2D 4 Scientific

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PLAXIS

Scientific Manual

2015


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TABLE OF CONTENTS

TABLE OF CONTENTS

1 Introduction 5

2 Deformation theory 7

2.1 Basic equations of continuum deformation 72.2 Finite element discretisation 8

2.3 Implicit integration of differential plasticity models 9

2.4 Global iterative procedure 10

3 Groundwater flow theory 13

3.1 Basic equations of flow 13

3.1.1 Transient flow 13

3.1.2 Continuity equation 14

3.2 Boundary Conditions 15

3.3 Finite element discretisation 16

3.4 Flow in interface elements 18

4 Consolidation theory 19

4.1 Basic equations of consolidation 19

4.2 Finite element discretisation 20

4.3 Elastoplastic consolidation 22

4.4 Critical time step 22

5 Element formulations 25

5.1 Interpolation functions of point elements 25

5.1.1 Structural elements 25

5.2 Interpolation functions and numerical integration of line elements 26

5.2.1 Interpolation functions of line elements 26


5.2.3 Derivatives of interpolation functions 31

5.2.4 Numerical integration of line elements 34

5.2.5 Calculation of element stiffness matrix 35

5.3 Interpolation functions and numerical integration of area elements 36

5.3.1 Interpolation functions of area elements 37


5.3.3 Numerical integration of area elements 40

5.4 Interpolation functions and numerical integration of volume elements 405.4.1 10-node tetrahedral element 40

5.4.2 Derivatives of interpolation functions 42

5.4.3 Numerical integration over volumes 43

5.4.4 Calculation of element stiffness matrix 43

5.5 Special elements 44

5.5.1 Embedded piles 44

6 Theory of sensitivity analysis & parameter variation 49

6.1 Sensitivity analysis 49

6.1.1 Definition of threshold value 506.2 Theory of parameter variation 51

6.2.1 Bounds on the system response 51

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7 Dynamics 53

7.1 Basic equation dynamic behaviour 53

7.2 Time integration 53

7.2.1 Critical time step 55

7.2.2 Dynamic Integration Coefficients 55

7.3 Model Boundaries 567.4 Viscous boundaries 57

7.5 Initial stresses and stress increments 57

7.6 Amplification of responses 58

7.7 Pseudo-spectral acceleration response spectrum for a single-degree-of-

freedom system 58

7.8 Natural frequency of vibration of a soil deposit 59

8 References 61

Appendix A - Calculation process 63

Appendix B - Symbols 65

4 Scientific Manual | PLAXIS 2015


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INTRODUCTION

1 INTRODUCTION

In this part of the manual some scientific background is given of the theories and

numerical methods on which the PLAXIS program is based. The manual contains

chapters on deformation theory, groundwater flow theory (PLAXIS 2D), consolidation

theory, dynamics as well as the corresponding finite element formulations and integrationrules for the various types of elements used in PLAXIS. In Appendix Aa global

calculation scheme is provided for a plastic deformation analysis.

In addition to the specific information given in this part of the manual, more information on

backgrounds of theory and numerical methods can be found in the literature, as amongst

others referred to in Chapter8. For detailed information on stresses, strains, constitutive

modelling and the types of soil models used in the PLAXIS program, the reader is

referred to theMaterial Models Manual.

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DEFORMATION THEORY

2 DEFORMATION THEORY

In this chapter the basic equations for the static deformation of a soil body are formulated

within the framework of continuum mechanics. A restriction is made in the sense that

deformations are considered to be small. This enables a formulation with reference to the

original undeformed geometry. The continuum description is discretised according to thefinite element method.

2.1 BASIC EQUATIONS OF CONTINUUM DEFORMATION

The static equilibrium of a continuum can be formulated as:

LT + b = 0 (2.1)

This equation relates the spatial derivatives of the six stress components, assembled in

vector, to the three components of the body forces, assembled in vector b. LT is thetranspose of a differential operator, defined as:

LT =

x

0 0 y

0 z

0 y

0 x

z

0

0 0 z

0 y

x

(2.2)

In addition to the equilibrium equation, the kinematic relation can be formulated as:

= L u (2.3)

This equation expresses the six strain components, assembled in vector , as the spatialderivatives of the three displacement components, assembled in vector u, using thepreviously defined differential operator L. The link between Eqs. (2.1)and(2.3) is formedby a constitutive relation representing the material behaviour. Constitutive relations, i.e.

relations between rates of stress and strain, are extensively discussed in theMaterial

Models Manual. The general relation is repeated here for completeness:

= M (2.4)

The combination of Eqs. (2.1),(2.3)and (2.4) would lead to a second-order partial

differential equation in the displacementsu.

However, instead of a direct combination, the equilibrium equation is reformulated in a

weak form according to Galerkin's variation principle: uT

LT + b

dV = 0 (2.5)

In this formulationurepresents a kinematically admissible variation of displacements.

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Applying Green's theorem for partial integration to the first term in Eq. (2.5)leads to: T dV =

uT b dV +

uT t dS (2.6)

This introduces a boundary integral in which the boundary traction appears. The three

components of the boundary traction are assembled in the vector t. Eq.(2.6) is referredto as the virtual work equation.

The development of the stress state can be regarded as an incremental process:

i = i1 + =

dt (2.7)

In this relation i represents the actual state of stress which is unknown and i1

represents the previous state of stress which is known. The stress increment is thestress rate integrated over a small time increment.

If Eq. (2.6)is considered for the actual statei, the unknown stressesi

can be eliminatedusing Eq. (2.7):

T dV =

uT bi dV +

uT ti dS

T i1 dV (2.8)

It should be noted that all quantities appearing in Eqs.(2.1)till (2.8) are functions of the

position in the three-dimensional space.

2.2 FINITE ELEMENT DISCRETISATION

According to the finite element method a continuum is divided into a number of (volume)

elements. Each element consists of a number of nodes. Each node has a number of

degrees of freedom that correspond to discrete values of the unknowns in the boundary

value problem to be solved. In the present case of deformation theory the degrees of

freedom correspond to the displacement components. Within an element the

displacement fielduis obtained from the discrete nodal values in a vector vusinginterpolation functions assembled in matrix N:

u = N v (2.9)

The interpolation functions in matrixNare often denoted as shape functions. Substitutionof Eq.(2.9) in the kinematic relation (Eq. 2.3)gives:

= L N v = B v (2.10)

In this relationBis the strain interpolation matrix, which contains the spatial derivatives ofthe interpolation functions. Eqs.(2.9) and (2.10) can be used in variational, incremental

and rate form as well.

Eq. (2.8)can now be reformulated in discretised form as:

B

T dV =

N

T bi dV +

N

T ti dS BT i1 dV



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DEFORMATION THEORY

(2.11)

The discrete displacements can be placed outside the integral:

T BT dV = T NT bi dV + T NT ti dST BT i1 dV

(2.12)

Provided that Eq. (2.12) holds for any kinematically admissible displacement variation

vT, the equation can be written as: BT dV =

NT bi dV +

NT ti dS

BT i1 dV (2.13)

The above equation is the elaborated equilibrium condition in discretised form. The first

term on the right-hand side together with the second term represent the current external

force vector and the last term represents the internal reaction vector from the previous

step. A difference between the external force vector and the internal reaction vectorshould be balanced by a stress increment .

The relation between stress increments and strain increments is usually non-linear. As a

result, strain increments can generally not be calculated directly, and global iterative

procedures are required to satisfy the equilibrium condition (Eq. 2.13) for all material

points. Global iterative procedures are described later in Section2.4,but the attention is

first focused on the (local) integration of stresses.

2.3 IMPLICIT INTEGRATION OF DIFFERENTIAL PLASTICITY MODELS

The stress increments are obtained by integration of the stress rates according to Eq.(2.7). For differential plasticity models the stress increments can generally be written as:

= De p (2.14)

In this relationDe

represents the elastic material matrix for the current stress increment.

The strain increments are obtained from the displacement increments vusing thestrain interpolation matrixB, similar to Eq. (2.10).

For elastic material behaviour, the plastic strain increment p is zero. For plastic

material behaviour, the plastic strain increment can be written, according toVermeer(1979), as:

p =

(1 )

g

i1 +

g

i

(2.15)

In this equation is the increment of the plastic multiplier and is a parameterindicating the type of time integration. For = 0 the integration is called explicit and for= 1 the integration is called implicit.

Vermeer (1979)has shown that the use of implicit integration ( = 1) has some majoradvantages, as it overcomes the requirement to update the stress to the yield surface in

the case of a transition from elastic to elastoplastic behaviour. Moreover, it can be proven

that implicit integration, under certain conditions, leads to a symmetric and positive



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differential matrix/ , which has a positive influence on iterative procedures. Becauseof these major advantages, restriction is made here to implicit integration and no attention

is given to other types of time integration.

Hence, for = 1 Eq. (2.15) reduces to:

p =

g

i (2.16)

Substitution of Eq. (2.16)into Eq. (2.14) and successively into Eq. (2.7)gives:

i = tr De

g

i with: tr = i1 + De (2.17)

In this relationtr is an auxiliary stress vector, referred to as the elastic stressesor trialstresses, which is the new stress state when considering purely linear elastic material

behaviour.

The increment of the plastic multiplier , as used in Eq. (2.17), can be solved from thecondition that the new stress state has to satisfy the yield condition:

f

i

= 0 (2.18)

For perfectly-plastic and linear hardening models the increment of the plastic multiplier

can be written as:

=f

tr

d+ h(2.19)

where:

d =

f

tr De

g

i (2.20)

The symbolhdenotes the hardening parameter, which is zero for perfectly-plastic modelsand constant for linear hardening models. In the latter case the new stress state can be

formulated as:

i = tr f

tr

d+ h De

g

i (2.21)

The -brackets are referred to as McCauley brackets, which have the followingconvention:

x = 0 for: x 0 and: x =x for: x>0

For non-linear hardening models the increment of the plastic multiplier is obtained using a

Newton-type iterative procedure with convergence control.

2.4 GLOBAL ITERATIVE PROCEDURE

Substitution of the relationship between increments of stress and increments of strain,



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DEFORMATION THEORY

= M, into the equilibrium equation (Eq. 2.13) leads to:

Kivi = fiex fi1in (2.22)

In this equationKis a stiffness matrix, vis the incremental displacement vector, fex is

the external force vector andfinis the internal reaction vector. The superscript irefers tothe step number. However, because the relation between stress increments and strainincrements is generally non-linear, the stiffness matrix cannot be formulated exactly

beforehand. Hence, a global iterative procedure is required to satisfy both the equilibrium

condition and the constitutive relation. The global iteration process can be written as:

Kj vj = fiex fj1in (2.23)

The superscriptjrefers to the iteration number. vis a vector containing sub-incrementaldisplacements, which contribute to the displacement increments of step i:

vi =

nj=1

vj (2.24)

wherenis the number of iterations within step i. The stiffness matrix K, as used in Eq.(2.23), represents the material behaviour in an approximated manner. The more accurate

the stiffness matrix, the fewer iterations are required to obtain equilibrium within a certain

tolerance.

In its simplest formKrepresents a linear-elastic response. In this case the stiffnessmatrix can be formulated as:

K =

BT De B dV (elastic stiffness matrix) (2.25)

whereDe is the elastic material matrix according to Hooke's law andBis the straininterpolation matrix. The use of an elastic stiffness matrix gives a robust iterative

procedure as long as the material stiffness does not increase, even when using

non-associated plasticity models. Special techniques such as arc-length control (Riks,

1979), over-relaxation and extrapolation (Vermeer & van Langen, 1989)can be used to

improve the iteration process. Moreover, the automatic step size procedure, as

introduced byVan Langen & Vermeer (1990), can be used to improve the practical

applicability. For material models with linear behaviour in the elastic domain, such as the

standard Mohr-Coulomb model, the use of an elastic stiffness matrix is particularlyfavourable, as the stiffness matrix needs only be formed and decomposed before the first

calculation step. This calculation procedure is summarised in Appendix A.



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GROUNDWATER FLOW THEORY

3 GROUNDWATER FLOW THEORY

In this chapter we will review the theory of groundwater flow as used in PLAXIS 2D. In

addition to a general description of groundwater flow, attention is focused on the finite

element formulation.

3.1 BASIC EQUATIONS OF FLOW

3.1.1 TRANSIENT FLOW

Flow in a porous medium can be described by Darcy's law which is expressed by the

following equation in three dimensions:

q=k

wgpw + wg

(3.1)

Where

=

x

y

z

(3.2)

q,k,gandware the specific discharge (fluid velocity), the tensor of permeability, theacceleration vector due to gravity Eq. (3.3)and the density of water, respectively.pw isthe gradient of the water pore pressure which causes groundwater flow. The term

wg is

used as the flow is not affected by the gradient of the water pore pressure in vertical

direction when hydrostatic conditions are assumed.

g=

0

g0

(3.3)

In unsaturated soils the coefficient of permeability kcan be related to the soil saturationas:

k=krelksat

(3.4)

where

ksat =

ksatx 0 0

0 ksaty 0

0 0 ksatz

(3.5)

In whichkrel is the ratio of the permeability at a given saturation to the permeability insaturated stateksat.



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3.1.2 CONTINUITY EQUATION

The mass concentration of the water in each elemental volume of the medium is equal to

wnS. The parametersnand Sare the porosity and the degree of saturation of the soil,respectively. According to the mass conservation, the water outflow from the the volume

is equal to the changes in the mass concentration. As the water outflow is equal to the

divergence of the specific discharge (T q), the continuity equation has the form(Song,1990):

T

krelwg

ksatpw + wg

=

t(wnS) (3.6)

By neglecting the deformations of solid particles and the gradients of the density of water

(Boussinesq's approximation), the continuity equation is smplified to:

T

krelwg

ksat

pw + wg

+ SmT

t n

S

Kw S

pw

pw

t= 0 (3.7)

Where

mT =

1 1 1 0 0 0

(3.8)

For transient groundwater flow the displacements of solid particles are neglected.

Therefore:

T

krelwg

ksatpw + wg

n

S

Kw S

pw

pw

t= 0 (3.9)

For steady state flow

pwt = 0

the continuity condition applies:

T

krelwg

ksatpw + wg

= 0 (3.10)

Eq. (3.10) expresses that there is no net inflow or outflow in an elementary area, as

illustrated in Figure3.1.

qx

qy

qy +q

ydy

qx + qx

dx

Figure 3.1 Illustration of continuity condition



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3.2 BOUNDARY CONDITIONS

The following boundary conditions are available in PLAXIS 2D:

Closed: This type of boundary conditions specifies a zero Darcy flux over the boundary

as

qxnx+ qyny = 0 (3.11)

wherenx andnyare the outward pointing normal vector components on the boundary.

Inflow: A non-zero Darcy flux over a boundary is set by a prescribed recharge value qand reads

qxnx+ qyny = q (3.12)

This indicates that the Darcy flux vector and the normal vector on the boundary are

pointing in opposite directions.Outflow: For outflow boundary conditions the direction of the prescribed Darcy flux, q,should equal the direction of the normal on the boundary, i.e.:

qxnx+ qyny = q (3.13)

Head: For prescribed head boundaries the value of the head is imposed as

= (3.14)

Alternatively prescribed pressure conditions can be given. Overtopping conditions for

example can be formulated as prescribed pressureboundaries.

p= 0 (3.15)

These conditions directly relate to a prescribed head boundary condition and are

implemented as such.

Infiltration: This type of boundary conditions poses a more complex mixed boundary

condition. An inflow value qmay depend on time and as in nature the amount of inflow islimited by the capacity of the soil. If the precipitation rate exceeds this capacity, ponding

takes place at a depth p,max and the boundary condition switches from inflow to

prescribed head. As soon as the soil capacity meets the infiltration rate the conditionswitches back.

= z+ p,max if ponding

qxnx+ qyny = q if z+ phi+ p,min

= z+ p,min if drying

(3.16)

This boundary condition simulates evaporation for negative values of q. The outflowboundary condition is limited by a minimum head p,minto ensure numerical stability.

Seepage: The water line option generates phreatic/seepage conditions by default. An

external head is prescribed on the part of the boundary beneath the water line, seepage



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or free conditions are applied to the rest of the line. The phreatic/seepage condition reads

= if zqxnx+ qyny = 0 if


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element theory please refer to the PLAXIS scientific manual, or (Bathe & Koshgoftaar,

1979), (Zienkiewicz, 1967). According to Eq. (3.1), the specific discharge is based on the

gradient of the groundwater pore pressure. This gradient can be determined by means of

theB-matrix, which contains the spatial derivatives of the interpolation functions, see thePLAXIS scientific manual. In the numerical formulation the specific discharge, q, is

written as:

q= krel

wksat

B pwn+ wg

(3.23)

where:

q=

qx

qy

and: ksat =

ksatx 0

0 ksaty

(3.24)

From the specific discharges in the integration points,q, the nodal dischargesQe

can be

integrated according to:

Qe =

BTq dV (3.25)

in whichBT is the transpose of the B-matrix. The termdVindicates integration over thevolume of the body.

Starting from the continuity equation Eq. (3.9) and applying the Galerkin approach and

incorporating prescribed boundary conditions we obtain:

H pwn Sd pw

ndt = qp

(3.26)

whereH,Sandqp

are the permeability matrix, the compressibility matrix and the

prescribed recharges that are given by the boundary conditions, respectively:

H =

NTkrelw

ksatN dV (3.27)

S =

NT

nS

Kw nS

pw

N dV (3.28)

qp =NTkrel

wksat wg dV NTq d (3.29)

qis the outflow prescribed flux on the boundary. The term d indicates a surface integral.

In PlaxFlow the bulk modulus of the pore fluid is taken automatically according to:

Kw

n =

3(u )(1 2u)(1 +)

Kskeleton (3.30)

Whereuhas a default value of 0.495. The value can be modified in the input programon the basis of Skempton's B-parameter. For material just switched on, the bulk modulus

of the pore fluid is neglected.Due to the unsaturated zone the set of equations is highly non-linear and a Picard



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scheme is used to solve the system of equations iteratively. The linear set is solved in

incremental form using an implicit time stepping schema. Application of this procedure to

Eq. (3.26) yields:

t H+ Spwn = t H pwn0 + tqp (3.31)andpwn0 denote value of water pore pressure at the beginning of a step. The parameter is the time integration coefficient. In general the integration coefficient can takevalues from 0 to 1. In PlaxFlow the fully implicit scheme of integration is used with = 1.

For steady state flow the governing equation is:

Hpwn = H pwn0 + qp (3.32)

Interface elements are treated specially in groundwater calculations. The elements can

be on or off. When the elements are switched on, there is a full coupling of the pore

pressure degrees of freedom. When the interface elements are switched off, there is noflow from one side of the interface element to the other (impermeable screen).

3.4 FLOW IN INTERFACE ELEMENTS

Interface elements are treated specially in groundwater calculations. The elements can

be on or off. When the elements are switched off, there is a full coupling of the pore

pressure degrees of freedom. When the interface elements are switched on, there is no

flow from one side of the interface element to the other (impermeable screen).



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CONSOLIDATION THEORY

4 CONSOLIDATION THEORY

In this chapter we will review the theory of consolidation as used in PLAXIS. In addition to

a general description of Biot's theory for coupled consolidation, attention is focused on

the finite element formulation. Moreover, a separate section is devoted to the use of

advanced soil models in a consolidation analysis (elastoplastic consolidation).

4.1 BASIC EQUATIONS OF CONSOLIDATION

The governing equations of consolidation as used in PLAXIS follow Biot's theory (Biot,

1956). Darcy's law for fluid flow and elastic behaviour of the soil skeleton are also

assumed. The formulation is based on small strain theory. According to Terzaghi's

principle, stresses are divided into effective stresses and pore pressures:

=

' + mp

steady + p

excess (4.1)

where:

=

xx yy zz xy yz zx

T and: m =

1 1 1 0 0 0

T (4.2)

is the vector with total stresses, ' contains the effective stresses,pexcessis the excesspore pressure andmis a vector containing unity terms for normal stress components andzero terms for the shear stress components. The steady state solution at the end of the

consolidation process is denoted aspsteady. Within PLAXISpsteady is defined as:

psteady = pinput (4.3)

wherepinput is the pore pressure generated in the input program based on phreatic lines

after the use of theK0 procedureor Gravity loading. Note that within PLAXIS

compressive stresses are considered to be negative; this applies to effective stresses as

well as to pore pressures. In fact it would be more appropriate to refer to pexcessandpsteadyas pore stresses, rather than pressures. However, the term pore pressure isretained, although it is positive for tension.

The constitutive equation is written in incremental form. Denoting an effective stress

increment as ' and a strain increment as , the constitutive equation is:

' = M (4.4)

where:

=

xx yy zz xy yz zx

T (4.5)

andMrepresents the material stiffness matrix. For details on constitutive relations, seetheMaterial Models Manual.



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4.2 FINITE ELEMENT DISCRETISATION

To apply a finite element approximation we use the standard notation:

u=N v p= N pn

= B v (4.6)

wherevis the nodal displacement vector, pn

is the excess pore pressure vector, uis thecontinuous displacement vector within an element andpis the (excess) pore pressure.The matrixNcontains the interpolation functions and B is the strain interpolation matrix.

In general the interpolation functions for the displacements may be different from the

interpolation functions for the pore pressure. In PLAXIS, however, the same functions are

used for displacements and pore pressures.

Starting from the incremental equilibrium equation and applying the above finite element

approximation we obtain:

BTd dV =

NTd b d V +

NTdt dS+ r0 (4.7)

with:

r0 =

NT b0 dV+

NT t0 dS

BT 0dV (4.8)

wherebis a body force due to self-weight and t represents the surface tractions. Ingeneral the residual force vector r0 will be equal to zero, but solutions of previous loadsteps may have been inaccurate. By adding the residual force vector the computational

procedure becomes self-correcting. The term dVindicates integration over the volume of

the body considered anddSindicates a surface integral.

Dividing the total stresses into pore pressure and effective stresses and introducing the

constitutive relationship gives the nodal equilibrium equation:

K dv + L d pn

= d fn (4.9)

whereKis the stiffness matrix,L is the coupling matrix andd fn is the incremental loadvector:

K = BT M B dV (4.10a)L =

BT m N dV (4.10b)

d fn =

NT d b d V +

NT dt dS (4.10c)

To formulate the flow problem, the continuity equation is adopted in the following form:

TR(wy psteady p)/ w mT t

+ n

Kw

p

t= 0 (4.11)



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whereR is the permeability matrix:

R=

kx 0 0

0 ky 0

0 0 kz

(4.12)

nis the porosity,Kwis the bulk modulus of the pore fluid and wis the unit weight of thepore fluid. This continuity equation includes the sign convention thatpsteady andpareconsidered positive for tension.

As the steady state solution is defined by the equation:

TR(wy psteady) / w = 0 (4.13)

the continuity equation takes the following form:

TRp/ w + mT t

nKw

pt

= 0 (4.14)

Applying finite element discretisation using a Galerkin procedure and incorporating

prescribed boundary conditions we obtain:

H pn

+ LT dv

dt S dpn

dt= q

n (4.15)

where:

H =

(N)T RN/ wdV, S = nKw NT N dV (4.16)andq

n is a vector due to prescribed outflow at the boundary. However within PLAXIS it is

not possible to have boundaries with non-zero prescribed outflow. The boundary is either

closed (zero flux) or open (zero excess pore pressure). In reality the bulk modulus of

water is very high and so the compressibility of water can be neglected in comparison to

the compressibility of the soil skeleton.

In PLAXIS the bulk modulus of the pore fluid is taken automatically according to (also see

Reference Manual):

Kwn

= 3(u )(1 2u)(1 +)

Kskeleton (4.17)

Whereuhas a default value of 0.495. The value can be modified in the input programon the basis of Skempton's B-parameter. For drained material and material just switched

on, the bulk modulus of the pore fluid is neglected.

The equilibrium and continuity equations may be compressed into a block matrix

equation:

K LLT S

d v

dt

dpndt

= 0 0

0 H v

pn

+

d fndt

qn

(4.18)

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A simple step-by-step integration procedure is used to solve this equation. Using the

symbol to denote finite increments, the integration gives:

K L

LT

S

v

pn

=

0 0

0 t H

v0

pn0

+

fn

t q

n

(4.19)

where:

S

= t H + S qn

= qn0

+ qn

(4.20)

andv0 andpn0 denote values at the beginning of a time step. The parameter is thetime integration coefficient. In general the integration coefficient can take values from 0to 1. In PLAXIS the fully implicit scheme of integration is used with = 1.

4.3 ELASTOPLASTIC CONSOLIDATION

In general, when a non-linear material model is used, iterations are needed to arrive at

the correct solution. Due to plasticity or stress-dependent stiffness behaviour the

equilibrium equations are not necessarily satisfied using the technique described above.

Therefore the equilibrium equation is inspected here. Instead of Eq. (4.9)the equilibrium

equation is written in sub-incremental form:

Kv + L pn

= rn (4.21)

wherernis the global residual force vector. The total displacement increment vis the

summation of sub-incrementsvfrom all iterations in the current step:

rn =

NTb dV +

NTt dS

BT dV (4.22)

with:

b=b0+ b and: t = t0 + t (4.23)

In the first iteration we consider = 0, i.e. the stress at the beginning of the step.Successive iterations are used on the current stresses that are computed from the

appropriate constitutive model.

4.4 CRITICAL TIME STEP

For most numerical integration procedures, accuracy increases when the time step is

reduced, but for consolidation there is a threshold value. Below a particular time

increment (critical time step) the accuracy rapidly decreases. Care should be taken with

time steps that are smaller than the advised minimum time step. The critical time step is

calculated as:

tcritical = H2

cv(4.24)



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where is the time integration coefficient which is equal to 1 for fully implicit integrationscheme, is a constant parameter which is determined for each types of element andHis the height of the element used. cv is the consolidation coefficient and is calculated as:

cv =

k

w1

K'+ Q

(4.25)

wherewis the unit weight of the pore fluid,kis the coefficient of permeability, K' is thedrained bulk modulus of soil skeleton and Qrepresents the compressibility of the fluidwhich is defined as:

Q=n

S

Kw S

pw

(4.26)

wherenis the porosity,S is the degree of saturation,pw is the suction pore pressure and

Kwis the elastic bulk modulus of water. Therefore the critical time step can be derived as:

tcritical = H2w

k

1

K'+ Q

(4.27)

For one dimensional consolidation (vertical flow) in fully saturated soil, the critical time

step can be simplified as:

tcritical = H2w

ky

1

Eoed+

n

Kw

(4.28)

in whichEoed is the oedometer modulus:

Eoed = E(1 )

(1 2)(1 +) (4.29)

is Poisson's ratio andEis the elastic Young's modulus.

For two dimensional elements as used in PLAXIS 2D, = 80and = 40for 15-nodetriangle and 6-node triangle elements, respectively. Therefore, the critical time step for

fully saturated soils can be calculated by:

tcritical =

H2w

80ky 1

Eoed +

n

Kw

(15 node triangles) (4.30)

tcritical = H2w

40ky

1

Eoed+

n

Kw

(6 node triangles) (4.31)

For three dimensional elements as used in PLAXIS 3D = 3. Therefore, the critical timestep for fully saturated soils can be calculated by:

tcritical = H2w

3k

1

Eoed+

n

Kw

(4.32)

Fine meshes allow for smaller time steps than coarse meshes. For unstructured mesheswith different element sizes or when dealing with different soil layers and thus different



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SCIENTIFIC MANUAL

values ofk,Eand, the above formula yields different values for the critical time step.To be on the safe side, the time step should not be smaller than the maximum value of

the critical time steps of all individual elements. This overall critical time step is

automatically adopted as theFirst time step in a consolidation analysis. For an

introduction to the critical time step concept, the reader is referred to Vermeer & Verruijt

(1981). Detailed information for various types of finite elements is given by Song (1990).



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ELEMENT FORMULATIONS

5 ELEMENT FORMULATIONS

In this chapter the interpolation functions of the finite elements used in the PLAXIS

program are described. Each element consists of a number of nodes. Each node has a

number of degrees of freedom that correspond to discrete values of the unknowns in the

boundary value problem to be solved. In the case of deformation theory the degrees offreedom correspond to the displacement components, whereas in the case of

groundwater flow (PLAXIS 2D) the degrees-of-freedom are the groundwater heads. For

consolidation problems degrees-of-freedom are both displacement components and

(excess) pore pressures. In addition to the interpolation functions it is described which

type of numerical integration over elements is used in the program.

5.1 INTERPOLATION FUNCTIONS OF POINT ELEMENTS

Point elements are elements existing of only one single node. Hence,the displacement

field of the elementu itself is only defined by the displacement field of this single node v:

u = v (5.1)

withu= (ux uy)T orv = (vx vy)

T (PLAXIS 2D) andu= (ux uy uz)T andv = (vx vy vz)

T

(PLAXIS 3D).

5.1.1 STRUCTURAL ELEMENTS

Fixed-end anchors

In PLAXIS fixed-end anchors are considered to be point elements. The contribution of

this element to the stiffness matrix can be derived from the traction the fixed-end anchor

imposes on a point in the geometry due to the displacement of this point (see Eq. ( 2.13)).

As a fixed-end anchor has only an axial stiffness and no bending stiffness, it is more

convenient to rotate the global displacement field vto the displacement fieldv such thatthe first axis of the rotated coordinate system coincides with the direction of the fixed-end

anchor:

v = R

v (5.2)

whereR denotes the rotation matrix. As only axial displacements are relevant, theelement will only have one degree of freedom in the rotated coordinate system. Thetraction in the rotated coordinate system t can be derived as:

t = Ds u (5.3)

whereDs denotes the constitutive relationship of an anchor as defined in the MaterialModels Manual. Converting the traction in the rotated coordinate system to the traction in

the global coordinate systemtby using the rotation matrix again and substituting Eq.(5.1) gives:

t = RT Ds Rv (5.4)



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SCIENTIFIC MANUAL

Substituting this equation in Eq. (2.13) gives the element stiffness matrix of the fixed-end

anchorKs

:

Ks =RT

Ds R

(5.5)

In case of elastoplastic behaviour of the anchor the maximum tension force is bound byFmax,tensand the maximum compression force is bound by Fmax,comp.

5.2 INTERPOLATION FUNCTIONS AND NUMERICAL INTEGRATION OF LINE

ELEMENTS

Within an element existing of more than one node the displacement field u = (ux uy)T

(PLAXIS 2D) oru = (ux uy uz)T (PLAXIS 3D) is obtained from the discrete nodal values

in a vectorv = (v1v2 ...vn)T using interpolation functions assembled in matrix N:

u = N v (5.6)

Hence, interpolation functionsNare used to interpolate values inside an element basedon known values in the nodes. Interpolation functions are also denoted as shape

functions.

Let us first consider a line element. Line elements are the basis for line loads, beams and

node-to-node anchors. The extension of this theory to areas and volumes is given in the

subsequent sections. When the local position,, of a point (usually a stress point or anintegration point) is known, one can write for a displacement component u:

u()=

ni=1

Ni()i (5.7)

where:

vi the nodal values,

Ni() the value of the shape function of nodeiat position,

u() the resulting value at positionand

n the number of nodes per element.

5.2.1 INTERPOLATION FUNCTIONS OF LINE ELEMENTS

Interpolation functions or shape functions are derived in a local coordinate system. This

has several advantages like programming only one function per element type, a simple

application of numerical integration and allowing higher-order elements to have curved

edges.

2-node line elements

In Figure5.1,an example of a 2-node line element is given. In contrast to a 3-node line

element or a 5-node line element in the PLAXIS 2D program, this element is not

compatible with an area element in the PLAXIS 2D or PLAXIS 3D program or a volumeelement in the PLAXIS 3D program. The 2-node line elements are the basis for



27/66


node-to-node anchors. The shape functions Nihave the property that the function valueis equal to unity at nodeiand zero at the other node. For 2-node line elements the nodesare located at =1and = 1. The shape functions are given by:

N1 = 1/2(1 ) (5.8)

N2 = 1/2(1 + )

2-node line elements provide a first-order (linear) interpolation of displacements.

Figure 5.1 Shape functions for a 2-node line element

3-node line elements

In Figure5.2,an example of a 3-node line element is given, which is compatible with the

side of a 6-node triangle in the PLAXIS 2D or PLAXIS 3D program or a 10-node volume

element in the PLAXIS 3D program, since these elements also have three nodes on aside. The shape functions Nihave the property that the function value is equal to unity atnodeiand zero at the other nodes. For 3-node line elements, where nodes 1, 2 and 3are located at = -1, 0 and 1 respectively, the shape functions are given by:

N1 = 1/2(1 ) (5.9)N2 = (1 + )(1 )N3 =

1/2(1 + )




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SCIENTIFIC MANUAL


3-node line elements provide a second-order (quadratic) interpolation of displacements.

These elements are the basis for line loads and beam elements.

5-node line elementsIn Figure5.3,an example of a 5-node line element is given, which is compatible with the

side of a 15-node triangle in the PLAXIS 2D program, since these elements also have

five nodes on a side. The shape functions Nihave the property that the function value isequal to unity at nodeiand zero at the other nodes. For 5-node line elements, wherenodes 1, 2, 3, 4 and 5 are located at = -1, -0.5, 0, 0.5 and 1 respectively, the shapefunctions are given by:

N1 = (1 )(1 2)(1 2)/6 (5.10)N2 = 4(1

)(1

2)(

1

)/3

N3 = (1 )(1 2)(1 2)(1 )N4 = 4(1 )(1 2)(1 )/3N5 = (1 2)(1 2)(1 )/6

5-node line elements provide a fourth-order (quartic) interpolation of displacements.

These elements are the basis for line loads and beam elements.


Structural line elements in the PLAXIS program, i.e. node-to-node anchors and beams

are based on the line elements as described in the previous sections. However, there are



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some differences.

Node-to-node anchors

Node-to-node anchors are springs that are used to model ties between two points. A

node-to-node anchor consists of a 2-node element with both nodes shared with the

elements the node-to-node anchor is attached to. Therefore, the nodes have three d.o.f.s

in the global coordinate system. However, as a node-to-node anchor can only sustain

normal forces, only the displacement in the axial direction of the node-to-node anchor is

relevant. Therefore it is more convenient to rotate the global coordinate system to a

coordinate system in which the first axis coincides with the direction of the anchor. This

rotated coordinate system is denoted as the x,y,z coordinate system and is similarto the (1, 2, 3)coordinate system used in theMaterial Models Manual. In this rotated

coordinate system, these elements have only one d.o.f. per node (a displacement in axial

direction).

The shape functions for these axial displacements are already given by Eq. (5.8). Using

index notation, the axial displacement can now be defined as:

ux = Niv

ix (5.11)

wherevixdenotes the nodal displacement in axial direction of nodei. The nodaldisplacements in the rotated coordinate system can be rotated to give the nodal

displacements in the global coordinate system:

v= RT

vix

i (5.12)

where the nodal displacement vector in the global coordinate system is denoted by vi =(vix viy)

T in the PLAXIS 2D program and vi = (vixviy viz)T in the PLAXIS 3D program

andR

denotes the rotation matrix. For further elaboration into the element stiffness

matrix see Sections5.2.3and5.2.5.

Beam elements

The 3-node beam elements are used to describe semi-one-dimensional structural objects

with flexural rigidity. Beam elements are slightly different from 3-node line elements in the

sense that they have six degrees of freedom per node instead of three in the global

coordinate system, i.e. three translational d.o.f.s (ux,uy,uz(3D only)) and three

rotational d.o.f.s (x,y,z).The rotated coordinate system is denoted as the (x,y,z) coordinate system and issimilar to the (1, 2, 3) coordinate system used in theMaterial Models Manual. The beam

elements are numerically integrated over their cross section using 4 (2x2) point Gaussian

integration. In addition, the beam elements are numerically integrated over their length

using 4-point Gaussian integration according to Table5.1. Beam elements have only one

local coordinate (). The element provides a quadratic interpolation (3-node element) ofthe axial displacement (see Eq. (5.9)). Using index notation, the axial displacement can

now be defined as:

u

x = Niv

ix (5.13)



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SCIENTIFIC MANUAL

wherevixdenotes the nodal displacement in axial direction of nodei.

As Mindlin's theory has been adopted the shape functions for transverse displacements

may be the same as for the axial displacements (see Eq. ( 5.9) and Eq. (5.10)). Using

index notation, the transverse displacements can now be defined as:

uw = Niwviw (5.14)

The local transverse displacements in any point is given by:

uw =

uy u

z

T (5.15)

The matrixNiw

for transverse displacements is defined as:

Niw

=

Ni() 0

0 Ni()

(5.16)

The local nodal transverse displacements and rotations of node iare given by viw:

viw =

viy v

iz

T (5.17)

whereviy andv

izdenote the nodal transverse displacements.

As the displacements and rotations are fully uncoupled according to Mindlin's theory, the

shape functions for the rotations may be different than the shape functions used for the

displacements. However, in PLAXIS the same functions are used. Using index notation,

the rotations can now be defined as:

w

= Niw

iw

(5.18)

where the matrixNiw

is defined by (5.16). The local rotations in any point is given by:

w

=

y

z

T (5.19)

The local nodal rotations of nodeiare given byiw

:

iw

=

iy

iz T (5.20)

whereiy and

izdenote the nodal rotations. For further elaboration into the element

stiffness matrix see Sections5.2.3and5.2.5.

Plate elements

The 3-node or 5-node plate elements are used to describe semi-two-dimensional

structural objects with flexural rigidity and a normal stiffness. Plate elements are slightly

different from 3-node or 5-node line elements in the sense that they have three degrees

of freedom per node instead of two in the global coordinate system, i.e. two translational

d.o.f.s (ux,uy) and one rotational d.o.f.s (z). The plate elements also have 3 d.o.f.s pernode in the rotated coordinate system, i.e.

one axial displacement (ux);



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one transverse displacement (uy);

one rotation (z).

The rotated coordinate system is denoted as the (x,y,z) coordinate system and issimilar to the (1, 2, 3) coordinate system used in theMaterial Models Manual. The plate

elements are numerically integrated over their height using 2 point Gaussian integration.In addition, the plate elements are numerically integrated over their length using 2-point

Gaussian integration in case of 3-node beam elements and 4-point Gaussian integration

in case of 5-node beam elements according to Table5.1.

The element provides a quadratic interpolation (3-node element) or a quartic interpolation

(5-node element) of the axial displacement (see Eq. (5.9)). Using index notation, the

axial displacement can now be defined as:

ux = Niv

ix (5.21)

wherev

ixdenotes the nodal displacement in axial direction of nodei.

As Mindlin's theory has been adopted the shape functions for transverse displacements

may be the same as for the axial displacements (see Eq. ( 5.9) and Eq. (5.10)). Using

index notation, the transverse displacementuycan now be defined as:

uy = Niv

iy (5.22)

whereviydenotes the node displacement perpendicular to the plate axis.

As the displacements and rotations are fully uncoupled according to Mindlin's theory, the

shape functions for the rotations may be different than the shape functions used for the

displacements. However, in PLAXIS the same functions are used. Using index notation,the rotationzin the PLAXIS 2D program can now be defined as:

z = Niiz (5.23)

whereizdenotes the nodal rotation. For further elaboration into the element stiffnessmatrix see Sections5.2.3and5.2.5.

5.2.3 DERIVATIVES OF INTERPOLATION FUNCTIONS

To compute the element stiffness matrix first the derivatives of the interpolation functions

should be derived.


As node-to-node anchors can only sustain axial forces, only the axial strains are of

interest:

= du/ dx. Using the chain rule for differentiation gives:

= du

dx =

du

d

d

dx (5.24)



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SCIENTIFIC MANUAL

where (using index notation)

du

d=

dNi

dvi (5.25)

and

dx

d=

dNi

dxi (5.26)

The parameterxi denotes the coordinate of the nodes in the rotated coordinate system.In case of 2-node line elements Eq. (5.26) can be simplified to:

dx

d=

L

2(5.27)

whereL denotes the length of the element in the global coordinate system. Inserting Eqs.

(5.25),(5.27)and (5.21) into Eq. (5.24) will give:

= Biv

ix (5.28)

where the rotated strain interpolation function Bi is given by:

Bi = 2

L

dNi

d(5.29)

Rotating the local nodal displacements back to the global coordinate system gives:

Bi =

2

L

dNi

d R (5.30)

Note that this strain interpolation function is still a function of the local coordinate as theshape functionsNiare a function of .

Beam elements

In case of axial displacements in the rotated coordinate system, the strain interpolation

matrix for beams can be derived from Eqs. (5.24) till(5.26). As node2 of a 3-node beam

element is located in the middle of the element by default, Eq. (5.26)can be simplified to

Eq. (5.27). So, the strain interpolation matrix in the global coordinate system for the

longitudinal displacements of beams is given by:

Bia

= 2

L

dNi

dR

(5.31)

To derive the shear forces in a Mindlin beam, a shear strain interpolation matrix is needed

to define the stiffness matrix. In contrast to the Bernoulli beam theory, the Mindlin beam

theory does not neglect the transverse shear deformations and thus the rotation ( ) of abeam cannot be calculated simply as a derivative of the transverse displacement:

1213

= duy

dx

z

duzdx

y = Biwviw Niwiw (5.32)



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where

Biw

=

dNidx

0

0 dNi

dx

(5.33)

andiw

is defined by Eq.(5.20). Using the chain rule for differentiation (Eq. (5.24)) and

the assumption of Eq. (5.27), Eq. (5.37)can be simplified to:

Biw

= 2

L

dNid

0

0 dNi

d

(5.34)

Using Eq. (5.12) to rotate local nodal transverse displacements and rotations to global

nodal displacements and rotations and inserting this equation in Eq. (5.41)gives:

Biw

= 2

L

dNi

d 0

0 dNi

d

R

(5.35)

Note that this strain interpolation function is still a function of the local coordinate as theshape functionsNiare a function of .

In case of bending moments, a curvature interpolation matrix is needed to define the

stiffness matrix. The curvature interpolation function describes the kinematic relationship

between curvatures and displacements:

=

2

3

=

dy

dxdzdx

= Biw

iw

(5.36)

Plate elements

In case of axial displacements in the rotated coordinate system, the strain interpolation

matrix for beams can be derived from Eqs. (5.24) till (5.26). As node2 of a 2-node beam

element is located in the middle of the element by default, Eq. (5.26)can be simplified to

Eq. (5.27). In case of 5-node beam elements (2D only), the nodes will also be equally

distributed along the length of the beam by default, simplifying Eq. (5.26)to Eq. (5.27).

So, the strain interpolation matrix in the global coordinate system for the longitudinaldisplacements of beams is given by:

Bi =

2

L

dNi

dR

(5.37)

In case of bending moments, a curvature interpolation matrix is needed to define the

stiffness matrix. The curvature interpolation function describes the kinematic relationship

between curvatures and displacements:

= 2

3

=

d2uz

dx 2

d2uy

dx 2

= Bi vi (5.38)



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SCIENTIFIC MANUAL

where

Bi

=

0

d2Niudx 2

d2Nidx 2

0

d2Niu

dx 2 0 0

d2Ni

dx 2

(5.39)

andviis defined by Eq.(5.20). Using the chain rule for differentiation twice gives:

d2N

dx 2 =

d

d

d

dxdN

d

d

dx =

d2N

d2

d

dx

2 (5.40)

As Eq.(5.27)still holds, Eq. (5.37)can be simplified to:

B

i = 4

L2

0

d2Niud2

d2Nid2

0

d2Niud2 0 0

d2Nid2

(5.41)

Using Eq. (5.12) to rotate local nodal transverse displacements and rotations to global

nodal displacements and rotations and inserting this equation in Eq. (5.41)gives:

Bi

= 4L2

0 d2Niu

d2d2Ni

d2 0

d2Niud2

0 0 d2Ni

d2

R (5.42)

Note that this strain interpolation function is still a function of the local coordinate as the

shape functionsNiare a function of .

5.2.4 NUMERICAL INTEGRATION OF LINE ELEMENTS

In order to obtain the integral over a certain line, the integral is numerically estimated as:

1

=1

F()dk

i=1

F(i)wi (5.43)

whereF(i) is the value of the functionFat positioniand withe weight factor for pointi.

A total ofksampling points is used. A method that is commonly used for numericalintegration is Gaussian integration, where the positions iand weightswiare chosen in aspecial way to obtain high accuracy. For Gaussian-integration a polynomial function of

degree 2k 1can be integrated exactly by using kpoints. The position and weightfactors of the integration are given in Table5.1. Note that the sum of the weight factors is

equal to 2, which is equal to the length of the line in local coordinates. The types of

integration used for the 2-node line elements and the 3-node line elements are shaded.



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Table 5.1 Gaussian integration

i wi max. polyn. degree

1 point 0.000000... 2 1

2 points 0.577350...(1/3) 1 33 points 0.774596...(0.6) 0.55555... (5/9) 5

0.000000... 0.88888... (8/9)

4 points 0.861136... 0.347854... 7 0.339981... 0.652145...

5 points 0.906179... 0.236926... 9 0.538469... 0.478628...0.000000... 0.568888...

5.2.5 CALCULATION OF ELEMENT STIFFNESS MATRIX


The element stiffness matrix of a node-to-node anchor is calculated by the integral (see

also Eq.2.25):

Ke =

BTDa B dV (5.44)

whereDa denotes the elastic constitutive relationship of the node-to-node anchor asdiscussed in theMaterial Models Manual.As the strain interpolation matrix is still a

function of the local coordinateit will make more sense to solve the integral of Eq.(5.44)in the local coordinate system. Applying the change of variables theorem tochange the integral to the local coordinate system gives:

Ke =

BT DaB

dx

ddV (5.45)

In case of a 2-node line element,dx/ d = L/2. This integral is estimated by numericalintegration as described in Section5.2.4. In fact, the element stiffness matrix is

composed of submatricesKeij whereiandjare the local nodes. The process ofcalculating the element stiffness matrix can be formulated as:

Keij

=

k

BTi

Da Bj

dx

dwk (5.46)

In case of elastoplastic behaviour of the anchor the maximum tension force is bound by

Fmax,tensand the maximum compression force is bound by Fmax,comp(PLAXIS 3D).



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SCIENTIFIC MANUAL

Beam elements

In case of axial forces, the element stiffness matrix is given by Eqs. (5.44)till (5.46). In

case of shear forces the stiffness matrix of a beam is calculated by the integral:

Ke =

BT

DbB

dV (5.47)

whereDb denotes the constitutive relationship of a beam in bending (see MaterialModels Manual):

Ds =

kGA 0

0 kGA

(5.48)

In case of bending moments the stiffness matrix of a beam is calculated by the integral:

Ke =

BT

DbB

dV (5.49)

whereDb denotes the constitutive relationship of a beam in bending (see MaterialModels Manual):

Db =

EI2 EI23

EI23 EI3

(5.50)

To solve the integral of Eq. (5.45) in the local coordinate system, the change of variablestheorem should be applied:

Ke =

BT

Db B

dx

ddV (5.51)

In PLAXIS 3-node beam elements, dx/d = L/2. This integral is estimated by numericalintegration as described in Section5.2.4. In fact, the element stiffness matrix is

composed of submatricesKeij whereiandjare the local nodes. The process ofcalculating the element stiffness matrix can be formulated as:

Ke

ij =

kB T

i Db

B

j

dx

d wk (5.52)

5.3 INTERPOLATION FUNCTIONS AND NUMERICAL INTEGRATION OF AREA

ELEMENTS

Areas and surfaces in PLAXIS 2D are formed by 6-node or 15-node triangular elements.

For the areas and surfaces in PLAXIS 3D only the 6-node triangular elements are

available. The interpolation functions and the type of integration of these elements are

described in the following subsections.

http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/http://2d-3-material-models.pdf/


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5.3.1 INTERPOLATION FUNCTIONS OF AREA ELEMENTS

6-node triangular elements

The 6-node triangles are one of the options for the basis for the soil elements in PLAXIS

2D and the basis for plate elements and distributed loads in PLAXIS 3D.

For triangular elements there are two local coordinates (and). In addition we use anauxiliary coordinate = 1 . 6-node triangular elements provide a second-orderinterpolation of displacements. The shape functions Nihave the property that thefunction value is equal to unity at nodeiand zero at the other nodes. The shapefunctions can be written as (see the local node numbering as shown in Figure5.4):

N1 =(2 1) (5.53)N2 =(2 1)N3 =(2 1)N4 = 4

N5 = 4

N6 = 4

=1.0

1 4 2

6

3

5

=0.0

=0.5

=1.0

=0.0 =0.5 =1.0

=0.5

=0.0

1

2 3

Figure 5.4 Local numbering and positioning of nodes () and integration points (x) of a 6-node

triangular element


The 15-node triangles are one of the options for the basis for soil elements in PLAXIS

2D. For triangular elements there are two local coordinates and. In addition we usean auxiliary coordinate= 1 . For 15-node triangles the shape functions can bewritten as (see the local node numbering as shown in Figure5.5):



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SCIENTIFIC MANUAL

N1 =(4 1)(4 2)(4 3)/6 (5.54)N2 =(4 1)(4 2)(4 3)/6N3 =(4 1)(4 2)(4 3)/6

N4 = 4(4 1)(4 1)N5 = 4(4 1)(4 1)N6 = 4(4 1)(4 1)N7 =(4 1)(4 2) 8/3N8 = (4 1)(4 2) 8/3N9 =(4 1)(4 2) 8/3N10 =(4 1)(4 2) 8/3

N11 = (4 1)(4 2) 8/3N12 =(4 1)(4 2) 8/3N13 = 32(4 1)N14 = 32(4 1)N15 = 32(4 1)

Figure 5.5 Local numbering and positioning of nodes of a 15-node triangular element


Structural area elements in the PLAXIS program, i.e. plates and interfaces are based on

the area elements as described in the previous sections. However, there are some

differences.



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Plate elements

Plate elements are different from the 6-node triangles which have three degrees of

freedom per node. As the plate elements cannot sustain torsional moments, the plate

elements have only 5 d.o.f.s per node in the rotated coordinate system, i.e.:

one axial displacement (u

x); two transverse displacements (uy andu

z);

two rotations (y and

z).

These elements are directly integrated over their cross section and numerically integrated

using 3 point Gaussian integration. The position of the integration points is indicated in

Figure5.6and corresponds with Table5.3.

5

=1.0

26

1

2

4

=1.0

=0.

5=0.0

=0.0

3

=0.

5

3

1

Figure 5.6 Local numbering and positioning of nodes () and integration points (x) of a 6-node plate

triangle.

Interface elements

Interface elements are different from the in the sense that they have pairs of nodes

instead of single nodes. The interface elements are numerically integrated using 6 point

Gauss integration. The distance between the two nodes of a node pair is zero. Each

node has three translational degrees of freedom (ux,uy,uz). As a result, interfaceelements allow for differential displacements between the node pairs (slipping and

gapping). The position and weight factors of the integration points are given in Table 5.2.

Table 5.2 6-point Gaussian integration for 12-node triangular elements

Point i i wi1 0.091576 0.816848 0.109952

2 0.091576 0.091576 0.109952

3 0.816848 0.091576 0.109952

4 0.108103 0.445948 0.223382

5 0.445948 0.108103 0.223382

6 0.445948 0.445948 0.223382

For more information seeDunavant (1985).



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SCIENTIFIC MANUAL

5.3.3 NUMERICAL INTEGRATION OF AREA ELEMENTS

As for line elements, one can formulate the numerical integration over areas as:

F(, )ddk

i=1

F(i, i)wi (5.55)

The PLAXIS program uses Gaussian integration within the area elements.


For 6-node triangular elements the integration is based on 3 sample points (Figure5.4).

The position and weight factors of the integration points are given in Table5.3. Note that

the sum of the weight factors is equal to 1.

Table 5.3 3-point Gaussian integration for 6-node triangular elements

Point i i wi

1 1/6 2/3 1/3

2 1/6 1/6 1/3

3 2/3 1/6 1/3


For 15-node elements 12 sample points are used. The position and weight factors of the

integration points are given in Table5.4Note that, in contrast to the line elements, the

sum of the weight factors is equal to 1.

Table 5.4 12-point integration for 15-node elements

Point i i i wi

1,2 & 3 0.063089... 0.063089... 0.873821... 0.050845...

4 .. 6 0.249286... 0.249286... 0.501426... 0.116786...

7..12 0.310352... 0.053145... 0 .636502... 0.082851...

5.4 INTERPOLATION FUNCTIONS AND NUMERICAL INTEGRATION OF VOLUME

ELEMENTS

The soil volume in the PLAXIS program is modelled by means of 10-node tetrahedral

elements. The interpolation functions, their derivatives and the numerical integration of

this type of element are described in the following subsections.

5.4.1 10-NODE TETRAHEDRAL ELEMENT

The 10-node tetrahedral elements are created in the 3D mesh procedure. This type of

element provides a second-order interpolation of displacements. For tetrahedral

elements there are three local coordinates (, and). The shape functionsNihave theproperty that the function value is equal to unity at node

iand zero at the other nodes.

The shape functions of these 10-node volume elements can be written as (see the local

node numbering as shown in Figure5.7):



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N1 = (1 )(1 2 2 2) (5.56)N2 =(2 1)N3 =(2 1)

N4 =(2 1)N5 = 4(1 )N6 = 4

N7 = 4(1 )N8 = 4(1 )N9 = 4

N10 = 4

1

2

3

4

1

2

3

4

5

6

7

8910

Figure 5.7 Local numbering and positioning of nodes () and integration points (x) of a 10-nodewedge element

The soil elements have three degrees of freedom per node: ux,uy anduz. The shapefunction matrixN

ican now be defined as:

Ni =

Ni 0 0

0 Ni 0

0 0 Ni

(5.57)

and the nodal displacement vector v is defined as:

v =

vix viy viz

T (5.58)

Table 5.5 4-point Gaussian integration for 10-node tetrahedral element

Point i i i wi

1 1/4-1/20

5 1/4-1/20

5 1/4-1/20

5 1/24

2 1/4-1/20

5 1/4-1/20

5 1/4+3/20

5 1/24

2 1/4+3/205 1/4-1/205 1/4-1/205 1/242 1/4-1/20

5 1/4+3/20

5 1/4-1/20

5 1/24



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SCIENTIFIC MANUAL

5.4.2 DERIVATIVES OF INTERPOLATION FUNCTIONS

In order to calculate Cartesian strain components from displacements, such as

formulated in Eq. (2.10), derivatives need to be taken with respect to the global system of

axes (x,y,z).

= Bivi (5.59)

where

Bi =

Nix

0 0

0 Niy

0

0 0 Ni

zNiy

Nix

0

0 Niz

Niy

Niz

0 Nix

(5.60)

Within the elements, derivatives are calculated with respect to the local coordinate system

(,,). The relationship between local and global derivatives involves the Jacobian J:

Ni

Ni

Ni

=

x

y

z

x

y

z

x

y

z

NixNiyNiz

= J

NixNiyNiz

(5.61)

Or inversely:

NixNiyNiz

= J1

Ni

Ni

Ni

(5.62)

The local derivativesNi/, etc., can easily be derived from the element shapefunctions, since the shape functions are formulated in local coordinates. The components

of the Jacobian are obtained from the differences in nodal coordinates. The inverse

JacobianJ1 is obtained by numerically inverting J.

The Cartesian strain components can now be calculated by summation of all nodal



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contributions:

xx

yy

zz

xy

yz

zx

=

i

Bi

vix

viy

viz

(5.63)

whereviare the displacement components in node i.

5.4.3 NUMERICAL INTEGRATION OVER VOLUMES

As for lines and areas, one can formulate the numerical integration over volumes as: F(, , )dd d

ki=1

F(i, i, i)wi (5.64)

The PLAXIS program uses Gaussian integration within tetrahedral elements. The

integration is based on 4 sample points. The position and weight factors of the integration

points are given in Table5.5. See Figure5.7for the local numbering of integration points.

Note that the sum of the weight factors is equal to 1/6.

5.4.4 CALCULATION OF ELEMENT STIFFNESS MATRIXThe element stiffness matrix,Ke, is calculated by the integral (see also Eq. 2.25):

Ke =

BT De B dV (5.65)

As it is more convenient to calculate the element stiffness matrix in the local coordinate

system, the change of variables theorem should be applied to change the integral to the

local coordinate system:

Ke = BT De B jdV (5.66)wherejdenotes the determinant of the Jacobian.

The integral is estimated by numerical integration as described in Section 5.4.3. In fact,

the element stiffness matrix is composed of submatrices Keij whereiandjare the localnodes. The process of calculating the element stiffness matrix can be formulated as:

Keij

=

k

BTi DeB

jj wk (5.67)

In case of plastic deformations of the soil only the elastic part of the soil stiffness will be

used in the stiffness matrix whereas the plasticity is solved for iteratively.



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SCIENTIFIC MANUAL

5.5 SPECIAL ELEMENTS

As special elements in PLAXIS embedded piles will be considered. Embedded piles are

based on the embedded beam approach(Sadek & Shahrour, 2004). Embedded piles

consist of beam elements to model the pile itself and embedded interface elements to

model the interaction between the soil and the pile at the pile skin as well as at the pilefoot.

5.5.1 EMBEDDED PILES

The embedded pile has been developed to describe the interaction of a pile with its

surrounding soil. The interaction at the pile skin and at the pile foot is described by

means of embedded interface elements. The pile is considered as a beam which can

cross a 10-node tetrahedral element at any place with any arbitrary orientation (Figure

5.8). Due to the existence of the beam element three extra nodes are introduced inside

the 10-node tetrahedral element.

x

y

z

x'

y'

z'

Figure 5.8 Illustration of the embedded beam element denoted by the solid line. The blank grey

circles denote the virtual nodes of the soil element.

Finite element discretisation

The finite element discretisation of the pile is similar to beam elements, as discussed in

Section5.2.2. The finite element discretisation of the interaction with the soil will be

discussed in this chapter. Using the standard notation the displacement of the soil usand



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the displacement of the pileupcan be discretised as:

us = Nsvs up = Npvp (5.68)

whereNs

andNp

are the matrices containing the interpolation functions of the soil

elements and the beam elements respectively (see Sections 5.4.1and5.2.2) andvsandvpare the nodal displacement vectors of the soil elements and the beam elementsrespectively.

Interaction at the skin

First, the interaction between the soil and the pile at the skin of the pile will be described

by embedded interface elements. These interface elements are based on 3-node line

elements with pairs of nodes instead of single nodes. One node of each pair belongs to

the beam element, whereas the other (virtual) node is a point in the 15-node wedge

element (Figure5.8). The interaction can be represented by a skin traction tskin. The

development of the skin traction can be regarded as an incremental process:

tskin = tskin0 + tskin (5.69)

In this equationtskin0 denotes the initial skin traction and tskin denotes the skin traction

increment. The constitutive relation between the skin traction increment and the relative

displacement increment is formulated as:

tskin =Tskinurel (5.70)

In this relationTskin denotes the material stiffness of the embedded interface element inthe global coordinate system. The increment in the relative displacement vector urel is

defined as the difference in the increment of the soil displacement and the increment of

the pile displacement:

urel = up us = Npvp Nsvs = Nrelvrel (5.71)

where

Nrel

=

NpN

s

(5.72)

and

vrel =

vpvs

(5.73)

Looking to the virtual work equation (Eq. 2.6)the traction increment at the pile skin can

be discretised as: uTrelt

skindS = vTrel

NT

relTskinN

reldSvrel = K

skinvrel (5.74)

In this formulation the element stiffness matrix Kskin represents the interaction between



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SCIENTIFIC MANUAL

Table 5.6 Newton-Cotes integration

i wi

2 nodes 1 13 nodes 1, 0 1/3, 4/34 nodes 1, 1/3 1/4, 3/45 nodes 1, 1/2, 0 7/45, 32/45, 12/45

the pile and the soil at the skin and consists of four parts:

Kskin =

Kskinpp Kskinps

Kskinsp

Kskinss

(5.75)

The matrixKskinpp

represents the contribution of the pile nodes to the interaction, the matrix

Kskinss

represents the contribution of the soil nodes to the interaction and the matrices

Kskinps

andKskin

sp are the mixed terms:

Kskinpp

=

NT

pTskin N

pdS (5.76a)

Kskinps

=

NTp

Tskin NsdS (5.76b)

Kskinsp

=

NTs

Tskin Np

dS (5.76c)

K

skin

ss =

N

T

s Tskin

NsdS (5.76d)

These integrals are numerically estimated using Eq. (5.43):

1

=1

F()dk

i=1

F(i)wi (5.77)

However, instead of Gauss integration Newton-Cotes integration is used. In this method

the pointsiare chosen at the position of the nodes, see Table 5.6.The type ofintegration used for the embedded interface elements is shaded. In case of plastic

deformations of the embedded interface elements only the elastic part of the interface

stiffness will be used in the stiffness matrix whereas the plasticity is solved for iteratively.

Interaction at the foot

The interaction of the embedded pile at the foot is described by an embedded interface

element. This interaction can be represented by a foot force vector ffoot

. Like the

development of the skin traction the development of the foot force is an incremental

process:

ffoot

= ffoot0 + f

foot(5.78)

In this equationffoot0 denotes the initial force and ffoot denotes the force increment at

the foot. The constitutive relation between the force increment at the foot and the relative



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displacement increment can be described by:

ffoot

= Dfooturel (5.79)

The symbolDfoot

denotes the material stiffness matrix of the spring element at the foot of

the embedded pile in the global coordinate system. As for the skin interaction the forceincrement at the foot of the pile can be discretised by means of the virtual work ((2.6)), as:

uTrelffoot

= vTrelNT

relDfoot N

relvrel = K

footvrel (5.80)

The stiffness matrix at the foot is represented byKfoot and consists of four parts:

Kfoot =

Kfootpp Kfootps

Kfootsp

Kfootss

(5.81)

In this equationKfootpp represents the contribution of the pile nodes,Kfootss represents the

contribution from the soil nodes andKfootps

andKfootsp

are the mixed terms:

Kfootpp

= NTp

Dfoot Np

(5.82a)

Kfootps

=NTp

Dfoot Ns

(5.82b)

Kfootsp

=NTs

Dfoot Np

(5.82c)

Kfootss

= NTs

Dfoot Ns

(5.82d)

In case of plastic deformations of the embedded interface element only the elastic part of

the interface stiffness will be used in thestiffness matrix whereas the plasticity is solved

for iteratively.



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SCIENTIFIC MANUAL



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THEORY OF SENSITIVITY ANALYSIS & PARAMETER VARIATION

6 THEORY OF SENSITIVITY ANALYSIS & PARAMETER VARIATION

This chapter presents some of the theoretical backgrounds of the sensitivity analysis and

parameter variation module. The chapter does not give a full theoretical description of the

methods of interval analysis. For a more detailed description you are referred to the

literature e.g.Moore (1966), Moore (1979), Alefeld & Herzberger (1983), Goos &Hartmanis (1985),Neumaier (1990),Kearfott & Kreinovich (1996)andJaulin, Kieffer,

Didrit & Walter (2001).

6.1 SENSITIVITY ANALYSIS

A method for quantifying sensitivity in the sense discussed in this Section and in the

SectionSensitivity analysis and Parameter variationof the Reference Manual is the

sensitivity ratioSR (EPA, 1999). The ratio is defined as the percentage change in outputdivided by the percentage change in input for a specific input variable, as shown in Eq.

(6.1):

SR =

f(xL,R) f(x)

f(x)

100%

xL,R xx

100%

(6.1)

f(x) is the reference value of the output variable using reference values of the inputvariables andf(xL,R) is the value of the output variable after changing the value of oneinput variable, whereasxandxL,Rin the denominator are the respective input variables.

For the sensitivity ratio, an input variable, xL,R, is varied individually across the entirerange requiring 2N+1 calculations,Nbeing the number of varied parameters considered.

An extension to the sensitivity ratio and a more robust method of evaluating important

sources of uncertainty is the sensitivity score,SS, which is the sensitivity ratioSRweighted by a normalized measure of the variability in an input variable, as given by Eq.

(6.2):

SS = SR (max xR min xR)x

(6.2)

By normalising the measure of variability (i.e. the range divided by the reference value),

this method effectively weights the ratios in a manner that is independent of the units ofthe input variable.

Performing a sensitivity analysis as described above, the sensitivity score of each

variable,SS,i, on respective resultsA,B, . . . ,Z, (e.g. displacements, forces, factor ofsafety, etc.) at each construction step (calculation phase) can be quantified as shown in

Table6.1(Sensitivity matrix). The total sensitivity score of each variable, SS,i, resultsfrom summation of all sensitivity scores for each respective result at each construction

step.

It should be noted, that the results of the sensitivity analysis appeared to be strongly

dependent on the respective results used and thus results relevant for the problem

investigated have to be chosen based on sound engineering judgment. Finally, the total



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SCIENTIFIC MANUAL

Table 6.1 Sensitivity matrix

Respective results

A B . . . Z

Input variables %

x1 SS,A1 SS,B1 . . . SS,Z1 SS,1 (x1)

x2 SS,A2 SS,B2 . . . SS,Z2 SS,2 (x2)...

......

......

......

xN SS,AN SS,BN . . . SS,ZN SS,N (xN)

Total relativesensitivity

0 0.05 0.15 0.20 0.25 0.30 0.35 0.40 0.450.10

x1

x2

x3

xN

(xN)

.

.

.

Threshold value

Figure 6.1 Total relative sensitivity in diagram form

relative sensitivity(xi)for each input variable is then given by Peschl (2004)as

(xi) = SS,i

N

i=1

SS,i

(6.3)

Figure6.1shows the total relative sensitivity of each parameter (xi)in diagram form inorder to illustrate the 'major' variables.

6.1.1 DEFINITION OF THRESHOLD VALUE

The benefit of such an analysis is twofold: Firstly, the results are the basis for a

decision-making in order to reduce the computational effort involved when utilizing a

parameter variation, i.e. at this end a decision has to be made (definition of a threshold

value), which variables (parameters) should be used in further calculations and which one

can be treated as deterministic values as their influence on the result is not significant(Figure6.1). Secondly, sensitivity analysis can be applied for example to design further

investigation programs to receive additional information about parameters with high



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THEORY OF SENSITIVITY ANALYSIS & PARAMETER VARIATION

sensitivity in order to reduce the uncertainty of the system response, i.e. the result may

act as a basis for the design of an investigation program (laboratory and/or in situ tests).

6.2 THEORY OF PARAMETER VARIATION

The parameter variation used in the PLAXIS parameter variation module refer to classical

set theory where uncertainty is represented in terms of closed intervals (bounds)

assuming that the true value of the relevant unknown quantity is captured (X[xmin , xmax]). In general, an interval is defined as a pair of elements of some (at leastpartially) ordered sets (Kulpa, 1997). An interval is identified with the set of elements

lying between the interval endpoints (including the endpoints) and using the set of real

numbers as the underlying ordered set (real intervals). Hence, all intervals are closed

sets. Thus, a (proper) real intervalx is a subset of the set of real numbers R such that:

x= [xmin , xmax]=

{x'

R

|xmin

x'

xmax

} (6.4)

wherexmin xmax andxmin = sup(x), xmax = inf(x) are endpoints of the interval x. Ingeneral,x' denotes any element of the interval x. If the true values of the parameters ofinterest are bounded by intervals, this will always ensure a reliable estimate

(worst/best-case analysis) based on the information available.

For the parameter variation, the input parametersxiare treated as interval numbers(xi,min/ xi,max) whose ranges contain the uncertainties in those parameters. The resultingcomputations, carried out entirely in interval form, would then literally carry the

uncertainties associated with the data through the analysis. Likewise, the final outcome

in interval form would contain all possible solutions due to the variations in input.

x x X

Figure 6.2 Total relative sensitivity in diagram form

6.2.1 BOUNDS ON THE SYSTEM RESPONSE

LetXbe a non-empty set containing all the possible values of a parameter xandy=f(x),f : X ->Ybe a real-valued function of x. The interval of the system responsethroughf, can be calculated by means of a function used in set theory. In fact, if x

belongs to setA, then the range of y is

f(A) = {f(x) :x A} (6.5)

Here, the setA is called the focal element. The basic step is the calculation of the system

response through functionfwhich represents here a numerical model. In general, thisinvolves two global optimisation problems which can be solved by applying twice the

techniques of global optimisation (e.g. Ratschek & Rokne (1988), Tuy (1998)) to find the

lower and upper bound,yminandymax, respectively, of the system response:

f(A) =[ymin , ymax] (6.6)



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SCIENTIFIC MANUAL

where

ymin = minxA

f(x) (6.7)

ymax

= maxxA

f(x) (6.8)

In the absence of any further information a so-called calculation matrix, can be

constructed by assuming independence between parameters x. A is the Cartesianproduct ofNfinite intervalsx1 ... xN(calculation matrix), therefore it is aN-dimensional box (interval vector) whose 2N vertices are indicated asvk,k= 1,...,2

N,

Nbeing the number of parameters considered. The lower and upper bounds ymin andymax of the system response can be obtained as follows:

ymin = mink f(k) :k= 1, ...2

N

(6.9)ymax = max

k

f(k) :k= 1, ...2

N

(6.10)

Iff(A) has no extreme value in the interior ofA, except at the vertices, Eqs. (6.9)and(6.10)are correct in which case the methods of interval analysis are applicable, e.g. the

Vertex method (Dong & Shah, 1987). If, on the other hand, f(A) has one or more extremevalues in the interior of A, then Eqs. (6.9)and(6.10)can be taken as approximations to

the true global minimum and maximum value.



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DYNAMICS

7 DYNAMICS

This chapter highlights some of the theoretical backgrounds of the dynamic module. The

chapter does not give a full theoretical description of the dynamic modelling. For a more

detailed description you are referred to the literatureZienkiewicz & Taylor (1991),Hughes

(1987),Das (1995), Kramer (1996), Haigh, Ghosh & Madabhushi (2005), Basabe & Sen(2007),Kelly, Ward, Treitel &

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