NON-SYMMETRICAL LARGE DEFORMATION OF A SHALLOW THIN SPHERICAL SHELL

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Some remarks on the theory of shallow spherical shells

Conclusions It is shown that Vlasov's explicit solutions for shallow spherical shells correspond to a very special boundary condition which does not usually occur in practice.A shell loaded by a normal loading p is fully discussed and it is shown that the discrepancy between these results and those obtained by Vlasov may be considerable. An asymptotic solution of the same problem is also given.Finally it is indicated how Geckeler's approximate equations can be derived from a suitable transformation of the linearized Marguerre equations.This paper was prepared under the support of the Argentine Council for Scientific and Technological Research.     

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.     

Nonlinear stability of spherical shallow shell under compound loads

Using a semi-analytical method,the nonlinear stability of a spherical shallow shellunder centrally distributed and concentrated loads is investigated in this paper.The longermanual calculation has been avoided when finding the approximate solution,and the P-Wmcharacteristic relation can be given analytically.     

On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections

Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton–Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh–Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov–Galerkin method.     

球形扁壳在冲击载荷作用下的超临界变形

本文利用Pogorelov提出的薄亮稳定性几何学理论,研究了球形扁壳在冲击作用下的超临界变形行为。这种方法是建立在实验观察中,壳结构的大变形是以近似于一种等距变换的方式产生的。首先,给出了壳体变形能的近似表达式,在此基础上,考虑了两种不同的在扁球壳顶部的冲击方式,利用能量原理,得到了描述运动的控制方程。从而给出扁球壳中心最大凹陷半径随冲击载荷变化的近似表达式,并将此结果与实验进行了比较,二者吻合的还是比较好的。     

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.     

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.     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

Non-symmetrical large deformation of a shallow thin conical shell

I.IntroductionItisimportanttoresearchnon-symmetricalquestionsofshallowcollicalshellsintheoryoronapplication.Asonekindofpressurevessel'sparts,shallowconicalshellsareverycommonlyusedillellgineerillgpractice,becausethedifficultyofmanul\lctul.illgthemis'small.AlthotlghwelookupmanyChineseandtbreignperiodicalswhicharctlblctobefound,wehavenotyeth'ulldarticlesanddocumentsfornon-symmetricalandnolllinearquestionsofshitllowconictllshells.Oval'rccelltyears,ProlbssorWangXinzhiandhiscolleagueshavedonealot…     

Analytical solution of cracked shell resting on elastic foundation

The elastostatic problem for cracked shallow spherical shell resting on linear elastic foundation is considered. The problem is formulated for a homogeneous isotropic material within the confines of a linearized shallow shell theory. By making use of integral transforms and asymptotic analysis, the problem is reduced to the solution of a pair of singular integral equations. The stress distribution obtained, around the crack tip, is similar to that of the elasticity solutions. The numerical results obtained agree well with those of previous work, where the elastic supports were neglected. The influences of the shell curvature and the modulus of subgrade reaction on the stress intensity factor are given.     

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...     

Snap-through buckling of eccentrically stiffened shallow spherical caps

The problem of snap-through buckling of a clamped, eccentrically stiffened shallow spherical cap is considered under quasi-statically applied uniform pressure and a special case of dynamically applied uniform pressure. This dynamic case is the constant load infinite duration case (step time-function) and it represents an extreme case of blast loading-large decay time, small decay rate.The analysis is based on the nonlinear shallow shell equations under the assumption of axisymmetric deformations and linear stress-strain laws. The eccentric stiff eners are disposed orthogonally along directions of principal curvature in such a way that the smeared mass, and extensional and flexural stiffnesses are constant. The stiffeners are also taken to be one-sided with constant eccentricity, and the stiffener-shell connection is assumed to be monolithic.The method developed in an earlier paper is employed. In this method, critical pressures are associated with characteristics of the total potential surface in the configuration space of the generalized coordinates.In addition, buckling of the complete thin eccentrically stiffened spherical shell under uniform quasi-statically applied pressure is considered, and these results are used to check the numerical answers. The complete spherical shell is stiffened in the same manner as the shallow cap.The results are presented in graphical form as load parameter vs initial rise parameter. Geometric configurations corresponding to isotropic, lightly stiffened, moderately stiffened and heavily stiffened geometries are considered. By lightly stiffened geometry one means that most of the extensional stiffness is provided by the thin shell. A computer program was written to solve for critical pressures. The Georgia Tech Univac 1108 high speed digital computer was used for this purpose.     

用Green函数求解扁球壳受环状线载和线偶作用的问题

本文从复变量形式的扁壳基本方程出发,通过建立复Green函数导出了在环状线载和线偶作用下扁球壳的位移和内力分布,通过积分可以求得轴对称的表面受变化分布载荷情况的解答,本文方法还可求得圆饭、圆柱壳等问题的解答,而且适用于各种轴对称边界条件。     

Thermoconductivity and thermoelasticity of thin isotropic shells heated by a moving heat pulse

The paper presents a solution to the problem of thermal conduction and thermoelasticity for a thin shallow spherical shell heated by a concentrated or local impulsive heat source moving over the shell surface. It is assumed that temperature is linearly distributed across the shell thickness and that the shell, on its sides, exchanges heat with the environment in accordance with Newton’s law of cooling. The Fourier and Laplace transforms are used to find an analytic solution. The dependence of the temperature field and stress/strain components on the type of heating and the form of heat source is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 85–92, November 2006.     

Buckling of thin spherical shells

Summary This paper deals with the buckling of a shallow spherical cap subjected to uniform edge moment and a clamped deep spherical shell under uniform pressure. The first problem is formulated in integral equations which are solved by an iterative procedure. The buckling moments are determined for a wide range of the shell geometrical parameter. The second problem is based on the concept that the highly deformed region around the apex is treated as a shallow spherical cap elastically supported by the rest of the shell. The stability of a thin sphere is treated as a special case. The results obtained in both problems are compared with existing solutions.

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Übersicht Es wird das Beulen sowohl für eine flache Kugelkalotte mit gleichförmigem Randmoment, als auch für eine tiefe Kugelschale unter gleichförmigem Druck untersucht. Die erstgenannte Aufgabe wird auf Integral-gleichungen zurückgeführt, die durch Iteration gelöst werden. Die Beulmomente werden für einen weiten Bereich der geometrischen Parameter der Schale bestimmt. Für die Lösung der zweiten Aufgabe wird angenommen, daß der stark verformte Teil der Schale in der Umgebung des zentralen Punktes als eine flache Kugelschale aufgefaßt werden kann, die elastisch von dem Rest der Schale getragen wird. Die Stabilität einer dünnen Kugel wird als Spezialfall betrachtet. In beiden Fällen werden die Ergebnisse mit vorhandenen Lösungen verglichen.

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The first problem in the analysis was sponsored by the National Research Council of Canada. The author is very grateful to Professor K. N. Tong for his illuminating suggestions regarding the second problem.     

On the nonlinear stability of a truncated shallow spherical shell under a concentrated load

In this paper, the axisymmetric nonlinear stability of a clamped truncated shallow spherical shell with a nondeformable rigid body at the center under a concentrated load is investigated by use of the modified iteration method. The analytic formulas of second approximation for determining the upper and lower critical buckling loads are obtained. This paper was read at The Third East China Symposium on Solid Mechanics, Jiuhuashan, October, 1986.     

集中载荷作用下变厚度开顶扁球壳的非线性稳定问题

首先应用逐步加载法将具有硬中心的开顶扁球壳在集中载荷作用下的非线性微分方程组线性化，然后利用样条配点法解线性微分方程组，得到了临界载荷的大小。     

Stress and deformation analysis from reduced-scale plastic-model testing

The use of scale models, which are made from plastic material, for stress and deformation analysis of missile nose-cone structures is discussed. The special strain-gage application and testing techniques, which are required because of the use of plastic materials, are detailed.The utilization of relatively inexpensive simplified models for the investigation of two specific design conditions is cited. The first case is a stress and deformation study of a thin, constant-thickness, shallow spherical shell which is supported by a circumferential line reaction and subjected to uniform external pressure. Comparisons are made with a recently published theoretical analysis of the problem.The second case is a particular design problem which is concerned with the determination of the stress and deformation in a variable-thickness, shallow spherical shell with several various-size cutouts. The shell is loaded with a varying external-pressure load which is reacted by a circumferential line load at the periphery. Influence curves for both stress and deformation are given.Some limitations of plastic-model testing are reviewed, and guides to successful use of the method are given.Paper was presented at 1959 SESA Spring Meeting held in Washington, D. C. on May 20–22.     

An analysis of shallow spheroidal shells by a semi-inverse contour method

This paper presents a linear analysis of a shallow prolate spheroidal shell with a planar elliptical boundary. The shell is subjected to a uniform load, q, and clamped along the boundary. The theory used in this paper is characterized by the well known Mushtari-Donnell-Vlasov equations which consist of a compatibility equation and an equilibrium equation where the normal displacement, w, and a stress function, φ, are the dependent variables.The method employed for the solution of this problem is developed in three major stages. The first stage involves the determination of w, under the assumption that the contours of w be ellipses concentric to the boundary. The second stage is devoted to the determination of a stress function φ, which, together with w, satisfies the MDV compatibility equation exactly. The third stage of the development is concerned with the computation of a loading, q^*, which, together with w and φ, satisfies the equilibrium equation exactly and which is nearly equal to the desired uniform loading q.