Shallow Shells Research Articles

Thermally induced vibration of photovoltaic honeycomb-based-thermoelectric hybrid device

This paper presents a comprehensive investigation of thermally induced vibration (TIV) of elastically constrained photovoltaic honeycomb-based-thermoelectric hybrid device under space thermal environment. Based on four-variable refined quasi-3D higher-order theory as well as virtual boundary spring technique, the equation of motion of the hybrid device is derived by using Lagrange equation. The nonlinear temperature profile is obtained via finite difference scheme and the TIV responses are solved by employing Newmark approach. The effects of boundary condition, size of honeycomb cell, damping effect, thickness of insulation layer and radius of curvature on TIV of the hybrid device are studied. Results show that the TIV response of the hybrid device can be greatly weakened by replacing the dense thermoelectric leg with honeycomb-based thermoelectric leg. Designing the honeycomb cell as re-entrant architecture possesses a better suppression effect on TIV response. The amplitude of TIV reduces with increasing the thickness-length ratio and decreasing the height-length ratio of honeycomb cell. A larger thickness of insulation layer can weaken the TIV response. The oscillation of TIV response can be eliminated by fabricating hybrid device as circular cylindrical and hyperbolic parabolic shallow shells. The stiffer the boundary conditions, the less prone to TIV behavior of the hybrid device. The introduction of damping effect can suppress TIV response, but it simultaneously causes the occurrence of thermal snap. In general, the research provides several design references to weaken the TIV response of the hybrid device under space heat flux.

Using Collocation with Radial Basis Functions in a Pseudospectral Framework for a New Layerwise Shallow Shell Theory

Using Collocation with Radial Basis Functions in a Pseudospectral Framework for a New Layerwise Shallow Shell Theory

Nonlinear vibro-acoustic analysis of an axially moving submerged circular cylindrical shell

In this study, the nonlinear vibro-acoustic dynamics and stability of doubly-clamped axially moving cylindrical shell are investigated. The exterior surface of the shell is in contact with fluid and subjected to oblique incident plane sound wave. Donnell’s nonlinear shallow shell theory is used to derive the nonlinear partial differential equation of the cylindrical shell for the radial motion. The Galerkin method is employed to discretize the equations of motion into the set of coupled nonlinear nonhomogeneous ordinary second order differential equations. Considering both driven and companion modes, Multiple Scales Method is used to obtain the response of the system. The effects of sound level, incident angle and axially velocity on the frequency response of the system are studied. The results show that, with increase in the shell velocity transmission loss of the circular shell increases. In addition, at low shell axial velocities simultaneous excitation of the driven and companion modes occurs.

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.

Free vibration and buckling behavior of porous orthotropic doubly-curved shallow shells subjected to non-uniform edge compression using higher-order shear deformation theory

Due to the continuous enhancement of manufacturing technology for porous materials, it is expected to be increasingly used in aerospace, aviation, and other engineering applications, where the necessity for lightweight structures is paramount. Also, the non-uniform edge loads caused by service conditions, complex adjacent structures, and external loads lead to localized stress concentrations within the shells. Localized stress concentrations result in a loss of load-carrying capacity in specific regions, causing step-by-step failure of the shell. This study investigates the buckling and free vibration behavior of porous orthotropic doubly-curved shallow shells with four different porosity distribution patterns subjected to non-uniform edge compressive loads. The equations of motion of doubly-curved shallow shells are established by using higher-order shear deformation theory and Hamilton’s principle. Then, the pre-buckling in-plane stress distributions of doubly-curved shallow shells are determined and appended to the equations of motion. These fundamental partial differential equations are solved by Galerkin’s method, and the critical buckling load and natural frequency formulations are achieved. By comparing with the results in published literature, the feasibility and accuracy of present formulations are validated. Finally, systematic parametric studies are carried out to discuss the effects of different non-uniform edge compression patterns, porosity distribution patterns, porosity coefficients, aspect ratios, arc length-to-thickness ratios, radius-to-arc length ratios, orthotropy ratios, and shell types on the buckling and free vibration responses of porous orthotropic doubly-curved shallow shells. Parametric studies indicate that the reduction or increment in critical buckling loads (CBLs) and fundamental natural frequencies (FNFs) of spherical shells (SSs) are more sensitive than those of hyperbolic paraboloidal shells (HPSs). The CBLs and FNFs of SSs are larger than those of HPSs. The maximum and minimum CBLs are achieved for the triangular and uniform edge compression, respectively. The porosity effect is independent of edge compression pattern types.

On Numerical Flexural Behaviour of Hybrid Composite Shallow Shell Panels with Experimental Validation

On Numerical Flexural Behaviour of Hybrid Composite Shallow Shell Panels with Experimental Validation

Dynamic buckling active control of FGM spherical shell with piezoelectric sensor and actuator layers

The mechanical snap-through phenomenon represents one of the buckling modes inherent in shallow shells, exemplifying the shell's reaction to external loads, which can potentially have detrimental effects on structural performance. This article delves into the study of this phenomenon within the context of spherical shells. Leveraging functionally graded material (FGM) in the main layer of this structure proves instrumental in enhancing the shell's performance against rapid mechanical loads. Furthermore, the strategic placement of piezoelectric layers, both symmetrically and asymmetrically, coupled with the implementation of a proportional and derivative control system, facilitates the control of the shell's response. The formulation of displacement field and electric potential is grounded in the first-order shear deformation theory (FSDT) and second-order distribution, respectively. The strong form of nonlinear dynamic and electrostatic equations is derived using Hamilton's principle. Solving these equations is achieved through the implementation of the generalized differential quadrature (GDQ) method in the spatial domain and Newmark in the time domain. Additionally, the Newton-Raphson method is employed to tackle the inherent nonlinearity of the problem. The Budiansky's criterion is employed for the assessment of the load necessary for the buckling of the shell. Following the validation of results against numerical, analytical, and laboratory outcomes, a set of parametric results is presented. These results underscore the remarkable impact of the control system in mitigating shell deformation under time-dependent mechanical loads, as well as in postponing the occurrence of the snap-through phenomenon and damping the vibrations.

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.

Uncertainty sensitivity analysis for vibration properties of composite doubly-curved shallow shells using Kriging method

This paper presents a Kriging based global sensitivity analysis (GSA) method for the frequency response of displacements of composite doubly-curved shallow shells. A unified solution is utilized to develop the dynamic vibration formulation using the First-order Shear Deformation Theory (FSDT) and the Rayleigh-Ritz method. Kriging surrogate model is employed to substitute the frequency response function (FRF) of displacements. Ten parameters including materials and geometrical dimension are considered as input uncertain variables. A variance-based GSA method for dynamic model is employed to quantify the influence of each uncertain parameter. In addition, to avoid the computational burden of Monte Carlo simulation method (MCS), the presented sensitivity indices are computed analytically based on the Kriging mode, which further improves computational efficiency. Based on the convergence studies and comparison with traditional methods, the accuracy and efficiency of the present method are validated. The results shows that the frequency response of displacements exhibits greater sensitivity to changes in width, and thickness is more influential than others in the example from this article. Finally, the presented numerical results demonstrate vibration characteristics of different types of shells and observation points, which can also serve as a reference for further study on uncertainty-propagation analysis.

Nonlinear thermo-mechanical dynamic buckling and vibration of FG-GPLRC circular plates and shallow spherical shells resting on the nonlinear viscoelastic foundation

Nonlinear thermo-mechanical dynamic buckling and vibration of FG-GPLRC circular plates and shallow spherical shells resting on the nonlinear viscoelastic foundation

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Precise analysis of the static bending response of functionally graded doubly curved shallow shells via an improved finite element shear model

Doubly curved shallow shells, common in aerospace and civil engineering, pose significant challenges in predicting bending responses due to their complex geometry and material properties. Therefore, this research focused on the development of an efficient eight-node quadrilateral isoparametric element model based on a new improved first-order shear deformation theory (IFSDT) to analyze the bending deflection and stresses distribution of Functionally Graded (FG) doubly curved shallow shells. The present IFSDT obviates the necessity of a shear correction factor, which has challenges in determining for FG doubly curved shallow shells, by adopting a simpler assumption about transverse shear stresses. This assumption guarantees that the model precisely predicts the parabolic shear stresses distribution throughout the thickness of the FG shell while also meeting the requirements for free traction conditions on the top and lower surfaces of the shell. In the present analysis, five distinct types of FG doubly curved shallow shells are modeled: flat plates, spherical shells, hyperbolic parabolic shells, cylindrical shells, and elliptical paraboloid shells. A power law is employed to describe the gradual changes in material properties through the thickness direction. Numerical results for displacements and stresses are obtained for various shell types, materials, power-law factors, radii of curvature, and boundary conditions under uniform and sinusoidal loads. Comprehensive comparison and convergence studies are conducted to assess the robustness and accuracy of the proposed numerical model. This study also contributes new numerical results that are valuable for future research. Ultimately, the proposed numerical model offers a robust and straightforward tool for engineers and researchers engaged in the design and analysis of complex shell structures.

Elastic–Plastic Buckling of Scaled Model of Radial Ring-Stiffened Shallow Spherical Shells Based on Energy Method

Stiffened shallow spherical shell has the advantages including high strength and light weight, which allows it to be widely applied in areas including aerospace, submarine and civil engineering. For the design of scaled model for elastic–plastic buckling of a large radially–circumferentially stiffened shallow spherical shell under uniformly distributed pressure and for the purpose of similarity prediction, the similarity relation of the elastic–plastic buckling coefficient of the shell is derived; then by using the dense stiffening theory and considering the dual nonlinearity of geometry and materials, the energy increment formula for the structure under uniformly distributed external pressure is established; next, based on the similarity transformation of the energy increment of the structural system, the generalized similarity condition and similarity relation of elastic–plastic buckling of the structure under uniformly distributed external pressure are identified. Based on the scaling and equivalence of tensile–compressive stiffness and bending stiffness, the distortion scaled model for the prototype is designed. Through numerical simulation, the distortion similarity prediction for elastic–plastic buckling under uniformly distributed pressure of the radially–circumferentially stiffened shallow spherical shell that has square-shaped dimple defects has been conducted; based on the derived similarity relation of elastic–plastic buckling of stiffened plate, the similarity prediction for the elastic–plastic buckling of orthogonally stiffened plate in the plane under unidirectional uniformly distributed pressure has been conducted. The results indicate that the load–displacement curve, buckling load, and buckling mode of the distortion similarity scaled model, combined with the established radial displacement scaling factor expression, the scaling relation of tension–compression stiffness, and the similarity relation of buckling mode, can effectively predict the elastic–plastic buckling characteristics of its prototype structure. The research results can provide theoretical reference for the design and test of the scaled model with distortion of elastic–plastic buckling of similar stiffened large shell structures.

Nonlinear thermo-mechanical static stability analysis of FG-TPMS shallow spherical shells

An analytical solution for the nonlinear static stability problem of functionally graded triply periodic minimal surface (FG-TPMS) shallow spherical shells is studied in the current research for the first time. Three common TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) with three models of functionally graded porosity distribution along the thickness are considered. The shallow spherical shells (shallow SSs) are subjected to combined thermo-mechanical loadings and rested on a nonlinear elastic foundation. The fundamental formulas are expressed based on the higher-order shear deformation theory (HSDT) and von Kármán's geometrical nonlinearities. Employing the Ritz energy minimization method, the explicit relationship between load and deflection is derived. Subsequently, the static stability behavior of FG-TPMS shallow SSs is investigated. Numerical illustrations are investigated to show the superior thermo-mechanical load-carrying performance of the FG-TPMS SSs compared to corresponding isotropic structures of the same weight. The significant effects of geometric parameters, nonlinear elastic foundation parameters, and the type of FG-TPMS structures on the nonlinear static stability behavior of shallow SSs are further considered.

Bayesian-informed fatigue life prediction in shallow shell structures with the dual boundary element method

This study introduces an innovative approach for statistically inferring the Equivalent Initial Flaw Size Distribution (EIFSD) in shallow shell structures with the aim of predicting fatigue life. The proposed approach integrates Bayesian inference and surrogate modelling techniques, and utilizes a long crack growth model. The EIFSD is crucial for estimating fatigue life and ensuring structural durability. To showcase the proposed methodology, a numerical example featuring a fuselage window structure under multi-axial loading, employing the Dual Boundary Element Method (DBEM) for crack growth modelling, is presented. Efficiency gains in the inference of the EIFSD are achieved through the development of surrogate models, notably a Co-Kriging model that significantly reduces computational time and outperforms a Kriging model in error reduction. The study explores two assumptions for inspection data sourcing, leading to enhanced methodological effectiveness across varying uncertainty levels. It considers the separate effects of Paris Law constants and loading conditions under increased uncertainty, presenting cases that demonstrate the Co-Kriging model’s high accuracy and computational efficiency. The proposed methodology’s robustness is further validated by assessing the impact of prior distribution quality on Bayesian inference, showing the method’s adaptability to high uncertainty while maintaining a good level of accuracy. This study also demonstrates an approach for fatigue life estimation of structures using the inferred EIFSD. By evaluating the structure up to a critical crack length, the methodology allows for the accurate estimation of fatigue life, which, in turn, facilitates the determination of optimal inspection intervals. Notably, under conditions of high uncertainty, the method showcased a maximum difference of 1.51% in the estimated fatigue life when compared to the true EIFSD, underscoring the technique’s high accuracy and reliability.

From elastic shallow shells to beams with elastic hinges by Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence

In this paper, we study the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-limit of a properly rescaled family of energies, defined on a narrow strip, as the width of the strip tends to zero. The limit energy is one-dimensional and is able to capture (and penalize) concentrations of the midline curvature. At the best of our knowledge, it is the first paper in the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence field for dimension reduction that predicts elastic hinges. In particular, starting from a purely elastic shell model with “smooth” solutions, we obtain a beam model where the derivatives of the displacement and/or of the rotation fields may have jump discontinuities. Mechanically speaking, elastic hinges can occur in the beam.

Shape prediction in continuous roll forming of sheet metal based on rigid rolls

The continuous roll forming based on rigid rolls (CRFRR) is an innovative process for rapid fabrication of doubly curved surfaces of sheet metal. The curved parts with different shapes and sizes can be obtained by adjusting the generatrix radius of the two rigid rolls or controlling the rolling reduction. In this paper, the principle of CRFRR is introduced and the process that a 3D surface of sheet metal with positive or negative Gaussian curvature is formed using two rigid rolls is analyzed based on the distribution of exit velocity. The exit velocity of material after the sheet metal passes through the roll gap is decomposed into two portions: a linearly distributed velocity and a very small additional velocity. On the basis of exit velocity decomposition and shallow shell theory, an analytical method for predicting the shape of formed surface in CRFRR process is proposed for the first time. The primary part of curvature of the formed 3D surface in longitudinal direction is determined by the linear portion of exit velocity and that in transverse direction is determined by the generatrix radius of the two rigid rolls. The curvature change in the final shape of 3D surface is induced by the additional portion of exit velocity and it is solved from equilibrium equation of shallow shell. The final curvature of the formed surface is acquired through superimposing the primary curvature and the curvature change. Numerical simulations and forming experiments have been carried out on convex and saddle-shaped parts, and the results confirm the validity and accuracy of presented method.

Vibrations of viscoelastic FG porous graphene-platelets reinforced doubly-curved shells via experimental characterisation of graphene platelets

This work explores the vibrations of viscoelastic functionally-graded (FG) porous graphene-platelets reinforced doubly-curved shallow shells via experimental characterisations of graphene platelets. In particular, the effects of ultrasonication time of graphene platelets, on the vibrations of graphene-platelets reinforced shells, are studied. This is of interest because graphene platelets are often ultrasonicated in solvents to achieve better dispersions during the fabrication of graphene-platelets/epoxy composite structures. In this study, geometric characteristics of the graphene platelets under different sonication times are quantitatively and qualitatively analysed through experimental characterisation techniques, including a laser diffraction particle size analyser, a scanning electron microscopy (SEM), and a scanning force microscopy (SFM). The geometric characteristics of graphene platelets are considered in the effective material properties using a micromechanics model of Halpin-Tsai. Different FG distributions of graphene platelets and closed-cell porosity are modelled. The Kelvin-Voigt viscoelastic model is used to consider the internal friction-induced energy dissipation for the polymeric composite shell. Hamilton’s variational principle and an assumed mode method are used to derive and solve the equations of motion, respectively, for the vibrations of the doubly-curved shells. The effects of graphene platelets sonication time, in combination with the effects of graphene platelets weight fraction, porosity coefficient, FG distributions, shell thickness ratio, and shell curvature radii, are scrutinised. The results show that increasing ultrasonication time reduces the shell’s out-of-plane dimensionless fundamental frequency as a result of the reduction in graphene platelets aspect ratio.

Vibration control of thin shallow paraboloidal shells with light-activated shape memory polymers

With the applications of thin-walled paraboloidal shell structure in aviation, aerospace and communication systems, smart-structure based active vibration control becomes essential to improve the performance of thin-walled parabolic structures. Light-activated shape memory polymer (LaSMP) is a novel actuator material that exhibits dynamic Young's modulus and strain properties under ultraviolet (UV) light exposures, which can be used for non-contact vibration control. This study focuses on investigating the vibration control performance of a shallow parabolic thin shell, with simply supported boundary conditions, with laminated LaSMP patches. Based on the LaSMP strain model, the vibration control effect of LaSMP patch on the thin shell is discussed. A finite element model of the simply supported parabolic thin shell is developed to obtain the natural frequencies. The modal expansion method is employed to investigate the vibration control of the shallow parabolic thin shell laminated with LaSMPs, and independent modal responses are calculated. According to the different control mode, the effects of LaSMP actuators on vibration control of shallow paraboloid shell are studied in three cases with varying actuation position and coverage area of LaSMP actuators. The results indicate that LaSMP can control the vibration of the thin shell by reducing the vibration amplitude and the final vibration control effect is determined by the mode shape, the initial LaSMP strain, and the location and area covered by LaSMP actuators. By establishing the model between the LaSMP actuator force and the attenuations in modal amplitude, this study provides an analytical tool for the application of LaSMPs in vibration control of flexible structures.

Post-buckling behaviour of corrugated-edge shells: Numerical insights

We provided the detailed buckling and post-buckling analysis of corrugated shallow spherical shells. Such shells are extensively used in civil and mechanical engineering. Here the main attention is paid to a corrugation as a key parameter affecting the instability. Three types of corrugation are introduced and discussed. The latter includes the following types: edge corrugated, half corrugated, and fully corrugated shells. Since the effect on instability is highly affected by the imperfections, a customised set of defects is proposed compared to a perfect structure. Several boundary conditions and load cases are analysed. An increase in the load multiplication factors highlights the effects of corrugation in shallow shells mechanics. Being inspired by the fascinating Nervi’s Flaminio dome in Rome, we introduce a novel way to improve the shells design against the instability effects.