Analysis of axisymmetric bore-type foundation in respect of plastic deformation/Ašiai simetrinio gręžininio pamato analizė įvertinant plastines deformacijas
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Abstract
In order to design efficient foundations, it is necessary to know exact behaviour of them and surrounding soil under the load. At present various numerical methods [1–11] are used to determine such response. The behaviour of axi-symmetric bored foundation is described in this paper. The finite element method is utilized in analysis of the foundation. Linear and non-linear properties of material are taken into account.
The investigation of properties of soil, predominating in Lithuania, and economical constructing of foundation gives preference to bored foundation [12]. Schematically this type of foundation can be depicted as a cylindrical body resting on soil (Fig 1). Geometrically the foundation can be described through the diameter d and the height h. F denotes the vector of the axisymmetric load. Such bored foundation has ratio h/d ≥ 2 and transmit part of the external load through their side surface to soil. It is very difficult to achieve a shear failure of soil mass for such a type of foundation, but the foundation may suffer significant deflections. It is, therefore, important to known the stress-strain state of soil for design purpose.
Various stress-strain models have been proposed for representing the behaviour of soil [14]. These range from very simple linear-elastic to complex elastic-plastic models. In general, the stress vector σ is related to the strain vector ε through the elasticity matrix [C] (1) [1,2]. The linear-elastic stress-strain model is the simplest. In this case matrix [C] is constant and history independent (2) [4]. More complex is the non-linear-elastic model. Two incremental linear-elastic approaches can be used to handle this problem [4]. In the first case a tangent and in the other—secant modulus are used. The described models imply that volume changes are induced by changes in mean normal effective stress alone, while shear strains are induced by shear stress alone. Investigation shows that volumetric strains are induced by changes in shear stress as well as by changes in the mean normal stress [4, 11]. This can be accounted for the dilatant-elastic stress-strain model. The incremental shear-induced volume change Δε_{v} can be expressed in terms of a tangent dilation parameter α_{ϵ} [10] according to (4), in which Δγ is the increment of maximum shear strain [4]. The dilatant-elastic materials lead to a three parameter stress-strain model in which the increments of volumetric and shear strain are related to the corresponding stress increments according to (5).
The most complex is the elastic-plastic stress-strain model. A basic assumption of elastic models is that the unloading path is identical to the loading path. This is generally not true for soils where the recoverable strain upon unloading is generally small. The recoverable strain is considered to be elastic, while the non-recoverable strain is considered to be plastic. There have been proposed various yield conditions to model those plastic properties of soils. Von Mises yield condition can be written (6) in terms of the second invariant of stress deviator J _{2} and yield stress Y(κ) from uniaxial tests [2]. For soils, concrete and other ‘frictional’ materials the Drucker and Prager law (7) is frequently used [2,13]. In this law hydrostatic press σ_{m} is incorporated, while c and Φ are the cohesion and angle of friction, respectively [8].
The problem is formulated and analysed by the finite element method. The region of a model is subdivided into discrete elements. The global system of equations to be solved is described by the equation (9), where [K(U)] denotes the global non-linear stiffness matrix, U is the unknown deflection vector and F is the vector of nodal forces [1,2]. The matrix [K(U)] and the vector F can be made up by adding up the element stiffness matrices [k _{e}(u _{e})] and the element nodal deflection vectors f _{e}, respectively. Therefore the problem can be mathematically described through the governing equation of the separate element (10), where u _{e} denotes the unknown element nodal deflection. The stiffness matrix can be determined from the principle of virtual work. This involves equating the work done by the internal stresses with that done by the nodal forces.
In a non-linear analysis the matrix [k _{e}(u _{e})] depends on the vector u _{e} and solution of a problem must be obtained throughout the complete history of incremental load application [1]. Time is a convenient variable tthat denotes different intensities of load applications. Equality of virtual work is expressed through displacement increments Δu _{e} in the time step Δt. In this case the relation of a governing equation is (11) where ^{t}p_{e} , denotes nodal point forces corresponding to the element stresses at time t. A solution of (11) may be subject to very significant errors, it is, therefore, necessary to iterate until the solution is obtained to sufficient accuracy. The incremental equations, used in the Newton-Raphson iteration, are (12) and (13), for i= 1, 2, 3,…with the initial conditions shown in (14).
The foundation and soil are considered a non-homogeneous deformable solid [2,4,9]. Two separate problems were formulated. The first problem deals with linear-elastic material properties. The other accounts for elastic-plastic properties of soil. Von Mises and Drucker-Prager yield conditions are applied. The formulation is geometrically linear and axisymmetric. The sketch of the model is depicted in Fig 2. The material properties of soil and concrete are chosen such, that predominate in Lithuania [15]. The ANSYS [5–8] computer code is used for the calculation.
Another problem is to choose geometric dimensions of a soil model because it is possible to analyse the finite size model by finite element method. The width of a model D=3.75 d is chosen from the investigations of other authors [2, 9]. Some analyses were carried out in order to determine the influence of the soil depth under the foundation H = 2—4.333 h on the deflection at the top of the foundation. In this case the linear-elastic model was applied. The results are depicted in Fig 3. The depth H =2.167 h was chosen for further analysis. The generated discrete model was estimated for the quality. The model was under the circular in plane with the radius r = 0.4 m uniform distributed surface loading p = 1.989 MPa [16]. The calculated stress and strain distribution in soil under the centre of loading are compared, as shown in Fig 4 and 5, respectively.
In case of non-linear analysis the load is applied in two stages. In the first stage the model is loaded by weight. The full external load p = 1.989 MPa is incrementally applied in the second stage. The history of load application is described through the time variable t. The history of the deflection at the top of the foundation is depicted in Fig 6. It seems that the deflection of linear model under the full 1 MN load and that of model, implemented with Drucker-Prager yield condition, differs by 11%. The distribution of accumulated equivalent plastic strain in the cross-section (Fig 7) of the model, utilizing the Drucker-Prager yield condition, shows that the biggest plastic deformation developed in the immediate contact with the foundation. The history of the principal stresses of the node under the foundation and Drucker-Prager and von Mises yield surfaces in principal stress space are depicted in Fig 8.
Our analysis of axisymmetric bored foundation allows to know stress-strain state near the foundation. We can see that the external force of great magnitude cause significant plastic deformation, which, in turn, leads to significant deflection of the foundations.
First Published Online: 26 Jul 2012
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