İnce dönel kabukların dönel simetrik olan ve olmayan yükler altında statik hesabı ile ilgili bir yaklaşım

Abstract

ÖZET Bu çalışmada, statik etkiler altındaki ince dönel kabuklarda oluşan kesit tesirleri ve yer değiştirmelerin çok az bilgisayar 2amanı ve hafızası gerektiren sayısal bir yaklaşımla hesabı önerilmiştir. İnce dönel kabukların denge, kinematik ve bünye denklemleri kullanılarak çıkarılmış olan iki bağımsız değişkenli kısmi diferansiyel denklemlerin bir doğrultudaki türevlerinin Kantorovich yöntemiyle veya Fourier serileriyle elimine edilmesiyle elde edilen birinci dereceden adi diferansiyel denklem takımının (kanonik denklemler) bir tahmin etme-düzeltme yöntemi ve yardımcı sayısal yöntemlerle çözümü tasarlanmıştır, tntegrasyon uzunluğunun kısa olmasında çözümün kararlı olduğu, ancak integrasyon uzunluğu arttıkça sayısal çözümün kararsızlaş- tığı gözlenmiştir. Söz konusu kararsızlığı gidermek için eğilme etkilerinin bazılarının veya hepsinin ihmal edilebileceği bölge veya bölgelerin tahminine dayanan bir deneme yanılma yöntemi önerilmiştir. Kuvvet metodunun bahsedilen deneme yanılma yaklaşımına bir alternatif olduğu ve ayrıca bu yaklaşımla beraber kullanılabilecek bir yöntem olduğu görülmüştür. I. bölümde ince dönel kabukların statik hesabı ile ilgili olarak yapılmış çalışmaların kısa özeti verilmiş ve bu çalışmada kullanılan temel fikirlerin açıklanmasına bir giriş yapılmıştır. II. bölümde lineer kabuk teorisinden; III. bölümde kısmi diferansiyel ince dönel kabuk denklemlerinin kanonik hale getirilişinden, elde edilmiş olan kanonik denklemlerin başlangıç değer yöntemleriyle çözümünde integ-rasyon boyunun uzun olması halinde karşılaşılan kararsızlıktan ve bu durumda kullanılabilecek kararlı çözüm yollarındı bahsedilmiştir. IV. bölümde iki noktalı bir lineer sınır değer probleminin başlangıç değer yöntemleri ile çözüm yollarından bahsedilmiştir. Sayısal uygulamalar ve sonuçların diğer çalışmalar sonucu elde edilenlerle karşılaştırılması V. bölümde, bu çalışmanın genel bir değerlendirilmesi VI. bölümde verilmiştir. Ek 1 kanonik denklemlerin mambran gerilme hal indeki karşılıklarını içermektedir. Ek 2' de bu çalışmada kullanılan sayısal yöntemler, Ek 3'de kullanılan bilgisayar programları hakkında açıklamalar yapılmıştır.

Ill AN APPROACH TO SOLVE PROBLEMS OF THIN SHELLS OF REVOLUTION UNDER STATICAL, AXIALLY SYMMETRICAL AND NONSYMMETRICAL LOADS SUMMARY In this study, a numerical approach which needs a very little computer effort is proposed for the determination of the stress-resultants, stress-couples and the displacements in the middle surface of thin shells of revolution subjected to static effects. The partial differential equations of thin shells of revolution (equilibrium, kinematic and constitutive equations), which have two independent variables., are reduced to a set of ordinary differential equations of first order (canonical equations) by eliminating the derivatives with respect to one of the independent variables by Kantorovich method, which needs the approximation of the behaviour of the dependent variables (stress-resultants »stress-couples and displacements which exist in the list of the natural boundary conditions of thin shells of revolution) in the direction of the independent variable with respect to which the derivatives of the dependent variables are eliminated. In cases the approximation of the behaviour of the dependent variables in one direction, in which the derivatives of the dependent variables are thought to be eliminated, is difficult or the behaviour in the mentioned direction is desired to be included in the computations more precisely, usage of Fourier series in the mentioned direction is suggested. The resultant canonical equations are tried to be solved numerically by a predictor-corrector method and auxiliary numerical methods (by direct integration). In case the integration length is short, the numerical procedure gives quite good results. InIV case Che integration length is long, a numerical unstab which makes the results and therefore seems to make the numerical algorithm unreliable is faced. The mentioned unstability is interpreted to be due to being high of tl norm of the coefficient matrix of the used canonical eqi tions and due to the nature of the bending effects. In order to increase the stability of the numerical solutii by initial value methods, a trial and error procedure wl depends on the estimation of the regions in which some < all of the bending effects are neglected is used. In the region where some or all of the bending effects are neglected, the norm of the coefficient matrix of the use canonical equations are decreased by replacing the terms including extensional rigidity, which exists in the equa because of the usage of an elasticity equation and is a very large number increasing the norm of the coefficient matrix a lot, by their equivalents obtained from the equilibrium equations corresponding to the cases where s or all of the bending effects are neglected. Additionall in case the mentioned region is a membrane one, all of t terms including bending rigidity are eliminated and ther the norm of the coefficient matrix is decreased even mor- The derived approximate canonical equations, depending 01 the negligence of the bending effects, are used with the general canonical equations, in which no bending effects are neglected, one at a time during the numerical solutic according to the mentioned estimation. At the intermedial regions, the variations of the terms of the canonical equations are made slowly not to cause a discontinuity which may be a reason of some extra bending effects artificially in the output. Getting use of the results of a trial of estimation, a better one is tried. Generally, two or three trials become sufficient to have quite good results. In case a bending effect exists only at one of the two boundaries and there is no loading between the tw boundaries, the direct integration (using initial valuemethods) gives stable solutions even for long integration lengths if the numerical computations aTe started from the boundary which is free from bending. This property brings the idea of using force method as another alternative to solve the static problems of thin shells of revolution with long integration lengths by initial value methods. Force method can also be a supportive technique to the mentioned trial and error approach especially in cases where two or more reasons of bending (discontinuity in geometry, discontinuity in loading, edge effects) exist in the problem. Some bending effects may be included in the list of the redundants of the force method and therefore the mentioned trial and error procedure may be applied more easily. An approximate definition of the words `short` and `long`, used for the integration lengths, are tried to be given. But a definite criterion for the boundary between short and long for all types of static problems of thin shells of revolution has not been found and it is concluded that the best way to answer such a question for a given problem is to apply the numerical procedure directly. In the first chapter, a summary of the past studies on the methods of calculating the stress-resultants, stress-couples and displacements in the middle surface of thin shells of revolution under static loads and an introduction to the explanation of the main points, considered in this thesis, to solve the static problems of thin shells of revolution by initial value methods numerically are given. Chapter II contains the linear theory of thin shells of revolution which reduces the problem of the determination. of the stresses and strains at any point in thin shellsVI of revolution (as is done in three dimensional elasticity) to the problem of the determination of the stress- resultants, stress-couples and displacements in the middle surface of thin shells of revolution and bases on the fact that the thicknesses of the thin shells of revolution are small in comparison to other dimensions. In this chapter, the assumptions used in linear theory of thin shells of revolution, equilibrium equations, kinematic equations, constitutive equations and the natural boundary conditions which should be used at the edges of thin shells of revolution are given. The form given by Kalnins to partial differential equations of thin shells of revolution with two independent variables, which contain the first order derivatives of the dependent variables with respect to one of the independent variables and various orders of derivatives of the dependent variables with respect to the other independent variable, and the reduction of the mentioned form to canonical equations by Kantorovich method or by Fourier series alternatively are given in Chapter III. Chapter III also includes the seed-thoughts of this thesis: a criticism on the reasons of the numerical unstability faced during the application of the initial value methods, the ways to be preserved from the mentioned unstability. Chapter IV mentions about some ways, including the one used in this study, to solve the canonical equations of two point linear boundary value problems of thin shells of revolution by initial value methods. The methods mentioned briefly are an approach which reduces the boundary value problems of thin shells of revolution to the solution of Riccati equations with initial values; a trial and error method which bases on the estimation of the unknown values in the boundary which is chosen to be the initial boundary and checking the results, after integration, at the ending boundary with the known values at this boundary and repeating this procedure again and again till the satisfactoryVII agreement of the results with the known boundary conditions at the ending boundary are obtained; a method which suggests to take the solution (displacements ( stress-resultants, stress couples) as the superposition of the effects of the initial conditions (homogeneous solution) on the effects of the loadings (particular solution), when the initial values are taken to be zero; a method, used in this study, which proposes the computation of the initial values of the linear boundary value problems with the help of the solution of the adjoint initial value problems, and performing the integration by using the computed initial values by the initial value methods. Numerical examples clarifying and checking the suggestions given during this. study and the comparisons of the results obtained here with the ones obtained by other studies are given in Chapter V. Chapter VI gives a discussion about approaches given in this thesis and some proposals for future researches. Appendix 1 contains the derivation of the canonical equations corresponding to the membrane state of stress. Similar to the application followed in Chapter III, a set of partial differential equations of the membrane state of stress containing the first order derivatives of the chosen dependent variables with respect to two independent variables are obtained and reduction of the mentioned partial differential equations to ordinary differential equations of first order are done by Kantorovich method. Appendix 1 also mentions about the variations of the obtained canonical equations when Fourier series, instead of Kantorovich method, are used alternatively. In Appendix 2, the numerical methods used in this study are given. Appendix 3 explains briefly the usage of the computer programs used during this study.