Kazı ve kazık problemlerinin sonlu elemanlar metodu yardımı ile çözümü

Abstract

Son yıllarda, özellikle son 15-20 yıl içerisinde gittikçe artan bir şekilde kullanılan ve her türlü yapının tasarımında, çözümünde, özellikle inşaat mühendisliği problemlerinde kullanılan sonlu elemanlar metodu; malzeme özelliklerinin değişkenliği ve sınır koşullarının farklılığı v.b. faktörleri kolaylıkla çözüme dahil eden, zeminin ve yapının davranışlarım de dikkate alan bir nümerik metoddur. Karmaşık, hesaplanması uzun zaman alan, kaya ve zemin mekaniği problemlerinin Çözümünde geleneksel metodlar yerine, her geçen gün daha da sıkça kullanılan yöntem; sonlu eleman metodudur. Bu tez çalışmasının amacı; kazı ve kazık problemlerinin sonlu elemanlar metodu ile gerilme, yerdeğiştirme analizlerinin yapılması esasına dayanmaktadır. Bu konu ile ilgili olarak, isimleri KAZI2 ve KAZI3, ve FORTRAN programala dili ile kodlanan iki adet bilgisayar programı geliştirilmiştir. KAZI2 programı; Düzlem gerilme şekil değiştirme problemlerine çözüm getirmektedir. Tez çalışmasında bu programla, bir kazak ve iki adet de temel-zemin etkileşimi ile ilgi olmak üzere üç adet sayısal ömek çözülmüştür. Bu örneklerle ilgili sonuçlar grafik olarak değerlendirip, tez sonuna ilave edilmiştir. KAZT3 programı; Zemin veya kaya kütlesinde yapılan kaza çalışması sonucu, zeminde oluşan gerilmelerin analizini yapmaktadır. Bu programla, değjşik şekilli sonlu elemanlar kullanılarak 3 adet sayısal örnek çözülmüştür, örneklerle ilgili sonuçlar grafik olarak değerlendirilip, tez sonuna ilave edildi. Sonlu elemanlar metodunu kullanan KAZI2 ve KAZI3 programlarından elde edilen gerilme, yerdeğiştirme analiz sonuçlan denge denklemleri ile karşılaştırılarak, sonuçlar kabul edilir hata ile denge denklemlerini sağladığı görüldü

In recent years, numerical methods have continued to expand and diversify into all the major fields of scientific and engineering studies. They have become popular due to rapid advancements in computer technology and its availability to engineer. In these methods, Finite Element Method (F.E.M) is the most popular method in engineering science. It has been applied to a large number of problems in widely different fields. It has now applications in a wide variety of field such as solid mechanics. Fluid mechanics, biomechanics, electricity and magnetism, heat transfer, semiconductor devices, etc. The method essentially involves dividing the body in smaller 'elements' of various shapes ( triangules or rectangules in two-dimensional cases and tetrahedron or 'brick' in three-dimensional cases, etc. ) held together at the 'nodes' which are corners of elements. The more number of element used to model the problem, the better approximation to the solution obtained. Displacement at the nodes are treated as unknowns and are calculated. The displacement at any point with an element are related to the displacement at the nodes by making certain assumptions. Displacements are fundamental variables. From the displacement field within the element, strain can be calculated. From the strains, using the stress-strain relationship, stresses can be calculated. The major disadvantage of the method is that considerable effort is required in preparing data for a problem and using a large high-speed digital computer. Preparing data a problem is particularly crucial in three-dimensional problems and has led to 'mesh generetion' programs. These program produce (to a large extent) the input data required for the Finite Element Method program. The method is also expensive in computer time. A large set of simultaneous equations ( Several hundreds to several thousands ) have to be solved to obtain solutions. The computer time goes up further if the problem is nonlinear, i.e. stress-strain relationship is not linear-which usually is the case. For a non linear problem, the set of simultaneous equations are required to be solved a lot of times. Inspite of the above disavantages, Finite Element Method ( F.E.M) has been extremely popular with engineers. Because the method gives lots of advantages. The methods strength lies in its generality and flexibility to handle all types of loads, sequences of construction, installation of supports etc. XUl In the past, soil and rock were considered essentially empirical disciplines. The enormous complexities encountered in natural states of geologic media can make analytical closed-form approaches very difrucult. Needless to say, a large number of simplifying assumptions were necessary to obtain the closed-form solution, it can not yield realistic solutions for problems involving such complexities as nonhomogeneous media, nonlineer material behavior, temporal variations in material properties, arbitrary geometries, discontinuities, and other factors imposed by geologic characteristics. Rock and soil mechanics have gained considerable importance in recent years due to a number of factors. One of these factors is large underground caverns which have considerable advantages over above ground structures for the storage of fuels like liquid petroleum gas, and other materials. Europe and U.S. A have man-made caverns of acpacities in excess of one million cubic metres. Also underground storage of radioactive nuclear waste is being considered in many countries. Others factors, human being started building underground powerstations, tube passageway, tunnels etc. In Turkey recent years, tunnelling and excavation problems have gained so improtance because of multistory building like tower, irrigation like G. A.P project, transportation like highways etc. Before the advent of the computers, the rock and soil structures were designed largely based on rules of thumb, experience and a trial and error procedure. Rules of thumb are invariably based on the past experience of the designer. They usually tend to be oversafe and are basically applicable to the situations similar to the ones for which they were developed. Engineers of today are many a time, faced with problems for which no past experience is available. It is also difficult to 'teach' past experience. The civil or mining engineering construction is usually a "one off " situation every time. The increased consciousness amongst the public regarding safety and economy has led the engineers to seek more rational solutions to the problems in rock mechanics related to civil and mining engineering. Analytical or closed form solutions are available for simpler situations or can be developed. However, they can in most cases be developed assuming rock as a linear elastic material which is a very drastic simplification. Numerical methods have, therefore, become very popular for solving problems in rock and soil mechanics. A number of numerical methods are available for solving problems of load deformation. By the term load-deformation problem, it means a problem in which a rock mass of arbitrary shape (this includes openings of arbitrary shape) is subjected to loads due to self weight, external forces, in situ stresses, temperature changes, fluid pressure, prestressing, dynamic forces, etc. and seek to find the deformation, strains and stresses throughout the rock and soil mass. XIV Developments of mathematical approaches of the various methods are limited but still useful for enginers with various backgrounds. Alternatives to numerical methods are also available as tool of civil engineering. The three alternatives most commonly used are: 1) Closed-form Methods 2) Analytical Methods 3) Numerical Methods Closed-form methods, analytical methods, and numerical methods differ in term of their capability to simulate actual conditions. Also different costs are associated with each method. A computational method should be used that best satisfies the specific need. If a simple problem is to be solved, a simple computational method may be sufficient. For a simple problem, use of a numerical method might mean an inefficient utilization of computational resources. If a complex problem is to be solved, the use of numerical method is most likely necessary. Sometimes, more than one approach method be suitable if consequently employed in different phases of the design for one engineering project. Numerical methods are applicable and used throughout engineering disciplines. Prevalent application to civil engineering problems are the analysis of stress, strain, and deformations. Also, the analysis of fluid flow and heat transfer through porous media is often performed by numerical methods. Tunnel engineering, for both civil and mining purposes, may involve all of these applications. Stress analysis problems for which no closed-form solution is available can be solved by methods of numerical approximation. These methods are for more powerful than closed-from solution, but take a little more time to set up and run. Most of the these time is o define the model and to prepare the input data. The time requirements are being reduced by automated mesh generators and user-friendly input control software, which are opening up numerical approaches at the non specialist engineer. The further time required for a computer program to iterate and converge on a solution is being greatly reduced with the advent of true desktop dedicated processors. Many of the numerical methods developed before the era of electronic computers are now adapted for use with these machines. Perhaps the best known is finite difference method. Other types of classical methods that have been adapted to modem computation are such residual methods as the method of least squares and such variational method as the Rite method. In contrast to the techniques mentioned above, finite element method is essentially product of the electronic digital computer age. Therefore, although the approach shares many of the features common to the previous numerical approximations, it possesses certain characteristics that take advantage of the special facilities offered by the high-speed computers. XV The finite element method is the most powerful and versatile technique for general stress analysis of surface and underground in rock and soil. It can accommodate two and three-dimensional situations, elastic, plastic and viscous materials, and can incorporate "no-tension" zones, joints, faults, and anisotropic behavior. The method is also used to solve problems of consolidation, heat flow. If the correct physical coupling differential equations can be written ( e.g. cooling water flow, heat flow, effective stresses, volume changes ), the finite element method can generate a solution. The general description the finite element method can be detailed in a step by step procedure. This sequence of steps describe the actual solution process that is followed in setting up and solving any equilibrium problem. 1. The first step is to divide the continuum or solution domain into the element. The continuum or solution domain is the physical body, structure, or solid being analysed. The element shape and sizes have to be convenient for the form of boundary and the property of continuity of the domain, sometimes it is not only desirable but also neceessary to use different types of elements in the same problem. Although the number and the type of elements to be used to in a given problem are matters of engineering judgement, the analist can rely on the experience of the others for quideliness. 2. The second step is to assign nodes to each element and to determine the necessary number of the nodal parameters. The choose type of interpolation function to represent the variation of the field variable over the element. The field variable may be a scaler, a vector or a high-order tensor. Generally, polynomials are often selected as interpolation function for the field variable. Beacause they are easy to integrate and differentiate. The degree of the polynomials chosen depends on the number of nodes and certain continuity requirements imposed at nodes and along the element boundaries. The magnitude (or magnitudes) of field variable as well as the magnitude of its derivatives at nodes may be taken as node parameters. 3. The step third: Once the finite elemnt model has been established ( that is once the elements and their interpolation functions have been selected. ), Users are ready to determine equations expressing the properties of the individual elements. The derivation of the finite element equation may be achieved by direct methods, variational methods, or the residuals methods. 4. The step fourth: If the local coordinates axes are used for determining the behavior of equation of the elements, they should be transformed to the global coordinate system 5. The step fifth: If there are any nodes apart from the comers, especially in the element, they should be compensated for. 6. The step sixth: Assembly of the algebraic equations for the overall discretized continuum. This process includes the assembly of overall or global stiffness matrix for the entire body from the individual element stiffness matrices, and the overall or global XVI force or load vector from the element nodal force vectors. The most common assembly technique is known as the "direct stiffness method" 7. The step seventh: In this step, the boundary conditions should be introduced into the system of the system behaviour equations. 8. The step eighth: The algebraic equations assembled in step six (system equations) are solved for the unknown nodes parameters. If the equations in the system are linear, a number of standart solution techniques can be used easily. But if the equations are nonlinear, their solution is more diffucult to obtain. 9. The step ninth: Computation of the elements nodal parameters (displacements ) and calculation of strains and stress. In certain cases the magnitudes of the primary unknows, that is the nodal displacements, will be all that are required for an engineering solution. More often, however, other quantities derived from the primary unknows, such as strains and/or stresses, must be computed. Finite element models can also cope with yielding behavior. The solution checks for yield and when detected, applies out-of-balance forces at the nodal points to restore the stresses in the element to the yield limit. Morover, it is economical in computer time and storage because larger elements can be used in regions of lesser interest. Furthermore, finite element models are capable of given a realistic result, particularly for complicated geomechanics situations with intricate geological conditions and openings of complex geometry. This thesis is divided into eight chapter. In the first chapter, some information is given about excavation problems in the soil and rocks and were mentioned about the thesis of aims. In addition, the same chapter desribes computer programs» KAZI2 and KAZI3, which have been developed for excavation and piles problems using by Finite Element Method. The second chapter describes the finite element method and gives history of the method, information about advantages and disavantages of the method and the field of problems that may be applied finite element method,were mentioned. Following chapter describes subdividing continuum domain for finite element method, alternative approaches to finite element formulation, briefly summary of the method step by step, and shape functions. The fourth and fifth chapters describe a general theory of the finite element method for one (bar or truss) and two dimensional problems (plane stress and plane strain). In addition, in the fifth chapter, a application was solved by handle for an example (wasn't used computer program). In the sixth chapter, the theory of an axisymmetric problem was explained and some figures were given to illustrate. XVll The seventh chapter gives some information about programs, KAZI2 and KAZI3, which can be used in stress and strain analysis of excavation and piles problems, these programs have been developed for this thesis. In addition, input data into the computer programs, KAZI2 and KAZI3, has been explained. In the last chapter, conclusion of the subject which is stresses analysis in the soil or rock mechanics, has been explained. End of the thesis, solution of the six practical problems were given and the results of these problems were given as graphics.