Shell Curvature Research Articles

Modeling process operations in the forming of parts made of flat metal-polymer composites

The forming of complex three-dimensional parts (tubular structures, sections, shells) from flat sheets of metal-polymer composites is examined. Results are reported from a series of experimental studies of the strain and strength characteristics of these composites, and features of the fabrication of shells of single and double curvature are analyzed. One method of solving the problem of bending shells of complex shape with large displacements (deflections) beyond the elastic limit of the material is proposed and substantiated empirically.

Transient stresses generated by low velocity impact on orthotropic laminated cylindrical shells

The transient stress response of an orthotropic laminated open cylindrical shell impacted by a solid striker is investigated. An analytic solution which includes both transverse shear and contact deformation is presented and used in this study. A higher-order shear deformation theory is employed and solutions for the loads, displacements and strains are obtained using Fourier series expansions. An analytic impact force function recently proposed by the authors is used to predict the contact force between the striker and the shell and this is incorporated into the solution. By using this solution, transient stress distributions in the shell during impact are analyzed and the locations of maximum stresses determined. In addition, the effects of shell thickness and curvature on the maximum stresses induced are examined and discussed.

A Higher-Order Theory for Vibration Analysis of Curvilinear Thick Shallow Shells with Constrained Boundaries

This article presents the vibration analysis of thick doubly curved shallow shells having curvilinear planform. The Gaussian curvature of shell varies from positive (such as spherical) to negative (such as hyperbolic paraboloidal). The boundaries are constrained with either soft-simply supported or fully clamped edges. A higher-order shear deformation theory, which includes transverse shear strain and rotary inertia, is developed to model the vibration characteristics of the shell. The inclusion of Lamé parameters in the present formulation accounts for the presence of shell curvature and yields cubic transverse shear strain distribution in contrast with the existing quadratic expressions. A set of versatile, globally continuous shape functions is adopted in the Ritz numerical procedure to approximate the displacement and rotation fields. A set of new results for a wide range of shell configurations is presented with some selected contour and three-dimensional displacement mode shapes.

Dynamic stability of stiffened laminated composite plates and shells subjected to in‐plane pulsating forces

AbstractThe dynamic stability of laminated composite stiffened or non‐stiffened plates and shells due to periodic in‐plane forces at boundaries is investigated in this paper. A three‐dimensional (3‐D) degenerated shell element and a 3‐D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in‐plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non‐stiffened plates and shells subjected to in‐plane pulsating forces at boundaries.

Forming of Woven Fabric Composites

Forming woven fabric composites into complex parts consisting of double curvature shells leads to complex redistribution and reorientation of fibers in the composite. The thermo-mechanical performance of the molded part is highly dependent on this rearrangement. Consequently, prediction of the properties of double curvature parts requires good understanding of the structural rearrangement in each layer of the composite laminate. This paper presents the results of a study on the prediction of the complex rearrangement that occurs during molding of complex components made of woven fabric prepregs. The deformation of the composite laminate to conform to a given geometry has been modeled using the pin-jointed net idealization and the volume conservation assumption. Experimental measurements of the fiber orientation and the ply thickness variation have also been carried out on molded parts with double curvature shape made of different fabric structures. The numerical prediction has been found to be in good agreement with experimental measurements.

Casimir surface forces on dielectric media in spherical geometry

This paper contains a calculation of the Casimir surface force density in spherical geomeuy under be different circumsmces: (i) The system is an infinitely thin, perfectly conducting shell, endowed with dispersive properties. The presence of dispersion means that the earlier expressions calculated by Boyer (1968) and others have to be generalized: in particular, it is possible to revisit the old semiclassical eleclron idea of Casimir (1956). (ii) The system consists of two different spherical shells, of the same type as above. In patticular. the non- dispersive Casimir surface force between two Rat plates is recovered as the leading term in the formalism when the curvatures of the shells go to zero. (iii) The system is a compact dielectric ball, surrounded by a vacuum. Gene4 formulae are given in all three cases, consistency checks carried out, and some simplifying approximations are given. All physical expressions, if necessary regularized by the Riemann Zeta function method, are clear-cut and finite.

Dynamic Stability Of Stiffened Laminated Composite Plates And Shells Subjected To In-Plane Pulsating Forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. The 3-D degenerated shell element and 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations for the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equations of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyze the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffening scheme, in-plane force type and the curvature of the cylindrical shells, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.

Flexural vibration of doubly-tapered cylindrical shallow shells

An approach based on the principle of energy is proposed to study the free flexural vibration of chordwise doubly-tapered cylindrical shallow shells. The variations in shell thickness are linear and symmetric. This topic is of practical interest, but one on which no previous work has been conducted. The analysis is performed using an efficient computational method based on the Ritz minimum energy approach. The in-plane and transverse displacement amplitude functions of the shells are approximated by sets of pb-2 shape functions with unknown coefficients. The pb-2 shape functions are basically a set of admissible functions composed of the product of a set of mathematically complete two-dimensional orthogonal polynomials and a boundary kinematically oriented basic function. The basic function is defined by the product of the equations of continuous piecewise boundary expressions of the shell planform each raised to an appropriate basic power corresponding to a free, simply supported or clamped edge, respectively. These pb-2 shape functions comply with the kinematic boundary conditions of the shells at the outset. Three classes of different shell configurations and boundary conditions are studied with selected mode shapes presented. A study on the ignoring of tangential inertia has shown little effect on the frequency response. However, the trend reveals a higher effect is evident as the shell curvature increases. The effects of symmetric thickness variation are reflected in the tabulated data, as well as the mode shape figures. Since no data for a doubly-tapered cylindrical shallow shell can be found in the open literature, the results presented in the current study can be used for future reference and comparison.

Design approaches to high power class III (inverse barrel stave) flextensional projectors

Inverse barrel stave or class III flextensional projectors provide the designer of flexural low frequency projectors with an alternative high efficiency, high power density, broadband source which is less sensitive to service life limitations due to long term creep associated with ceramic stacks in other flextensional types. Other benefits arise from the stress-biased concave stave shell which undergoes a reversal in the sign of static stresses as the projector descends in depth resulting in greater shell strength margins allocated to dynamics. Results from combined electroacoustic modeling, finite element analysis and measurements show that, unlike convex shell flextensionals, shell wall thickness has only a very weak influence on resonance frequency and coupling. Instead, the R/L ratio (defined as the radius of curvature of the stave shell normalized by the ceramic stack length) exerts a surprisingly strong influence on the projector’s frequency response and performance metrics. [Work supported by SPAWAR and NCCOSC, San Diego.]

Elastoplastic dynamic instability of long circular cylindrical shells under pure bending

This paper presents a numerical study which is concerned with the prediction of the response and instabilities in long circular cylindrical shells under dynamic pure bending. Of particular interest is the response of such shells, bent into the plastic range of the material, and the various instability characteristics of the shells under dynamic bending (sudden step load). It was found that the major deformation characteristic of the shells is essentially similar to that observed in the static bending when the applied moment is much smaller than the critical dynamic moment. However, when the applied moment is close to the critical dynamic moment, the ovalization of the shell cross-section was found to be localized over a length of several shell diameters in the central region, even though the response of the shell curvature was shown to be still stable in this case. When the applied moment reaches the critical dynamic moment, the response of the shell curvature was shown significantly increasing with time and the shell buckled catastrophically. For thicker shells, it was found that the development of localized ovalization of the shell cross-section is the major factor that causes shell dynamic instability. For thinner shells, however, besides the localized ovalization, the bifurcation induced by short wavelength ripples on the compressed side of the shell was also observed in the initial buckling patterns. After the bifurcation, the initial buckling pattern was replaced by the final postbuckling mode characterized by a localized sharp cupping in the centre of the shell.

Acoustic coupling to membrane waves on elastic shells

The interaction of an acoustic field with a smooth thin shell in a fluid is described by the superposition of a background field plus membrane waves on the shell. The former is defined by a local impedance condition, which accounts for the inertia of the shell, but takes no account of the in-surface, membrane effects. The shell’s flexural stiffness turns out to be of secondary importance. The bulk of the paper deals with the coupling mechanism between the acoustic field and the supersonic membrane waves, both longitudinal and shear. The coupling is mediated by the shell curvature, and vanishes when the curvature vanishes. Ray methods are used to express the membrane waves by curved wave fronts with amplitudes subject to a transport equation over the curved shell surface. The coupling, and decoupling or launching, then reduces to solving an ordinary differential equation for the unknown ray amplitude. In essence, the transport equation is forced, or ‘‘beaten’’ by the locally phase-matched background field. Explicit expressions are obtained for the coupling and detachment coefficients on arbitrarily curved regions. These are combined, using ray theory for the propagation over the shell, to give the scattered field due to rays traveling over the shell. The general results are explicitly tested on the cylinder and sphere, for which the ensemble of surface rays can be summed into a resonance form, and numerical comparisons are made with the exact results for these canonical geometries.

On gibbs-phenomenon-free fourier solution for finite shear-flexible laminated clamped curved panels

A hitherto unavailable analytical solution to the boundary-value problem of static response and free vibration of a rigidly clamped arbitrarily laminated (of which symmetric/antisymmetric cross-ply and angle-ply laminations are special cases) shear-flexible doubly-curved shell of rectangular planform is presented. A boundary-continuous-displacement based double Fourier series approach, designed to avoid Gibbs phenomenon, is developed to solve the boundary-value problems, involving five highly coupled linear partial differential equations with constant coefficients, resulting from Sanders' FSDT (first-order shear deformation theory)-based formulation that also includes surfaceparallel and rotatory inertias. Extensive numerical results that are presented in this study include (i) convergence characteristics of computed deflections, moments and natural frequencies, and (ii) effects of length-to-thickness ratio, radius-to-length ratio, fiber orientation angle, lamination sequence and shell geometry on the response quantities of interest. Also investigated is the highly complex interaction among bending-stretching type coupling effect, membrane action due to shell curvature, and the effects of transverse shear deformation, rotatory inertias and surface-parallel inertias. Additionally, these results are used in assessing the accuracy of their recently obtained FSDT-based boundary-discontinuous counterparts.

Instability in a cylindrical panel subjected to normal pressure: Bifurcation vs nonlinear analyses

Cylindrical shell panels subjected to external pressure are seen in thin semi-monocoque aerospace applications and in the thick-wall pressure hulls of deep-sea submersibles. Both cases are weight-sensitive designs, and therefore, laminated constructions due to their high strength and stiffness-to-weight ratios appear advantageous. In this paper, both linear bifurcation and nonlinear analyses are presented for a wide range of geometries. For shells of high curvature, i.e. a Batdorf parameter, Z, greater than 100, the bifurcation load predicts the onset of instability very accurately as compared to a nonlinear analysis. On the other hand, shallow shells indicate a bifurcation instability where none exists in the nonlinear analysis. The response for some curvatures is significantly altered where transverse shear flexibility apparently has a major influence.

The elastic response of orthotropic laminated cylindrical shells to low-velocity impact

This paper presents an analysis of laminated open cylindrical shells subjected to impact loading. A spring-mass model is developed to determine the contact force between the shell and impactor during impact. Based on this model, an analytic function describing the contact force is derived in terms of the material properties and mass of the shell and impactor, as well as the impact velocity. An analytic solution to predict the response of laminated shells subjected to impact is presented. This solution includes both contact deformation and transverse shear deformation. The present analysis is used to predict the response of laminated shells to impact loading and to study the effects of different impact conditions and shell size and curvature on the contact force and central deflection of the shell. The analysis is verified by comparing the present results with those reported by others.

Finite element buckling analysis of sandwich shells

A four-noded quadratic isoparametric flat shell element with six degrees of freedom per node is developed in order to study buckling behavior of sandwich shells under nodal concentric loading conditions. The relationship between the shell's curvature and resultant buckling strength, as well as buckling mode shapes of the shell has been investigated.

Coupling and launching of supersonic waves in elastic shells

The interaction of an acoustic wave with a smooth thin shell in a fluid is considered. The coupling mechanism between the acoustic field and the supersonic membrane waves, both longitudinal and shear, is discussed in detail. The coupling is mediated by the shell curvature, and vanishes when the curvature vanishes. Ray methods are used to express the membrane waves by curved wave fronts with amplitudes subject to a transport equation over the curved shell surface. The coupling, and decoupling or launching, then reduces to solving an ordinary differential equation for the unknown ray amplitude. In essence, the transport equation is forced, or ‘‘beaten’’ by the locally phase‐matched background field. Explicit expressions are obtained for the coupling and detachment coefficients on arbitrarily curved regions. These are combined, using ray theory for the propagation over the shell, to give the scattered field due to rays traveling over the shell. The general results are explicitly tested on the cylinder and sphere, for which the ensemble of surface rays can be summed into a resonance form, and numerical comparisons are made with the exact results for these canonical geometries. [Work supported by ONR.]

Convolution formulation of leaky wave contributions to scattering by plates and by cylinders and shells of variable curvature

A novel high‐frequency formulation is investigated that approximates the leaky wave amplitude at the scatterer in terms of a spatial convolution of the local incident wave pressure and a one‐sided line response function h(s≥0)=−jα exp(−αs+ikls). Here, s is the propagation distance along the flat or curved surface, α is the reciprocal of the attenuation length, kl the real part of the wave number, and j=1 for equal fluid loading on both sides of a plate but j=2 for one‐sided fluid loading of a shell or for Rayleigh waves on a solid. Application to plane waves incident on cylindrical surfaces (empty shell or solid) of slowly varying curvature yields the following far‐field amplitude from a leaky ray propagating a distance S on the surface: pl=−2αpinc(2πa1a2/ kr)1/2 exp(−αS+iπ/4+iη), where a1 and a2 are the radii of curvature at the launching and detachment regions and η is a geometrical phase accumulation. When a1=a2, the coupling coefficient Gl for a circular cylinder derived previously is recovered. The result can be modified to situations where α varies weakly with curvature. [Work supported by ARL:UTIR&D Program and by ONR.]

Diffraction effects from the joining of two curved plates

Acoustic scattering from a fluid‐loaded shell consisting of two joined curved plates is considered. Asymptotic approximations are made based upon the assumption that the fluid wavelength is much shorter than the smaller radius of curvature. A plane wave in the fluid incident upon a line weld at which the shell properties and curvature are discontinuous, but the tangent is continuous, is considered. The problem simplifies by first assuming that the angle of incidence with respect to the tangent plane at the weld is not near the critical angle for supersonic membrane waves on either side of the weld. The discontinuity then generates a diffracted wave field in the fluid which is determined by solving a Wiener–Hopf problem. This background wave field in turn forces the ordinary differential equations describing the membrane waves. From their solution, diffraction coefficients are obtained for the membrane waves generated on each curved plate by the incident acoustic field. When the incident wave is close to one of the critical angles it couples directly to the shell waves via curvature, and the interaction of this launched membrane wave with the discontinuity results in scattered membrane wave fields with diffraction coefficients determined in a manner similar to the previous case. [Work supported by ONR.]

Crack propagation in cylindrical shells

Results of a theoretical study of dynamic steady crack propagation in a circular cylindrical shell are presented. Equations of thin circular cylindrical shells are utilized for theoretical investigation. The semi-analytic crack tip stress solutions to shell equations are obtained by the perturbation method as well as an iterative method. Plots of internal stresses in the shell with a stationary as well as a steadily running crack are presented. These results are compared with those related to flat plates. It is concluded that the shell curvature has a pronounced effect on increasing the values of crack tip stresses. It would also qualitatively affect the variation of stress field around the crack tip. In some cases, the curvature action is shown to be even more pronounced than the inertia effects.

An analytic model for the vibrations of rectangular shells of variable curvature and thickness

A flexible and powerful model is developed for analyzing the vibration of a rectangular platform shell whose curvature and thickness are arbitrary. The shape of the shell can have almost any conceivable representation, since the curvature and thickness are represented by bicubic polynomials of the centerline arc length. The vibration model includes the effects of shear deformation, rotary inertia, and centerline extension. The equations of motion are solved by an alternative form of the Rayleigh–Ritz method. The resulting integral formulas for the stiffness and mass matrix elements are evaluated by a set of simple computer routines that do symbolic manipulations of algebra and calculus. Predictions of the natural frequency for the several shell geometries are compared to published data, with good agreement. A parameter study shows the dependence of the resonance frequencies and mode shapes on the curvature and the thickness of the shell.