Düzgün yayılı yük etkisindeki betonarme plakların sonlu elemanlar yöntemi ile doğrusal olmayan hesabı
Finite elements method for the non-linear analysis of reinforced concrete plates subjected to uniformly distributed loads
Yazar: KUTLU DARILMAZ
Danışman: PROF. DR. NAHİT KUMBASAR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Yapı Ana Bilim Dalı
Derece: Yüksek Lisans, 1997
Konu: İnşaat Mühendisliği = Civil Engineering
Dizin: Betonarme levhalar = Reinforced concrete plates ; Sonlu elemanlar yöntemi = Finite element method

Özet:
Yapı sistemleri, dış etkiler artarak belirli bir değere ulaşınca kullanılamaz duruma gelirler, yani göçerler. Bir yapının göçmesi, kırılma, burkulma, büyük yerdeğiştirme, büyük çatlak gibi olayların birinin veya birkaçının onaya çıkmasıyla oluşur. Göçme sırasında yapı, genellikle malzeme ve geometri değişimi bakımından veya her iki bakımdan da doğrusal davranış göstermediğinden sistemin doğrusal olmayan teoriye göre hesaplanması gerekmektedir. Yüksek Lisans Tezi olarak sunulan bu çalışma, düzlemine dik etkiyen düzgün yayıh yük etkisindeki betonarme plakların doğrusal olmayan analizini içermektedir. Birinci bölümde konu ile ilgili bilgiler verildikten sonra çalışmada izlenen yol ve çalışmanın amacı hakkında açıklamalar yapılmaktadır. İkinci bölüm düzlemine dik etkiyen düzgün yayıh yük etkisindeki betonarme plakların doğrusal olmayan hesabına ayrılmıştır. Bu bölümde döşeme sistemlerinin sonlu elemanlar yöntemi ile doğrusal olmayan hesabına ilişkin geliştirilen hesap yöntemi açıklanmıştır. Yapılan varsayımlar, yöntemin dayandığı temeller, eğilme ve burulma etkileri altında akma durumuna erişim ve yöntemin matematiksel formülasyonu verilmektedir. Üçüncü bölümde kullanılan yönteme ilişkin akış diyagramı ve hesaplama adımları açıklanmıştır. Dördüncü bölümde farklı geometri ve sınır koşullarına sahip betonarme döşeme elemanlarının geliştirilen yöntemle hesabı yapılmaktadır. Bulunan sonuçlar kırılma çizgileri yöntemiyle elde edilen çözümlerle karşılaştırılmıştır. Beşinci ve son bölümde sonuçlar değerlendirilmiştir.

Abstract:
Structures subjected to load obviously have to be made safe against failure, and so it is vitally important to determine their strength through tests and calculations. The analysis and design methods of non-linear structures gain much importance since these methods yield both more realistic and economical solutions. The elastic-plastic design methods enable engineers to design structures by taking advantage of the load carrying capacity of the material beyond elastic limit. Therefore the development of methods that take into account the material nonlinearity is of great importance. An elastic analysis of a reinforced concrete slab gives no indication of its ultimate load carrying capacity and further analysis have to be made to reach a realistic and economical solutions. In this MSc thesis the non-linear analysis of orthogonally lightly reinforced concrete slab subjected to uniformly distributed load is presented. The analysis was carried out by a group of computer programs named EPLATE, STIFFCH, LOADrNC, respectively. In developing the program EPLATE the finite element method has been used. The approach named the displacement or stiffness method associated with the finite element method. This approach assumes the displacements of nodes as unknowns of the problem. For example compatibility conditions require that elements connected at a common node, along a common edge before loading, remain connected at that node and edge after loading and deformations satisfied. During the development of the analysis program the below steps are carried out: i- Selection of element type and dividing the the plate system into a system of finite elements with associated nodes called substitute structure. The elements should be small enough to obtain usable results and yet large enough to reduce computational effort. Throughout this thesis a rectangular element type is chosen. The number of elements is optimized by comparing the displacements and internal forces produced by EPPLATE with an another finite element program named SAP90. This process is carried out up to the close enough results. ii- Selection of displacement function. Throughout this thesis polynomials are used as displacement functions since they are simple to work within finite element formulation. For a plate element the displacement function is a function of coordinates in its plane. iii-Defining the stress-strain relationships. In this thesis orthotropic stress-strain relationships are used. iv-Assembling the element stiffness matrix and equations. By using the principle of virtual work the element behaviour matrix has been assembled. vnv- Assembling the system stiffness matrix and global equations. By using the the method of superposition the individual element behaviour matrixes and equations assemble the system stiffness matrix and global equations. vi-Applying the boundary conditions and solving the equations. The two dimensional system stiffness matrix has been assembled as one dimensional system matrix and global system equations are solved by an elimination method. The unknowns w, 8*. 0y have been obtained. vii-Obtaining the element strains and stresses. The internal forces have been obtained in terms of displacements and rotations determined by stress-strain relationships. In the derivation of Kirchoff s small deflection plate theory, the number of independent elastic constants was two, E and v. If we assume that the principal directions of orthotropy coincide with the x and y coordinate axes, it becomes evident that four elastic constants Ex, Ey, vx and vy are required for the description of the orthotropic stress-strain relationships: e-'i:-v'-i; y = - where the shear modulus Gxy of the orthotropic material can be expressed in terms of Ex and Ey as follows: G = fi^K ov = * 2.(l + A/v~vJ (sx+Vy-Ey) 1 - v -v x y °y = ~i İEv+Vx > 1 - v. v., y.(ey+v8-Bs) x 'y * = Gxv-y Expression of the strains in terms of deflection substitutes these relations mx=-Dx my =-Dy ö2w d2w I K.OK cy J 8 w o w ox. j mxy=-2-Dt ( 02 ^ o w ^Sx-oy, where Dx and Dy are the flexural rigidities of the orthotropic plate while (l->x-vy) d,=Vd7v- ^ represents its torsional rigidity. For an orthotropic plate of uniform thickness, the torsional rigidity can be written as viaThe moment-curvature has a non-linear character for a reinforced concrete section due to undergoing phenomena of concrete plasticization and cracking. A non-linear model algorithm developed by modification of Huber-Mises- Hencky's hypothesis to take into account the non-linearity of reinforced concrete, assuming the principle of decrease of stiffness. The criteria formulation takes the below form for reinforcement bars working uniaxially v*=yjmj + m xy 'ö/5îy + mxy where ; _ M, M.. my= M. M. ms>"M iwy The below figure represents a graphic illustration of the surface of local failure in the slab. The principle of decrease of stiffness has taken into account as an exponential function D, = Dw.(l-v*) Dy = Doy-(l-v^) where Dx, Dy are plate stiffness for directions x and y and Dox, Doy are initial plate stiffnesses. Parameter \\i (v|/>l) is a quantity dependent exclusively on material properties and the geometrical shape of the section. For small efforts v, stiffness IXdecreases very slowly, and the section works fairly elastically. For effort v goes to 1 the section's stiffness decreases to zero which means the formation of plasticization The ultimate bending moment is reached when tensile reinforcement yields It is commonly assumed that the yield moment in one direction is quite independent' of the reinforcement at right angles and whether the latter reinforcement yields or not The ultimate torsinal moment is assumed to be the smallest yield moment corresponding to average of reinforcement in both directions on the top and bottom face. In other words the related reinforcement which gives the ultimate torsional moment may be the top or bottom reinforcement's average. The contribution of the concrete in the torsional moment capacity of section is ignored. An exact solution for the ultimate strength of a slab can be found only rarely but it is possible to determine upper and lower bounds to the true collapse load The yield line method of analysis by assuming the mechanism of collapse and using energy method approach gives an upper bound to the ultimate load carrying capacity of a reinforced concrete slab. By using the equilibrium of separate parts, the ultimate load has been evaluated..~*\ r nAt failure, plastic deformations occur along yield lines where the reinforcement has yielded, while the parts into which the slab is divided by the yield lines are only deformed elastically; since the elastic deformations can be ignored in comparison with the plastic ones, the individual parts of the slab can be regarded as plane, and their intersections, the yield lines, as straight lines, with a good degree of approximation. It is thus assumed that deformation occurs only in the yield lines, consisting of relative rotation of the two adjoining parts of the slab about axes whose location depends upon the supports. Each part may be regarded as plane, and so it will be seen that the yield line between two parts of a slab must pass through the point of intersection of their axes of rotation. For a part of a slab supported along its edge, the axis of rotation must lie along the edge, and for a part supported ona column, the axis must pass over the column; other wise it may lie in any direction. The yield lines are assumed to occure due to the yielding of the reinforcement. If the reinforcement at a particular element yields plastic sections representing the yield line, are formed at the equivalent elements which are in the same direction with the reinforcement. The pattern of plastic sections formed in the finite element mesh system resemble the yield lines of a reinforced concrete slab. The slabs, that are taken as examples have been designed to the TS500 ultimate load design approach.