3d 4 scientific

PLAXIS

Scientific Manual

2015

Build 7649

TABLE OF CONTENTS

TABLE OF CONTENTS

1 Introduction 5

2 Deformation theory 72.1 Basic equations of continuum deformation 72.2 Finite element discretisation 82.3 Implicit integration of differential plasticity models 92.4 Global iterative procedure 10

3 Groundwater flow theory 133.1 Basic equations of flow 13

3.1.1 Transient flow 133.1.2 Continuity equation 14

3.2 Boundary Conditions 153.3 Finite element discretisation 163.4 Flow in interface elements 18

4 Consolidation theory 194.1 Basic equations of consolidation 194.2 Finite element discretisation 204.3 Elastoplastic consolidation 224.4 Critical time step 22

5 Element formulations 255.1 Interpolation functions of point elements 25

5.1.1 Structural elements 255.2 Interpolation functions and numerical integration of line elements 26

5.2.1 Interpolation functions of line elements 265.2.2 Structural elements 285.2.3 Derivatives of interpolation functions 315.2.4 Numerical integration of line elements 345.2.5 Calculation of element stiffness matrix 35

5.3 Interpolation functions and numerical integration of area elements 365.3.1 Interpolation functions of area elements 375.3.2 Structural elements 385.3.3 Numerical integration of area elements 40

5.4 Interpolation functions and numerical integration of volume elements 405.4.1 10-node tetrahedral element 405.4.2 Derivatives of interpolation functions 425.4.3 Numerical integration over volumes 435.4.4 Calculation of element stiffness matrix 43

5.5 Special elements 445.5.1 Embedded piles 44

6 Theory of sensitivity analysis & parameter variation 496.1 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

PLAXIS 2015 | Scientific Manual 3

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7 Dynamics 537.1 Basic equation dynamic behaviour 537.2 Time integration 53

7.2.1 Critical time step 557.2.2 Dynamic Integration Coefficients 55

7.3 Model Boundaries 567.4 Viscous boundaries 577.5 Initial stresses and stress increments 577.6 Amplification of responses 587.7 Pseudo-spectral acceleration response spectrum for a single-degree-of-

freedom system 587.8 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

INTRODUCTION

1 INTRODUCTION

In this part of the manual some scientific background is given of the theories andnumerical methods on which the PLAXIS program is based. The manual containschapters on deformation theory, groundwater flow theory (PLAXIS 2D), consolidationtheory, dynamics as well as the corresponding finite element formulations and integrationrules for the various types of elements used in PLAXIS. In Appendix A a globalcalculation scheme is provided for a plastic deformation analysis.

In addition to the specific information given in this part of the manual, more information onbackgrounds of theory and numerical methods can be found in the literature, as amongstothers referred to in Chapter 8. For detailed information on stresses, strains, constitutivemodelling and the types of soil models used in the PLAXIS program, the reader isreferred to the Material 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 formulatedwithin the framework of continuum mechanics. A restriction is made in the sense thatdeformations are considered to be small. This enables a formulation with reference to theoriginal 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 invector , 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:

= Lu (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 the MaterialModels 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 partialdifferential equation in the displacements u.

However, instead of a direct combination, the equilibrium equation is reformulated in aweak form according to Galerkin's variation principle:

uT(LT + b

)dV = 0 (2.5)

In this formulation u represents 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 threecomponents 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 state i , the unknown stresses i can be eliminatedusing Eq. (2.7):

T dV =uT bi dV +

uT t i dS

T i1 dV (2.8)

It should be noted that all quantities appearing in Eqs. (2.1) till (2.8) are functions of theposition 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 ofdegrees of freedom that correspond to discrete values of the unknowns in the boundaryvalue problem to be solved. In the present case of deformation theory the degrees offreedom correspond to the displacement components. Within an element thedisplacement field u is obtained from the discrete nodal values in a vector v usinginterpolation functions assembled in matrix N:

u = N v (2.9)

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

= LN v = B v (2.10)

In this relation B is the strain interpolation matrix, which contains the spatial derivatives ofthe interpolation functions. Eqs. (2.9) and (2.10) can be used in variational, incrementaland rate form as well.

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

)T dV = (N )T bi dV + (N )T t i dS (B)T i1 dV


DEFORMATION THEORY

(2.11)

The discrete displacements can be placed outside the integral:

T

BT dV = T

NT bi dV + T

NT t i dS T

BT i1 dV

(2.12)

Provided that Eq. (2.12) holds for any kinematically admissible displacement variationvT , the equation can be written as:

BT dV =

NT bi dV +

NT t i dS

BT i1 dV (2.13)

The above equation is the elaborated equilibrium condition in discretised form. The firstterm on the right-hand side together with the second term represent the current externalforce vector and the last term represents the internal reaction vector from the previousstep. 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 aresult, strain increments can generally not be calculated directly, and global iterativeprocedures are required to satisfy the equilibrium condition (Eq. 2.13) for all materialpoints. Global iterative procedures are described later in Section 2.4, but the attention isfirst 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 relation De represents the elastic material matrix for the current stress increment.The strain increments are obtained from the displacement increments v using thestrain interpolation matrix B, similar to Eq. (2.10).

For elastic material behaviour, the plastic strain increment p is zero. For plasticmaterial behaviour, the plastic strain increment can be written, according to Vermeer(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 inthe case of a transition from elastic to elastoplastic behaviour. Moreover, it can be proventhat 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 attentionis 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 relation tr is an auxiliary stress vector, referred to as the elastic stresses or trialstresses, which is the new stress state when considering purely linear elastic materialbehaviour.

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 multipliercan be written as:

=f(tr)

d + h(2.19)

where:

d =(f

)tr De

(g

)i (2.20)

The symbol h denotes the hardening parameter, which is zero for perfectly-plastic modelsand constant for linear hardening models. In the latter case the new stress state can beformulated as:

i = tr f

(tr)d + hDe(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 aNewton-type iterative procedure with convergence control.

2.4 GLOBAL ITERATIVE PROCEDURE

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


DEFORMATION THEORY

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

K i v i = f iex f i1in (2.22)

In this equation K is a stiffness matrix, v is the incremental displacement vector, f ex isthe external force vector and f in is the internal reaction vector. The superscript i refers tothe step number. However, because the relation between stress increments and strainincrements is generally non-linear, the stiffness matrix cannot be formulated exactlybeforehand. Hence, a global iterative procedure is required to satisfy both the equilibriumcondition and the constitutive relation. The global iteration process can be written as:

K j v j = f iex f j1in (2.23)

The superscript j refers to the iteration number. v is a vector containing sub-incrementaldisplacements, which contribute to the displacement increments of step i :

v i =nj=1

v j (2.24)

where n is 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 accuratethe stiffness matrix, the fewer iterations are required to obtain equilibrium within a certaintolerance.

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

K =

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

where De is the elastic material matrix according to Hooke's law and B is the straininterpolation matrix. The use of an elastic stiffness matrix gives a robust iterativeprocedure as long as the material stiffness does not increase, even when usingnon-associated plasticity models. Special techniques such as arc-length control (Riks,1979), over-relaxation and extrapolation (Vermeer & van Langen, 1989) can be used toimprove the iteration process. Moreover, the automatic step size procedure, asintroduced by Van Langen & Vermeer (1990), can be used to improve the practicalapplicability. For material models with linear behaviour in the elastic domain, such as thestandard Mohr-Coulomb model, the use of an elastic stiffness matrix is particularlyfavourable, as the stiffness matrix needs only be formed and decomposed before the firstcalculation 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. Inaddition to a general description of groundwater flow, attention is focused on the finiteelement 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 thefollowing equation in three dimensions:

q =kwg

(pw + wg

)(3.1)

Where

=

xyz

(3.2)q, k , g and w are 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 isused as the flow is not affected by the gradient of the water pore pressure in verticaldirection when hydrostatic conditions are assumed.

g =

0

g0

(3.3)In unsaturated soils the coefficient of permeability k can be related to the soil saturationas:

k = krel ksat (3.4)

where

ksat =

ksatx 0 0

0 ksaty 0

0 0 ksatz

(3.5)In which krel is the ratio of the permeability at a given saturation to the permeability insaturated state ksat .


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3.1.2 CONTINUITY EQUATION

The mass concentration of the water in each elemental volume of the medium is equal townS. The parameters n and S are the porosity and the degree of saturation of the soil,respectively. According to the mass conservation, the water outflow from the the volumeis equal to the changes in the mass concentration. As the water outflow is equal to thedivergence of the specific discharge (T q), the continuity equation has the form (Song,1990):

T [krelwg

ksat(pw + 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

(SKw Spw

)pwt

= 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

ksat(pw + wg

)] n

(SKw Spw

)pwt

= 0 (3.9)

For steady state flow(pwt

= 0)

the continuity condition applies:

T [krelwg

ksat(pw + wg

)]= 0 (3.10)

Eq. (3.10) expresses that there is no net inflow or outflow in an elementary area, asillustrated in Figure 3.1.

qx

qy

qy + qy dy

qx + qx

dx

Figure 3.1 Illustration of continuity condition



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 boundaryas

qxnx + qyny = 0 (3.11)

where nx and ny are 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 arepointing 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 forexample can be formulated as prescribed pressureboundaries.

p = 0 (3.15)

These conditions directly relate to a prescribed head boundary condition and areimplemented as such.

Infiltration: This type of boundary conditions poses a more complex mixed boundarycondition. An inflow value q may 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, pondingtakes place at a depth p,max and the boundary condition switches from inflow toprescribed head. As soon as the soil capacity meets the infiltration rate the conditionswitches back.

= z + p,max if pondingqxnx + qyny = q if < z + p,max

> 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,min to ensure numerical stability.

Seepage: The water line option generates phreatic/seepage conditions by default. Anexternal 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 < z

< z

= z if < z

qxnx + qyny > 0(3.17)

The seepage condition only allows for outflow of groundwater at atmospheric pressure.For unsaturated conditions at the boundary the boundary is closed.

Alternatively a water line may generate a phreatic/closed condition if the upper part of theline is replaced by closed conditions. This condition is written as{

= if zqxnx + qyny = 0 if < z

(3.18)

The external head may vary in a time dependent way, however the part that remainsclosed is derived from the initial setting.

Infiltration well: Inside the domain wells are modelled as source terms, where Qspecifies the inflowing flux per meter.

Q = Q (3.19)

As the source term in the governing equation simulates water flowing in the system, thesource term is positive for a recharge well.

Extraction well: A discharge rate Q simulates an amount of water leaving the domain

Q = Q (3.20)

The source term in th e governing equation is negative for a discharge well.

Drain: Drains are handled as seepage boundaries. However, drains may be locatedinside the domain as well. The condition is written as{

= z if Q < 0Q = 0 if < z

(3.21)

A drain permits water leaving the modelling domain at atmospheric pressure. The drainitself does not generate a resistance against flow.

Initial conditions are generated as a steady state solution for a problem with a given set ofboundary conditions.


The groundwater pore pressure in any position within an element can be expressed interms of nodal values:

pw = Npwn (3.22)

where N is the vector with interpolation functions. For more information on the finite



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 thegradient of the groundwater pore pressure. This gradient can be determined by means ofthe B-matrix, which contains the spatial derivatives of the interpolation functions, see thePLAXIS scientific manual. In the numerical formulation the specific discharge, q, iswritten as:

q =krelw

ksat(B pwn + wg

)(3.23)

where:

q =

qxqy

and: ksat = ksatx 0

0 ksaty

(3.24)From the specific discharges in the integration points, q, the nodal discharges Qe can beintegrated according to:

Qe =

BTq dV (3.25)

in which BT is the transpose of the B-matrix. The term dV indicates integration over thevolume of the body.

Starting from the continuity equation Eq. (3.9) and applying the Galerkin approach andincorporating prescribed boundary conditions we obtain:

H pwn Sdpwndt

= qp

(3.26)

where H, S and qp

are the permeability matrix, the compressibility matrix and theprescribed recharges that are given by the boundary conditions, respectively:

H = (N)T krel

wksat

(N) dV (3.27)S =

NT(nSKw n S

pw

)N dV (3.28)

qp

= (N)T krel

wksat wg dV

NT q d (3.29)

q is 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:

Kwn

=3(u )

(1 2u)(1 + )Kskeleton (3.30)

Where u has 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 modulusof 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 inincremental form using an implicit time stepping schema. Application of this procedure toEq. (3.26) yields:

(t H + S

)pwn = t H pwn0 + tqp (3.31)

and pwn0 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 canbe on or off. When the elements are switched on, there is a full coupling of the porepressure 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 canbe on or off. When the elements are switched off, there is a full coupling of the porepressure degrees of freedom. When the interface elements are switched on, there is noflow from one side of the interface element to the other (impermeable screen).


CONSOLIDATION THEORY

4 CONSOLIDATION THEORY

In this chapter we will review the theory of consolidation as used in PLAXIS. In addition toa general description of Biot's theory for coupled consolidation, attention is focused onthe finite element formulation. Moreover, a separate section is devoted to the use ofadvanced 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 alsoassumed. The formulation is based on small strain theory. According to Terzaghi'sprinciple, stresses are divided into effective stresses and pore pressures:

= ' + m(psteady + pexcess

)(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, pexcess is the excesspore pressure and m is a vector containing unity terms for normal stress components andzero terms for the shear stress components. The steady state solution at the end of theconsolidation process is denoted as psteady . Within PLAXIS psteady is defined as:

psteady = pinput (4.3)

where pinput is the pore pressure generated in the input program based on phreatic linesafter the use of the K0 procedure or Gravity loading. Note that within PLAXIScompressive stresses are considered to be negative; this applies to effective stresses aswell as to pore pressures. In fact it would be more appropriate to refer to pexcess andpsteady as 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 stressincrement as ' and a strain increment as , the constitutive equation is:

' = M (4.4)

where:

=(xx yy zz xy yz zx

)T (4.5)

and M represents the material stiffness matrix. For details on constitutive relations, seethe Material Models Manual.


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

u = N v p = N pn

= B v (4.6)

where v is the nodal displacement vector, pn

is the excess pore pressure vector, u is thecontinuous displacement vector within an element and p is the (excess) pore pressure.The matrix N contains the interpolation functions and B is the strain interpolation matrix.

In general the interpolation functions for the displacements may be different from theinterpolation functions for the pore pressure. In PLAXIS, however, the same functions areused for displacements and pore pressures.

Starting from the incremental equilibrium equation and applying the above finite elementapproximation we obtain:

BTd dV =

NTdb dV +

NTdt dS + r0 (4.7)

with:

r0 =

NT b0 dV +

NT t0 dS

BT 0 dV (4.8)

where b is 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 computationalprocedure becomes self-correcting. The term dV indicates integration over the volume ofthe body considered and dS indicates a surface integral.

Dividing the total stresses into pore pressure and effective stresses and introducing theconstitutive relationship gives the nodal equilibrium equation:

K dv + Ldpn

= df n (4.9)

where K is the stiffness matrix, L is the coupling matrix and df n is the incremental loadvector:

K =

BT M B dV (4.10a)

L =

BT mN dV (4.10b)

df n =

NT db dV +

NT dt dS (4.10c)

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

TR(w y psteady p )/ w mT t

+nKw

pt

= 0 (4.11)



where R is the permeability matrix:

R =

kx 0 0

0 ky 0

0 0 kz

(4.12)n is the porosity, Kw is the bulk modulus of the pore fluid and w is the unit weight of thepore fluid. This continuity equation includes the sign convention that psteady and p areconsidered positive for tension.

As the steady state solution is defined by the equation:

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

the continuity equation takes the following form:

TRp / w + mT t n

Kw

pt

= 0 (4.14)

Applying finite element discretisation using a Galerkin procedure and incorporatingprescribed boundary conditions we obtain:

H pn

+ LTdvdt S dpn

dt= q

n(4.15)

where:

H =

(N)T RN / w dV , S =

nKw

NT N dV (4.16)

and qn

is a vector due to prescribed outflow at the boundary. However within PLAXIS it isnot possible to have boundaries with non-zero prescribed outflow. The boundary is eitherclosed (zero flux) or open (zero excess pore pressure). In reality the bulk modulus ofwater is very high and so the compressibility of water can be neglected in comparison tothe compressibility of the soil skeleton.

In PLAXIS the bulk modulus of the pore fluid is taken automatically according to (also seeReference Manual):

Kwn

=3(u )

(1 2u)(1 + )Kskeleton (4.17)

Where u has 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 switchedon, the bulk modulus of the pore fluid is neglected.

The equilibrium and continuity equations may be compressed into a block matrixequation: K L

LT S

dvdtdp

ndt

= 0 0

0 H

vpn

+ df ndt

qn

(4.18)


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A simple step-by-step integration procedure is used to solve this equation. Using thesymbol to denote finite increments, the integration gives: K L

LT S

vp

n

= 0 0

0 t H

v0pn0

+ f nt q

n

(4.19)where:

S = t H + S qn

= qn0

+ qn

(4.20)

and v0 and pn0 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 atthe correct solution. Due to plasticity or stress-dependent stiffness behaviour theequilibrium equations are not necessarily satisfied using the technique described above.Therefore the equilibrium equation is inspected here. Instead of Eq. (4.9) the equilibriumequation is written in sub-incremental form:

K v + L pn

= rn (4.21)

where rn is the global residual force vector. The total displacement increment v is thesummation of sub-increments v from all iterations in the current step:

rn =

NTb dV +

NT t 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 theappropriate constitutive model.

4.4 CRITICAL TIME STEP

For most numerical integration procedures, accuracy increases when the time step isreduced, but for consolidation there is a threshold value. Below a particular timeincrement (critical time step) the accuracy rapidly decreases. Care should be taken withtime steps that are smaller than the advised minimum time step. The critical time step iscalculated as:

tcritical =H2

cv(4.24)



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 and His the height of the element used. cv is the consolidation coefficient and is calculated as:

cv =

kw

1K '

+ Q(4.25)

where w is the unit weight of the pore fluid, k is the coefficient of permeability, K ' is thedrained bulk modulus of soil skeleton and Q represents the compressibility of the fluidwhich is defined as:

Q = n(

SKw Spw

)(4.26)

where n is the porosity, S is the degree of saturation, pw is the suction pore pressure andKw is the elastic bulk modulus of water. Therefore the critical time step can be derived as:

tcritical =H2wk

(1K '

+ Q)

(4.27)

For one dimensional consolidation (vertical flow) in fully saturated soil, the critical timestep can be simplified as:

tcritical =H2wky

(1

Eoed+

nKw

)(4.28)

in which Eoed is the oedometer modulus:

Eoed =E (1 )

(1 2)(1 + ) (4.29)

is Poisson's ratio and E is the elastic Young's modulus.

For two dimensional elements as used in PLAXIS 2D, = 80 and = 40 for 15-nodetriangle and 6-node triangle elements, respectively. Therefore, the critical time step forfully saturated soils can be calculated by:

tcritical =H2w80ky

(1

Eoed+

nKw

)(15 node triangles) (4.30)

tcritical =H2w40ky

(1

Eoed+

nKw

)(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+

nKw

)(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


SCIENTIFIC MANUAL

values of k , E and , 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 ofthe critical time steps of all individual elements. This overall critical time step isautomatically adopted as the First time step in a consolidation analysis. For anintroduction 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).


ELEMENT FORMULATIONS

5 ELEMENT FORMULATIONS

In this chapter the interpolation functions of the finite elements used in the PLAXISprogram are described. Each element consists of a number of nodes. Each node has anumber of degrees of freedom that correspond to discrete values of the unknowns in theboundary value problem to be solved. In the case of deformation theory the degrees offreedom correspond to the displacement components, whereas in the case ofgroundwater flow (PLAXIS 2D) the degrees-of-freedom are the groundwater heads. Forconsolidation problems degrees-of-freedom are both displacement components and(excess) pore pressures. In addition to the interpolation functions it is described whichtype 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 displacementfield of the element u itself is only defined by the displacement field of this single node v :

u = v (5.1)

with u = (ux uy )T or v = (vx vy )T (PLAXIS 2D) and u = (ux uy uz )T and v = (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 ofthis element to the stiffness matrix can be derived from the traction the fixed-end anchorimposes 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 moreconvenient to rotate the global displacement field v to the displacement field v such thatthe first axis of the rotated coordinate system coincides with the direction of the fixed-endanchor:

v = Rv (5.2)

where R

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)

where Ds denotes the constitutive relationship of an anchor as defined in the MaterialModels Manual. Converting the traction in the rotated coordinate system to the traction inthe global coordinate system t by using the rotation matrix again and substituting Eq.(5.1) gives:

t = RTDs R

v (5.4)


SCIENTIFIC MANUAL

Substituting this equation in Eq. (2.13) gives the element stiffness matrix of the fixed-endanchor K s:

K s = RTDs R

(5.5)

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

5.2 INTERPOLATION FUNCTIONS AND NUMERICAL INTEGRATION OF LINEELEMENTS

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

(PLAXIS 2D) or u = (ux uy uz )T (PLAXIS 3D) is obtained from the discrete nodal valuesin a vector v = (v1 v2 ...vn)T using interpolation functions assembled in matrix N:

u = N v (5.6)

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

Let us first consider a line element. Line elements are the basis for line loads, beams andnode-to-node anchors. The extension of this theory to areas and volumes is given in thesubsequent 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 node i at position ,

u() the resulting value at position and

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. Thishas several advantages like programming only one function per element type, a simpleapplication of numerical integration and allowing higher-order elements to have curvededges.

2-node line elements

In Figure 5.1, an example of a 2-node line element is given. In contrast to a 3-node lineelement or a 5-node line element in the PLAXIS 2D program, this element is notcompatible 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



node-to-node anchors. The shape functions Ni have the property that the function valueis equal to unity at node i and zero at the other node. For 2-node line elements the nodesare located at = 1 and = 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.

x=-1 x=1

N1 N2

N

1 2x

Figure 5.1 Shape functions for a 2-node line element


In Figure 5.2, an example of a 3-node line element is given, which is compatible with theside of a 6-node triangle in the PLAXIS 2D or PLAXIS 3D program or a 10-node volumeelement in the PLAXIS 3D program, since these elements also have three nodes on aside. The shape functions Ni have the property that the function value is equal to unity atnode i and 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 + )

x=-1 x=1

N1N2 N3

N

1 2 3x

x=0



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.


In Figure 5.3, an example of a 5-node line element is given, which is compatible with theside of a 15-node triangle in the PLAXIS 2D program, since these elements also havefive nodes on a side. The shape functions Ni have the property that the function value isequal to unity at node i and 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 )/3N3 = (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 beamsare based on the line elements as described in the previous sections. However, there are



some differences.

Node-to-node anchors

Node-to-node anchors are springs that are used to model ties between two points. Anode-to-node anchor consists of a 2-node element with both nodes shared with theelements the node-to-node anchor is attached to. Therefore, the nodes have three d.o.f.sin the global coordinate system. However, as a node-to-node anchor can only sustainnormal forces, only the displacement in the axial direction of the node-to-node anchor isrelevant. Therefore it is more convenient to rotate the global coordinate system to acoordinate system in which the first axis coincides with the direction of the anchor. Thisrotated coordinate system is denoted as the x, y, z coordinate system and is similarto the (1, 2, 3) coordinate system used in the Material Models Manual. In this rotatedcoordinate system, these elements have only one d.o.f. per node (a displacement in axialdirection).

The shape functions for these axial displacements are already given by Eq. (5.8). Usingindex notation, the axial displacement can now be defined as:

ux = Ni vix (5.11)

where vix denotes the nodal displacement in axial direction of node i . The nodaldisplacements in the rotated coordinate system can be rotated to give the nodaldisplacements in the global coordinate system:

v=RT

vix

i (5.12)

where the nodal displacement vector in the global coordinate system is denoted by v i =(vix viy )T in the PLAXIS 2D program and v i = (vix viy viz )T in the PLAXIS 3D programand R

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

matrix see Sections 5.2.3 and 5.2.5.

Beam elements

The 3-node beam elements are used to describe semi-one-dimensional structural objectswith flexural rigidity. Beam elements are slightly different from 3-node line elements in thesense that they have six degrees of freedom per node instead of three in the globalcoordinate system, i.e. three translational d.o.f.s (ux , uy , uz (3D only)) and threerotational 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 the Material Models Manual. The beamelements are numerically integrated over their cross section using 4 (2x2) point Gaussianintegration. In addition, the beam elements are numerically integrated over their lengthusing 4-point Gaussian integration according to Table 5.1. Beam elements have only onelocal coordinate (). The element provides a quadratic interpolation (3-node element) ofthe axial displacement (see Eq. (5.9)). Using index notation, the axial displacement cannow be defined as:

ux = Ni vix (5.13)


SCIENTIFIC MANUAL

where vix denotes the nodal displacement in axial direction of node i .

As Mindlin's theory has been adopted the shape functions for transverse displacementsmay be the same as for the axial displacements (see Eq. (5.9) and Eq. (5.10)). Usingindex 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 uz

]T (5.15)

The matrix Niw

for transverse displacements is defined as:

Niw

=

Ni () 00 Ni ()

(5.16)The local nodal transverse displacements and rotations of node i are given by viw :

viw =[viy v

iz

]T (5.17)

where viy and viz denote the nodal transverse displacements.

As the displacements and rotations are fully uncoupled according to Mindlin's theory, theshape functions for the rotations may be different than the shape functions used for thedisplacements. However, in PLAXIS the same functions are used. Using index notation,the rotations can now be defined as:

w

= Niwiw

(5.18)

where the matrix Niw

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 node i are given by iw

:

iw

=[iy

iz

]T (5.20)

where iy and iz denote the nodal rotations. For further elaboration into the element

stiffness matrix see Sections 5.2.3 and 5.2.5.

Plate elements

The 3-node or 5-node plate elements are used to describe semi-two-dimensionalstructural objects with flexural rigidity and a normal stiffness. Plate elements are slightlydifferent from 3-node or 5-node line elements in the sense that they have three degreesof freedom per node instead of two in the global coordinate system, i.e. two translationald.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 );



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 the Material Models Manual. The plateelements are numerically integrated over their height using 2 point Gaussian integration.In addition, the plate elements are numerically integrated over their length using 2-pointGaussian integration in case of 3-node beam elements and 4-point Gaussian integrationin case of 5-node beam elements according to Table 5.1.

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, theaxial displacement can now be defined as:

ux = Ni vix (5.21)

where vix denotes the nodal displacement in axial direction of node i .

As Mindlin's theory has been adopted the shape functions for transverse displacementsmay be the same as for the axial displacements (see Eq. (5.9) and Eq. (5.10)). Usingindex notation, the transverse displacement uy can now be defined as:

uy = Ni viy (5.22)

where viy denotes the node displacement perpendicular to the plate axis.

As the displacements and rotations are fully uncoupled according to Mindlin's theory, theshape functions for the rotations may be different than the shape functions used for thedisplacements. However, in PLAXIS the same functions are used. Using index notation,the rotation z in the PLAXIS 2D program can now be defined as:

z = Ni iz (5.23)

where iz denotes the nodal rotation. For further elaboration into the element stiffnessmatrix see Sections 5.2.3 and 5.2.5.

5.2.3 DERIVATIVES OF INTERPOLATION FUNCTIONS

To compute the element stiffness matrix first the derivatives of the interpolation functionsshould be derived.


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

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

=du

dx=

du

dddx

(5.24)


SCIENTIFIC MANUAL

where (using index notation)

du

d=

dNid

vi (5.25)

and

dx

d=

dNid

xi (5.26)

The parameter xi 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=

L2

(5.27)

where L 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:

= Bi vix (5.28)

where the rotated strain interpolation function Bi is given by:

Bi =2LdNid

(5.29)

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

Bi

=2LdNid

R

(5.30)

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

Beam elements

In case of axial displacements in the rotated coordinate system, the strain interpolationmatrix for beams can be derived from Eqs. (5.24) till (5.26). As node 2 of a 3-node beamelement is located in the middle of the element by default, Eq. (5.26) can be simplified toEq. (5.27). So, the strain interpolation matrix in the global coordinate system for thelongitudinal displacements of beams is given by:

Bia

=2LdNid

R

(5.31)

To derive the shear forces in a Mindlin beam, a shear strain interpolation matrix is neededto define the stiffness matrix. In contrast to the Bernoulli beam theory, the Mindlin beamtheory does not neglect the transverse shear deformations and thus the rotation () of abeam cannot be calculated simply as a derivative of the transverse displacement: 12

13

= du

y

dx z

duzdx y

= Biw viw Niw iw (5.32)



where

Biw

=

dNidx 00 dNi

dx

(5.33)and

iwis 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

=2L

dNid 00 dNi

d

(5.34)Using Eq. (5.12) to rotate local nodal transverse displacements and rotations to globalnodal displacements and rotations and inserting this equation in Eq. (5.41) gives:

Biw

=2L

dNid 00 dNi

d

R

(5.35)

Note that this strain interpolation function is still a function of the local coordinate as theshape functions N i are a function of .

In case of bending moments, a curvature interpolation matrix is needed to define thestiffness matrix. The curvature interpolation function describes the kinematic relationshipbetween curvatures and displacements:

=

23

= d

y

dxdzdx

= Biw iw (5.36)Plate elements

In case of axial displacements in the rotated coordinate system, the strain interpolationmatrix for beams can be derived from Eqs. (5.24) till (5.26). As node 2 of a 2-node beamelement is located in the middle of the element by default, Eq. (5.26) can be simplified toEq. (5.27). In case of 5-node beam elements (2D only), the nodes will also be equallydistributed 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

=2LdNid

R

(5.37)

In case of bending moments, a curvature interpolation matrix is needed to define thestiffness matrix. The curvature interpolation function describes the kinematic relationshipbetween curvatures and displacements:

=

23

= d

2uzdx 2

d2uy

dx 2

= Bi vi (5.38)


SCIENTIFIC MANUAL

where

Bi

=

0d2Niudx 2

d2Nidx 2

0

d2Niudx 2

0 0d2Nidx 2

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

d2Ndx 2

=dd

ddx

dNd

ddx

=d2Nd2

(ddx

)2 (5.40)

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

Bi

= 4L2

0d2Niud2

d2Nid2

0

d2Niud2

0 0d2Nid2

(5.41)Using Eq. (5.12) to rotate local nodal transverse displacements and rotations to globalnodal displacements and rotations and inserting this equation in Eq. (5.41) gives:

Bi

= 4L2

0d2Niud2

d2Nid2

0

d2Niud2

0 0d2Nid2

R (5.42)Note that this strain interpolation function is still a function of the local coordinate as theshape functions N i are 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

=1F () d

ki=1

F (i ) wi (5.43)

where F (i ) is the value of the function F at position i and wi the weight factor for point i .A total of k sampling points is used. A method that is commonly used for numericalintegration is Gaussian integration, where the positions i and weights wi are chosen in aspecial way to obtain high accuracy. For Gaussian-integration a polynomial function ofdegree 2k 1 can be integrated exactly by using k points. The position and weightfactors of the integration are given in Table 5.1. Note that the sum of the weight factors isequal to 2, which is equal to the length of the line in local coordinates. The types ofintegration used for the 2-node line elements and the 3-node line elements are shaded.



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 (seealso Eq. 2.25):

K e =

BTDa B dV (5.44)

where Da denotes the elastic constitutive relationship of the node-to-node anchor asdiscussed in the Material Models Manual. As the strain interpolation matrix is still afunction of the local coordinate it 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:

K e =

BT DaBdx

ddV (5.45)

In case of a 2-node line element, dx /d = L/2. This integral is estimated by numericalintegration as described in Section 5.2.4. In fact, the element stiffness matrix iscomposed of submatrices K eij where i and j are the local nodes. The process ofcalculating the element stiffness matrix can be formulated as:

K eij

=k

B TiDa B

j

dx

dwk (5.46)

In case of elastoplastic behaviour of the anchor the maximum tension force is bound byFmax ,tens and the maximum compression force is bound by Fmax ,comp (PLAXIS 3D).


SCIENTIFIC MANUAL

Beam elements

In case of axial forces, the element stiffness matrix is given by Eqs. (5.44) till (5.46). Incase of shear forces the stiffness matrix of a beam is calculated by the integral:

K e =

BTDbB

dV (5.47)

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

Ds =

kGA 00 kGA

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

K e =

BTDbB

dV (5.49)

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

Db =

EI2 EI23EI23 EI3

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

K e =

BTDb B

dx

ddV (5.51)

In PLAXIS 3-node beam elements, dx/d =L/2. This integral is estimated by numericalintegration as described in Section 5.2.4. In fact, the element stiffness matrix iscomposed of submatrices K eij where i and j are the local nodes. The process ofcalculating the element stiffness matrix can be formulated as:

K eij

=k

B Ti

Db Bj

dx

dwk (5.52)

5.3 INTERPOLATION FUNCTIONS AND NUMERICAL INTEGRATION OF AREAELEMENTS

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 areavailable. The interpolation functions and the type of integration of these elements aredescribed in the following subsections.



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 PLAXIS2D 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 N i have the property that thefunction value is equal to unity at node i and zero at the other nodes. The shapefunctions can be written as (see the local node numbering as shown in Figure 5.4):

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

N5 = 4

N6 = 4

z=1.0

1 4 2

6

3

5

h=0.0

h=0.5

h=1.0

x=0.0 x=0.5 x=1.0

z=0.5

z=0.0

1

2 3

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


The 15-node triangles are one of the options for the basis for soil elements in PLAXIS2D. 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 Figure 5.5):


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)/6N4 = 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) 8/3N11 = (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 onthe area elements as described in the previous sections. However, there are somedifferences.



Plate elements

Plate elements are different from the 6-node triangles which have three degrees offreedom per node. As the plate elements cannot sustain torsional moments, the plateelements have only 5 d.o.f.s per node in the rotated coordinate system, i.e.:

one axial displacement (ux );

two transverse displacements (uy and uz );

two rotations (y and z ).

These elements are directly integrated over their cross section and numerically integratedusing 3 point Gaussian integration. The position of the integration points is indicated inFigure 5.6 and corresponds with Table 5.3.

5

h=1.0

26

1

2

4

x=1.

0

x=0.

5h=0.0

x=0.0

3

h=0.

5

3

1

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

Interface elements

Interface elements are different from the in the sense that they have pairs of nodesinstead of single nodes. The interface elements are numerically integrated using 6 pointGauss integration. The distance between the two nodes of a node pair is zero. Eachnode has three translational degrees of freedom (ux , uy , uz ). As a result, interfaceelements allow for differential displacements between the node pairs (slipping andgapping). 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 wi

1 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 see Dunavant (1985).


SCIENTIFIC MANUAL

5.3.3 NUMERICAL INTEGRATION OF AREA ELEMENTS

As for line elements, one can formulate the numerical integration over areas as: F (, )d d

ki=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 (Figure 5.4).The position and weight factors of the integration points are given in Table 5.3. Note thatthe 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 theintegration points are given in Table 5.4 Note that, in contrast to the line elements, thesum of the weight factors is equal to 1.

Table 5.4 12-point integration for 15-node elements

Point i i i wi1,2 & 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 VOLUMEELEMENTS

The soil volume in the PLAXIS program is modelled by means of 10-node tetrahedralelements. The interpolation functions, their derivatives and the numerical integration ofthis 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 ofelement provides a second-order interpolation of displacements. For tetrahedralelements there are three local coordinates (, and ). The shape functions Ni have theproperty that the function value is equal to unity at node i and zero at the other nodes.The shape functions of these 10-node volume elements can be written as (see the localnode numbering as shown in Figure 5.7):



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

12

3

4

1

2

3

4

56

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 and uz . The shapefunction matrix N

ican now be defined as:

N i =

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 wi1 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+3/20

5 1/4-1/20

5 1/4-1/20

5 1/24

2 1/4-1/20

5 1/4+3/20

5 1/4-1/20

5 1/24


SCIENTIFIC MANUAL

5.4.2 DERIVATIVES OF INTERPOLATION FUNCTIONS

In order to calculate Cartesian strain components from displacements, such asformulated in Eq. (2.10), derivatives need to be taken with respect to the global system ofaxes (x,y,z).

= Biv i (5.59)

where

Bi

=

Nix

0 0

0 Niy

0

0 0 Niz

Niy

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 :

NiNiNi

=x

y

z

x

y

z

x

y

z

NixNiyNiz

= JNixNiyNiz

(5.61)

Or inversely:NixNiyNiz

= J1NiNiNi

(5.62)

The local derivatives Ni /, etc., can easily be derived from the element shapefunctions, since the shape functions are formulated in local coordinates. The componentsof the Jacobian are obtained from the differences in nodal coordinates. The inverseJacobian J1 is obtained by numerically inverting J .

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



contributions:

xx

yy

zz

xy

yz

zx

=i

Bi

vix

viy

viz

(5.63)

where vi are 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 (, , ) d d d

ki=1

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

The PLAXIS program uses Gaussian integration within tetrahedral elements. Theintegration is based on 4 sample points. The position and weight factors of the integrationpoints are given in Table 5.5. See Figure 5.7 for 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 MATRIX

The element stiffness matrix, K e, is calculated by the integral (see also Eq. 2.25):

K e =

BT De B dV (5.65)

As it is more convenient to calculate the element stiffness matrix in the local coordinatesystem, the change of variables theorem should be applied to change the integral to thelocal coordinate system:

K e =

BT De B jdV (5.66)

where j denotes 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 K eij where i and j are the localnodes. The process of calculating the element stiffness matrix can be formulated as:

K eij

=k

BTiDeB

jj wk (5.67)

In case of plastic deformations of the soil only the elastic part of the soil stiffness will beused in the stiffness matrix whereas the plasticity is solved for iteratively.


SCIENTIFIC MANUAL

5.5 SPECIAL ELEMENTS

As special elements in PLAXIS embedded piles will be considered. Embedded piles arebased on the embedded beam approach (Sadek & Shahrour, 2004). Embedded pilesconsist of beam elements to model the pile itself and embedded interface elements tomodel 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 itssurrounding soil. The interaction at the pile skin and at the pile foot is described bymeans of embedded interface elements. The pile is considered as a beam which cancross a 10-node tetrahedral element at any place with any arbitrary orientation (Figure5.8). Due to the existence of the beam element three extra nodes are introduced insidethe 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 greycircles 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 inSection 5.2.2. The finite element discretisation of the interaction with the soil will bediscussed in this chapter. Using the standard notation the displacement of the soil us and



the displacement of the pile up can be discretised as:

us = Ns vs up = Np vp (5.68)

where Ns

and Np

are the matrices containing the interpolation functions of the soilelements and the beam elements respectively (see Sections 5.4.1 and 5.2.2) and vs andvp are 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 describedby embedded interface elements. These interface elements are based on 3-node lineelements with pairs of nodes instead of single nodes. One node of each pair belongs tothe beam element, whereas the other (virtual) node is a point in the 15-node wedgeelement (Figure 5.8). The interaction can be represented by a skin traction tskin. Thedevelopment of the skin traction can be regarded as an incremental process:

tskin = tskin0 + tskin (5.69)

In this equation tskin0 denotes the initial skin traction and tskin denotes the skin traction

increment. The constitutive relation between the skin traction increment and the relativedisplacement increment is formulated as:

tskin = T skinurel (5.70)

In this relation T skin denotes the material stiffness of the embedded interface element inthe global coordinate system. The increment in the relative displacement vector urel isdefined as the difference in the increment of the soil displacement and the increment ofthe pile displacement:

urel = up us = Npvp Nsvs = N rel v rel (5.71)

where

Nrel

=[N

pN

s

](5.72)

and

v rel =

vpvs

(5.73)Looking to the virtual work equation (Eq. 2.6) the traction increment at the pile skin canbe discretised as:

uTrel tskin dS = vTrel

NT

relT skin N

reldSv rel = K

skinv rel (5.74)

In this formulation the element stiffness matrix K skin represents the interaction between


SCIENTIFIC MANUAL

Table 5.6 Newton-Cotes integration

i wi2 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:

K skin =

K skinpp K skinpsK skin

spK skin

ss

(5.75)The matrix K skin

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

K skinss

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

K skinps

and K skinsp

are the mixed terms:

K skinpp

=

NTpT skin N

pdS (5.76a)

K skinps

=

NTpT skin N

sdS (5.76b)

K skinsp

=

NTsT skin N

pdS (5.76c)

K skinss

=

NTsT skin N

sdS (5.76d)

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

1

=1F () d

ki=1

F (i ) wi (5.77)

However, instead of Gauss integration Newton-Cotes integration is used. In this methodthe points i are 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 plasticdeformations of the embedded interface elements only the elastic part of the interfacestiffness 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 interfaceelement. This interaction can be represented by a foot force vector f foot . Like thedevelopment of the skin traction the development of the foot force is an incrementalprocess:

f foot = f foot0 + ffoot (5.78)

In this equation f foot0 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



displacement increment can be described by:

f foot = Dfoot urel (5.79)

The symbol Dfoot denotes the material stiffness matrix of the spring element at the foot ofthe 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:

uTrel ffoot = vTrel N

TrelDfoot N

relv rel = K

foot v rel (5.80)

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

K foot =

K footpp K footpsK foot

spK foot

ss

(5.81)In this equation K foot

pprepresents the contribution of the pile nodes, K foot

ssrepresents the

contribution from the soil nodes and K footps

and K footsp

are the mixed terms:

K footpp

= NTpDfoot N

p(5.82a)

K footps

= NTpDfoot N

s(5.82b)

K footsp

= NTsDfoot N

p(5.82c)

K footss

= NTsDfoot N

s(5.82d)

In case of plastic deformations of the embedded interface element only the elastic part ofthe interface stiffness will be used in thestiffness matrix whereas the plasticity is solvedfor iteratively.


SCIENTIFIC MANUAL


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 andparameter variation module. The chapter does not give a full theoretical description of themethods of interval analysis. For a more detailed description you are referred to theliterature e.g. Moore (1966), Moore (1979), Alefeld & Herzberger (1983), Goos &Hartmanis (1985), Neumaier (1990), Kearfott & Kreinovich (1996) and Jaulin, Kieffer,Didrit & Walter (2001).

6.1 SENSITIVITY ANALYSIS

A method for quantifying sensitivity in the sense discussed in this Section and in theSection Sensitivity analysis and Parameter variation of the Reference Manual is thesensitivity ratio SR (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 and f (xL,R) is the value of the output variable after changing the value of oneinput variable, whereas x and xL,R in 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, N being the number of varied parameters considered.

An extension to the sensitivity ratio and a more robust method of evaluating importantsources of uncertainty is the sensitivity score, SS , which is the sensitivity ratio SRweighted 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 eachvariable, SS,i , on respective results A,B,. . . ,Z , (e.g. displacements, forces, factor ofsafety, etc.) at each construction step (calculation phase) can be quantified as shown inTable 6.1 (Sensitivity matrix). The total sensitivity score of each variable, SS,i , resultsfrom summation of all sensitivity scores for each respective result at each constructionstep.

It should be noted, that the results of the sensitivity analysis appeared to be stronglydependent on the respective results used and thus results relevant for the probleminvestigated have to be chosen based on sound engineering judgment. Finally, the total


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 relative sensitivity

0 0.05 0.15 0.20 0.25 0.30 0.35 0.40 0.450.10

x1

x2

x3

xN

a(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,iNi=1

SS,i

(6.3)

Figure 6.1 shows 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 adecision-making in order to reduce the computational effort involved when utilizing aparameter variation, i.e. at this end a decision has to be made (definition of a thresholdvalue), which variables (parameters) should be used in further calculations and which onecan be treated as deterministic values as their influence on the result is not significant(Figure 6.1). Secondly, sensitivity analysis can be applied for example to design furtherinvestigation programs to receive additional information about parameters with high


THEORY OF SENSITIVITY ANALYSIS & PARAMETER VARIATION

sensitivity in order to reduce the uncertainty of the system response, i.e. the result mayact 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 classicalset 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 elementslying between the interval endpoints (including the endpoints) and using the set of realnumbers as the underlying ordered set (real intervals). Hence, all intervals are closedsets. Thus, a (proper) real interval x is a subset of the set of real numbers R such that:

x =[xmin , xmax] = {x ' R|xmin x ' xmax} (6.4)

where xmin xmax and xmin = 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 parameters xi are 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 theuncertainties associated with the data through the analysis. Likewise, the final outcomein 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

Let X be a non-empty set containing all the possible values of a parameter x and y =f (x), f : X -> Y be a real-valued function of x . The interval of the system responsethrough f , can be calculated by means of a function used in set theory. In fact, if xbelongs to set A, then the range of y is

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

Here, the set A is called the focal element. The basic step is the calculation of the systemresponse through function f which represents here a numerical model. In general, thisinvolves two global optimisation problems which can be solved by applying twice thetechniques of global optimisation (e.g. Ratschek & Rokne (1988), Tuy (1998)) to find thelower and upper bound, ymin and ymax , respectively, of the system response:

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


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 beconstructed by assuming independence between parameters x . A is the Cartesianproduct of N finite intervals x1 ... xN (calculation matrix), therefore it is aN-dimensional box (interval vector) whose 2N vertices are indicated as vk , k = 1,. . . ,2N ,N being 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, ...2N

}(6.9)

ymax = maxk

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

}(6.10)

If f (A) has no extreme value in the interior of A, except at the vertices, Eqs. (6.9) and(6.10) are correct in which case the methods of interval analysis are applicable, e.g. theVertex 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 tothe true global minimum and maximum value.


DYNAMICS

7 DYNAMICS

This chapter highlights some of the theoretical backgrounds of the dynamic module. Thechapter does not give a full theoretical description of the dynamic modelling. For a moredetailed description you are referred to the literature Zienkiewicz & Taylor (1991), Hughes(1987), Das (1995), Kramer (1996), Haigh, Ghosh & Madabhushi (2005), Basabe & Sen(2007), Kelly, Ward, Treitel & Alford (1976) and Pradhan, Baidya & Ghosh (2004).

7.1 BASIC EQUATION DYNAMIC BEHAVIOUR

The basic equation for the time-dependent movement of a volume under the influence ofa (dynamic) load is:

M u + C u + K u = F (7.1)

Here, M is the mass matrix, u is the displacement vector, C is the damping matrix, K isthe stiffness matrix and F is the load vector. The displacement, u, the velocity, u, and theacceleration, u, can vary with time. The last two terms in the Eq. (7.1) (K u = F )correspond to the static deformation.

Here the theory is described on the bases of linear elasticity. However, in principle, allmodels in PLAXIS can be used for dynamic analysis. The soil behaviour can be bothdrained and undrained. In the latter case, the bulk stiffness of the groundwater is addedto the stiffness matrix K , as is the case for the static calculation.

In the matrix M , the mass of the materials (soil + water + any constructions) is taken intoaccount. In PLAXIS the mass matrix is implemented as a lumped matrix.

The matrix C represents the material damping of the materials. In reality, materialdamping is caused by friction or by irreversible deformations (plasticity or viscosity). Withmore viscosity or more plasticity, more vibration energy can be dissipated. If elasticity isassumed, damping can still be taken into account using the matrix C. To determine thedamping matrix, extra parameters are required, which are difficult to determine from tests.In finite element formulations, C is often formulated as a function of the mass andstiffness matrices (Rayleigh damping) (Zienkiewicz & Taylor, 1991; Hughes, 1987) as:

C = R M + R K (7.2)

This limits the determination of the damping matrix to the Rayleigh coefficients R andR . Here, when the contribution of M is dominant (for example, R = 102 andR = 103) more of the low frequency vibrations are damped, and when the contributionof K is dominant (for example, R = 103 and R = 102) more of the high-frequencyvibrations are damped. In the standard setting of PLAXIS, R = R = 0.

7.2 TIME INTEGRATION

In the numerical implementation of dynamics, the formulation of the time integrationconstitutes an important factor for the stability and accuracy of the calculation process.Explicit and implicit integration are the two commonly used time integration schemes.


SCIENTIFIC MANUAL

The advantage of explicit integration is that it is relatively simple to formulate. However,the disadvantage is that the calculation process is not as robust and it imposes seriouslimitations on the time step. The implicit method is more complicated, but it produces amore reliable (more stable) calculation process and usually a more accurate solution(Sluys, 1992).

The implicit time integration scheme of Newmark is a frequently used method. With thismethod, the displacement and the velocity at the point in time t + t are expressedrespectively as:

u t+t = u t + u t t +((

12

)u t + u t+t

)t2 (7.3a)

u t+t = u t +((1 ) u t + u t+t)t (7.3b)

In the above equations, t is the time step. The coefficients and determine theaccuracy of the numerical time integration. They are not equal to the and for theRayleigh damping. In order to obtain a stable solution, the following condition must apply:

0.5, 14

(12

+ )

2 (7.4)

The user is advised to use the standard setting of PLAXIS, in which the Newmarkscheme with = 0.25 and = 0.50 (average acceleration method) is utilised. Othercombinations are also possible, however.

Implementation of the integration scheme in PLAXIS

Eq. (7.3) can also be written as:

u t+t = c0u c2 u t c3 u tu t+t = u t + c6 u t + c7 u t+t

u t+t = u t + u (7.5a)

or as:u t+t = c0u c2 u t c3 u tu t+t = c1u c4 u t c5 u tu t+t = u t + u (7.5b)

where the coefficients c0 .. c7 can be expressed in the time step and in the integrationparameters and . In this way, the displacement, the velocity and the acceleration atthe end of the time step are expressed by those at the start of the time step and thedisplacement increment. With implicit time integration, Eq. (7.1) must be obtained at theend of a time step (t + t):

M u t+t + C u t+t + K u t+t = F t+t (7.6)

This equation, combined with the expressions (5.5) for the displacements, velocities and


DYNAMICS

accelerations at the end of the time step, produce:(c0 M + c1 C + K

)u = F t+text + M

(c2u t + c3u t

)+ C

(c4u t + c5u t

) F tint

(7.7)

In this form, the system of equations for a dynamic analysis reasonably matches that of astatic analysis. The difference is that the 'stiffness matrix' contains extra terms for massand damping and that the right-hand term contains extra terms specifying the velocity andacceleration at the start

3d 4 scientific

Documents

structural elements

special elements

interface elements

time integration

deformation theory

various types of elements

consolidation theory

groundwater flow theory