Asma köprüler ve titreşim araştırmaları

Abstract

Bu tez çalışmasında, asma köprülerle ilgili genel bilgiler verilmiş ve bu bilgiler ışığında asma köprülerin hesabında kullanılan teorilerden bahsedilmiş ve tezin asıl çalışma alanı asma köprülerin doğal frekans değerlerinin elde edilme yöntemleri olmuştur. Çalışmanın 1. Bölümünde bir asma köprüyü oluşturan kablolar, askılar, mesnet semerleri, tabiiye, kuleler ve ankraj kütleleri hakkında genel bilgiler verilmiştir. 2. Bölümde ise asma köprülerde kablo hesaplarından bahsedilmiştir. Önce kablolar için en genel durum olan "basit kablo" konusu gözönüne alınıp, kablo şekilleri anlatılmıştır, sonra ise basit kablonun çeşitli yük durumları karşısında davranışı incelenmiştir. Asma köprü kablo hesaplarıyla ilgili olarak Rankine Teorisi, Elastik Teori, Defleksiyon Teorisi ve Lineerleştirilmiş Defleksiyon Teorileri konu edilerek, bu teorilerle ilgili formüller elde edilmiştir. 3. Bölümde ise sadece basit kablo için ve eğilmeye çalışan komple köprü için çeşitli bilim adamları tarafından elde edilen düşey frekans formülleri ve "Rocard" tarafından elde edilen burulmaya çalışan komple köprünün burulma frekans formülleri verilmiştir. Ülkemizde bulunan Boğaziçi ve Fatih Sultan Mehmet Köprülerinin çevresel etkilerden oluşan titreşim araştırmaları 4. ve 5. Bölümde konu edilmiştir, bu bölümler iki köprünün mod şekilleri ve frekans değerleri ölçümlerinin nasıl yapıldığı ve bu ölçümlerden elde edilen değerlerin çeşitli değerlerle karşılaştırılması anlatılmıştır. Asma köprülerin yanal rüzgar etkisinde salınımları ve bu etki nedeniyle oluşan burulma, eğilme ve birleşik titreşimler konuları 6. Bölümde işlenmiştir. Son bölüm olan 7. Bölümde önce basit kirişi, basit kablo ve kablo çekme kuvvetinin yatay bileşeni için doğal düşey frekans formülleri ve bu formüllerin Boğaziçi Köprüsü için değerleri elde edilmiştir. Bu değerler Boğaziçi Köprüsü için 4.Bölümde elde edilen değerlerle karşılaştırılarak, değerlendirme yapılmıştır. Ayrıca denge denklemleri yardımıyla farklı durumlar için asma köprülerin doğal frekans formülleri bulunmuştur. Bilgisayar programlan yardımıyla Boğaziçi Köprüsü'nün mod şekilleri ve frekans değerleri bulunup, karşılaştırma ve değerlendirme yapılmıştır.

In this thesis, some general knowledge about suspension bridges has been given and depending this knowledge, has been explained some theories that are used to solve suspension bridge problems. Main subject in this thesis is, investigations of natural frequencies and mode shapes of a suspension bridge. In chapter 1, the parts of a suspension bridge (deck, cables, hangers, towers, anchorages, bearing saddles) has been explained. Main parts of a suspension bridge are cables, toqers and deck. A deck of a suspension bridge should have been constructed either as al lattice girders or a box-girder. For lang spans, stiffenig girders are mostly constructed Pratt (N) or Warren (V) type. The towers of modern bridges are built up of external plates, riveted or welded together, or are built up of reinforced conerete. Chapter 2 is devoted to the theories that are used to solve suspension bridge problems. Firstly "the simple cable" has been explained, a single cable suspended between two points is the simplest possible kind of suspension bridge. In this case, the initial problem is to determine the form adopted by the cable when loaded solely by its own weight to find the tension in the cable at any point along its length. These are also made for "the parabolic cable" and for "the heterogenous cable". Beyond these the response of a simple cable under different loadings has been explained. Rankine Theory is the first theory of the suspension bridge proper that is, a bridge comprising a roadway slung from suspension cables and stiffexed in some measure by longitidunial girders at the road level. There are three basic assumptions in Rankine Theory. These are; a) Under the dead loading on the bridge the cable is parabolic and stiffening girder is unstressed. b) That any live loading applied to the girder is so distributed by it to the cable to cany a uniformly distributed loading across its whole span. xvi c) The total live load divided by the span. Beyond these, in Chapter 2, the response of two pinned and three pinned girder under a uniform load and a single concentrated load has been discussed about Rankine Theory The Elastic Theoiy is one of the theory is about suspension bridges which has only one really new feature when compared with the Rankine Theory. Assumptions are made in the Rankine Theory, also is excepted in the Elastic Theory. In place of third assumption of Rankine relating to the value of q, the uniform distributed loading acting on the cable, is made the assumption that q depends in magnitude upon the elastic stiffnes of the cable in tension and the stiffening girder in bending In other words, the cable is treated as inverted elastic parabolic arch, under uniform loading, with a suspended a elastic beam. In this chapter, under the subject of Elastic Theory; examination of Elastic Theory in terms of the cable loading q, influence lines for bending moments and shear forces, girder deflections, temperature effects etc. has been explained. In the Elastic Theory, bending moment is given as below; M=n + h.y (1) In this formula, it was assumed that the deflection (v) of the girder was negligiable compared with ordinates y of the initial shape but in the Deflection Theoiy, it is seen that the presence of v gives rise to a change M that, because of multiplier (H + h). So M becomes; M = M= j.1 + h.y + (H + h) v (2) The fundamental equation of the Deflection Theoiy is given as below; EI^-(H + h£=p + h? (3) dy dx" dx" In linearised Deflection theory, the horizontal tension h is neglected as it will be small compared with the initial tension H.So (2.132) d4v d2v d2v EI-T-H-r-^P + 1^4 (4) dx dx" dx xvu By using Tie Analogy method (4) has been rewritten as; d4v d2v E,s?-H^=p-^ _h_ H (5) Chapter 3 is devoted to the survey of natural frequencies of suspension bridges. Steinmann, Pugsley, Bleich have studies about the natural frequency of complete bridge in flexure From the studies of Pugsley natural frequency formulas corresponding first and third modes of complete bridge in flexure are given as below Til 1 Y 2V2 Vd d2} 1 1-3- V1 + 327TR t J (6) ^--û&lı-°4)^+m"2R (7) After Pugsley, Steinmann investigated the natural frequency formula corresponding second mode and for inextensible cable he had given this formula; 1!,=:^/! Vl + 78.37t2R " 2V2 Vd (8) Bleich's work on this problem allowed for elastic extensibility of the cable and he had a solution which can be written as; ru = -^ J- Vl + 69.37TR 2 2V2Vd (9) From work about complete bridge in torsion, Rocard gave an equation which is the torsional equivalent of the flexural equation and its solution gives for the circular frequency in torsion as below. w. b2y fl6rc4EI. TT^I 2tc2GI.y 4.k __ + V w.L4 2d; + =- w.kd2.L2 (10) XVI II Chapter 4 is devoted to the ambient vibration survey of Bosporus Suspension Bridge. Traffic and wind excitation has been used to obtain the dynamic characteristics of the first Bosporus Suspension Bridge. Structural symmetry and the absence of suspended side-spans allowed attention to be focused on the mani spain and the Asian tower. In this chapter, vertical, torsional, lateral modes of main span and tower is given. Also a detailed comparison is given between measured modes and calculated modes which are obtained by three dimensional finite element model. In Chapter 5, ambient vibration survey of the Fatih Sultan Mehmet Suspension Bridge is explained. The measurement methods of ambient accelerations due to dynamic excitation by wind and traffic are explined. Also obtained natural frequencies, mode shapes, damping ratios for vertical, lateral,torsional in the deck and tower are given in this chapter. Similiar to Chapter 5, detailed comparison is given between measured modes and calculated modes. Oscillations of suspension bridges under lateral winds are investigated in Chapter 6. In this chapter explanations are given about static actions, flexural oscillations, torsional oscillations and coupled oscillations. A suspension bridge has three types of oscillatory motion. These are given below; a) Purely flexural oscillations of the bridge, with each cross - section of the bridge deck moving up and down in a vertical plane, every point in it having the same amplitude of motion b) Purely torsional oscillations of the bridge, with each cross-section of the bridge deck oscillating angularly in its own plane abast an axis at or close to the mid line along the roadway. c) Coupled flexural - torsional oscillations of the bridge, with each cross - section of the bridge deck undergoing in general both vertical and angular motions. In Chapter 8, some solutions have been found for different situations and between made. These comparisons are shown at the tables. By the investigations about natural frequencies of simple cable and simple beam, it is seen that cables of a suspension bridge was being more effective, on the natural frequency of complete bridge in flexure, than the deck. When the horizontal component of cable tension is applied to bearings of simple beam, the natural frequency values of this beam in flexure will be about the natural frequency values of complete bridge in flexure. Using equation (5), new equation about the natural frequency of a bridge under live load, p has been written as below; OD Under own weight, the natural frequency of a suspension bridge is investigated and the equation below is written; XIX Wi=^Aı^ - (12) ' MM For complete bridge in flexure, two-dimensional finite element mesh is formed. This is solved by structural analysis programme and for twelve modes natural frequencies is obtained, A detailed comparison is given between obtained values and measured values before.