Doubly-curved Shells Research Articles

Static Analysis of Doubly-Curved Shell Structures of Smart Materials and Arbitrary Shape Subjected to General Loads Employing Higher Order Theories and Generalized Differential Quadrature Method

The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for nonuniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories.

Seismic performance of buildings supported by a shallow doubly-curved shell raft foundation

Foundation transmits loads of superstructure to soil layers. Also, seismic loads are transmitted to the superstructure via the foundation. It has been shown that the characteristics of the foundation affect the seismic response of the superstructure. The influence of shell foundations on the seismic performance of structures has not been investigated in the literature. In this paper, the seismic response of 5, 10, and 15-story steel moment-resisting structures supported by a shallow doubly-curved shell raft foundation is compared to the seismic response of these structures supported by a pile-raft foundation. It was assumed that structures are founded on soft-clayey soil. Soil-foundation-structure systems were analyzed under the influence of four earthquake excitations using fully nonlinear time-history analysis. The numerical results generally show that the maximum displacement decreases while the maximum shear force increases in each story of structures with shell foundation compared to structures with pile-raft foundation. Moreover, the results reveal the better seismic performance of structures supported by shell foundation in terms of the maximum inter-story drift compared to structures supported by pile-raft foundation.

Generalized higher order layerwise theory for the dynamic study of anisotropic doubly -curved shells with a mapped geometry

An innovative generalized Layer-Wise (LW) approach is proposed for the modal analysis of anisotropic doubly-curved shells with an arbitrary shape by employing the Generalized Differential Quadrature (GDQ) method. The geometrical description of shell structures can be traced from both the differential geometry and mapping technique based on a proper definition of the blending functions.The kinematic field is described in each lamina by employing several interpolating polynomials in a generalized setting, while developing a LW theory that involves different orders of kinematic expansion. The Equivalent Single Layer (ESL) generalized formulation is, thus, derived as particular case of the LW theory. Angle-ply generally anisotropic laminated shells are here investigated, as well as layups with a thick foam softcore. Several numerical examples are illustrated, including different boundary conditions of the mapped geometry, different geometric curvatures for both unmapped and mapped paths, along with a variety of interpolating polynomials and expansion orders of the kinematic field. The accuracy of the proposed model is well-established by a comparative evaluation between our solutions in terms of frequencies and mode shapes and predictions based on classical 3D finite elements.

Wave dispersion characteristics of graphene reinforced nanocomposite curved viscoelastic panels

This paper tries to investigate the wave phenomenon in nanoshell made of graphene nanoplatelets (GNPs) reinforced nanocomposites. The second-order shear deformation theory in curvilinear coordinate is utilized to develop the doubly-curved shell as a continuous structure and the general nonlocal strain gradient theory is adopted to calculate nonlocality and strain gradient size-dependency. The effective material properties are estimated through the Halpin–Tsai micromechanical model and a modified rule of mixture. A virtual work of Hamilton statement is employed to obtain the governing motion equations and then a harmonic solution based on an analytical procedure is developed to find the wave response. Afterwards, a parametric study is performed to analyze the effects of the linear spring and damper coefficients, small-scale parameters, different curves, radius to thickness ratio, GNP's weight fraction as well as its number of layers on the phase velocity response of the GNPs reinforced doubly curved nanoshell with various shape panels resting on a visco-Winkler medium.

The clamps of sheet blank movement trajectory simulation under the shell part stretch forming process

The main object for analyzing the results of modeling and optimizing the shaping method by the skin is the graphs of the change in thickness at characteristic or selected points on the surface of the shell of double curvature. The article discusses two variants of the method, which have the same cyclic diagram of step movements of the main and auxiliary working cylinders of the stretch-tightening press RO-630-11. Both options contain kinematic techniques that implement the following two effects. The first is the effect of isometric bending of the shell surface after unloading the sheet blank at the third step motion of the press according to the cycle diagram. The second is the effect of the forced position of a flat and sufficiently long sheet blank in rectilinearly arranged discrete clamping devices of the press. This provided a plastic deformation of the middle part of the sheet blank without localizing the thinning deformation and the alignment of its values in different zones of the sheet blank.

On the forced mechanics of doubly-curved nanoshell

This paper is determined to study the forced vibration response of doubly-curved shells including different shape panels. A power-full higher-order shear-deformation theory in curvilinear coordinate is developed to model the doubly-curved nano-size shell. Furthermore, a general nonlocal strain gradient theory is employed in order to catch up with both phenomena of small-scale behaves. The nanoshell is made of advanced composite materials whose effective material properties vary continuously through the z-axis. After estimating the effective material properties utilizing a modified power-law, a virtual work of Hamilton statement is applied over the theories to obtain both governing equations as well as boundary conditions. Afterwards, an analytical technique based upon double Fourier series is exploited to satisfy conditions in edges. The numerical examples are presented to reveal the effect of the power-law index, porosity coefficient, elastic medium, aspect and length-to-thickness ratios and small-scale parameters, highlighted by loading time interval, on the dynamic response (i.e., transverse deflection and stresses) of spherical- elliptical, hyperbolic as well as cylindrical nano-panels.

Higher-order modeling of anisogrid composite lattice structures with complex geometries

The work investigates the vibration behaviour of anisogrid composite lattice shell structures, typically formed by a system of geodesic unidirectional composite ribs. A homogenization approach is here embedded within an Equivalent Single Layer (ESL) formulation for doubly-curved shells, accounting for the geometric contribution of cell patterns. The lattice shells are modelled as anisotropic homogenized continuous structures characterized by effective stiffness parameters. The governing equations of motion are derived using the Higher-order Shear Deformation Theories (HSDTs) of shells, whereas the Generalized Differential Quadrature (GDQ) method is applied to determine the fundamental frequencies with a reduced computational effort. A piecewise field variable assumption is considered for a proper description of zigzag interfacial effects between the lattice core and the external skins. The reliability and efficiency of the proposed numerical strategy is verified by comparison with finite element-based predictions. A systematic analysis aims at studying the effect of different geometric and stiffness parameters, e.g. the number of ribs, their cross-sectional dimensions, orientation and spacing, on the magnitude of the fundamental frequencies for some lattice structural members that could be of great interest for design purposes in the aerospace or automotive engineering practice.

The closed-form solutions for buckling and postbuckling behaviour of anisotropic shear deformable laminated doubly-curved shells by matching method with the boundary layer of shell buckling

The closed-form solutions for buckling and postbuckling behaviour of anisotropic shear deformable laminated doubly-curved shells by matching method with the boundary layer of shell buckling

Metric transformation of the surface of a tight punch for forming sheet shells of double curvature

For the first time, the problem of parametric reduction of the circumscribing surface of the shell of double curvature to the main axes and to the planes of symmetry defined by Gaussian coordinates is considered in detail. The solution of this problem is essential for ensuring the conditions of symmetrical shaping of the double-curvature circumflex shells by tight fit. The tight punch is presented in the form of a technological electronic layout, containing metric information of the surface of double curvature and storing information of the geometric alignment of the skin that extends to the outer circumferential surface of the unit. In this case, the circumferential surface of the shell of double curvature remains invariant with respect to the coordinate transformation.

Improvement of the processes of forming tight shells of double curvature

The article analyzes the process of shaping double curvature shells made of aluminum sheet material used for covering the outer contour of an aircraft. The main rejection signs of casings made by the method of shaping by wrapping are considered. The main problems arising during the shaping of double-curvature shells by wrapping are presented, and the ways of their solution are considered and proposed.

Unified wavelet finite element formulation for static and vibration analysis of laminated composite shells

This study presents, for the first time, a unified wavelet finite element formulation for static and free vibration analysis of laminated composite shells, combining the wavelet finite element method and general shell theory. The governing equations of the proposed unified wavelet finite element formulation are derived from general shell theory and first-order shear deformation theory (FSDT) using the principle of minimum total potential energy. The formulated kinematics of laminated shells can be applied to flat plates, cylindrical shells, and doubly-curved shells with the selection of appropriate Lamé coefficients with surface metric tensor and radii of curvature parameters. With the excellent approximation of B-spline wavelets on the interval (BSWI) for structure analysis, BSWI scaling functions are used as interpolating functions to construct wavelet finite elements for laminated shells. The accuracy and efficiency of the proposed BSWI method are assessed through numerical comparisons with analytical 3-D elasticity solutions and other reference solutions in the literature for static and free vibration analysis of laminated composite plates, cylindrical shells, and doubly-curved shells.

Application of Haar wavelet discretization method for free vibration analysis of inversely coupled composite laminated shells

In this paper, the free vibration behavior of inversely coupled composite laminated shells (ICCLSs) are studied using Haar wavelet discretization method (HWDM) for the first time. Two doubly-curved shells are inversely coupled on both end of cylindrical shell, in which the doubly-curved shell contains the elliptical, parabolic and hyperbolic shells. The individual shells are coupled by the continuous condition. The boundary condition is modeled by using the artificial spring technique. Based on the first order shear deformation theory (FSDT), the displacement field of coupled shell are established. All displacement functions containing the boundary and continuous conditions are expanded by Haar wavelet series in the meridional direction, and by trigonometric series in the circumferential direction. The boundary and continuous condition functions are contained to satisfy the constants expressed by Haar wavelet series integral form in the function of main system. The convergence and accuracy of current method for the proposed structure are verified by comparison with the finite element method (FEM). The results show that this method is very reasonable for analyzing the free vibration of ICCLSs. Then, the effects of geometric dimensions, fiber orientation angle and lamination type on the natural frequency of ICCLSs are investigated. In end, the new results on natural frequencies and mode shapes of ICCLS are presented, and they will be benchmark data for further research.

Vibration characteristics of piezoelectric functionally graded carbon nanotube-reinforced composite doubly-curved shells

This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube (CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton’s principle. Navier’s solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.

An approach to the solution of nonlinear forced vibration problem of structural systems reinforced with advanced materials in the presence of viscous damping

In this study, the nonlinear forced vibration of composite structural systems such as plates, panels and shells reinforced with advanced materials in the presence of linear viscous damping is investigated. Hamilton principle and von Kármán-type nonlinear theory are used to obtain the theoretical model of double-curved shells reinforced by carbon nanotubes (CNTs). The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method. By using the multiscale method, the frequency-amplitude relation and nonlinear forced vibration frequency of structural systems are obtained for the first time. Since double-curved shells can be transformed into other structural systems such as spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panel in special cases, the expressions for nonlinear frequencies can also be used for them. In additional, the backbone curve and the nonlinear frequency/linear frequency ratio are determined as a function of the amplitude in primary resonance for the first time. The results are verified by comparing the reliability and accuracy of the proposed formulation with those in the literature. Finally, a systematic study is aimed at controlling the influence of nonlinearity and types of distribution of CNTs on the frequencies and their quantitative and qualitative variation in the presence of external excitation and viscous damping.

Eigenfrequencies of prestressed variable stiffness composite shells

The introduction of the variable stiffness concept has broadened the design space for high-performance lightweight composite structures. In particular, when considering prestressed dynamically excited aerospace components, a wider design space allows designers to find more effective solutions with higher overall stiffness and fundamental frequency. In this context, an efficient and versatile Ritz method for the eigenfrequency analysis of prestressed variable stiffness laminated doubly-curved shells is presented. First-order shear deformation theory is considered without further assumptions on the shallowness or on the thinness of the structure. A rational Bézier surface representation is adopted for the description of the shell, allowing general orthogonal surfaces to be represented. The proposed approach is validated by comparison with published benchmark results and finite element solutions, showing great accuracy with 65% to an order of magnitude fewer variables. Due to the reliability and efficiency of the Ritz method for eigenfrequency analysis, original parametric studies are carried out, showing the flexibility given by the variable stiffness concept for finding trade-off solutions for prestressed shell components.

Higher order formulations for doubly-curved shell structures with a honeycomb core

Anisotropic doubly-curved shells reinforced with a honeycomb core are innovative structures for applications in civil, biomedical, and aerospace engineering. In this context, the homogenization technique represents one of the simplest way for analyzing such complex structures. A proper formulation must be capable to give accurate results for any cell configuration and/or curved shape. In the present work an innovative model is proposed, based on an Equivalent Single Layer (ESL) approach and higher order theories, for an accurate estimation of the vibrational response of plates, panels and shells, whose results are compared with predictions from a classical Finite Element Method (FEM). The work starts with a comparative study performed on aluminum sandwich plates with hexagonal, rectangular and re-entrant cells. Then, a sensitivity analysis evaluates the dynamic response of single- and doubly-curved panels with different cell typologies. The fundamental equations are tackled numerically by resorting to the 2D Generalized Differential Quadrature (GDQ) method. The influence of the kinematic assumptions throughout the thickness on the dynamic response of shells is investigated, accounting for different Representative Volume Element (RVE) deformation effects within the homogenized model. In all the analyses, cell units are analyzed by means of different geometric angles, thin and thick cores, as well as classic and double thickness vertical walls or commercial honeycomb cores.

Nonlinear free and forced vibration of porous piezoelectric doubly-curved shells based on NUEF model

In this paper, the nonlinear free and forced vibration of porous piezoelectric doubly-curved shells resting on visco-Pasternak foundation is performed within the framework of the nonuniform electric field (NUEF) model and von Kármán geometric nonlinearity assumption. Three types of porosity distribution are adopted to study the effect of porosity on the nonlinear dynamic responses of the current system. According to Hamilton’s principle, the nonlinear governing differential equations are obtained. For the nonlinear free vibration of porous piezoelectric doubly-curved shells, an analytical solution is achieved by utilizing the harmonic balance method. For the nonlinear forced vibration of the present system, the numerical solution is obtained by means of Runge–Kutta method. Through numerical examples, the relationship between nonlinear free vibration responses and nonlinear forced vibration responses is explored under different amplitudes of external excitation, width to thickness ratios, types of porosity distribution, electric field models and elastic foundation parameters. The main novelty of the present study is that the nonlinear frequency ratio that is calculated based on the nonlinear free vibration, is in excellent agreement with the resonant frequency ratio which is determined on the basis of the nonlinear forced vibration.

Thermal Buckling Analysis of Porous Conical Shell on Elastic Foundation

In this research, the thermal buckling analysis of a truncated conical shell made of porous materials on elastic foundation is investigated. The equilibrium equations and the conical shell`s stability equations are obtained by using the Euler`s and the Trefftz equations .Properties of the materials used in the conical shell are considered as porous foam made of steel, which is characterized by its non-uniform distribution of porous materials along the thickness direction. Initially, the displacement field relation based on the classical model for double-curved shell is expressed in terms of the Donnell`s assumptions. Non-linear strain-displacement relations are obtained according to the von Karman assumptions by applying the Green-Lagrange strain relationship. Then, performing the Euler equations leads obtaining nonlinear equilibrium equations of cylindrical shell. The stability equations of conical shell are obtained based on neighboring equilibrium benchmark (adjacent state). In order to solve the stability equations, primarily, due to the existence of axial symmetry, we consider the cone crust displacement as a sinusoidal geometry, and then, using the generalized differential quadrature method, we solve them to obtain the critical temperature values of the buckling Future. In order to validate the results, they compare with the results of other published articles. At the end of the experiment, various parameters such as dimensions, boundary conditions, cone angle, porosity parameter and elastic bed coefficients are investigated on the critical temperature of the buckling.

Higher order theories for the vibration study of doubly-curved anisotropic shells with a variable thickness and isogeometric mapped geometry

Anisotropic materials are widely used in many complex structures and panels, whose modelling can represent a challenging issue for engineers, especially when variable geometries are involved. In the present work we propose a novel strategy, based on the Equivalent Single Layer (ESL) approach, to analyze doubly-curved shells with constant and/or varying thicknesses made of completely anisotropic materials. An isogeometric mapping procedure is applied to define the reference surface, thus enabling to study a wide range of structural shapes. Different higher order theories are here applied in a unified setting to describe the kinematic field, where the governing equations of the problem are realized applying the Hamiltonian Principle, and solved numerically in a strong form with the 2D Generalized Differential Quadrature (GDQ) method. Several examples are illustrated for validation purposes of the proposed method, whose results are compared with predictions from 3D finite elements. A large parametric investigation is also performed to assess the sensitivity of the vibration response of generally-curved shell structures to different stacking sequences, material typologies, and boundary conditions.

Fabrication of doubly-curved CFRP shell structures with control over fiber directions

We introduce a fabrication method of doubly-curved shell structures by CFRP with control over fiber directions. Doubly-curved surfaces are tessellated into structured quadrilateral patches according to the purpose of use. One can design the tessellation in the parameter space, and map it to the 3D parametric surface, or use the orthogonal net of curves on surfaces such as lines of curvature, principal stretch lines, and principal stress lines. These patches, which we call generalized principal patches, are flattened onto a plane, and connected one by one by aligning the equi-length adjacent edges using translations and rotations forming generalized principal strips. Carbon fiber tows are placed onto these strips with one stroke by tailored fiber placement embroidery machines so that the fibers are not disconnected within strips. Preforms are stacked layer by layer into a mold of particular parts, and VaRTM or L-RTM method is employed to fabricate CFRP parts. We demonstrated the effectiveness of our proposed method by manufacturing an automobile hood and a marine propeller blade.