Shell Problems Research Articles

Use of coherent node clusters as coarse grid in 2-Level Schwarz solver in finite element solid and structural mechanics

The Coherent Nodal Cluster (CoNC) model-reduction technique is used to construct an algebraic transformation from nodal degrees of freedom to generalized degrees of freedom for compact (coherent) clusters of nodes. The novel idea here is to construct a coarse-grid preconditioner for a conjugate gradient solver based on the CoNC technique, and integrate it into a two-level Schwarz algorithm. The finite element grid is divided into overlapping partitions for the fine-grid part of the preconditioner (level 1). Grouping of the nodes into clusters, independent of the fine-grid partitioning, is employed to develop a global Galerkin approximate solver by setting up a transformation matrix between the original degrees of freedom and generalized degrees of freedom associated with polynomial basis functions within each cluster (level 2). Performance of the proposed methodology in terms of strong and weak scaling is tested on compressible and nearly incompressible elasticity in three dimensions, and general three-dimensional shell problems.

GEOMETRICLY NONLINEAR BENDING OF FUNCTIONAL-GRADIENT SHALLOW SHELLS ON AN ELASTIC FOUNDASHION

The paper considers the problem of geometrically nonlinear bending of shallow elements of structures made of functional gradient materials (FGM) under the influence of various transverse loads. The shallow shells under consideration can have arbitrary plan shapes and be in contact with an elastic base of the Winkler-Pasternak type. The mathematical formulation is made within the framework of classical geometric-nonlinear theory. To linearize the nonlinear system of differential equations of equilibrium, the sequential load method in combination with Newton's method is used. The effective mechanical properties of FGM vary along the thickness and are calculated according to the power law. The use of the theory of R-functions made it possible to build the necessary systems of coordinate functions for shells with arbitrary geometry and support conditions. The proposed approach is implemented in software, tested, and applied to solve the problems of bending shallow shells of complex plan shapes with holes. The influence of the elasticity coefficients of the base, the gradient index in the distribution of metal and ceramic particles, as well as other parameters on the deflections of structural elements, was studied.

Fresh Concrete Properties Required for Vertical Slip-form Concrete

Abstract: The guidelines available in the present concrete mix design standard IS-10262-2019 [2] are silent about the special mix proportion criteria for slip-form concrete. The slip-form concrete is subjected to dynamic loads in the fresh state due to the movement of the slip-form panel over the fresh concrete. In current practice, particularly in power plant industries where RCC chimneys are required, concrete mix design is often carried out by a third party based solely on hardened concrete design strength criteria. However, such proportioning guidelines as per IS-10262-2009 [2] are suitable only for static concrete pouring operations. For slip-form concrete, special criteria for fresh concrete is needed, like adequate shear strength of the cover concrete to be self-sustaining against the lifting friction of the slip-form panel during its movement, as well as to prevent surface problems in the RCC chimney shell during construction using the slip-form technique. In this research work, addressing real execution problems of RCC chimney shell surface issues, it has been concluded that the greater the shear strength of fresh concrete, the more resistant it will be to the lifting friction of the slip-form panel, resulting in better surface quality of the chimney shell and fewer surface problems. The research also concluded that various properties of concrete ingredients directly influence the shear strength of the concrete and plays an important role in preventing surface problems of the chimney shell during construction.

Assessment of a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells

Abstract Static response of simply supported functionally graded (FG) shallow shells using a new higher-order shear and normal deformation theory is focused in this article. The effects of transverse strains and stresses on the bending response of FG shell are considered by the present theory. The current theory considers the impacts of transverse normal and shear deformations that meet the requirements for traction-free boundary conditions. The virtual work principle is applied to the mathematical formulation of the present theory. The simply supported doubly curved shallow shell problems under the static transverse load are analyzed using Navier’s solution technique. To verify the existing theory, the current results are, whenever possible, compared with those that have already been published. Additionally, a few benchmark results are presented in this article that will be helpful to researchers in the future.

On the applicability limits of the structurally orthotropic shell model in problems of calculating the buckling under axial compression of waffle cylindrical shells

The article considers the solution of the buckling problem of a waffle cylindrical shell under axial compression using the Euler (bifurcation) approach. Two models are used: a model based on the numerical integration method and a model based on the finite element method. The first model assumes that the stiffeners are structurally an orthotropic shell obeying the Kirchhoff-Love hypotheses, with the assumption of "smearing". The second model uses a tetrahedron element with ten nodes. The study is based on experimental data from tests of one of the waffle cylindrical shell samples under axial compression. The research results show that with an increase in the waffle shell rib thickness, a discrepancy between the calculation results of the two models is observed. With the optimal value of the k parameter, characterizing the ratio of the rib thickness to the rib pitch, equal to 0.035, such a discrepancy is estimated to be about 5%. At the same time, the calculation using the model of a structurally orthotropic shell leads to an underestimation of the critical load compared to the finite element model. This provides an estimate of the loss of stability of a waffle cylindrical shell with a safety margin. For parameter k between 0.02 and 0.05, the difference between the results from the two computational models for critical loads does not exceed 11%.

Large deformation problem of hyperbolic shallow shells with bimodular effect: A perturbation‐variation method

AbstractAs an important analytical method, the perturbation‐variation method combines the advantages of perturbation method and variational method, and is widely used for the solution of nonlinear plate and shell problems. In this study, the perturbation‐variation method is used to solve the large deformation problem of a flexible hyperbolic thin shallow shells with different moduli in tension and compression (bimodular effect). First, we establish the governing equations expressed in terms of three displacement components, in which the bimodular effect of materials is considered in physical equations and the large deformation of shell is considered in geometrical equations. Next, we use the perturbation‐variation method to solve the governing equations established, obtaining the perturbation‐variation solution up to third‐order. During the application of the method, the displacements are expressed into perturbation formulas of all levels first, and the unknown constants and functions in the perturbation formulas are determined by the extreme value conditions of the functional established (variational principle), hence the perturbation‐variation method gets its name from this practice. The numerical results from FEM basically agree with the perturbation‐variation solution obtained. The method proposed may be extended into the solution of similar partial differential equations established in applied fields.

Semi-analytical layerwise solutions for FG MEE complex shell structures

This paper presents a unique analytical layerwise solution for functionally graded magneto-electric-elastic shell with complex geometry. The middle surface of the shell is modeled by a parametric equation and the Lamé parameter and radii of curvature is modeled by Differential Geometry. The mechanical displacements, electrical and magnetic potentials are written in term of a simple cosine layerwise based on a unified formulation. The highly coupled differential equations are discretized by the Chebyshev-Gauss-Lobatto grid points and solved numerically via the Differential Quadrature Method (DQM). Lagrange interpolation polynomials are employed as the basis functions. The highly coupled differential equations are solved for shells subjected to different loads and boundary conditions. Since extremely few results on this topic is available in the literature, benchmarks complex shell problems and their solutions are introduced in this paper for the very first time.

Mathematical modelling of stability problems for thin transversally graded cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected stability problems for such shells. As a tool of modelling we shall apply the tolerance averaging technique. The second aim is to derive and discuss a new mathematical averaged asymptotic model. This model will be formulated using the consistent asymptotic modelling procedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love second-order theory of thin elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in circumferential direction, whereas equations of the proposed models have continuous and slowly-varying coefficients. Moreover, some of coefficients of the tolerance model equations depend on a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental cell-independent, but also the new additional cell-dependent critical forces can be derived and analysed.

Nonlinear resonant responses of a novel FGM sandwich cylindrical shell structure for supersonic flight vehicles based on 1:1 internal resonance between conjugate modes

A new design scheme for the functionally graded material (FGM) sandwich cylindrical shell structure made of ceramic-FGM-carbon fiber composites is proposed, which is geared towards supersonic flight vehicles and has high specific stiffness and high-temperature resistance. We focus on analyzing the nonlinear resonant responses of the novel FGM sandwich cylindrical shell subjected to external excitation and aerodynamic force. Firstly, the mechanical properties of the novel FGM sandwich material are calculated, and then the nonlinear dynamic model of the FGM sandwich cylindrical shell is established based on the first order shear deformation theory (FSDT) and Hamilton's principle. Due to the axisymmetric property of the cylindrical shell, a 1:1 internal resonance between the driven and companion modes always exists in the perfect cylindrical shell. Taking this into consideration, the nonlinear resonant response problems of the FGM sandwich cylindrical shell are numerically simulated by combining the Galerkin method and the pseudo-arc length continuation method. Finally, the effects of complex parameters such as external excitation, aerodynamic force, aspect ratio, gradient index, and skin-core-skin ratio on the nonlinear resonant responses of the FGM sandwich cylindrical shell are investigated.

Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm

Abstract This research develops a data-driven methodology for structural instability problems with highly nonlinear, difficult, noisy, and small data. A fast analysis and prediction (FAP) approach for instability problems of thin shells is first proposed. This approach contains two phases: the fast numerical analysis and the pure prediction utilizing artificial neural networks (ANNs) incorporated with the Bayesian regularization (B-R) algorithm as follows: (1) in Phase 1 (the fast numerical analysis), post-buckling analysis is conducted utilizing a minor amount of load steps. The load–displacement relation achieved from Phase 1 is not exact because of the small number of load steps utilized; (2) in Phase 2 (the prediction), the loads and deflections achieved from Phase 1 were employed as the data for training ANNs. The trained networks, including the load and displacement networks, were employed to fast predict loads and deflections at any step of the post-buckling analysis. After utilizing Phase 2, a smooth, complete and exact load–displacement curve was achieved. In Phase 1, the available formulation for post-buckling analysis of thin shells in the literature was utilized. Five popular types of instabilities chosen to confirm the effectiveness and exactness of the FAP were snap-through, snap-back, softening–hardening, kink instabilities, and delamination buckling and post-buckling of composites. The high exactness and effectiveness of the FAP were confirmed in the numerical verification section. The present approach saves a huge computation compared to the other ones. It was found that ANNs incorporated with the B-R algorithm have notable advantages compared to numerous neural networks. The proposed approach is applicable to simulations or experiments where data are “expensive”, highly nonlinear, difficult, and limited. Utilizing the proposed approach for these fields can dramatically save time and money.

An extension of composite bending strain κMITC3+ scheme in analysis of shell structures reinforced with rib stiffeners

In this paper, the composite bending strain κMITC3+ approach is extended for analyzing the shell structural domain in the rib-stiffened shell problems. Developed to achieve enhanced performance and emphasize a softer bending behavior, this novel element employs a polynomial projection technique to construct an assumed composite bending strain field. This technique originates from the Hu-Washizu three-filed principle and the orthogonality condition. In the membrane component context, an improved Allman-like triangular element is utilized to remove the spurious energy mode commonly associated with classical Allman's theory. To address the shear-locking phenomenon within Timoshenko's beam model for the rib stiffeners, a selective/reduced integration approach is adopted. This involves using a standard two-point Gaussian rule for accurate bending strain computation and a one-point Gaussian rule for shear strain computation. The numerical examples showcased in our study unequivocally highlight the distinct superiority of the κMITC3+ element.

Application of von-Karman model for laminated composite singly curved shells with non-uniform boundary conditions

The industrial applications of singly curved shells significantly increased in modern engineering structures and several of those have non-uniform boundary conditions. A detailed relative performance study for the non-uniformly restricted shells can popularize their use further among practicing engineers. The literature review confirms that there is dearth of research studies in this area. This paper aims to fill up that void. A finite element code of C0 continuity combining von-Karman strains, first order shear deformation theory and eight noded curved elements is proposed. Wide variety of shell problems are solved for different boundary conditions, laminations, radii of curvatures, thickness, and plan dimensions. Relative nondimensional deflections and fundamental frequencies are studied for varying boundaries. It is concluded that although the performances of shells are comparable for different boundaries, the shells with CCCC edges offer better performances. These shells should have longer span along the arch direction, a symmetric 16°/-16°/-16°/16° laminate, side/thickness = 100, and radius of curvature/plan = 0.75 to achieve minimum deflection and maximum fundamental frequency.

Natural frequencies and modes of parametric vibrations of reservoir shell with shape imperfections

Development of software on the basis of the finite element method led to intensive creation of the numeral methods for the decision of static and dynamic problems of the thin shells. The finite element model of the thin shell has an infinite number of freedom degrees and natural frequencies, so solving the dynamics problem of the shell is difficult. If behavior of natural vibrations of the shell was researched, it is possible to talk about shell internal properties which take a great place at the forced vibrations including parametric one. It is important to take into account influence of the constant component of parametric load on natural frequencies and modes of the shell. Nowadays forming of an effective model of parametric vibrations of the shell with shape imperfections and choice of the most dangerous imperfections model remain relevant. Influence of real and modelled imperfections on natural frequencies and modes of reservoir shell parametric vibrations excited by axial load and on shell stability loss was investigated in this article. The finite element models of the shell was formed by software NASTRAN. The modelled shape imperfections as a lower buckling form of perfect shell under static pressure were presented. The real imperfections as the deviations of the shell wall from the vertical were obtained by theodolite surveying. The natural frequencies and modes of the imperfect shell taking into account the its previous stress state from action of the constant component of the parametric load were received by the Lanczos method. Investigations showed that the real imperfections of shell a little influenced on natural frequencies and modes of parametric vibrations. These decreased the critical loads on the first natural frequency on 0,58 % and increased on the second natural frequency shells on 0,79 %, but qualitatively changed the form of stability loss on these frequencies. Modelled imperfections had a greater but not considerable influence on natural frequencies and modes of the shell. But the modelled imperfections of the shell under constant component of parametric load considerably increased the critical load on the first and second natural frequency accordingly on 12,3 % and 5,26 % and changed the shell forms of stability loss. So the modelled imperfections of the shell in the form of the regular circular half-waves increased shell carrying capacity, but not decreased. This positive effect takes place at constructing of cylinder shells from the corrugated rolled sheets.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

Mode-II stress intensity factors for special-shaped cylindrical shells in torsion

Special-shaped cylindrical shells in torsion are encountered in engineering applications, which will induce the Mode-II singular stress fields next to circumferential crack tips. The Mode-II stress intensity factor (SIF) represents the strength of these singular stress fields and is an important fracture parameter. The crack problems for the cylindrical shells are typical finite boundary value issues. It is very difficult to get the exact solutions by using the classical theory of elasticity. In present article, based on the concepts of the conservation law and the elementary strength theory, a hybrid methodology to determine the Mode-II SIFs for circumferential cracked special-shaped cylindrical shells in torsion is suggested. Due to the introduction of elementary mechanics, the corresponding errors are also inevitably carried into the Mode-II SIFs. However, the finite element analysis indicates that the error of the Mode-II SIFs calculated by present method is acceptable.

Nonlinear resonant responses and chaotic dynamics of three-dimensional braided composite cylindrical shell

The advanced three-dimensional braided composite structure has excellent integrity, which can effectively avoid the problems of interlayer stress jump and interface debonding that may occur in traditional composite laminated structures. However, most of the existing references on the three-dimensional braided composite structure are fragmented, and there is little research on their vibrations. Taking the cylindrical shell of a rocket as the object and considering the complex flight environment, a general framework for the vibration problems of the three-dimensional braided composite shell is proposed for the first time. The equivalent mechanical parameters of the three-dimensional braided composite material are calculated based on the volume averaging method. Then, the nonlinear dynamic model of the three-dimensional braided composite cylindrical shell is established, and the natural vibration characteristics are analyzed. Under the case of 1:1 internal resonance, the pseudo-arc length continuation method is used to solve the nonlinear resonant responses of this nonlinear system. Finally, we apply the fourth-order Runge-Kutta algorithm to study the chaotic dynamics of the three-dimensional braided composite cylindrical shell model. This framework presented in this paper can be extended to the study of nonlinear vibrations and chaotic dynamics of other three-dimensional braided composite structures.

Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

SummaryWe present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.

On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates

The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates

On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates