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Yatak katsayısı ve temel yapılarına uygulanması

Yatak katsayısı ve temel yapılarına uygulanması

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Develioğlu, İlker N.

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu çalışmada, esasları Winkler tarafından verilmiş olan yatak katsayısı kavramı incelenmiştir. Yatak katsayısının deneysel olarak ve diğer zemin özelliklerine bağlı olarak ampirik bağıntılarla ne şekilde elde edilebileceği anlatılmıştır. Yatak katsayısı temeli üzerine kurulmuş olan çeşitli hesap yöntemlerinden kısaca bahsedilmiş ve bu hipotezden yararlanılarak sonlu elemanlar yönteminin kullanılması ile geliştirilmiş olan bir bilgisayar programı verilmiştir. Yalak katsayısının zeminlere ait fiziksel tanımlayıcı bir özellik olmayışı sebebi ile uygulamada ortaya çıkan tereddütlerin giderilmesi amacı ile orta sıkılıkta bir kum zemin üzerine oturmuş sürekli bir temel değişik yatak katsayısı değerleri dikkate alınarak incelenmiş, yatak katsayısı değişiminin tasarımda kullanılacak moment ve ani temel oturmalarına etkisi gösterilmiştir.

Almost all problems in foundation engineering are concerned with stresses and deformations in the soil mass due to the boundary conditions and body forces. The elasticity and plasticity theories have been widely used to solve these problems. And even elasto plastic theories were utilized in order to obtain more realistic results. Nevertheless, none of the proposed methods could be succeeded at all. The reality lies behind these fails is mainly based on the soil itself which is not easily modeled by any of the theories. Soil is not a material that can be defined exactly such as a concrete, steel, plastic or any other material used in civil engineering. A lot of researcher has been dealing with not only engineering properties of soils but also the application methods of those properties to engineering systems. Because of the lack of resources and huge commercial competition, they will continue to investigate the most realistic model. The aim of this study is to describe concept of subgrade modulus, determination methods, ite relationship with the elastic properties of soil and to develop a user friendly computer program which uses concept of subgrade modulus. The main module of this computer program is relied on a computer program developed and given by Bowles [5]. A short summary of this study was mentioned below with the same sequence. The modulus of subgrade, ko, is only a conceptual relationship with the pressure and soil movement. This approach was presented by Winkler in 1867 [2], and has been widely used since then. The foundation material representation which he adopted is now called a Winkler foundation. It is also called one parameter model since it describes foundation behavior just with a proportionally constant k. Before getting into more details regarding determination and using of subgrade modulus, basic concepts of Winkler approach and recent improvements over his theory should be discussed firstly. According to Winkler theory only the loaded area settles while adjacent fields stay unchanged. So it assumes the soil as a disconnected medium. But in real situation, when a load applied to the surface of a linearly elastic half-space not only loaded area deflects but also unloaded adjacent areas moves down with diminishing displacements with distance as the soil is a connected continuum. A linearly elastic continuum is described by two material properties, Young's modulus E and Poisson's ratio t>. Some researchers studied on disability of Winkler approach for describing soil behavior, may be the most attractive one is Vlasov and Leontiev's model. It enables deflections outside the loaded region to be effected and permits both deflections and moments to be matched. Various improvements along this line have been suggested by a number of authors including Wieghardt, Filonenko-Borodich, Hetenyi and Pasternak [2]. All their additional assumption to Winkler's, the tops of the soil springs forming the ground surface are tied together by a stretched elastic string or membrane or shear beam. The tension, S, in this string is the second soil property. Vesic showed with as a result of his experiments that Winkler model represents the behavior of soil fairly well. In this study half of the attention was paid to the determination of subgrade modulus with direct experimental studies or with using other experimental date. Soil modulus, is usually obtained with plate loading tests and laboratory experiments. And sometimes other experimental data such as Standard penetration test results or Holland penetration test results can be used to get information about value of soil modulus. As a recent development, using of a dilatometer test data proposed by Gabr, Lunne and Powell to obtain subgrade modulus was described in this study. Also, in chapter 2, a number of empirical relationships were established among subgrade modulus and elastic properties of soil and geometric structure of foundation including equations of Terzaghi, Terzaghi and Peck, Vesic, Broms, Kögler and Scheidig. The meaning of horizontal and vertical modulus was also described in the same section. If the soil response to a beam resting on its surface is that of a linear spring, then its behavior against an embedded beam or pile can be represented with two springs, one at the front and the other at the rear of the pile. This assumption ignores the shearing reactions along the pile sides. It shows that in an isotropic soil the basic subgrade reaction coefficient ko to be used in pile studies is twice the value that would be employed for the same member acting as a beam at the surface of the same soil. Concepts for constant or variable k, soil modulus was discussed more detailed in chapter 2. As a summary, for both sands and clays, the elastic modulus does not vary with the width of the piles. For stiff clay, the coefficient may be assumed to be constant with depth. And it was shown that subgrade modulus can vary with the depth of pile. Subsequent chapters are concerning with the fundamentals of finite element method, and its use for the foundation engineering together with the subgrade modulus. XI Use of elastic and plastic theories together or separately for solving foundation problems has great difficulties for practical purposes. The main disadvantages of these approaches are determining the soil structure interaction and behavior of the soil under different load cases because of the computational difficulties. The solutions to this type of problems have been obtained easily by the finite element method. According to the finite element method, soil mass is assumed as if it consists of a finite number of discrete elements interconnected at a finite number of nodal points. The properties of the elements are adjusted so that the assemblage of elements behaves in the same manner as the original continuum. Behavior of every discrete element is known with using various techniques of structural analysis. The basic steps of finite element solution can be summarized as followings; 1st step - Dividing of the continuum into finite elements, 2nd step - Building stiffness and load terms of an arbitrary element with respect to a convenient local coordinate system. 3rd step - Development of a transformation matrix ( it is also called constraint matrix) to transform the stiffness matrix from the local coordinate system to a generalized coordinate system. 4th step - Generation of the final stiffness matrix for the entire assemblage of elements, incorporating the boundary forces, body forces and deflections. 5th step - Solution of the resulting linear simultaneous equations for the unknown nodal forces. 6th step - Evaluation of subsidiary element quantities such as stresses in the displacement method. Generally, below mentioned three conditions must be satisfied in the theory of finite element analysis in order to develop the stiffness matrix equation. 1. Deformations of adjacent elements must be compatible. 2. The forces acting on a finite element must be in equilibrium. 3. The displacements of each element as a result of the applied forces must be consistent with the physical properties of the material. xu So far, the finite element method has been defined briefly. In this study, finite element method was used to solve problems of beam on an elastic subgrade and laterally loaded piles. A combined footing can be analyzed by the conventional rigid method or beam on elastic foundation method. The second, beam on elastic foundation method was developed by Winkler for computing rail road deflections under vehicles' loads. Later, Heteny developed Winkler's equations for a load at any point along a beam measured from left end. Each method is called classical solution of beam on elastic subgrade. The most efficient technique to solve this kind of problems is the finite element method. It is easy to account for boundary conditions, beam weight and nonlinear effects, including footing separation. The classical solutions of beam on elastic foundation type problems have several disadvantages over the finite element method, such as; - Assumes weightless beam. In reality state weight is a factor when footing tends to separate from soil. - Difficult to remove soil effect when footing tends to separate from soil. - Difficult to account for boundary conditions of known rotation or deflection at selected point. - Difficult to apply multiple types of loads to a footing. - Difficult to change footing properties of moment of inertia, thickness and width of beam. - Difficult to allow to change subgrade reaction along footing. The data structure and fundamentals of given computer program were described in chapter 5, As mentioned above, program was initially developed by Bowles. The program was modified for entering data more easily and adding some abilities to control over program. Bowles originally created his program with using FORTRAN IV language. It is translated into Basic programming language and using C++ programming language to establishing menus. A working copy of programs stored in a 3.5" 1.44 Mb MS DOS formatted diskette was attached. It should be added that no special computer configuration is necessary for running this program. In chapter 6, a continuous footing problem was solved thanks to given computer program with several values of subgrade modulus. The foundation soil assumed as a medium dense sand. Minimum and maximum values of subgrade modulus for medium dense sand were taken from table 2.6. This range of values divided into four parts and results were obtained for these four values. Moment diagrams and settlement curves were shown for each result data. Xlll As a conclusion, it can be seen from the discussion in previous sections that for a uniform, homogeneous linearly elastic subgrade subjected loads applied through foundation beams, the calculation of displacements, moments and stresses involves lengthy and difficult computations. Especially when layers of material of different properties are involved, only theoretical solutions are available. Because of this situation, design engineers turned their attention to the Winkler foundation representation, since it offered simpler mathematical relations. It is shown that moments and stresses in a beam are quite independent from selected values of subgrade modulus. According to the example given here, computed moments and soil stresses did not hugely changed despite the wide selection range of subgrade modulus. So, Winkler approach is not a sensitive method to describe soil behavior and it should be used for initial design values or for the foundations which have not a great importance such as ordinary building foundations. It should be expressed that in case of a multi story building or dynamically loaded foundations, behavior of foundation has to be defined more realistic and precise form than the Winkler approach.

Almost all problems in foundation engineering are concerned with stresses and deformations in the soil mass due to the boundary conditions and body forces. The elasticity and plasticity theories have been widely used to solve these problems. And even elasto plastic theories were utilized in order to obtain more realistic results. Nevertheless, none of the proposed methods could be succeeded at all. The reality lies behind these fails is mainly based on the soil itself which is not easily modeled by any of the theories. Soil is not a material that can be defined exactly such as a concrete, steel, plastic or any other material used in civil engineering. A lot of researcher has been dealing with not only engineering properties of soils but also the application methods of those properties to engineering systems. Because of the lack of resources and huge commercial competition, they will continue to investigate the most realistic model. The aim of this study is to describe concept of subgrade modulus, determination methods, ite relationship with the elastic properties of soil and to develop a user friendly computer program which uses concept of subgrade modulus. The main module of this computer program is relied on a computer program developed and given by Bowles [5]. A short summary of this study was mentioned below with the same sequence. The modulus of subgrade, ko, is only a conceptual relationship with the pressure and soil movement. This approach was presented by Winkler in 1867 [2], and has been widely used since then. The foundation material representation which he adopted is now called a Winkler foundation. It is also called one parameter model since it describes foundation behavior just with a proportionally constant k. Before getting into more details regarding determination and using of subgrade modulus, basic concepts of Winkler approach and recent improvements over his theory should be discussed firstly. According to Winkler theory only the loaded area settles while adjacent fields stay unchanged. So it assumes the soil as a disconnected medium. But in real situation, when a load applied to the surface of a linearly elastic half-space not only loaded area deflects but also unloaded adjacent areas moves down with diminishing displacements with distance as the soil is a connected continuum. A linearly elastic continuum is described by two material properties, Young's modulus E and Poisson's ratio t>. Some researchers studied on disability of Winkler approach for describing soil behavior, may be the most attractive one is Vlasov and Leontiev's model. It enables deflections outside the loaded region to be effected and permits both deflections and moments to be matched. Various improvements along this line have been suggested by a number of authors including Wieghardt, Filonenko-Borodich, Hetenyi and Pasternak [2]. All their additional assumption to Winkler's, the tops of the soil springs forming the ground surface are tied together by a stretched elastic string or membrane or shear beam. The tension, S, in this string is the second soil property. Vesic showed with as a result of his experiments that Winkler model represents the behavior of soil fairly well. In this study half of the attention was paid to the determination of subgrade modulus with direct experimental studies or with using other experimental date. Soil modulus, is usually obtained with plate loading tests and laboratory experiments. And sometimes other experimental data such as Standard penetration test results or Holland penetration test results can be used to get information about value of soil modulus. As a recent development, using of a dilatometer test data proposed by Gabr, Lunne and Powell to obtain subgrade modulus was described in this study. Also, in chapter 2, a number of empirical relationships were established among subgrade modulus and elastic properties of soil and geometric structure of foundation including equations of Terzaghi, Terzaghi and Peck, Vesic, Broms, Kögler and Scheidig. The meaning of horizontal and vertical modulus was also described in the same section. If the soil response to a beam resting on its surface is that of a linear spring, then its behavior against an embedded beam or pile can be represented with two springs, one at the front and the other at the rear of the pile. This assumption ignores the shearing reactions along the pile sides. It shows that in an isotropic soil the basic subgrade reaction coefficient ko to be used in pile studies is twice the value that would be employed for the same member acting as a beam at the surface of the same soil. Concepts for constant or variable k, soil modulus was discussed more detailed in chapter 2. As a summary, for both sands and clays, the elastic modulus does not vary with the width of the piles. For stiff clay, the coefficient may be assumed to be constant with depth. And it was shown that subgrade modulus can vary with the depth of pile. Subsequent chapters are concerning with the fundamentals of finite element method, and its use for the foundation engineering together with the subgrade modulus. XI Use of elastic and plastic theories together or separately for solving foundation problems has great difficulties for practical purposes. The main disadvantages of these approaches are determining the soil structure interaction and behavior of the soil under different load cases because of the computational difficulties. The solutions to this type of problems have been obtained easily by the finite element method. According to the finite element method, soil mass is assumed as if it consists of a finite number of discrete elements interconnected at a finite number of nodal points. The properties of the elements are adjusted so that the assemblage of elements behaves in the same manner as the original continuum. Behavior of every discrete element is known with using various techniques of structural analysis. The basic steps of finite element solution can be summarized as followings; 1st step - Dividing of the continuum into finite elements, 2nd step - Building stiffness and load terms of an arbitrary element with respect to a convenient local coordinate system. 3rd step - Development of a transformation matrix ( it is also called constraint matrix) to transform the stiffness matrix from the local coordinate system to a generalized coordinate system. 4th step - Generation of the final stiffness matrix for the entire assemblage of elements, incorporating the boundary forces, body forces and deflections. 5th step - Solution of the resulting linear simultaneous equations for the unknown nodal forces. 6th step - Evaluation of subsidiary element quantities such as stresses in the displacement method. Generally, below mentioned three conditions must be satisfied in the theory of finite element analysis in order to develop the stiffness matrix equation. 1. Deformations of adjacent elements must be compatible. 2. The forces acting on a finite element must be in equilibrium. 3. The displacements of each element as a result of the applied forces must be consistent with the physical properties of the material. xu So far, the finite element method has been defined briefly. In this study, finite element method was used to solve problems of beam on an elastic subgrade and laterally loaded piles. A combined footing can be analyzed by the conventional rigid method or beam on elastic foundation method. The second, beam on elastic foundation method was developed by Winkler for computing rail road deflections under vehicles' loads. Later, Heteny developed Winkler's equations for a load at any point along a beam measured from left end. Each method is called classical solution of beam on elastic subgrade. The most efficient technique to solve this kind of problems is the finite element method. It is easy to account for boundary conditions, beam weight and nonlinear effects, including footing separation. The classical solutions of beam on elastic foundation type problems have several disadvantages over the finite element method, such as; - Assumes weightless beam. In reality state weight is a factor when footing tends to separate from soil. - Difficult to remove soil effect when footing tends to separate from soil. - Difficult to account for boundary conditions of known rotation or deflection at selected point. - Difficult to apply multiple types of loads to a footing. - Difficult to change footing properties of moment of inertia, thickness and width of beam. - Difficult to allow to change subgrade reaction along footing. The data structure and fundamentals of given computer program were described in chapter 5, As mentioned above, program was initially developed by Bowles. The program was modified for entering data more easily and adding some abilities to control over program. Bowles originally created his program with using FORTRAN IV language. It is translated into Basic programming language and using C++ programming language to establishing menus. A working copy of programs stored in a 3.5" 1.44 Mb MS DOS formatted diskette was attached. It should be added that no special computer configuration is necessary for running this program. In chapter 6, a continuous footing problem was solved thanks to given computer program with several values of subgrade modulus. The foundation soil assumed as a medium dense sand. Minimum and maximum values of subgrade modulus for medium dense sand were taken from table 2.6. This range of values divided into four parts and results were obtained for these four values. Moment diagrams and settlement curves were shown for each result data. Xlll As a conclusion, it can be seen from the discussion in previous sections that for a uniform, homogeneous linearly elastic subgrade subjected loads applied through foundation beams, the calculation of displacements, moments and stresses involves lengthy and difficult computations. Especially when layers of material of different properties are involved, only theoretical solutions are available. Because of this situation, design engineers turned their attention to the Winkler foundation representation, since it offered simpler mathematical relations. It is shown that moments and stresses in a beam are quite independent from selected values of subgrade modulus. According to the example given here, computed moments and soil stresses did not hugely changed despite the wide selection range of subgrade modulus. So, Winkler approach is not a sensitive method to describe soil behavior and it should be used for initial design values or for the foundations which have not a great importance such as ordinary building foundations. It should be expressed that in case of a multi story building or dynamically loaded foundations, behavior of foundation has to be defined more realistic and precise form than the Winkler approach.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

##### Anahtar kelimeler

Jeoteknik,
Yatak katsayısı,
Geotechnics,
Subgrade modulus