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

element nodal points (joints), as shown by the two-dimensional 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.

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

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

1 The soil has a nonlinear stress-strain relationship

2 Different element types must be used to represent the pipe and the soil

3 It may be necessary to allow movement between the soil and the walls of the pipe, requiring the use of

an interface element

4 Very flexible pipes may involve large displace-ments 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 stress-strain 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 stress-strain 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.

Figure 28-3 Hyperbolic representation of a stress-strain curve [after Duncan et al (1980)].

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 = tangent elastic modulus

Pa = atmospheric pressure

K = an elastic modulus constant

n = elastic modulus exponent

σ1 = major principal stress

σ3 = minor principal stress (confining pressure),

and

Rf = failure ratio

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)]

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, do 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

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

When using the finite element method to solve

geotechnical engineering problems the nonlinear

stress-strain conditions are generally

accommo-dated 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

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

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

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

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 two-force element Both beam and bar elements are called one-dimensional elements, their