Geometry - Idealisation for numerical modelling as before for physical modelling

2.5 Idealisation for numerical modelling as before for physical modelling

2.5.1 Geometry

• try to represent 3-dimensional effects as 2-dimensional effects (cheaper, quicker)

→ may reproduce as a plane strain or axisymmetric problem by use of symmetry or asymmetry

• consider and idealise boundaries: soil/structure

• draw outline section and plan with material/boundaries

→ (too much vertical deflection at rollers indicates boundaries may not be far enough away?)

• create mesh → nodes and elements

→ avoid large jumps in element size to < 3x (FEM) or < 1.5x (FDM)

→ refine mesh in regions of high strain but beware infinite stress concentrations

→ limit number of nodes and elements according to complexity, typically...

* non-critical structure 150~200 elements

* dam with deep foundation 300~400 elements (or more)

Some finite element types:

→ 2 dimensional triangular constant strain

→ 2 dimensional quadrilateral linear strain

→ 3 dimensional hexahedral cubic strain

→ interface elements: relative movement between elements (progressive slip on piles)

→ bar elements: capacity for tension (soil reinforcement) /compression (props)

→ beam elements: capacity for axial force and bending moments

(structural inclusions)

→ infinite elements: models unbounded area e.g. in dynamics where fixed boundary would reflect waves 2.5.2 Mesh design

This has been shown in the past to influence the results obtained and the major guidelines will be presented in more detail in chapter 3. Several examples have been shown on pages 2 - 4.

ADAPTIVE MESH REFINEMENT (e.g. El-Hamalawi, 1997) can be used to enrich and subdivide mesh as regions of high strain develop - so that mesh choice does not precon-dition outcome of the analysis.

2.5.3 Structure Material

• use 'drained' properties (not much pore pressure in steel or concrete!)

• linear elastic (although can use linear elastic-perfectly plastic if trying to 'fail' structure)

• much stiffer than soil so beware of numerical instabilities (sometimes need double precision in FEM or need more time steps/finer mesh in FDM)

Equivalence

• row of piles as a sheet pile wall - equivalent bending rigidity (EI)_wall = n (EI)individual piles + (EI)soil between piles

• similar equivalence when modelling cylindrical sand drains as a 2-dimensional sand drain wall.

Principles of numerical modelling 2 - 10

2.5.4 Loading and construction effects

• in-situ stresses defined initially

• loads primarily - normal and shear forces (tractions) on elements

• excavation and fill: construction sequences for embankment and retaining wall

• superposition of layers of soil or concrete (for geometric purposes) for subsequent removal

• displacement and rotation fixities (x, y, z, θ) - either the soil or structure can be moved relative to rest of mesh (Bransby analysis p. 5)

• pore pressure fixities - can be use to set up excess pore pressures, drains or free water surfaces.

Figure 2.10: Equivalence

= (EI)soil between piles

+ n(EI)_piles

(EI)_wall

1 m 10 kN/m²

5 kN/m 5 kN/m

2 kN/m2 1 kN/m 1 kN/m

Figure 2.11: Normal surface loading

Reality Model

3 2 1

1 2 3

Excavation Fill

Figure 2.12: Excavation and fill

Use pile response to various loadings as examples:

• axial loading: shaft friction and end bearing (3-dimensional to axisymmetry) see page 12

e.g. the pile behaviour is a function of…..f ( E_pile / G , l / r_o , ν ) where

E_pile is the pile Young's modulus G is soil shear modulus

l is the pile length r_o is the pile radius

ν is the Poisson's ratio

• lateral thrust/loading due to embankment surcharge (3-dimensional to plane strain)

• piled abutment (3-dimensional to plane strain) Axisymmetry

Driven pile installation? not so good numerically (unless dynamic analysis); better in the centrifuge

• spherical / cylindrical cavity expansion

• remoulds soil around pile with massive strains

• changes stress history

• wish-in-place pile is normally adopted

• it is artificially possible to change soil properties adjacent to pile or use interface elements

Bored pile installation? not so good for centrifuge modelling; better in numerical analysis

• remove soil elements and replace with bentonite (relax circumferential stresses)

• tremie concrete (heavy liquid) to reload excavated cylindrical hole circumferentially

• replace concrete as a heavy liquid by hardened concrete.

Principles of numerical modelling 2 - 12 Figure 2.13: Modelling an axially loaded pile

What is the vertical deformation pattern in the soil and in the pile due to the axial load on the pile?

Pile

Plan

θ r

Plan axisymmetry r and θ plane: z common

Axial load

End bearing δz_s?

δz_p?

Pile

Shaft friction component

Normal, uniformly distributed load

End bearing component Pile Soil

z r

Soil

z ^r

Circular ‘footing’ load

Mesh

Section Soil

Soil

Shaft friction Axial loading on a long flexible cylindrical pile

r z

Plane strain

Piled full-height bridge abutment Piled full-height bridge abutment

δu

Embankment Soft

Stiff

Piled full-height bridge abutment Piled full-height bridge abutment

Ellis PhD, 1997

δup

arching

Figure 2.14: Piled full-height bridge abutment

δ^u

Plan

When pile is displaced laterally relative to the soil, what is the relative soil-pile movement?

Select half space: PLANE STRAIN x and y plane: z common

Pile Soil

Section x

Lateral thrust

Figure 2.15: Idealisation for lateral thrust on a single row of piles from embankment loading

Principles of numerical modelling 2 - 14

2.5.5 Ellis (1997): Piled full-height abutment: 3D problem as 2D....

• behaviour of soil under embankment of most interest

• relative soil-pile displacement less critical than this

• additional component of lateral thrust caused by arching is critical

• model row of piles as a wall of equivalent bending rigidity

• overlay soil and 'pile' wall with interaction law with relative soil-pile movement

• soil may be displaced past 'pile' wall so lateral thrust on piles added to equilibrium equation.

Figure 2.17: The finite element mesh with vertical drains (Ellis, 1997)

sand

embankment

Figure 2.16: Finite element analysis: contours of horizontal stress (Ellis, 1997)

piled abutment wall

kPa

low sress sand

embankment hig

h stress

soft clay

low sress

soft clay

SAND SAND

CLAY

2.5.6 Soil

Why can we not use simple back of the envelope calculations probably based on elastic analyses?

• simple elastic models do not reproduce key aspects of soil behaviour

• we can select more appropriate soil models for design, to account for

→ pre-yield stiffness

→ yield and failure criteria

• OK for more complex analysis because computing power in design offices is growing

Must select?

• type of analysis

• model of soil behaviour - or - constitutive model.

Type of analysis

• steady state (time-independent)

→ steady state seepage

→ static load-deformation problems.

• transient (time-dependent)

→ consolidation

→ dynamic loading (earthquakes, wave action)

→ contaminant transport processes

→ creep.

Drained analysis

• no excess pore pressure - highly permeable soils

• all the loads will be transferred to the soil skeleton: effective stress

• long-term condition - mostly interested in displacements.

Undrained analysis - low permeability soils

• loads will be carried by both soil skeleton and pore pressure

• no volume change - very large bulk modulus K compared to shear modulus G: K>>G

• short-term stability - mostly interested in (total) stresses - undrained failure of clays?

• avoid using equal size elements if the solution is oscillating or use higher order elements, or

• set ν_u= 0.49 with a short time step within a consolidation analysis.

Principles of numerical modelling 2 - 16

Consolidation analysis (Biot's equations) - more time consuming

• transition from undrained condition to drained condition

• check the movement of the system with time

Which do you want to choose for your analysis?

Figure 2.18: Influence of the Poissons ratio on the settlement of a strip footing (Potts & Zdraykovic, 1999)

time

settlement

undrained

Figure 2.19: Settlement of a footing in time

2.6 References

1. M.S.S. Almeida, Stage constructed embankment on soft clay. PhD thesis, University of Cambridge. 1984.

2. M.F. Bransby, Piled foundations adjacent to surcharge loads. PhD thesis, University of Cambridge. 1995.

3. A. El-Hamalawi, Adaptive refinement of finite element meshes for geotechnical analysis. PhD thesis, University of Cambridge. 1997.

4. E.A. Ellis, Soil-Structure interaction for full-height piled bridge abutments constructed on soft clay. PhD thesis, University of Cambridge. 1997.

5. D.M. Potts, L. Zdravkovic, Finite Element Analysis in Geotechnical Engineering. Vols.

1 & 2. Thomas Telford, London.1999.

6. H.G. Poulos, Experiences with soil-structure interaction in the Far East. 2^nd Int.

Conference on Soil Structure Interaction in Urban Civil Engineering. Zürich, 2002.

7. L. Prandtl, Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden, Zeitschrift für angewandte Mathematik und Mechanik, 1921, 1(1), 15-20.

8. M.F. Randolph and G.T. Houlsby, The limiting pressure on a circular pile loaded laterally in cohesive soil. Géotechnique, 1984, 34, No. 4, pp. 613-623.

Modelling in Geotechnics

Numerical Modelling Finite Element Method

Dr. Jitendra Sharma

3 Finite Element Method (FEM) in Geotechnical Engineering

3.1 Introduction

The importance of a carefully planned and executed experimental modelling can not be overstated. However, experimental modelling can be expensive and time-consuming and is normally 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 as accurately as possible. These parameters are then used as input to either limit equilibrium based programs (e.g. slope stability, bearing capacity, etc.) to predict failure loads (ultimate limit state) or a numerical analysis program (e.g. finite element method, finite difference method, etc.) to predict the deformation under working load conditions (serviceability limit state). In this chapter, we will focus on one of the most popular numerical analysis technique used in geotechnical engineering – the finite element method or FEM. The aim of this 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 reader may well consult one of the numerous books that deal with the mathematics and the numerical techniques used in the FEM, e.g. Zienkiewicz and Taylor (1989).

3.2 Numerical methods used in geotechnical engineering

Figure 3.1: Various ways of solving a geotechnical engineering problem

As stated in the beginning of this course, there are several different ways of finding solutions to a geotechnical engineering problem. These are summarized in Figure 3.1. In this section, we will focus on the numerical methods. One of the characteristic features of the numerical methods is that they usually involve solving a set of simultaneous partial differential 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, an iterative numerical approach is used. There are five major types of numerical methods used in geotechnical engineering – the finite element, the finite difference, the boundary element, the discrete element and the combined boundary/finite element. The way the PDEs are formulated and solved differs for each of these methods.

Solution of Geotechnical Problems

Finite Element Method (FEM) in Geotechnical Engineering Page 3 - 2

3.3 What is FEM?

Figure 3.2: Discrete vs. continuous problem

Before introducing the concept of the FEM, let us first explore the difference between a discrete and a continuous system. For a discrete system, an adequate solution can be obtained using a finite number of well-defined components. Such problems can be readily solved even with rather large number of components, e.g. the analysis of a building frame consisting of beams, columns and slabs (Figure 3.2). For a continuous system, such as a soil layer, the sub-division is continued infinitely so that the problem can only be defined using the mathematical fiction of infinitesimal. Depending on the level of complexity involved, there are two ways of solving such a problem. Simple, linear problems can be solved easily by mathematical manipulation. Solution of complex, non-linear problems involves discretization of the problem into components of finite dimensions (Figure 3.2) and then using a numerical method such as the FEM.

The most distinctive feature of the FEM that separates it from other numerical methods is the division of a given domain into a set of simple subdomains, called finite elements. Any geometric shape that allows computation of the solution or its approximation, or provides necessary relation among the values of the solution at selected points, called nodes, of the subdomain, qualifies as a finite element. Such a subdivision of a whole into parts has two advantages:

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

3.3.1 Historical Background

The idea of representing a given domain as a collection of discrete parts is not unique to the FEM. It was recorded that ancient Greek mathematicians estimated the value of π by noting that the perimeter of a polygon inscribed in a circle approximates the circumference of the circle. 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 approximate solution or comprehension of the whole, by studying the constituent parts of the whole is vital to almost all investigations in science, humanities, and engineering. The FEM is an outgrowth of the familiar procedures such as the frame analysis and the lattice analogy for 2- and 3-dimensional bodies. Its application is not exclusive to engineering. It has been used in other fields such as mathematics & physics. One of the earliest examples of its use was in mathematics by R. Courant who used it for the solution of equilibrium and vibration problems (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 plane stress 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 the advent 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 cheap computing power at their disposal, FEM is the first choice for many engineers and scientists embarking on the analysis of a wide variety of engineering problems – from designing a new ergonomic shoe sole to designing a supersonic fighter aircraft. Its use in the field of bioengineering, for example, the modelling of knee prosthesis or stress analysis of brain oedema, is also fast becoming popular.

3.3.2 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 in order 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 a problem.

3.3.3 Approximation of the Circumference of a Circle

Consider the problem of determining the perimeter of a circle of radius R (Figure 3.3).

Ancient mathematicians estimated the value of the circumference by approximating it by line segments, whose lengths they were able to measure. The approximate value of the circumference 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 the finite element analysis of any problem.

Finite Element Method (FEM) in Geotechnical Engineering Page 3 - 4

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 3.3b).

2. Element equations: A typical element is isolated and its required properties, i.e. its length, are computed by some appropriate means. Let h_e be the length of the element Ω^e in the mesh. For a typical element Ω^e, h_e is given by (see Figure 3.3c):

(3.1) where R is the radius of the circle and θe < π is the angle subtended by the line segment at the centre of the circle. The above equations are called element equations. Ancient mathematicians most likely made measurements, rather than using (3.1) to find h_e.

Figure 3.3: Approximation of the circumference of a circle by line elements

Assembly of element equations and solution: The approximate value of the circumference (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 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.

(a) (b)

(c)

Approximation of the circumference of a circle by line elements:

(a) Circle of radius R; (b) Uniform and non-uniform meshes used to represent the circumference of the circle;

Element

Node R

θ

h

(3.2)

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

(3.3)

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

(3.4)

We can estimate the error in the approximation and show that the approximate solution P_n converges 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 as closely as we wish by a finite number of piecewise-linear functions. As the number of elements is increased, the approximation improves, i.e. the error in the approximation decreases.

3.4 Basic formulation of the FEM

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

3.4.1 Interconnected elastic springs

d b

c d₁

d₂

d₃

d₄

2 T_a

T_b W₂ T_d

Equilibrium at Node 2

Finite Element Method (FEM) in Geotechnical Engineering Page 3 - 6 Figure 3.4: 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 k_a, k_b, k_c and k_d. Therefore, the tensions in these springs are given by:

(3.5) where e_a, e_b, e_c and e_d 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:

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

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

(3.7)

7. Equilibrium (at node 2, see Figure 3.4):

(3.8) which on rearrangement, results in:

(3.9) 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:

(3.10)

The matrix on the left-hand-side is called the global stiffness matrix. Equation (3.10) can be written in matrix notation as:

Kd = W

These simultaneous equations can be solved by elimination and values of displacements can 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 k_e is the stiffness of one particular spring):

10.

(3.11)

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

(3.12) 3.4.2 A plane truss element

Figure 3.5: A plane truss element

In this section, we will apply the same principles of compatibility, material behaviour and equilibrium to a one-dimensional plane truss element (Figure 3.5). The formulation is now

In document Modelling in Geotechnics (Page 33-175)