Numerical Methods in Soil Mechanics 28.PDF

Numerical Methods in Soil Mechanics 28.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.

Anderson, Loren Runar et al "ANALYSIS OF BURIED STRUCTURES BY THE FINITE ELEMENT METHOD" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 CHAPTER 28 ANALYSIS OF BURIED STRUCTURES BY THE FINITE ELEMENT METHOD Introduction The finite element method was introduced as a tool for engineering applications by Turner et al (1956) for the solution of stress analysis problems related primarily to the aircraft industry Since that time the finite element method has become a useful and accepted tool in many areas of civil engineering Applications in geotechnical engineering include static and dynamic stress analysis of various soil and soil-structure systems, seepage analysis including groundwater modeling, and consolidation analysis including both magnitude and rate of settlement Stress analysis applications in geotechnical engineering for static and dynamic loading was introduced in the late 1960s and early 1970s and included such applications as: static analysis of stresses and movements in embankments [Kulhawy, et al (1969); Duncan, (1972); Kulhawy and Duncan, (1972)]; earthquake stress analysis of embankments [Clough and Chopra, (1966)]; earthquake response analysis [Idriss, et al (1974)], and soil-structure interaction [Clough, (1972)] Katona, et al (1976) pioneered the application of the finite element method for the solution of buried pipe problems Their FHWA-sponsored project produced the well-known public domain computer program CANDE (Culvert ANalysis and DEsign) CANDE has been upgraded several times and is now available for use on a PC Others also made early contributions in the use of the finite element method for buried structures problems, Katona, (1982); Leonards, et al (1982); Sharp, et al (1984); and Sharp, et al (1985); TRB Record 1008 The basic idea behind the finite element method for stress analysis is that a continuum is represented by a number of elements connected only at the ©2000 CRC Press LLC element nodal points (joints), as shown by the twodimensional representation of a buried pipe in Figure 28-1 A structural analysis of the finite element assemblage can be made in a manner similar to the structural analysis of a building The process involves solving for the nodal displacements and then, based on the nodal displacements, the stresses and strains within each element of the assemblage can be determined The elements shown in Figure 28-1 are the basic structural units of the soil-pipe continuum, just as beams and columns are the basic structural units of building frames Each element is continuous and stresses and strains can be evaluated at any point within the element The major difference between the analysis of a continuum and a framed structure is that even though the finite element representation of a continuum is only connected to adjacent elements at its nodal points, it is necessary to maintain displacement compatibility between adjacent elements Special shape functions are used to relate displacements along the element boundaries to the nodal displacements and to specify the displacement compatibility between adjacent elements Once the continuum has been idealized as shown on Figure 28-1, an exact structural analysis of the system is performed using the stiffness method of analysis [Zienkiewicz, (1977); Gere and Weaver, (1980); Dunn, Anderson and Kiefer, (1980)] Note in Figure 28-1 that only half of the soil pipe system is represented The analysis results for the other half of the pipe can be obtained by symmetry as long as the geometry, properties, and loading conditions are symmetrical The boundary conditions along the line of symmetry must be properly established to model the full system behavior Taking advantage of symmetry significantly reduces the size of the problem that must be solved as discussed in the next section Figure 28-1 Mesh representing symmetric pipe-soil system ©2000 CRC Press LLC Most geotechnical engineering applications can be solved using a two-dimensional idealization as shown in Figure 28-1 However, there are some problems that must recognize the three-dimensional nature of the problem Figure 28-2 shows a finite element representation of a buried cylindric al tank The model was used to investigate the development of leaks in the tank at the junction between the cylindrical walls and the end plates on the tank Again, symmetry was used to minimize the size of the problem Basic Principles of the Finite Element Analysis Equation 28.1 represents the equilibrium equations, in matrix form, for each node of a finite element assemblage such as the one shown on Figure 28-1 After applying boundary conditions (identifying nodes with fixed or restricted movement) the system of equations given by Equation 28.1 can be solved for the unknown nodal displacements represented by the vector {d} These displacements can in turn be used to evaluate element stresses and strains [K] {d} = {f} (28.1) where [K] = the global stiffness matrix {d} = the nodal displacement vector and {f} = the nodal load vector The stiffness matrix [K] relates the nodal displacements to nodal forces The elements of the martix are functions of the structural geometry, the element dimensions, the elastic properties of the elements, and the element shape functions The size of the stiffness matrix depends on the number of degrees of freedom at each node and the number of nodes Thus, the more nodes that are used to represent a contiuum, the larger the system of simultaneous equations that must be solved Taking advantage of symmetry, as discussed in the previous section, can significantly reduce the number of equations that must be solved A complete derivation ©2000 CRC Press LLC of the finite element method for soil-structure interaction problems is presented by Nyby (1981) A finite element analysis of a soil-structure interaction system, such as a buried pipe, is different from a finite element analysis of a simple linearly elastic continuum in several ways The soil has a nonlinear stress-strain relationship Different element types must be used to represent the pipe and the soil It may be necessary to allow movement between the soil and the walls of the pipe, requiring the use of an interface element Very flexible pipes may involve large displacements for which the solution may be geometrically nonlinear Nonlinear soil properties The stress strain behavior of soil is nonlinear Therefore, large load increments can lead to significant errors in evaluating stress and strain within a soil mass The stress-strain relationship should be determined from the results of laboratory tests on representative soil samples Duncan et al (1980) suggested a method for describing the stressstrain characteristics of soil using hyperbolic parameters They also presented typical values for soil that can be used if the results of laboratory tests are not available Care should always be exercised when using “typical” values The Duncan soil model is often used for geotechnical engineering applications (Duncan, et al 1980) The original development of hyperbolic stress-strain theory that is used by the Duncan soil model was presented by Kondner and Zelasko (1963) The soil model assumes that the stressstrain properties of soil can be modeled using a hyperbolic relationship A thorough discussion of the Duncan soil model is presented in Duncan et al (1980) Figure 28-3 shows a typical nonlinear stress-strain Figure 28-2 Mesh for a buried tank which requires three-dimensional iterations ©2000 CRC Press LLC Figure 28-3 Hyperbolic representation of a stress-strain curve [after Duncan et al (1980)] ©2000 CRC Press LLC c urve and the corresponding hyperbolic transformation that is often used as a convenient w ay to represent the stress-strain properties (Duncan et al 1980) In Figure 28-3, the value of the initial tangent modulus Et is a function of the confining pressure Figure 28-3 shows the change in the tangent modulus that occurs as strain increases For a given constant value of confining pressure, the value of the elastic modulus is a function of the percent of mobilized strength of the soil, or the stress level As the stress level approaches unity (100% of the available strength is mobilized) the value of the modulus of elasticity approaches zero The Mohr-Coulomb strength theory of soil indicates that the strength of the soil is also dependent on confining pressure, as shown in Figure 28-4 Figure 28-5 shows the logarithmic relationship between the initial tangent modulus versus confining pressure The soil model combines the relationship of variation of initial tangent modulus with confining pressure and the variation of elasticity with stress level to evaluate the tangent modulus of elasticity at any given stress condition The equation that is used to evaluate the modulus of elasticity as a function of confining pressure strength is: where Et = Pa = K = n = σ1 = σ3 = Rf tangent elastic modulus atmospheric pressure an elastic modulus constant elastic modulus exponent major principal stress minor principal stress (confining pressure), and = failure ratio ©2000 CRC Press LLC The soil model as presented in Duncan et al (1980) also uses a hyperbolic model for the bulk modulus The hyperbolic relationship for the bulk modulus is similar to the initial elastic modulus relationship, where the bulk modulus is exponentially related to the confining pressure This particular soil model does not allow for dilatency of the soil during straining The equation that is used which relates the bulk modulus to confining pressure is: where B = the bulk modulus Kb = bulk modulus constant, and m = bulk modulus exponent The two equations given above are used to evaluate the strain-dependent elasticity parameters that are required in the stiffness matrix Poisson's ratio and the shear modulus are both computed based on equations developed in classical theory of elasticity Shear failure is tested by evaluating the stress level before the modulus of elasticity is computed If the stress level is computed to be more than 95% of the strength, the modulus of elasticity is computed based on a stress level of 0.95 This result is a low modulus of elasticity The bulk modulus is unaffected, thus modeling a high resistance to volumetric compression in shear A test must also be performed to evaluate if tension failure has occurred when computing the elastic parameters If the confining pressure is negative, then the soil element is in tension failure, and the elastic parameters need to be set to very small values, thus simulating a tension condition Figure 28-6 shows a stress-strain curve for a soil sample in a triaxial shear test The loading sequence for the sample was to increase the vertical stress until the sample had undergone initial strain, then to unload the sample, and Figure 28-4 Variation of strength with confining pressure [after Duncan et al (1980)] Figure 28-5 Variation of initial tangent modulus with confining pressure [after Duncan et al (1980)] ©2000 CRC Press LLC Figure 28-6 Deviator stress versus strain for a triaxial soil sample showing primary loading, unloading, and reloading finally to reload the sample until failure In Figure 26-6 it can be seen that the sample has a nonlinear stress-strain response on primary loading The unloading and reloading characteristics below the previous maximum past pressure, however, not follow the initial primary curve; they show an inelastic response After reloading beyond the maximum past pressure, the stress-strain curve again follows the initial nonlinear primary loading curve Duncan et al (1980) discuss the behavior of soil on unloading and reloading in comparison with that on primary loading The soil stiffness is reported to be 1.3 to 3.0 times greater when in the overconsolidated range Volumetric strain is reported to be unaffected by stress history Triaxial testing for unloading and reloading generally shows that the magnitudes of the hyperbolic constant and exponent depend on whether the soil is in primary loading or unloading and reloading It is necessary to model stress history for each soil element in a finite element analysis in order to model initial deformation of the pipe due to compaction Some of the soil elements should respond in the rebound range because of compaction until the surcharge ©2000 CRC Press LLC pressures exceed compactive loading pressures For other applications, such as pipe rerounding, soil elements must respond appropriately as the pipe rerounds when internal pressure is added to the loading sequence Because the soil is much stiffer in the rebound range, the pipe deformation is dependent on the stress history of the soil Not all of the soil elements in the finite-element mesh will respond to the rebound range at any given time as the pipe rerounds or as the compaction loads are modeled Thus, it is necessary to monitor the stress history of each soil element during the analysis and to use appropriate stiffness parameters depending on the current stresses of each element The stress history of the soil elements can be monitored by evaluating the position of the center of Mohr’s circle for each element (the average stress) The average stress at any load increment is compared with the maximum average stress from previous increments If the average current stress is less than the maximum previous stress, the soil elastic modulus is computed by using the unloading parameters Likewise, the soil elements are monitored in the rebound range and will convert to the primary loading curve when the average current stresses exceed the maximum past average stress This method allows simulation of soil element response for any soil element on either rebound or primary loading, depending on the loading conditions and soil response Two additional soil parameters are required as inputs for the Duncan soil model that account for the behavior of the soil in the unloading and reloading range, and the maximum past effective stress (preconsolidation stress or compaction stress) must also be specified at their common nodal points In some cases, however, it may be necessary to allow slip to occur between the pipe and the soil This can be accommodated in the finite element analysis by placing "interface" elements between the pipe nodes and the soil element nodes These interface elements have essentially no size, but kinematic ally allow movement between nodes when a specified friction force is exceeded When using the finite element method to solve geotechnical engineering problems the nonlinear stress-strain conditions are generally accommodated by adding loads in increments and adjusting the soil properties according to the magnitude of the strain If an embankment is being constructed, it is necessary to follow the construction sequence by adding the soil layers in increments and then adjusting the soil properties after each layer is added Each new layer is first represented as a load to determine the increase in stress within the embankment After determining the stress increase from the new layer, additional elements are added to the finite element mesh to represent the new material This allows the FEA program to follow the nonlinear stress-strain properties of the soil The stiffness matrix [K] in Equation 28-1 is a function of the material properties, the geometry of the element and the shape functions that are used to describe the stress-strain behavior at the edges of the elements The stiffness matrix is initially formed using the beginning soil properties and geometry of the element As loads are added to the soil structure the soil deforms and the soil properties change, and thus, the stiffness matrix must be adjusted to reflect the new soil properties Geometric Nonlinearity As described above in the section on nonlinear soil properties, the stiffness matrix [K] in Equation 28.1 is a function of the material properties of each element, the geometry of the element and the element shape function As the soil deforms under added loads the geometry of the finite element mesh changes If these changes are small (small displacement theory) the stiffness matrix does not have to be reformulated after each load inc rement However, if the pipe is very flexible the deformations can be large and it is necessary to reformulate the geometry of the finite element mesh after each loading increment This is referred to as geometric nonlinearity Construction of the Stiffness Matrix The stiffness matrix is composed of several parts One component is a constitutive matrix relating stress to strain through the elasticity parameters Another component relates element strains to nodal displacements through the strain-displacement matrix This matrix is computed based on element types, shape functions, and nodal coordinates It is not within the scope of this report to derive the above mentioned relationships Interface Elements In the finite element analysis of buried pipes, the pipe is generally modeled using beam elements in which shear, moment, and thrust can be represented at the ends of each element The nodes of the pipe elements are connected to the adjacent soil elements ©2000 CRC Press LLC Beam, bar, and soil elements each have their own particular stiffness matrices This comes about due to basic engineering mechanics principles A beam element is a three-force element and a bar is a twoforce element Both beam and bar elements are called one-dimensional elements, their strain- displacement matrix is derived based on the appropriate shape functions and their cross-sectional area, length, and angle of inclination A soil element is a two-dimensional element It does not transmit moment stresses The strain-displacement matrix is derived based on the coordinates of each node which comprise the element, and the shape functions that are used to describe the deformation characteristics of the soil elements A finite element analysis of a continuum involves the solution of the system of equations shown as Equation 28.1 If the continuum is a linear elastic system and the displacements are small then the stiffness matrix, [K], in Equation 28.1 can be generated once and then used over and over again for solving the problem for different loading conditions The load vector {f} changes for different loading conditions but the stiffness matrix remains the same Furthermore, the load of interest can be applied in one increment since the material properties are linearly elastic However, if the material properties are non-linear (strain dependent) then the stiffness matrix changes as the load increases and it is not possible to apply the load to the system in one large increment The load must, therefore, be added in increments and the stiffness matrix modified with each loading increment If the problem involves large displacement, which may be the case for very flexible pipes, then the geometry component to the stiffness matrix must change as the system deforms The global stiffness matrix for the entire system is made up from contributions from all of the elements in the assemblage Beam and bar elements tend to be linear elastic and the soil elements are non-linear (strain dependent) The individual element stiffness matrices are inserted into the overall global stiffness matrix by considering equilibrium at each node in the finite element assemblage Required Input Execution of a finite element program requires the user to prepare data that includes the mesh ©2000 CRC Press LLC geometry, material properties and loading conditions Some programs have an automatic mesh generation option that significantly reduces the effort required to input the data The data that are required include nodal coordinates, element data, material properties, interface properties, nodal link properties, construction sequence information, preexisting element stresses, strains, and displacements, and external loading information Boundary conditions are input The boundary conditions indicate whether a node is free to displace in the x or y direction or in rotation Material information Material information for each specified material type must be input Soil, structure (bars and beams), and interface materials will each have separate input requirements The material properties that are needed for bar elements include the cross-sectional area, modulus of elasticity, and weight/unit length Beam materials also require moment of inertia, shear area , Poisson's ratio, and distances from extreme fibers to the neutral axis Soil material properties that are needed include all the parameters that are used in the Duncan soil model described in a previous section Additionally, Mohr-Coulomb strength information, unit weight, and the lateral earth pressure coefficient are generally necessary for each soil type The nodal link elements behave like springs Normal and shear spring coefficients and orientation angles are required for the nodal link elements Interface elements are similar to nodal links except that spring coefficients are nonlinear Normal and shear spring coefficients, a modulus exponent, wall friction angle, and an adhesion value are necessary on each interface element material type Construction information The construction sequence is modeled by adding layers of soil elements or placing the structure (beam elements) Soil layer construction information is indicated by specifying the sequence at which the layers are to be added relative to sequences of adding external loads Structural placement information is indicated by specifying the soil layer that immediately follows the structure placement and the largest soil layer number that is adjacent to the structure Soil layer data that are needed for each layer are the largest and smallest soil element numbers within the soil layer, largest and smallest newly placed node numbers within the soil layer, and a series of nodes which define the top surface of the new layer Preexisting stresses An important feature when solving non-linear problems, such as soil-structure interaction problems, is the ability to specify preexisting stresses These stresses may be in the soil, structure, or interface elements Preexisting stresses are stresses which are already in place before any construction layers or external loading forces are added to the system For some problems it may also be important to specify preexisting strains or displacements The "preexisting stress" concept is very convenient when performing a series of analyses The use of preexisting stresses, strains, and displacements essentially defines the stress condition for the preexisting elements Construction sequences, therefore, need only be modeled once for a given mesh and soil configuration Afterwards, the preexisting stresses resulting from that construction simulation can be input for the entire mesh, and the subsequent analyses can be performed by adding only combinations of external loads to the mesh includes a summary of the input data and the results of the analysis in terms of stresses, strains, and displacements The input summary includes element and node information, material properties, construction and load sequencing, preexisting element information, and initial stresses used for estimating the initial elastic parameters Nodal displacements include the total displacements for the x, y and rotation components, and the incremental displacements and rotations for each particular loading increment The structural responses that are listed include the moment, shear, and thrust for each node of each structural member The listing contains the incremental structural forces and the total structural forces from the accumulated incremental forces The soil element strain information includes the soil element strains in x and y direction and the shear strain The principal strains can also be listed for each element External loads External loads are generally input as either concentrated loads or uniform loads Their input is rather simple Each loading sequence must have the number of concentrated loads and number of uniform loads that are to be used Concentrated loads are specified by denoting the node number which will receive the load, and the x and y components of the point load Uniform loads are specified for each element that will receive the uniform load Soil element stresses that can be output include the horizontal and vertical stresses, shear stresses, and principal stresses The angle of orientation of the origin of planes with respect to the principal plane, the ratio of major to minor principle stress, and stress levels can also be printed out for each element The stress levels that are output indicate the stress condition of each element If the computed stress is greater than the strength of the material, then the element has undergone a local shear failure, and the elastic parameters would have required adjustment If tension stresses were computed in the soil, the the element would have undergone a tension failure The element elasticity parameters would again need to be adjusted to allow the displacements that would occur for a soil element in tension Output Summary Typical output from a finite element analysis The finite element method is a powerful tool for ©2000 CRC Press LLC stress analysis of complex systems It has been particularly useful in geotechnical engineering for the solution of a wide variety of problems including soilstructure interaction problems such as the analysis of buried structures Its application in solving soilstructure interaction problems requires an understanding of basic engineering mechanics and an understanding of soil behavior Judgement is required in conducting and reviewing the results of finite element analysis of geotechnical problems Comparing the results of a finite element analysis solution with measurements made on a physical system is important whenever possible The power of the method lies in the ability to solve complex systems and in being able to look at many different loading conditions and system configurations However, never accept the results at face value without a thorough critical review REFERENCES J.M Duncan, P Byrne, K.S Wong, and P Mabry Strength, stress-strain and bulk-modulus parameters for finite element analyses of stresses and movements in soil masses Geotechnical Engineering Report UCB/GT/80-01 University of California, Berkeley, 1980 F.H.Kulhawy, J.M.Duncan, and H.B.Seed (1969), Finite element analysis of stresses and movements in embankments during construction Geotechnical Engineering Report TE-69-4 University of California, 1969 M.G.Katona, J.B.Forrest, R.J.Odello, andJ.R.Allgood (1976), CANDE—a modern ©2000 CRC Press LLC approach for the structural design and analysis of buried culverts, Report FHWA-RD-77-5 FHWA, U.S Department of Transportation, 1976 G.A.Leonards, T.H.Wu, and C.H.Juang (1982), Predicting performance of buried conduits Report FHWA/IN/JHRP-81\3 FHWA, U.S Department of Transportation, 1982 K.D.Sharp, F.W Kiefer, L.R.Anderson, and E.Jones (1984), Soils Testing Report for applications of finite element analysis of FRP pipe performance: Soils Testing Report Buried Structures Laboratory, Utah State University, Logan, UT, 1984 I.S.Dunn, L.R.Anderson, and F.W.Kiefer, (1980) Fundamentals of Geotechnical Analysis, Wiley, 1980 B.W.Nyby and L.R.Anderson (1981), Finite element analysis of soil-structure interaction Proceedings of the International Conference on Finite Element Methods (H.Guangqian and Y.K.Cheung, eds.) Science Press, Beijing China, 1982 K.D.Sharp, L.R.Anderson, A.P.Moser, and M.J.Warner (1984), Applications of finite element analysis of FRP pipe performance Buried Structures Laboratory, Utah State University, Logan, UT, 1984 ... engineering problems the nonlinear stress-strain conditions are generally accommodated by adding loads in increments and adjusting the soil properties according to the magnitude of the strain... listing contains the incremental structural forces and the total structural forces from the accumulated incremental forces The soil element strain information includes the soil element strains in. .. soilstructure interaction problems requires an understanding of basic engineering mechanics and an understanding of soil behavior Judgement is required in conducting and reviewing the results of finite