Shell Curvature Research Articles

Snap-through of functionally graded graphene origami-enabled auxetic metamaterial doubly curved nonlinear shells

The lightweight design of thin-walled curved structures as spherical shells are frequently implemented in architecture, aerospace, mechanical, automotive, nuclear, and defense structures. Under the transverse loads, shells may largely deform and hence snap from one equilibrium position to the other. Thus, this work aims to develop a mathematical model and computational solution to investigate the nonlinear bending and snap-through behavior of doubly curved auxetic metamaterial shell subjected to transversal loading, for the first time. The shell is composed of several layers through the shell thickness, each layer is manufactured from a copper (Cu) matrix reinforced with a specified weight fraction of graphene origami auxetic metamaterial (GOAM). The mechanical properties of the GOAM shell panel are presented and described by functions of the GOAM volume fraction and folding degree. Three types of GOAM-distributions are considered, which are U-type, X-type, and O-type. The theoretical framework of Kirchhoff–Love hypotheses for thin shells and von Karman type nonlinearity are used to derive the governing equations. The differential quadrature method (DQM) is implemented to discretize the space domain and convert the nonlinear partial differential equation to nonlinear algebraic equations in terms of displacement field. An efficient incremental iterative procedure is developed to solve the nonlinear equations and predict the snap-through behavior. A model conversion and validation with isotropic spherical shell is considered. Several numerical results are conducted considering the effects of changing GOAM content, distribution pattern, folding degree, and shell curvature and thickness. For panels exhibiting snap-through behavior, increasing the shell curvature leads to higher snap-through limiting load; however, the instability gap is enlarged.

Curved Three‐Director Cosserat Shells with Strong Coupling

AbstractContinuum‐based shell models are an established approach for the simulation of thin deformables in computer graphics. However, existing research in physically‐based animation is mostly focused on shear‐rigid Kirchhoff‐Love shells. In this work we explore three‐director Cosserat (micropolar) shells which introduce additional rotational degrees of freedom. This microrotation field models transverse shearing and in‐plane drilling rotations. We propose an incremental potential formulation of the Cosserat shell dynamics which allows for strong coupling with frictional contact and other physical systems. We evaluate a corresponding finite element discretization for non‐planar shells using second‐order elements which alleviates shear‐locking and permits simulation of curved geometries. Our formulation and the discretization, in particular of the rotational degrees of freedom, is designed to integrate well with typical simulation approaches in physically‐based animation. While the discretization of the rotations requires some care, we demonstrate that they do not pose significant numerical challenges in Newton's method. In our experiments we also show that the codimensional shell model is consistent with the respective three‐dimensional model. We qualitatively compare our formulation with Kirchhoff‐Love shells and demonstrate intriguing use cases for the additional modes of control over dynamic deformations offered by the Cosserat model such as directly prescribing rotations or angular velocities and influencing the shell's curvature.

An analytical study of thermal environment’s effect on the free vibration of FGM double curved FG shells

The main objectif of this work is to examine the thermal environment’s effect of on the free vibration of double curvature shells made of functionally graded materials. A new mathematical model based on higher order shear deformation theory, is developed to incorporate the influence of dependent and independent thermal stresses, which can be uniform, linear, nonlinear, or sinusoidal on the mechanical properties of the double curvature shell. Two material couples, ( S i 3 N 4 / SUS 304 , ZrO 2 /Ti‐6Al‐4V ) , are considered as progressive materials, accounting for temperature-dependent and temperature-independent heat conduction and material properties in the thickness direction. The temperature field is assumed to be uniform across the shell surface but varies only in the thickness direction. The accuracy of the analytical results is validated by comparing them with existing literature results for functionally graded material (FGM) shells with infinite radii under dependent and independent temperature conditions. The study aims to demonstrate the thermal effects on material composition, as well as various specific geometric shapes such as plates, cylinders, spheres and ellipses and how different temperature laws influence the frequencies of vibrating double curvature FGM shells. Both existing theories accurately predict temperature-independent and temperature-dependent vibration responses of the shells. By verifying and comparing the results with the literature, a better agreement is achieved, thereby paving the way for further research in this field.

Vibrations of viscoelastic FG porous graphene-platelets reinforced doubly-curved shells via experimental characterisation of graphene platelets

This work explores the vibrations of viscoelastic functionally-graded (FG) porous graphene-platelets reinforced doubly-curved shallow shells via experimental characterisations of graphene platelets. In particular, the effects of ultrasonication time of graphene platelets, on the vibrations of graphene-platelets reinforced shells, are studied. This is of interest because graphene platelets are often ultrasonicated in solvents to achieve better dispersions during the fabrication of graphene-platelets/epoxy composite structures. In this study, geometric characteristics of the graphene platelets under different sonication times are quantitatively and qualitatively analysed through experimental characterisation techniques, including a laser diffraction particle size analyser, a scanning electron microscopy (SEM), and a scanning force microscopy (SFM). The geometric characteristics of graphene platelets are considered in the effective material properties using a micromechanics model of Halpin-Tsai. Different FG distributions of graphene platelets and closed-cell porosity are modelled. The Kelvin-Voigt viscoelastic model is used to consider the internal friction-induced energy dissipation for the polymeric composite shell. Hamilton’s variational principle and an assumed mode method are used to derive and solve the equations of motion, respectively, for the vibrations of the doubly-curved shells. The effects of graphene platelets sonication time, in combination with the effects of graphene platelets weight fraction, porosity coefficient, FG distributions, shell thickness ratio, and shell curvature radii, are scrutinised. The results show that increasing ultrasonication time reduces the shell’s out-of-plane dimensionless fundamental frequency as a result of the reduction in graphene platelets aspect ratio.

Study on the shape prediction of doubly curved shallow shell in flexible rolling process

Flexible rolling (FR) is a novel process for manufacturing doubly curved shallow shells with various shapes based on a pair of small-diameter bendable rolls. In this paper, the fundamental reason for bi-directional bending of the sheet metal in FR is explained. For the purpose of predicting the forming radii in the transverse and longitudinal directions more effectively, the exit velocity is divided into linear velocity and interference velocity, with the latter so small as to be considered to cause only elastic deformation of the sheet. The theoretical calculation model of the pre-curvature radius resulted from linear velocity is established. Using the mechanical theory of the doubly curved shallow shell, the transverse and longitudinal curvature changes caused by the interference velocity are derived and solved by the analytical approach, which are added to the transverse and longitudinal pre-curvatures respectively so that the actual curvature radii are obtained. In order to verify the reliability of the proposed shape prediction method of FR, numerical simulations and forming experiments for convex and saddle-shaped parts are carried out. Through measurement and comparison, it is found that the simulated and experimental results are extremely consistent with the theoretical data. Besides, the effect of important parameters on elastic deformation is investigated by means of the deduced formulas.

Supportless 3D-printing of non-planar thin-walled structures with the multi-axis screw-extrusion additive manufacturing system

Thin-walled structures, including the hollow tubes and curvature shells, are integral to a number of engineering applications. Manufacturing those structures with the recently emerged additive manufacturing scheme, such as the screw-extrusion additive manufacturing (SEAM), can greatly improve the manufacturing efficiency and reduce the overall manufacturing cost. Yet, the support structure is often required to print such structures, adding undesired weight to the otherwise light-weight structures. In this work, the SEAM is integrated with a 6-axis industrial robot to enable the dynamic reorientation of the nozzle, which stays perpendicular to the deposition surface. As a result, the geometrically-complex hollow tubes and arc-like thin shells can be directly printed without any support structure. In addition, by selectively designate the infill directions and starting points, a light-weight honeycomb-infilled curved shell can be directly printed without any additional processing. Mechanical tests have also shown the conformally-printed curved parts exhibit higher strength than the conventionally-printed counterparts. The result from this work can hopefully incentive new insights in developing novel strategies for 3D-printing functional thin-walled structures.

ПРЕОБРАЗОВАНИЕ СХЕМЫ ФОРМООБРАЗОВАНИЯ ОБТЯЖКОЙ НА ПРИНЦИПАХ СИММЕТРИИ

We assume that the concept of symmetry plays an important role in improving the theory and practice of the forming process by covering load-bearing thin-sheet shells of double curvature, which guarantees the production of contour-forming parts of the skin of modern aircraft with an exact geometric shape and minimal variation in thickness. To understand our assumption, it is necessary to remember that the complexity of the theory of plastic shells lies in the geometric and physical nonlinearity and the need to take into account shear stresses and deformations along the surface of the shell and its variability in curvature, therefore there is still no unified approach to the analysis of the formative processes of covering from sheet material. This assumption applies only to thin-sheet shells of double curvature, providing a different isometric position (curved surfaces one on top of the other) of the open surface as a whole, and not closed like a sphere. Therefore, the process of shaping a sheet blank into a shell part of an aircraft has some form of symmetry with respect to a given set of surface transformations, unless certain properties or relationships, the so-called invariants of the set of transformations, change when they are performed.

On issue of stress and curvature of shell of continuous cast billet

On issue of stress and curvature of shell of continuous cast billet

Multi-objective optimization for snap-through response of spherical shell panels

Composite spherical shell panels are commonly used for the lightweight design of thin-walled structures in the structure, architecture, aerospace engineering. Depending on the shell panel length, thickness, curvature, and stacking sequence, the load-deflection response under lateral load can be either a completely stable equilibrium path or an unstable snap-through path. In this work, the response of simply supported cross-ply composite spherical shell panels subjected to lateral static loads is studied and optimized. The spherical shell exhibiting snap-through response is optimized to (a) maximize the load required to initiate snap-through and (b) minimize the unstable region. Meanwhile, the spherical shell exhibiting continuous response is optimized to (a) maximize the softening-hardening load and (b) minimize the shell weight. The decision variables for both constraint multi-objective optimization (CMOO) problems include layup sequence, thickness, and the radius of curvature to side length ratio. The nonlinear governing equations are derived based on Kirchhoff–Love hypotheses for thin shells and Hamilton's principle. A novel semi-analytical method is presented based on Bernstein polynomials combined with a fast iterative approach to compute the objective features of the responses. Benefiting from the high efficiency and robustness of the proposed approach, the CMOO problems are solved by the metaheuristic Multi-Objective Artificial Hummingbird Algorithm (MOAHA). Composite spherical shell panels with various geometry and lamination configurations are considered to validate the performance of the proposed method. The optimization of geometric parameters (shell curvature and thickness) in addition to the staking sequence provide a high degree of tailoring of the shell performance to meet various objectives. These parameters are taken as design variables that must be optimized to achieve some objectives and satisfy arbitrary nonlinear constraints.

Buckling of Stiffened Heterogeneous Shells Taking into Account Material Creep

The authors propose a mathematical model of shell structures taking into account the linear theory of hereditary material creep. This model is based on the total potential strain energy functional. Shells are reinforced with stiffeners, and the contact between the stiffener and the shell skin along a strip is taken into account. The Ritz method is applied to the functional, and a system of algebraic or integro-algebraic equations is found. The resulting system is first solved as a linear system (without account for the terms reflecting material creep), and the state of the structure under the specified load is determined. Then the iterative method is used to solve the creep problem for a given value of time. The paper presents the results of studying stiffened shallow shells of double curvature made of reinforced concrete under uniformly distributed load. The buckling of shells occurs over time as a result of the development of creep strains. Values of the buckling load as a result of material creep are found. It is shown that stresses are redistributed over the shell field over time and the maximum stress is observed near the contour of the structure.

Damage performance and failure analysis of reinforced concrete doubly curved shells under seismic excitations considering curvature and thickness variation

Most of the past investigations on the structural response of the doubly curved shell (DCS) roofs under seismic excitation were limited to linear dynamic analysis, whereas nonlinear dynamic analysis was rarely introduced, making it difficult to understand their dynamic behavior and draw general conclusions. Therefore, in the current study, linear dynamic analysis (LDA) and nonlinear dynamic analysis (NLDA) are conducted in tandem to fill the gap in the literature and give a complete perception of how these roofs respond to earthquakes. To this end, an innovative automatic finite element-based design algorithm is developed to obtain the steel reinforcement in preparation for the NLDA. Accordingly, the current study exploits the multi-layered shell element incorporated with the multi-axial concrete damage plasticity model (implemented via VUMAT subroutine) and the developed design algorithm (via Python script) to perform the NLDA and accurately capture the structural response and damage performance of these roofs. Unlike most common structures, the results demonstrated that DCS is highly affected by the vertical component of the earthquake rather than the horizontal component. The NLDA revealed that the shell exhibits severe damage and a high displacement response under strong seismic excitations, which could not be captured via the LDA. Additionally, the failure analysis is performed for the studied roof, considering the variation in shell curvature and thickness. This analysis shows a notable decrease in the demand-capacity ratio and steel rebar’s stress with increasing the shell curvature. Consequently, according to the proposed performance levels, the shell’s state changed from the initial failure to the elastic state. In contrast, slight changes in the overall behavior are observed due to the variation in the shell thickness. Finally, a new formula is proposed to predict the damage induced in the DCS under seismic excitations.

Dynamic response of spherical shells subjected to the underwater impulse

The dynamic response of spherical shells under impact loads is investigated experimentally. The analysis focuses on the effects of flyer velocity, shell thickness, and shell curvature radius on shell deformation. The spherical shells are subjected to underwater shock waves of different intensities using a projectile-impact-based underwater shock tube. The dynamic response of the spherical shell is captured by the method of three-dimensional digital image correlation with high-speed photography. The response mechanism of the spherical shells is analyzed by the method of experiment and numerical simulations. The results show that the residual displacement of the center point of the target plate has a nonlinear relationship with the thickness and curvature radius of the target plate under the same underwater impact load. The slope of the residual displacement-curvature radius curve of the center point of the target plate increases with the decrease of the thickness under the same thickness, and the slope of the residual displacement-thickness curve of the center point of the target plate increases with the increase of the curvature radius under the same curvature radius. Quantitative structure–load–performance relation is carried out to facilitate the advanced study of metallic structures and guide the impact resistance design of spherical shells in marine structures.

Three-dimensional Non-linear Finite Element Analysis of Impact Response and Damage in Laminated Composite Shells

The impact response and the impact-induced damage in curved composite laminates subjected to transverse impact by a metallic impactor are studied using three-dimensional non-linear transient dynamic finite element formulation. A layered version of isoparametric eight-noded hexahedral element with incompatible modes is developed which incorporates geometrical non-linearity based on total Langragian approach. The non-linearity of both strain displacement relation and contact loading are simultaneously solved using Newton-Raphson incrementaliterative method. Impact-induced damages (matrix cracking and delamination) are predicted using appropriate three-dimensional stress-based failure criteria. Some example problems of graphite/epoxy laminated cylindrical shells with variation of important parameters such as impactor velocity, shell curvature, laminate dimension and fibre orientation of plies are solved and the influence of geometrical non-linear effect on both impact response and resulting damages is demonstrated.

Seismic evaluation cylindrical concrete shells

Abstract The reasons for concrete roof shells’ apparent seismic resistance have been subject to limited research but they have been shown to be inherently resilient to earthquakes. Shells constructed of concrete exhibit high structural efficiency and can therefore be made very thin. As a result of their relatively lightweight nature, thin shell structures are implicitly resistant to earthquake forces. The shell structure is typically designed so that it performs optimally under gravity loads, which are carried mainly by membrane action over the shell surface. As earthquakes induce unexpected horizontal forces, concrete shell structures can be damaged by bending stresses. By studying 8 cm-thick concrete roof shells using parametric analysis, this research shows that small and midsized (span <30 m) thin concrete roof shells can indeed be intrinsically earthquake resistant. These structures have high geometric stiffness and low mass, which results in fundamental frequencies far higher than those of realistic seismic events. Under earthquake excitation, these characteristics result in elastic shell behavior, without exceeding the maximum concrete strength. A shallow shell exhibits greater stress in response to earthquake vibrations caused by the vertical components than by horizontal components. Further, by increasing the rise and curvature of large shells, the fundamental frequency increases and the damaging effect of vertical earthquake vibration is reduced. The aim of this study in general is to show the analysis and the effect of earthquakes on cylindrical concrete shells.

Flügge shell theory and solution for the vibration analysis of a thin elastic cylindrical panel of varying curvature

In this paper, the free vibration behavior of isotropic and orthotropic cylindrical shell panels with variable curvature subjected to certain ends boundary conditions is analyzed by the transfer matrix technique and solved by the Romberg integration approach. Flügge’s shell theory is employed to model the governing three-dimensional equations of vibration and Fourier’s approach is used to deform the displacement fields as trigonometric functions in the longitudinal direction. As a result of a semi-analytical procedure is conducted, the coupled governing equations of panel problem can be written in a matrix partial differential equation of first order with variable coefficients and reduced from the two-dimensional problem to one-dimensional one which is solved numerically as an initial-value problem. The proposed model is applied to get the natural frequencies and mode shapes for the vibration behavior of two-side simply-supported cylindrical panels with simply-supported, free-free, clamped-free and clamped-clamped ends boundary conditions lying at their curved edges. The sensitivity of the vibration behavior to the variety of curvature, panel parameters and material orthotropy of shell is studied.

Design method of combined stiffened pressure hulls with variable-curvature characteristic

This study presents a combined pressure hull design method with variable-section ribs. This method for determining the length-radius ratio of the pressure hull's bearing unit is derived based on the stiffness-matching characteristic of the shell and stiffener. The bearing mechanism of variable curvature stiffened shell is discussed through theoretical derivation and numerical simulation. The design scheme of variable cross-section and the equal-stiffness rib is proposed to address the limitations of outer envelope size and inner space utilization of the pressure hull. Its buckling load is determined by the energy method. The design approach for a new type of stiffened pressure hull is summarized. In addition, the method is verified by the design example and the experiment. The results revealed that the combined variable curvature shells are a superior pressure hull configuration. Compared to the cylindrical stiffened shell, the buckling load of the pressure hull with variable cross-section ribs can be improved significantly (about 26.7%). At the same time, space utilization remains at the original level. The stiffened design technique described in this research based on elastic buckling theory can accurately determine the length-radius ratio of the bearing unit, relying on the design objective with high feasibility.

Hygro-thermo-mechanical analysis of sandwich shallow shells considering the effects of transverse normal strain

The static response of softcore doubly-curved sandwich shallow shells under hygro-thermo-mechanical loading is presented herein using a fifth-order shell theory which considers the effects of transverse normal strains and stresses at the same time. The present theory accounts for non-linear variations of displacements. The strain-displacement relations take into account only small curvatures following a shallow shell description. The present theory ensured the traction-free boundary conditions at the top and the bottom faces of the shell. The governing equations along with the natural and essential boundary conditions are derived using the principle of virtual work. The governing equations are solved by using Navier’s method. Three layered sandwich shallow shells of double curvature subjected to hygro-thermo-mechanical loading are analyzed. The accuracy and efficacy of the present theory are proved by applying it to the benchmark solutions available in the literature and then the present theory is extended for the analysis of softcore sandwich shells of double curvature. Since an exact 3D elasticity solution is not available for many such problems; this paper can be serving as a benchmark reference solution for future researchers working in this area.

Higher-order static and free vibration analysis of doubly-curved FGM sandwich shallow shells

Plenty of research papers are available on the static and free vibration analysis of single-layer FG shells, however, literature on the analysis of FGM sandwich shells of double curvature is limited. Especially, the authors have not found any paper on the analysis of hyperbolic and elliptical paraboloid FGM sandwich shells. Therefore, FGM sandwich shallow shells with double curvature are analyzed in this study for the static and free vibration conditions. Face sheets of sandwich shells are made up of functionally graded material whereas the core is made up of isotropic material. A functionally graded material considered herein is a combination of alumina and aluminum. Sandwich shallow shells on rectangular planform are modeled using various equivalent single-layer shell theories via unified formulation considering the effects of shear deformation and rotary inertia. Different shell theories recovered from the present unified formulation satisfy the transverse shear stress-free conditions on the lower and upper surfaces of the shell. Hamilton's principle is applied to the present unified formulation for establishing equations of motion. Solutions to free vibration and static problems of simply-supported sandwich shells are obtained using Navier's technique. The numerical results of frequencies, displacements, and stresses are obtained for various types of shells, different sandwich schemes, different values of the power-law factor, and radii of curvature. The results available in the literature are used for the comparison of the present results and found in good agreement with those. However, many new and useful results are also reported in this paper for the reference of readers.

Effects of different geometric patterns on free form gridshell structures

<p class="JIASSAbstract">Gridshells are commonly known as structures with the shape and rigidity of a double curvature shell consisting of a grid, not a continuous surface. In recent decades, these structures have attracted significant attention. The impact of various geometric patterns on free form gridshell structures is investigated here to demonstrate the necessity of collaboration between structural and architectural characteristics in enhancing structure efficiency. To that goal, a framework is proposed where three shells are first designed, and then six geometric patterns are formed on them. The main factors for evaluation of gridshells are decreasing the steel weight as an economic index and decreasing the displacement as a structural index, also, finite element method is used for structurally analyzing the gridshells, and the generated gridshells are compared to each other based on the mentioned indices. For the optimization process, an approach is suggested to find the most optimum gridshell, then numerical results show the efficiency of the proposed alternative approach.</p>

Membrane structures: application experience and development prospects

Introduction. At present, studying an increase in the applicability of steel structures in all segments of the construction industry, particularly in residential construction, is relevant. Membrane systems are high-tech and economical structures. The behavior of large-span membrane structures is studied in great detail in practice. However, membranes, combined with medium and small spans, remain understudied.
 
 Materials and methods. Findings of basic theoretical, experimental and numerical studies of membrane systems, used in structural analysis and design, are provided. An advanced membrane slab design, designated for metal-framed multistory residential buildings, is described. Its flat support contour is rectangular in plan; it has a membrane and a monolithic concrete slab over it.
 
 Results. Lira-SAPR software package was used to conduct numerical studies of the membrane slab. The studies took account of its structural, geometrical and physical nonlinearity, and the results were compared with an experimental study. The distribution of forces in its support contour is typical for rectangular membrane structures with positive curvature shells.
 
 Conclusions. Numerical analysis of a full-scale structure and a tested model having a monolithic reinforced concrete slab on the membrane proved the long-term benefits of this structure. The results of numerical analysis demonstrate the qualitative convergence with the experimental data. The variable thickness of the slab resting on the membrane, collaboration of the slab, the membrane and the support contour, the physical nonlinearity of the system elements and, above all, the reinforced concrete slab must be taken into account to reduce the qualitative discrepancy between numerical and experimental results.