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A flugge-type theory for the analysis of anisotropic laminated non-circular cylindrical shells
Abstract
For a generally anisotropic laminated thin elastic non-circular cylindrical shell, subjected to a combined loading, the equations of motion of a second approximation Flugge-type theory are derived and expressed in terms of the shell middle-surface displacement components. As an application, for the free vibration problem of a cross-ply laminated non-circular cylindrical shell subjected to S2 simply supported edge boundary conditions, these equations are solved by employing the method of Galerkin. For a family of regular antisymmetric cross-ply laminated oval shells, numerical results are obtained and discussed. Comparisons are also made between some of the obtained results and corresponding results obtained from the solution of the quasi-shallow shell Donnell-type equations of motion.
... It may be concluded from the literature that few contributions are available concerning with free vibration analysis of anisotropic laminated non-circular cylindrical shells compared to those of isotropic case, and they are cited here. The free vibration of laminated non-circular case has been analyzed employing classical theory (Soldatos and Tzivanidis, 1982;Soldatos, 1984;Hui and Du, 1986;Suzuki et al., 1994), and using first-order shear deformation theory (Noor, 1973;Kumar and Singh, 1995;Suzuki et al., 1996). The theory assuming parabolic variation of thickness shear for the study of composite non-circular shells has been attempted (Soldatos, 1987;Kumar and Singh, 1996). ...
... The theory assuming parabolic variation of thickness shear for the study of composite non-circular shells has been attempted (Soldatos, 1987;Kumar and Singh, 1996). The Galerkin procedure was employed in the work of Soldatos and Tzivanidis (1982), Soldatos (1984), Soldatos (1987) and Hui and Du (1986) whereas the power series expansion method was adopted in the work of Suzuki et al. (1994Suzuki et al. ( , 1996. Noor (1973) solved the problem using multilocal difference discretization method while the energy approach was applied by Singh (1995, 1996). ...
Here, the dynamic analysis of laminated cross-ply composite non-circular thick cylindrical shells subjected to thermal/mechanical load is carried out based on higher-order theory. The formulation accounts for the variation of the in-plane and transverse displacements through the thickness, abrupt discontinuity in slope of the in-plane displacements at the interfaces, and includes in-plane, rotary inertia terms, and also the inertia contributions due to the coupling between the different order displacement terms. The strain–displacement relations are accurately accounted for in the formulation. The shell responses are obtained employing finite element approach in conjunction with direct time integration technique. A detailed parametric study is carried out to bring out the effects of length and thickness ratios, eccentricity parameters and number of layers on the thermal/mechanical response characteristics of non-circular shells.
... Then Eqs. (45) and (49) result in the following formulae for parameters b and λ 1 : ...
The problem on buckling of a thin laminated non-circular cylindrical shell under action of axial compressive forces non-uniformly distributed along edges is considered. It is assumed that some layers are made of a “soft” material so that the reduced (effective) shear modulus for the entire package is much less than the reduced Young's modulus. The differential equations based on the generalized hypotheses of Timoshenko and including the effect of transverse shears are used to predict the buckling of laminated cylinders regardless a number of layers and their mechanical properties. Using the asymptotic method, the buckling modes are constructed in the form of functions rapidly decaying far away from some generatrix at the reference surface. It is shown that accounting transverse shears strongly effect on the buckling modes and corresponding critical buckling forces. In particular, the preferable buckling form for a medium-length thin laminated cylinder with a low reduced shear modulus (as compared with the reduced Young's modulus) is found to be a system of small dents in the axial direction, whose amplitudes decay in the circumferential direction without oscillations; whereas the buckling of a shell with a relatively large reduced shear modulus may occur with formation of waves in both the axial and circumferential directions. As an example, the buckling of cylindrical sandwiches assembled from the ABS-plastic and magnetorheological elastomer with variable shear modulus under different levels of an applied magnetic field is examined
... Similar studies were carried out by Armenakas and Koumousis [41] for the vibration of simply-supported elliptic cylinders. Soldatos [42] presented natural frequencies of anisotropic laminated elliptic Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/tws cylinders using the Flügge shell theory. ...
We use the generalized differential quadrature method (GDQ) and shell theories of different order to study free vibrations of laminated cylinders of oval and elliptic cross-sections. In the GDQ method partial derivatives of a function at a point are expressed as weighted sums of values of the function at several neighboring points. Thus, strong forms of equations of motion are analyzed. It is found that the computed frequencies rapidly converge with an increase in the number of grid points along the oval or elliptic circumference defining the cross-section of the mid-surface of the cylinder. For a clamped-free elliptic cylinder the converged frequencies match well with the corresponding experimental ones available in the literature. Furthermore, the lowest ten frequencies computed with either an equivalent single layer theory or a layer wise theory of first order and using shear correction factor are accurate.
... Further, Swaddiwudhipong [5,6] studied the bending problem of elliptic paraboloid shells. Subsequently, Soldatos [7] and Soldatos and Tzivanidis [8] investigated the free vibrations of multilayered closed oval shells using classical shell theory, whereas Noor [9], and Kumar and Singh [10] investigated the same shell structures employing the shear deformation theory. Suzuki et al. [11] presented results for non-circular elliptical shells based on classical shell theory. ...
A variety of elliptical/parabolic dome type structures are used in important aerospace and civil structural systems such as underwater vehicles, stadium covers, exhibition halls, auditoriums and museum halls. For the analysis of such structures, a shear deformable four-noded finite element based on a hybrid/mixed assumed stress is presented in this paper. The element called iHES (improved Hybrid and Enhanced Shell element) is developed assuming the most general arbitrary orthogonal coordinate system. The element is based on a first-order shear flexible formulation and essentially consists of a combination of drilling degrees of freedom with assumed stress and enhanced strain techniques. Using the element developed here, a detailed parametric study of anisotropic elliptical and parabolic shells of various configurations is carried out to investigate the effects of aspect and height ratios as well as layer lay-up schemes.
... Although imperfections are another issue, the natural frequencies of the perfect oval cylinders exhibited good agreement with the study of Soldatos. 3 Suzuki et al. 5 used Love shell theory, Hamilton's principle, and the Rayleigh-Ritz technique to investigate the natural frequencies for a set of symmetrically laminated cross-ply elliptical cylinders with simple supports. They studied the influence of the number of layers and the stacking sequence on the frequencies. ...
Hamilton’s principle coupled with the Rayleigh–Ritz technique is used to compute the fundamental frequencies of simply supported thin-walled fiber-reinforced composite cylinders with elliptical cross sections. Owing to the decreased geometric stiffness resulting from less curvature, it is expected that the normal displacement component of the vibratory motion will be larger in the flatter regions of the cross section than that in the more curved regions. Accordingly, in the Rayleigh–Ritz formulation, the normal displacement component of the vibratory motion is modulated with circumferential location to represent this characteristic by using a so-called shape factor. A number of simplifications in the analysis lead to a hierarchy of expressions for the fundamental frequency, including the one termed Lo’s approximation. The so-called large and small cylinders, as measured by cylinder circumference and with wall laminates [±θ/0/90]2S and[±θ/0/90]S, respectively, θ in the range of 0 to 90°, are considered. It is demonstrated that the comparisons with finite element calculations are good, particularly for Lo’s approximation. Then, parameter studies using Lo’s approximation are conducted to illustrate the dependence of the fundamental frequency on fiber angle θ, cross-sectional geometry, cylinder circumference, and cylinder length. It is shown that for cylinders of the same circumference, an elliptical cylinder has a lower fundamental frequency than a circular one and that difference is quantified. However, the dependence of the fundamental frequency on other geometric parameters and fiber angle is much the same for cylinders with elliptical cross sections as for circular cylinders.
The aerospace industry often uses cylindrical shells with elliptical cross-section, which are manufactured from composite material using a filament winding method. During the fabrication process or operation of the structure, there is a probability of shape imperfection in the form of deviation from a circular cross-section. The vibration analysis of such structures containing fluid requires an in-depth study to determine the performance characteristics affecting their life cycle. In this article we develop a mathematical formulation and present the corresponding finite element algorithm for determining the natural frequencies of vibrations of layered composite elliptical cylindrical shells filled with fluid. The problem is solved in a three-dimensional formulation by the finite element method. The curvilinear surface of the shell is represented as a set of flat rectangular segments, in which the relations of the classical laminated plate theory are fulfilled. The membrane displacements are described using bilinear Lagrange shape functions. The deflection in the direction normal to the lateral surface and the rotation angles are approximated by incompatible cubic Hermite polynomials. Small vibrations of an ideal compressible fluid are described in the framework of the acoustic approximation by a wave equation for hydrodynamic pressure, which, together with the boundary conditions and the impermeability condition on the wetted surface, is transformed to a weak form. The verification of the developed numerical algorithm is carried out by comparing the obtained natural frequencies of vibration with the known data presented in the literature for layered composite circular cylindrical shells. A number of examples are considered to evaluate the influence of geometrical dimensions of the structure, boundary conditions at the shell edges and the ratio of ellipse semi-axes. New quantitative and qualitative dependencies have been established, and the possibility of the natural frequency control through the selection of parameters of composite material has been shown.
In this paper, a unified analysis model is proposed for the first time to study the free vibration of laminated composite elliptic cylinders with general boundary conditions including the classical boundary, elastic boundary and their combinations. The theoretical model is established by means of the modified variational principle and multilevel partition technique based on the first-order shear deformation theory. The interface continuity and boundary constraints are enforced by using the coupling and boundary spring technique. On the basis of that, the displacement components of each shell domain are expanded in the form of double Jacobi polynomials along the meridional and circumferential direction. The convergence and comparison analysis for laminated composite elliptic cylinders subject to different classical boundary conditions is conducted to show the reliability and accuracy of the present method. To make the research topic understood better, some mode shapes are also depicted. The present solutions show stable and rapid convergence characteristics, and the natural frequencies and mode shapes agree well with the Finite Element Analysis results. Some new vibration results and parameterized results are presented and may be as the reference data by other researchers in the future.
The free vibration problem of cross-ply laminated oval cylindrical shells has been studied on the basis of classical thin shell theories [1,2]. For the results presented in reference [1], Donnell-type quasi-shallow shell approximations were used. The corresponding study made in reference [2] was based on Flügge’s second order approximations. However, it is wellknown that laminated composite thin walled structures are very sensitive in thickness shear deformation. Therefore, consideration of thickness shear effects into the free vibration analysis of cross-ply laminated oval cylindrical shells [3,4] is essential.
A non-circular shell cross-section with flat sides and circular arc corners is analyzed using the theorem of minimum potential energy (MPE) and finite element (FE) analysis for internal pressurization loading. A 2-D shape, utilizing plane strain assumptions is examined first. The shape is then extended to three dimensions with both clamped and simply supported boundary conditions. Potential energy expressions for both the 2-D and 3-D structures are developed, and include first-order transverse shear deformation effects. In the MPE method, the unknown displacements are represented by power series. Displacements and stresses are calculated for a composite sandwich construction. The MPE and FE solutions are compared, and a trade study on the shell length and corner radius is preformed with FE analysis. In the 2-D case, excellent agreement is found between the MPE method, other analytical methods, and finite element analyses. In the 3-D case the MPE method agreed well with the FE solution for short shells, but had poor agreement for long shells. In the FE trade study, a unique bending boundary layer behavior is discovered, and the overall response of the shell is investigated. © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
This paper deals with the free-vibration behaviour of anisotropic laminated angle-ply non-circular cylindrical shells using finite element approach. The formulation is based on first-order shear deformation theory. The present model accounts for in-plane and rotary inertia effects. A detailed study has been carried out to highlight the effects of shell geometry, cross-sectional properties, lay-up and ply-angles on the natural frequencies of different types of modes of vibration of non-circular elliptical shell structures.
This paper deals with the free vibration behavior of anisotropic laminated angle-ply non-circular cylindrical shells using finite element approach. The formulation is based on first-order shear deformation theory. A detailed study is carried out to highlight the effects of lap-up and ply-angle on the natural frequencies pertaining to different types of modes of vibrations of simply supported non-circular shell structures.
A cylindrical composite shell with a non-circular, symmetric cross-section of flat sides and circular arc corners is analyzed using the abaqus finite element program. A trade study on the effects of various corner radii and shell lengths is performed. The response of the shell to constant internal pressurization is studied, with particular attention given to the bending boundary layer (BBL) near the ends. It is found that the extent of the BBL in the non-circular case is 2.5–4 times longer than that predicted by the classical equation, however, the ‘intensity’ of the bending boundary layer is reduced. An unusual ‘compounding’ effect in boundary layer response for short non-circular shells is described.
A hitherto unavailable analytical solution to the boundary value problem of free vibration response of shear-flexible antisymmetric cross-ply laminated cylindrical panels is presented. The equivalent single layer approach based on a first order shear deformation theory including rotary and in-plane inertias is incorporated into the shell formulation. The characteristic equations of the panel are defined by five highly coupled second and third order partial differential equations in five unknowns, i.e., three displacements, and two rotations. A recently developed solution methodology, based on a boundary-continuous double Fourier series approach, is utilized to solve the eigenvalue problem. Numerical results presented for various parametric effects such as length-to-thickness ratio, radius-to-thickness ratio, aspect ratio, and major-to-minor modulus ratio, etc., should serve as a bench mark for future comparison. A four-node shear-flexible finite element is selected to compare the results with the present solution.
Hitherto unavailable analytical solutions to the boundary-value problems of static response and free vibration of an arbitrarily laminated doubly-curved panel of rectangular planform are presented. Four classical shallow shell theories (namely, Donnell, Sanders, Reissner and modified Sanders) have been utilized in the formulation, which generates a system of one fourth-order and two third-order partial differential equations with constant coefficients. A novel double Fourier series approach has been developed to solve this system of three partial differential equations with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the solutions is ascertained by studying the convergence characteristics of the lowest two natural frequencies, deflections and moments of angle-ply panels, and also by comparison with the available FSDT-based analytical and CLT-based Galerkin solutions. Also presented are comparisons of deflections and moments of antisymmetric angle-ply cylindrical panels, computed using the four classical shallow shell theories considered. Comparisons with the available FSDT (first-order shear deformation theory)-based analytical solutions are presented for the purpose of establishing the upper limit (with respect to the thickness-to-length ratio) of validity of the present CLT (classical lamination theory)-based solutions for angle-ply panels. Also studied is the highly complex interaction of bending-stretching type coupling effect with the effects of transverse shear deformation, rotatory inertias, inplane inertias, and membrane action due to shell curvature. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as radius-to-length ratio, length-to-thickness ratio and angle of fiber orientation.
This paper deals with the free flexural vibration characteristics of anisotropic laminated angle-ply elliptical cylindrical shells using finite element approach. The formulation is based on first-order shear deformation theory. The present model accounts for in-plane and rotary inertia effects. A detailed study is carried out to highlight the effects of shell geometry, cross-sectional properties, lap-up and ply-angle on the natural frequencies pertaining to different types of modes of vibrations of non-circular shell structures.
Buckling analysis of laminated cylindrical shells with noncircular cross section of arbitrary closed shape is presented. The equations, in terms of the normal displacement and Airy stress function, of the Donnell type, are derived via the Hu-Washizu mixed formulation. The curvature, which is a function of the circumferential coordinate, is expanded in Fourier series. The circumferential dependence is eliminated by a combination of Fourier expansion and Galerkin's method. The resulting ordinary differential equations are then reduced to matrix equations by the use of finite differences. The configurational aspect is investigated parametrically. Unlike the circular cylindrical shell, coupling of the wave number in the circumferential direction is significantly high.
This article presents a review of the research work related to the mechanical behavior of non-circular cylindrical shells and shell segments. To this end, after a brief reference to the basic nomenclature that is mainly used, it initially provides quite a general framework for most of the relevant governing equations employed in the relevant literature. It proceeds with a review of the corresponding dynamic analyses, which are primarily grouped according to the geometrical configuration of the noncircular shell considered and secondarily according to the type of the mathematical model employed. These deal with the dynamics of closed cylindrical shells and open cylindrical panels based on classical (CST) or transverse shear deformable shell theories (SDST). The static analyses reviewed next are divided according to the nature of the physical problem considered and deal with small as well as with large deflections of statically loaded non-circular cylindrical shells. These include both linearized and geometrically nonlinear elastic stability analyses as well as the very few relevant studies that assumed an elastic-plastic response of the shell material constitution. This review article contains 196 references.
The analytical (exact in the limit) or strong (or differential) form of solutions to the bench-mark problems of (i) axisymmetric angle-ply circular cylindrical panels of rectangular planform and (ii) circumferentially complete circular cylindrical shells, subjected to transverse load and with SS2-type simply-supported boundary conditions prescribed at the edges, are presented. The problems investigated, which were hitherto thought to be incapable of admitting analytical solutions, have been solved, utilizing a recently developed novel boundary-discontinuous double Fourier series approach, for three kinematic relations, which are extensions of those due to Sanders. Love and Donnell to the first-order shear deformation theory (FSDT). Numerical results presented for two-layer square antisymmetric angle-ply panels, which demonstrate good convergence, and show the effects of fiber orientation and thickness on the static response of these panels, should serve as baseline solutions (in the context of FSDT) for future comparison with various approximate weak forms of solutions with either local (e.g. finite element methods) or global supports (e.g. Raleigh-Ritz. Galerkin).
This paper is concerned with the problem of free vibrations of homogeneous isotropic non-circular cylindrical shells, including the effects of thickness shear deformation and rotatory inertia. For this problem the equations of motion of two first approximation shell theories are derived. Both theories are transverse shear deformable analogues of the classical Love-type theory. The first theory involves thickness shear correction factors while the second one assumes a parabolic variation for thickness shear strains and stresses, with zero values at the inner and outer shell surfaces. The equations of both theories are solved, for the case of a simply supported non-circular cylindrical shell and, as an application, the free vibration problem of a simply supported oval cylindrical shell is considered. From comparisons made between corresponding numerical results based on both theories, as well as the classical Love-type theory, a superiority of the theory assuming parabolic variation of thickness shear is concluded.
The free vibrational characteristics of composite noncircular cylindrical shells are investigated in this paper. The shells are composed of layered media of different material properties. The thickness of each layer is considered to be constant. First order composite shell theory, which includes the effects of shear deformation and rotary inertia, is used in the formulation. A combination of Bezier functions and beam functions is used to describe the displacement fields along the circumference and longitudinal directions, respectively, of the shell surface. The shell is modelled using a number of curved cylindrical panels. Displacement (C 0), slope (C 1) and curvature (C 2) continuities between the panels are enforced by proper blending of the Bezier curves. Numerical results are included for a circular sandwich shell and a two-layer cross-ply oval cylinder that provide excellent agreement with those from the literature. The natural frequencies of clamped oval sandwich cylinders made of stiff outer layers and light middle layer are also presented.
Here, free vibrations and transient dynamic response analyses of laminated cross-ply oval cylindrical shells are carried out. The formulation is based on higher order theory that accounts for the transverse shear and the transverse normal deformations, and includes zig-zag variation in the in-plane displacements across the thickness of the multi-layered shells. The contributions of inertia effect due to in-plane and rotary motions, and the higher order function arising from the assumed displacement models are included. The governing equations obtained using Lagrangian equations of motion are solved through finite element approach. A detailed parametric study is conducted to bring out the influence of different shell geometry, ovality parameter, lay-up and loading environment on the vibration characteristics related to different modes of vibrations of oval shell.
This paper studies sound and vibration transmission across a sandwich beam made of anisotropic materials. In our previous study, we have found that there is a significant increase in the transmission loss for the sandwich beam with anisotropic materials as compared with isotropic ones. This paper presents an extensive numerical study of the effects of damping, thickness of the laminae and density of the material on the sound transmission loss. This work may eventually lead to a new way of designing sandwich structures with high vibration and noise isolation performance.
This paper deals with a numerical method for the free vibrational analysis of laminated deep shells. The strain-displacement relations are obtained for a general laminated shell geometry described by orthogonal curvilinear coordinates. Parabolic variation of transverse shear stresses along the thickness and the effects of rotary inertia are included in the formulation. The displacement fields are represented by Bezier patches. The shape and size of these patches are controlled by certain arbitrary points called control points. Owing to the special characteristics of these control points, the treatment of displacements, slopes, curvatures, etc., at a particular edge becomes very simple. Hence, the enforcement of boundary conditions along the edges is straightforward. Ritz-type solution procedure is used for the eigen-analysis of the shell structure. Numerical examples involving laminated spherical, conical, and cylindrical shells are investigated in detail. Such shell geometries usually have planes of symmetry; hence, only one-quarter of the shell is analyzed in this study. Good convergence of the natural frequencies is observed by using eight-order Bezier functions. The results are compared with the existing sources in the literature. The influences of material strength and number of layers on the natural frequencies are also examined.
The influence of the laminate configuration on the performance of various composite plate, panel, and shell structures has received extensive attention in the literature. Of the papers contributed to the stability of anisotropic laminated shells, the majority pertains to the stability and optimum design problem of laminate composite circular cylinders. This study extends the development of Hutchinson (1968) to include oval cylinders made of anisotropic composite laminates without imposing any restriction on the lamination scheme. The shell analysis is based on Donnell's shallow shell theory and the post-buckling b-coefficient is employed to indicate the sensitivity of laminated oval cylinders to asymmetric geometric imperfections. The intention of this work, however, is to examine the effects of the laminate configuration, as well as, the eccentricity of the oval cross-section on the buckling and initial post-buckling behavior of laminated composite oval cylinders.
The results from semianalytical predictions and experiments are used to study the response of composite cylinders with elliptical cross sections loaded axially to a significant percentage of their buckling load. The semianalytical approach is based on the methods of Marguerre, Rayleigh-Ritz, and Kantorovich. The radius of curvature and the displacements are approximated by expansions in harmonic series in the circumferential arc-length coordinate, and the coefficients of the displacement series are unknown functions of x which are solved for using the finite-difference method. The primary features of the predicted response are first described. Then the experiments are described and results for elliptical cylinders with varying degrees of orthotropy are compared with predictions. Where appropriate, calculations based on the analysis of circular cylinders are compared with the semianalytical calculations for the ellipse. Correlation between experiments and predictions is good, and it is demonstrated that despite the noncircular cross section, many responses of an ellipse are very similar to the axisymmetric response of circular cylinders subjected to an axial load. The similarity is independent of the degree of orthotropy of the elliptical cylinder.
Here, the free vibration characteristics of thick laminated composite non-circular cylindrical shells are analyzed using higher-order theory. The formulation accounts for the variation of the in-plane and transverse displacements through the thickness, abrupt discontinuity in slope of the in-plane displacements at the interfaces, and includes in-plane, rotary inertia terms, and also the inertia contributions due to the coupling between the different order displacement terms. The accurate strain–displacement relations are used for the evaluation of strain energy. The governing equations are solved employing the finite element procedure. Detailed study is made to highlight the influences of length and thickness ratios, eccentricity parameters, ply-angles and number of layers on the free vibration characteristics of non-circular shells.
The assessment of classical lamination shell theory and first-order shear deformation theory is presented for simply supported finite circular cylindrical hybrid shell with cross-ply composite laminate as elastic substrate under electromechanical static load. Navier-type solutions are obtained and used in threedimensional equilibrium equations and transverse strain—displacement relation to obtain transverse stress components and improved value of deflection. These solutions are assessed by comparison with the threedimensional solution. The error in the two-dimensional shell theories increases as the shell becomes thicker and it is more for the patch loads in comparison to the uniformly distributed and sinusoidal loads.
Hitherto unavailable analytical solutions to the boundary-value problem of moderately thick general cross-ply laminated doubly-curved panels of rectangular planform, subjected to various boundary conditions, are presented. The five highly coupled second-order linear partial differential equations, that characterize the deformation of such laminates are solved in Part I of the paper using a recently developed double Fourier series based approach, together with the SS1-, SS2- and SS4-types of simply-supported and C4-type of clamped boundary conditions prescribed at all four edges. The issues of derivation of the linear algebraic equations arising from these boundary conditions, together with an efficient method of solving the complete system of linear algebraic equations, convergence characteristics and other numerical results are addressed in the accompanying Part II of this investigation.
A non-circular shell cross-section with flat sides and circular arc corners is analyzed using the theorem of minimum potential energy. Two-dimensional, plane strain assumptions are utilized, and the potential energy (PE) expression for the structure is developed, including first-order transverse shear deformation effects. The unknown displacements are represented by power series, and the PE expression is rewritten in terms of the summation convention for the power series. The variation of the PE expression is taken, leading to a linear system of equations that is solved for the unknown power series coefficients. With the displacements determined, stresses are calculated for a composite sandwich construction. Excellent agreement is found with other analytical methods and with finite element analyses.
The free vibration problem of thin elastic cross-ply laminated circular cylindrical panels is considered. For this problem, a theoretical unification as well as a numerical comparison of the thin shell theories most commonly used (in engineering applications) is presented. In more detail, the problem is formulated in such a way that by using some tracers, which have the form of Kronecker's deltas, the stress-strain relations, constitutive equations and equations of motion obtained produce, as special cases, the corresponding relations and equations of Donnell's, Love's, Sanders' and Flugge's theories. By using a closed form solution, obtained for simply supported panels, a comparison of corresponding numerical results obtained on the basis of all of the aforementioned shell theories is attempted.
A theoretical analysis is presented for determining the free vibra tional characteristics of thin-walled, circular cylindrical shells with layers of anisotropic elastic material arbitrarily laminated either sym metrically or unsymmetrically about the shell middle surface. An arbitrarily laminated, anisotropic version of Love's first-approximation shell theory is used to formulate the coupled equations of motion. An exact solution with a classical checkerboard nodal pattern is found for the case of a shell with specially orthotropic layers arbitrarily laminated and with freely supported ends. For a boron/epoxy composite cylinder, the significant effect of omitting bending-stretching coupling is demonstrated and various lamination arrangements are investigated. Also, a general solution is presented for the axisymmetric modes of an arbitrarily laminated anisotropic shell. Finally, an approximate solution, using a combination of two helical-nodal-pattern modes, is obtained for the unsymmetric modes of the same general class of shell with a supported boundary condition.
A theoretical analysis of the buckling problems of heterogeneous aeolotropic cylindrical shells under combined axial, radial, and torsional loads is presented. Four boundary conditions at each end of the cylinder are satisfied for the case of both ends hinged or that of both ends clamped. Classical thin shell theory of small deflection is followed. Because only six elastic coefficients are required out of the usual 21 for a general aeolotropic body, it is possible to solve Flugge's differential equations of equilibrium by assuming suitable functions for the displacements of the middle surface. By the superposition of these solutions, a general solution that satisfies the boundary conditions can be reached. If the thin shell is laminated from layers of different materials, the resultant forces and moments of an element are integrated from layer to layer by considering that the six elastic coefficients are piecewise continuous. Orthotropic and isotropic materials are particular cases of this analysis. © 1963 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
This paper presents an analysis of the deflections of and stresses in a short noncircular cylindrical shell of uniform wall thickness whose median-surface cross section is described analytically by a simple expression corresponding to a family of doubly symmetric ovals. The cylinder is under a uniform lateral load and is simply supported at its edges. The small deflection analysis considered is based upon a series solution of appropriate differential equations of shell theory which leads ultimately to infinite sets of algebraic equations, truncated forms of which are considered. Numerical values of the significant stresses and displacements for points of the oval cylinder, which are 5 percent of the axial length and 2.5 percent of the circumferential length a part, have been calculated for an oval cross section with a major-minor axis ratio of 1.10.
The free vibration of an oval cylindrical shell of finite length was investigated with the aid of the kinematic relations of the first-order shell theory of Sanders. Transverse and in-plane inertia terms were retained throughout. A method incorporating a type of eigenfunction expansion into Hamilton's principle, was developed and found to be far more convenient than a parallel Fourier analysis. In addition to the determination of the natural frequencies and deformation characteristics, attention was focused on the influence of various types of simple support and clamped end conditions. Two modes of deformation corresponding to a ″higher″ and a ″lower″ frequency were observed for every pair of axial and circumferential wave numbers, depending upon the degree of circumferential symmetry in the deformation pattern.
A study was made of the free vibration frequencies and mode shapes for freely supported oval cylindrical shells. Cross section curvatures were expressed in terms of a single eccentricity parameter that allowed a wide range of doubly symmetric ovals to be studied. Kinematic equations employing both the Love and the Donnell assumptions from thin shell theory were used in this study and results of the two formulations were compared. Little difference was observed between the results obtained from the two theories for a wide range of shell configurations. Comparisons were also made between the results obtained from this study and those from two previous approximate analyses. It was found that one of the approximate analyses (a Rayleigh-Ritz technique) was quite accurate for all ranges of eccentricities studied. The other approximate analysis (a perturbation technique) was found to be reliable for ovals with eccentricities in the range (-0.5 ≤ ∈ ≤ 0.5). A study was also made to determine the effects of eccentricity of oval cross sections. The frequencies and mode shapes were found to vary significantly with increasing eccentricities. Irregularities in the frequency vs wave-number curves and a localized “cupping” in the region near the minimum frequency were observed. In-plane inertias were retained yielding the expected three frequencies for each combination of longitudinal and circumferential wave numbers. However, unlike the unstiffened circular cylinder, more than one set of three natural frequencies and associated mode shapes were found for some combinations of longitudinal and circumferential wave numbers. However, although the wave numbers (i.e., number of crossings) were the same in these cases, the wave shapes were obviously different. © 1971 by the American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
In this paper a theoretical solution is developed for the torsion on a round thin-walled tube for which the walls become unstable. The results of this theory are given by a few simple formulas and curves which cover all cases. The differential equations of equilibrium are derived in a simpler form than previously found, it being shown that many items can be neglected. The solution obtained is ''exact'' for the two extreme cases when the diameter to length ratio is zero and infinite, and is a good approximation for intermediate cases. The theory is compared with all available experiments, including about 50 tests made by the author. The experimental-failure torque is always smaller than the theoretical-buckling torque, averaging about 75% of it, with a minimum of 60%. As the form of the deflection checks closely with that predicted by theory and the experiments cover a great range of shapes and materials, this discrepancy can reasonably be ascribed largely to initial eccentricities in actual tubes.
Recent investigations by Stein and by Fischer on the influence of edge conditions on the critical load of cylindrical shells are here extended to cover six additional combinations of boundary conditions. The results show that drastic reductions of the critical load for cylinders with lateral support of the edges are obtained only if the edges are free in the tangential direction. For other boundary conditions, this reduction is never more than about 20 percent. Consequently, the results of this investigation alone cannot explain the well-known discrepancy between theory and test data. However, the importance of the choice of boundary conditions for practical analysis is clearly demonstrated.
The equations of motion, derived from a Love-type theory, are presented for laminated filament-wound cylindrical shells in which each layer is permitted an arbitrary fixed fiber orientation. A general method of solution is established, based upon the use of a complex finite Fourier transform. The frequency spectra of free natural vibrations are investigated for numerous single, bi- and tri-layered clamped or simply supported generally orthotropic shells. The effect of fiber orientation on the frequency response is found to be quite considerable in certain composite shells.
The vibrational response of orthotropic composite cylindrical shells subjected to axial compression is examined, following a refined Love-type theory. Results obtained are compared with those predicted by Donnell-type theory, as found in the literature.
Important effects due to shell lay-up, length to radius ratio, radius to thickness ratio and fiber reversal are noted from the calculations performed for a number of double and triple-layered shells.
The geometry of the middle surface lines of curvature of a thin conical shell, whose cross-section is bounded by a certain closed convex plane curve, is studied. Then, for such a shell, several sets of linear and nonlinear equations of motion are derived in terms of its middle surface orthogonal line-of-curvature coordinate system. As an application of the presented analysis, the free vibration problem of thin circular and elliptical frustums is investigated by means of linear Donnell-type equations of motion. These equations are expressed in terms of the shell middle surface displacement components and they are solved approximately by means of Galerkin's method. Numerical results are presented for frustums with clamped both edges.
The free vibration problem of a thin composite cross-ply laminated non-circular cylindrical shell subjected to an axial compression is studied. The equations of motion are derived, in the framework of the Donnell-type theory, in terms of the shell middle surface displacement components. The differential equations of motion have variable coefficients and they are solved by employing Galerkin's method. As an application, the problem of the free vibrations and buckling of cross-ply laminated oval cylinders is studied. Numerical results for antisymmetric and unsymmetric cross-ply laminated shells of graphite-epoxy are presented and discussed.
A refined Love-type theory of motion is established for orthotropic composite cylindrical shells. An extensional-rotational dynamic coupling effect is shown to exist, expressed by R1 inertia terms. An extendedversion of the theoryis formulated to account for dynamic stability problems involving time-dependent and non-conservative forcesThe frequency spectra of free natural vibrations are investigated for numerous layered shells, using Love- and Donnell type theories, including the effects of R1 terms. Heterogeneity is found to considerably affect the results for the natural frequencies; for certain shells produced of a fixed amount of materials, differing only in their arrangement, a suitable composition raises the lowest frequency by a factor of 1·50.A study of the error involved in a Donnell-type theory is carried out. For length-to-radius ratios of about 5 the resulting first lowest frequency may be higher by a factor of 1·10 than the one given by the present Love-type theory. However, when higher frequencies are considered this factor may go down to 0·66. These deviations are, in several instances, associated with different predictions of the corresponding lowest characteristic mode shapes. Higher errors, strongly depending on shell heterogeneity, are noted as the length-to-radius ratios increase beyond 5.
A theory is derived in which all strains vanish for any rigid-body motion in contrast to the results of Love's theory. Expressions for the stress resultants and couples which satisfy the homogeneous equilibrium equations are given in terms of three stress friction. The special forms of the equations of the now theory in the case of a circular cylinder are given in an appendix.
This paper reports an experimental and analytical vibration study of elliptical cylindrical shells having a wide range of cross-sectional eccentricities. Vibration tests were conducted on four thin-shell, isotropic, clamped-free cylinders of equal length, perimeter, and thickness with eccentricities ranging from zero (circular cylinder) to 0.916, corresponding to major-tominor axis ratios from 1 to 2.5, respectively. Mode shapes were obtained through the use of a noncontact, inductance-type proximity sensor that could be moved automatically over the shell surface. Measured frequencies and mode shapes are compared with analytical frequencies and mode shapes obtained by application of a Rayleigh-Ritz type of vibration analysis featuring multiterm circumferential and longitudinal modal expansions. Results show generally good agreement between experiment and analysis. Reductions in frequency by as much as 44% were found for an eccentricity of 0.916. © 1971, American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
Stress and displacement analysis of a simply supported noncircular cylindrical shell under lateral pressure
- Ramano
Structural behaviour of composite materials
- Tsai