Shell Equations Research Articles

Structural Mechanics of cylindrical fish-cell zero Poisson’s ratio metamaterials

In this paper, a novel cylindrical metamaterial exhibiting zero Poisson’s ratio in two different directions is introduced. Detailed CAD modelling of a curved Fish-Cells necessary for numerical and experimental analysis are presented. High-fidelity finite element models are developed to assess the homogenisation studies of Poisson’s ratio, Young’s modulus and torsion behaviour, demonstrating the curvature effect and independency of the mechanical behaviour of cylindrical Fish-Cells metamaterial from tessellation numbers. Experimental analysis is performed to validate the zero Poisson’s ratio, deformation and fracture mechanism discussed in numerical simulations. Moreover, buckling and modal behaviours of the cylindrical Fish-Cells metamaterials are studied and compared with equivalent shell models.

Research on Dynamics Behaviors of Two Coaxial Circular Cylindrical Thin Shells Constructed with Functionally Graded Materials

The paper reports the dynamic characteristics of two coaxial circular cylindrical thin shells (CCTSs) containing internal or annular flowing fluid in which one shell is constructed with functionally graded materials (FGMs). A new set of shell equations are firstly used to analyse the vibration of fluid-conveying thin shells. Two cases, i.e. inner shell or outer shell is constructed with FGMs, are calculated by adopting the Galerkin’s method. The results show that as the volume fraction index increases, the natural frequencies don’t always decrease which depend on the positions of FGM shell and flowing fluid of two coaxial CCTSs. The paper indicates that the effects of FGM shell cannot be neglected for the design of FGM shells conveying fluid.

Analysis of nonlinear vibrations of thin cylindrical shells subjected to supersonic flow

This paper presents a numerical model for analysis of the aeroelastic stability of a thin cylindrical shell subjected to external supersonic airflow including consideration of the effect of geometric nonlinearity of the shell on the dynamic behaviour of the structure. The nonlinear stiffness matrix is derived based on a new formulation developed for thin cylindrical shells, which accounts for the shell curvature effect in the circumferential direction of the displacement field, and the effect of coupling between the different nonlinear modes on the dynamic behaviour of the shell. Nonlinear strain–displacement relationships are inferred from Novozhilov’s theory. The linear and nonlinear mass and stiffness matrices are calculated by adopting the displacement functions derived from exact solutions of linear Sander’s theory equilibrium equations for thin cylindrical shells. The classic finite element method is employed to derive the global mass and stiffness matrices. The aerodynamic damping and stiffness matrices are evaluated using the first–order potential (piston) theory. The initial stiffening effect due to radial pressure and/or axial loading is also taken into account. The governing equations of motion are derived using the Lagrange method and solved numerically with the help of a direct iterative method. Numerical studies are conducted to illustrate the effects of different parameters including internal pressure, nonlinear coupling, circumferential wave number, radius-to-thickness ratio, length-to-radius ratio and boundary conditions on flutter boundaries and on the nonlinear dynamic behaviour of the cylindrical shell at the onset of flutter and during the flutter stage. Good agreement is found between the results obtained using the presented approach and those published in the literature.

Effects of the geometrical shapes on buckling of conical shells under external pressure

The purpose of this study is to investigate the effects of geometrical shapes on the buckling of conical shells under uniform external pressure. A curved conical shell, straight conical shell, and equivalent cylindrical shell were respectively designed with constant mass. Two curved conical shells, two straight conical shells and two cylindrical shells made of 201 austenitic stainless steel were fabricated and tested. Measurements were made to the wall thickness, diameter, axial length, and geometry of each shell and the material properties of the corresponding sheets. The conical shells were destroyed by slow pressure, resulting in good repeatability. According to measured results, the buckling of the tested shells was further studied numerically. The results reveal that the load-carrying capacity of the conical shell was significantly improved when the meridional shape was in the form of a circular arc.

Approximate resolving equations of mathematical model of a curved thin-walled cylinder

The resolving equations for mathematical model of the stress-strain state of a curved thin-walled cylinder were derived. This model is based on Koiter’s–Vlasov’s theory of moment shells. A method was proposed for approximate solution of a mathematical model equations on the basis of a sequential asymptotic expansion of unknown functions into a small parameter series and representation of the expansion coefficients in the form of Fourier series. Using this method, a one-dimensional statement of the problem was obtained. Limitations on parameters in the shell equations are indicated for which such problem transformation is possible. For the mathematical model of the curved thin-walled cylinder one-dimensional equations in two different formulations were obtained. Conditions determining applicability limits of the constructed one-dimensional mathematical models were proved. Numerical experiments were carried out and it was found that the constructed one-dimensional mathematical model approximates the original problem with high accuracy. From an applied point of view, curved thin-walled cylinder simulates a pipeline section.

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration.

Acoustic insulation characteristics of sandwich composite shell systems with double curvature: The effect of nature of viscoelastic core

A viscoelastic model is proposed in this approach to determine the sound transmission loss coefficient of a sandwich shell system with double curvature. The structure is composed of a double-walled composite shell subjected to a viscoelastic core. Investigating the efficient impresses of rotary inertia and shear deformation, vibration equations of both outer and inner shells are extracted within the framework of shear deformation shallow shell theory. Besides, the Zener mathematical model is used for viscoelastic material, which is based on a spring connected in series with a parallel mixture of spring and dashpot. This model presents the dynamic response in the whole frequency domain at which shear modulus and bulk complex modulus are frequency dependent. Since the performed studies on the sound transmission loss of this kind of structures are insignificant, the outcomes of plate models with a viscoelastic core are used to provide a reliable sound transmission loss comparison. The results show that the applied strategy can improve the acoustic characteristics of the system at high frequencies compared to that of a single-layer one with the same mass. This issue is more highlighted while the thickness of the viscoelastic layer enhances, which confirms the positive performance of the viscoelastic materials in this range of frequency, particularly in the resonant frequency. In addition to the curvature effect on acoustic features, the vibration response of the system is configured based on various frequencies and materials.

Dynamic responses of corrugated cylindrical shells subjected to nonlinear low-velocity impact

Owing to high capacity of energy absorption, high compressive strength, high stiffness to weight ratio, good ductility and other excellent characteristics, corrugated structures are widely used in aerospace, civil, petrochemical engineering and other industries. These structures are often subjected to low-velocity impact during operation, which has a destructive effect on the strength and stiffness of engineering structures. However, since the process of low-velocity impact is a strongly nonlinear time-varying process, it is difficult to obtain the impact force. The present study proposes a novel computational method for nonlinear impact force in the local contact area, and analyzes the dynamic response of corrugated structures. Firstly, a new equivalent shell model of corrugated cylindrical shells is established. Afterwards, the governing equations are obtained by using the Love thin shell theory and Hamilton principle, and then turned into ordinary differential equations by the Galerkin procedure, upon which numerical solutions are achieved by the Duhamel integration and small-time increment technique. Then, numerical examples are given to illustrate validity and efficiency of the proposed method. Finally, the influence of key parameters on impact response of different types of corrugated shells is analyzed.

Free Vibration of Composite Cylindrical Shells Based on Third-Order Shear Deformation Theory

The focus of this study is to analyse the free vibration of cylindrical shells under third-order shear deformation theory (TSDT). The constitutive equations of the cylindrical shells are obtained using third-order shear deformation theory (TSDT). The surface and traverse displacements are expected to have cubic and quadratic variation. Spline approximation is used to approximate the displacements and transverse rotations. The resulting generalized eigenvalue problem is solved for the frequency parameter to get as many eigenfrequencies as required starting from the least. From the eigenvectors, the spline coefficients are computed from which the mode shapes are constructed. The frequency of cylindrical shells is analysed by varying circumferential node number, length dimension, layer number, and different materials. The authenticity of the present formulation is established by comparing with the available FEM results.

Flexural wave dispersion characteristics of imperfect Ti-6Al-4V foam circular cylindrical shells in a thermal environment

The present paper is mainly focused on analyzing the flexural wave dispersion of imperfect Ti-6Al-4V foam circular cylindrical shells in a thermal environment. The pores were supposed to be distributed across the thickness (z-direction) in the form of three different patterns as follows: symmetric porosity distribution (SPD), asymmetric porosity distribution (ASPD) and uniform porosity distribution (UPD). Besides, various kinds of temperature variations, including sinusoidal, linear and uniform temperature variations, were studied. The strain-displacement relationship of the shell was derived based on the first-order shear deformable theory (FSDT) of shells. Hamilton’s principle was also applied to obtain the governing equations of metal foam shells which were then solved using an analytical method. Finally, influences of different parameters including circumferential wave number, different kinds of temperature variation, temperature change, radius to thickness ratio , types of porosity distribution across the thickness, porosity coefficient and mode number on the variations of phase velocity and wave frequency were investigated and the results were illustrated and discussed.

Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein–Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabès and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity.

Effect of boundary condition and variable shell thickness on the vibration behavior of grid-stiffened composite conical shells

In this research, it is aimed to investigate the effect of boundary condition and variable shell thickness on the vibration behavior of grid-stiffened composite conical shells. In order to fabricate the stiffeners and shell for the experimental study, continuous and woven glass fibers have been used during a filament winding process, respectively. Due to the non-uniform distribution of the stiffeners in a stiffened conical shell, the equivalent thickness varies along the longitudinal axis. The stiffener and shell structures have been reduced into an equivalent shell using smeared method. The governing equations have been solved in order to evaluate the natural frequencies of the grid-stiffened conical shell by the use of first-shear deformation theory (FSDT) along with the power series method. In order to validate the analytical solutions, an experimental modal analysis have been carried out on a real grid-stiffened composite conical shell with uniform shell thickness. In addition, the finite element solution has been provided for further validations. Furthermore, the effects of various geometric parameters, variable shell thickness and boundary conditions have been discussed.

Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

Rapid and Application-Tailored Assessment Tool for Biogenic Powders from Crustacean Shell Waste: Fourier Transform-Infrared Spectroscopy Complemented with X-ray Diffraction, Scanning Electron Microscopy, and Nuclear Magnetic Resonance Spectroscopy.

Due to their chemical composition, richness in calcium carbonate, chitin, proteins, and pigments, and nanoporous structure, crustacean shell waste shows great potential for a wide variety of applications. Large quantities of waste shells are produced annually, meaning that they can be considered a renewable source of ecofriendly biogenic materials, which can be turned into value-added byproducts. In this paper, an IR-based technique is developed to differentiate various biogenic powders originated from crude or food-processed crustacean shells. The validity of the method is supported by cross-checking with XRD, NMR, and SEM–EDX analyses. Our goal was to determine changes in properties of waste crab shells after the two most common treatments, deproteinization and milling. We discovered that deproteinization with NaOH could be tracked from the IR absorbance intensity ratio of the υ(CH2,3) and υasym(CO32–) bands while milling time less influenced this ratio but induced changes in powder particle size distribution and morphology. The relative organic/inorganic ratio was different for different colored shells. Unexpectedly, waste shells stored for an average of 6 months or more were found to contain hydrated calcium carbonate (monohydrocalcite), which was absent in equivalent fresh shell composition. Deproteinization caused changes in mechanical properties of shells, making them more brittle, which resulted in a larger fraction of fine particles after powdering.

An Analytical Solution for the Vibration and Far-Field Sound Radiation Analysis of Finite, Semisubmerged Cylindrical Shells

The vibration response and far-field sound radiation of a semisubmerged, finite cylindrical shell with low-frequency excitation are studied. The solution to this problem can be divided into two steps. The first step is to apply the wave propagation approach to determine the vibration response of the cylindrical shell. In the cylindrical coordinate system, the Flügge shell equations and Laplace equation are used to describe the cylindrical shell and surrounding fluid so that the vibration responses of the shell can be addressed analytically. The fluid free surface effect is taken into account by applying the sine series to force the velocity potential on the free surface to be zero. Furthermore, compared with the FEM (the finite element method), the present method is not only reliable but also effective. In the second step, the far-field sound radiation is solved by the Fourier transform technique and the stationary phase method in accordance with the vibration responses of the shell from the previous step. The boundary element method is applied to validate the reliability of the acoustical radiation calculation. The circumferential directivity of far-field sound pressure is discussed, and it is found that the maximum value of the sound pressure always appears directly under the structure when the driving frequencies are relatively low. Besides, in consideration of simplicity and less computation effort, the present method can be used for the rapid prediction of the vibration and far-field sound pressure of a semisubmerged cylindrical shell with low-frequency excitation.

Equivalent Shell Model of Elastic Gridshells Including the Effect of the Geometric Curvature

In this work, an equivalent continuum of a barrel gridshell is introduced. Constitutive identification procedures based on periodic homogenization are provided in the literature for this purpose, based on a flat Representative Element Volume (REV), notwithstanding that the geometry of the structures concerned is curved. Therefore, the novelty of the present study is the selection of a curved REV to obtain the equivalent elastic constants. The numerical validation of the identification procedure is made comparing gridshell response to that of the equivalent shell under homogeneous load conditions. Finally, in order to highlight the effect of the curved geometry on the constitutive law of the continuum, the response of the proposed model is also compared to that of a continuum obtained from a flat REV.

An Analytical Approach for the Nonlinear Free Vibration Analysis of Thin-Walled Circular Cylindrical Shells

This paper presents a new formulation combining the nonlinear theory of Novozhilov with the classical finite element method for the purpose of evaluating the vibratory characteristics of thin, closed and isotropic cylindrical shells. The theory developed in this paper is able to include the shell curvature effect in the circumferential direction of the orthogonal displacements and considers the impact of initial geometric imperfections on the dynamic response of the system. The formulation first takes a general form by expressing the shell displacements as an alliance between the generalized coordinates and spatial functions. Nonlinear kinematic relationships are inferred from Novozhilov’s theory. The equations of motion as well as the expressions of the mass, linear and nonlinear stiffness matrices are derived through the Lagrange method by considering the coupling between the different modes. An application of this model is illustrated in a further step, by adopting the displacement functions derived from exact solutions of linear Sanders’ theory equilibrium equations for thin cylindrical shells. The governing equations of motion are solved with the help of a direct iterative method. Linear and nonlinear frequencies are validated by comparison with the results in the literature. The relative nonlinear frequencies are determined as a function of vibration amplitudes and then compared with published results for several cases of shells. Excellent agreement is observed between the results derived from this theory and those found in the literature. The effect of different parameters including axial and circumferential wave number, length-to-radius ratio, thickness-to-radius ratio and various boundary conditions, on the nonlinear frequencies of cylindrical shells is investigated.

Structural Fire Analysis of One‐storey Steel‐framed Buildings with Steel Claddings

AbstractTraditionally the structural stability of steel building frames is ensured by frame action or bracing elements. Recent European research projects have demonstrated that significant cost savings can be achieved, if cladding panels forming the building envelope are used to provide stability. The purpose of the paper is to investigate the possible stabilization effect of steel structures by steel claddings in fire, without additional wind bracings using Finite Element Method. The Finite Element analyses for full‐scale steel‐framed buildings with steel claddings under fires are still challenging due to large number of elements in the steel claddings. In this paper, the equivalent orthotropic shell element is used to model the trapezoidal roof sheeting, and composite shell element with soft core is used to model the sandwich panel in the cool compartment. Due to the delamination of inner steel face in fire, the sandwich panels in the fire compartment can be either neglected or modeled with two‐layer composite shell ignoring the inner steel sheet. The comparative study is performed, and the results suggest that the difference is small and the sandwich panels in the fire compartment can be neglected. The stabilization effect of steel claddings in fire is studied by the comparison of deformation and force responses between buildings with traditional steel bracings and with steel cladding only. In case of ISO fire, the fire resistance of entire one‐storey building with steel claddings only is 22% longer than the one with traditional steel bracings.

A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

Manufacturing and Characterization of Tube-Filled ZA27 Metal Foam Heat Exchangers

This study investigates the heat transfer performance of a novel ZA27 metal foam heat exchanger. An open-celled metal foam is combined with a thin-walled copper tube in a single-step casting process. The heat transfer between two separated water streams flowing through the copper tube and foam, respectively, is measured and compared to an equivalent shell tube heat exchanger arrangement. Heat transfer enhancement of up to 71% and a heat transfer rate exceeding 30 kW are observed and attributed to the increased surface area of the metallic foam. However, overall performance was limited by the inefficient heat transfer between the internal mass stream and the copper tube.