Krişsiz döşemelerde düşey yükler altında iç kuvvet dağılımı yapı sistemlerinin hesap yöntemlerinin karşılaştırılması

Abstract

ÖZET Yüksek lisans tezi olarak sunulan bu çalışmanın birinci bölümünde kirişsiz döşemelerde düşey uniform yük altında çeşitli kolon boyutlarına bağlı olarak eğilme momentlerinin değişimi Sonlu Elemanlar Yöntemi ile incelenmiştir. ikinci bölümde Yapı Sistemlerinin Hesap Yöntemlerinin Karşılaştırılması oluşturmaktadır. Bu bölümde, günümüzde en çok kullanılan yöntemler gözönünde bulundurularak seçilen örnek bir düzlem sistemin boyutlandırılmasında değişik yükleme durumları için farklı hesap yöntemleri kullanılmış böylece yöntemler değişik açılardan karşılaştırılmışlardır. Bulunan sonuçlara bağlı olarakta, en elverişsiz kesit zorlarına göre betonarme sistemi kesit hesabı yapılmıştır. Bu çalışmada son olarak, sistemin hareketli yüklere göre hesaba bir örnek olmak üzere iki kesitte M, N, T kesit tesirlerine ait tesir çizgileri endirekt deplasman yöntemi ile çizilmiştir.

SUMMARY APPLICATION OF THE FINITE ELEMENT METHOD IN ANALYSIS OF THE SLABS WITHOUT BEAMS SUBJECTED TO VERTICAL LOADS AND COMPARISON OF METHODS OF STRUCTURAL ANALYSIS This study which is submitted as Master Thesis, consist of two parts : 1- Application of the finite element method in analysis of The Slabs Without Beams Subjected to vertical Loads 2-Comparison of Methods of Structural Analysis The finite element technique is relatively new and highly useful method for stress analysis of structural contmua. The method relies strongly on the matrix formulation of structural analysis which had been introduced a few years earlier, mainly as a result of the increasing use of electronic computer. The finite element method is essentially a process through which a continium with infinite degrees of freedom can be approximated to by an assemblage of subregions (or elements) each with a specified but now finite number of unknowns. Further, each such element interconnects with others in a way familiar to engineers dealing with discrete structural or electrical assemblies. The finite element method, like any other numerical analysis technique, can always be modified to be more efficient and easier. As the method is applied in larger and more complex problems, it becomes increasingly important that the solution process remains economical. It is obvious that studies on the best modification of the finite element method to solve Xlllsimultaneous linear and non-linear equations will certainly continue. In the first part of this thesis, an thin plate subjected to uniform vertical load is analysed by the Finite Element Method and the stresses of the joints which are calculated for different support types, are compared with the Turkish Design Code. The numerical results obtained in the course of the study are presented. The basic steps for deriving a finite solution to an equilibrium can be summarized as : - The boundry and the solution domain is subdivided by imaginary lines (or surface) into a finite number of discrete sized subregions or finite elements. - A discrete number of nodal points are established with the imaginary mesh that divides the region. - Evaluation of element stiffness and load terms - Assembly of element stiffness and load terms into an overall stiffness matrix and load vector. - Solution of the resulting linear simultaneous equations for the unknown nodal forces. In this study, firstly, the system is divided into 16 rectangular elements. Every element is numbered. Element stiffness matrix is formed for every element. Secondly, the system stiffness matrix is formed by adding the nodal parameters of elements in the same coordinate. System action matrix is formed by taking loads which effect nodal points. After the system stiffness matrix is formed, the displacements are found by solving the equation shown as follows : [S] [d] + [Po]=0 A constant matrix is formed to find the stresses. This constant matrix is multiplied by the displacements of xivevery element and the stresses, found for every element. If there is two or more elements at a nodal point, the average of stresses are calculated. The stresses which have been found are called as Mx, My and Mxy. After that, the proportion of the stresses Mx and My which have been distributed to column strip and middle strip, are found and they are compared with Turkish Design Code. It has been observed that an increation of reinforcement should be formed along the column strip. In the second part of the thesis, the analysis of a three-span reinforced concrete plan frame subjected to various external effects is presented. Different analysis methods have been used for each external loading. Thus, the application and comparison of these methods have been illustrated. The preliminary cross-sectional dimensions of frame have been determined through the utilization of the Slope-deflection Method. In the preliminary design of the structural system, realistic member sizes can be obtained by decreasing the characteristics strengths of material proportionally since only the dead loads and live loads are considered. In symetrical structures, only half of the unknowns when the loads are symetrical or antisymetrical can be taken into account. Using this symmetry, the results are obtained in an easier way. In the chapter numbered 3.4. of this part, the structure is analysed by the Matrix Displacement Method for dead loads acting on. In the Matrix Displacement Method, the unknowns are the joint translations and rotations. This method is more convenient for those systems having high degree of statical indeterminancy. In the other words, using this method for systems having more members meeting at joints, enables the designer to deal with less unknowns. Although the band width of simultaneous equation is limited and there is no elasticity in choosing the unknowns, generation of stiffness matrix is usually practical because of localized effect, i.e, a displacement of joint effects only the members meeting at the given joint. Thus, the formulation of The Matrix Displacement Method is easier and this method is more suitable for computer programing. xvIn the chapter numbered 3.5. of this part, the structure is analysed by Matrix Force Method for live loads Pi and P2. In Matrix Force Method, the unknowns are the end forces of the members which forms the structure. In this method, a number of forces, equal to the number of unknowns (the degree of indeterminacy) are released. Each release can be made by removing of either support reactions or internal forces. In this method, analysis can be made with lesser unknowns for the system, having more members in a frame. Further, it is possible to obtain equations with sufficient stability and with low band width due to the freedom in choosing unknowns. Forming these equations are systematically formed, however they can be formed automatically because of the elasticity in choosing unknowns. In the chapter numbered 3.6. of this part, the structure is analysed by the Moment Distribution (Cross) Method for lateral loads and uniform temperature change. The uniform temperature change has been taken into account as an external effect on the structure. Uniform temperature change is the temperature change at the centerline of members. As it is known, the analysis of statically indeterminate structures generally requires the solution of the simultaneous equations which correspond to the joint rotations by successive iterations. In the chapter numbered 3.7. of this part, the structure is analysed by Slope-deflection Method for different support settlements as an external effect. In Slope-deflection Method, the unknowns are the rotation of joints and the independent relative displacements of members. The linear simultaneous equations can be obtained automatically. At the end of these calculations, the dimensions of critical cross-section obtained from the preliminary analysis are checked under the most unsuitable loading conditions. These loading conditions are several combinations of different external effects in certain proportions according to Turkish Design Code. In this study, it is observed that the most unsuitable loading condition is obtained from the following combinations : xvi1.4G + 1.6P G : Dead Weight P : Live Load Finally, In the chapter numbered 3.8. of this part, the influence lines for bending moment, axial force and shear force of two given section are obtained by means of the Indirect Displacement Method which is an efficient and reliable method. In the third part of this thesis, the results obtained in the first and second parts of the study are given. xvi i