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Elektronik tablolarda sonlu farklar yöntemiyle plak ve üçmoment denklemiyle sürekli kiriş çözümü

Başlık çevirisi mevcut değil.

  1. Tez No: 75390
  2. Yazar: BATUHAN ÇALİN
  3. Danışmanlar: PROF. DR. MEHMET BAKİOĞLU
  4. Tez Türü: Yüksek Lisans
  5. Konular: İnşaat Mühendisliği, Civil Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1998
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: İnşaat Ana Bilim Dalı
  12. Bilim Dalı: Yapı Mühendisliği Bilim Dalı
  13. Sayfa Sayısı: 171

Özet

ÖZET sürekli kirişler ve plaklar Bu çalışmada, elektronik tablolarda bulunan döngüsel başvurudan (circular reference) yararlanarak sayısal çözümü lineer lineer denklem takımının çözümüne indirgenebilen mühendislik problemlerinden sürekli kirişler ve plaklar elektronik tablolar yardımıyla çözülmüştür. Problemin sayısal algoritması kurulduktan sonra çözüm için programlama bilgisi gerekmemektedir. Sadece karışık problemler için mantıksal fonksiyonların bilinmesi yeterli olacaktır. Bunlara ek olarak çözüm genelde şablonlar ve tablolar halinde yapıldığı ile yapıldığı için sonuçlar koordinat noktalarına göre düzenlenmiş tablolar şeklinde elde edilmektedir. İki serbest değişkenli çözüm fonksiyonlarında, sonuçların tablolar halinde elde edilmesi yorumlamada büyük kolaylık sağlamaktadır. Ayrıca tablo şeklinde elde edilen sonuçlara ait grafiklerin anında gerek üç boyutlu gerek konturlu halinde çizilmesi bu tip çözümlere üstünlük sağlamaktadır. Bu çalışmada, yukarıda bahsedilen yöntemle sürekli kirişlerde Clayperon (Üç Moment) denklemleri kullanılarak 20 açıklıklı ve her açıklıkta farklı atalet momentine sahip olan sürekli kirişlerin mesnet momentleri ve mesnet tepkileri hesaplanmıştır. Problemin çözümünde her açıklık üzerinde 4 farklı veya aynı tip yükleme yapılabilmektedir. Sonuçlar her yüklemede elde edilen değerlerin süperpozisyonuyla ortaya çıkmaktadır. Giriş verileri olarak açıklık, atalet momenti oranı, yüklemelere ait karakteristik değerler, yükleme tipi, taşıyıcı sistemin sınır koşullarının belirtilmesi gerekmektedir. Sürekli kirişlerin hesabına ek olarak elastik zemine oturan veya oturmayan ve çeşitli yükleme ve mesnetlenme şartlarında dikdörtgen plakların analizi de diferansiyel denklemlerin sonlu farklar metoduna göre yazılması sonucu oluşturulan lineer denklem takımının çözümü elektronik tabloda iterasyon yardımıyla yapılmıştır. Burada plağın çökmesi, iç kuvvetleri ve mesnet reaksiyonları genelde yeter yakınsaklık sağlanarak hesaplanmıştır. Oluşturulan tabloda dış yük, kenarların birbirine oranı, plağın ne şekilde mesnetlendiği, poisson oranı, zemin yatak katsayısı ve aralık oranı tablonun giriş verileridir.

Özet (Çeviri)

SUMMARY THE SOLUTION OF RECTANGLE PLATE PROBLEMS WITH FINITE DIFFERENCE METHOD AND THE SOLUTION OF CONTINUOUS BEAMS WITH EQUATION OF THREE MOMENTS The aim of this study is to solve the engineering problems that the solution can be reduced to a numerical solution of linear equations by using circular references in electronic spreadsheets. A formula that is written to a cell in electronic spreadsheet can be written with a reference to a value or formula that is written in another cell. When a formula refers to its own cell, either directly or indirectly, it is called a circular reference. If it is wanted this can be made continuously between the cells. To calculate such a formula, the spreadsheet must calculate each cell involved in the circular reference once by using the results of the previous iteration. The repeated calculation of a circular reference until a specific numeric condition is called iteration. Iteration stops, if the change between iterations or the number of iterations are reached to the values in the settings for iteration. This values can be changed, whenever wanted. If the equations in engineering problems is written in a form of circular references, they can be solved by iteration method. In the algorithm of the problem the cells are planned like the values at the points and the formula that connects a cell to another values in the cells is written to this cell. For this operation the user don't need to know any programming languages. Only for some complicated problems they have to know any logical functions (IF, OR, NOT, AND). The solutions are obtained generally in tables that are arranged according to the co-ordinates, because they are made in tables and pattern. This enables to interpret problems with two variables. Also three or two dimensional graphics of this results that are given in tables can be drew easily. In this study the solutions of thin plates and continuous beams in electronic spreadsheets are examined. 1. Bending Problem of Plates by Finite Differences Many structural mechanics problems involve differential equations, either ordinary, required for beam shears, moments, slopes, deflections, and buckling of columns; or partial, required for bending of plates and vibration of beams. Direct solution of these equations can be achieved only for simple conditions, when the load distribution, sectional properties and boundary conditions are easily represented by mathematical17 expressions. If these conditions become involved, exact solution sometimes becomes impossible and one powerful tool in solving these equations is the method of finite differences. In using numerical methods to solve differential equations, it is necessary to express the derivatives of a function by the difference expressions of the function at finite intervals. At each pivotal point of the intervals, an equation expressing the differential equation by the finite differences can be written. Numerical solution by finite difference method generally requires replacing the derivatives of a function by difference expressions of the function at the nodes. The differential equation governing the displacement (or stress) is applied in a difference form at each node, relating the displacement at the given node and nodes in its vicinity to the external applied load. This provides a sufficient number of equations for the displacements (or stresses) to be determined. In this thesis, some engineering problems such bending of plates and analysis of continuous beams are solved numerical easily by electronic sheets in MS Excel 7.0. If it is assumed that - x - y - w --- q(x,y) - Ka4 x = _ y = _5 wq(x,y) = -, K- a' J b D q0 d The differential equation of plate becomes d4w _ 2 d4w 4 34w -, -. a-4,2a ^^“+a ^T=q(x3y)-Kw ox ox oy dy The mixed finite differences with respect to x and y are obtained by taking the first partial difference in one direction and then taking the of the first in other direction. If the derivatives of the differential equation are expressed in finite differences, we obtain the finite difference form of the derivative V4w.18 q(x,y)=....Ax4 Internal forces such as Mx and My bending moments, Mxy twisting moments, Qx and Qy shear forces, Vx, Vy and R reactions on supported edges can be expressed by finite differences easily. It's insufficient to apply the mesh of plate, because if the mesh is applied to the boundary or a close point to the boundary some other fictitious points are required. This point are have to be expressed in terms of other displacement values. This is done by boundary conditions of plate. The distance of the fictitious point, when the boundary is clamped or simply supported, is X. The distance from the boundary increases to 2X, if the boundary condition at the edge is free. Program consists of three stages. In the first stage of the program are given the data, i.e. external forces, code number of the support conditions, Poisson ratio, space between nodes in x direction A*, ratio of the edges a/b a is referred to the edge in x direction, b is referred to the edge in y direction. Support codes are 1 for clamped support, -1 for simply support and 0 for free edge. In the second stage, program begins to solve the problem with iteration. In this stage the number of iteration and the change value have to be given. Now iteration can start. After iteration is finished, calculated displacements, internal forces and reactions, can be compared with analytical solutions. The results are compared with the analytical solution. It's recognised that the difference between analytical solutions and the solutions with finite differences increases parallel with the a ratio. To reduce the differences between results it's recommended that, if the plate is symmetric the co-ordinate axes can be changed and the smaller a value is used. 2. Analysis of Continuous Beams with Equation of three Moments Clayperon developed a method of analysis for the reconstruction of a bridge in 1849 and presented this procedure for analysing continuous beams. The condition of19 compatibility of the angle of rotation for one point on the beam is used the governing equation. This equation relates the internal moments at three points on the beam to the applied loading, the material behaviour, and the cross-sectional properties of the beam members. The points chosen for obtaining the internal moments are the supports. The angles of rotation at this points due to the three moments Mi, Mm, Mi+1 By superposition the angles of rotation to the left and right of point i 6ı- Rı+ Rı'+ Ri ' 62- R2+ R2f+ R2”A,a, 1 M..L. M:L; 0 =_L_L+ _L_L_L + __L_L 1 EJi Lj 6EJi 3EJi A2a2 1, Mi+1Li+1 MjLi+1“1 - + + EJi+ı Li+ı 6EJi+ı 3EJi+i For continuity of slope at point I Qr=-02, Thus, J ^T T N Li-5-Mi_1+2Mi j0t j 0 L: +T^L + Li+1^Mi+1=-Wi^Lr£i+iyfi-L T 1 T I+1 ”1+1 T 1+1 İT 1 ' ' T U; Jj+ı J Jj J; Ji+1 i+1 which is the three-moment equation. Such an equation can be written for each intermediate support of a continuous beam; and if the ends of the beam are simply supported, the number of equations is equal to the number of statically indeterminate moments at the supports. Hence, all unknowns can be found from these equations.20 If the left end of the beam is built in at the first support, we have an additional condition starting that the left end of the beam does not rotate. It is seen that a built-in end introduces an additional unknown moment, but at the same time we have an additional equation. Hence, the number of unknowns is again equal to the number of equations. This program can solve continuos beam with 20 spans and can be made 4 loads. It can also find the reactions at the supports. As input data must be typed the support conditions, span number, length of the spans, type of the load, characteristics of the load, ratio of the moments of inertia. The results are shown as a table and it's easy to interpret.

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