8 Finite Element Method (FEM) in Geotechnical Engineeringkhoavan.nguyen.free.fr/documents/8-FEM...

ETH Zürich Modelling in GeotechnicsInstitute of Geotechnical Engineering

8 Finite Element Method (FEM) in Geotechnical Engineering

8.1 IntroductionThe importance of a carefully planned and executed experimental modelling can not beoverstated. However, experimental modelling can be expensive and time-consuming and isnormally used only for high-cost and high-risk projects. For “normal” projects, site investi-gation is undertaken in combination with laboratory testing to obtain soil parameters asaccurately as possible. These parameters are then used as input to either limit equilibriumbased programs (e.g. slope stability, bearing capacity, etc.) to predict failure loads (ultimatelimit state) or a numerical analysis program (e.g. finite element method, finite differencemethod, etc.) to predict the deformation under working load conditions (serviceability limitstate). In this chapter, we will focus on one of the most popular numerical analysistechnique used in geotechnical engineering – the finite element method or FEM. The aim ofthis chapter is to learn how to apply the FEM in solving a geotechnical engineering problem.The emphasis is on the application and not on the formulation of the FEM. A curious readermay well consult one of the numerous books that deal with the mathematics and thenumerical techniques used in the FEM, e.g. Zienkiewicz and Taylor (1989).

8.2 Numerical methods used in geotechnical engineering

Figure 8.1: Various ways of solving a geotechnical engineering problem

As stated in the beginning of this course, there are several different ways of findingsolutions to a geotechnical engineering problem. These are summarized in Figure 8.1. Inthis section, we will focus on the numerical methods. One of the characteristic features ofthe numerical methods is that they usually involve solving a set of simultaneous partialdifferential equations (PDEs). Since soil is essentially a non-linear elasto-viscoplastic,three-phase material, direct solution of the set of PDEs is often impossible. Therefore, aniterative numerical approach is used. There are five major types of numerical methods usedin geotechnical engineering – the finite element, the finite difference, the boundary element,the discrete element and the combined boundary/finite element. The way the PDEs areformulated and solved differs for each of these methods.

Solution of Geotechnical Problems

Solution of Geotechnical Problems

Empirical, Based on Experience“Exact” or

Closed Form Numerical

Finite Element

Boundary Element

FiniteDifference

Limit Equilibrium

Finite/Boundary Element

DiscreteElement

Finite Element Method (FEM) in Geotechnical Engineering Page 8 - 1


8.3 What is FEM?

Figure 8.2: Discrete vs. continuous problem

Before introducing the concept of the FEM, let us first explore the difference between adiscrete and a continuous system. For a discrete system, an adequate solution can beobtained using a finite number of well-defined components. Such problems can be readilysolved even with rather large number of components, e.g. the analysis of a building frameconsisting of beams, columns and slabs (Figure 8.2). For a continuous system, such as asoil layer, the sub-division is continued infinitely so that the problem can only be definedusing the mathematical fiction of infinitesimal. Depending on the level of complexityinvolved, there are two ways of solving such a problem. Simple, linear problems can besolved easily by mathematical manipulation. Solution of complex, non-linear problemsinvolves discretization of the problem into components of finite dimensions (Figure 8.2) andthen using a numerical method such as the FEM.

The most distinctive feature of the FEM that separates it from other numerical methods isthe division of a given domain into a set of simple subdomains, called finite elements. Anygeometric shape that allows computation of the solution or its approximation, or providesnecessary relation among the values of the solution at selected points, called nodes, of thesubdomain, qualifies as a finite element. Such a subdivision of a whole into parts has twoadvantages:

1. It allows accurate representation of complex geometries and inclusion of dissimilar materials.

2. It enables accurate representation of the solution within each element, to bring out local effects (e.g. large gradients of the solution).

Discrete Problem

Semi-infinite Continuum

A finite element

Discretization

Continuous Problem



8.3.1 Historical Background

The idea of representing a given domain as a collection of discrete parts is not unique to theFEM. It was recorded that ancient Greek mathematicians estimated the value of π by notingthat the perimeter of a polygon inscribed in a circle approximates the circumference of thecircle. They predicted the value of π to accuracies of almost 40 significant digits by repre-senting the circle as a polygon of finitely large number of sides. Searching for approximatesolution or comprehension of the whole, by studying the constituent parts of the whole isvital to almost all investigations in science, humanities, and engineering. The FEM is anoutgrowth of the familiar procedures such as the frame analysis and the lattice analogy for2- and 3-dimensional bodies. Its application is not exclusive to engineering. It has beenused in other fields such as mathematics & physics. One of the earliest examples of its usewas in mathematics by R. Courant who used it for the solution of equilibrium and vibrationproblems (Courant, 1943). However, Courant did not call his method the FEM. It was R.W.Clough who first coined the term finite element in 1960 when he applied the FEM to planestress analysis (Clough, 1960).

During the early days of the digital revolution, due to the excessive cost of using the bulky,not-so-easy-to-use mainframe computers, the FEM remained in the hands for those “elite”people of science who had access to this rather expensive computing power. Only after theadvent of the personal computer and the smaller, more manageable and efficient minicom-puters, did it manage to break the barriers. Now, with tremendous amount of rather cheapcomputing power at their disposal, FEM is the first choice for many engineers and scientistsembarking on the analysis of a wide variety of engineering problems – from designing anew ergonomic shoe sole to designing a supersonic fighter aircraft. Its use in the field ofbioengineering, for example, the modelling of knee prosthesis or stress analysis of brainoedema, is also fast becoming popular.

8.3.2 The fundamental steps of the FEM

The three fundamental steps of the FEM are:

1. Divide the whole into parts (both to represent the geometry as well as the solution of the problem).

2. Over each part, seek an approximation to the solution as a linear combination of nodal values and approximation functions.

3. Derive the algebraic relations among the nodal values of the solution over each part, and assemble the parts to obtain the solution of the whole.

We will consider the example of the approximation of the circumference of the circle inorder to understand each of these three steps. Although this is a trivial example, it illus-trates several (but not all) ideas and the steps involved in the finite element analysis of aproblem.

8.3.3 Approximation of the Circumference of a Circle

Consider the problem of determining the perimeter of a circle of radius R (Figure 8.3).Ancient mathematicians estimated the value of the circumference by approximating it byline segments, whose lengths they were able to measure. The approximate value of thecircumference is obtained by summing the lengths of all the line segments that were used.Let us now outline the steps involved in computing an approximate value of the circum-ference of the circle. In doing so, we will also learn about certain terms that are used in thefinite element analysis of any problem.



1. Finite element discretization: First, the domain (i.e. the circumference of the circle) is represented as a collection of a finite number of n subdomains, namely, line segments. This is called discretization of the domain. Each subdomain (i.e. the line segment) is called an element. The collection of elements is called the finite element mesh. The elements are connected to each other at points called nodes. In the present case, we discretize the circumference into a mesh of five (n = 5) line segments. The line segments can be of different lengths. When all elements are of same length, the mesh is said to be uniform; otherwise, it is called a non-uniform mesh (see Figure 8.3b).

2. Element equations: A typical element is isolated and its required properties, i.e. its length, are computed by some appropriate means. Let he be the length of the element

Ωe in the mesh. For a typical element Ωe, he is given by (see Figure 8.3c):

(8.1)

where R is the radius of the circle and θe < π is the angle subtended by the line segment atthe centre of the circle. The above equations are called element equations. Ancientmathematicians most likely made measurements, rather than using (8.1) to find he.

Figure 8.3: Approximation of the circumference of a circle by line elements

3. Assembly of element equations and solution: The approximate value of the circum-ference (or perimeter) of the circle is obtained by putting together the element properties in a meaningful way; this process is called the assembly of the element

(a) (b)

(c)

Approximation of the circumferenceof a circle by line elements: (a) Circle of radius R; (b) Uniform and non-uniform meshes used to representhe circumference of the circle; (c) a typical element.

Element

Node

R

θ e

h e



equations. It is based, in the present case, on the simple idea that the total perimeter of the polygon (assembled elements) is equal to the sum of the lengths of individual elements:

(8.2)

Then, Pn represents an approximation to the actual perimeter, p, of the circle. If the mesh isuniform, i.e. he is the same for each element in the mesh, θe = 2π/n, and we have

8.3)

4. Convergence and error estimate: For this simple problem, we know the exact solution:

(8.4)

We can estimate the error in the approximation and show that the approximate solution Pnconverges to the exact solution p in the limit as n → ∞.

In the summary, it is shown that the circumference of a circle can be approximated asclosely as we wish by a finite number of piecewise-linear functions. As the number ofelements is increased, the approximation improves, i.e. the error in the approximationdecreases.

8.4 Basic formulation of the FEM

In this section, the basic formulation of the FEM will be introduced using three simpleexamples: (1) a system of interconnected elastic springs; (2) a one-dimensional plane trusselement; and (3) a constant strain triangular finite element.



8.4.1 Interconnected elastic springs

Figure 8.4: A system of interconnected springs

1. In this system, linear elastic springs are the finite elements.

2. From a structural mechanics point-of-view, the structure is statically indeterminate.

3. Let the stiffnesses of individual springs be ka, kb, kc and kd. Therefore, the tensions in these springs are given by:

(8.5)

where ea, eb, ec and ed are extensions of springs a, b, c and d, respectively.

4. Let us now invoke three fundamental principles of structural mechanics: compatibility, material behaviour and equilibrium for the calculation of the displacement of each spring. These three principles are applied in the order of compatibility – material behaviour – equilibrium.

5. The compatibility equations are:

(8.6)

where d1, d2, d3 and d4 are displacements of nodes 1, 2, 3 and 4, respectively. Here, weare making sure that the system does not fall apart, i.e. springs remain connected with eachother.

6. Material behaviour can be expressed using spring stiffnesses as:

(8.7)

a

d

b

c

d1

d2

d3

d4

1

2

3

4

2Ta

Tb TdW2

Equilibrium at Node 2



7. Equilibrium (at node 2, see Figure 8.4):

or

8.8)

which on rearrangement, results in:

(8.9)

8. Similar equations can be written for other nodes, giving four linear simultaneous equations in d1, d2, d3 and d4 that can be expressed in matrix form as:

(8.10)

The matrix on the left-hand-side is called the global stiffness matrix. Equation (8.10) canbe written in matrix notation as:

Kd = W

These simultaneous equations can be solved by elimination and values of displacementscan be obtained. From the values of displacements, the force in each spring can be calcu-lated.

9. The global stiffness matrix K consists of the sum of matrices of the following form (where ke is the stiffness of one particular spring):

(8.11)

This matrix is called the element stiffness matrix. It relates the nodal displacements to theforces exerted on each spring at nodal points. One of these matrices is added into theglobal stiffness matrix for each spring in the system.

(8.12)



8.4.2 A plane truss element

Figure 8.5: A plane truss element

In this section, we will apply the same principles of compatibility, material behaviour andequilibrium to a one-dimensional plane truss element (Figure 8.5). The formulation is nowmore complex than that for a simple system of linear elastic springs. You may have noticedthat in the case of linear elastic springs, each node was allowed to move in only y-direction,i.e. up or down. Here, each of the two nodes of the plane truss elements has two degreesof freedom, i.e. it can move in both x- and y-direction. However, as we shall see, thegeneral solution procedure remains the same regardless of the increased complexity.

In calculating the strains in this element, we are only interested in the displacements alongthe direction of the element. It is, therefore, logical to define a system of axes x’-y’ that islocal to the element, with x’-axis coincident with the direction of the element.

1. Let us first apply the condition of compatibility, i.e. the element should not break in the middle. Mathematically, it can be expressed in terms of the equation for displacement at a distance x’ along the element:

(8.13)

2. To obtain the element stiffness matrix, we need to write this expression in terms of the degrees of freedom dx1, dy1, dx2 and dy2. This is achieved by noting that

(8.14)

from simple geometric consideration.

y

x

y’

x’

dy2

dx2

dy1

dx1

α

Length = L

1

2

Finite Element Method (FEM) in Geotechnical Engineering - 8


3. Making this substitution, we obtain:

(8.15)

4. The strains inside the element can now be related to nodal displacements using a matrix that is obtained by differentiating equation (8.15) with respect to x’. This matrix is called the B matrix in the FEM formulation and is given by:

In the matrix notation, the strain matrix is now written as:

(8.16)

where ae is the vector of nodal displacements – right-hand-side matrix in equation (8.15).

5. Assuming the plane truss element to be linear elastic, the stress inside the element can now be expressed in terms of nodal displacements as:

(8.17)

where D is the matrix of material behaviour or constitutive matrix for the element. In thiscase, it simply reduces to the Young’s modulus of the plane truss element, E.

6. The principle of virtual work can now be used to find the nodal forces Fe that are in equilibrium with this state of internal stress. A set of virtual nodal displacements applied to the element accompanies a set of virtual strains within the element according to the relation:

(8.18)

The principle of virtual work gives:

(8.19)

7. Substituting for σ and , we obtain:

(8.20)

ˆ



From the above equation, can be cancelled out to give:

(8.21)

where K is the element stiffness matrix. For our plane truss element, it can be shown to begiven by:

(8.22)

where A is the cross-sectional area of the plane truss element, C = cosα and S = sinα.

8. For a typical plane truss problem, the forces acting on the nodes are known. Hence, equation (8.21) can be solved by first inverting the K matrix and then solving the resulting simultaneous equations for nodal displacements.

8.4.3 A constant strain triangular finite element

After having successfully formulated the FEM for the solution of two one-dimensionalproblems, we move to the formulation of a two-dimensional constant strain triangular finiteelement. Figure 8.6 shows the simplest triangular finite element used for two-dimensionalcontinuum analysis.

Figure 8.6: A constant strain triangular finite element

Tea

dy2

dx2

dy3

dx3

dy1

dx1

h

h

3 1

2

x

y



1. Each of its three nodes has two degrees of freedoms and the terms dx1, dy1, dx2, dy2, dx3, and dy3 denote the nodal displacements. In this case, the unknown variation of the displacement within the element adds to the complexity of the problem. Here, we are going to assume that this variation is linear, i.e.

and

(8.23)

Since the strain is the first derivative of the displacement, it will be constant within theelement. Hence, the element is called a finite element.

2. The coefficients c0, c1, etc. in equation (8.23) are obtained by substituting the coordi-nates of the three nodal points into these expressions. In this case, too, we assume a local coordinate system with origin at node 3 and x-axis along side 3-1 and y-axis along side 3-2. Solving the resulting sets of simultaneous equations, we obtain:

and

(8.24)

3. Equation (8.24) can be written in matrix notation as:

(8.25)

where N is the matrix of shape functions for the finite element and is given by:

(8.26)

4. Now, we can formulate the B matrix by partially differentiating the N matrix with respect to x and y as:

(8.27)

Here, the first row denotes strain in x-direction, second row denotes strain in y-direction andthe third row denotes the shear strain in the x-y plane.



5. Assuming plane strain conditions, the element stiffness matrix D can be easily obtained from Hooke’s law as:

(8.28)

where E is the Young’s modulus and ν is the Poisson’s ratio for the material.

6. Formulating the element stiffness matrix K is now a simple task of calculating the matrix product BTDB times the area of the element (h2/2) since the terms of all these matrices are constant. K is given by:

(8.29)

where a = 1 – ν; b = 0.5 – ν and c = 1.5 – 2ν.

Although the above three examples illustrate the basic idea of the FEM, there are severalother features that are either not present or not apparent from the discussion of theseexamples. These are summarized below:

1. Depending on its shape, a domain can be discretized into a mesh that contains more than one type of element. For example, in the discretization of an irregular two-dimen-sional domain, one can use a combination of triangular and quadrilateral finite elements. However, if more than one type of element is used, one of each kind should be isolated and its equations developed. All the commercial FEM software take this into account and therefore, it is not a problem to mix element types during an analysis.

2. The governing (simultaneous) equations are generally more complex than those considered in these three examples. They are usually partial differential equations. In most cases, these equations cannot be solved over an element for two reasons. First, they do not permit exact solution. Second, the discrete equations obtained cannot be solved independent of the remaining elements because the assemblage of the elements is subjected to certain continuity, boundary and/or initial conditions.

3. The number and location of nodes in an element depend on (a) the geometry of the element, (b) the degree of polynomial approximation, and (c) the integral form of the equations. This point is elaborated further in the section dealing with types of finite elements.

4. There are three sources of errors in a solution obtained by the FEM: (a) those due to the approximation of the domain; (b) those due to the approximation of the solution; and (c) those due to numerical computations. The estimation of these errors is not a simple



matter. The accuracy and convergence of a FEM solution depends on the differential equation, the integral form and the element used. Accuracy refers to the difference between the exact solution and the solution obtained by the FEM whereas conver-gence refers to the accuracy as the number of elements in the mesh is increased. This point is discussed in detail later in the chapter.

8.5 Approximations, accuracy and convergence in the FEM1. Engineers sometimes regard the finite elements in a mesh as being connected only at

the nodal points in the mesh. This is not a good conceptual picture of how the elements behave. Straining of finite elements results in a deformation pattern similar to that shown in Figure 8.7a rather than that shown in Figure 8.7b (i.e. there are no gaps that open up at the element boundaries). This is because the polynomials or shape functions that approximate the distribution of displacement are chosen in such a way that there is a continuity of displacements within the elements as well as between the adjoining elements.

Figure 8.7: Continuity of displacements in adjoining finite elements

2. Although strains will be continuous within a finite element, there will usually be a discon-tinuity of strains between adjacent elements. Some approximation (e.g. a smoothing zone as shown in Figure 8.8) is necessary so that the terms being integrated become continuous.

(a) (b)



Figure 8.8: The use of smoothing zone at element boundaries

3. The stress field within an element will be continuous but may not satisfy the equations of equilibrium. Except for very simple problems, stresses on either side of element boundaries will not be equal. Equilibrium is satisfied, however, in an average sense through the equilibrium equations at nodal points where the resultant forces equivalent to internal stress field balance the resultant forces due to external traction and body forces. The extent to which the local stresses appear not to be in equilibrium with the external forces gives some indication of the accuracy of the solution.

4. Before applying the FEM to solve real problems, it is advisable to test its accuracy by solving certain benchmark or validation problems for which an exact or closed-form solution exists. An error-free, robust FEM program should be able to reproduce the exact solution accurately. One of the most popular benchmark problem in geotechnical engineering is the calculation of undrained collapse load (qu) of a circular foundation on soft clay of uniform undrained shear strength (su) – qu = (π+2) su.

Dis

plac

emen

t uS

trai

n du

/dx

Rat

e of

str

ain

d2u/

dx2

∆

‘Smoothing’ zone

-∞



5. In addition to testing the accuracy of the FEM, a convergence test should be carried out for a given problem for which we do not have an exact solution. It involves conducting three or more FEM analyses with progressively finer mesh. Convergence is achieved when further refinement of mesh does not result in a significant increase in the accuracy of the solution (Figure 8.9).

Figure 8.9: Testing the convergence by progressive mesh refinement

8.6 Geotechnical finite element analysis

Most of the commercially available FEM programs are written with structural/mechanicalapplications in mind. These programs cater for materials that can be produced undercontrolled conditions and therefore, have well-defined physical or mechanical properties,e.g. metals, plastics, polymers, concrete, etc. The most important material in a geotechnicalanalysis is the soil. A soil’s physical or mechanical properties have to be measured insteadof being specified or specially fabricated. These properties vary enormously from site tosite and can be profoundly affected by factors such as sampling techniques, specimenhandling and preparation, characteristics of the measurement and data acquisitiontechniques. Therefore, the constitutive modelling takes the centre stage in a geotechnicalFEM program. The three phase (soil-water-air) nature of soil makes realistic constitutivemodelling of soil a formidable task. Since the shear strength of a soil at a given pointdepends on the effective stress at that point, the stress-strain response of a soil is highlynon-linear. For a geotechnical finite element analysis, the FEM program should have thefollowing features:

1. Material models that are capable of modelling non-linear stress-strain behaviour and that include options for undrained analysis (short-term behaviour), drained analysis (long-term behaviour), most importantly, coupled consolidation analysis.

qu

No. of Elements24 48 96

‘Exact’ solutionMesh A - 24 Elements

Mesh B - 48 Elements

Mesh C - 96 Elements



2. The ability to specify non-zero in-situ stresses.

3. The ability to add or remove elements during the analysis (for modelling the construction or excavation, respectively).

8.6.1 Plane strain and axisymmetric problems

While a three-dimensional finite element analysis is frequently used in structural ormechanical applications, it is rarely used in geotechnical engineering. Most of the geotech-nical problems can be assumed to be either plane strain or axisymmetric without signif-icant loss of the accuracy of the solution.

1. Plane strain problems: The characteristic feature of a plane strain problem (Figure 8.10) is that one dimension – in this case the dimension along the z-axis – is considerably greater than the other two dimensions. As a result, the strains in the direction of z-axis can be assumed to be zero. Therefore, we only have to solve for strains in the x-y plane and the problem reduces to a plane strain problem. For plane strain problems, the numerical integration is performed for a unit section (1 unit length) along the z-axis. Typical examples of plane strain geotechnical problems are embankments, retaining walls, tunnels (at sections sufficiently away from the head of the tunnel).

Figure 8.10: A plane strain problem

Axisymmetric problems: For an axisymmetric problem, both the structure and the loadingexhibit radial symmetry about the central vertical axis (Figure 8.11). Consequently, thecircumferential strains can be ignored in the solution and the problem reduces to a two-dimensional problem in a vertical radial plane. Keep in mind that the problem can only bereduced to an axisymmetric problem when both the structure and the loading are symmetricabout the central vertical axis. If one of the two does not exhibit radial symmetry, either theproblem has to be treated as a three-dimensional problem or techniques involving FastFourier Transforms (FFTs) have to be used. The numerical integration for an axisymmetricproblem is performed from zero to 2p, i.e. for the entire horizontal circular cross-section.Typical examples of axisymmetic geotechnical problems are pile foundation subject tovertical concentric loads, excavation of vertical shafts of circular cross-section, consoli-dation around a vertical drain.

y

xz



Figure 8.11: An axisymmetric problem

8.6.2 Different types of finite elements

There are many different types of finite elements available for use with a geotechnical FEMprogram. These elements can be classified based on either the dimensions of the problemor the order of the element. They can also be classified on the basis of whether the coupledconsolidation formulation is adopted or not.

1. 1-D, 2-D and 3-D elements (Figure 8.12): 1-D and 2-D elements are used mainly for the plane strain and axisymmetric problems. 3-D elements are used only for the truly three-dimensional problems.

• Typical 1-D elements include: (a) bar elements for the modelling of struts, geotextilereinforcement, ground anchors and any other structural element that is not capable ofresisting flexure, and (b) beam elements for the modelling of retaining walls, tunnellinings and any other structural element requiring flexural rigidity.

• Typical 2-D elements include (a) triangles and quadrilaterals for the modelling of soil andstructural components of significant dimensions, and (b) slip elements for modelling ofsoil-structure interface behaviour.

• Typical 3-D elements are hexahedrons and tetrahedrons for the modelling of soil andstructural components. Some FEM programs also have 3-D slip elements for modellingof soil-structure interface behaviour.

CL

r

y



Figure 8.12: 1-, 2- and 3-D elements

2. First-, second- and fourth-order elements (Figure 8.13): The order of the element is determined by the order of the polynomial used as the shape function.

• For a first-order element, a first-order polynomial, i.e. a straight line, is used as shapefunction. The constant strain triangle in the example above is a first-order element. Amesh containing only first-order elements requires a large number of elements for a suffi-ciently accurate solution.

• For a second-order element, a quadratic or second-order polynomial is used as shapefunction. As a result, the strain within the element is distributed linearly. Hence, theseelements are also called linear strain elements. Such elements usually have one or moremid-side nodes in addition to the vertex nodes. One does not need to use a largenumber of second-order elements in order to achieve sufficient accuracy.

• For a fourth-order element, a quartic or a fourth-order polynomial is used as shapefunction. The strains, therefore, have a cubic variation within the element and theelement is often called a cubic-strain element. Such elements have several mid-sidenodes as well as nodes inside the element in addition to the vertex nodes. It is notcommon to use such elements for a routine geotechnical analysis. Their use is limited tospecial situations such as testing a new constitutive model, unit cell radial consolidationproblems.

dy2

dx2

dy1

dx1

(a) Two-noded bar element

dx2dy1

dx1

dy2

θ1

θ2

(b) Two-noded beam element

(c) 2-D element

dy1

dx1

dy2

dx2

dy3

dx3

(e) 3-D elements

t

LSoil

Structure

(d) 2-D slip element



Figure 8.13: First-, second- and fourth-order finite elements

3. Consolidation elements (Figure 8.14): These elements are required when the FEM program adopts a coupled consolidation formulation. In a coupled-consolidation formu-lation, the excess pore pressures are treated as unknowns. Any variation in the magnitude of excess pore pressure at a given point is reflected simultaneously in the magnitude of effective stress at that point. In addition to the standard displacement nodes, consolidation elements have pore pressure nodes where the value of excess pore pressure is calculated. For second-order elements, pore pressure nodes are normally superimposed on vertex displacement nodes of the element. For higher-order elements, pore pressure nodes also exist inside the elements.

Figure 8.14: Consolidation element

Displacement

x

Displacement

x

Displacement

x

(a)

(b)

(c)

(a) First-order element

(b) Second-order element

(c) Fourth-order element

+ =

DisplacementElement

Pore PressureElement

ConsolidationElement



8.7 Techniques for modelling non-linear stress-strain response

The basic formulation of the FEM described in Section 8.4 is applicable only for materialsthat obey linear stress-strain laws (Figure 8.15a). However, as mentioned above, thestress-strain behaviour of a soil is highly non-linear (Figure 8.15b) and therefore, forsolution of geotechnical engineering problems the fundamental equation of the FEM(equation 8.21) cannot be used in its present form. First, the non-linear stress-strain curveshould be approximated by a set of interconnected straight lines (i.e. it is made piecewiselinear) and then an incremental form of equation 8.21 is used. This approach is illustrated inFigure 8.16. Depending on the degree of non-linearity, the imposed loading (ordisplacement) is divided into sufficient number of increments and equation 8.21 is solvedfor each increment in succession. This is the simplest way of modelling a non-linearmaterial. The trick here is to make sure that the piecewise linear approximation does notdrift from the true stress-strain curve by a certain tolerable amount. However, the appli-cation of this method is limited to material models that have a well-defined yield function,e.g. models based on critical state soil mechanics theory. This method is not suitable forelastic-perfectly plastic models such as the Mohr-Coulomb model. The reason for this isthat the yield function and the failure criterion are one and the same for such models andthere is no other way of detecting the yielding of the material than to cross (and go out of)the failure envelope (Figure 8.17a). Such a stress state is not admissible and will result ininternal forces that are not in equilibrium with external forces. Therefore, the stress statemust be corrected back to the failure criterion. This can be achieved in several differentways. The following two methods are commonly used in a geotechnical FEM software:

1. Tangential stiffness approach with carry over of unbalanced load

2. Modified Newton-Raphson method

Figure 8.15: Linear and non-linear materialbehaviour

Figure 8.16: Piecewise linear approximation ofnon-linear material behaviour

σ

σ

ε

ε

(a) Linear stress-strain response

(b) Non-linear stress-strain response

σ

ε

σ

εIncrement No. 1 2 3 4 5 6

(a) Non-linear stress-strain response

(b) Piecewise linear approximation

E4



8.7.1 Tangential stiffness approach with carry over of unbalanced load

This approach is illustrated in Figure 8.17b. In this approach, the global stiffness matrix iscomputed based on the tangential stiffness at the beginning of an increment, say from 0 toa displacement d1 as shown in figure 8.17b. In other words, the stress-strain response isnow considered linear for this increment and is represented by the tangent drawn at thestarting point of the increment. The internal load at the end of this increment (∆P1) is nolonger in equilibrium with external load and this out-of-balance load (∆PC1) is re-applied tothe finite element mesh at the beginning of the next increment (from displacement d1 to d2).It is obvious that the accuracy of the solution will suffer considerably if the magnitude of theout-of-balance load is rather large. The accuracy of the solution can be assessed byexamining the global equilibrium error (percent difference between the sum of externalloads and sum of internal forces) at the end of each increment. For elastic-perfectly plasticmodels, this error should never be allowed to go beyond 15 to 20%. To achieve this goal, asufficiently large number of increments should be used. Another alternative is to divideeach increment into 5 or 10 sub-increments (Figure 8.17c). This will ensure that themagnitude of out-of-balance load for each sub-increment is small.

Figure 8.17: Methods of modelling non-linear material behaviour

8.7.2 Modified Newton-Raphson method

It is also known as the quasi Newton-Raphson method. In this method, similar to thetangential stiffness approach, the stiffness matrix is computed based on the tangentialstiffness at the beginning of an increment. However, the out-of-balance load is not carriedover to the next increment. Instead, an iterative procedure shown in Figure 8.17d isfollowed. The out-of-balance load (∆PC1) is re-applied to the mesh and the resulting incre-

∆τc11

2

yieldsurface

τ

σ ( )∫ ∆=∆ vol11 dP cc τTB

(a) Stress state correction

∆Pc1∆P1

∆P2

d1 d2

P

d

(b) Tangential stiffness approach

Sub-increment 1 2 3

P

d

(c) Use of sub-increments to applyout-of-balance load

∆P1

d

P

d1

(d) Modified Newton-Raphson method



mental displacements are added to the current displacements. If further yielding takesplace during the application of ∆PC1 then a second set of out-of-balance load (∆PC2) arecalculated and the above procedure is repeated until convergence is reached, i.e. theresulting incremental displacements or the out-of-balance load is less than a presettolerance. The main advantage of this procedure is that the stiffness matrix is computedonly at the beginning of an increment. However, rather large number of iterations requiredto achieve convergence compensates the savings on computation time thus achieved.Also, the method may fail to converge for some highly non-linear problems.

8.8 Techniques for modelling excavation and construction

8.8.1 Excavation

Geotechnical activities that involve excavation can be broadly classified into three maincategories: trenches, shafts and tunnels. Trenches can be rather small, e.g. for laying of adrainage pipe (Figure 8.18a), or big and deep, e.g. for the construction of basement carpark (Figure 8.18b). The effect of excavating a small trench on surrounding soil and struc-tures is not so great and, therefore, such a problem is rarely analyzed using the FEM.However, a deep excavation can result in significant ground movements capable ofdamaging the surrounding structures. It is, therefore, not surprising that its design almostinvariably involves conducting a few FEM analyses. The length and the width for a typicaldeep excavation are comparable and hence, it is a 3-D problem. However, a 3-D FEManalysis is rarely used and often, the problem is assumed to be a plane strain problem.Excavation of a shaft is modelled similar to a trench or a deep excavation; the onlydifference is that axisymmetric conditions are assumed.

Figure 8.18: Trenches and deep excavations

1 m

2 m

5~10 m 4~8

m6~

12 mStruts

Diaphragm Wall

(a)

CL

(b)



Figure 8.19: Excavation of a tunnel and its 2-D FEM approximations

The construction of tunnels is an urban necessity. It involves excavation of soil using atunnel boring machine (TBM) and installation of permanent lining for the excavated section.Excavation of a tunnel causes ground loss as well as stress relief at the face of the tunnel,resulting in significant surface settlements. These surface settlements can result in signif-icant damage to nearby structures. Tunnel excavation is also a 3-D problem (Figure 8.19).However, it is quite common to assume plane strain conditions (representing a verticaltransverse section sufficiently away from the face of the tunnel) and to use a volume lossparameter that takes into account the 3-D effect in an approximate manner. To study theground movements ahead of the tunnel face, a vertical longitudinal section is consideredand plane strain conditions are assumed.

For both the deep excavation and the tunnel, the modelling of excavation is achieved in thesame way – by removing the elements from the mesh. Here, it is worth noting that the bodyforces within the element are composed of both soil (effective stress) and water (porepressure) as shown in Figure 8.20. When an element is removed, both the soil and waterbody forces are removed. For the deep excavation, it represents an excavation that is dry,i.e. not filled with water. For an excavation in a clayey soil, this means that there arenegative pore pressures (pore suction) on the inner boundaries of the excavation. Unlesssome support in the form of a retaining wall is provided, the soil will eventually lose itssuction and the excavation will collapse. However, such removal of body forces is notrealistic for certain situations, e.g. installation of a diaphragm wall. The trench for adiaphragm wall is filled with either water or bentonite slurry. In this situation, one must eitherre-apply the water body forces or apply body forces corresponding to the bentonite slurryon the inner boundaries of the excavation.

Tunnel Face

GroundSurface

(a) Vertical Transverse Section

(b) Vertical Longitudinal Section

Unsupported Heading



Figure 8.20: Stress changes during modelling of an excavation

8.8.2 Construction

The word “construction” in geotechnical engineering usually means placing one or morelayers of soil over existing or made-up ground, e.g. construction of a highway embankmenton soft clay. The placing of a layer is modelled either by adding elements to the existingmesh or by applying pressure at the boundaries (Figure 8.21). The latter approach givessatisfactory solution provided the newly placed layers are not expected to undergo anyshear deformation. If this is not the case, the technique of adding elements to the meshshould be used. An element that is added is assumed to be unstressed and the self-weightof the element is the only contributor to the body forces of that element. For this reason, theadded elements must either have elastic properties or have a small non-zero value ofapparent cohesion c’ if elastic-perfectly plastic model is used. A constitutive model thatrequires specification of a stress history, e.g. Cam-clay or other critical state models, isunsuitable for modelling of added elements.

Figure 8.21: Techniques for modelling layered construction of an embankment

(a) Excavation (b) Stresses acting before excavation

(c) Stresses after excavation (zero) (d) Net effect of excavation

Effective stress + pore pressure

Second LayerFirst Layer

Soft Clay

EmbankmentCL CL

(a) by adding elements (b) by pressure loading



Compaction is an integral part of a geotechnical construction activity and its effects shouldideally be included in the modelling. However, the effects of compaction are difficult toquantify in terms of stresses. In addition, the soil that is being compacted is usually partiallysaturated. These factors make the modelling of compaction activity quite complicated andtherefore, most commercial geotechnical FEM software simply ignore it.

8.9 Advantages and drawbacks of the FEM

8.9.1 Advantages

1. It is relatively easy to use and, therefore, it is one of the most popular methods for advanced geotechnical modelling.

2. There are many commercial FEM programs available that are capable of geotechnical modelling (discussed later).

3. Since each element’s properties are modelled and evaluated separately, it is quite easy to incorporate non-homogenous ground conditions such as layers of different soils.

4. Any shape of domain can be modelled with the possibility of including holes, gaps, etc.

5. Boundary conditions can be applied easily.

6. It is possible to couple different physical phenomena such as diffusion and thermal conduction within the same formulation. This is possible because all of these phenomena can be described by the Laplacian equation.

7. Construction and excavation of soil layers in geotechnical engineering can be done easily by adding or removing elements from the mesh (discussed later).

8.9.2 Drawbacks

1. While an FEM program is relatively easy to use, interpretation of its output can be a formidable task and usually requires considerable expertise and experience.

2. It is not suitable for highly non-linear problems or problems that involve large strains, e.g. cone penetration test, consolidation of a hydraulic fill or a clay slurry. For such problems, a finite difference formulation incorporating fast Lagrangian analysis procedure is more suitable.

3. It is also not suitable for the modelling of brittle materials that exhibit discontinuities in the form of cracks, faults and fissures, e.g. rock. For such materials, a discrete element formulation is more suitable.

8.10 Some popular commercial FEM programs

8.10.1 ABAQUS

ABAQUS is a general-purpose FEM program that contains many useful features:

• Static stress-displacement, transient dynamic stress-displacement, heat transfer, masstransport and steady-state transport analyses.

• Coupled formulations that include: Biot’s consolidation theory, thermo-mechanicalcoupling, thermo-electrical coupling, fluid flow-mechanical coupling, stress-massdiffusion coupling, piezoelectric and acoustic-mechanical coupling. The most importantof these from a geotechnical point-of-view is Biot’s consolidation theory.



• Dynamic stress-displacement analysis, determination of natural modes and frequencies,transient response via modal superposition, steady-state response resulting fromharmonic loading, response spectrum analysis, and dynamic response resulting fromrandom loading. These features are mainly for earthquake or other dynamic applications.

• It has a huge library of all finite elements developed in the literature such as 1-D, 2-D and3-D continuum elements, shell, membrane, pipe, beam and elbow elements, springs,dashpots, joint, interface and infinite elements. User-defined elements can also be used.

• Similarly, it has an impressive collection of constitutive models including general elastic(linear and non-linear), elasto-plastic, elasto-viscoplastic, hyper and hypo-elasticmodels. Constitutive models that are useful for geotechnical analysis are von Mises,Mohr-Coulomb, Drucker-Prager, Extended Drucker-Prager (non-associated flow), CamClay, Modified Cam Clay, Capped Drucker-Prager (Cam Clay with Extended Drucker-Prager for use in tunnel excavation), and strain-rate dependent plastic laws.

• User-defined constitutive models can also be incorporated with the help of a subroutineinterface.

• It is possible to simulate excavation and construction.• It can deal with large strain and large deformations.• Presently, it is the only commercial program except ZSOIL (described below) that can

deal with partially saturated soils.• It can model seepage problems with phreatic surfaces and capillary effects.• It even allows cracks and rock joints to be modelled and it can model creep, too.• It can perform adaptive mesh refinement for undrained and drained problems only.• Operating System: Windows NT, UNIX, Sun Solaris and a host of other systems running

mainly on multiprocessor or parallel computers.• It is very expensive but a cheaper, educational version with limited capabilities is

available for teaching and research use.• More information can be obtained from http://www.abaqus.com/

8.10.2 SAGE CRISP

SAGE CRISP has evolved from CRISP – CRItical State Program – developed by theCambridge University Soil Mechanics Group in the 1970s and 80s. CRISP was one of thefirst FEM programs dedicated to geotechnical analysis. In the early 1990s, SAGEEngineering Ltd., UK developed the pre- and post-processors for this program and beganmarketing the program by the name SAGE CRISP. The main features of SAGE CRISP areas follows:

• It can perform static stress-displacement and coupled consolidation analyses in one-,two- and three-dimensions. At present, there is no facility to do dynamic analysis but thedevelopers of SAGE CRISP are in process of incorporating this facility.

• Its element library includes 1-D, 2D and 3D continuum, bar, beam and interfaceelements.

• Almost all of its constitutive models cater for geotechnical applications. These includegeneral elastic (linear and non-linear), anisotropic elastic, elastic-perfectly plastic withvon Mises, Tresca, Drucker-Prager, Mohr-Coulomb failure criteria, Cam Clay, ModifiedCam Clay, Schofield, 3-Surface Kinematic Hardening (for small-strain modelling) andhyperbolic (Duncan and Chang type) models.

• It can model excavation and construction.



• It does not include a large-strain formulation but it can deal with large strain problems inan approximate manner by using large number of increments and updating of geometryat the end of each increment.

• Operating System: Windows (older versions of CRISP run under MS DOS).• More information can be obtained from http://www.crispconsortium.com/

8.10.3 PLAXIS

PLAXIS is a geotechnical FEM program developed by PLAXIS BV of the Netherlands. Itsname is a combination of PLane strain and AXISymmetric. As the name suggests, it canonly do 1-D and 2-D analyses although a 3-D version is being developed. Its featuresinclude:

• 1-D and 2-D static stress-displacement and coupled consolidation analyses.• The element library consists of 1-D and 2-D continuum, beam, spring and interface

elements.• Its library of constitutive models includes general elastic (linear and non-linear) aniso-

tropic elastic, Mohr-Coulomb (associated as well as non-associated flow), Soft soil (CamClay), Soft soil creep and Hardening soil (hyperbolic) models.

• It can model excavation and construction. In addition, it can do analysis of tunnelexcavation that incorporates a volume loss parameter that represents the contractionaround tunnel lining due to overcut by the tunnel boring machine and the loss of pressureat the face of the tunnel.

• It can deal with large strain and large deformation situations and can also model creep.• It is able to select the optimum number of increments needed for efficient convergence of

non-linear problems.• It is able to model seepage problems involving phreatic surfaces and capillary effects.• It allows for incorporation of safety factors into an analysis of, for example, foundations

or slopes.• Operating System: Windows.• More information can be obtained from http://www.plaxis.nl/

8.10.4 ZSOIL

ZSOIL is a geotechnical FEM program developed by Zace Services AG, Switzerland. Itsfeatures include:

• 1-D and 2-D static stress-displacement and coupled consolidation analyses.• Elements include 1-D and 2-D continuum, beam, spring, shell, cable and interface

elements.• Constitutive models include general elastic (linear and non-linear), anisotropic elastic,

elastic-perfectly plastic with Mohr-Coulomb and capped Drucker-Prager failure criteria,and Hoek-Brown models.

• It can model excavation and construction.• It is able to deal with large strain and large deformation problems.• It can model seepage problems involving phreatic surfaces and capillary effects.• It can model partially saturated flow problems and problems involving creep.• Operating System: Windows.• More information can be obtained from http://www.zace.com/



8.11 Guidelines for the use of FEM in geotechnical engineeringThere are no shortcuts for learning to use the FEM effectively. One becomes an FEMexpert by experience and a lot of hard work. However, the following guidelines will makesure that one has a good start to the learning endeavour.

• Use smaller elements in regions where the rate of change of stress with distance isgreater. This happens, for example, near the edges of a loaded area, near a re-entrantcorner in the mesh or where adjacent parts of a mesh have significant differences instiffness (e.g. soil reinforcement, retaining wall, pile foundation) as shown in Figure 8.22.Note that some of these situations result in stress concentrations where the stressestend to infinity. The smaller we make the elements near the concentration, the higher arethe stresses. Sometimes, a stress concentration will “spoil” the solution locally, leading tooscillation of stresses. Here, it is worth remembering that infinite stress concentrationsare mathematical fiction that may be unimportant in describing real behaviour andtherefore, it is often advisable to ignore them.

• When increasing the element size from area of interest to the far boundaries, avoidincreasing the element size by more than a factor of 2 between adjacent elements.

• Wherever possible, make use of symmetry of the problem (if any) - it will save both yoursand the computer’s time.

• Keep the triangular elements as equilateral as possible and the quadrilateral elements assquare as possible.

Figure 8.22: Areas of FEM domains that require finer elements

• Avoid using curved elements as interior edges between elements in a mesh - only usethem at external boundaries or internal boundaries (e.g. inside of a tunnel) if absolutelynecessary.

• Where you place the boundary of a mesh can make a big difference to the outcome ofthe analysis. If you are unsure of the boundary effect, try two different meshes – one witha close and the other with a far boundary.

Re-entrantCorner

Edge of theLoaded Area

Interface betweenpile and soil

Stress Concentration

Zone of Interest



• When modelling an axisymmetric undrained problem where collapse is expected (e.g.cylindrical cavity expansion in a pressuremeter test), use only fourth-order (cubic strain)elements.

• Always check that the in-situ stresses specified at the start of the analysis are inequilibrium. If the in-situ stresses are not in equilibrium, either you have not inputconsistent values of soil unit weight or have not applied correct fixity conditions to one ormore of mesh boundaries.

• When using elastic-perfectly plastic constitutive models with either a Mohr-Coulomb or aDrucker-Prager failure criterion, specify a small value for c’ (0.1 or 1 kPa) even if thematerial has c’ = 0 kPa. This will ensure that the initial state of stress for the material isnot on the failure surface.

• In order to model the incompressibility of a saturated soil under undrained conditions, aPoisson’s ratio (ν) of 0.5 should ideally be used. However, if ν = 0.5 is input into an FEManalysis, it will result in serious ill-conditioning of the equations. The reason for this isthat the bulk modulus (K) of the soil approaches infinity as ν → 0.5. In such situations, ν= 0.49 usually gives satisfactory results.

• Treat pore pressure boundary conditions with respect. They are the most likely source ofdisaster in a geotechnical FEM analysis. Before applying these boundary conditions,make sure that you fully understand the ground water conditions for your problem. MostFEM programs treat any mesh boundary as impermeable by default. Setting the excesspore pressure to zero on a boundary means that the boundary is now able to drain.However, the task is not complete by just “switching on” the pore pressure boundary. Itseffect must be felt by the adjacent elements in the next time step. Otherwise, oscillationof pore pressures can occur. The minimum time step required for this purpose can becomputed based on the parabolic isochrone solution to the consolidation equation asshown in Figure 8.23.

Figure 8.23: Minimum time-step for dissipation of excess pore pressures

• When modelling excavation or construction by removing or adding elements to themesh, respectively, use several layers of elements and remove/add these elements layerby layer, applying each layer over several increments. This will ensure that the stiffnessof the soil being removed or added is correctly modelled.

y

uumax

L

y

uumax

Oscillation of pore pressuredue to insufficient time-steptcL v12= or

vc

Lt

12

2

min =



8.12 Concluding remarksIn the beginning of the chapter, we stated that site investigation and laboratory testing areused to obtain the input soil parameters for an analysis using FEM. As a geotechnicalengineer, one must never forget that soil is an extremely difficult material to characterize.Sampling disturbances, poorly controlled laboratory experiments, failure to interpret theresults from laboratory tests in a scientific manner are some of the factors that introduceerrors and uncertainty in the values of soil parameters. Therefore, the results of a FEManalysis must always be critically examined by comparing them with the results of anotherFEM analysis of a successfully completed project in similar ground conditions. Otherwise,one is likely to fall victim to the simplest equation of them all: Garbage In = Garbage Out !!

8.13 References1. Clough, R.W. (1960). The finite element method in plane stress analysis. Proc. Second

Conference on Electronic Computation, ASCE, Pittsburgh.

2. Courant, R. (1943). Variational methods for the solution of problems of equilibrium andvibrations. Bulletin of American Mathematics Society, Vol.49.

3. Zienkiewicz, O.C. and Taylor, R.L. (1989). The Finite Element Method, Vol. 1, BasicFormulation and Linear Problems, McGraw-Hill, London.


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