Burkulma yükünü en büyük yapan malzeme yayılışı

Abstract

ÖZET Bu çalışmada; yapı sistemlerinde verilmiş bir malzeme miktarı için elastik burkulma yükünü en büyük yapan sürekli malzeme yayılışını veren genel şart denklemi çıkartılmış ve çeşitli sistemlere uygulanmıştır. Çalışma sekiz bölümden oluşmaktadır. Birinci bölümde konu tanıtılmakta ve daha önce bu konuda yapılan çalışmalar özetlenmektedir. ikinci bölümde, yapı sistemlerinin ikinci mertebe he¬ sabı ve burkulma yüklerinin bulunması için gerekli tanım ve varsayımlar verilmekte ve sistemlerin burkulma yönünden sınıflandırılmaları yapılmaktadır, ikinci bölümün son kıs¬ mında, Rayleigh oranı virtüel iş teoreminden yararlanarak genel olarak elde edilmektedir. Çalışmanın üçüncü bölümünde, yapı sistemlerinde burkul¬ ma yükünü en büyük yapan sürekli malzeme yayılışını elde et¬ mek için gerekli olan genel şart denklemi çıkartılmış ve daha sonra bu gerek şartın aynı zamanda yeter olduğu da gösteril¬ miştir. Bu şart, sistemdeki dış etkilerin ve şekil değiştir¬ melerin özelliklerinden bağımsız olarak çıkartılmıştır. Mal¬ zeme yayılışının tayini ve burkulma yükünün hesabı için bir sayısal ardışık yaklaşım yöntemi geliştirilmiştir. Daha sonra dış etkiler, şekil değiştirmeler ve kesit şekilleri için yapılan bazı kabullerle, genel olan gerek ve yeter şart denkleminin özel durumları elde edilmiş ve doğru eksenli çubuklarla, dikdörtgen ince plakların yanal burkul¬ ması hallerine uygulanmıştır.II Dördüncü bölümde, sayısal çözümde kullanılan sonlu elemanlar yöntemi anlatılarak, çubuk ve dikdörtgen plak elemanlarının birinci ve ikinci mertebe rijitlik matrisleri' elde edilmiştir.. Çalışmanın beşinci bölümünde, yalnız.eğilme şekil de¬ ğiştirmeleri göz önüne alınan doğru eksenli çubuklarda de¬ ğişik mesnet halleri ve dış etkiler için çözümler yapılarak en büyük burkulma yükünü veren malzeme yayılışları bulunmuş¬ tur. Bundan başka, eğilme ve kayma şekil değiştirmelerinin birlikte hesaba katılması hali için de çözümler yapılmış ve sonuçlar, yalnız eğilme şekil değiştirmeleri göz önünde tu¬ tularak bulunan sonuçlarla karşılaştırılmıştır. Altıncı bölümde, dikdörtgen plakların, verilmiş bir malzeme miktarı için burkulma yüklerini en büyük yapan mal¬ zeme yayılışları tayin edilmiştir. Bu çözümler plakların değişik mesnet halleri, kenar oranları ve dış etkiler için yapılmıştır. Bulunan sonuçlar tablo ve şekillerle verilmiş¬ tir. Yedinci bölümde, çalışmanın dördüncü, beşinci ve altın¬ cı bölümlerinde verilen sayısal sonuçların elde edilmesi için hazırlanan elektronik hesap makinası programlarının ayrıntıları verilmiştir. Bu programlar FORTRAN IV dilinde kodlanmıştır. Çalışmanın sekizinci' bölümünde, elde edilen sonuçlar açıklanmıştır. s

S U M M A R Y Öne of the majör aims of engineering is to find the most convenient solution för the design of structural systems. A successful.design requires.the complete satisfaction of two opposite conditions, namely safety and economy. in some engineering problems öne of these conditions may become more important than the other. in certain structures, in order to provide sufficient stiffness against static and dynamic effects, the safety condition may dominate över bhe cost as in the design of nuclear power plants. However, in some other problems another aspect, namely weight may have greater importance, For example in designing aerospace vehicles ör long-span -. structures,.weight becomes öne of.the majör parameters. in these types of structures the principle of the minimization of weight is more important than that of cost. in recent years there has been a number of studies investigating the minimization problem of a component of the displacement vector, ör the maximization of the buckling load för a given âmount of material. in the study of this kind of design problem secondary conditions of the type mentioned above have also been taken into consideration as much as possible. in this study the design problem of continuous systems which provide a maximum buckling load fpr a given âmount of material, has been investigated.IV The contents of this thesis are divided into eight chapters. The description of the problem, the fundamental concepts and existing literatüre on this subject are outlined in the first chapter. The second chapter contains the second order theory and the definitions and assumptions which are used in the determination of the buckling load. The Rayleigh quotient is obtained using the principle of virtual work in the mpst general form in matrix notation. in the third chapter the conditional equation which provides the maximum buckling load för a given amount of material is obtained. This conditional equation is completely independent of the external effects and deformations. The acting loads can be shown as follows M-PM.' vhere fP is the parameter of buckling load and [n] is the matrix of the ratio of the loads acting on the system. The loads are considered in the most general form. The weight of the system is also taken as an external load. Thus the matrix [n] is not a constant matrix. The given amount of material V may be eacpressed in the form V - Fds, t waere s is the system variable. in a system composed of o;ıe dimensional members, ds is an element of the length. 0& the other hand in two dimensional problems, such as p! ateş, ds stands för dxdy, which is an element of the ar^.a. F is density (per unit length, area ör volume depmding on the dimensionality of the system.).V The problem is to de t er mine the maximum value of the load parametre j/3^. Therefore it can be treated as a variational problem. But there is a constraint condition, because the amoımt of the material is constant. in this case the f irst variation of the function ( 5^. - XV) muş t be equal to zero, i.e., 6(£k -XV) =0 Here X is introduced as a Lagrange's multiplier. Using the above relation, the necessary condition has been found in Eq.(3.14), vhich makes the buckling load mıadımm f ör a given amount of material. As mentioned above, this condition has been obtained in the most general form. Other conditions obtained in previous vork are only special cases of the general condition given by Eq.(3.14). it is also shown that this condition is sufficient. This is given by the expression (3.26). / The main problem is to find the function Fwhich describes the optimal distribution of the material. The buckling load and the buckling mode must also be determined. These unknovns can be obtained by using the folloving equations vhich are : the condition_given by Eq. (3.14), the differential equilibrium equations due to the acting loads and the boundary conditions. Except in some special cases the sölution of the system is extremely difficult. in this study a numerical method of successive approKİmations is used to determine the distribution of the material and the buckling load. in the last section of the third chapter, some special forms of the generalized `necessary and sufficient condition` have been obtained by introducing some assumptions on the form of the cross-section, the displacements and the external loads. Eq.(3.14) takes a very simple form upon introduction the following assumption on the cross-sectional rigidity D and the cross-section function F,VI D.-k/1 -. vhere k,- and Pi are constants. The condİtion which is Sâs sterASSJSM çş- MjuS-j£Si?s -sr,s: s1;^^ of the plates. in this study, the finite element method has been used in the analysis of the system. As the aım of thıs study is to obtain the distribution of materıal whıch provides the masunum buckling load, the cross-sect1On of the svstem will be variable.. Thış varıatıon of the cîoss-seSion can be accötnplished by dividing the system into sufficiently small finite elements. in the fourth cSpter! the first and second order stiffness ^r.ces of the finite elements of hars and plates, are obtamed. These are then used in the numerical analysıs. During the construction of the second °rder.s;if^f matrirof the *.ectangular plate element, the appUed loads p! p are considered to be uniformly dıstrıbuted S^r VH-» pö «SLSrs^L.ı». forces. in chapter five, the »necessary and suffici«* condİtion` obtained in chapter three xs applıed to slender Srs. Two cases are investigated.. in the fırst case only the bending deformations are taken ınto account. in the second casl shear deformations have also been ««.£.£ together with the bending deformations. in the fırst case, th! axial load P is considered as actıng on the top point of the bar and supports are taken.a^fed^ene^lts hinged-clamped, clamped-free, clamped-clamped. The results obtained are compared with the system hav^ng constant «oss-section and eaual volume. For example in sımply sSported coîuls with circular cross-section this comparıson sK thtt same buckling load gives 15% relative materıal.VII profit. Similar calculatiohs are also carried out f ör columns subjected to more general loads, whose supports are hinged-hinged and clamped-free. The systems which have small shear stiffnesses have been considered and the effect of the shear deformation on the section variation and buckling load has been investigated in this part as well. in composite columns shear deformations have a great effect on the solution. in this case the value of buckling load decreases. Compared to the system vhere only the bending deformation effects have been taken into consideration, it is only a half. in chapter six, some parametrical investigations have been carried out för rectangular thin.plates. Ali the solutions are found for simply supported rectangular plates with sides a,b subjected to uniformly distribüted Px compressive forces, and the.optimal variations of plate thickness obtained.1 The comparison of a sguare plate having constânt thickness with a plate having optimal thickness variation subjected to the satne buckling load, gives 10,5^%^ relative material prof it. Also the maximiim buckling load and thickness variation are found for a sinply supported plate by changing the ratios a/b and a = ?x/Py. Here the plate is eonsidered as loaded ^ by the forces Px, P in the direetions x aad y -j* - respectively. The makimum buckling loads obtained are»' çonsiderably greater than the buckling load of â flafâe^ç ©f equal volume, with constânt übJUskness. TM «elalâ$â increase is 67% for a/b = l eaaâ o?f l * :': s, J '`-^ £, Similar calculations have also been canda* pöÇ ^fr the plates which are supported sİJ^le-f«e4 elı^te*»f^ ^ :,, and clâmped-simple in the x aad y dirmetieaos g _ j respectively. The results obtained are outliaed İM tatpii and figüres.The finite element method and the numerical method of succesive approximations that has been developed iş applied to the solution öf plate and bar systems using computer programs. The details of these programs are given in chapteT seven. The programs are coded in FOKTKAN IV, and are independent of the.form of the cross-sections, supports and the symmetty jşroperties of the system. The results are sınmnarized in chapter eight.