Yapı sistemlerinin dinamik dış etkiler altındaki davranışlarının incelenmesi
Başlık çevirisi mevcut değil.
- Tez No: 75328
- Danışmanlar: PROF. DR. ERKAN ÖZER
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- 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: 129
Özet
ÖZET Yüksek lisans tezi olarak sunulan bu çalışmada, dinamik dış etkiler ve deprem etkileri altındaki yapı sistemlerinin lineer ve lineer olmayan davranışları incelenmiştir. Sekiz bölüm ve ek bölümünden oluşan çalışmanın birinci bölümünde konunun tanıtılması, konu ile ilgili çalışmaların gözden geçirilmesi, çalışmanın amacı ve kapsamı yer almaktadır. İkinci bölüm, yapı sistemlerinin dinamik dış etkiler altındaki davranışlarının incelenmesine ayrılmıştır. Bu bölümde, tek ve çok serbestlik dereceli sistemler için ayrı ayrı olmak üzere, dinamik dış etkiler ve deprem yüklemelerine ait hareket denklemleri elde edilmiştir. Ayrıca, kayma çerçevesi idealleştirilmesi yapılan çok katlı çerçeve sistemlerde yatay rijitlik matrisinin nasıl belirleneceği açıklanmıştır. Üçüncü bölümde, ilk olarak dinamik dış etkiler altındaki yapı sistemlerinin hesabı için pratikte uygulanmakta olan yöntemler özetlenmiş, daha sonra bu yöntemlerden zaman artımı yöntemi açıklanmıştır. Bu çalışmada uygulanan yöntemde, geleneksel zaman artımı yönteminden farklı olarak, yerdeğiştirme, hız ve ivme farklarına ait rekürans formülleri kullanılmakta; böylece ardışık yaklaşım yapılmasına gerek kalmamaktadır. Bu bölümde, sözkonusu formülasyona ait bağıntılar da elde edilmiştir. Yöndeğiştiren yükler altındaki iç kuvvet-şekil değiştirme bağıntıları ve davranış modellerine ayrılan dördüncü bölümde, özellikle ideal elastoplastik davranış modeli ve bu modelin çerçeve sistemlere uygulanması açıklanmaktadır. Beşinci bölümde, bu çalışma kapsamı içinde hazırlanan SPEC bilgisayar programından yararlanarak, Erzincan 1992 depremi kuzey-güney ve doğu-batı bileşenleri için elde edilen yerdeğiştirme,hız ve ivme spektrumları verilmiştir. Altıncı bölümde, bu çalışmada oluşturulan algoritmanın pratik uygulamaları için hazırlanan EPFOR bilgisayar programının dayandığı varsayımlar, çalışma düzeni, giriş ve çıkış bilgileri açıklanmıştır. Sayısal uygulamalara ayrılan yedinci bölümde, iki taşıyıcı sistem modelinin deprem etkileri altındaki lineer ve lineer olmayan davranışları incelenmiş ve sayısal sonuçlar değerlendirilmiştir. Sekizinci bölümde, bu çalışmada elde edilen sonuçlar açıklanmıştır. Çalışmanın Ek A bölümünde, Erzincan 1992 depremi için beşinci bölümde verilen spektrum eğrilerine ait sayısal değerler topluca yeralmaktadır.. xııı
Özet (Çeviri)
LINEAR AND NON-LINEAR BEHAVIOR OF STRUCTURAL SYSTEMS SUBJECTED TO DYNAMIC EFFECTS SUMMARY As the dwelling areas in our country are generally located on earthquake hazardous zones, the earthquake resistant and economical design of building structures becomes more important. The basic components of earthquake resistant and economical design are as follows: a) realistic prediction of earthquake effects which may act on structures during their life times, b) determination of performance criteria and necessary characteristics of structures which optimally satisfy the safety and economy requirements under these effects, c) developing advanced methods of analysis which enable the engineers to determine the actual dynamic behavior of structures. In this study, dynamic responses of single and multi degree of freedom systems which are made of either linear-elastic or ideal elastic-plastic material and subjected to dynamic loads and seismic effects are investigated by using the time-history analysis method based on a numerical integration technique. The thesis consists of eight chapters. In the first chapter, after introducing the subject and related works, the scope and objectives of the study are explained. The main objective of this study is to investigate the linear and non-linear dynamic responses of structures under seismic loads. For this purpose, computer programs based on the method of time increments are developed and numerical examples are given to illustrate the method. In the second chapter, the equations of motion for single and multi degree of freedom systems subjected to dynamic external forces or earthquake excitations are derived. (a) Figure 1 Single Story Frame (b) XIVThe system considered is shown schematically in Fig. 1. It consists of a mass m concentrated at the roof level, a massless frame that provides stiffness to the system and a viscous damper that dissipates vibrational energy of the system. The number independent displacements required to define the displaced positions of all the masses, relative to their original position is called the number of degrees of freedom (DOF's) for dynamic analysis. The static analysis problem has to be formulated with three DOF's, such as lateral displacement and two joint rotations to determine the lateral stiffness of the frame. In contrast, the structure has only one DOF, such as lateral displacement, for dynamic analysis if it is idealized with mass concentrated at one location. Thus we call this is a single degree of freedom (SDF) system. Two types of dynamic excitation are considered : 1) external force f acting in the lateral direction, 2) earthquake-induced ground motion dg(t). In both cases d denotes the relative displacement between the mass and the base of the structure. Figure la shows the idealized one-story frame subjected to an externally applied dynamic force p(i) in the direction of the DOF. This notation indicates that the force p varies with time t. The external force is taken to be positive in the direction of the x-axis, and the displacement d(t), velocity d(t), and acceleration d(t) are also positive in the direction of the x-axis. The elastic forces (fs) and damping force (fD) are acting in the opposite direction because they are internal forces that resist the deformation and velocity, respectively. The equation of motion under the force p(t) for a single degree of freedom system becomes,... md(t)+cd(t)+sd(t) = p(t) (1) where c is the viscous damping coefficient and s is the lateral stiffness coefficient, respectively. This is the equation of motion governing the deformation or displacement d(t) of the idealized stucture of Fig la, which is assumed to be linearly elastic and subjected to external dynamis forces of p(t). For inelastic systems, the equation of motion is md(t)+cd(t)+fs(d,d) = p(t) (2) where fs(d,d) is the force fs coresponding to deformation d and depends on the history of deformation and on whether the deformation is increasing (positive velocity) or decreasing (negative velocity). xvFigure lb shows the behavior of SDF system subjected to earthquake-induced motion of the base. The displacement of the ground is denoted by dg (/), the total displacement of the mass by d,(t) and the relative displacement between the mass and ground by d(t) at each instant of time. These displacements are related by d,{t) = d{f) + dg{t) (3) The equation of motion governing the relative displacement d(t) of a single degree of freedom system, subjected to ground acceleration dg{t) is m d(t) +cd(t)+s d{t) =-m d At) (4) for linearly elastic case, and m d(t) + cd(t)+fs(d,d) =-m d(t) (5) for inelastic case. The equation of motion for a linearly elastic multi degree of freedom system subjected to earthquake excitation can be written in matrix form as (6) where [U] is a vector of order N with each element equals to unity. Equation (6) contains N differential equations governing the relative displacements d(f) of a linearly elastic MDF system subjected to ground acceleration ds (t). The stiffness matrix [S] in Eq.(6) refers to the horizantal displacements d and can be obtained through the static condensation method by eliminating the rotational and vertical DOF's of nodes. Hence, this matrix is known as the lateral stiffness matrix. The stiffness matrix [s] is constant for linear-elastic systems. However, for inelastic systems, the stifness matrix changes with displacement and velocity. The third chapter outlines the general analysis methods applied to structural systems subjected to time varying effects such as earthquake and wind. A group of these methods are called as time-stepping methods. After stating the basic assumptions, the governing equations of the method are given for single and multi degree of freedom systems. xviAnalytical solution of the equation of motion for a SDF system is usually not possible if the excitation - applied force p(t) or ground acceleration i?s(7) -varies arbitrarily with time or if the system is nonlinear. Newmark developed a family of time-stepping methods based on the following equations d(t + At)=d(t) + At d(t) + ((0.5- p)At2)d(t) + (/3 At2) d(t + At) (7a) d(t + At)='d(t) + ((\-y)At)d(t) + iyAt)d(t + At) (7b) The parameters f3 and y define the variation of acceleration over a time step and determine the stability and accuracy characteristics of the method. Newmark' s equation with y = \l2 and /? = l/6 corresponds to the assumption of linear variation of acceleration, over a time step. These two equations, together with the equation of motion written for t + At, provide the basis for computing d(t + At), 'dit + At),d(t + At) at time t + At from the known d(t),d(t) and d(t) values attune /. Iteration is required to implement these computations because the unknown 'dit + At) appears in the right side of Eq.(7). For linear systems it is possible to modify Newmark's original formulation, however, to permit solution of Eqs.(7) and (1) without iteration. In this thesis linear acceleration method is adopted for solving the equation of motion. In the formulation of the method, the incremental quantities of displacement, velocity and acceleration are considered as unknowns, to avoid iteration. The incremental quantities and the equations of motion at times / and t + At are m d (t) + c d (t) +sd(t)= p(t) (8)... md(t + At) +c d (t + At)+sd(t + At) = p{t + At) (9) Ad = d (t + At) -d(t) (10a) Ad = d{t + At)-d(t) (10b) Ad = d(t + At)-d(t) (10c) While the incremental form is not necessary for analysis of linear systems, it is introduced because it provides a convenient extension to nonlinear systems. The incremental equation of motion m Ad+c Ad+s Ad = -mAds, (11)is obtained by subtracting Eq.(8) from Eq.(9). Similarly, the following expressions are obtained through Eqs. (7) and (10) for ^ = 1/2 and £ = 1/6“6 6 * Ad^^Ad-^-d-ld (12) At2 At 3 ' At ”Ad=-Ad-3d--d (13) At 2 The substitution of Eqs. (12) and (13) into Eq. (1 1) gives K Ad=AP (14) where and“ 6 m 3 c.... K= - r + - + s (15) At2 At ”I 6 ]. A/ ** Af^-mA^ + i - m + 3cW+(3w + - c) d (16) 8 A/* 2 With J£ and A P known from the system properties m, c and s, and d. and J values at the beginning of the time step, the incremental displacement is computed from Ad=- (17) AP Once Ad is known, Ad and A*â can be computed from Eqs.(12) and (13) respectively, and d(t + At), 'd{t + At),'d{t + At) from Eqs.l0a,b,c. The acceleration can also be obtained from the equation of motion at / + At 1 d(t + At) =- (p(t + At)-cd(t + At)-s d{t + At)) (18) m In the fourth chapter, various behavior models of material for cyclic and repeated loadings are presented. Characteristics and mathematical formulations of the models investigated in this study are given as in the following.Figure 2 Linearly Elastic Behavior Model The linearly elastic behavior model, for which the loading curve coincides with the unloading curve, may be expressed by a linear equation as V = sd (19) When the material behavior is assumed to be ideal elastic-plastic, the loading and unloading curves consist of linear segments, as follows: A' A: AB: A'B': V = sd for -dy2=- £ = -^- (23) m ima d(t) + 2£y d(t)+ cû2 d{t) = -dg (/) (24) In the application of the time-stepping method, which is explained in chapter two, the following formulation is used: Ad + 2Çco Ad+û)2 &d = -&d (25) &d=^Ad--'d(t)-3d(t) (26) At2 At Ad=-Ad~3d(t)-~d(t) (27) At 2 xxK= 6 +öj2 + 6^ (28) Ar At AP = -Ad]+ j- + 6Çü)\'d(t) + (3 + ÇcoAt)d(t) (29) KAd = Ap (30) The basic relationships which exist between the spectrum values of displacement, velocity and acceleration are as follows: S^^^d^ğJ)^ (31) SV{Ç,T) = d{t,Ç,T)^ = û)Sd(Ç,T) (32) SMJ) = dit&T)^ + Ut(t) = Sd{ğ,T) (33) In order to illustrate the application of the algoritm and computer program (SPEC) developed for the construction of response spectrum, the displacement, velocity and acceleration spectrum diagrams obtained for east-west and north-south components of Erzincan 1992 earthquake are presented. Most buildings are expected to deform beyond the linear-elastic limit, when subjected to strong ground motion. Thus the earthquake response of building structures deforming in the inelastic range is of central importance in earthquake engineering. In the sixth chapter, it has been aimed to study the linear and nonlinear behavior of structural systems subjected to dynamic forces and earthquake excitations. For this purpose, a computational procedure has been developed for the linear and non-linear analyses of planar frame structures exposed to earthquake excitations. The time-history analysis procedure is based on the time-stepping method explained in chapter three. Also, a computer program named as EPFOR has been developed for the practical applications of the procedure and coded in FORTRAN 77 programming language. The program is executed in a batch program structure. The following assumptions and limitations are imposed throughout the study. a) The structural system is a single- or multi-story plane frame with masses concentrated at floor levels. b) The shear frame idealization is made. In this idealization, the lateral stiffness of each story is assumed to be proportional to the sum of lateral stiffness of story columns. Besides, the interection between stories is ignored. c) Damping force at each floor is considered to be proportional to the floor velocity. xxid) Axial deformations of columns are neglected. e) Only linear-elastic and ideal elastic-plastic behavior models are considered in the development of the computer program. However, other nonlinear behavior models can be easily incorporated in the program. The main purpose of EPFOR computer program is to determine the complete dynamic behavior of linear-elastic and ideal elastic-plastic structural systems under real and simulated ground motions. The output data covers the story lateral displacement and story shear calculated for each time increment. Two numerical examples are given in the seventh chapter. In these examples, the structural systems which have been designed in accordance with the current Turkish Standards and the Seismic Code, are examined in detail. In example one, a single-story steel plane frame is analyzed by considering both linear-elastic and ideal elastic-plastic behavior models. In example two, the linear-elastic and ieal elastic-plastic behavior of a five-story steel plane frame subjected to earthquake acceleration of dg(t) is investigated. The results are organized as composed of the following: a) the total and relative story drifts, b) the story shears, c) the relation between story shear and story drift as the structure goes through several cycles of motion. In addition, using the results of earthquake response analysis, story and system ductilities are discussed. The eighth chapter covers the conclusions. The basic features of the time-stepping procedure developed in the study, the evaluation of the results numerical investigation and the possible future extensions of the study are presented in this chapter.
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