Irregular Shells Research Articles

On the exact equilibrium conditions of irregular shells reinforced by beams along the junctions

The exact, resultant equilibrium conditions for irregular shells reinforced by beams along the junctions are formulated. The equilibrium conditions are derived by performing direct integration of the global equilibrium conditions of continuum mechanics. New, exact resultant static continuity conditions along the singular curve modelling reinforced junction are presented. The results do not depend on shell thickness, internal through-the-thickness shell structure, or material properties of shell and beam elements. In this theoretical approach, the beam’s kinematics is represented by the elastic Cosserat curve. Kinematically, the six-parameter model of shell structures coincides with the Cosserat curve model of the beam. The presented method can be easily applied to cases of connection of three or four shell elements with the reinforcement along the junction.

Block-Layer Approach for the Analysis of the Stress–Strain State of Three-Layer Irregular Cylindrical Shells of Rotation

Block-Layer Approach for the Analysis of the Stress–Strain State of Three-Layer Irregular Cylindrical Shells of Rotation

Examination of selected failure criteria with asymmetric shear stresses in the collapse analysis of laminated shells

The paper is concerned with failure analysis of composite shells performed with the usage of the nonlinear 6-parameter shell theory with drilling rotation degree of freedom. This special theory embodies naturally unlimited translations and rotations and is suitable for analysis of irregular shells for instance with various, particularly orthogonal, intersections. The presence of the drilling rotation is inherently accompanied with asymmetry of in-plane shear stresses and strains which must be taken into account in the composite failure criteria. In the paper in general two failure measures are applied: Hashin criterion and Tsai-Wu criterion which are appropriately modified to accommodate the stress asymmetry. It is shown, that depending on the used failure criterion, it is possible to obtain slightly different damage zones and values of the first ply failure and ultimate load. These values are compared with those obtained experimentally.

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.

Model for Layer-by-Layer Analysis of the Stress-Strain State of Three-Layer Irregular Shells of Revolution of Double Curvature

Based on the layer-by-layer analysis approach, the construction of a model of two types of finite elements (FE) of natural curvature (two-dimensional FE of the momentary bearing layers and three-dimensional FE of the filler) is considered for refined investigation of the stress-strain state (SSS) in layers of three-layer generally irregular shells of revolution of double curvature. The presented model-building algorithm allows one to avoid errors due to the discontinuity of generalized displacements on the docking surfaces of the FE of the bearing layers and the aggregate, to analyze the SSS for all three coordinates to which the shell is assigned, and to obtain a solution in an updated statement for various shell shapes and boundary conditions of the layers, as well as violations of their continuity. At the same time, the aggregate layer can be modeled in thickness by the required number of finite elements, which allows one to take into account the change in geometric characteristics, physical and mechanical properties, and SSS parameters not only in the meridional and circumferential coordinates, but also in the thickness of the shell and aggregate layer. The proposed approach significantly expands the range of problems solved in the refined formulation and allows us to carry out the calculation with a high degree of accuracy and detail. As an example, the calculation of the SSS of a three-layer animated casing with cut-outs is given. The influence of the size of the cut-outs on the stress – strain state in the layers of the three-layer shell of rotation of double curvature is studied.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Block finite-element approach to building refined models of layer-by-layer analysis of the stress-strain state of three-layer irregular shells

This work offers a block finite element approach to building refined models of layer-by-layer analysis of stress-strain state of three-layer generally irregular cylindrical shells.

A Model for Refined Calculation of the Stress-Strain State of Sandwich Conical Irregular Shells

A Model for Refined Calculation of the Stress-Strain State of Sandwich Conical Irregular Shells

Structural design of irregular curved lattice shells in China

Irregular curved lattice shell structures are proving increasingly popular in China. However, the behaviour of irregular shells can be substantially different from regular shells in terms of strength, stiffness and stability characteristics as well as joint detailing. This paper reviews a number of projects in China in which irregular shells have been used. The structural system, force transfer path, vulnerable locations and strengthening measures are discussed. This is followed by a case study on a 45 m span greenhouse building featuring novel drum joints in the shell.

Block finite-element model of layer-by-layer analysis of three-layer stress–strain state in general irregular shells of rotation of double curvature

This study proposes a finite-element block approach to building a new, refined model for layer-by-layer analysis of the stress–strain state of generally irregular sandwich shells of revolution with double curvature. A core material model is developed for the first time for such shells, based on more precise statements compared to those of similar common models; it allows the avoidance of the discontinuity of generalized displacements on the surfaces of an interface with base layers and switching to simpler models depending on the problem statement. Using the proposed model, it is possible to create an allowance for the changes in the properties and parameters of the stress–strain state in all the three coordinates, to which the shell is assigned, and to obtain a solution within the specified statement for different shell shapes and boundary conditions of layers, including in the case of discontinuity.

Robust four-node elements based on Hu\u2013Washizu principle for nonlinear analysis of Cosserat shells

Mixed 4-node shell elements with the drilling rotation and Cosserat-type strain measures based on the three-field Hu–Washizu principle are proposed. In the formulation, apart from displacement and rotation fields, both strain and stress resultant fields are treated as independent. The elements are derived in the framework of a general nonlinear 6-parameter shell theory dedicated to the analysis of multifold irregular shells. The novelty of the developed elements stems from the fact that the measures of assumed strains and stress resultants are asymmetric. The original interpolation of drilling and bending components of strains and stress resultants is proposed. In the formulation of new mixed elements, two variants of the interpolation of membrane components are used and interpolation of the independent fields is defined in the natural or skew coordinates. Accuracy and efficiency of the developed elements are tested in several linear and nonlinear numerical examples. It is shown that smaller number of independent parameters in the interpolation of membrane components gives more accurate results. The proposed mixed 4-node elements enable the use of large load steps in nonlinear computations. Moreover, they require significantly less equilibrium iterations than other shell elements formulated in the 6-parameter shell theory.

Structural intensity assessment on shells via a finite element approximation.

Structural intensity on plates or shells can provide insights on how the vibrational energy is transmitted throughout a sample. Its assessment via experimental deflections are widely documented in the case of plates, which just requires the computation of spatial derivatives of out-of-plane displacements or velocities and a knowledge of the sample's material properties. However, if the structural intensity is to be assessed on arbitrary shells, a more elaborate data processing is required. The in-plane displacements become relevant terms and the spatial derivatives along a predefined local coordinate system need to be computed. Here, a method from which experimental data is interpolated on a finite element mesh is proposed. First, the global displacements and shape of a sample's outer-surface are measured. These data are then projected on a quadratic mesh, where the Kirchhoff plate theory is invoked for the individual elements. The data differentiation is computed via quadratic shape functions, from which the strains and structural intensity are estimated. Through the obtained vibrational energy results on the basis of measured displacement and shape data and by validating the method via a numerical simulation, the proposed work has shown to be a reliable tool to assess energy transmission on irregular shells.

Block Finite-Element Model of Layer-by-Layer Analysis of the Stress−Strain State of Three-Layer Generally Irregular Shells of Double-Curvature Revolution

The block finite-element approach is proposed for constructing a new corrected model for layer-by-layer analysis of the stress−strain state of three-layer generally irregular shells of double-curvature revolution. For the first time, a filler model has been developed for such shells, which is based on more accurate settings than those in similar conventional models; in this case, one succeeds in avoiding the breaking of generalized displacements on the interfaces with the carrier layers and, depending on the conditions of the problem, passes to simpler models. The model constructed makes it possible to take into account the change in the properties and parameters of the stress−strain state along all three coordinates to which the shell is assigned and to obtain for the first time the solution in a corrected formulation for various shell shapes and boundary conditions of the layers, as well as for their discontinuities.

2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis

We propose 2-D Cosserat type orthotropic constitutive equations for laminated shells for the purpose of initial failure estimation in a laminate layer. We use nonlinear 6-parameter shell theory with asymmetric membrane strain measures and Cosserat kinematics as the framework. This theory is specially dedicated to the analysis of irregular shells, inter alia, with orthogonal intersections, since it takes into account the drilling rotation degree of freedom. Therefore, the shell is endowed naturally with 6 degrees of freedom: 3 translations and 3 rotations. The proposed equations are formulated from the statement of the generalized Cosserat plane stress with additional transverse shear components and integrated over the shell's thickness using the equivalent single layer approach (ESL). The resulting formulae are implemented into the own Fortran code enabling nonlinear shell analysis. Some numerical results are presented to show the performance of the proposed approach.

Structural intensity on shell structures via a Finite-Element-Method approximation

The study of energy transmission on plate-like structures is widely documented in literature. Its analysis requires the computation of the spatial derivatives of the out-of-plane displacement field of the studied sample and a priori knowledge of its material properties. However, if the structural intensity is to be assessed on irregular shells, such a study requires a more elaborate data processing. In addition to in-plane displacements, also the spatial derivatives along the sample’s local coordinates are needed. For this purpose, a method was developed to approximate both the experimental displacement data and the spatial coordinates of a given arbitrary shell using a Finite-Element-Method model. After measuring the global displacement fields and their corresponding spatial coordinates for a given sample, the data was transferred to a shell model whose basis functions were properly defined in accordance to the application’s demands and whose individual elements were assumed to behave in accordance with the Kirchhoff plate theory. The proposed method was able to process the experimental displacement and shape data of shells structures and proved to be a reliable tool to assess energy transmission.

Factors affecting biodiversity on hermit crab shells

This study explores the abundance, diversity and assemblage structure of epifauna on the shells used by two hermit crab species (Pagurus bernhardus and P. pubescens) in the Arctic (Svalbard and Northern Norway) and investigates the biotic and physical drivers of such patterns. Contrary to our expectations, we found that location (which reflects the variability in environmental conditions and the local species pool of potential colonizers) is a key determinant not only in the cold, ice-scoured, glacier-dominated Arctic shallows of Svalbard but also in boreal Norwegian fjords, where other factors were hypothesized to be more important. Depending on region, shell area and identity were of lesser magnitude, with larger and more irregular shells containing more diverse assemblages. Crab host species also played a role (P. pubescens-inhabited shells supported larger number of individuals and higher diversity than those of P. bernhardus) but this effect might be species or region specific. In this study, no effect of crab gender could be detected. The study indicated that epifaunal assemblages of hermit crab shells are influenced by complex set of factors that interact together to different degree at various locations.

Core-shell nanoparticles by silica coating of metal oxides in a dual-stage hydrothermal flow reactor.

γ-Fe2O3@SiO2, TiO2@SiO2, and α-Fe2O3@SiO2 core-shell nanoparticles were synthesized by a surfactant-free hydrothermal method in a continuous flow dual-stage reactor. Uniform silica shells of 2 nm thickness were obtained when grown on γ-Fe2O3 and TiO2 nanoparticles while thicker and more irregular shells were deposited on α-Fe2O3 due to low pH.

Elastoplastic law of Cosserat type in shell theory with drilling rotation

Within the framework of six-parameter non-linear shell theory, with strain measures of the Cosserat type, we develop small-strain J2-type elastoplastic constitutive relations. The relations are obtained from the Cosserat plane stress relations assumed in each shell layer, by through-the-thickness integration employing the first-order shear theory. The formulation allows for unlimited translations and rotations. The constitutive relations are expressed in terms of the following material data: Young’s modulus, Poisson’s ratio, Cosserat shear modulus, characteristic length and material parameters pertaining to plasticity. Numerical examples are presented to show the correctness of the proposed model and capacity to analyze regular and irregular shells.

О возможности применения сетчатых оболочек с несимметрично уложенными нитями в качестве приводов управляемой упругой деформации

О возможности применения сетчатых оболочек с несимметрично уложенными нитями в качестве приводов управляемой упругой деформации

The SAURON project - IX. A kinematic classification for early-type galaxies

Two-dimensional stellar kinematics of 48 representative elliptical (E) and lenticular (S0) galaxies obtained with the SAURON integral-field spectrograph reveal that early-type galaxies appear in two broad flavours, depending on whether they exhibit clear large-scale rotation or not. We define a new parameter ⁠, which involves luminosity-weighted averages over the full two-dimensional kinematic field as a proxy to quantify the observed projected stellar angular momentum per unit mass. We use it as a basis for a new kinematic classification: early-type galaxies are separated into slow and fast rotators, depending on whether they have λR values within their effective radius Re below or above 0.1, respectively. Slow and fast rotators are shown to be physically distinct classes of galaxies, a result which cannot simply be the consequence of a biased viewing angle. Fast rotators tend to be relatively low-luminosity galaxies with MB≳−20.5. Slow rotators tend to be brighter and more massive galaxies, but are still spread over a wide range of absolute magnitude. Three slow rotators of our sample, among the most massive ones, are consistent with zero rotation. Remarkably, all other slow rotators (besides the atypical case of NGC 4550) contain a large kpc-scale kinematically decoupled core (KDC). All fast rotators (except one galaxy with well-known irregular shells) show well-aligned photometric and kinemetric axes, and small velocity twists, in contrast with most slow rotators which exhibit significant misalignments and velocity twists. These results are supported by a supplement of 18 additional early-type galaxies observed with SAURON. In a companion paper (Paper X), we also show that fast and slow rotators are distinct classes in terms of their orbital distribution. We suggest that gas is a key ingredient in the formation and evolution of fast rotators, and that the slowest rotators are the extreme evolutionary end point reached deep in gravitational potential wells where dissipationless mergers had a major role in the evolution, and for which most of the baryonic angular momentum was expelled outwards. Detailed numerical simulations in a cosmological context are required to understand how to form large-scale KDCs within slow rotators, and more generally to explain the distribution of λR values within early-type galaxies and the distinction between fast and slow rotators.