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Elastik zemine oturan ızgara sistemlerinin hesabı ve yapı sistemlerinin hesap yöntemlerinin karşılaştırılması

Analysis of grid systems on elastic foundation comparision of methods of structural analysis

  1. Tez No: 66751
  2. Yazar: GÖKHAN YAZGAN
  3. Danışmanlar: PROF. DR. AHMET IŞIN SAYGIN
  4. Tez Türü: Yüksek Lisans
  5. Konular: İnşaat Mühendisliği, Civil Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1997
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: İnşaat Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Yapı Deprem Mühendisliği Bilim Dalı
  13. Sayfa Sayısı: 142

Özet

ÖZET Yüksek Lisans Tezi olarak sunulan bu çalışma iki ana bölümden oluşmaktadır.“Elastik Zemine Oturan Izgara Sistemlerin Hesabı”ve“Yapı Sistemlerinin Hesap Yöntemlerinin Karşılaştırılması”. Birinci bölümde, elastik zemine oturan örnek bir yapı temelinin elastik zemine oturan kirişlerinde burulma rijitliği ve burulmaya karşı elastik zemine etkilerinin dikkate alınması veya terk edilmesi halinde verilen düşey ve deprem yükleri altında moment diyagramları çizilmiştir. İkinci bölümde, yapı sistemlerinin hesap yöntemleri, örnek olarak seçilen üç açıklıktı bir düzlem çerçeve üzerinde çeşitli yükleme durumları için farklı hesap yöntemleri kullanılarak karşılaştırılmıştır. Önce Açı Yöntemi yardımıyla yapının ön boyutlandırılması yapılmıştır. Daha sonra sırayla, yapı yükleri için Açı Yöntemi, P1, P2, P3 ilave yükleri için Matris Deplasman Yöntemi, W (deprem) yükü için Cross Yöntemi, düzgün sıcaklık değişmesi için Matris Kuvvet Yöntemi ve mesnet çökmeleri için de Cross Yöntemi kullanılarak kesit tesirleri bulunmuştur. En elverişsiz iç kuvvetler düzenlenen bir süperpozisyon tablosu yardımıyla bulunmuş ve ayrıca seçilen iki kesite ait M, N, T tesir çizgileri Endirekt Deplasman Yöntemi kullanılarak çizilmiştir. ix

Özet (Çeviri)

SUMMARY This study which is submitted as Master Thesis, consists of two parts. 1. Analysis of Grid Systems on Elastic Foundation 2. Comparison of Methods of Structural Analysis First section of this study which is presented as Master Thesis, has been constituting of solution of grid base system of a building resting on elastic foundation, under vertical and lateral loads. The grid system has been analyzed by either ignoring or considering the torsional rigidities of beam and elastic foundation effects on the beams resting on the elastic foundation. Beams on elastic foundations are widely used as structural elements in engineering application. Assuming that the base is consisted of closely independent linear springs Winkler provided the simplest representation of a continuous elastic foundation. The relation between the pressure and the deflection of foundation surface, both parallel to the z axis is given as r(x) = c.w(x) where c is the modulus of subgrade reaction. The bar element of beam has two nodes. At each node one displacement, two rotations, one shear force, one twisting, one bending moments are defined as a total of six unknowns. When there is no load on it, differential equation belonging to elastic curve of i-j prismatic element given in Figure 1.3 is, d“w r(x) C Solution for this homogenous equation is W = C,.Coshyx. CosYx + C2.Sinhyx.Cosyx + C3 Coshyx.Sinyx+ C4Sinhyx.SinyxHere, d, C2, C3, C4 are the integration constants; y is a constant that may be defined as._4CH ~”V4EI 4EI Bending moment and shearing force d2w M(x) = -EI T(x) = -EI dx2 d3w dx3 with the help of above equations the [k] unit displacement matrix which connects all end forces to the end displacements of bars. In the calculation of unit displacement matrix, the terms of kl 1, kl2, kl4, kl6, k44 and k46 have been calculated using the following equations. _EI?, SinhylCoshyl -SinylCosyl 11 ~ I Y Sinh2yl -Sin2yl k“=-r2rl EI CoshylSinyl SinhylCosyl 1 ' Sinh2yl-Sin2yl k,4~l2 {y} ~Sinh2yl-Sin2yl _EI 2Sinh2yl+Sm2yl kl6”“l2AYU Sinh2yl -Sin2yl - -d( l \3 SinhYİ CoshY' + S”1“/1 CosT' I3 Sinh2yl -Sin2yl _ EI. 3 Coshyl Sinyl + Sinhyl Cosyl Sinh2yl -Sin2yl If ground elastic to bending (twisting) on beams setting on elastic ground is considered in solution, terms belonging to unit displacement matrixes; may be found by the help of the above formulas. XIk* = k^GJ^ÜCh(X1) k35-k53-GJSh(X1) When ground effects elastic to twisting is left, the terms has been taken as ”0". Unit displacement matrixes found by the help of the above formulas, have been transferred to a common coordinate axis selected for all sticks combining at knot points. The matrix of system rigidity is obtained by placing the unit displacement matrixes which is produced before hand for x axis set into the locations at the system rigidity matrix. Superimposed units are added on. On the example given is this thesis, the symmetrical characteristic of the building is taken into account while producing the system rigidity matrix. Hence, the system rigidity matrix has been arranged for both ways by considering the fact that the lateral load which produces symmetrical axis for vertical loads which also built antimetrical axis. Besides, in case that the torsional effects would be avoided, because the unit of k33, k35, k53 and k55 have been equitated to zero value, the alteration of system rigidity matrix had been taken into the consideration. Using the equation [S] [d] = [Po] where, [S] : system stiffness matrix [d]: displacement matrix [P0] : the nodal loads matrix we obtain the displacement matrix [d]. By the help of the formulas that is [PJ^tkUtd^ + M^td]^ [P^M^td^+M^td]^ form these edge displacements, it has been passed to edge end forces by calculating the system matrix by the help of transferred unit displacement matrixes. Moreover, depending on these end forces; with the help of formula, Mmiddle = (Shyl+Sinyl)' yl. yl Sh7'Sm y _ ( yl yl yl yl.(P4-P6)+ Sh-i-Cos-r+Chi-.Sin^- L(P2-Pt) y v 4 *' V 2 2 2 2 xuthe momentums in the midlest of clearance of bars can be easily found. According to the end forces and clearance middle momentums the system momentum diagrams have been illustrated. This study results the following: - When the torsional rigidities are ignored, no significant behavioural change is expected in the system, at the same time the diffusion rate of bending moments remained almost the same during the application. - When the foundation system is idealized as a grid frame, for the seismic loads the longitudinal and transverse beam moments considerably decrease with respect to the continous beam idealization. In the first chapter of the second section, the analysis of a three span reinforced concrete plane frame subjected to various external effects is presented. Various structural methods have been used for each different external effect. Thus, the applications 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. As a result of this part, a suitable result is obtained in pre-designing of structural systems as decreasing the characteristic resistance of material at the same proportions while only the dead loads are considered. In symmetrical structures, only half of the unknowns when the loads are symmetrical or antisymmetrical can be taken into account. Using this symmetry, the results are obtained in an easier way. In the second chapter of this section, the structure is analyzed by Slope Deflection Method for dead weight acting on the structure. 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. In the third chapter of this section, the structure is analyzed by Matrix displacement Method for live loads PI, P2, and P3. In the Matrix Displacement Method, the unknowns are the joint translations and relations. This method is more useful for the systems having high degrees of statically indeterminacy. In other words, if systems having various members meeting at joints of the systems, this method is advantageous since it operates with lesser unknowns. Although the band width of simultaneous equation is limited and there is no elasticity in choosing the unknowns, generation of the stiffness matrix is usually not difficult for the members meeting at the given joint. Thus, it is easy to formulate the Matrix Displacement Method and besides all these, this method is more suitable for computer programming. The linear simultaneous equations are obtained automatically and solved by the help of computer. In the fourth chapter of second section, the structure is analyzed by the Cross Method for lateral loads. The unknowns in this method, are relations of joints and independent translations of the member end, as is in the Slope Deflection Method. xuiIn the fifth chapter of this section, the uniform temperature changes have been taken into account as an external effect on the structure. Uniform temperature change is considered as the temperature change acting at the centerline of the members. Because of this effect, some internal forces acting on the cross-sections of statically indeterminate structure occur. To determine these forces the structure is analyzed by the Matrix Force Method. The unknowns are the forces acting at the ends of the members, which have formed the structure. In this method, first a member force released which are equal to number of unknowns in the Force Method (the degree of indeterminacy). Each release can be made by the removal of the either support reactions or internal forces. Due to this property, the analysis can be made it lesser unknowns for the systems, having more members in a frame. In addition, it is possible to obtain equation is stable, by means of freedom in choosing unknowns. These equations, however, are written systematically even they can be desired automatically. In the sixth chapter of this section, the structure is solved for different support settlements. The structure is analyzed by Cross Method again. The unknowns in this method, are the rotations of joints and independent translations of the member ends, as it is mentioned above. At the end of these calculations, the dimensions of the critical cross-sections, which are obtained from the preliminary analysis, are checked and compared under the most unsuitable loading conditions. These loading conditions are some combinations which consider different external effects acting in certain proportions according to Turkish Design Code ?1.4G+1.6Q *G+Q+E *G+1.2Q+1.2T Where; G: Dead weight Q: Live load E: Lateral load T: Uniform temperature change and support settlement load Consequently, in the second section, 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. xiv

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