Fotoelastisite'de sınır eleman yönteminin ayırma metodu olarak kullanılması (2 cilt)
Application of boundary elements methods in photoelasticityas a seperation method
- Tez No: 21800
- Danışmanlar: PROF. DR. YALÇIN AKÖZ
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Fotoelastisite, Sınır elemanlar yöntemi, Photoelasticity, Boundary element method
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Deneysel bir yöntem olan fotoelasti siteden elde edi len verilerle asal gerilme farkları ve asal gerilme doğrultularına ait bilgiler doğrudan elde edilmektedir. Asal gerilmeleri veya gerilme bileşenlerini elde etmek için ilave deneyler veya bölgenin herhangi bir nokta sında asal gerilme toplamlarını veren çözümlerin yapıl ması gerekmektedir. Ayırma yöntemleri dediğimiz işlem leri uyguluyarak gerilme bileşenlerini tek tek elde e- debi liriz. Bu çalışmada Sınır Eleman Yönteminin fotoelasti sitede ayırma yöntemi olarak kullanılması araştırılmış ve prob lemlerin çözümü için Fortran kodlama dilinde bir bilgi sayar programı hazırlanmıştır. Giriş bölümünde f otoelastisi te, ve sınır eleman yön temine ait çalışmalar gözden geçirilmiş ve incelenen problem açıklanmıştır. İkinci bölümde ışık, ışığın vektörel karekteri, pola rize ışık, pol ar i zer 1 er, dalga plakası ve f otoelastisi te için uygulanan Jones vektörleri tanıtılmıştır. Üçüncü bölümde polariskop ve Jones matrisleri anlatı larak polariskop elemanları için Jones matrisleri anla tılmıştır. Dördüncü bölümde düzlem el ast i site teorisi için foto elasti site teorisi anlatılmıştır. Beşinci bölümde fotoelasti site deneylerinden elde edilen verilerin analizi açıklanmıştır. Altıncı bölümde kullanılan ayırma metodları açıklan mış ve bunlar hakkında bilgi verilmiştir. Yedinci bölümde sınır eleman yönteminin ayırma metodu olarak kullanılabileceği açıklanmış ve bu metoda ait bilgisayar programı verilmiştir. Sekizinci bölümde karşılıklı iki kenar ortay nokta sından eşit ve karşıt tekil kuvvetlere maruz kare levha- hanın çözümü yapılarak gerilme bileşenleri elde edilmiş tir. Yine aynı problemin yanlarına yarıçapları sabit kalmak üzere derinliği değişen çentik açılarak sınır elemanları yöntemiyle çözümleri yapılmıştır.
Özet (Çeviri)
SUMMARY APPLICATION OF BOUNDARY ELEMENTS METHODS IN PHOTOELASnCITYAS A SEPERATION METHOD In case the geometric configuration of boundary of a structure is irregular, solution of the system will be troublesome. In this condition photoelasticity solution will be available. Brewster observed that a loaded glass sheat produces color lines which are seen by polar i scope and this observation of him constituted the base of the photoel asti ci ty. Neumann and Maxwell had been the first theoriers of photoelasticity. After their work, the optical coefficient of various materials had been calculated. In 1912 Mesnager had used photoelasticity for the first time in solving engineering problems by calculating the tension on a glass model. Coker has developed many practical investigations and techniques which made the engineering applications possible. After the works by Hetenyi, Drucker, Frocht, Midlin, Jesop and Kuske had been developed various photoel asti c measurements and experimental techniques. If the stressed model is examined by the polar i scope, the isoclonic lines and isochromatic lines are observed. The differ ance of the principle stresses and their oriantation at a certain point of material can be calculated by using two kinds of data lines. Also, isoclinic fringe pattern which gives the angle between x axis and principal stress direction at a certain point is used to calculate the shear stress, a -a t = * 2 sina© CEq.lD xy 2 The differance of the principle stresses is given by the following formula N.f a -a = - CEq.SZ) 1 2 h N appearing in the above equation is the isochromaticpat. tern number h is the thickness of material and term f is the optical coefficient of the material. Stresses in the zone can not be found directly by the photoelastic experimental data, however, those are on the boundary can be. If the boundary is free, then the principal stress normal to the boundary will be zero, and therefore the other principal stress can be determined directly from CEq. 23. If the boundary is not free and normal load which is applied is known, tangential boundary stress would be interpreted by following equation. stresses need a third relation. The ways for obtaining elements of the stress tensor are called the seperation methods. Seperation methods can be classified in three groups: semi -theoretical methods, another experimental methods or numerical methods. Amoung them, numerical methods has an appear ent advantage: once a numerical method is set then the whole solution in the zone can be obtained with the ai d of a computer. In this study, the boundary elements method is used as a seperation method in photoelasticity. The boundary elements method is more capable than other numerical methods. This separation method depends on the solution of Laplace equation. 72C» +<y 3 = O CEq. 4} * y If we denote the first invariant of plane stress state by u Cff +a 3 = Co- +a 3 = u CEq. 53 x y l 2 then CEq. 43 becomes v"Zu =0 CEq. 63 Function u is determined by solving the CEq. 63 in the domain Q, corresponding to certain conditions given V»ion the boundary F. Fig. [1.1]. Two dimensional problem will be solved in the domain O with two types of boundary conditions. vu = b in domain Q u - u q » q on T Essential conditions i on F Natural conditions 2 CEq.73 y-r,u=u -> x Fi gur e E 1. 1 3 Notati on If the both sides of the first equation in CEq. 7D is multiplied by weighting function w and integrated in the domain, we can write the equilibrium equation can be written as follows: J CV2u fcö w dO = 0 CEq. 8> D If the CEq. 8D is integrated by parts twice, du JCV2u -b3w dD = JCv- w3u -bwD dO + J w ^- dr no r -r; « £ dr ? ° CEq.SO is determined. For the aim of study b is unimportant. Omitting b the following equation is obtained. dw du o Jcv2 wDu do = J u §£ dr - J w f£ dr CEq. 103 The CEq.lCO being the starting point of the boundary elements method has a prime importance. Assume that the boundary of a dimensional domain has been divided into vinN elements. Figure 1 1.23 element Fi gur e C 1. 2 3 Boundar y El ement By using the Divergence theorem for left the side of the CEq. IID. N N. u. = } q. f w dT - ) u, f dw dn dr CEq.llD J = i J j = i J wher e, c. = 1 /2 when p is on boundar v c. = 1 when p is on internal point There are two types of integrals, which are shown in CEq.l2D, to calculate the values of the function u on the elements. These can be written in the matrix form as follows: G, <,'! w dr CEq. 1 2D »u'J -dr If point p, which has t nod, is on the boundary and if the CEq.l2D is substituted in the CEq. 113 we get N N -2~ Ut I*<iui- I G.. q. CEq. 133 J = i 3 = i IXWhere, H.. = < 'İS t /' H. I aqes j CEq. 143 H.. + -4- i «= j If we rearrange the CEq. 133, we get H u = G q CEq. 153 the expression of CEq. 153 in matrix form. The boundary conditions corresponding to our problem are predetermined value of u by photoelasticity. In order to calculate the values of q on boundary, we can set another equati on A. Y = F CEq. 163 Y is vector of q. After solving CEq. 163, then all unknown values on boundary will be determined. The value of the function u at internal point t can be solved after finding the function q on boundary. N N u. = ) G.. q. - V H. u. CEq. 173 Assume that, the differ ance of the principal stresses is equal to p, p = a -a CEq. 183 r 1 2 M The principal stresses can be calculated by using CEq. 53 and CEq. 183. u+p CEq. 183After finding the principal stresses and using the pirinci pal stresses directions C©D which were determined from isoclinic fringe patterns, the elements of stress tensor can be solved in the domain. 1 a - mju. Cu + pcos2©D x a i a - a,' Cu - pcos2Ğ5 t =_* p sinae CEq.l9D xy a ^ ^_ For boundary elements method as a separation method in photoel asti ci ty can be refered to I2K In that study the deep beam problem having concentrated load and uniformly distributed load at top edge have been solved by using separetely using the boundary elements method. In this study the stress distribution in a square plate which is loaded by concentrated forces on natural edges along symmetry axis is investigated. The analysis has been carried out by photoel asti city in which the baundary elements method is a employed as a seperation method. The results are obtained by a computer programme which is written is FORTRAN code and are compared with the results obtained by övünç in which the same problem is solved by the complex function techinique. The same plate, which has a several depths of the grooveses in place of both sides of the plate and has a constant radii, is solved. The results obtained above solution are compared with the results of the square plate. XI
Benzer Tezler
- Sıkı geçme durumunda oluşan gerilmelerin incelenmesi
Investigation of the stresses of the state of press-fit
HARUN KARAKAYA
Yüksek Lisans
Türkçe
1998
Makine MühendisliğiPamukkale ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
DOÇ. DR. MUZAFFER TOPÇU
- Tipik bir prefabrik çerçevelerde elastik birleşim redörünün sayısal ve deneysel olarak belirlenmesi
A Numerical approach to defina the flexural stipness of a scarfed joint in a typical prefabricated R.C. frame
İLKNUR KARSLIOĞLU
- Levhada delik ve çentik probleminin fotoelastisite yöntemi ile incelenmesi
Analyzing hole and notch problems in plate with photoelasticity method
YÜCEL ATAÇ
Yüksek Lisans
Türkçe
2000
İnşaat MühendisliğiYıldız Teknik Üniversitesiİnşaat Mühendisliği Ana Bilim Dalı
PROF. SİNAN ÇAĞDAŞ
- Fotoelastisite yöntemi ile deneysel gerilme analizi
Başlık çevirisi yok
TURAN KARABÖRK
Yüksek Lisans
Türkçe
1993
İnşaat MühendisliğiAnadolu Üniversitesiİnşaat Ana Bilim Dalı
YRD. DOÇ. DR. İLKER BEKİR TOPÇU
- Bir gemi lumbuzuna gelen gerilme dağılımlarının fotoelastisite yöntem ile incelenmesi
Başlık çevirisi yok
HAMİ GÜRTUNCA
Yüksek Lisans
Türkçe
1989
Gemi MühendisliğiDokuz Eylül ÜniversitesiDeniz Teknolojisi Ana Bilim Dalı
DOÇ. DR. ONUR SAYMAN