Order Shell Theory Research Articles

A GFDM based computational model for the analysis of tunnels under gravitational loadings

Based on an impedance relation, a new computational model is developed for the analysis of tunnels under gravitational loading. The shape of tunnel is arbitrary. The impedance matrix, representing the soil resistance, is evaluated through integration of the equations of elasticity by generalized finite difference method (GFDM). The rigidity matrix of tunnel ring is constructed by using two types of elements: plate and shell elements. The equations of these elements are evaluated through integration of the equations of higher order plate and shell theories again by GFDM. These element equations accommodate not only axial and bending deformations, but, also shear deformations in the ring. In the study, the rotational joint deformation is simulated through the use of rotational spring model and the slippage is considered by modifying the impedance matrix so that the tangential soil resistance force between the ring and soil medium vanishes. Some numerical results are presented, in nondimensional forms, for circular and square tunnels where the following three cases are considered for circular tunnels: a) the soil medium is infinite, the change in σv (vertical (in situ) compressional stress) in the vicinity of tunnel is disregarded b) the soil medium is infinite, the change in σv is taken into account c) the soil medium is halfspace (HS). In connection with the case of (a), a comparison is presented for the ring flexibility curves of radial displacement and section forces of circular ring with those reported in literature. In view of the interpretations of the results and comparisons, we think, the proposed model may be used reliably in the analysis of tunnels.

Mathematical Modeling of Anisotropic Hyperelastic Cylindrical Thick Shells by Incorporating Thickness Deformation and Compressibility with Application to Arterial Walls

This paper is devoted to mathematical modeling of anisotropic hyperelastic thick cylindrical shells like arteries by taking into account the volume compressibility and through-the-thickness deformation. To describe the hyperelastic behavior of this kind of shells and extract their constitutive relations, the modified anisotropic (MA) model is employed, which is able to characterize compressible behavior of hyperelastic materials like soft tissues. By considering the arterial segment as a thick cylindrical shell, the higher order thickness deformation shell theory together with nonlinear Green’s strains are exploited to express its deformations and capture the thickness stretching effect. The shell is then discretized via Lagrange equations of motion to achieve the responses of the arterial wall. Numerical results are given in two sections. First, the presented formulation is validated by conducting a comparative study on a single-layer compressible pressurized shell. Close agreement between the outcomes of the developed model with those provided by finite element (FE) simulation signifies the necessity of incorporating thickness deformation in the higher order shell theory. In the next step, by considering the artery as a two-layer cylindrical shell comprising the media (the middle layer of the artery) and the adventitia (the outer layer), mechanical responses of an arterial wall subjected to an internal pressure are appraised through a number of parametric studies.

Influences of fiber reinforced polymer layer on the dynamic deflection of concrete pipes containing nanoparticle subjected to earthquake load

AbstractA numerical solution for dynamic analysis induced by earthquake load in concrete pipes containing SiO2 nanoparticles warped by fiber reinforced polymer (FRP) layer is utilized in the present paper on the basis of Mori–Tanaka model. The FRP layer is used as reinforcement for the concrete pipe. Utilizing the cylindrical higher order shell theory, the concrete pipe is mathematically simulated and the related governing relations are derived based on. Utilizing numerical methods of Newmark and Galerkin, the dynamic displacement of the concrete pipe is obtained. The influence of FRP layer and its thickness, volume fraction and agglomeration of nanoparticle, boundary conditions and length to radius ratio of the concrete pipe is studied on the dynamic deflection of the concrete pipe. Numerical outcomes illustrate that the FRP layer can reduces the dynamic deflection significantly.

Accurate compact solution of fluid-filled FG cylindrical shell inducting fluid term: Frequency analysis

Shell motion equations are framed with first order shell theory of Love. Vibration investigation of fluid-filled three layered cylindrical shells is studied here. It is also exhibited that the effect of frequencies is investigated by varying the different layers with constituent material. The coupled and uncoupled frequencies changes with these layers according to the material formation of fluid-filled FG-CSs. A cylindrical shell is immersed in a fluid which is a non-viscous one. These equations are partial differential equations which are usually solved by approximate technique. Robust and efficient techniques are favored to get precise results. Use of acoustic wave equation is done to incorporate the sound pressure produced in a fluid. Hankel’s functions of second kind designate the fluid influence. Mathematically the integral form of the Lagrange energy functional is converted into a set of three partial differential equations.

Structural performance of submerged ring support FG shell using numerical ananlysis

In this study, the cylindrical shell submerged in a fluid and surrounded by ring supports. The use of acoustic wave equation is done to incorporate the sound pressure produced in a fluid. Hankel's functions of second kind designate the fluid influence. Mathematically the integral form of the Lagrange energy functional is converted into a set of three partial differential equations. Shell motion equations are framed first order shell theory due to Love. These equations are partial differential equations which are usually solved by approximate technique. The transverse constraints produced ring supports are assumed by the polynomial functions possessing degree equal to the number of ring supports. The frequencies with ring supports against wave number, length-to-radius ratio and height-to-radius ratio are investigated. The frequency analysis versus wave number for simply supported cylindrical shells submerged in a fluid with ring supports is given for different types of configuration. The variations of frequencies against the positions of the ring supports are furnished for not submerged and submerged cylindrical shells. It is observed that vibration frequencies increase and decreases as the positions of a ring support is increased. Programming is written in MATLAB codes to solve the frequency equation for the computation of frequencies of shells submerged in a fluid along with ring supports. The frequency result of submerged cylindrical shell is less than with the results of not submerged cylindrical shell. Robust and efficient technique produced the valid results.

Free vibration analysis of the macro-micro-nano plates and shells made of a material with functionally graded porosity: A closed-form solution

This article deals with introducing a closed-form solution for considering the vibration of macro-micro-nano plates and shells made of unit material with functionally graded porosity (FGP). Classic continuum theory beside modified nonlocal strain gradient theory are used at macro and micro/nano scales, respectively. The governing equations are obtained in the frame work of third order plate and shell theories. In the solution procedure, a differential operator is defined and by using it, five equations of motion are converted to the five algebraic equations. Then, by satisfying the equations of boundary conditions and achieving the nonzero solution, the frequencies are extracted.

Popular benchmarks of nonlinear shell analysis solved by 1D and 2D CUF-based finite elements

This research work deals with the analysis of elastic shell structures in the large displacement and rotation field adopting one-dimensional (1D) and two-dimensional (2D) unified models. Namely, higher order beam and shell theories accounting for geometrical nonlinearities are formulated by employing a unified framework based on the Carrera unified formulation (CUF) and a total Lagrangian approach. Thus, a finite element (FE) approximation is used along with a Newton-Raphson method and an arc-length path-following approach to perform nonlinear analyses. Low- to higher order beam and shell theories are used to evaluate the nonlinear equilibrium path, and results are compared between the two models, with reference solutions coming from literature or 2D and three-dimensional (3D) models from NASTRAN. Convergence analyses show how CUF 1D models are able to describe the geometrical nonlinear behavior of analyzed structure with a lower number of degrees of freedom (DoFs) than 2D and 3D models.

Dynamic response of functionally graded graphene nanoplatelet reinforced shells with porosity distributions under transverse dynamic loads

This article investigates forced vibrational characteristics of a porous nanocomposite shell reinforced by graphene platelets (GPLs) under radial dynamic loads. Material properties of the shell dependents on uniform and non-uniform distributions of GPLs and porosities. Description of the material properties of such nanocomposites accounting for porosity and GPL content is performed via Halpin-Tsai micromechanical rule. The dynamic equations of the nanocomposite shell are based on first order shell theory and a semi-analytical approach is implement to solve these equations and obtain dynamic deflections due to applied load. It is elucidated that GPL and porosity content can significantly affect the dynamic response of a nanocomposite shell. Actually, the resonance frequency of a nanocomposite shell can be increased by increasing the GPL percentage.

Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory

In this paper, a variational approach for the wave dispersion in anisotropic doubly-curved nanoshells is presented. To study the doubly-curved nanoshell as a continuum model, a new size-dependent higher-order shear deformation theory is introduced. In order to capture the small scale effects, nonlocal strain gradient elasticity theory has been implemented. The present model incorporates two scale coefficients to examine the wave characteristics much accurately. Based on Hamilton’s principle, the governing equations of the doubly-curved nanoshells are obtained. These equations are solved via analytical approach. From the best knowledge of authors, it is the first time that present formulation is used to investigate the wave dispersion in anisotropic doubly-curved nanoshells. Also, it is the first time that small scale effects are considered in doubly-curved nanoshells made of anisotropic materials. Unlike the classical (scaling-free) model, the presented nonlocal strain gradient higher-order model shows a good calibration with the experimental frequencies and phase velocities. It is demonstrated that the material properties, nonlocal-strain gradient parameters and wave number have remarkable influences on wave frequencies and phase velocities. Presented results for wave dispersion can serve as benchmarks for future analysis of doubly-curved nanoshells.

Analysis of sound transmission loss through thick-walled cylindrical shell using three-dimensional elasticity theory

In this paper, the transmission loss (TL) of a thick-walled shell under obliquely plane incident wave is investigated considering three dimensional (3-D) theory of elasticity. Governing equations of the thick shell have been derived in radial, axial, and circumferential directions. Then, Helmholtz decomposition is used to solve the equations. Therefore, the displacement field is considered in terms of Lame potential functions. A comparison of the present method results with those obtained from classical shell theory (CST), first and third-order shear deformation theories (FSDT, TSDT) for this special case of the thin shell indicates an excellent agreement. Moreover, the comparison of present method with these theories for a thick shell having R/h=20 reveals good conformity at low frequencies. However, at the high frequencies these classical and higher order shell theories encounter insufficient accuracies as a result of increasing of the rotational terms as well as shear wave effects. The results of the present method show that with thickening the shell, the critical frequency is getting closer to the ring frequency. Consequently, the mass-low controlled region is going to be completely removed which leads to significant enhancement of the TLs. Finally, the obtained numerical results demonstrate that the transmission loss is considerably decreased at the frequencies greater than the resonance frequency by increasing Mach number of the external mean flow.

On numerical analysis of axisymmetric thick circular cylindrical shells based on higher order shell theories by segmentation method

Numerical analysis is carried out in this paper for thick circular cylindrical shells using higher order shell theory. In orthogonal curvilinear coordinate system, all equations are derived with the inclusion of the additional quadratic and cubic terms in the Taylor’s series expansions of the both in-plane as well as the transverse displacement components for the improvement of bending behaviour of the shell. Assuming ( h/ R)2 ≪ 1, a rigorous formulation involving the reduction of a three-dimensional elasticity problem to a two-dimensional one, based partly upon the Reissner’s variational principle, is presented. These equations are algebraically manipulated to be in the form of a coupled system of first-order differential equations in terms of the intrinsic dependent variables. These are then solved by a segmentation method – numerical integration technique for various combinations of material and geometric parameters. The theory is shown to result in a partial differential equation system of sixteenth order.

The Laminated Composite Thermosetting Pipe

The thermosetting pipes are eligible to be used in oil transportation where resistance to crude oil, paraffin build-up as well as ability to withstand relatively high pressures is required. Lamination design of such system could provide better strength and stability for internal and external loadings in both the circumferential and longitudinal directions. To this aim, a straightforward design and modeling of thermosetting pipe is developed via using first order shell theories. Apart from possessing an accurate operational condition, on account of its simplicity the proposed simulation seems also very suitable for further developing and prototyping purposes. Finally, it has been shown that the proposed thermosetting pipe, which partially attains some classical characteristic of both offshore and onshore pipeline by a different line of reasoning, is may serve as a reference in designing thermosetting pipe.

Free vibration analysis of open thin deep shells with variable radii of curvature

In the present paper, free vibration of a thin open curved shell with parabolic curvature was studied. This shell has a curvature with variable radius in one direction. The equations of motion of this shell were inferred by first order shell theory. According to perpendicular nature of loading on shell of marine structures, the assumptions of Donnell–Mushtari–Vlasov can be used with an acceptable level of accuracy and the in-plane displacement along shell straight direction “x” can be neglected as compared to the displacement in two other directions. The natural frequencies and mode shapes related to the first five vibrational modes were extracted using semi-analytical methods including power series method, Galerkin method and beam function method. The results of the semi-analytical methods were validated against those obtained by using the finite element method. Out of the studied semi-analytical methods, Galerkin method was found to have an appropriate convergence in both natural frequency and mode shape. Adopting eight terms of the response series, Galerkin method has an appropriate convergence compared with the results of finite element.

Assessment of different higher order theories for low-velocity impact analysis of fibre-metal laminate cylindrical shells

Assessment of different higher order shell theories for low-velocity impact response analysis of simply-supported fibre-metal laminate circular cylindrical shells is achieved. A new two-degree-of-freedom spring-mass model is adapted and used for contact force history estimation. The Fourier series method is used to solve the governing equations of fibre-metal laminate shell. The mechanical behaviour of fibre-metal laminate is assumed to be elastic. The impact behaviour is assumed to be quasi-static. In order to investigate the effect of the thickness flexibility on the impact response analysis of fibre-metal laminate cylindrical shells, different 2D and 3D higher order shell theories are evaluated. Also, effects of some parameters including metal volume fraction and fibre-metal laminate layup on the impact response are investigated. The results indicate that regardless of the fibre-metal laminate layup, as the value of metal volume fraction increases, the influence of the thickness flexibility on the impact response of fibre-metal laminate cylindrical shells is more significant. Also, as the metal volume fraction increases, the impulse (area under contact force–time) remains almost unchanged. In fact, the fundamental frequency of the shell governs both the peak value of the contact force as well as the contact duration. It is shown that Al/G 2/1 is the best layup to minimise the shell deflections due to impact load and to maximise the fundamental frequency.

Free vibrations of laminated composite shells with uniformly distributed attached mass using higher order shell theory including stiffness effect

In this paper, free vibrations of a cross-ply composite shell with or without a uniformly distributed attached mass are analyzed using higher order shell theory. The results of free vibrations without distributed attached mass are validated by previous literatures. The stiffness effect of this distributed attached mass are also considered and compared with those well-known published results in which this effect is ignored. Various results for composite shells under a variety of conditions such as variations in the thickness of the shell, variation in the thickness of the distributed attached mass, variation in the radii of curvatures and various elasticity moduli are presented in this paper. In some cases, to verify the novel results, first-order shear deformation theory (FSDT) is also used. In this paper, parameters which influence the natural frequencies of the shells with attached mass including the stiffness of the mass are investigated. Parameters which are investigated in this paper are thickness of the shell, thickness of the distributed attached mass, elasticity moduli of the distributed attached mass and radius of curvatures of shells. Increasing the thickness or elasticity moduli of the distributed attached mass will increase the fundamental natural frequency of the shell. The effect of the stiffness of the distributed attached mass is decreased by decreasing the radii of curvatures or increasing the thickness of the shells.

Exact analysis for static response of cross ply laminated smart shells

Models and analytical solutions are formulated and developed for the static behavior of cross ply smart laminated shells with extension piezoelectric laminae. The models are based on a rigorous first order shell theory. The state space approach is used to find exact solutions for the static response of cross ply spherical, cylindrical and doubly curved shells with various boundary conditions. The smart shells possess two parallel edges simply supported and the remaining ones having any possible combination of boundary conditions: free, clamped or simply supported. Deflections of cross ply laminated shells incorporating piezoelectric layers are determined. Numerical results of six layer laminates are generated to investigate the shell static behavior. The exact solutions for deflections can be used as benchmarks for approximate solutions such as Rayleigh–Ritz and finite element methods.

Comparative dynamic and static studies for cross-ply shells based on a deep thick shell theory

Exact solutions for free vibration and static deflection of cross-ply laminated cylindrical panels and circular cylindrical shells are obtained using deep first order shell theory. The state space approach has been used to generate the exact solutions for different sets of boundary conditions. Comparison between the results has been made using the deep first order shell theory and the less accurate one, where the 1 + z/ R terms have been approximated by unity. The effects of various parameters including shear deformation, shallowness of the shell, orthotropicity ratio, lamination scheme and boundary condition on shell frequencies and centre deflections are studied. The fundamental frequencies of the deep shell theory are compared with 3D theory of elasticity available in the literature.

Thermo-Mechanical Analysis Of Functionally Graded Shells

The thermo-mechanical analysis of a simply supported, functionally graded shell is considered in this work. Refined shell theories are considered to account for grading material variation in the thickness direction. The governing thermodynamical equations are derived from the Principle of Virtual Displacements. The distribution of the temperature field T(z) is not assumed linear in the thickness direction of the layered shells and Fourier's heat conduction equation is solved to provide T(z). Classical and higher order shell theories are implemented in cases of both an equivalent single layer and layer-wise variable description by referring to Carrera's Unified Formulation. The numerical results show temperature, displacement and stress distributions along the thickness direction. Different volume fractions of the metallic and ceramic constituents as well as different shell thickness ratios and orders of expansion are analyzed. These are in good agreement with the quasi-3D solution obtained considering mathematical layers with constant properties in the FGM layer and using high orders of expansion.

Vibration of elliptic cylindrical shells: Higher order shell theory

The equations for the free vibration of an elliptic cylindrical shell of constant thickness were derived using a Ritz approach. A higher order shell theory is employed that includes the effects of shear deformation, rotary inertia, and symmetric and antisymmetric thickness stretch deformations. The frequency-wavenumber spectrum has seven branches: flexural, extensional, torsional, two thickness shear, and two thickness stretch. The resulting seven coupled algebraic equations are symmetric and positive definite. The shell has a constant thickness, h, finite length, L, and is "simply supported" at its ends, (z=0,L), where z is the axial coordinate. The elliptic cross-section is defined by the shape parameter, a, and the half-length of the major axis, l. The modal solutions are expanded in a doubly infinite series of comparison functions in terms of circular functions in the angular and axial coordinates. Numerical results for the natural frequencies were obtained for two values of h/l and L/l, and various shape parameters, including the limiting case of a simply supported cylindrical shell (a approximately 100).

Transient thermal response in thick orthotropic hollow cylinders with finite length: High order shell theory

Abstract The transient thermal response of a thick orthotropic hollow cylinder with finite length is studied by a high order shell theory. The radial and axial displacements are assumed to have quadratic and cubic variations through the thickness, respectively. It is important that the radial stress is approximated by a cubic expansion satisfying the boundary conditions at the inner and outer surfaces, and the corresponding strain should be least-squares compatible with the strain derived from the strain-displacement relation. The equations of motion are derived from the integration of the equilibrium equations of stresses, which are solved by precise integration method (PIM). Numerical results are obtained, and compared with FE simulations and dynamic thermo-elasticity solutions, which indicates that the high order shell theory is capable of predicting the transient thermal response of an orthotropic (or isotropic) thick hollow cylinder efficiently, and for the detonation tube of actual pulse detonation engines (PDE) heated continuously, the thermal stresses will become too large to be neglected, which are not like those in the one time experiments with very short time.