Shallow Shell Theory Research Articles

Analytical Investigation of Dynamics of Inflatable Parabolic Membrane Reflector

An analytical method for computing the dynamic characteristics of a lenticular inflatable parabolic reflector supported by an inflatable torus is presented. The system is modeled by a combined structure consisting of two pressurized paraboloidal membranes with an elastic circular ring support. Equations of motion for the transverse free vibration of the pressurized paraboloidal membrane are derived first by using the nonlinear theory of shallow shells of revolution. Then, the analytic solutions for those equations of motion are formulated. Afterward, according to the force balance and the displacement continuity between the membranes and the circular ring, the displacements of the ring are calculated as boundary conditions of the paraboloidal membranes. As a result, the frequency equation and analytic mode shape functions of the inflatable reflector are obtained, and the influences of support stiffness on the dynamic characteristics of the reflector are analyzed. The validity of these analytical modal solutions is verified by finite-element simulations.

Free vibration analysis of joined conical shells: Analytical and experimental study

Natural frequencies and mode shapes of two joined isotropic conical shells are presented in this study. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder–plates or cone–plates, conical and cylindrical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell and Hamilton׳s principle. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions and all combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations and also modal testing. The effects of semi-vertex angles and meridional lengths on the natural frequency and circumferential wave number of joined shells are investigated.

Modelling of orthorhombic quasicrystal shallow shells

Abstract The Reissner–Mindlin theory is applied to shallow shells with material properties of orthorhombic quasicrystals (QC). A quasi-periodic arrangement of atoms through the thickness of the shell is considered. Due to the intrinsic characteristics of QCs two excitations need to be described under dynamic loads, namely, phonon (elastic waves) and phason (long wavelength fluctuations) fields. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation shallow shell theory depends only on the in-plane coordinates on the mid-surface of the shell. A weak formulation for the set of governing equations in the Reissner–Mindlin theory is transformed into local integral equations on local subdomains on the mid-surface of the shell by using a unit-step test function. Nodal points are randomly spread on the mid-surface of the shell and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving Least-Squares (MLS) method is employed for the implementation. The influences of the shell curvature and the coupling parameter of quasicrystals on the shell deflection are investigated.

The stiffness variation of a micro-ring driven by a traveling piecewise-electrode.

In the practice of electrostatically actuated micro devices; the electrostatic force is implemented by sequentially actuated piecewise-electrodes which result in a traveling distributed electrostatic force. However; such force was modeled as a traveling concentrated electrostatic force in literatures. This article; for the first time; presents an analytical study on the stiffness variation of microstructures driven by a traveling piecewise electrode. The analytical model is based on the theory of shallow shell and uniform electrical field. The traveling electrode not only applies electrostatic force on the circular-ring but also alters its dynamical characteristics via the negative electrostatic stiffness. It is known that; when a structure is subjected to a traveling constant force; its natural mode will be resonated as the traveling speed approaches certain critical speeds; and each natural mode refers to exactly one critical speed. However; for the case of a traveling electrostatic force; the number of critical speeds is more than that of the natural modes. This is due to the fact that the traveling electrostatic force makes the resonant frequencies of the forward and backward traveling waves of the circular-ring different. Furthermore; the resonance and stability can be independently controlled by the length of the traveling electrode; though the driving voltage and traveling speed of the electrostatic force alter the dynamics and stabilities of microstructures. This paper extends the fundamental insights into the electromechanical behavior of microstructures driven by electrostatic forces as well as the future development of MEMS/NEMS devices with electrostatic actuation and sensing.

Re-interpreting simultaneous buckling modes of axially compressed isotropic conical shells

Elastic stability of shell structures under certain loading conditions is characterized by a dramatically unstable postbuckling behavior. The presence of simultaneous ‘competing’ buckling modes (corresponding to the same critical buckling load) is understood to be largely responsible for such behavior. In this paper, within the framework of linear bifurcation eigenvalue analysis and Donnell shallow shell theory, the presence of simultaneous buckling modes in axially compressed isotropic cones is determined using the semi-analytical method of Galerkin. The results are presented in the plane of the dimensionless reciprocal meridional and circumferential buckling half wavelengths, and are compared with the locus of simultaneous buckling modes of axially compressed cylinders, described by the so-called Koiter circle. By using an optimizing procedure, it is shown that the cluster of simultaneous buckling modes in cones is well described by the Koiter circle of an equivalent cylinder with appropriate length and radius. Such optimizing values of length and radius allow us to gain some insight into the simplifying treatment of the buckling of cones through the concept of equivalent cylinder.

Active optics with a minimum number of actuators

Abstract Optics for astronomy implies powerful developments of active and adaptive optics methods applied to instrumentation from X-rays to the near infrared for the design of telescopes, spectrographs, and coronagraph planet finders. This presentation particularly emphasizes the development of active optics methods. Highly accurate and remarkably smooth surfaces from active optics methods allow new optical systems that use highly aspheric and non-axisymmetric – freeform – surfaces. Depending on the goal and performance required for a deformable optical surface, elasticity theory analysis is carried out either with small deformation thin plate theory, large deformation thin plate theory, shallow spherical shell theory, or the weakly conical shell theory. A mirror thickness distribution is then determined as a function of associated bending actuators and boundary conditions. For a given optical shape to generate, one searches for optical solutions with a minimum number of actuators.

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.

Nonlinear vibration of functionally graded cylindrical shells embedded with a piezoelectric layer

This paper addresses the nonlinear vibration problem of simply supported functionally graded (FG) cylindrical shells with embedded piezoelectric layers. The governing differential equations of motion of the FG cylindrical shell are derived using the Lagrange equations under the assumption of the Donnell׳s nonlinear shallow-shell theory. A semi analytical approach, wherein the displacement fields are expanded by means of a double mixed series based on linear mode shape functions for the longitudinal, circumferential and radial variables, is proposed to characterize the nonlinear response of the cylindrical shell. The large-amplitude response and amplitude frequency curves of the cylindrical shell are obtained by using the proposed approach. Finally, the effects of excitation force and applied voltage on the vibration behavior of the cylindrical shell are investigated.

Multimode interaction in axially excited cylindrical shells

Cylindrical shells exhibit a dense frequency spectrum, especially near the lowest frequency range. In addition, due to the circumferential symmetry, frequencies occur in pairs. So, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to a complex dynamic behavior. So, the aim of the present work is to investigate the dynamic behavior and stability of cylindrical shells under axial forcing with multiple equal or nearly equal natural frequencies. The shell is modelled using the Donnell nonlinear shallow shell theory and the discretized equations of motion are obtained by applying the Galerkin method. For this, a modal solution that takes into account the modal interaction among the relevant modes and the influence of their companion modes (modes with rotational symmetry), which satisfies the boundary and continuity conditions of the shell, is derived. Special attention is given to the 1:1:1:1 internal resonance (four interacting modes). Solving numerically the governing equations of motion and using several tools of nonlinear dynamics, a detailed parametric analysis is conducted to clarify the influence of the internal resonances on the bifurcations, stability boundaries, nonlinear vibration modes and basins of attraction of the structure.

Vibration analysis of curved graphene ribbons based on an elastic shell model

An orthotropic elastic shell model is developed to study the vibration characteristics of curved graphene ribbons (CGRs). The effect of a small length scale is incorporated in the formulations using the gradient elasticity theory. Novozhilov's linear shallow shell theory is used and it is assumed that CGR is simply supported. Analytical solution to the equations is proposed to obtain the frequencies of CGRs. The vibrational properties of CGRs are investigated with respect to the variations of various parameters. Results indicate significant dependence of natural frequencies on the curvature change as well as the modes being considered.

Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties

This paper presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded cylindrical shells subjected to thermal and axial compressive loads. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The governing equations are established within the framework of classical thin shallow shell theory taking both geometrical nonlinearity in von Karman–Donnell sense and initial imperfection into consideration. Thermal stability analysis also incorporates the effects of tangential edge constraints. A Galerkin procedure is applied to derive expressions of load-deflection relations from which the thermal buckling loads and postbuckling curves of the shells are obtained by an iteration. Effects played by material and geometrical properties, tangential stiffness, imperfection and buckling modes are discussed.

Free vibration analysis of joined cross-ply laminated conical shells

Abstract The present study deals with vibrational behavior of two joined cross-ply laminated conical shells. The natural frequencies and mode shapes are investigated. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder-plates or cone-plates, cylindrical and conical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell type and Hamilton’s principle. The appropriate expressions among stress resultants and deformations are extracted as continuity conditions at the joining section of the cones. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions. All combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations. The effects of semi-vertex angles, meridional lengths and shell thicknesses on the natural frequency and circumferential wave number of joined shells are investigated.

Non-linear thermo-elastic and buckling analysis of FGM shallow arches

The paper presents a theoretical investigation on the non-linear thermo-elastic behaviour of pin-ended functionally graded material (FGM) circular shallow arches. Temperature dependency of constituents is taken into account and the arch is subjected to a uniform temperature field. Classical arches theory along with the non-linear shallow shell theory of Donnell are postulated as the basic assumptions. The virtual displacement principle and calculus of variation techniques are implemented to obtain the equilibrium equations. An analytical solution is presented to trace the primary equilibrium path of a pin-ended arch under in-plane thermal loading. Afterwards, closed form expressions are yielded to obtain the radial and axial displacement, stress, strain and bending moment of the arch as a function of temperature parameter. Adjacent equilibrium criterion is used to extract the stability equation associated to the non-linear primary equilibrium path. A closed form solution is presented to estimate the fundamental thermal bifurcation points of the arch. Illustrative results examine the role of the various involved parameters such as power law index, opening angle and length to thickness ratio. Numerical results reveal that, in most cases, critical buckling temperature difference of the FGM shallow arches are too high, even for thin class of arches.

Effects of modal coupling on the dynamics of parametrically and directly excited cylindrical shells

Cylindrical shells exhibit a dense frequency spectrum, especially in the lowest frequency range. In addition, due to the circumferential symmetry, frequencies occur in pairs. So, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to multiple internal resonances. The aim of the present work is to investigate the dynamic behavior and stability of cylindrical shells under lateral and axial forcing with equal natural frequencies. The shell is modeled using the Donnell nonlinear shallow shell theory. A consistent modal solution for this problem is deduced and used to discretize the equations of motion by applying the Galerkin method. A parametric analysis is conducted to clarify the influence of the modal interaction among these nonlinear vibration modes on coexisting solutions, bifurcations, resonances curves and stability boundaries of the shell.

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.

Stress Analysis of Reeled Composite Pipelines Based on Shallow Shell Theories

The instabilities in reeled pipelines during spooling of (initially straight) or laying of (initially bent) pipeare investigated based on shell theories. A combination of composite materials was studied and number of bending stability cases is discussed. It was demonstrated that composite pipeline exhibits more stability than steel pipes. The bending stability of spooled pipeline is examined in detailed and in particular, the case on the diameter drum of the reel lay system. Moreover, results on the strain energy release in the reel lay system are presented, extending the findings of previous works on controlling the spooling or laying mechanism.

Stability analysis of joined isotropic conical shells under axial compression

In this study, the buckling of two joined isotropic conical shells under axial compression and simply supported boundary conditions is investigated. The governing equations are obtained using thin-walled shallow shell theory of Donnell-type and theorem of minimum potential energy. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved by assuming trigonometric response in circumferential and series solution in meridional direction. The results are validated in comparison with the available results in the literature. Effects of semi-vertex angles and meridional lengths on the buckling load and typical mode shape and circumferential wave number are investigated. In addition, the application of the results of this study on the load carrying capacity of these structures is presented.

Sound transmission through a double-panel construction lined with poroelastic material in the presence of mean flow

This paper investigates the sound transmission characteristics through a system of double-panel lined with poroelastic material in the core. The panels are surrounded by external and internal fluid media where a uniform external mean flow exists on one side. Biot's theory is used to model the porous material. Three types of constructions—bonded–bonded, bonded–unbonded and unbonded–unbonded—are considered. The effect of Mach number of the external flow on the sound transmission over a wide frequency range in a diffuse sound field is examined. External mean flow is shown to give a modest increase in transmission loss at low frequency, but a significant increase at high frequency. It is brought out that calculations based on static air on the incidence side provide a conservative estimate of sound transmission through the sandwich structure. The acoustic performance of the sandwich panel for different configurations is presented. The effect of curvature of the panel is also brought out by using shallow shell theory.

Optical design and active optics methods in astronomy

Optical designs for astronomy involve implementation of active optics and adaptive optics from X-ray to the infrared. Developments and results of active optics methods for telescopes, spectrographs and coronagraph planet finders are presented. The high accuracy and remarkable smoothness of surfaces generated by active optics methods also allow elaborating new optical design types with high aspheric and/or non-axisymmetric surfaces. Depending on the goal and performance requested for a deformable optical surface analytical investigations are carried out with one of the various facets of elasticity theory: small deformation thin plate theory, large deformation thin plate theory, shallow spherical shell theory, weakly conical shell theory. The resulting thickness distribution and associated bending force boundaries can be refined further with finite element analysis. Keywords: active optics, optical design, elasticity theory, astronomical optics, diffractive optics, X-ray optics