Thin Shell Theory Research Articles

Magneto-thermoelastic coupled resonance characteristics of a rotating functionally graded ferromagnetic cylindrical shell under double harmonic line loads

The magneto-thermoelastic coupled resonance of a rotating functionally graded (FG) ferromagnetic cylindrical shell subjected to double harmonic line loads in magnetic and temperature fields is investigated. In accordance with Donnell's thin shell theory on the physical neutral surface, the nonlinear geometric equations are determined. Utilizing the thermoelastic constitutive relationships, combining with the gradient characteristics and temperature dependence of physical parameters, the governing equations of kinetic energy and strain energy are given. Based on the magnetoelasticity theory, and considering the nonlinear magnetization of ferromagnetic metal, the calculation formulas of Lorentz force and magnetizing force acting on the shell are deduced. The magneto-thermoelastic coupled dynamic equations are derived by applying Hamilton principle and discretized by Galerkin method. Subsequently, considering the different resonance forms, the multiple scales method and Lyapunov stability theory are employed to obtain the theoretical solutions and stability discriminants. Numerical examples are carried out to plot the characteristic diagrams with different parameters, which are used to analyze the characteristics of nonlinear coupled resonance. Furthermore, the primary resonance and combination resonance are selected to analyze the bifurcation and chaotic behavior. Results show that the variation of different parameters affects obviously the resonance amplitude and stability. The changes of amplitude and position of hard excitation can lead to resonance region drift. Amplitude, frequency, and position of excitation have significant effects on bifurcation and chaos. The motion state of cylindrical shell is transmitted from periodic response to chaotic response through the accumulation of period-doubling bifurcation.

Geometrically nonlinear bending of stiffened composite skewed cylindrical shells under transverse pressure

The literature review on relative performances of laminated composite skewed shells confirms that research reports on bending performances of moderately thin, stiffened, laminated composite skewed cylindrical panels, using the geometrically nonlinear approach, are not available. This paper aims to fill that deficiency and proposes a finite element code combining eight-noded, doubly curved elements with modified Sanders’ first approximation theory for thin shells and von Kármán-type nonlinear strains. Correctness of the proposed geometrically nonlinear bending formulation for skewed shells are verified through solutions of benchmark problems. The deflections, force, and moment resultants are reported for different skew angles, laminations, stacking sequences, radius of curvature, plan dimension ratios, and stiffener properties like orientations, numbers, and eccentric positions. The results are discussed critically which reveals that shells having curved edges free and straight edges clamped fabricated using 0°/90°/0° laminate offer the best performances. The biaxial stiffeners, nx = 7, ny = 7, show the minimum deflections and stress resultants. The skewed shells offer greater deflections and hence, must be avoided in industrial practices.

Free vibration analysis of barrel-shaped sandwich shells with auxetic honeycomb core using modified thick shell theory

Modern engineering extensively employs complexly curved shell structures, presenting mathematical challenges in analyzing their mechanical behavior due to geometric intricacies. The recent trend of incorporating advanced lightweight materials into these structures drives the need to develop improved theories and methods. This study presents an investigation into the free vibration analysis of barrel-shaped sandwich (BSS) shells using a modified thick shell theory (MTST). These sandwich shells feature functionally graded (FG) ceramic-metal layers on both inner and outer surfaces, with a lightweight honeycomb core. Notably, the honeycomb layer's thickness surpasses that of the inner and outer layers, necessitating the application of thick shell theory for accurate analysis. However, previous studies have often used classical theories for thin shells, and those applying thick shell theory sometimes neglected the curvature effects in the fundamental equations and rotary inertia components. Thus, this research employs the MTST to overcome these limitations. The study includes comparative validations and explores the impact of material parameters and curvature radii on the natural frequencies of these barrel-shaped sandwich shells through numerical examples.

Sound radiation from a cylindrical shell in an underwater waveguide

Sound radiation from cylindrical structures immersed in an underwater acoustic waveguide, that is, a fluid layer with an upper free surface and lower rigid floor, is a challenging engineering problem to analyze due to the complex interaction of the radiation from the excited structure and its subsequent propagation within an acoustic waveguide. To contribute to the understanding of this phenomenon, an analytical model is presented. The model involves an infinitely-long three-dimensional cylindrical shell that is point-excited and submerged in a perfect underwater waveguide, where the reflections off the free surface and rigid floor have no absorption. The equations of motion of the shell are given by thin shell theory with heavy fluid loading. The image-source method is applied directly to the acoustic boundary conditions to account for the vibroacoustic coupling of the shell and waveguide. Together, these techniques combine into an analytical framework that can evaluate both the vibration of the shell and its acoustic radiation. Results include the far-field radiated sound pressure with observations of the relative and combined effect of the waveguide boundaries.

Theoretical analysis for free vibration of submerged conical motor stator

The conical rim-driven motor stator is a structure with variable thickness and non-standard boundaries, which can be treated as an orthotropic conical shell. To prevent resonance and reduce electromagnetic noise, a modal analysis of the stator is necessary. This paper presents an analytical model for the free vibration of the conical stator in both onshore and underwater environments. Due to the relatively large thickness of the motor stator, the first-order shear deformation theory (FSDT) is employed. The fluid loading is calculated by discretizing the conical shell into a series of narrow cylindrical shells featuring diverse radii and thickness. The resulting equations are solved employing the transfer matrix method (TMM), which is capable of different types of boundary conditions (BCs). The convergence and accuracy of the proposed method are validated through a comparative analysis involving numerical simulations and outcomes derived from the thin shell theory. The importance of considering shear deformation and rotatory inertia in determining the natural frequency of motor stators is demonstrated. Finally, the impact of fluid loading on the vibration behavior of stators with various geometric parameters and boundary conditions are investigated and discussed. The proposed model presents a precise and convenient approach for analyzing the free vibration of submerged conical motor stators, and can be extended to conventional cylindrical motor stators through appropriate degradation procedures.

A numerical study of the multicriteria parametric optimization of the displacement and weight of a two-connected conical shell of minimal surface under thermal and power loading

Various methods and approaches are used to solve the problem of optimal design. Problems of this type are used for the optimal design of spatial coverings, which in modern architecture are progressive structures that are of interest to the whole world. Spatial coatings allow to give the building architectural expressiveness, cover large spans and combine load-bearing and enclosing functions.
 The hulls of minimal surfaces are the hulls of negative Gaussian curvature. In general, the geometry of minimal surfaces cannot be described analytically. To solve this problem, which is associated with solving the differential equation that describes the minimal surface in general, under appropriate boundary conditions, one has to use methods of numerical analysis. This approach allows us to build a point frame of the shell, which is a matrix of discrete solutions of functions that searches for a given minimum surface. In view of this, its geometric characteristics can be obtained only in numerical form.
 The basic relations of the linear theory of thin elastic shells of minimal surface are highlighted in the numerical study of parametric optimization. The mathematical justifications of how the external load is perceived and the process of creating the stress-strain state of the shells of minimal surfaces are highlighted.
 For the numerical implementation of the multicriteria parametric optimization of a double-connected conical shell of minimal surface, a finite element model was built from plate finite elements in the amount of 4824 pieces and 4896 nodes. The perimeter is rigidly clamped to the ground disk. The thermal force load is set, which consists of a combination of static and temperature loads.
 The developed methodology shows quite good results that coincide with the works of other authors and makes it possible to use two types of optimization for one research object simultaneously. The first stage is shape optimization, and the second stage is multicriteria parametric optimization.
 This methodology makes it possible to perform optimization processes in an automated mode, which is an important applied task for construction and applied mechanics.

A convergence criterion for sound transmission of single-, double-, and triple-walled cylindrical shell based on Love's, Donnell's, and Flügge's shell theories

Different sound transmission loss (STL) analytical methods based on Love's, Donnell's and Flügge's cylindrical thin-shell theories have been widely used. However, there are no available convergence criterion and formulas of sound transmission losses for multi-layer cylindrical shell structures. The current convergence determination approach is to select several typical frequencies (low, middle, and high frequencies) for convergence analysis, and then estimate the convergence of all other frequencies. This typically results in sound transmission loss jumps, leading to inaccurate sound transmission loss results in the high-frequency region. In this study, a novel convergence criterion and formulas are developed for calculating sound transmission loss mode numbers in the complete convergence zones of single-, double-, and triple-walled cylindrical shells based on Love's, Donnell's, and Flügge's thin shell theories, which can be used to directly determine the truncation iterations for sound transmission losses in all frequency ranges.

A new combined asymptotic-tolerance model of thermoelasticity problems for thin uniperiodic cylindrical shells

AbstractThe objects of consideration are thin linearly thermoelastic Kirchhoff–Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential direction (uniperiodic shells). The aim of this contribution is to formulate and discuss a new averaged mathematical model for the analysis of selecteddynamic thermoelasticity problemsfor the shells under consideration. This so-called combined asymptotic-tolerance model is derived by applying the combined modelling including the consistent asymptotic and the tolerance non-asymptotic modelling techniques, which are conjugated with themselves into a newprocedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation. For the periodic shells, the starting equations have highly oscillating, non-continuous and periodic coefficients, whereas equations of the proposed model have constant coefficients dependent also on a cell size.

Analytical method for deformation and stress calculation of bell-and-spigot joints in PVC-U pipes

The stable operation of water supply pipelines relies significantly on the structural safety of pipe joints. A novel analytical pipe joint structural calculation method based on the Novozhilov thin shell theory was proposed to calculate the deformation and stress of bell-and-spigot joints in unplasticized polyvinyl chloride (PVC-U) pipes under internal pressure. An equivalent stress index was established to investigate and evaluate the safety performance of pipe joints. The effects of various factors, including spigot insertion depth, the clearance between the bell and spigot, internal pressure, and different pipe nominal outside diameter (DN) and nominal pressure (PN) series, on the maximum equivalent stress of pipe joints, were thoroughly analyzed. The results indicate that the maximum equivalent stress under nominal pressure is located on the inner surface of the bell and consistently exceeds the design stress. The influence of spigot insertion depth on the maximum equivalent stress is minimal. However, as the clearance value between the bell and spigot decreases and the internal pressure increases, the maximum equivalent stress increases significantly. Pipes with smaller DN and PN series result in a higher maximum equivalent stress at the joint. The analytical method indicates that increasing pipe thickness and reinforcing the bell with continuous carbon fiber filament winding are two potentially effective methods to enhance the mechanical performance of the joint and reduce the maximum equivalent stress. This study's findings offer valuable theoretical support for designing and selecting joints in PVC-U pipelines.

Study of Gap Flow Effects on Sound Transmission Through a Micro-Perforated Sandwich Cylindrical Shell Excited by a Plane Wave

This paper presents a theoretical model to study sound transmission through a micro-perforated double-walled sandwich cylindrical shell with emphasis on the potential of internal gap flow to improve the sound insulation performance. This model adopts Donnell’s thin shell theory to govern the shell motions and a simplified model based on Biot’s theory to describe sound propagation in the porous lining. The mean acoustic particle velocity model with the grazing flow effect is applied to define the coupling condition between the fluid medium and the perforated shell. Transmission loss (TL) through the configuration with one gap flow is numerically calculated using the mode superposition method when subjected to an oblique plane wave in the presence of an external mean flow. Results reveal that the gap flow in the opposite direction to the external flow would elevate the mid-frequency TL. Gap depth has a complicated effect on TL since it influences the aerodynamic damping of the gap flow and sound dissipation in the porous layer. The average TL in 20–20[Formula: see text]000[Formula: see text]Hz is then optimized with a genetic algorithm, and an increase of 17.82 dB is finally achieved. The possibility of tuning the structural sound insulation performance by the gap flow has been verified.

Simulation and Optimization of Hemispherical Resonator's Equivalent Bottom Angle for Frequency-Splitting Suppression.

As an inertial sensor with excellent performance, the hemispherical resonator gyro is widely used in aerospace, weapon navigation and other fields due to its advantages of high precision, high reliability, and long life. Due to the uneven distributions of material properties and mass of the resonator in the circumferential direction, the frequencies of the two 4-antinodes vibration modes (operational mode) of resonator in different directions are different, which is called frequency splitting. Frequency splitting is the main error source affecting the accuracy of the hemispherical resonator gyro and must be suppressed. The frequency splitting is related to the structure of the resonator. For the planar-electrode-type hemispherical resonator gyro, in order to suppress the frequency splitting from the structure, improve the accuracy of the hemispherical resonator gyro, and determine and optimize the equivalent bottom angle parameters of the hemispherical resonator, this paper starts from the thin shell theory, and the 4-antinodes vibration mode and waveform precession model of the hemispherical resonator are researched. The effect of the equivalent bottom angle on the 4-antinodes vibration mode frequency value under different boundary conditions is theoretically analyzed and simulated. The simulation results show that the equivalent bottom angle affects the 4-antinodes vibration mode of the hemispherical resonator through radial constraints. The hemispherical resonator with mid-surface radius R=15 mm and shell thickness h=1 mm is the optimization object, and the stem diameter D and fillet radius R1 are experimental factors, with the 4-antinodes vibration mode frequency value and mass sensitivity factor as the response indexes. The central composite design is carried out to optimize the equivalent bottom angle parameters. The optimized structural parameters are: stem diameter D=7 mm, fillet radii R1=1 mm, R2=0.8 mm. The simulation results show that the 4-antinodes vibration mode frequency value is 5441.761 Hz, and the mass sensitivity factor is 3.91 Hz/mg, which meets the working and excitation requirements wonderfully. This research will provide guidance and reference for improving the accuracy of the hemispherical resonator gyro.

Lagrange multipliers at the boundary in the inextensional bending theory of thin elastic shells

The theory of inextensional deformations of shells is extended by introducing Lagrange multipliers at the boundary to supplement those operating in the interior. These are associated with a constraint on the metric on the closure of the shell midsurface. The additional multipliers allow for a richer set of external actions than those admitted by the conventional theory. In particular, they balance concentrated forces in the tangent plane of the surface applied at corners of the boundary.

Predicting the wrinkling in AA5052 seamless tubes manufactured by free bending forming technology

Wrinkling is a significant defect during forming of thin-walled metallic tubes via free-bending forming (FBF) technology. Thus, it's quite important to predict the wrinkling in the tubes to improve the quality of the final products. Accordingly, new theoretical and numerical frameworks were proposed for predicting the wrinkling of the tubes manufactured via FBF technology. The theoretical wrinkling prediction model is based on the energy criteria and the thin shell theory. The numerical prediction model was constructed on the ABAQUS/Implicit FE code. Besides, The Knoop microhardness test characterized the anisotropy of the tube. The tube's imperfection was introduced to the numerical model through pure bending buckling analysis. The bending experiments were conducted to verify the results obtained from the proposed theoretical and numerical frameworks and showed good agreement with them. Furthermore, it was noticed that the additional axial force in the FBF process reduces the tube's ability to resist wrinkles. The larger additional axial force makes tubes more sensitive to imperfection in free bending than in other bending technologies. The numerical prediction model with the perfect tube exhibits good accuracy in predicting the initiation of the tube wrinkling in free bending but not in terms of the wrinkling shape. The model considering the imperfection can predict the onset and development of wrinkling properly.

Traveling Wave Characteristics of a Rotating Functionally Graded Laminated Cylindrical Shell with General Boundary Conditions

The traveling wave modes of rotating functionally graded laminated cylindrical shells with general boundary conditions are studied by using different admissible displacement functions. Firstly, the effective temperature-dependent material properties of the functionally graded materials (FGMs) are described by the Voigt model and the Sigmoid-type volume fraction. Subsequently, considering Love’s thin shell theory, the motion equations of the FGMs shell subjected to the Coriolis force, the centrifugal inertial force, the hoop initial tension and the internal thermal force are derived. Thereafter, the displacement functions are expanded by six different sets of admissible displacement functions. The Rayleigh–Ritz method is utilized to derive the eigenvalue equations of the rotating FGMs laminated shells. In the end, the convergence rate and computational efficiency of the six polynomial functions are compared and parameter study is carried out. Results indicate that the Legendre orthogonal polynomials (LOP) are the best ones in terms of convergence rate and computational efficiency for predicting the traveling wave modes. The patterns of [Formula: see text] expressed by Sigmoid law are more sensitive to the modal frequency of the shell than the ceramic volume fraction exponents. The stiffness of the axial springs plays a more significant role in the traveling wave modes than the stiffness of other springs.

Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

A unified vibration modeling of open cylindrical shell-rectangular plate coupling structures based on the dynamic stiffness method

This paper presents a unified vibration modeling for free and forced vibration analysis of coupled open cylindrical shell-plate structures (COSPS). In the model, the coupled structure is first split into several open cylindrical shells and rectangular plates, and then based on Kirchoff's thin plate theory and Flügge's thin shell theory, the dynamic stiffness (DS) matrix of each substructure is separately established by applying the generalized superposition method and the projection method. Subsequently, according to the continuity and equilibrium conditions of the coupled boundary, the coordinate transformation matrix of each substructure is derived. After obtaining fundamental DS matrices and their coordinate transformation matrices, global DS matrices of various COSPS are assembled using a strategy similar to the finite element method (FEM) without repeating the theoretical derivation. To verify the convergence and reliability of the current formulation, free vibration and forced vibration analysis of three types of coupled structures are carried out, and the results are compared with those from published works and FEM solutions. In addition, an experimental model of a coupled structure is established and an experimental test is performed. The comparison results show that the proposed model is reliable and effective, and its modeling process is more direct and convenient. This work not only greatly expands the application scope of the DSM but also provides a new idea for the vibration analysis of COSPS.

Nonlinear low-velocity impact of graphene platelets reinforced metal foams cylindrical shell: Effect of spinning motion and initial geometric imperfections

A nonlinear analysis is presented for low-velocity impact behavior of a graphene platelets-reinforced metal foams (GPLRMF) cylindrical shell with spinning motion in thermal environment, in which the initial geometric imperfection is incorporated. Taking three different graphene platelets (GPLs) distribution patterns into account, the GPLs reinforcement is either uniformly-distributed or functionally-graded along the shell thickness direction. The temperature-dependent micromechanical model is applied to estimate the material properties of GPLs reinforced composites. The nonlinear Donnell thin shell theory is utilized to derive the motion equations, in which von Kármán-type of kinematic nonlinearity is included. The influences of geometrical imperfections, spinning velocity, different boundary conditions, GPLs distribution patterns, foam distribution types, foam coefficient, GPLs weight fraction, temperature changes, the impactor' radius and initial velocity, prestressing force and damping coefficient on the low-velocity impact problems are discussed using the Runge-Kutta method in detail.

Assessing the effect of mechanical deformation of the panasonic NCR18650B lithium-ion power cell housing on its fire safety

This paper considers the deformation properties of the body of the lithium-ion power cell (LIPC) Panasonic NCR18650B (LiNi0.8Co0.15Al0.05O2) exposed to the action of static load at various techniques of fixing the cell. Determining the properties of LIPCs under appropriate conditions makes it possible to fill the gap in existing studies, which will further ensure the safety of their use. Based on the results of experimental studies, the LIPC rigidity and temperature indicators were determined in accordance with the applied load. The most dangerous variant, from the point of view of fire danger, of applying a static load on the cell has been established. It was experimentally established that, on average, the Panasonic NCR18650B LIPC housing can withstand a load of about 80 kg·s/cm2 (or 7.84 MPa) without further ignition. An increase in pressure force in the range exceeding 85–90 kg·s/cm2 leads to an irreversible chain thermochemical reaction, which, within 2–3 seconds, leads to LIPC ignition. Compressing the LIPC evenly along its lateral surface showed the occurrence of combustion at the load on the cell equal to 150 kg·s/cm2. The average temperature of the cell during combustion caused by the deformation of the housing is 350–450 °C, and the maximum value is registered in the range of 580–680 °C. The mathematical model built on the basis of the mathematical theory of thin shells adequately describes the stressed-strained state of the cylindrical body of cells under the action of a force concentrated and distributed load. The estimation model is satisfactorily verified by experimental results, making it possible to improve the strength and rigidity of LIPC housing by choosing the appropriate steel grade for its body, the geometric dimensions, and the structural technique of its fastening.

Influence of Non-uniform Parameter of Bolt Joint on Complexity of Frequency Characteristics of Cylindrical Shell

Bolt connection is one of the main fixing methods of cylindrical shell structures. A typical bolted connection model is considered as a tuned system. However, in the actual working conditions, due to the manufacturing error, installation error and uneven materials of bolts, there are always random errors between different bolts. To investigate the influence of non-uniform parameters of bolt joint, including the stiffness and the distribution position, on frequency complexity characteristics of cylindrical shell through a statistical method is the main aim of this paper. The bolted joints considered here were simplified as a series of springs with random features. The vibration equation of the bolted joined cylindrical shell was derived based on Sanders’ thin shell theory. The Monte Carlo simulation and statistical theory were applied to the statistical analysis of mode characteristics of the system. First, the frequency and mode shape of the tuned system were investigated and compared with FEM. Then, the effect of the random distribution and the random constraint stiffness of the bolts on the frequency and mode shape were studied. And the statistical analysis on the natural frequencies was evaluated for different mistuned levels. And some special cases were presented to help understand the effect of random mistuning. This research introduces random theory into the modeling of bolted joints and proposes a reference result to interpret the complexity of the modal characteristics of cylindrical shells with non-uniform parameters of bolt joints.

Investigation on Thermal Buckling of Functionally Gradient Material-Coated Cylindrical Shell Considering Material Properties Varying With Temperature

Abstract Although the thermal buckling problem of functionally gradient material (FGM) cylindrical shells has been investigated for many years, its theoretical solution is rarely reported when considering the material properties varying with temperature, and the existing commercial software also cannot directly solve the critical temperature rise of thermal buckling. Therefore, the theoretical solution of critical temperature rise was first derived for the FGM-coated cylindrical shell with temperature-dependent material properties based on the Donnell thin shell theory. And then, a stepped layer discrete finite element (FE) model was developed by integrating the bisection method into a user subroutine to calculate the critical temperature rise. The results show that the theoretical solutions are in good agreement with the numerical ones, and find out the temperature has a relatively large negative effect on the thermal buckling resistance of the FGM-coated cylindrical shell. Finally, the influence factors on the critical temperature rise were discussed in detail, and some suggestions have been formed to improve the calculation accuracy. This work not only provides a theoretical calculation formula but also develops an FE numerical method to calculate the critical temperature rise of the FGM-coated cylindrical shell, which will help the engineers to design the FGM-related structures easily.