简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

Oscillating laminar fluid flow in a cylindrical elastic pipe of annular cross-section

The coupled elastohydrodynamic problem based on the dynamic equations for a viscous incompressible fluid and for two closed finite-length cylindrical elastic shells, inner and outer, described using the Kirchhoff-Love hypotheses is formulated and solved with the corresponding boundary conditions for harmonic variation of the pressure at the inlet and outlet of an elastic annular pipe. From the solution of this problem the flow parameters and the elastic shell displacements are found. The amplitude and phase frequency characteristics and resonant frequencies of the shells are found. The cases of shells simply supported and with fixed ends are considered. The effect of the support mode and the fluid characteristics on the resonant frequencies and the amplitude frequency characteristics of the shells is investigated.     

METHOD OF GREEN＇S FUNCTION OF CORRUGATED SHELLS

By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green＇s function. To solve the integral equations, expansion method was used to obtain Green＇s function. Then the integral equations were reduced to the form with degenerate core by expanding Green＇s function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton＇ s iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example,elastic characteristic of shallow corrugated shells with spherical taper was studied.Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results.     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

On modified displacement version of the non-linear theory of thin shells

We discuss the non-linear theory of thin shells expressed in terms of displacements of the shell reference surface as the only independent field variables. The formulation is based on the principle of virtual work postulated for the reference surface. In our approach: (1) the vector equilibrium equations are represented through components in the deformed contravariant surface base, and using the compatibility conditions the resulting tangential equilibrium equations are additionally simplified, (2) at the shell boundary the new scalar function of displacement derivatives is defined and new sets of four work-conjugate static and geometric boundary conditions are derived, as well as (3) for prescribed shell geometry all non-linear shell relations are generated automatically by two packages set up in Mathematica. The displacement boundary value problem and the associated homogeneous shell buckling problem are generated exactly without using any additional approximations following from errors of the constitutive equations. Both problems are extremely complex and available only in the computer memory. Such an approach allows us to account also for those a few supposedly small terms which may be critical for finding the correct buckling load of shells sensitive to imperfections. This approach is used in the accompanying paper by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On refined analysis of bifurcation buckling for the axially compressed circular cylinder. International Journal of Solids and Structures, 46, 3111–3123.] to perform the refined numerical analysis of bifurcation buckling for the axially compressed circular cylinder.     

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob-lem. The mode shape differential equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa-tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

Circular shallow spherical shells with central circular hole under simultaneous actions of arbitrary unsteady temperature field and arbitrary dynamic normal load

In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.     

Nonlinear strain gradient elastic thin shallow shells

The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedure. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient shallow thin shell theories. Those terms highly increase the stiffness of the structures. When the curvature of the shallow shell becomes zero, the governing equilibrium for the plates is derived.     

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF SIMPLY-SUPPORTED TRAPEZOIDAL SHALLOW SPHERICAL SHELL ON WINKLER FOUNDATION

The idea of Green quasifunction method is clarified in detail by considering a free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem.This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation, the irregularity of the kernel of integral equations is avoided.Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

A perturbation method for nonlinear vibrations of imperfect structures: Application to cylindrical shell vibrations

A perturbation method is used to analyse the nonlinear vibration behaviour of imperfect general structures under static preloading. The method is based on a perturbation expansion for both the frequency parameter and the dependent variables. The effects on the linearized and nonlinear vibrations caused by geometric imperfections, a static fundamental state, and a nontrivial static state are included in the perturbation procedure.The theory is applied in the nonlinear vibration analysis of anisotropic cylindrical shells. In the analysis the specified boundary conditions at the shell edges can be satisfied accurately. The characteristics of the analysis capability are shown through examples of the vibration behaviour of specific shells. Results for single mode and coupled mode nonlinear vibrations of shells are presented. Parametric studies have been performed for a composite shell.     

Static and dynamic buckling modes of a cylindrical shell under external pressure

We consider the problem of static and dynamic buckling modes of thin shells under external hydrostatic pressure. If the statement of the problem uses the linearized equations of motion obtained in the moderately large bending theory of shells according to the classical or refined model, then part of terms related to the external load in these equations are assumed to be conservative, and the other terms are assumed to be nonconservative. In this connection, we study four statements of the elastic stability problem for a cylindrical shell with hinged faces. The first of them is the statement of the static boundary value problem in the sense of Euler, where the action of external pressure is assumed to be conservative. The second statement is used to study small vibrations near the static equilibrium by a dynamic method for the same conservative load. The third and fourth statements of the problem correspond to the action of a nonconservative load and are similar to the first and second statements, respectively. They use the linearized equations of equilibrium and motion constructed earlier in a consistent version on the basis of a Timoshenko type model and allowing one to reveal all classical and nonclassical shell buckling modes.     

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.     

On the influence of geometric imperfections on the stability and vibration of thin-walled shell structures

The stability equations presented in this paper—formulated using the Ljapunov criterion—are characterized by the fact that any fundamental motion state of imperfect shells can be put into them in order to examine its kinetic stability. Using the established stability equations, specific analyses of the individual effects on the kinetic buckling loads can be performed. In view of the significance of the applied disturbance concept, it is intended to incorporate imperfections into the theory for perfect structures in a form as general as possible and to use them as disturbance-induced quantities of the perfect initial geometry.In order to be able to distinctly analyse the effects of quite different individual influences on the kinetic buckling loads, the respective portions are as well separated to prepare the numerical implementation. In doing so, a compact equation of work is obtained which—using the tensor calculus—permits a clear and very effective numerical implementation. The use of stability equations valid for shells of any geometry and suitable for a large variety of imperfections is demonstrated for a circular cylindrical shell with a bulge-type imperfection as an example.     

Solution of elastic-plastic shallow shell problems by the boundary element method

The boundary integral equations for elasto-plastic problems of shallow shells are established by using the fundamental solutions of shallow shells derived previously. The strains and stress-resultants in the plastic region are used as unknown variables. The simultaneous nonlinear equations of these variables and unknown boundary values are established and solved by direct iteration method.     

Buckling and postbuckling behavior of half-loaded shallow spherical shells

Applying the Galerkin procedure to Marguerre's shallow shell equations, reasonably accurate solutions are obtained for the buckling and postbuckling behavior of clamped shallow spherical shells subjected to external pressure distributed over half the surface.For the geometrical parameter λ smaller than 4.5, the half-loaded shells do not buckle. The buckling loads for 6 ≦ λ ≦ 15 lie about 17–30% below those of uniformly loaded shells. However, the lowest loads after buckling are higher than those for uniformly loading. As λ keeps increasing, the buckling mode shows more and more waves along the circumferential direction. The amplitude of the asymmetrical deflection component Δ₁ becomes remarkably large immediately after buckling. After passing the postbuckling lowest equilibrium point, the sinusoidal deflection disappears and the deflection is changed such that only one location on the shell surface, including the non-loaded area, has been deflected almost symmetrically. The effects of initial imperfections are not so great as for the uniformly loaded case.     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

Analysis of creep deformation and creep damage in thin-walled branched shells from materials with different behavior in tension and compression

A constitutive model for describing the creep and creep damage in initially isotropic materials with different properties in tension and compression has been applied to the modeling of creep deformation and creep damage growth in thin-walled shells of revolution with the branched meridian. The approach of establishing the basic equations for axisymmetrically loaded branched shells under creep deformation and creep damage conditions has been introduced. To solve the initial/boundary-value problem, the fourth-order Runge–Kutta–Merson’s method of time integration with the combination of the numerically stable Godunov’s method of discrete orthogonalization is used. The solution of the boundary value problem for the branched shell at each time instant is reduced to integration of the series of systems of ordinary differential equations describing the deformation of each branch and the shell with basic meridian. Some numerical examples are considered, and the processes of creep deformation and creep damage growth in a shell with non-branched meridian as well as in a branched shell are analyzed. The influence of the tension–compression asymmetry on the stress–strain state and damage evolution in a shell with non-branched meridian as well as in a branched shell with time are discussed.     

NONLINEAR DYNAMICAL BIFURCATION AND CHAOTIC MOTION OF SHALLOW CONICAL LATTICE SHELL

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.     

Effects of imperfections on the buckling response of compression-loaded composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling and postbuckling response of three unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells with different orthotropic and quasi-isotropic shell-wall laminates are presented. The results identify the effects of traditional and non-traditional initial imperfections on the non-linear response and buckling loads of the shells. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The non-traditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, non-uniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. A high-fidelity non-linear shell analysis procedure that accurately accounts for the effects of these traditional and non-traditional imperfections on the non-linear responses and buckling loads of the shells is described. The analysis procedure includes a non-linear static analysis that predicts stable response characteristics of the shells and a non-linear transient analysis that predicts unstable response characteristics.