the dynamic behaviors of the nanocomposite structure.

Active vibration control of nonlinear stiffened FG cylindrical shell under periodic loads

Active control of nonlinear vibration of stiffened functionally graded (SFG) cylindrical shell is studied in this paper. The system is subjected to axial and transverse periodic loads in the presence of thermal uncertainty. The material composition is considered to be continuously graded in the thickness direction, also these properties depend on temperature. The relations of straindisplacement are derived based on the classical shell theory and the von Karman equations. For modeling the stiffeners on the cylindrical shell surface, the smeared stiffener technique is used. The Galerkin method is used to discretize the partial differential equations of motion. Some comparisons are made to validate the SFG model. For suppression of the nonlinear vibration, the linear and nonlinear control strategies are applied. For control objectives, the piezoelectric actuator is attached to the external surface of the shell and the thin ring piezoelectric sensor is attached to the middle internal surface of shell. The effect of PID, feedback linearization and sliding mode control on the suppression of vibration for SFG cylindrical shell is presented.

Nonlinear dynamics of functionally graded graphene nanoplatelet reinforced polymer doubly-curved shallow shells resting on elastic foundation using a micromechanical model

The paper focuses on the nonlinear vibration of functionally graded graphene nanoplatelet reinforced composite doubly curved shallow shells resting on elastic foundations. The graphene nanoplatelet reinforced composites are assumed to be distributed uniformly and functionally graded through the thickness. The material properties are assumed to be temperature-dependent and are estimated through the Halpin–Tsai micromechanical model, while the Poisson’s ratio, density mass, and thermal expansion are implemented by the rule of mixtures. The mathematical formulation is developed based on the classical shell theory and Von Karman-Donnell geometrical nonlinearity assumption. The dynamical responses of a simply supported functionally graded-graphene nanoplatelet reinforced composite doubly curved shallow shells are obtained by employing the Airy’s stress function and the Galerkin’s method. The responses of nonlinear vibration as time history, frequency-amplitude curve, phase plane graphs, and Poincare maps are carried out in this paper. In addition, the effects of the environment, graphene nanoplatelets weight fraction, graphene nanoplatelets distribution patterns, and thickness-to-length ratio are scrutinized. The obtained results are also compared and validated with those of other studies.

A multiscale analysis for free vibration of fuzzy fiber-reinforced nanocomposite conical shells

Abstract In this paper, the free vibration analysis of fuzzy fiber reinforced (FFRC) nanocomposite truncated conical shell is investigated. The FFRC constructional feature is that the uniformly aligned carbon nanotubes (CNTs) are radially grown on the circumferential surfaces of unidirectional carbon fibers. Using a micromechanical model based on the simplified unit cell (SUC) method, the effective material properties of the FFRC conical shells are evaluated. The thin-walled classical shell theory and Hamilton's principle are used to extract the governing equations, and the Ritz method is used to solve the problem. The model predictions are compared with other numerical results available in the literature and the correctness of the proposed theoretical method is attested. Some novel results, including the vibration results of FFRC conical shell accompanied with different combinations of boundary conditions and different material and geometrical properties are presented. The results reveal that the FFRC conical shell vibration behavior is strongly dependent on the material properties, volume fractions of two reinforcements, geometrical characteristics and boundary conditions.

Isogeometric iFEM Analysis of Thin Shell Structures.

Shape sensing is one of most crucial components of typical structural health monitoring systems and has become a promising technology for future large-scale engineering structures to achieve significant improvement in their safety, reliability, and affordability. The inverse finite element method (iFEM) is an innovative shape-sensing technique that was introduced to perform three-dimensional displacement reconstruction of structures using in situ surface strain measurements. Moreover, isogeometric analysis (IGA) presents smooth function spaces such as non-uniform rational basis splines (NURBS), to numerically solve a number of engineering problems, and recently received a great deal of attention from both academy and industry. In this study, we propose a novel “isogeometric iFEM approach” for the shape sensing of thin and curved shell structures, through coupling the NURBS-based IGA together with the iFEM methodology. The main aim is to represent exact computational geometry, simplify mesh refinement, use smooth basis/shape functions, and allocate a lower number of strain sensors for shape sensing. For numerical implementation, a rotation-free isogeometric inverse-shell element (isogeometric Kirchhoff–Love inverse-shell element (iKLS)) is developed by utilizing the kinematics of the Kirchhoff–Love shell theory in convected curvilinear coordinates. Therefore, the isogeometric iFEM methodology presented herein minimizes a weighted-least-squares functional that uses membrane and bending section strains, consistent with the classical shell theory. Various validation and demonstration cases are presented, including Scordelis–Lo roof, pinched hemisphere, and partly clamped hyperbolic paraboloid. Finally, the effect of sensor locations, number of sensors, and the discretization of the geometry on solution accuracy is examined and the high accuracy and practical aspects of isogeometric iFEM analysis for linear/nonlinear shape sensing of curved shells are clearly demonstrated.

Effective Model of Load-Bearing Layers for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Cylindrical Irregular Shells of Revolution

The use of effective approximations that increase the rate of convergence of numerical results when constructing a finite element model of bearing layers for a more accurate layer-by-layer analysis of the stress-strain state of three-layer irregular cylindrical shells is considered. It is believed that the carrier layers are sufficiently thin and rigid, and two-dimensional finite elements of natural curvature, constructed on the basis of the classical theory of moment shells, are used to simulate the stress-strain state. It is assumed that the aggregate layer can be modeled in thickness by the required number of three-dimensional finite elements, which allows one to take into account the change in geometric and physical-mechanical characteristics, as well as the parameters of the stress-strain state, not only along the meridional and circumferential coordinates, but also along the thickness of the shell and the aggregate layer. The finite element approximations considered allow one to reduce the order of systems of equations, i.e. to reduce the dimensionality of the problems being solved in comparison with the traditionally used approximations, which is especially important for layer-by-layer analysis of layered-heterogeneous structures. The high convergence rate of the numerical results obtained using the considered finite element model of the bearing layers is confirmed by a comparison with other known finite elements.

Postbuckling Behavior of Carbon-Nanotube-Reinforced Composite Toroidal Shell Segments Subjected to Thermomechanical Loadings

Motivated by evident absence of studies on doubly curved closed nanocomposite shells under complex thermomechanical loading conditions, this paper presents an analytical investigation on the buckling and postbuckling behavior of the carbon-nanotube-reinforced composite toroidal shell segment surrounded by an elastic foundation, exposed to preexisting thermal loadings and subjected to axial compression, external pressure, and combined mechanical loads. Carbon nanotubes are reinforced into the matrix phase through uniform or functionally graded distributions. The material properties of the constituents are assumed to be temperature dependent, and the effective properties of the nanocomposite are estimated by an extended rule of mixture. Two temperature conditions and various situations of mechanical loads are considered. Basic equations are established within the framework of the classical shell theory taking into account the geometrical nonlinearity and surrounding medium-shell interaction. Multiterm solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to obtain nonlinear load-deflection relations from which buckling loads and postbuckling paths are determined. Numerical examples are carried out, interesting discussions are given, and beneficial and deteriorative effects of various factors are analyzed.