Zemin mühendisliğinde gerilme-şekil değiştirme davranışının sonlu elemanlar yöntemiyle incelenmesi

Abstract

Son yirmi yıldır gittikçe artan bir şekilde kullanılan ve her türlü yapının tasarımında ve çözümünde, özellikle inşaat mühendisliği problemlerinde kullanılan sonlu elemanlar yöntemi; malzeme özelliklerinin çeşitliliği ve sınır koşullarının farklılığı vb. faktörleri güçlükle karşılaşmadan çözüme katmaya olanak tanıyan, zeminin ve yapının davranışım da dikkate alan bir sayısal analiz yöntemidir. Geoteknik mühendisleri tarafından karmaşık ve hesaplanması uzun problemlerin çözümünde geleneksel yöntemler yerine her geçen gün daha sıkça kullanılan bu yöntemin teorisi üzerine son yıllarda önemli çalışmalar yapılmıştır. Ancak, sonlu elemanlar yönteminin teorik güçlüklerinin üstesinden gelmek için yeni matematiksel karakteristikler geliştirmek veya matematiksel temelini güçlendirmek bu tezin yapılış amacının dışındadır. Bu çalışmada ana amaç, geoteknik mühendisleri tarafından yaygın bir kullanım alam bulan sonlu elemanlar metodunun, uygulamacılara ve mühendislere yöntemin teorisi hakkında genel bir bilgi vererek çözüm tekniğinin tanıtılmasıdır. Bu maksatla sonlu elemanlar yönteminin ana prensipleri anlatılmış ve uygulamacı geoteknik mühendisleri için bilinmesi gerekli olan bölgelendirme prensibine ve modellemeye önem verilmiştir. Tez çalışmasının ikinci bölümünde ise sonlu elemanlar metodunu kullanan bir bilgisayar yazılım programı ile İstanbul Hafif Raylı Sistemi Mevhibe İnönü Tüneli 'nin ve Bilecik-Adapazarı Devlet Yolu İkilemesi Hanlıköy Köprülü Kavşağı 'nın gerilme-şekil değiştirme analizleri yapılmıştır. SIGMA/W Programı kullanılarak her iki problem detaylı olarak incelenmiş; hemen her türlü analiz yapılarak bulunan sonuçlar değerlendirilmiş ve tartışılmıştır.

The past decade has witnessed a tremendous growth of numerical methods for solution of problems in engineering science. The popularity and versatility of these techniques have been greatly enhanced by the availability of the large high-speed digital computer. 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 difficult. Needless to say, a large number of simplifying assumptions were necessary to obtain the closed-form solutions. Although this approach has provided useful solutions for many practical situations, it cannot yield realistic solutions for problems involving such complexities as non homogeneous media, nonlinear material behavior, in situ stress conditions, spatial and temporal variations in material properties, arbitrary geometries, discontinuities, and other factors imposed by geologic characteristics. Developments of mathematical approaches of the various methods are limited but still useful for engineers with various backgrounds. Alternatives to numerical methods are also available as tools of geotechnical 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 terms 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, the 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 methods is most likely necessary. Sometimes, more than one approach may be suitable if consequently employed in different phases of the design for one geotechnical project. Stress analysis problems for which no closed-form solution is available can be solved by methods of numerical approximation. These are for more powerful than closed-form solutions, but take a little more time to set up and run. Most of this time is to 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 to 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. One distinctive characteristic of numerical methods is a discretization of the problem to be solved. By comparison, closed-form and analytical methods do not require such discretization. Discretization is necessary if the problem to be analyzed is very complex or if true conditions shall be modeled with high accuracy. Discretization typically requires a large number of equations to describe the individual elements and their interrelations. Consequently, computers are used. The use of computer to solve discretized problems is implied in the term "Numerical Method". The trend to more and more complex and hence realistic model has created difficulties. Programs which can address simple linear elastic or linearly viscoelastic problems are widely understood and widely distributed. However, the complex formulations are often idiosyncratic and inadequately documented, and their behavior is only understood in detail by the person who wrote and uses the program regularly. The casual user runs a serious risk that the results may be misleading. Complex problems should therefore be referred to an analytical specialist. Numerical methods are applicable and used throughout engineering disciplines. Prevalent applications 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. The descriptions given so far have shown that numerical analysis can play an important role as a research tool in studying earth retaining structures. Its use as a design tool, however, has been somewhat limited; some of the reasons for this minimal design role may be: VII 1) Satisfaction with conventional methods 2) Difficulties with older numerical methods 3) Cost 4) Character of the geotechnical engineer Unconventional problems and development of more economical designs are clearly areas where numerical analysis tool can play a role, assuming that the analysis can be performed for a reasonable cost and with a minimum of oversimplifying assumptions. The growing numbers of successful, documented numerical analysis of practical problems suggest that these possibilities can be realized. This is particularly true for the finite element method. Given the advantages of newer numerical methods and proof that they can be used with confidence, under appropriate circumstances the geotechnical engineer should find them attractive accessories to his/her conventional design and his/her engineering judgment. It remains clear, however, that the use of numerical methods should never preclude the generous use of engineering judgment, because while the results obtained may be more detailed and comprehensive, simulations of the captious elements of nature and man's work are always highly idealized. 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 modern computation are such residual methods as the method of least squares and such variational methods as the Ritz method. In contrast to the techniques mentioned above, the 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. 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 seepage, consolidation, and heat flow. If the correct physical coupling differential equations can be written (e.g. coupling water flow, heat flow, effective stresses, volume changes) the finite element method can generate a solution. VIII The basis of the finite element method is the representation of a body or structure by an assemblage of subdivisions called "finite elements". These elements are considered interconnected at joints which are called "nodals" or "nodal points". Simple functions are chosen to approximate the distribution or variation of the actual displacements over each finite element. Such assumed functions are called "displacement functions" or "displacement models". The unknown magnitudes or amplitudes of the displacement functions are the displacements (or the derivatives of the displacements) at the nodal points. Hence the final solution will yield the approximate displacement at discrete locations in the body, the nodal points. A displacement model can be expressed in various simple forms, such as polynomials and trigonometric functions. Since polynomials offer ease in mathematical manipulations, they have been employed commonly in finite element applications. The general description of the finite element method can be detailed in a step- by-step procedure. This sequence of steps describes the actual solution process that is followed in setting up and solving any equilibrium problem. Although the present summary is based upon a procedure developed for structural mechanics applications, it can be generalized to other fields as well. The following six steps summarize the finite element analysis procedure: 1) Discretization of the continuum: The continuum is the physical body, structure, or solid being analyzed. Discretization may be simply described as the process in which the given body is subdivided into an equivalent system of finite elements. The finite elements may be triangles, quadrilaterals for two-dimensional continuum, tetrahedra, rectangular prisms, or hexahedra for a three-dimensional continuum. 2) Selection of the displacement models: The assumed displacements functions or models represent only approximately the actual or exact distribution of the displacements. For example, a displacement function is commonly assumed in a polynomial form, and practical considerations limit the number of terms can be retained in the polynomial. 3) Derivation of the finite element equations: The derivation of the finite element equations may be achieved by direct methods, variational methods, or the residuals methods. 4) Assembly of the algebraic equations for the overall dicretized 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 force or load vector from the element nodal force vectors. The most common assembly technique is known as the "direct stiffness method". IX 5) Solution for the unknown displacements: The algebraic equations assembled in step 4 are solved for the unknown displacements. 6) Computation of the elements strains and stress from the nodal displacements: In certain cases the magnitudes of the primary unknowns, that is the nodal displacements, will be all that are required for an engineering solution. More often, however, other quantities derived from the primary unknowns, 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. Moreover, 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. The finite element method, although powerful, requires the entire continuum to be divided into elements, which is slow and expensive for larger models. Complex mine layouts, for example, are seldom analyzed by finite elements. They would require a prohibitively large number of elements end considerable time and computer storage to achieve a realistic solution. This limitation in many cases can be overcome by one of the boundary-discretization methods. Boundary-element and finite-element methods can be combined to give an economical and elegant hybrid way of solving complex geomechanic problems. A comprehensive description of the fundamentals of finite element method is beyond the scope of this thesis. Numerical procedures are now accepted and employed in practice and are included in the research and teaching at most universities. However, there is no unified treatment of the theory and applications available at this time. This thesis is divided into three parts: first two chapters contain introduction and numerical methods, following two chapters describe a general theory of the finite element method, and the remaining chapters describe a finite element software program-SIGMA/W- that can be used in stress-strain analysis of geotechnical problems. Solutions of two practical problems are also given in the thesis. This study presents the results of an analytical study of an embankment for a bridge approach, which was placed on very soft clay subgrade. Hanlıköy Interchange Bridge is a part of Bilecik-Adapazan Highway. Granular fill behind a piled bridge abutment, may generate sufficient horizontal soil movement to cause distress in the piles. This form of loading can also lead to significant movement of X the structure. To avoid these damages lots of alternatives were considered. The purpose of this study was to compare the alternative solutions. The parametric studies show that the comparison of the alternatives Bridge Approach Support Piles (BASP) construction technique is the best solution. Evaluation of tunnel stability in soil and rock conditions can be readily performed using the finite element method of stress analysis. This paper is concerned with interpretation of the stresses and displacements of Mevhibe İnönü Tunnel which is a part of Istanbul Light Rapid Transit System at KM 6+500. The tunnel, which passes through much difficult conditions, is being constructed by the use of New Austrian Tunneling Method (NATM), which offers advanced technology and cost-efficiency. The first part of the study shows the results of the initial stability analysis and the comparison of the predicted and measured deformations. The second part is a back analysis with modified parameters, which give similar predicted and measured deformations. The progress of the tunnel excavation has been modeled by a more detailed series of sub-steps: in-situ, load deformation, pre-relaxation of bench and invert, and excavation were considered in five steps. Back analysis were carried out using the same finite element computer software program SIGMA/W and the same finite element mesh as for the first analysis. Except the moduli of elasticity all other parameters were kept constant. A comparison of displacements determined from SIGMA/W program show that they are in good agreement with the results of the tunnel designer.