Shell Problems Research Articles

FORMATION OF A MATHEMATICAL MODEL AND ALGORITHM FOR IMPLEMENTATION OF CALCULATION MAIN PIPELINE - OF SHELL STRUCTURE

The paper considers methods of numerical calculation of elements of shell structures - main pipelines under various types of loading. Refined equations of motion for cylindrical shell structures are given. To solve boundary value problems of thin-walled structures, the Bubnov-Galerkin method, finite differences using the sweep method is used. As an example, the solution of the boundary value problem of a cylindrical shell under static loading by the finite difference method is given. And also the results of the study of the numerical convergence of the calculated values.

Die Casting Die Design and Process Optimization of Aluminum Alloy Gearbox Shell.

Taking an aluminum alloy gearbox of an automobile as an example, according to its structural characteristics, the parting surface was determined, and the initial gating system was designed by using 3D modeling software UG. Based on Magmasoft software, the numerical simulation of the filling and solidification process was carried out to determine the best gating system scheme. The cooling system and core pulling structure were designed, and the parameter design process of the aluminum alloy gearbox shell in the die-casting process was introduced. Aiming at the leakage problem of the gearbox shell in the bench and road test after assembly, the cause was found through numerical simulation and industrial CT analysis, and the problem was solved by adding high-pressure point cooling at the corresponding position of the leakage, and the correctness of the optimization was verified. It provides an effective method for the die-casting production of the transmission housing and the analysis and solution of product defects, which improves the product quality and shortens the production cycle.

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.

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

Natural frequency analysis of shells of revolution based on hybrid dual-mixed [formula omitted]-finite element formulation

A newly-developed, dimensionally reduced, hp-type axisymmetric shell finite element model is extended to linear elastodynamic problems of thin shells of revolution. The hp-shell finite element relies on the hybridized version of a three-field dual-mixed variational formulation, the application of which dictates the obligate usage of the inverse three-dimensional constitutive relation for homogeneous and isotropic materials, thereby ensuring the volumetric locking-free characteristic of the shell model at theory level. The fundamental fields are the a priori non-symmetric stress tensor, the displacement vector, the infinitesimal rotation vector and the hybrid variable defined on the element interfaces. Since the dimensional reduction process guided by this hybrid-mixed formulation does not necessitate the use of any classical kinematic assumptions appearing in the scientific literature, the inverse 3D Hooke’s law does not have to be modified. The numerical performance of the shell finite element is analyzed comprehensively for natural frequency computations of clamped-free and simply supported, silicone, conoid, spherical and hyperboloid shells of revolution. From their relative error convergence behaviors it follows that the extended hybrid-mixed shell finite element is not sensitive to the decrease of the slenderness ratio, namely providing reliable, uniformly stable numerical results for both h- and p-approximation. From theoretical point of view, the beneficial properties of the hybrid-mixed hp shell finite element model are as follows: (i) this is effectively applicable to modeling not only extremely thin but also moderately thick shell structures with transverse shear deformations, as well as (ii) both the through-the-thickness variation and the membrane stress normal to the shell mid-surface are retained as independent variables, making it much easier to upbuild shell model for contact problems. From numerical point of view, the nice feature of the hybrid-mixed hp shell finite element is that the global flexibility matrix of the system can be inverted block-wise at element level at cheap computational cost during the assembling procedure because of the hybridization technique.

Equivalent-Single-Layer discontinuous Galerkin methods for static analysis of multilayered shells

An original formulation for the elastic analysis of multilayered shells is presented in this work. The key features of the formulation are: the representation of the shell mean surface via a generic system of curvilinear coordinates; the unified treatment of general shell theories via an Equivalent-Single-Layer approach based on the through-the-thickness expansion of the covariant components of the displacement field; and an Interior Penalty discontinuous Galerkin scheme for the solution of the set of governing equations. The combined use of these features enables a high-order solution of the multilayered shell problem. Several numerical tests are presented for isotropic, orthotropic and multilayered shells with different geometrical configurations and boundary conditions, including the case of a non-smooth geometry. Comparisons with analytical solutions and finite-element simulations show the high-order accuracy as well as the capability and robustness of the proposed formulation, which can be a valuable tool for the analysis of generally-curved multilayered shells.

Searching for admissible thrust surfaces in axial-symmetric masonry domes: Some first explicit solutions

Determining the internal forces acting within a masonry dome or vault loaded by an assigned distribution of external actions is a problem that is still open, as evidenced by even recent contributions to the scientific literature on the topic. The present work intends to address this issue by proposing a method for determining admissible distributions of stresses in vaults and masonry domes. The problem is solved analytically with the aim of obtaining, when possible, explicit expressions for the internal forces. Although applicable in principle to arbitrarily distributed loads, the procedure adopted herein for searching for statically admissible internal forces is described in detail for the case of distributed and concentrated vertical loads. The analysis is performed by building suitable analytical solutions to the so-called “direct” and “inverse” problems of a thin shell in which bending forces are nil and only membrane forces are present. The solutions thusly obtained are applied to vaults and domes by making use of the so-called “thrust surface” concept, which represents a natural generalization of the thrust line for masonry arches. According to Heyman’s hypothesis for masonry, when the thrust surface is entirely contained within the vault thickness, a corresponding statically admissible stress field can be found. Thrust surfaces corresponding to admissible stress fields are determined by means of an expressly developed iterative procedure that begins by assigning an initial shape to the thrust surface. Then, by suitably using the solutions of the inverse problem, the shape of the thrust surface is modified so that the corresponding stresses become, when possible, statically admissible. By using the well-known theorems of limit analysis, both the mechanical and geometric safety coefficients are assessed for vaults and existing domes. As an example, the proposed procedure is applied to three practical case studies: a hemispherical dome of constant thickness, the dome of the Rome Pantheon, and the dome of Bernini’s Santa Maria Assunta Church in Ariccia (Rome).

Modeling large deformations of thin-walled SMA structures by shell finite elements

Many shape memory alloy (SMA) applications, such as biomedical devices, electromechanical actuators, and elastocaloric cooling devices, are based on thin-walled flat or shell-like structures. An advanced design of such structures requires the development of an efficient and accurate numerical tool for simulations of very thin and curved SMA structures that may experience large deformations and even buckling upon thermo-mechanical loading. So far, finite element models for finite strain deformations of SMA structures have been based on 3D solid formulations, which are relatively inefficient for solving (thin) shell problems. In this paper, we present a finite element model for the analysis of shape memory alloy shells. Our model is based on a 7-parameter, large-rotation, one-director shell formulation, which takes into account a fully three-dimensional form of the constitutive equations for the isothermal transformations of isotropic superelasticity, as well as the shape-memory effect in a simplified way. In fact, we present three 4-node shell finite elements for SMAs. Two of them use the assumed natural strain concepts for the transverse shear strains, through-the-thickness normal strain, and membrane strains. The third element is a combination of the assumed natural strain and the enhanced assumed strain concepts, applied to satisfy the zero through-the-thickness-normal-stress condition for thin geometries to a high degree of accuracy. After a detailed description of the SMA finite element models for shells in the first part of the paper, numerical examples in the second part illustrate the approach. Compared to 3D solid SMA formulations, our results show excellent accuracy, even with a significantly reduced number of degrees of freedom, which consequently translates into a reduction in the computational time.

Analytical solution of stress in functionally graded cylindrical/spherical pressure vessel

For the exact solution of the stress in the functionally graded (FG) cylindrical/spherical pressure vessel, this paper presents a unified form of the basic equations of the FG hollow cylinder/spherical shell by introducing parameter, and then, the axial/spherical symmetry mechanical problems of FG hollow shells are studied under three different boundary conditions, respectively. Assume that the Young's modulus of the material changes with thickness distribution of cylinder/shell in the form of power functions and the value of Poisson’s ratio is constant, the analytical solutions of the displacement and stress of the FG hollow cylindrical/spherical pressure vessel are derived under different boundary conditions. By comparing the analytical solutions of the FG hollow cylinder/spherical shell obtained in this paper with the existing classical theoretical solutions and numerical solutions, the correctness of the analytical solution given in this paper is verified. In the numerical discussion, the distributions of displacement, radial stress and circumferential stress of the FG hollow cylinder/spherical shell under different gradient parameters and different size conditions are given, respectively. Finally, based on the difference numerical method, the stress distribution of FG hollow cylinder/spherical shell with the Young's modulus and Poisson's ratio along thickness distribution in the form of power functions is analyzed numerically. The results show that gradient parameter and geometric size have a great influence on the mechanical response of FG structure under different boundary conditions.

A solution to the problem of stability of thin-walled steel cylindrical shells

Introduction. It is necessary to improve methods of analysis of the limit states, occurring when a thin-walled shell is in the elastoplastic domain, to use these cylindrical shells as elements of heavily loaded products of construction and machine building industries. Materials and methods. The problem of stability of a circular thin-walled cylindrical shell, made of steel 45 GOST 1050-2013, that takes the load induced by pure compression and axial torsion, has been studied. Besides, experimental and theoretical components of the problem have been analyzed. Experimental facility SN-EVM was applied to perform an experiment test and analyze its findings in terms of different versions of the theory of plasticity used to solve shell stability problems beyond the elastic limit. The co-authors emphasize the unavailability of any definition of the criterion of stability loss under combined loading based on experimental dependences that were identified earlier. The results of the experiment were compared with the results of the theoretical study. The analysis of the shell stability in the case of complex subcritical loading are based on the A.A. Ilyushin theory of stability, in which plasticity functions are taken according to V.G. Zubchaninov’s approximations. Results. The problem was solved using the software programme, developed by the co-authors. The software solves the bifurcation problem of a cylindrical shell with regard for the complex nature of deformations at the moment of stability loss in the case of exposure to complex subcritical loading, commensurable processes and the trajectory that has the form of circular arcs. It has been shown that the proposed method of analysis and approximations describe the real stress-strain state of shells that feature low flexibility in respect of a complex pattern of deformation and characterize a stable state of the material beyond the elastic limit. Conclusions. The theoretical strength and deformability analysis of a cylindrical shell and its experimental studies demonstrate sufficient convergence which proves their reliability. This conclusion will allow to improve the process of design of structural elements made of materials that have complex mechanical properties.

An efficient quadrature method for vibration analysis of thin elliptical plates with continuous and discontinuous edge conditions

Mapping an irregular domain into a square one is a common technique in analyzing problems of plates and shells with irregular shapes. For the irregular shape without four corners such as the elliptical shape, the difficulty arises that the Jacobian determinant is zero at the corner points. An efficient quadrature method is presented to analyze the transverse vibration of thin plates with an elliptical shape. To circumvent the above-mentioned difficulty, Gauss quadrature is used in numerical integration. Besides, derivative degrees of freedom are not used, and a boundary point is modeled by two nodes separated by a very small distance. Since the nodes are not coinciding with integration points, a way indirectly using the differential quadrature law is employed to derive the explicit formulas to ease the programming. A convergence study is performed. Free vibration of elliptical plates with continuous and discontinuous edge conditions is analyzed to demonstrate the efficiency of the developed rotation-free weak-form quadrature method.

Describing law of motion of flexible inextensional shell in gravity force field

This work makes use of Navier-Stokes equations to describe an analytical method of finding the motion speed of a flexible inextensional shell falling down to the ground from a preset height and determines the duration of this fall. The soft shell in question is a fabric body of aerodynamic shape or an item of clothes, an airborne vehicle element, etc. Analytical relations are presented for the speed at which the shell moves in the air, taking account of the air resistance and the shell fall duration. The boundary problem of the soft shell vertically falling in the air is solved.

New benchmark free vibration solutions of non-Lévy-type thick rectangular plates based on third-order shear deformation theory

Higher-order shear deformation theories are generally adopted to study the dynamic behavior of substantially thick plates, of which Reddy’s third-order deformation theory (TSDT) is most extensively used in view of its conciseness and high accuracy. However, due to the intractability in solving the governing higher-order partial differential equations, analytic solutions of TSDT-based thick plate problems are limited to very few special cases, i.e., Navier-type or Lévy-type solutions. Since analytic solutions can not only serve as benchmarks for validation of various methods, but are also useful for structural designs, especially at the early design stage, developing new analytic methods that are applicable to more common non-Lévy-type thick plates is of much importance. In this paper, for the first time, a novel symplectic superposition method, which was proposed originally for thin/moderately thick plate problems, is extended to free vibration problems of thick rectangular plates based on the TSDT. Besides yielding Lévy-type solutions, new non-Lévy-type analytic free vibration solutions are obtained. Comprehensive benchmark natural frequencies and mode shapes are tabulated/plotted, all of which are well validated by the 3D theory-based solutions from the finite element method (FEM) and the literature, if any. The present method is implemented in the symplectic space under the Hamiltonian system framework, rather than in the Euclidean space under the Lagrangian system framework as implemented in conventional methods, providing a new route to more analytic solutions for complex plate and shell problems that have not been explored due to mathematical challenge.

On the Problem of Solvability of Nonlinear Boundary Value Problems for Arbitrary Isotropic Shallow Shells of the Timoshenko Type with Free Edges

On the Problem of Solvability of Nonlinear Boundary Value Problems for Arbitrary Isotropic Shallow Shells of the Timoshenko Type with Free Edges

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells

The generalized finite integral transformations method in linear and nonlinear static problems for shallow shells

The Ritz method application for stress and deformation analyses of standard orthotropic pressure vessels

Currently, the design of pressure vessels is based mainly on the simplified calculations described in the technical standards. Despite the numerous theories found in various valuable monographs and papers, the possibility of precise analytical solutions of static problems of shells including bending effects remains poor. The only alternative to such calculations is the finite element method. The presented paper focuses on the application of the Ritz method for the stress and deformation analyses of standard pressure vessels assuming orthotropic model of the material. The strain energy and the potential energy of external forces are derived for an arbitrary shell structure. Three different displacement functions are considered within the study. The results are compared to the solutions obtained with a use of the finite element method. The convergence of the results is obtained for wide range of the shell thickness. Additionally, the outcome of the study is referred to the design method of ellipsoidal and supposedly equivalent torispherical dished ends described in the technical standards, indicating significant ambiguities.

A shell superelement for mechanical analysis of cylindrical structures

This paper aims at developing a new cylindrical shell element, called shell superelement. The element is defined based on the classical shell theory, and it consists of eight nodes each with six degree-of-freedoms (dofs). In this element, the trigonometric shape functions are incorporated along the angular direction of element while polynomials were used in other two directions. Therefore, there is no need for meshing a shell structure with cylindrical geometry through the angular direction. This property helps an engineer to deal with complicated analyses on cylindrical shell structures with less number of dofs. At the end, the defined element is used in the stress analysis of two different classical shell problems and the results are compared with the ones reported in the literature, and obtained by means of shell elements in a commercial software package.

Assessment of the effect of negative Poisson’s ratio on the thermal postbuckling of temperature dependent FG-GRMMC laminated cylindrical shells

Auxetic materials have recently emerged as new types of advanced materials with unique material properties that conventional materials do not possess. In this paper, we examine the effect of in-plane negative Poisson’s ratio (NPR) on the thermal postbuckling behavior of graphene-reinforced metal matrix composite (GRMMC) laminated cylindrical shells. The shell consists of GRMMC layers arranged in a piece-wise functionally graded (FG) pattern and is subjected to a uniform thermal load surrounded by an elastic medium. Based on the molecular dynamics simulation results, it is noted that the temperature-dependent material properties of GRMMCs can be expressed as a nonlinear function of temperature. The thermal postbuckling problem of shells is modeled under the framework of the Reddy’s third order shear deformation theory and solved by using a singular perturbation technique in conjunction with a two-step perturbation approach. Numerical investigations are carried out for the postbuckling of (10/-10/10/-10/10)S and (10/-10/10)S shells with in-plane NPR. It is found that the FG-X pattern can enhance the buckling temperature and the thermal postbuckling strength of the shells. The anomaly is that the thermal buckling load and postbuckling strength of UD (10/-10/10)S shell are slightly higher than those of UD (10/-10/10/-10/10)S GRMMC laminated cylindrical shell.

A nonlinear geometric couple stress based strain gradient Kirchhoff–Love shell formulation for microscale thin-wall structures

We present a nonlinear Kirchhoff–Love micro-shell element based on isogeometric analysis (IGA) and couple stress theory. Higher-order NURBS functions are exploited for analyzing the strain gradient effect which automatically fulfill the higher-order continuity requirements. We express the strain gradient elastic formulation in natural curvilinear coordinates, which leads to an efficient numerical tool to examine geometric nonlinearities of thin micro-shell structures. The presented IGA formulation is verified through comparisons to analytical solution, experimental data as well as other popular benchmark problems of nonlinear geometric shells. We believe that the presented formulation is particularly suitable for analyzing two-dimensional materials at larger length scales, which are commonly studied at nanoscale.

On the Existence of Solutions of Nonlinear Boundary Value Problems for Inhomogeneous Isotropic Shallow Shells of the Timoshenko Type with Free Edges

On the Existence of Solutions of Nonlinear Boundary Value Problems for Inhomogeneous Isotropic Shallow Shells of the Timoshenko Type with Free Edges