Çok katlı düzlem çerçevelerin yaklaşık ikinci mertebe hesabı
Başlık çevirisi mevcut değil.
- Tez No: 39432
- Danışmanlar: PROF.DR. NAHİT KUMBASAR
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 100
Özet
Bu çalışmanın amacı, taşıyıcı sistemlerin iç kuvvet ve deformasyonları- nm matris deplasman metodu ile çözülmesi ve bu konu ile ilgili bir program geliştirilmesi üzerine dayalıdır. Hiperstatik sistemlerin hesap metodlarım, başlıca 3 ana bölüme ayırabi¬ liriz; a- Kuvvet Metodu b- Deplasman Metodu c- Başlangıç Değerleri Metodu(Travers Metodu) Diğer metodlar bunlara ait denklemlerin kuruluş ve çözülüşündeki farklar¬ dan kaynaklanmaktadır. Bu çalışmamda deplasman metodu seçilmiştir. Yön¬ temin amacı, dış etkilerden meydana gelen uç kuvvetleri ile uç deplasman¬ larını hesaplamaktır. Çünkü bunlara bağlı olarak iç kuvvetler, yer değiştirme¬ ler, şekil değiştirmeler bulunabilir. Tezin amacı, sistemi önce birinci mertebe teorisiyle çözmek, sonra elde edilen normal kuvvetlere bağlı olarak, ikinci mertebe teorisiyle hesaplamak ve burkulma yükünü tespit etmektir. Tezin birinci bölümünde, matris deplasman yöntemini seçmemizin sebepleri ana hatlarıyla açıklanmıştır. ikinci bölümde, yöntemin tanımı ve özellikleri belirtilmiştir. Üçüncü bölümde, 1.ci mertebe teorisi, dördüncü bölümde ise 2.ci mertebe teorisine göre yöntemin nasıl uygulandığı anlatılmıştır. Beşinci bölümde ise sistemin burkulma yükünün nasıl tayin edileceği açıklanmıştır. Altıncı bölümde bilgisayar programının içeriği, akış diyagramı, yedinci bölümde programla çözülen uygulamalar gösterilmiştir. Sekizinci bölümde ise sonuçlar ve öneriler yer almaktadır.
Özet (Çeviri)
in structural engineering, both safety and economic factors are considered in structure design. Therefore, these two basic factors considerably effect eachother. Safety factor was the most important aspect in the design of structure due to the indeterminacy of the real behaviour of structures, before the use of computer technotogy in structural engineering. The behaviour of structures is determtned more predsdy, because of the devetopment of structural analysis methods and computer technoiogy. So, the problem of economical design becomes more important. For this reason, structural engineers use the design methods which consider both material and geometrical non-linearities. in this study, matrbc disptacement method is prefered. The unknowns are displacements and rotations of the joints. This method is more useful tor the systems having high degrees of statically indeterminacy. in other words, method enabtes to operate wfth tesser unknowns in systems having more members unrted at joints forming the system. Also, this method can be appfied to every kind of support systems such as planar trusses, planar frames, planar beams. Although the bandwidth of simuttaneous equation is limited and there is no flexibility in choosing the unknowns, equations can be obtained automatically these equations enabte to have a good stability. But when the equations are not good in stability retuming is nestricted, because of lackness elasticity in choosing unknowns. The displacement of a joint effect only the members meeting at the given joint Therefore, it is easy to formulate the Matrix Displacement Method which is availabte tor computer programming. This study which has been carried out for M.Sc. thesis, consists of 8 chapters. in the first chapter, the reasons why we prefered matrix displacement method, are explained generally. in the second chapter, the principals of the method are mentioned.in the third chapter, the solution of a planer framed structure in accordance to the linear theory ( first order theory ) wfth matrix displacement method is explained. in the fourth chapter, the solution of a planar framed structure in accordance to the second order theory wrth matrix displacement method is explained. A main difference between second order theory and fırst order theory is that the equilibrium equations are written at the structures which are deforrned, in the second order theory. Therefore, when we evaluate the stiffness matrix, vve conslder the effect of axial forces. But, we don't know the value of axial forces at the begining. So we estimate the axial forces step by step until we get the nearest values. in the fifth chapter, the buckling toads are calculated in the planar framed structures. Here, we use 2 methods; a- Definite Method (Determinant Criterion Method) b- Approximate Calculation Method We use determinant criterion method in systems which do not defomn before buckling (undeformed systems). We use approximate calculation methods in systems which defoma before buckling (deformed systems). in the sbcth chapter, the program which is vvritten in basic language is presented. We explain how to solve the planar framed structures by the program. Especially, ftow chart diagrams and the data files are indicated. in the seventh chapter, the applications which are solved by the program are explained. in the eighth chapter, the resutts and the comments are expressed. Many structures of engineering interest may be considered as an assemblage of line (one-dimensional) members, that is parts whose lengths are large compared wrth their other dimensions. The locus of the centroids of the cross section of a line member is called axis of the line member. This axis can be a straight line ör a curve. Moreover, the dimensions and orientation of the cross sections of a line member can change along its length. A line member is referred to as prismatic ör cylindrical when the dimensions and orientation of its cross sections do not change along its length. Structures made up of line members joined together are called framed struc¬ tures. l n this thesis,we limit our attention to framed structures whose membershave a constant cross section. Moreover, we consider primarily structures wfth straight-line members. Curved-line members of framed staıctures can be approximated by a model consisting of a series of straight-line elements(see Fiği) Straight-line member Curved-line memberFigüre 1 Approrimation of a curved- line member by a series of straight-line elements. The directions of the principal centroidal axes of the cross sections of each straight-line elements are considered constant and egual to those of the cross section at its öne end. it is evident that as the number of straight-line elements of this model increases, the model's properties approach those of the actual curved member. in general, framed structures have a three dimensional configuration. Often, however, for purposes of analysis and design, a framed structure may be broken down into planar parts vvhose response can be considered as two-dimensional. They are called planar framed structures. The axes of their member lie in öne plane and moreover they are subjected to extemal disturbances (such as forces and moments) which do not induce movement of their particles in the direction normal to their plane. Furthermore.one princi¬ pal centroidal axis of the cross section of a planar structure is normal to its plane. There are several assumptions that form the basis of the mechanics of materials theories which are employed in the analysis of framed structures. Some of them are as follow; 1- in the mechanics of materials theories, the components of stress at any point of a cross section of a line member are expressed in terms of thecomponents of the resultant force and moment acting on this cross section by simple relations. 2- In the mechanics of materials theories, it is assumed that plane sections normal to the axes of a member remain plane after deformation. Consequently, the movement of a cross section of a member of a planar framed structure, due to its deformation, is specified by the two components (in the plane of the structure)of the displacement vector of its centroid and by its component of rotation about the axis normal to the plane of the structure (see Fig.2). The components of the displacement vector of centroid of a cross section of a member of a structure are referred to as the components of translation of this cross section. Undeformed plane section *1 Deformed plane section u/A), U3(A) = component of translation e (A) component of rotation Figure 2 Components of displacements of a cross section of a planar structure. In the analysis of framed stuctures we are interested in establishing ; 1- The components of internal forces and moments acting on their cross sections 2- The components of translational and rotation of certain of their cross sections. The components of internal forces and moments acting on the cross sections of a member of a framed structure and the components of translational and rotation of its cross sections are functions only of its axial coordinate. When XII-analyzing framed structures using the methods presented in this study, the framed structures are subdivided into line elements whose ends are imagined as being connected to a number of points called nodes. The ends of an element are refered to as its nodal points. Thus, a line element extends between two nodes, and it is either a member of the structure or a portion of a member of the structure. The nodes of a structure are its joints, its supports, the free ends of its members, and any other points that we may choose along the length of its members. As a rule, we choose the smallest number of nodes required for the analysis of a structure. For instance, the smallest number of nodes for the beams of Fig. 3 is the sum of its support points(1,3 and 6), the two points on each side of the internal rollers, and the point where the external force is applied.The latter has been chosen because we want the external forces to act at nodes of the beam. The nodes and the elements of a structure are numbered consecutively and the number of each element is placed in a circle, as shown in Fig.3. Moreover, the ends of each element are denoted by i and j (i being the end of the element connected to the node having the smallest number). It is preferable that the nodes of framed structures are numbered so that the difference between the numbers of the nodes at the ends of each element is as small as possible. After the nodes and the elements of a structure are numbered, their connectivity can be expressed, for example as shown in Table 1 for the structure of Fig.3. We refer each framed structure to a right-handed rectangular system of axes ( cartesian axes ) xj, x2,x3 called the global axes of the structure. Moreover, we refer each element of a structure to a right-handed rectangular system of axes x1( x2, x 3 called its local axes. As the local axes of an element, we choose the set of axes whose origin is the centroid of the cross section at the end i of the element ; its x^ axis is directed along the axis öf the element from its end i to its end j; its x2 and x3 are the principal centroidal axes of the cross section at the end i of the element see Fig.4. -xm-Figüre 4 Global axes of a planar frame and local axes of element 3 Table 1 Connectivity of the Elements and Nodes of the thestructureoffig. 3The stiffness equations for an element: We can express the matrix of nodal actions of an element as {A}={AE} + {AR}(1) where {AE} is the matrix of nodal actions of the element when subjected only to its nodal displacements, and { AR) is the matrix of nodal actions of the element when subjected to the given external disturbances with its ends fıxed. { AR} is called the matrix of fixed-end actions of the element. We express the local components of the nodal displacements as a linear combination of its nodal displacements. That is, {AE> = [K]{D}(2) Relations (2) are called the stiffness equations for the element.The matrix [K] is called the local stiffness matrix for the element, its terms are called the local stiffness coefficients for the element.The local stiffness coefficient K^ represents the nodal action Am (the action in the m th row of the matrix {AE}) of the element when it is subjected only to a nodal displacement Dn=1, while ali other nodal displacements vanish ( Dn is the displacement in the n th row of the matrix {D}). Local stiffness matrix for an element of a planar beam ör a planar frame: The physical significance of the stiffness coefficients of the second column of the stiffness matrix for a general planar element can be established by considering such an element subjected only to the components of nodal actions which are required to induce the following nodal displacements (see Fiğ 5).Where L = Length of element A = area of cross section of element l = moment of inertia of cross sectton of element about its x local axis E= modulus of elasticfty of material from which element is made [ 1 ].
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