Shallow Shells Research Articles

Nonlinear thermo-mechanical analysis of ES double curved shallow auxetic honeycomb sandwich shells with temperature-dependent properties

It is the first time the nonlinear thermo-mechanical analysis (buckling and postbuckling) of imperfect eccentrically stiffened (ES) double curved shallow shells made of auxetic material under uniform external pressure, axial compressive and thermal loads has been studied by adopting analytical method. The double curved shallow sandwich shell is constructed by an auxetic honeycomb core layer with negative Poisson’s ratio and two isotropic homogeneous skin layers. In the present study, the outer surface of the structure is reinforced by a system of FGM stiffeners and taking the influence of pores in the stiffeners into account to increase the load capacity of the structure. Besides, the material properties of the shells and stiffeners are both temperature-dependent. The first-order shear deformation theory is applied to derive equilibrium and compatibility equations of double curved shallow auxetic shell, considering both the geometrical nonlinearity in the Von Karman sense and initial geometrical imperfections and shells placed on the Winkler – Pasternak foundation. The Galerkin and stress function methods are proposed to determine the critical load value and the deflection– load curve by taking the root form of the two-term rotation angles to give the Galerkin integral an equilibrium equation. Several comparisons with results in other literature are made to examine the reliability and accuracy of the method used in the present study. Finally, the effects of thermal environment, reinforced stiffeners, elastic foundation, pores and parameters of auxetic core layer on the buckling and postbuckling loads are evaluated in this paper. The numerical and graphical results show that the reinforced stiffeners, auxetic core, temperature and porous distribution have significantly influenced on the buckling and postbuckling loads of ES double curved shallow auxetic honeycomb sandwich shells.

Forced vibration analyses of FGP shallow shells with variable curvature

Forced vibration analyses of FGP shallow shells with variable curvature

Accurate Results for Free Vibration of Doubly Curved Shallow Shells of Rectangular Planform (Part.1)

A method is presented for determining the free vibration frequencies of doubly curved, isotropic shallow shells under general edge conditions and is used to obtain accurate natural frequencies for wide range of geometric parameters. Based on the shallow shell theory applicable to thin thickness shells, a method of Ritz is extended to derive a frequency equation wherein the displacement functions are modified to accommodate arbitrary sets of edge conditions for both in-plane and out-of-plane motions. In numerical computation, convergence is tested against series terms and comparison study is made with existing results by other authors. Twenty one sets of frequency parameters are tabulated for a wide range of shell shape and curvature ratio to serve as data for future comparison and practical design purpose.

Experimental verification of multi-fibre hybridization influence on dynamic deflection and stress values of curved composite panel

The finite-element time-dependent deflection and stress responses of the shallow composite panels subjected to variable mechanical loadings (uniformly distributed load and sinusoidally distributed load) are reported in this article. The study reveals the influence of the advanced fibre hybridization (glass–carbon–kevlar) on the dynamic responses and establishes the accuracy of the numerical responses by comparing them with the experimental values. The numerical steady-state deflections of the hybrid composite structure are evaluated using a generic mathematical model derived through Reddy's third-order shear deformation in conjunction with the finite-element technique and Newmark's time integration scheme. The experimentally evaluated mechanical properties of the fabricated composites are utilized in the computations. The validity of the computed solutions is ascertained with in-house experimental (transient deflection) results. Besides, the numerical model is extended to assess the parametric dependence (aspect ratio, thickness ratio, curvature ratio, geometry, hybrid schemes and support conditions) of the dynamic deflection/stress responses of hybrid composite shallow shell panels subjected to variable mechanical loading types.

Nonlinear thermal flutter analysis of variable angle tow composite curved panels in supersonic airflow

The advanced Automated Fibre Placement (AFP) technologies make the synthesis of Variable Angle Tow (VAT) composites possible, which provides an extended space for the design of lightweight structures. At the same time, the variable stiffness feature inherent in tow-steered composite panels also poses a huge challenge to solve the mechanical problems including the panel flutter problem. In this article, the nonlinear thermal flutter characteristics of an infinitely long VAT composite curved panel in supersonic airflow is investigated for the first time. The proposed panel model is based on the Donnell’s shallow shell theory and accounts for the von Kármán’s geometrical nonlinearity. The aerodynamic load induced by the supersonic airflow passing over the panel surface is constructed from the first-order piston theory, whilst the thermal load imposed on the curved panel follows the quasi-steady thermal stress theory. The aerothermoelastic equations of motion that govern the nonlinear thermal flutter problem are determined via Hamilton’s variational principle. The Galerkin’s method is applied to convert the partial differential governing equation with variable coefficients into a set of ordinary differential equations, and the resulting nonlinear equations dependent upon time are solved through the fourth order Runge–Kutta method. Several qualitative tools, including the time history, phase portrait, Poincaré map and bifurcation diagram, are employed to study the dynamic behaviours, and a variety of attractors, both regular and chaotic, are observed. The accuracy and effectiveness of the proposed Galerkin procedure are validated by comparing with those available in literature. Effects of height-rise parameter, temperature parameter and fibre orientation angle on the nonlinear flutter responses of VAT composite curved panels in thermal environments are discussed in details.

Photo-Activated Snap-Through of Nematic Shallow Spherical Shells

Photo-Activated Snap-Through of Nematic Shallow Spherical Shells

Accurate Results for Free Vibration of Doubly Curved Shallow Shells of Rectangular Planform (Part.2 Thickness effect)

This paper presents a follow-up study of a previous work that deals with the free vibration of moderately thin isotropic shallow shells under general edge conditions. The same semi-analytical method is used in this study for identical shape and degree of curvature in doubly curved geometry, and accurate natural frequencies are tabulated for a wide range of the shell edge conditions. Emphasis is made, however, to present the frequency parameters for the shallow shells with very thin thickness (representative length/shell thickness=100). In numerical experiments, convergence test is made against series terms in the case of very thin shallow shells. Twenty-one sets of frequency parameters are tabulated for three shell shapes (spherical, cylindrical and hyperbolic paraboloidal shells) and two curvature ratios. These two papers (Part.1 and 2) will constitute the accurate standard in the area of shallow shell vibration of rectangular planform and serve for future comparison and practical design purpose.

Mechanism study and power transmission feature of acoustically stimulated and thermally loaded composite shell structures with double curvature

Because of the modification in environmental conditions, thermal stresses may induce dynamic instability in a composite shell model. Therefore, according to the double Fourier series solution, an analytical strategy is proposed in this study for the first time to acoustically determine the vibration response of a doubly curved composite shell in a thermal environment. The main aim is to show the effect of thermal loads on sound transmission loss (STL). The motion equations of the structure integrated with thermal stresses are derived considering shear deformation shallow shell theory (SDSST) through Hamilton’s principle. Before verifying the outcomes, the convergence of the STL curve is also checked to prove that the obtained results are reliable. Parametric studies are performed to either show the temperature effect on acoustic characteristics or propose a 3D configuration of thermo-acoustic pressures through the system. Based on the temperature variations, it is found that there is an opposite trend between the increase of this term and the STL of structure. This issue also causes to move the dip point of the STL spectrum to lower frequencies. To manifest the fact of these behaviors, the natural frequency ratio against temperature rise is also plotted.

On the free vibration problem of thin shallow and non-shallow shells using tensor formulation

Linear free vibration of thin shells with rectangular base such as parabolic conoids, elliptic paraboloids and hyperbolic paraboloids are investigated. These shells are described by non-orthogonal curvilinear coordinates hence, Koiter theory, which uses a tensor formulation, is applied to accurately describe the strain–displacement relations. Four different boundary conditions are considered: simply supported with movable edges, simply supported with immovable edges; fully clamped with movable edges and fully clamped with immovable edges. Using an energy approach, Rayleigh–Ritz method is applied to obtain a set of ordinary differential equations of motion and Fourier series and Legendre polynomials are used to describe the displacements fields of the shell. Both, shallow and non-shallow shells were studied and obtained results of natural frequencies and vibration modes were compared with those from Maguerre theory and FEM using Abaqus, as can be seen, a good agreement was observed with FEM showing the efficacy of the applied theory. When comparing natural frequencies to those from Marguerre theory, there are substantial differences, indicating that lack of use of the tensor formulation which is not appropriate to shells with non-orthogonal coordinates.

Load-bearing capacity of rigid-plastic reinforced shallow shells and plates

The extreme problem of finding upper (kinematic) limit load of rigid-plastic fiber-reinforced spherical shallow shells and plates, which are circular in plan, is formulated and solved. The structures are clamped and have rigid insert. Two-layer shell model is used. The materials of components are rigid-perfectly plastic and asymmetric in tension and compression. The binder is isotropic or orthotropic. The structural model of composition takes into account two-dimensional stress state in the components. The piecewise-linear yield curves are constructed for radial-circumferential and spiral reinforcement. Comparisons results with other studies are given and influence of reinforcement parameters on limit load is investigated.

Semi-analytical solutions for buckling and free vibration of composite anisogrid lattice cylindrical panels

In this study, highly accurate approximate closed-form solution is presented for buckling and free vibration of clamped composite anisogrid lattice cylindrical panels. Using theory of orthotropic cylindrical shallow shells based on the continuous model of lattice structures, the system of governing partial differential equations (PDEs) of motion is derived. The resulted PDEs are converted to a couple of systems of ordinary differential equations (ODEs) by employing of the Extended Kantorovich Method (EKM). Semi-analytical solutions are presented for both buckling and free vibration of anisogrid lattice panels via iterative application of infinite power series solution for the sets of ODEs. Efficiency of the presented method in terms of accuracy, stability, fast convergence and low computational cost for prediction of the critical buckling loads and natural frequencies are examined. Accuracy of the predictions for various cases are studied by comparing with available numerical and analytical results. Moreover, it is shown that the method can be used to obtain accurate solutions for buckling and free vibration of composite lattice rectangular plates by assuming large values for the radius together with small values for the angle of the panel. This implies stability of the method for large or small values while providing accurate predictions.

Free vibration and stability of hybrid nanocomposite-reinforced shallow toroidal shells using an extended closed-form formula based on the Galerkin method

This paper presents an analytical solution for free-vibration and stability analyses of hybrid-composite toroidal shells. The considered hybrid material incorporates the composition of fibers and nanocomposite-reinforced matrix (NCRM) such as carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs). The governing differential equations are extended based on the first-order transverse shear deformation theory (FSDT) and are solved by employing the approximation of Fourier series and Galerkin method. The considered hybrid composite material in the parametric studies are consisted of three layers in which the polymeric matrix are reinforced by CNTs in some layers and reinforced by GPLs in the other layers.

A Mathematical Analysis Method for Bending Problem of Clamped Shallow Spherical Shell on Elastic Foundation

A mathematical analysis method is employed to solve the bending problem of slip clamped shallow spherical shell on elastic foundation. Using the slip clamped boundary conditions, the differential equations of the problem are simplified to a biharmonic equation. Using the [Formula: see text]-function, the fundamental solution and the boundary equation of the biharmonic equation, a function is established. This function satisfies the homogeneous boundary condition of the biharmonic equation. The biharmonic equation of the slip clamped shallow spherical shell bending problem on elastic foundation is transformed to Fredholm integral equation of the second kind by using Green’s formula. The vector expression of the integral equation kernel is derived. Choosing a suitable form of the normalized boundary equation, the singularity of the integral equation kernel is overcome. To obtain the numerical results, the discretization of the integral equation of the bending problem is conducted. The treatment of singular term in the discretization equation is to use the integration by parts. Numerical results of rectangular, trapezoidal, pentagonal, L-shaped and concave shape shallow spherical shells show high accuracy of the proposed method. The numerical results show fine agreement with the ANSYS finite element method (FEM) solution, which shows the proposed method is an effective mathematical analysis method.

Nonlinear Static and Dynamic Buckling Analysis of Imperfect Stiffened Double-Layer FGP Shallow Spherical Shells Embedded Within Elastic Foundations

In this work, an analytical method is presented to analyze the nonlinear static and dynamic buckling of imperfect stiffened double-layer functionally graded porous (FGP) shallow shells subjected to external pressure. The shallow spherical shell consists of two layers, where upper layer of shell is nonporous functionally graded (FG) material and the lower layer of shell is FG porous. This spherical shell is resting on elastic foundations, which consist of two-term Winkler–Pasternak’s model. Two FG porous distributions types consisting of symmetric porosity distribution (SPD) and nonsymmetric porosity distribution (NSPD) are investigated. The porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Utilizing the classical theory of shells considering geometrical nonlinearity based on von Kármán–Donnell relations and Hooke’s law, the governing equations of stiffened double-layer FG porous spherical shells are extracted. To discretize the partial differential equation of shell and obtaining the system equations of motion, the Galerkin method is applied. Regarding the discretized equation of motion, the explicit expressions for dynamic and static critical buckling load are determined. To investigate the dynamic behavior, the equation of motion is solved via a numerical method named Runge–Kutta approach, and then a presented approach by Budiansky–Roth is employed to obtain a critical load for the nonlinear dynamic buckling. The influence of various porosity distributions and coefficients of the shallow shell, elastic foundations, and geometrical characteristics on the nonlinear static and dynamic buckling analysis of imperfect stiffened double-layer FG porous shallow shells is examined.

Dynamic Instability of Shallow Shells Interacting with a Three-Dimensional Potential Gas Flow

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

Computer simulation of stress strain state of functionally graded sandwich plates and shallow shells of the complex shape resting on elastic foundation

Computer simulation of stress strain state of functionally graded sandwich plates and shallow shells of the complex shape resting on elastic foundation

Theoretical and numerical studies on multi‐physical issues of space parabolic membrane antennas

Front-fed parabolic reflectors are among the most commonly used antennas in the industry. While in spaceborne applications, membrane reflectors are very promising due to their lightweight and foldable features. However, considering the large size, small thickness and low stiffness, solar radiation and microwave radiation will have considerable influences on the antennas’ shape accuracy as well as the radiation characteristics. In this article, a theoretical approach is presented to solve the multi-physical effects of the parabolic antenna. The deformation of the reflector is derived by the shallow shell theory, taking into account the solar pressure, the microwave pressure and the thermal effects due to solar and microwave heating. The far-field electromagnetic radiation pattern is then obtained by considering the deformation of the reflector. On the other hand, a numerical approach combining the finite element method, the multi-level fast multipole method, and the large element physical optics is also presented. Numerical examples suggest good agreement between the theoretical and numerical results. The methods have been applied into the analysis of design models in the Space Solar Power Station project. Also, these approaches can be directly extended into other space membrane reflector antennas.

Probing the buckling of pressurized spherical shells

The prediction of the critical buckling conditions of shell structures is plagued by imperfection sensitivity. Non-destructive testing through point-load probing has been recently proposed to map the stability landscape of cylindrical shells. However, the counterpart procedure for spherical shells is still debatable. Here, we focus on the mechanical response of pressurized spherical shells containing a single dimple–like defect to a point probe. Combining experiments, finite element modeling, and existing results from classic shell theory, we characterize the nonlinear force-indentation response of imperfect shells at different pressurization levels. From these curves, we seek to identify the critical buckling pressure of the shell. In particular, the indentation angle is varied systematically to examine its effect on the probing efficacy. We find that the critical buckling point can be inferred non-destructively by tracking the maxima of the indentation force–displacement curves, if the probe is implemented sufficiently close to the defect. When probing further away from the defect, the test fails in predicting the onset of buckling since the deformation due to indentation remains localized in the vicinity of the probe. Using FEM simulations and shallow shell theory, we quantify the characteristic length associated with this localized deformation, both in the linear and nonlinear regimes. Our results demonstrate the limiting conditions of applicability for the usage of probing as a non-destructive technique to assess the stability of spherical shells.

Meshless analysis for cracked shallow shell

A moderated thick double curved shallow shell with functionally graded materials subjected to static and dynamic loads are investigated by the meshless methods. Shear deformable plate theory is applied and the governing equation with respect to the middle surface is formulated in the Cartesian coordinate system with five degrees of freedom. The numerical solutions of the partial differential equations are obtained by the Finite Block Method in both strong form formulation (Point Collocation Method) and weak form formulation (Meshless Local Petrov-Galerkin) with Lagrange series interpolation and mapping technique. Two-dimensional plane-stress and plate bending problems are coupled in the governing equations. The second-order partial differentials are employed in the algorithm of strong form of governing equations and the first-order derivatives are required only in the weak form. The stress resultant intensity factors are evaluated by the Crack Opening Displacement (COD) and by the Variation Principle Method (VPM) techniques. The Laplace transform method and Durbin's inverse technique are utilized to deal with the crack problems under dynamic load. Comparisons have been made with analytical and numerical solutions with the finite element method in order to demonstrate the accuracy and convergence of the meshless approaches.

DEFORMATION AND BUCKLING ANALYSIS OF SHALLOW CONICAL SHELLS BASED ON IMPROVED MODEL OF WIND LOAD USING ENVIRONMENT ANSYS

The process of deformation and buckling of shallow thin-walled elastic conical shells has been investigated for the case of significantly non-uniform stress-strain state due to the action of wind load based on improved model of pressure application schema to the surface of shallow shell and for hinged hedge of border. An improved model of wind load was based on data presented in terms [5, 6] and was a logical continuation of previous investigation of wind action on shallow conical shells based on model of first approach [3]. Deformation and buckling process investigation has been carried out using software ANSYS which effectivity was approved by the fact of being used by NASA for its aerospace projects. A model of shallow conical shell has been made using four-corner finite element SHELL 281 with 8 nodes that let us obtain not only symmetrical relatively to the axis of rotation buckling form but an asymmetrical too. Two types of computation have been made during numerical modeling – linear bifurcation computation with determination of linear pressure qcr value and corresponding to it buckling form, and computation of geometrically non-linear problem of deformation with determination of limit pressure qlim and corresponding buckling form. Obtained buckling forms have been compared to the deformed shape of shell surface when aerodynamic computations have been carried out using software ANSYS. An estimation analysis has been made for case of application of improved model of wind load in comparison to the previous investigation according to the values of baring capacity and buckling shape coherence during resolution of static tasks and comparison to the results of aerodynamic solution. An analysis of base parameter influence has been carried out for the model of first approach and current improved model according to the bearing capacity value and local extremums on schema of pressure intensity distribution of wind load. Specific moments of deformation process computations based on improved model using environment ANSYS have been mentioned and of further analysis on the basis of improved model with it specifics have been given too.