Donnell Equations Research Articles

Critical circumferential wavelength of elastic buckling of longitudinal compressed thin-walled cylindrical shells

The classical theory of elastic critical buckling stress works well for slender columns and thin flat plates under compression; however, the situation is different for longitudinally compressed thin-walled circular cylindrical shells, and the issue has plagued us despite considerable efforts over the last 100 years. We noticed that all such buckling analyses thus far, both linear and nonlinear, in terms of the main philosophy, inherited and were confined to Euler’s pioneering solution for the slender column model that focuses on the longitudinal buckling deformation mode and should be classified as the ‘longitudinal open-loop’ eigenmode because the deformations of the two longitudinal ends are physically independent of each other. In view of this, for an ideal linear-elastic buckling model of a thin-walled perfectly circular cylindrical shell under uniform longitudinal compression on the foundation of the longitudinal open-loop eigenmode solution, it is also necessary to consider a ‘circumferential closed-loop’ eigenmode simultaneously to physically avoid violating the reality of its ideal periodic deformation on the entire perimeter and to mathematically redefine the biunique and precise relationship for each distinct eigenmode by the critical circumferential wavelength. Originating from such a case study, the mathematical uniqueness issue hidden in the general solution of the Donnell equation is further discussed. The authenticity of the competing eigenmode characterized by the Koiter circle is also discussed. Furthermore, a preliminary attempt was conducted to interpret the discrepancy between theoretical and experimental buckling loads, mainly initiated by the characteristic type of longitudinally generated circumferential local inward displacement in initial geometric imperfections, using the insights herein.

ОЦЕНКА ВИБРАЦИОННЫХ ВОЗДЕЙСТВИЙ МАССИВНЫХ ДЕТАЛЕЙ НА ЦИЛИНДРИЧЕСКУЮ ЧАСТЬ КОТЛОВ ВАГОНОВ-ЦИСТЕРН НА ОСНОВЕ ТЕОРИИ ОБОЛОЧЕК

The problems related to the vibration effects of bulky elements on the cylindrical shell of tank car boilers are considered. To do this, an approach based on the shell theory is proposed. Donnell equations are used to define the loading during vibrations.
 Equations based on the theories of depressed cylindrical shells and on the polytechnic theory of shells are used. In order to confirm the reliability of the results obtained according to the proposed approach comparative analysis is performed. The rigidity of the attached cylindrical shell is estimated, for which experimental values of deformation under the local load are obtained.
 The results of calculations for the local single load of a closed cylindrical shell by means of finite element method (FEM) are given. These results show a satisfactory conformity with the data obtained experimentally (based on calculations). Using the proposed technique, the authors evaluated rigidity of the boiler cylindrical part where bulky elements are installed and this makes it possible to define the values of natural frequencies and the level of dynamic addition for various design parameters. Geometrics for tetra axial and eight axial tank cars are used as examples. Additionally, gain-frequency characteristics of deformations, accelerations and dynamic coefficients in the places of bulky elements are presented. Study results can be used to improve the design and the conditions for using tank car boilers.

Buckling Properties of a Subsea Function Chamber for Oil/Gas Processing in Deep Waters

Due to the complexity of installations and connections of subsea production equipment and the massive structures involved in a challenging environment, the failure of subsea production equipment could induce enormous loss to the safety and reliability of structures in addition to the cost of the oilfield development. One of the challenges that the subsea production structures face, as it moves to ultra-deep water and polar underwater equipment, is to design subsea shell structures capable of withstanding high external pressures. Hence, a subsea function chamber (SFC) has been lately proposed as a viable solution, which has a high level of safety and reliability, and a technique for the subsea production system. This paper presents a general and efficient buckling and collapse analysis strategy. In this work, the SFC is composed of cylindrical and hemispherical shaped steel material. Initial imperfection-based nonlinear buckling analysis has been carried out to investigate the buckling and risks associated with different thicknesses of the structure. Linear and nonlinear static buckling analyses have been carried out using ABAQUS software. By introducing the nonlinear properties of materials, the nonlinear numerical model of SFC is established. The effects of the thickness of different models and the number of stiffeners on the buckling modes are discussed. The wall thickness is calculated by the Donnell equation and Timoshenko’s classical method. It has been found that the classical solutions given by the Donnell and Timoshenko equations are more accurate for structures with larger lengths and diam. The thickness and number of stiffeners have a great influence on the ultimate buckling external pressure load of SFC structure

Acoustic scattering from an infinitely long cylindrical shell with periodic internal lengthwise ribs.

The acoustic scattering from an infinitely long cylindrical shell with periodic lengthwise ribs is studied. The shell motion is described by the Donnell equations, and the lengthwise rib is modeled as an elastic beam whose motion is decomposed into longitudinal and flexural vibrations. A circumferential mode expansion is used to obtain numerical results for the scattering sound field. The backscattering characteristics in the far-field can be explained by the resonance and interference phenomena. It is shown that subsonic flexural waves can be generated and radiated by the ribs. Due to the periodical distribution of the ribs, there exist multi-order flexural Bloch waves in the circumferential direction. The multi-order flexural Bloch waves can form standing circumferential waves, which lead to a complex acoustic resonance. The attachments of the ribs to the shell can reflect an acoustic wave directly and the reflected wave will interfere with the specular reflection, which is dominant in the frequency-angle spectra with an increasing number of ribs. Furthermore, the flexural wave and flexural Bloch waves can radiate through the attachments and interfere with the specular reflection. However, the interference fringes in the frequency-angle spectra caused by the flexural wave and flexural Bloch waves are coincident at broadside.

Asymptotic Separation and Synthesis of a Stress State and Thermal-Force Analogues in the Theory of Shells

An approach to calculating the stress-strain state during axisymmetric deformation of domes and bottoms by synthesizing two stress states, namely, momentless and edge effect is generalized to the case of arbitrary deformation of cylindrical and conical shells. It is proposed to construct the stress state on the basis of approximate equations describing the so-called elementary stress states: ground state (semi-momentless theory), edge effect, stress state with high variability (Vlasov–Donnell equations), and the equations of bending or tangential states, which are analogous to the equations of the plane problem and problem on bending plates. Thermal-force analogies between stress states under force and thermal actions are established.

Nonlinear Postbuckling of Eccentrically Oblique-Stiffened Functionally Graded Doubly Curved Shallow Shells Based on Improved Donnell Equations

The nonlinear buckling and postbuckling of eccentrically oblique-stiffened doubly curved shallow functionally graded shells is investigated based on improved Donnell equations. The improved Lekhnitskii smeared stiffeners technique is employed to found the stiffness matrix of the stiffened shells. The shells are reinforced by eccentrically oblique stiffeners with an arbitrary inclination angle. Using the Galerkin method, an analytical approximate solution for the deflection of reinforced FGM doubly curved shallow shells is obtained. The influence of geometrical parameters, oblique stiffeners, and temperature on the postbuckling behavior of the shells is analyzed.

Stability analysis of generally laminated conical shells with variable thickness under axial compression

The buckling of generally laminated conical shells having thickness variations under axial compression is investigated. This problem usually arises in the filament wound conical shells where the thickness changes through the length of the cone. The thickness may be assumed to change linearly through the length of the cone. The fundamental relations for a conical shell with variable thickness applying thin-walled shallow shell theory of Donnell-type and theorem of minimum potential energy have been derived. Nonlinear terms of Donnell equations are linearized by the use of adjacent-equilibrium criterion. Governing equations are solved using power series method. This procedure enables us to investigate all combinations of classical boundary conditions. The results are verified in comparison with Galerkin method and the available results in the literature. Effects of thickness function coefficient, semi-vertex angle, lamination sequence, length to diameter ratio, and initial thickness of the cone on the buckling load are investigated. It is observed that these parameters have considerable effects on the critical buckling load of a conical shell.

Vibroacoustic behavior of an infinitely long cylindrical shell with periodic internal lengthwise ribs.

The vibroacoustic behavior of an infinitely long cylindrical shell with periodic lengthwise ribs is studied. The shell motion is described by the Donnell equations, and the lengthwise rib is modeled as an elastic beam whose motion is decomposed into longitudinal and flexural vibrations. Analytical expressions are obtained for the shell motion via a circumferential mode expansion based on the periodicity in the circumferential direction and the Fourier transform in the longitudinal direction. Furthermore, the far-field radiated pressure is obtained via the stationary-phase approach. The shell vibration and sound radiation are analyzed using discrete circumferential wavenumbers. Multi-order flexural Bloch waves exist in the circumferential direction when the cylindrical shell is equipped with periodic lengthwise ribs. The supersonic components of the flexural Bloch waves radiate efficiently and lead to acoustic radiation resonances in the far field.

Temperature dependence of the excitonic spectra of monolayer transition metal dichalcogenides

We theoretically study the temperature dependence of the excitonic spectra of monolayer transition metal dichalcogenides using the O′Donnell equation, \({E_g}(T) = {E_g}(0) - S\langle \hbar \omega \rangle [cloth(\frac{{\hbar \omega }}{{2{k_B}T}} - 1)]\) . We develop a theoretical model for the quantitative estimation of the Huang–Rhys factor S and average phonon energy \(\langle \hbar \omega \rangle \) based on exciton coupling with longitudinal optical and acoustic phonons in the Frohlich and deformation potential mechanisms, respectively. We present reasonable explanations for the fitted values of the Huang–Rhys factor and average phonon energy adopted in experiments. Comparison with experimental results reveals that the temperature dependence of the peak position in the excitonic spectra can be well reproduced by modulating the polarization parameter and deformation potential constant.

Analytical and Numerical Investigations on Buckling of an Axially Compressed Cylindrical Panel with Specific Boundary Condition

In this paper, the stability of cylindrical panels with circular cross section in elastic form is estimated using both analytical and Numerical methods. The Donnell equations for linear buckling of cylindrical shells are assumed and finally the equations are resulted to numerical values with the use of Galerkin method. Boundary condition in this case is such that the two curved edges are simply supported and two straight edges are free. The effects of geometrical parameters such as thickness, radius, length and central angle on buckling of cylindrical shells have been investigated both numerically and analytically. Also, the results of both methods are presented in figures.

Discussion on “Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations” by D.H. Bich and N. Xuan Nguyen, Journal of Sound and Vibration 331(25) (2012) 5488–5501

Discussion on “Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations” by D.H. Bich and N. Xuan Nguyen, Journal of Sound and Vibration 331(25) (2012) 5488–5501

Reply to: Discussion on “Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations” by D.H. Bich and N. Xuan Nguyen, Journal of Sound and Vibration 331 (25) (2012) 5488–5501

Reply to: Discussion on “Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations” by D.H. Bich and N. Xuan Nguyen, Journal of Sound and Vibration 331 (25) (2012) 5488–5501

Acoustic scattering from the finite periodically ribbed two concentric cylindrical shells

Acoustic scattering from the submerged finite periodically ribbed two concentric cylindrical shells, insonified by an incident plane wave, is studied. Motion of the shell is described by Donnell equations, while the motion of the disks is described by thin plate motion equations and plane ströss equations ignoring the effects of axial forces on the disks. Through angle-frequency spectra, besides the elastic waves of the shells, the contributions of the fluid between the inner and outer shells and the interaction of the inner and outer shells with the disks play an important role in the far scattering field. There are fine echo waves such as Bragg scattering and fluid-added waves excited on the angle-frequency spectra. Finally the theories are verified by experiments.

Acoustic scattering from a double periodically bulkheaded and ribbed finite cylindrical shell

The sound scattering from a finite vacuumed cylindrical shell simultaneously loaded with the bulkheads and the ribs are studied. The motions of the shell are described by the Donnell equations. Each rib is modeled as a rod of circular curvature where only the normal force is applied. The motions of the bulkheads are decomposed into the out-plate bending motions and the in-plate extensional motions, which are independent of each other. The backscattered sound field is analyzed via the frequency-angle spectra. Compared against the backscattered sound from a ribbed cylindrical shell, the Bloch-Floquet waves disappear at low frequencies but still exist at mid and high frequencies. The Bragg scattering excited by the ribs is unchanged at mid and high frequencies. Besides the ribs' Bloch-Floquet waves and Bragg scattering, there exists the Bragg scattering excited by the bulkheads. The backscattering from a cylindrical shell loaded with just the bulkheads is calculated. When the spacing and the thickness of the bulkheads are equal to those of the ribs, the crests of the Bragg scattering and the Bloch-Floquet waves on the frequency-angle spectra are almost the same as those of the ribbed cylindrical shell. The resonances of the bulkheads are also excited in this case.

Postbuckling of functionally graded cylindrical shells based on improved Donnell equations

This paper presents an analytical approach to investigate the buckling and postbuckling of functionally graded cylindrical shells subjected to axial and transversemechanical loads incorporating the effects of temperature. Material properties are assumed to be temperature independent, and graded in the thickness direction accordingto a simple power law distribution in terms of the volume fractions of constituents. Equilibrium equations for perfect cylindrical shells are derived by using improved Donnell shell theory taking into account geometrical nonlinearity. One-term approximate solution is assumed to satisfy simply supported boundary conditions and closed-form expressions of buckling loads and load-deflection curves are determined by Galerkin method. Analysis shows the effects of material and the geometric parameters, buckling mode, pre-existent axial compressive and thermal loads on the nonlinear response of the shells.

Response of a tank under blast loading – part II: experimental structural response and simplified analytical approach

The problem of dynamic buckling of thin cylindrical tanks subjected to an external blast caused by an accidental external gaseous detonation is studied in this paper. A study is performed on flexible cylinders at a reduced scale to quantify the structural response. Simplified semi-analytical models based on Donnell's equations and critical imperfection amplification thresholds are used to provide damage predictions. Numerical results are compared to experimental ones which shows a qualitative good agreement. Finally, a reliability analysis on a real atmospheric oil tank is performed and compared with damage diagrams in the form of pressure impulse curves and safety recommendations. Dans cet article, on s'intéresse au flambage dynamique de réservoirs métalliques soumis à une onde de souffle issue d'une détonation de gaz. Une campagne d'essais est réalisée à échelle réduite. Une approche semi-analytique basée sur le modèle de coque surbaissée de Donnell et l'amplification d'imperfections est utilisée pour déterminer des courbes critiques de flambage. La confrontation des résultats expérimentaux et de la modélisation simplifiée étant satisfaisante, une analyse de fiabilité et une comparaison aux seuils forfaitaires est réalisée sur un cas réel.

Capillary buckling of a thin film adhering to a sphere

We present a combined theoretical and experimental study of the buckling of a thin film wrapped around a sphere under the action of capillary forces. A rigid sphere is coated with a wetting liquid, and then wrapped by a thin film having an approximately cylindrical shape. The equilibrium of the film is governed by the competing effects of elasticity and capillarity: elasticity tends to keep the film developable while capillarity tends to curve it in both directions so as to maximize the area of contact with the sphere. In the experiments, the region of contact between the film and the sphere has cylindrical symmetry when the sphere radius is small, but destabilizes to a non-symmetric, wrinkled configuration when the radius is larger than a critical value. We combine the Donnell equations for near-cylindrical shells to include a unilateral constraint with the impenetrable sphere, and the capillary forces acting along a moving edge. A non-linear solution describing the axisymmetric configuration of the film is derived. A linear stability analysis is then presented, which successfully captures the wrinkling instability, the symmetry of the unstable mode, the instability threshold and the critical wavelength. The motion of the free boundary at the edge of the region of contact, which has an effect on the instability, is treated without any approximation.

An extension of Karman–Donnell's theory for non-shallow, long cylindrical shells undergoing large deflection

Conventionally cylindrical shells are often treated as shallow shells whose governing equation, according to Karman–Donnell's theory, can be approximated by those of thin plates. In this paper Karman–Donnell's theory for shallow shells is extended for long cylindrical shells undergoing large, nonlinear flexural deflection. The kinematic relations between the changes of curvature and the displacement are derived and the governing equations are established by considering the influence of the initial curvature of the cylindrical shells. In particular, the extended Karman–Donnell's theory is applied for the failure analysis of infinitely cylindrical shell under lateral pressure . A regional collapse mode is identified to occur in the shell with a longitudinal span proportional to (radius 3 /thickness) (1/2) and a transverse profile of dog bone shape. It is found that the buckling pressure of the shell is in proportion to (thickness/radius) 3 and converges to the classic solution given by Timoshenko and Gere (1961) . A comparison to the previous works indicates that ignoring the effect of the initial curvature will result in an overestimate of the buckling pressure for 33%. It shows that the initial curvature of long cylindrical shells has significant influence on the load carrying capacity and the extended Karman–Donnell's equations give very accurate predictions. ►The nonlinear kinematic relations of curvature change vs deflection for non-shallow, cylindrical shells are deduced.►The Karman–Donnell’s theory is extended by considering the effect of the initial curvature in circumferential direction. ►Post-buckling of infinitely long, cylindrical shells subjected to the lateral external pressure are analyzed.►A regional collapse mechanism is identified with a longitudinal span proportional to (radius 3 /thickness) (1/2) .►It shows that the initial curvature of long cylindrical shells has significant influence on the load carrying capacity.

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.

Acoustic scattering from a finite periodically bulkheads in cylindrical shell

Acoustic scattering from a submerged finite cylindrical shell with periodically bulkheads, insonified by an incident plane wave, is studied. The motion of shell is described by Donnell equations, while the motion of bulkheads is described by thin plate motion equations and plane stress equations with considering axial force circumferential force radial force and bending moment simultaneously. Through angle-frequency spectrum besides the surface elastic waves, Bragg waves and Bloch-Floquet waves for the periodically bulkheads, the interactions between bulkheads resonance can be seen. Finally the theory is verified by experiment. Bragg waves, partial Bloch-Floquet waves and resonance of bulkheads highlight are all accordant with theory.