Elastoplastik şekil değiştiren sistemlerin burkulma sonrası davranışlarının incelenmesi
Başlık çevirisi mevcut değil.
- Tez No: 75445
- Danışmanlar: PROF. DR. ERKAN ÖZER
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Burkulma (fen), Elastoplastik, Şekil değiştirme, Buckling, Elastoplastic, Strain
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Yapı Ana Bilim Dalı
- Bilim Dalı: Yapı Analizi ve Boyutlandırma Bilim Dalı
- Sayfa Sayısı: Belirtilmemiş.
Özet
Yüksek Lisans Tezi olarak sunulan ve“ Elastoplastik Şekildeğiştiren Sistemlerin Burkulma Sonrası Davranışlarının İncelenmesi ”ni konu alan bu çalışma beş bölümden oluşmaktadır. Birinci bölümde, konunun ve ilgili çalışmaların tanıtılması, çalışmanın amacı ve kapsamı yer almaktadır. İkinci bölümde, bu çalışmada kullanılan yük artımı yönteminin dayandığı varsayımlar, yöntemin esasları, matematik formülasyonu ve yöntemin uygulanmasında izlenen yol açıklanmakta ve söz konusu yöntem yardımıyla tek katlı ankastre çelik düzlem bir çerçevenin burkulma sonrası davranışı incelenmektedir. Üçüncü bölümde, ikinci mertebe etkilerinin önemli olduğu narin yapı sistemlerini temsil etmek üzere seçilen tek serbestlik dereceli bir taşıyıcı sistem modeli üzerinde parametrik araştırma yapılmaktadır. Parametrik araştırmada gözönüne alınan taşıyıcı sistem modeli, düşey ve yatay kuvvetler etkisindeki bir konsol kiriştir. Basitlik açısından, sistemin tüm şekildeğiştirmelerinin mesnedindeki bir şekildeğiştiren elemanda toplandığı varsayılmaktadır. Dördüncü bölümde, sekiz katlı tek açıklıklı çelik bir düzlem taşıyıcı sistem modeli çeşitli yöntemlerle boyutlandırılmakta ve boyutlandırılan sistemlerin sabit düşey yükler ve artan yatay deprem kuvvetleri altında birinci ve ikinci mertebe teorilerine göre hesabı yapılarak elde edilen sonuçlar, dayanım ve süneklik açılarından karşılaştırılarak değerlendirilmektedir. Beşinci bölüm, bu çalışmada elde edilen sonuçların değerlendirilmesine ayrılmıştır.
Özet (Çeviri)
The use of elastic-plastic analysis and design methods, which take into account the strength of structural materials beyond the linear-elastic limit, results in more slender structures and therefore more economic solutions. However, due to the large displacements which develop in. slender, structures, the second order effects, gain importance and many times, this non-linear effect should also be considered.. The recent developments in the analysis and design methods, of materially and geometrically non-linear structures help engineers to examine the actual structural behaviour under static, effects and. due. to determine the real collapse safety. A more effective evaluation of this possibility requires the development of these methods in a manner to include the dynamic-characterised earthquake effects and the application of these methods to the earthquakes resistant building design. In this study, the post-buckling. behaviour of materially and. geometrically non-linear structures and the earthquake resistant design of slender structures are investigated in detail. The thesis consists of five chapters. In the first chapter,, after introducing, the subject and. the. related works^ the. scope and objectives of the study are explained. The aim of this study is, a- to expand the applicability of the bad. increments method, developed, lea; the analysis of materially and geometrically non-linear systems in a manner to include the post-buckling behaviour, b- by the use of the load increments method, to examine numerically and parametrically the various alternative approaches which can be applied to earthquake resistant design under certain, ductility level, c- to evaluate the results obtained in numerical studies. The method, of investigation followed, in this study is composed, of these, steps. a- The description of load increments method which is developed for the second- order analysis of elastic-plastic, structures and the expansion of the method, to include the post-buckling behaviour.b- The comparison of first- and second-order elastic-plastic theories on a simple model with single degree, of freedom and parametric research, of various design methods which are based on the second order elastic-plastic theory. Cr Numerical investigations, conducted, on. a. multi-story steel planar frane. d- Evaluation of the numerical results. In the second chapter,, the assumptions,, basic principles and. mathematical formulation of the load increments method used in this investigation are presented and the corresponding. analysis, procedure is. explained. Furthermore^ by the use of this method, the post-buckling behaviour of a single-story steel portal frame is examined. The following, assumptions and. limitations are imposed, in the development of the method. a- The internal force-deformation relationships for steel frame elements under bending combined with axial force are assumed to be ideal elastic-plastic. h- Non-linear deformations are assumed, to be accumulated at. plastic sections while the remaining part of the structure behaves linearly elastic. This assumption is the extension of classical, plastic, hinge hypothesis which, is limited to planar elements subjected to simple bending. c- Yield (failure) conditions may be expressed, in terms of bending, moments and axial force. In this study, the effects of shear forces and torsional moment on the yield conditions are neglected. d- The plastic deformation vector is assumed to be normal to the yield surface, in the case of biaxial bending combined with axial force. e- The second-order theory may be applied to the analysis of slender structures under high axial forces. In the second-order theory, the equilibrium equations are formulated for the deformed configurations while the effect of geometrical changes on the compatibility equation is ignored. f- Changes m me direction of loads due to deflections are assumed to be negligible. g- The structure is composed of straight prismatic members with constant axial forces. The members which do not meet these requirements can be divided into smaller straight and prismatic segments with constant axial forces. h- Distributed loads may be approximated by sufficient number of statically equivalent concentrated loads. In the load increments method used in this study,, the structure is analysed under factored constant gravity loads and monotonically increasing lateral loads. Thus, at the end of this analysis,, the factor of safety against earthquake loads is determined under the anticipated safety factor for gravity loads.In this method the structure is analysed for successive lateral load increments. At the end of each, load increment, the. state, of internal forces, at a. certain critical. section reaches the limit state defined by the yield condition, that is, a plastic section forms. Since. the yield vector is assumed to be normal to the yield, curve^ the plastic deformation components may be represented by a single plastic deformation parameter which, is introduced, as. a new unknown for the next load, increment. Besides, an equation is added to the system of equations to express the incremental yield condition for the last formed, plastic section-. This equation is. linear^ because the yield surface is approximated to be composed of linear regions. Since the system, of equations corresponding, to the previous load, increment has already been solved, the solution for the current load increment is simply obtained by the elimination of the new unknown. In the second order elastic-plastic theory, the structure generally collapses at the second-order limit load due to the lack of stability. This situation is checked by testing.the determinant value of the extended system of equations. If the magnitude of determinant is less than or equal to zero the second-order limit load is reached. During the investigation of the post-buckling behaviour of the system, the analysis is continnet by tracing the formation of the plastic sections and negative lateral load increments, which cause the formation of each, plastic section is determined^ Thus, the part of the load parameter-displacement relationship which corresponds to the buckling, behaviour is obtained. In some cases, the structure may he considered, as. being, collapsed due to large deflections and excessive plastic rotations. The analysis method described herein allows, for the detection of collapse load, caused. by these reasons. At each step of the load increments method, a structural system with several plastic sections, is analysed, for lateral load increment. In the mathematical formulation of the method, two groups of unknowns are considered,, such as a- nodal displacement components, b- plastic deformation parameters, of plastic sections. The equations are also considered in two groups a- The equlibrium equations, of nodes in the directions of nodal displacement components. b- The incremental yield conditions of plastic sections which express, that the states of internal forces at plastic sections remain on the yield surface during a load increment.In the third chapter, parametric investigations are carried, out on a. single degree of freedom structural system selected to represent slender structural systems for which the second-order effects, are significant. The structural system considered in the parametric investigation is a cantilever column subjected to vertical and lateral, loads Ear simplicity,, it is assumed that deformations of the system are accumulated on a deformable element located at the support, as shown in Figure 1. N rigid element (El=°°) deformable element Figure 1 System. andParameters An ideal elastic-plastic behaviour model is adapted for the deformable element. Therefore, under constant axial force, the bending moment-rotation (M-0) relationship is considered to be composed of two line segments as defined below, Figure 2.., OA: M = R>.0 for 0-<Q<>9i AB: M = Mr for a>9i (D (N=No) Figure 2. M-6 Relationship of Deformable Element The typical load parameter-displacement (P-5) diagrams of the structure, obtained through the first- and second-order theories are shown in Figure 3. P Figure 3 Load Parameter - Displacement (P-S) Diagrams XVThe P-8 relationships for the examined structural system can be derived as in the following. In the first order.theory,, before the loads, reach the first-order limit load, the equilibrium equation and the bending moment-rotation relationship for the deformable. element are M = Ph (M : base moment ) ( 2 ) Consequently, the following relationship exists, between the displacement. S=Gh and the load parameter P h2 OA: P = k8 (*=77) (4) When the bending moment at the deformable. element reaches the limiting value of h/L,, the first-order limit load and the corresponding- displacement becomes n M,, Pa Pa. = -r- and &i = ~fr- ( 5 ) h k Then, the load parameter remains, constant AA': P = PL1 (6) as the displacement increases. In the second-order theory,., before the second-order limit load, the equilibrium equation can be stated as M = Ph + N6 (7) Therefore, the following P-5 relationship is obtained OB: P = k8(H>) (8) where. NS N 1 Ph h k (9) In case that the bending moment at the. deformable element is. equal to the. plastic moment, as M = PL2h + N6l2 =M, ( 10 ) XVIthe second-order limit load and the corresponding displacement are found to be Pl2 = Pli(M>) and 8^ = 81,1 = -^ (H) Beyond the second, order limit loa4- increasing, displacements- correspond to decreasing load parameters This part of the P-S relationship can be determined by the use. of the equilibrium equation, writtenfor the. mechanism M = Ph.+ N8=M* (12) In this case, the load parameter - displacement relationship becomes BB': P = Pn-k<tS (S>Si4 (13) The P-5 relationships obtained for the single degree of freedom system are given in Figure 4. Pu= Pli=Pu(1-0) -- T RRST-ORDER SECOND-ORDER El=°° 0=_LJL k h <5, 6u=^L=6^ k R(Tw Figure 4 P-S Relationships, for First- and Second-Order. Theories A parametric investigation is carried out on a cantilever column subjected to vertical and lateral loads^ The aim of this investigation is to evaluate and compare various design methods, m terms of strength and ductility. The structural system and the analysis model are given in. Figure 5.H P=cN- N El h V%7~ P=cM- rigid element (E1=qo) deform able element a) System b) Analysis. Model Figures Structural System. and. Analysis Model The cross-sectional geometry and dimensions are shown in Figure 6. d=2b*- i=2w b=20w Figure 6 Cross-sectional Geometry and Dimensions As seen from the figure that, the cross-sectional geometry is defined in terms of web thickness w. In numerical investigations, it is. assumed that an infinite range, of. steel plates is available i.e. the web thickness w change continuously. In the development of the analysis, made! shown, in. Figure 5.h,. it is. also assumed that the deformations are accumulated on a single deformable element and the remaining parts of. the system behave as. infinitely rigid. The system parameters considered in the numerical investigations are the height of the system h, the vertical loat N and the coefficient c - N (14) which ewpresses the ratio of the lateral earthquake load to the. vertical load.. Eight different systems are defined for various values of the parameters. The systems, and parameters are summarised in Table L XVUITable 1 Systems and Parameters These structural systems, are < -according. to three different approaches. In design calculations, it is considered that lateral load P represents the earthquake efects. Therefore,, the allowable stress for vertical loads + earthquake (HD) taken as ojT = 1.33x1400 = 1862 kg/cm2 for Fe37 steuctural stell. In the following, paragraphs^ the design methods, adapted in this, inyestigatin are explained Design- 1 First-order Design. In this methods the, structural designed, according, to system, is the first-order theory. Thus, it is aimed to obtain a reference solution for the evaluation and comparison of the results obtained, by other methods. In this approach, the following formula is used for stress check M N m °=W+T-°m (15) whereM andN are the bending moment and a*Tl forra Design - 2 Design as per TS648 In this, design, methods the system, is analysed, in accordance with the first-order theory, The second-order effects are taken into account through the application of formula (16) giyen in. TS.648. XIXOd, Cno.Oİ Ota»,- n °*x (1.0- - )o* ?^1.00 (16) Oib Ohx, ^ + - <L00 Gbem Ob (17) are used. Design - 3 Design, with Fictitious. Forces This approach is a method of successive approximations. In this method, in order to represent the^ second order effects^ fictitious lateral forces are applied, to the system. These forces are so determined that the positive and negative areas between the P-5 curve of the first-order theory obtained, for Design- I and. the second order P-5. curve obtained for Design-3 are equal, as shown in Figure 7. In the numerical, investigations carried out in. this study, R structural behaviour factor is taken as R=8. In this methods the stress checks for the internal, forces calculated under vertical and lateral loads and lateral fictitious forces are performed using the formula M N (15) Design- 1 First- ofder Design-3.Second*- opder <5. <5u _L(<5U+R6.) R(5. 6 Z Figure 7 The Equivalence of The Areas Under P-5 Curves The following conclusions can be reached by examining the numerical results of parametric investigations. a- In the. systems considered in. parametric investigations, the «ji parameter which represents the second-order effects is between <|x = 0.062 -0.248 XXcovering a wide range of systems in terms of structural second-order effects. b- In the systems considered,, the. ratio of the second-order limit load, to the firstrorder limit load varies between Pl2/Pl>= 0.752 -0.938 depending on the value of parameter (|> c- Second-order limit load parameters for structural systems, designed, in accordance with TS648 standard are PL2- 1,907-2.596 and always exceed the first order limit load obtained for Design - 1. This, result indicates that, for the systems investigated, the TS648 standard provides sufficient safety against second-order effects. d- However,, the second order P-S relationships obtained as the result of Design - 2 for all systems remain below the diagrams obtained for Design - 3 which is based on the conservation of the consumed eartquake energy.. Therefore, it is concluded that the ductility level corresponding the structural behaviour factor of R=8 cannot be reached thouglu the design based on TS.6.48 standard. e- In the systems designed in accordance with TS648 standards the ratio of the positive energy consumption to the negative energy consumption is between AWV AW" = 0.202 -0.917 f- In the comparison, of result of Design - 2, it is observed that the difference in consumed earthquake energy is between 0.004 - 0.078 depending on the system parameters. g- For Design - 3 in which, the required safety against, second order effects and sufficient ductility level corresponding to R structural behaviour factor are provided, the ratio of displacement which is taken as the base for calculation of. fictitious forces representing the second-order effects to the working load displacement varies between 3 16* = 2.80 -3.14 ~ 3.00 This result shows that,, for the investigated structural systems, the fictitious forces calculated by considening the horizontal displacement that is approximately equal three times the working.load displacement provide sufficient safety and ductility level against second-order effects. XXIlathe fourth, chapter, a single-bay,. eight story steel plane frame is designed by various methods and analysed under constant vertical loads and monotonically increasing lateral earthquake forces. Then,the results ohtained. by the firsts and second-order theories are compared and evaluated in terms of strength and ductility. The structural is considened to be. built in. sismic. zone No. L The local soil group of Z2 and structural behaviouar factor of R=8 are anticipated in the design and analysis. The structural is made of fe37 structural steel with a yield stress of 0a =2.4 t/cm2. The structure is designed by three different approaches : Design - 1 First-order Design Design - 2 Design as per TS648 Design - 3 Design, with. Fictitious Forces In this numerical investigation, the Design - 3, is applied as follows. Firstly, the structural system is designed according to the first order theory (Design - 1) and 8o relative story drifts developed under working loads are calculated. Then,, six.different designed, are performed, by taking, into, account. 1,2^_,6 times the relative story drift, i.e., 80, Z80,...G80 respatinely. It is noticed that, due to. the fact that. the structural is a discrete, system and the size of rolled cross-sections does not change continonsly^ some bean and columa cros-section are abtained for a- 2So and 38o b- 48o,.5So and 68o thus, the three different set of bean and column cross-sections obtained for Design-3 correspond to average story drifts of 80, 2.5So and 58o, respeatively. The following, conclusions are reached through the numerical investigation. a- The ratio of second-order limit load to the first-order limit load is PL2/Pli= 0.751 for design 1. b- The Second. order limit load, parameter obtained for the structural which is designed in compliance with TS648 standard, takes the value of PL2 = 1.942 xxiand is below the first-order limit load obtained for Design - 1. This result shows that, for the structural, inmestigated,. the design according, to TS.64& standard cannot provide the required safety and ductility against second-order effects. c- Similarly it is. found, that the second-order, limit loads, which, are calculated for 80 and 2.58o in the same system, are below the anticipated value. d- On the other handy. in. the design performed by considering, the fictitious, forces, five times the 80 displacement, sufficient result can be obtained in terms of both safety against, second-order limit load and ductility level. e- The design performed in compliance with TS648 standard (Design - 2), it is observed that the variation in the consumed earthquake energy takes the value of 0.122 The sixth, chapter covers the conclusions The basic features, of the investigation performed in the study and the general evaluation of numerical results are presented in this, chapter. xxii
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1998
İnşaat Mühendisliğiİstanbul Teknik ÜniversitesiYapı Ana Bilim Dalı
PROF. DR. ERKAN ÖZER
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