Şerit temellerin taşıma kapasitesi

Abstract

Şerit temellerin taşıma kapasitesi hakkında, günümü ze kadar bir çok farklı çalışmalarda bulunulmuş dolayı sıyla değişik yöntemler geliştirilmiştir. Bu tez çalışmasında, elastik olduğu farzedilen ve mükemmel olarak da plastik olan yarı belirli kütleli zemin yüzeyindeki tek bir şerit temelin sınır taşıma kapasitesi üzerinde durulmuş ve bu konu ile ilgili değişik metodlar incelenmiştir. Bu yöntemlerden sınırdenge veya plastik denge olarak adlandırılan yöntemler geleneksel olarak zeminin taşıma kapasitesi hakkında yaklaşık çözümler elde etmek için kul lanılmıştır. Bu yaklaşımın örnekleri Terzaghi(1943) ve Meyerhof (1951) "un çözümleridir. Yöntem muhtemelen bir slip-line alanının oluşturul masında en iyi yaklaşım olarak tanımlanabilir ve genellik le olduğu varsayılan bir hasar yüzeyi gerektirir. Hasar yüzeyi tarafından kuşatılmış ilgili zemin içersindeki gerilme dağılımı hakkında sağlıklı kabuller yapmak gerek lidir. Böylece etkiyen kuvvetler cinsinden taşıma kapasite si hesabı için bir denge denklemi yazılabilir. Sürekli sistemler mekaniğinin hiçbir denklemi iç ve ya dış hasar yüzeyinin herhangi bir yerinde y&teri kadar tatminkar değildir. Çünkü gerilme dağılımı hasar yüzeyi ile uyumlu bir gerilme dağılımı ve tatminkâr edici denge için gerilme sınır koşullarından ve eğilme fonksiyonunun varlığından sözedemez. Sınır denge tekniği, sınır analizinin ve uç model tekniğinin temel felsefesinden faydalanmasına rağ men sınır denge çözümün bir uç model olmadığı bir hasar yüzeyi kabulü yapılır ve en az bir yanıt aranır. VI Bütün analiz yöntemleri mükemmel plastiklik düşüncesinden faydalanmasına rağmen değişik analitik metodlarla ilgili çözümler arasındaki ilişki zemin mekaniği alanında genel kullanışı olmayan özel kavramlar ve termonoloji gerekti rir. Slip-line çözümü, basit bir model ve bundan dolayı da ger çek bir çözümdür. Ağırlığı olan zeminler için slip-line çözümlerinin çoğu tam çözüm olarak gösterilmemiştir. Düzgün pürüzsüz temel ler için bu çözümlerin tam çözüm olarak kullanılacağı söylenebilir. Pürüzlü derin temeller için sınır analiz çözümleri iyi sonuçlar vermektedir.

The problem under consideration in this work is the determination of ultimate bearing capacity of a single, strip footing bearing on a plane surface of a semi-infinite mass of soil that is assumed to be elastic-perfectly plastic material. It is further assumed that the force acted on the footing is normally and centrally loaded and increased until penetration occurs as a result of plastic flow in the soil. The load required to produce the complete plastic flow or failure of the soil support is called the critical load or the total ultimate bearing capacity. The average critical load per unit of area, qo is called the bearing capacity of the soil. The value of the bearing capacity of a soil depends not only on the mechanical properties of the soil but also on the size of the loaded area, its shape and its location with the reference to the surface of the soil. The investigation is. limited to the bearing capacity, of strip footings on horizontal bearing areas for cohesive material with internal friction, and for special cases of purely cohesive and cohesionless materials. The term "strip footing" is applied to a footing whose length is very long comparison with its width. It has a uniform width which essentially gives rise to a two- dimensional plane strain condition. In most part of this work, the soil is assumed to be an isotropic, homogeneous, and elastic-perfectly plastic material which obeys the Coulomb yield condition and the associated flow rule. The limit analysis method here in does not consider the deformation of the and the solutions obtained are essentially the same as that assuming the soil to be rigid-perfectly plastic material. VIII In this work, First a brief description of the salient features of the limit analysis and limit e equilibrium methods relevant to bearing capacity problems of footings will be given. Then we solve in succession the following problems: (a) Computation of the ultimate bearing capacity of a footing on a cohesive ponderable soil (c-A-Y soil). (b) Computation of the ultimate bearing capacity of a footing on a cohesionless ponderable soil ( 51- Y soil) and (c) Computation of the ultimate bearing capacity of a footing on a cohesive imponderable (weightless) (c-A soil). Slip-line solutions are frequently used in various of this work for comparison with the. theoretical analysis solutions. The so-called limit equilibrium or plastic equilibrium method has traditionally been used to obtain approximate solutions for the bearing capacity of soils. Examples of this approach are the solutions of Terzaghi (1943) and Meyerhof (1951). The method can probably best be described as an approximate approach to the construction of a slip-line field and generally entails an assumed failure surface. It is necessary to make sufficient assumptions about the stress distribution within the soil domain bounded by the failure surface such that an equation of equilibrium, in terms of resultant forces, may be written for bearing capacity determination. None of the equations of continuum mechanics are explicitly satified every where inside or outside of the failure surface. Since the stress distribution is not defined precisely everywhere inside of the assumed failure surface, one cannot say definitely that a stress distribution compatible with the assumed failure surface and satisfying equilibrium, stress boundary conditions and the yield function exists. Although the limit equilibrium technique utilizes the basic philosopy of the upper-bound theorem of limit analysis, that is, a failure surface is assumed and a least answer is sought, a limit equilibrium solution may not be an upper bound. The method basically gives no consideration to soil kinematics, and equilibrium conditions are satisfied only in a limited sense. IX It is chear then that a solution obtain using the limit equilibrium method is not necessarily an upper or a lower bound. However, any upper-bound limit analysis solution will obviously be a limit equilibrium solution, In the following computations, the upper-bound technique of limit analysis is employed to generate approximate solutions to the bearing capacity problems. The lower-bound technique of limit analysis is not considered, but the computer method presented by Lysmer (1970) can be applied and may give good lower-bound solutions. In which contributions to the bearing capacity from different soil and loading parameters are summed. These contributions are represented by the expression: q0« cNc + qNg -f-Y i BNy where the bearing capacity factors, Nc, N, and Ny represent the effects due to soil cohesion c, surface loading, q and soil unit weight respectively. These parameters N are all functions of- the angle of internal friction, Terzaghi's quasiempirical method assumed that these effect are directy supporposable, whereas the soil behavior in the plastic range is nonlinear and thus superposition does not hold for general soil bearing capacities. Meyerhof (1951), using Terzaghi's concept, presents numerical results for shallow and deep footings by assuming failure mechanisms for the footing and by presenting results in the form of bearing capacity factors, N. The reason for using the simplified method (superpositon method) is largely due to the mathematical difficulties encountered when the conventional limit equilibrium method is used. Since the investigation of the influence of the weight of a soil on the plastic equilibrium of a footing has not yet passed beyond the stage of formulating the differential equations and integrating numerically for a given individual problem the general bearing capacity problem can best solved in two stages: The first stage is essentially based on an extension of the analytical work of Prandtl (1920) and Reissner (1924) ; X this assumes a weightless material and gives the first part of the bearing capacity cNc - qN^ in closed form expressions. The second stage is a semigraphical treatment based on an extension of the work of Ohde(1938), or the numerical integration scheme of Sokolovskii (1965), or the graphical method of de Jong (1957) j this takes the weight of the material into account and gives the second part of the bearing capacity Y( ^B)NY in table or chart form. The upper-bound technique of limit analysis is used herein to develop approximate solutions for the bearing capacity of strip footing on a general cohesive soil with weight. Analytical solutions are first obtained for smooth and rough and surface and subsurface footings. Numerical results are then calculated and compared with existing slip-line and limit equilibrium solutions. The limit analysis solutions for smooth, surface footings are shown to compare favorably with slip-line solutions. Meyerhof's solutions and the limit analysis solutions for rough, subsurface footings are shown to agree remarkably well. The upper-bound technique of limit analysis is used here to develop approximate solutions for the bearing capacity of cohesive soils with weight. Solutions are presented for smooth and rough and surface and subsurface footings (shallow and deep). Soil is treated as a perfectly plastic medium with the. associated flow rule. The limit analysis solutions for smooth surface footings are shown to compare fovorably with slip-line solutions. Meyerhof's solutions and the limit analysis solutions for rough subsurface footings are shown to agree to agree remarkably well. The problem of the indentation of semi-infinite medium by a smooth, rigid wedge under conditions of plane strain was first solved by Hill et al. (1947) for a perfectly plastic Tresca material. An approximate method of solution to this prolem was given later by Hodge (1950). Following the work of Hill et al., Shield (1953) obtained the solution of the same problem for the more general case of Coulomb material. Paslay at al. (1968) presented a solution of this problem using a special yield surface which was believed to be particularly appropriate for rock subjected to high hydrostatic pressure such as those found during deep driling below the earth's surface. The essential feature of this problem is that the plastic region changes in such a way that its configuration XI always retains geometrical similarity to some inital state. The simplest examples of this type of deformation other than the wedge footing considered above are problems of the expansion of cylindrical and spherical cavities in unbounded space, beginning from zero radius. The problem considered here is limited to the case of symmetric, rigid (non-deformable) wedge. Friction at the contact surface is neglected (the surface is lubricated). A number of other problems of this type of geometric similarity (oblique indentation by a rigid wedge, compression of a wedge by a rigid plane, etc.) have been studied by Hill (1950) and other authors for a perfectly plastic Tresca material. The upper-bound technique of limit analysis is used herein to develop approximate solutions for the bearing capacity of cohesive soils with weight. Solutions are presented for smooth and rough and surface, shallow and deep footings. Soil is treated as a perfectly plastic medium with the associated flow rule. The ümit analysis solutions for smooth surface footings are shown to compare favorably with slip-line solutions. Meyerhof's solutions and the limit analysis solutions for rough, shallow and deep footings are shown to agree remarkably well. It has been shown that the upper-bound technique of limit analysis can predict bearing capacities of cohesive ponderable soils with internal friction to within a reasonable degree of accuracy, for I ranging from 0° to 40° and G ranging from 0 to 5. At the least, it can be said that the results compare favorably with existing limit equilibrium solutions. The most forceful argument for the adoption of the limit analysis method is the fact that is rational basis allows it to be conveniently extended to more complex bearing capacity problems. For example, as demonstrated here, the limit analysis method can be easily adopted to the solution of layered soils. More important, the method can be used to obtain upper and lower-bound solutions of the three dimensional bearing capacity problem where exact solutions of the equations of plasticity are all but impossible except for the most elementary of problems.