Shallow Shells Research Articles

An analytical solution of boundary problems of shell deformation mechanics

This paper proposes an algorithm for the analytical (with a controllable error) solution of boundary-value problems of shell deformation mechanics. Decomposition of partial differential equations is per-formed. The obtained ordinary differential equations in matrix form are solved in the form of converging matrix series in the corresponding coordinates. Atthe point of intersection of the coordinates of ordinary differential equations, the algebraic equations of their connection are satisfied. The error control of the solution of the boundary value problem is carried out by comparing the results when the grid of the de-composition of equations is refined. The study solved the problems of deformation a hinged cylindrical shell and a shallow shell fixed on the edges loaded with external pressure.

Simulation of Viscoelastoplastic Behavior of Shallow Shells with Account for Strain Rate of the Material

Simulation of Viscoelastoplastic Behavior of Shallow Shells with Account for Strain Rate of the Material

Mechanical response of aluminum foam sandwich structure under impact load

This study investigates the mechanical response of aluminum foam sandwich panels, sandwich cylindrical shells, and sandwich shallow shells under impact loads. First, a finite element model of the sandwich panel was established, and an impact load was applied. The numerical results were compared with theoretical and experimental results to verify the model’s effectiveness. Second, the energy absorption efficiency and overall deformation of sandwich panels, sandwich cylindrical shells, and sandwich shallow shells under the same impact load were studied. The research shows that the energy absorption performance of the sandwich shells is better than that of the sandwich panels, and the overall deformation is less than that of the sandwich panels. The effect of increasing panel thickness on the two types of sandwich shell studies is based on this basis. The conclusions describe that increasing the panel thickness will significantly reduce the structure’s energy absorption efficiency and deformation. Finally, the effect of single-and double-layer structure on the impact resistance of sandwich shells was studied when the total thickness of the sandwich structure was unchanged. The results show that compared with the single-layer structure, the energy absorption efficiency, overall deformation, and contact force between the projectile and structure of the double-layer structure will be reduced.

Thermo-Mechanical Transient Flexure of Glass-Carbon-Kevlar-Reinforced Hybrid Curved Composite Shell Panels: An Experimental Verification

The time-dependent deflection values of the flat/curved hybrid fibre (Glass–Carbon–Kevlar)-reinforced composite panels under the influence of the mechanical loading (UDL and SDL) and the elevated thermal environment are analyzed in this research. To compute the numerical responses, a higher-order kinematic model is prepared considering the effect of single/double curvature of the shell panel. The weak form of the governing equation is established through the variational technique. The steady-state deflections are obtained by solving the discrete form of governing equation of transient vibrations using finite element method (FEM) by employing Newmark’s time integration method in MATLAB environment. The isoparametric Lagrangian element (nine-noded) is adopted to discretize the system equation of the hybrid panels. The validity of the numerical solutions, computed using the experimental hybrid composite properties, is verified with the experimental (in-house) results as well as with the benchmark results available in the published literature. Finally, the parametric study of dynamic deflection characteristics of hybrid composite shallow shell panels is carried out by varying the parameters such as curvature ratio, support conditions, hybrid schemes and type of loading in the elevated thermal environment and the results are discussed in detail.

Thermo-elastic damped nonlinear dynamic response of the initially stressed hybrid GPL/CNT/fiber/polymer composite toroidal shells surrounded by elastic foundation

This paper aims to investigate the geometrical nonlinear response of the hybrid composite toroidal shells subjected to harmonic time-dependent external pressure and constant in-plane compressive loadings. The hybrid composite incorporates some layers made of the fibres and the matrix in which the polymeric matrix in some layers is reinforced by carbon nanotubes (CNTs) and in the other layers by graphene nanoplatelets (GPLs). The fundamental equations are extended based on the first-order shear deformation theory (FSDT) and are solved by introducing the harmonic functions to estimate the deformation field and employing Galerkin method. Finally, the obtained nonlinear dynamic differential equation of motion is numerically solved using fourth-order Runge–Kutta method. The extended formulation is verified by comparing the obtained results corresponding to some problems with those are achieved in the other studies using the various methods. Then, an extended parametric study is carried out by using the present method incorporating the effects of thermo-mechanical, geometrical, and loading factors on the forced vibration of the shell. The obtained results show that the novelty of the present study can be justified by some advantages. Firstly, it enables a robust tool for analysis of GPL/CNT/fiber/polymer hybrid composite shallow toroidal shell segments in the geometrical nonlinear state with no time consuming computational efforts. Secondly, the present formulation is so simple while it has not restriction for thin or moderately thick shells.

Nonlinear static analysis of thin shallow and non-shallow shells using tensor formulation

In this study, a large-amplitude nonlinear static analysis of thin shallow and non-shallow shells is performed. Simply supported linear homogeneous elastic spherical panels and parabolic conoidal shells subjected to concentrated and distributed external loads are considered. To model the shells, the non-orthogonal tensor theory is applied and, to describe the strain–displacement relations of the shells, two approximations of Koiter’s theory are employed. To obtain the set of nonlinear equilibrium equations, a large-dimensional model is obtained by expanding the field displacements using Fourier series and a reduced order model is generated by using the Karhunen–Loève method. The proper orthogonal modes (POMs) of Karhunen–Loève method are acquired from a numerical model performed via Finite Element Analysis (FEA). The nonlinear static paths are compared with literature and with results obtained from FEA. The tensor formulation applied in this work has proven to be efficient when dealing with both shallow and non-shallow shells which are described by non-orthogonal curvilinear coordinates on nonlinear regime.

Seismic performance of buildings supported by a shallow doubly-curved shell raft foundation

Foundation transmits loads of superstructure to soil layers. Also, seismic loads are transmitted to the superstructure via the foundation. It has been shown that the characteristics of the foundation affect the seismic response of the superstructure. The influence of shell foundations on the seismic performance of structures has not been investigated in the literature. In this paper, the seismic response of 5, 10, and 15-story steel moment-resisting structures supported by a shallow doubly-curved shell raft foundation is compared to the seismic response of these structures supported by a pile-raft foundation. It was assumed that structures are founded on soft-clayey soil. Soil-foundation-structure systems were analyzed under the influence of four earthquake excitations using fully nonlinear time-history analysis. The numerical results generally show that the maximum displacement decreases while the maximum shear force increases in each story of structures with shell foundation compared to structures with pile-raft foundation. Moreover, the results reveal the better seismic performance of structures supported by shell foundation in terms of the maximum inter-story drift compared to structures supported by pile-raft foundation.

The Refined Model of the Elastic-Plastic Dynamic Behavior of Reinforced Curved Panels Sensitive to Strain Rate

The initial-boundary value problem of dynamic elastic-viscoplastic deformation of flexible curved panels (shallow shells) with plane -cross and spatial reinforcement structures is formulated. The inelastic behavior of the materials of the composition components is described by the constitutive equations of the theory of plastic flow with isotropic hardening, and their sensitivity to strain rate is taken into account. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The used kinematic and dynamic two-dimensional relations and the corresponding boundary conditions make it possible to describe, with varying degrees of accuracy, the mechanical bending behavior of shallow composite shells. This takes into account the possible weak resistance of such reinforced panels to transverse shears. In the first approximation, the used two-dimensional equations, the initial and boundary conditions degenerate into the relations of the traditional non-classical Ambartsumyan theory. For the numerical integration of the formulated nonlinear dynamic problem, an algorithm of time steps is applied, based on the use of an explicit scheme of the cross type. The elastoplastic and elastic-viscoplastic behavior of the reinforced cylindrical shallow shells under transverse dynamic loads generated by an air blast wave is investigated. Metal-composite and fiberglass thin-walled constructions are considered. It is shown that the refusal to take into account the dependence of the plastic properties of the components of the composition on the rate of their deformation does not allow adequately describing the inelastic dynamic behavior of both metal-composite and fiberglass shallow shells. It is shown that in the calculations of even relatively thin reinforced cylindrical panels (with a relative thickness of 1/50), the use of the Ambartsumyan theory leads to completely unacceptable results in comparison with the refined bending theory. It has been demonstrated that even for relatively thin curved fiberglass panels, replacing the traditional flat -cross reinforcement structure with a spatial structure with obliquely laid fiber families can significantly reduce not only the intensity of deformations in the binder, but also the maximum deflection values in modulus. For metal-composite shallow shells with a weakly expressed anisotropy of the composition, the positive effect of the indicated replacement of reinforcement structures is practically not manifested.

Vibration and nonlinear dynamic response of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio

This paper reports a study on natural frequency, nonlinear dynamic response and frequency amplitude relation of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio. The nonlinear motion equations and the compatibility equation are retrieved from first order shear deformation theory including the effect of von Karman nonlinearity terms, elastic foundations and initial imperfection. The system of dynamic governing equations is obtained by utilizing the Airy stress function and Bubnov–Galerkin procedure. The nonlinear dynamic response is also obtained by applying the 4th-order Runge–Kutta method. The frequency–amplitude relations of the shell is determined by using the assumption of inertia caused by a rotation. Besides, the analytical method and the finite element method are proposed to calculate the vibrational frequencies. Five types of FG-CNTRC laminate shell, three types of the structures and different thermal environments with the shell’s material properties depending on the temperature are also considered in this study. The numerical results are presented, verified with available studies in the literature. Finally, the effects of elastic foundations, initial imperfection, and temperature-dependence, typing double curved shallow shell, angle of lamination against the shell x-axis on the natural frequency and nonlinear dynamic response of FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio are discussed.

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Analysis of electrical energy harvesting from piezoelectric integrated shallow conical composite shells in metastable configurations using mixed formulation

This article presents a study of energy harvesting from smart composite Belleville springs, which are truncated conical shallow shells bonded with piezoelectric patches. When the springs subject to mechanical loading–unloading cycles, electrical energy is generated from the integrated piezoelectric elements via the direct piezoelectric effect. A mixed strong and weak form mathematical model composed of strain energy and electric enthalpy is formulated to analyze the electro-elastic performance of the shells. Undergoing snap-through and snap-back buckling when loaded and unloaded in compression, the smart Belleville springs in metastable configurations exhibit unprecedented complex electromechanical behaviors, which can be applied to enhance energy scavenging. The developed model is firstly validated with the finite-element method using ABAQUS™ and the responses from both approaches are in excellent agreement. Parametric studies on the spring geometry are systematically conducted to determine geometrical effects on stored electrical energy in the piezoelectric patches. The numerical results obtained from the mixed-form model show that snap buckling in the metastable conical shells can be used to harvest a relatively large amount of electrical energy. Conclusions of this study lay down a design guideline for energy extraction from the smart composite Belleville springs with a variety of geometric and kinematic constraints.

Nonlinear thermo-mechanical static analysis of toroidal shells made of nanocomposite/fiber reinforced composite plies surrounded by elastic medium

This article aims to evaluate the geometrical nonlinear behavior of toroidal shell segments surrounded by elastic medium under two types of axial edge-loading and external pressure, including the thermal effects. The considered shells are assumed to be made of some composite plies incorporating three constituents: carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs) as nanocomposites, fibers and polymeric resin. The considered nanocomposites are assumed to be randomly oriented and functionally distributed in the resin. The governing equations are extended based the first-order shear deformation theory (FSDT) for the moderately thick shells. Also, an analytical solution using Galerkin method is proposed which yields to a closed-form load–deflection relationship. The present formulation has two important advantages which justify its novelty. It provides a robust tool for geometrical nonlinear static analysis of hybrid GPL/CNT/fiber/resin composite shallow toroidal shells with no computational efforts. Although the present method incorporates thin and moderately thick shells with a complicated thermo-mechanical specifications, it has been extended simpler than the other related formulations proposed in the previous studies. In order to validate the applicability of the extended formulation, some benchmark examples from the existing literature are considered and the corresponding equilibrium paths are compared with those are obtained by using the present method. Also, the parametric studies incorporate the effects of various geometrical, mechanical, and thermal parameters on the nonlinear equilibrium path of the proposed shells. The verification procedure and the performed parametric studies show that the proposed method and extended formulation enable a strong tool for the geometrical nonlinear analysis of such shells.

Nonlinear dynamic analysis of nonlocal composite laminated toroidal shell segments subjected to mechanical shock

The present study deals with the nonlinear forced vibration of composite shallow nano toroidal shell segments. The improved Donnell shell theory is used with the nonlinear geometric strains, i.e., von Karman, Stein, and McElman’s assumptions. A coupled nonlinear partial differential equations using nonlocal theory, including the influence of nano-scales, is derived. The boundary conditions are simply supported and clamped–clamped, and the nano toroid shell is subjected to impulse radial dynamic excitation. The linear solution is verified by comparing natural frequencies with the available results using Galerkin’s method. The approximated linear mode shapes of the free vibration, which satisfy the boundary conditions, are used as the assumed multi-mode displacement functions required to solve the nonlinear vibration. Using Hamilton’s principle, the nonlinear equations of motion in generalized coordinates are solved using the displacement field by applying Galerkin’s and Runge–Kutta’s methods. Using the presented model, the effects of layer stacking, toroidal shell segments curvature (convex and concave toroidal shells), length scale parameter, and material properties are studied in detail, and some conclusions are surveyed. It is observed that the nonlocal parameter demonstrates a considerable effect on the dynamics behavior characteristics response of the toroidal shell segments. Furthermore, different fiber orientation effects, displacement fields, time response, and geometry parameters, i.e., thickness, and length are studied. Results indicate that natural frequencies significantly depend on the toroid dimensions, boundary conditions, layer stacking, curvature change, and considered wave numbers.

Potential well evolution and metastable dynamics of bistable asymmetric laminated composite square shallow shell under external and parametric excitations

This paper studies the occurrence conditions of the metastable chaos by analyzing the potential well evolution from the double potential wells to triple potential wells in the bistable asymmetric laminated composite square shallow shell under the foundation excitation. Based on the dependence of the potential well types on the parameters of the potential function, the critical condition is derived for the metastable chaos in the bistable asymmetric laminated composite square shallow shell. In the theoretical researches, when the parametric excitation is large enough to exceed the critical value, it is found that the shapes of the potential curves change from the double potential wells to triple potential wells. By contrast, the external excitation only affects the depth variation of the potential well, but not the type of the potential well. It is verified that the larger parametric excitation leads to the metastable chaos of the bistable asymmetric laminated composite square shallow shell. The results of numerical simulation are consistent with the theoretical analysis. The parametric excitation is the key factor leading to the metastable phenomena in the bistable asymmetric laminated composite square shallow shell. The influences of the external and parametric excitations on the metastable dynamics of the bistable asymmetric laminated composite square shallow shell are studied based on the results of numerical simulations. It is also found that there is no the direct jumping path between the upper-mode and lower-mode oscillations once the metastable chaos occurs.

Nonlinear Dynamic Analysis of Porous Graphene Platelet-Reinforced Composite Sandwich Shallow Spherical Shells

Nonlinear Dynamic Analysis of Porous Graphene Platelet-Reinforced Composite Sandwich Shallow Spherical Shells

A unified modeling method for dynamic analysis of GPLs-FGP sandwich shallow shell embedded SMA wires with general boundary conditions under hygrothermal loading

In this research, the dynamic analysis of graphene platelets (GPLs) reinforced functionally graded porous (FGP) sandwich shallow shell embedded shape memory alloy (SMA) wires with general boundary conditions under hygrothermal loading is conducted. Four different porosity distributions of FGP reinforced with four various GPLs dispersion patterns are considered. Based on the Brinson formulation, the constitutive equation of the SMA is established. The energy functional of the sandwich shallow shell is constructed by employing the Hooke’s law and the first-order shear deformation theory (FSDT). Ultimately, the dynamic characteristics of the sandwich shallow shell are obtained by solving the energy functional with the Rayleigh-Ritz method. After the verification of accuracy and reliability, parametric studies on the effects of material properties, geometrical parameters, and boundary conditions are investigated.

Local buckling of thin-walled circular hollow section under uniform bending

This article presents a semi-analytical finite strip method based on Marguerre’s shallow shell theory and Kirchhoff’s assumption. The formulated finite strip is used to study the buckling behavior of thin-walled circular hollow sections (CHS) subjected to uniform bending. The shallow finite strip program of the present study is compared to the plate strip implemented in CUFSM4.05 program for demonstrating the accuracy and better convergence of the former. By varying the length of the CHS, the signature curve relating buckling stresses to half-wave lengths is established. The minimum local buckling point with critical stress and corresponding critical length can be found from the curve. Parametric studies are performed to propose approximative expressions for calculating the local critical stress and local critical length of steel and aluminum CHS.

A second‐order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations

AbstractThis article develops an efficient second‐order reduced multiscale (SORM) method to study the nonlinear shell structure with orthogonal periodic configurations. The heterogenous shell structure is periodically distributed in orthogonal curvilinear coordinate systems. At first, the nonlinear problems for the shell structure are introduced, and the detailed higher‐order nonlinear multiscale formulas based on the asymptotic homogenization approach are given including microscale unit cell functions, effective material parameter and the homogenized equation. Also, since it requires a large number of computation cost to solve the nonlinear multiscale problems by the traditional high‐order homogenization methods, the novel reduced order multiscale model is constructed. Further, according to the reduced‐order multiscale models and higher‐order nonlinear formulas, an effective SORM algorithm is provided for studying the nonlinear shell structures. The main characteristics of the proposed algorithm are that the novel reduced forms established to investigate the nonlinear shell structures and an efficient higher‐order homogenized solution evaluated by postprocessing that does not need higher‐order continuities of the homogenization solutions. Finally, according to some typical nonlinear examples including block structures, cylindrical shell and double‐curved shallow shell, the availabilities of the SORM algorithm are confirmed.

A FSDT meshfree method for free vibration analysis of arbitrary laminated composite shells and spatial structures

In this paper, a FSDT meshfree method based on the 3D continuous shell theory and moving-least squares approximation is proposed for the first time to investigate the free vibration behavior of arbitrary laminated composite shells and spatial structures. The novelty lies in that the resultant meshfree model has completely general geometric and kinematic descriptions, which are more readily applied to complex laminated shell geometries. In the formulation, the geometrical representation is based on the mapping technique without simplifying assumptions on the shell geometry, whereas the moving least-squares (MLS) approach and the first shear order theory (FSDT) are employed to build displacement fields. By using the mapping technique, a 3D arbitrary curved surface is expanded in a two-dimensional space, and the displacement approximation is established based on a set of nodes scattered in the Converted coordinate system. The simple Gauss integral are also formed in the Converted coordinate system. The essential boundary conditions are imposed by the full transformation method. According to Hamilton principle, the meshfree governing equations of free vibration of arbitrary laminated shells are obtained. The convergence, accuracy and applicability of the proposed method are demonstrated for the laminated shells of different geometrical shell shapes involving the square plates, shallow shells, open cylindrical shells, conical shells and corrugated plates with different boundary conditions and lamination schemes.

Static response analysis of shallow spherical shell under local support of magnetorheological fluid (MRF)

Shallow spherical shell is a typical engineering structure with excellent pressure bearing capacity. To resist the machining deformation induced by cutting loads, magnetorheological fluid (MRF) is adopted to support the shallow spherical shell. However, the static response of shallow spherical shell under MRF support has a close relationship with the applied magnetic field, which has not been characterized quantitatively and essentially. In this article, the static response model of shallow spherical shell under local MRF support was established. A numerical solution of the static response model was proposed. After solving the model, the relationship between the static response of shell and the magnetic field was quantitatively characterized. From comparison, the predicted deformation results using the developed model were consistent with the FE simulations and experiment tests. In engineering, the model can be used to ensure the thickness accuracy of shallow spherical shell.