-Non Linear Computation in Plaxis

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PLAXIS Standard Course Universti Teknologi Malaysia, Johor

May 14-16,2007

Presented by GOUW Tjie-Liong

[email protected] 1

PLAXISFINITE ELEMENT CODE FOR SOIL AND ROCK ANALYSES

Non-linear Computations

in PLAXIS

presented by:

GOUW Tjie-Liong, Ir., M.Eng, ChFCSenior Geotechnical ConsultantEmail: [email protected]

Source of Lecture Notes:

Paul Bonnier

PLAXIS BV


Contents

• Calculation

• Local level

• Global level

• Load advancement




May 14-16,2007


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Calculation

• Initial situation

• Geometry

(mesh, loads, boundary conditions)

• Material parameters

• Initial stresses and pore pressures


Calculation

• Calculation phases

• Calculation types

• Plastic• Consolidation

• Phi-c reduction (safety analysis)




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Calculation

• Calculation phases

• Loading input

• Staged construction

• Switch on/off parts of geometry

(soil, structural elements)

• Switch on/off loads (change input values)

• Change pore pressures

• Total Multipliers (L.A. Ultimate level)

• Incremental Multipliers (L.A. no. of steps)


Calculation

Displacements Loads

Strains Stresses

Model(s)Parameters

Phase

• Find equilibrium in

final situation

• Subdivide in load

steps

Load step

• Find equilibrium for

load increment




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Calculation

Loads and Multipliers

• Applied load = Load multiplier x Input load

• Standard setting: Input load = 1 stress unit

• Actual load is specified before each

calculation

• Staged construction : Input load

• Total incr. Multipliers : Load multipliers


Calculation

load increment

load

Total load

dis lacement or settlement

u

A

q




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Calculation

Most calculations: Staged Construction

• Total load in last calculation step

Some calculations: Total multipliers

• Total load in last calculation step

• SMdisp, SMloadA, SMloadB, SMweight

Very few calculations: Incremental multipliers

• Additional load in one calculation step

• Mdisp, MloadA, MloadB, Mweight, Msf


u

A

q

Phase

q

∆u

Load-displacement curve for point A

∆q

1

Elasticstiffness

Ste s

Calculation

Global system of equations P u K = PuK ∆∆ = PuK δδ =or or




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PK u 1

δδ −=

uuu δ∆∆ +=

uB∆ε∆ =

)(D pe0c

ε∆ε∆σσ −+=

∫ = dVBP

ct

in σ

inex PP ≈

2)PP(K u inex1 →−= −

δ

Calculation

Basic algorithm per step

• additional displacements 1:

• total increments 2:

• strain increment 3:

• determine stresses 4:

• internal reaction 5:• equilibrium ?? 6:

• No: next iteration 7:


Calculation

• Output

• Per step/phase:

• displacements, stresses, forces• Load-displacement curves




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Local level

6-node triangle 15-node triangle

x x

xstress point (σ, ε)

node (ux, uy)

x-axis

y-axis x x

xx

xx xx

x

xx

x

6-node elements 15-node elements

Plane strain

Working load

Failure loads

Phi-c reduction

Axi-symmetry


Local level

f=0

c

σ3 σ1

φ

τ

σ

c

σ3 σ1

φ

τ

σ

Mohr Coulomb Mohr Coulomb

τ

ε

f=0

f<0

f<0

Mohr Coulomb

Elastic pointf<0

p

0ε =&

Plastic pointf=0

p0ε ≠&

τ

ε

f<0 Advanced models

Advanced plastic model

f=0

f=0

f=0

• Elastic and plastic stress point for Mohr-Coulomb

model




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Local level

Cap (Advanced models only)-σ1

Normal plastic pointf<0

-σ3 Apex point

-σ 1

f<0

-σ 3

Tension point

• Apex stress point and tension stress point


eq

c

||||

||||c

eqc

Error Local σ

σσ −=

Local level

Local error Constitutive stress σc:

Stress that follows from the

constitute model (Mohr- Coulomb)

Equilibrium stress σeq:

Stress that is in equilibrium with the

external load

Inaccurate point: Local error > Tolerated error

Standard setting: To lerated error = 0.01




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Global level

uuo

q0+∆q

q0

∆ue

∆up

• Initial stress method or Elastic stiffness

method


Global level

uuo

q0+∆q

q0

A BOverrelaxation

A

+=

A

B

Standard setting: 1.2

Absolute maximum: 2.0

Low friction angles (<20°): 1.5 acceptable

• Initial stress method + Over relaxation




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Global level

uuo

q0+∆q

q0

Arc

q0+∆q

Arc-length control

• Initial stress method + arc-length control


Global level

uuo

q

e

unbalance

=|q

e-q

c|

qc

qe = “external“ load

(including weights)

qc = internal reaction forces

(integral of σc)

Convergence requirement:

Global error = Tolerated error

Standard setting:

Tolerated error = 0.01

| || |

| || |

load

unbalance Error Global =

• Global error




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Global level

| || |

||| |

load

unbalance Error Global =

||||

||||c

eqc

Error Local σ

σσ −=

Convergence requirements

• Accuracy reached if:

• Global error = Tolerated error

• Inaccurate stress points = 3 + (pl. soil points)/10

• Inaccurate interface points = 3 + (pl. intf.

points)/10


Load advancement

u

qScaling up

Scaling down

Scaling up

• Converged within desired minimum number of iterations:

• Scaling up by a factor 2

• Not converged within desired maximum number of iterations:

• Scaling down by a factor 2




May 14-16,2007



Load advancement

u

q

Reaching prescribed

ultimate level

X

u

q

Reaching final step

X

u

q

Failure beforereachingprescribed ultimate

level

X

No. of steps procedure: • Plastic calculation, incr. Multipliers• Phi – c reduction (safety analysis)

Ultimate level procedure: Plastic calculation: • Staged construction• Total multipliers

Documents

-Non Linear Computation in Plaxis