Simetrik konsollu köprü ayaklarının temel ivmeleri altında dinamik ve spektral analizi
Başlık çevirisi mevcut değil.
- Tez No: 75248
- Danışmanlar: PROF. DR. ZEKİ HASGÜR
- 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: İnşaat Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 75
Özet
ÖZET Bu çalışmada, aktif deprem kuşağında yer alan ülkemizde de yaygın olarak kullanılan, köprü ayağı üzerindeki çift simetrik konsollu yapının; temel ivmeleri altında dinamik karşılıkları adım adım integrasyon yöntemiyle incelenmiştir. Öncelikle bir model seçilmiş ve bu modelin dinamik analizi yapılmıştır. 1. ve 2. Bölümlerde; zaman tanım alanı yaklaşımı, altı serbestlik derecesi ile modellendirilmiş düşey ve yatay zemin ivme bileşenlerine maruz sistem tanıtılmıştır. Bu tip köprüler, özellikle yer hareketinin düşey bileşeninden daha çok etkilenirler. 1995 yılında Kobe'de meydana gelen (Hyogo - ken Nanbu) depremi, bu tip köprülerin kolonlarında büyük hasarlar meydana getirmişti. Köprü ayağı üzerindeki çift simetrik konsollu yapının dinamik analizi yapılmıştır. Dinamik analiz için; genelleştirilmiş Jacobi yöntemi ve adım - adım integrasyon yöntemi izlenmiştir. Aynı yapı spektral analiz yöntemi ile de irdelenmiştir. 3. Bölümde; ikinci bölümde genelleştirilmiş Jacobi yöntemi ve adım - adım integrasyon yöntemi ile bulunan formülasyonlan izleyen bir program geliştirilmiştir. Deprem ivme kayıtlarım (yatay + düşey) kullanarak; deprem sırasında kuvvetli yer ivmeleri etkisinde kaldığı düşünülen sistemin; yerdeğiştirme, hız ve mutlak ivme karşılıklarım hesaplayan, yerdeğiştirme değerlerini kullanarak her mod için kuvvet vektörünü hesaplayan ve sonuç olarak taban devirme momenti, taban kesme kuvveti, konsol momenti ve konsol kesme kuvvetinin zaman geçmişinin yapısal karşılıklar olarak hesaplayan bir program geliştirilmiştir. Maksimum deprem etkileri adım adım integrasyon yöntemi ile hesaplanmıştır. Aynı zamanda bir seçenek olarak, spektral analiz geliştirilerek bu program içerisinde yatay ve düşey depremin ayrı ayrı etkimesi göz önüne alınarak yapının maksimum kuvvet ve momentlerinin spektrum karşılıklarının karelerinin karekökü alınarak adım adım integrasyon değerleri ile karşılaştırılmıştır. 4. Bölümde; sayısal örnek 1995 Dinar depreminin gerçek yatay ve düşey ivme kayıtlan kullanılarak, Yeni Haliç Köprüsü'nün çift simetrik konsollu P3 ayağına ait veriler uygun şekilde değiştirilmiş, yatay ve düşey deplasmanlar, taban devirme momenti ve taban kesme kuvveti, konsol momenti ve konsol kesme kuvvetinin zaman geçmişi ve spektral değerlerinin yapısal karşılıkları olarak hesaplanmıştır. 5. Bölümde; sonuçlar ve öneriler sunulmuştur. Sonuçlardan bazıları: * Çift simetrik konsollu köprü ayaklan, yer hareketinin düşey bileşeninden daha çok etkilenirler. * Programın çalıştırılabilmesi için seçilen modelde matrisin frekans determinantının pozitif olması gerekmektedir şeklindedir. Konsol boyu arttıkça periyod artmaktadır. ix
Özet (Çeviri)
DYNAMIC ANALYSIS AND SPECTRAL ANALYSIS OF BRIDGE PIERS WITH SYMMETRIC CANTILEVERS UNDER THE BASE ACCELERATIONS SUMMARY In this research work, frequently used bridge type structure with double symmetric cantilevers over its pier is analysed by dynamic method. The time domain approach is utilized for the six (6) degrees of freedoom modelled system subjected to vertical (ay(t)) and horizontal (ax(t)) components of the recorded ground accelerations in the Dinar earthquake. For the frequency domain approach smilar modal in Ref. 1 was applied using El Centra earthquake. In that case modified constant power spectral and cross spectral densities were utilazed. In these cases structural dynamic responses mainly the maximum displacements, maximum cantilever shear, maximum base shear, maximum cantilever moment and maximum overturning moment are calculated employing the data of real structure, New Golden Horn Bridge. Cantilever type structures, especially in the bridges are most sensitive to the vertical base acceleration of the ground motion. Recently 1995 Hyogo-ken Nanbu (Kobe) earthquake gave huge damage to the groupe of piers of the this type bridges. In the first, the second and the third chapters; the system that subjected to vertical (ay(t)) and horizontal (ax(t)) components of the ground accelerations are modelled in the time domain approach is utilized for the six (6) degrees of freedoom is introduced and some information is given about these system. The structure is modelled as a T shaped structure and cantilevers with uniform mass mTper unit length from the superstructure with flexural stiffness Eli and pier with uniform mass m2 per unit length with flexural stiflhess EI2 in its plane and it is discretized with six (6) lumped masses as illustrated in Figure 1. According to this model, translations of the masses are defined as six degrees of freedom system in x and y directions and rotational freedoms are ignored. The analytical model of the bridge type structure with double symmetric cantilevers over its pier is analysed by dynamic method where dynamic analysis is carried out considering both horizontal (ax(t)) and vertical (ay(t))components of ground motions. The generalized Jacobi method and the step-by-step integration methods are used to dynamic response analysis of the system.The modal analyses are carried out first establishing flexibility and mass matrices. Then the orthonormalized eigenvectors and eigenfrequencies are obtained using generalized Jacobi method and they are given as modal matrix O and o vectors respectively. Ll LI i4 li' E, II, Ll cu OJ UJ n j E, II, Ll ru OJ OJ nj ?>ax(t) a y ( X ) Figure 1 The model of the bridge pier with symmetric cantilevers. When free vibration is under consideration, the strucure is not subjected to any external excitation (force or support motion) and its motion is governed only by the initial conditions. There are occasionally circumstances for which it is necessary to determine the motion of the structure under conditions of free vibration, but this is seldom the case. Nevertheless, the analysis of the structure in free motion provides the most important dynamic properties of the structure which are the natural frequencies and the corresponding modal shapes. We begin by considering both formulations for the equations of motion, namely, the stiffness and the flexibility equations. For the stiffness equation with force vector {F} = {0}, we have; [ [K] - cû2[M] ] {a} = {0} (1) XIwhich for the general case is set of n homogeneous (right-hand side equal to zero) algebraic system of linear equations with n unknown displacements a, and an unknown parameter a2. The formulation of equation (1) is an important mathematical problem known as an eigenproblem. Its nontrivial solution, that is, the solution for which not all a* = 0, requires that the determinant of the coefficient matrix {a} to be equal to zero; in this case, [K] - ö2[Mİ = 0 (2) In general, equation (2) results in a polynomial equation of degree n in terms of©2 which should be satisfied for n values of o2. This polynomial is known as the characteristic equation of the system. For each of these values of }“ = Generalized stiflhess of the structure, P* = {}nT [P] {0}n = Generalized force of the structure, {q} = Relative displacement vector. Equation (3.) should turn to equation (3. a.), adapting to our system; q”n(t) + 2^tDnq,n(t) + (Dn2q“(t) = Pa!{fJ n=l, 2,....,6 (3. a) M. Ç”= Cg = Damping ratio 2M“on ©n = Ka = Natural frequencies \| M”xnThe step-by-step integration method has been used for the dynamic response analysis of the system. Structures are usually designed on the assumption that the structure is linearly elastic and that it remains linearly elastic when subjected to any expected dynamic excitation. However, there are situations in which the structure has to be designed for an eventual excitation of large magnitude such as strong motion earthquake or the effects of nuclear explosion. In these cases, it is not realistic to assume that the structure will remain linearly elastic and it is then necessary to design the structure to withstand deformation beyond the elastic limit. The simplest and most accepted assumption for the design beyond the elastic limit is to assume an elastoplastic behavior. In this type of behavior, the structure is elastic until the restoring force reaches a maximum value (tension or compression) at which it remains constant until the motion reverses its direction and returns to an elastic behavior. Among the many methods available for the solution of the nonlinear equation of motion, probably one of the most effective is the step-by-step integration method. In this method, the response is evaluated at successive increments At of time, usually taken of equal length of time for computational convenience. At the beginning of each interval, the condition of dynamic equilibrium is established. Then, the response for a time increment At is evaluated approximately on the basis that the coefficients k and c remain constant during the interval At. The nonlinear characteristics of these coefficients are considered in the analysis by reevaluating these coefficients at the beginning of each time increment. The response is then obtained using the displacement and velocity calculated at the end of the time interval as the initial conditions for the next time step. Assuming that the acceleration varies linearly during the time step between the initial and final values of u o and u \, equations of acceleration (linear), velocity (quadratic), displacement (cubic) are; u“(x) = u ”o + (o“i - u”o) * t / h u'(t) = o'o + u“o * T + (u”ı - u“o) * t2 / (2 * h) u(x) = uo + u'o * x + u”o * x2 / 2 + (u“i - u”“) * t3 / (6 * h) Expressing the effective static equilibrium equation for the linear acceleration analysis by k~d * ui = p~ia the effective stiffness and loading are given by k~d = k + (3c/h) + (6m/h2) xiiip~id = pi + m * (6uo / h2 + 6u'o / h + 2u”0) + c * (3u0 / h + 2u'0 + u“0h / 2) When the displacement u'ı has been calculated, the velocity at the same time is given by u'ı = 3 * (ui - uo) / h - 2o o - u”o * h / 2 The linear acceleration method is only conditionally stable, but this factor is seldom important in analysis of SDOF systems and it is apparent that assuming a linear variation of acceleration during each step will give approximation of the true behavior. Also, in the third chapter; a Fortran program, to calculate displacement, velocity and absolute acceleration response values with the double precision for the both horizontal and vertical ground motions, is developed. After calculating the modal displacement in normal coordinates we obtain the force vector for every n01 normal mode. Herein, the overturning moment, base shear force, cantilever moment and cantilever shear force contributed by n* normal mode can be obtained from the force vector. fln(t) f2n(t). f3n(t) N S-(t) Ut ) = on2[m]{0}nUn(t) n=l,2,,6 (4) And also the spectrum analysis of the structure, having the acceleration response spectrum of Dinar Earthquake was illustrated by evaluating the response of the structure. Maximum total responses (the modal maximum displacements, the modal maximum forces, the modal maximum shears and the modal maximum moments) cannot be obtained in general, by merely adding the modal maxima because these maxima usually do not occur at the same time. In most cases, when one mode achieves its maximum response, the other modal responses are less than then- individual maxima. Therefore, although the superposition of the modal spectral values obviously provides an upper limit to the total response, it generally over estimates this maximum by a significant amount. From the spectral values, the maximum response is the square root of the sum of the squares of the maximum modal responses. In the fourth chapter; in case of modified constant power spectral and cross spectral densities for the Dinar earthquake the variances of structural dynamic responses mainly overturning moment, base shear force, cantilever moment and cantilever shear force are calculated employing the data of real structure, New Golden Horn Bridge. The method of analysis mentioned before is applied to the viaduct on with 20.546 m. height on the New Golden Horn Bridge way and the dimensions, distributed masses xivand flexural rigidities shown in Table 1. The pier is hollow single cell (the box section) shaped and the beams are supported on the cross-head cantilever section. Table 1 Some information about the ground motion record of Dinar earthquake is presented in the Table 2.; Table 2 Ground Motion Records In the sixth chapter; the results are discussed and some main characteristics observed from the graphs are fabulated. As a result; * Cantilever type structures, especially in the bridges are most sensitive to the vertical base acceleration of the ground motion. * Extremely long bridges of medium length spans, long-span bridges and very high bridges are susceptible to the acceleration forces. * The results of the step by step integration method and spectrum analysis are discussed. * Step by step integration method gives the exact solution, but the spectrum analysis makes an approximate analysis based on the ground motion response spectra. XV
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