INTRODUCTION INTO FINITE INTRODUCTION INTO FINITE ELEMENT NONLINEAR ELEMENT NONLINEAR

ANALYSESANALYSESDoc. Ing. Vladimír Ivančo, PhD.Technical University of Košice

Faculty of Mechanical EngineeringDepartment of Applied Mechanics and Mechatronics

HS Wismar, June 2009

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CONTENS1. Introduction

1.1 Types of structural nonlinearities

1.2 Concept of time curves

2. Geometrically nonlinear finite element analysis

3. Incremental – iterative solution

3.1 Incremental method

3.2 Iterative methods

4. Material nonlinearities

5. Examples

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Linear static analysis - the most common and the most simplified analysis of structures is based on assumptions:

• static = loading is so slow that dynamic effects can be neglected

• linear = a) material obeys Hooke’s lawb) external forces are conservative

c) supports remain unchanged during loadingd) deformations are so small that change of the structure configuration is neglectable

1. Introduction 1.1 Sources of nonlinearities

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Consequences:• displacements and stresses are proportional to loads, principle of superposition holds

• in FEM we obtain a set of linear algebraic equations for computation of displacements

FdK where

K – global stiffness matrix

d – vector of unknown nodal displacements

F – vector of external nodal forces

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1. Geometric nonlinearities - changes of the structure shape (or configuration changes) cannot be neglected and its deformed configuration should be considered.

2. Material nonlinearities - material behaves nonlinearly and linear Hooke’s law cannot be used. More complicated material models should be then used instead e.g.

Nonlinear analysis – sources of nonlinearities can be classified as

nonlinear elastic (Mooney-Rivlin’s model for materials like rubber), elastoplastic (Huber-von Mises for metals, Drucker-Prager model to simulate the behaviour of granular soil materials such as sand and gravel) etc.

3. Boundary nonlinearities - displacement dependent boundary conditions. The most frequent boundary nonlinearities are encountered in contact problems.

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FdK FdR )(we obtain a set of nonlinear algebraic equations

Consequences of assuming nonlinearities in FEM:

Instead of set of linear algebraic equations

Consequences of nonlinear structural behaviour that have to be recognized are:

1. The principle of superposition cannot be applied. For example, the results of several load cases

cannot be combined. Results of the nonlinear analysis cannot be scaled.

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2. Only one load case can be handled at a time.

3. The sequence of application of loads (loading history) may be important. Especially, plastic deformations depend on a manner of loading. This is a reason for dividing loads into small increments in nonlinear FE analysis.

4. The structural behaviour can be markedly non-proportional to the applied load. The initial state of stress (e.g. residual stresses from heat treatment, welding etc.) may be important.

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1.2 Concept of time curvesIn order to reflect history of loading, loads are associated with time curves.

Example - values of forces at any time are defined as

111 ftF 222 ftF

where f1 and f2 are nominal (input) values of forces and 1 and 1 are load parameters that are functions of time t.

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For nonlinear static analysis, the “time” variable represents a pseudo time, which denotes the intensity of the applied loads at certain step.

For nonlinear dynamic analysis and nonlinear static analysis with time-dependent material properties the “time” represents the real time associated with the loads’ application.

The most common case – all loads are proportional to time:

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2. Geometrically nonlinear finite element analysisExample – linearly elastic truss

0sin PN

.0 PL

uhN

Luh

sin

0, yiF

0, yiF

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Condition of equilibrium

.0 PL

uhN

0AEN

0

0

LLL

220 haL 22 )( uhaL

where

engineering strain

axial force

0A cross-section of the truss

Initial and current length of the truss are

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To avoid complications, it is convenient to introduce new measure of strain – Green’s strain defined as

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20

2

2 LLL

G

2220 haL 222 )( uhaL

In our example is

hence

2

00020

22222

21

22

Lu

Lu

Lh

Lhauhuha

G

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0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 0,1 0,2 0,3 0,4 0,5 0,6

Dl / l 0

stra

inGreen's strainLogarithmic strainEngineering strain

Example of different strain measures

)1ln()ln(0

log

LL

Logarithmic strain (true strain)

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The stress-strain relation was measured as

EAN

0

When using Green’s strain the relation should be

GGGG

EEE

211

21 2

GG EE

*

211

211

*

EE

This means that constitutive equation should be

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If strain is small (e.g. less than 2%) differences are negligible

211

*

EE

The new modulus of elasticity is not constant but it depends on strain

DL / L0 G (MPa) (MPa) (MPa) G(MPa)

0,0000 0,0000 0,000000 21 000 21 000 0 0

0,0050 0,0050 0,005013 21 000 20 948 105 105

0,0100 0,0100 0,010050 21 000 20 896 210 210

0,0150 0,0150 0,015113 21 000 20 844 315 315

0,0200 0,0200 0,020200 21 000 20 792 420 420

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Assuming that strain is small, we can write

GG AEAEAEN 00*

0 and after substituting into equation

we can derive the condition of equilibrium in the form

.0 PL

uhN

PuhuhuLAE

22330

0 232

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Consequence of considering configuration changes - relation between load P and displacement u is nonlinear

PuhuhuLAE

22330

0 232

Generally, using FEM we obtain a set of nonlinear algebraic equations for unknown nodal displacements d

FdR )(

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is tangent stiffness matrix

FdR )(

3. Incremental – iterative solutionAssumption of large displacements leads to nonlinear equation of equilibrium

dKdRddRdRddR d)(d)()d( T

FFddR d)d(

dRK

T

FdK dd T

For infinitesimal increments of internal and external forces we can write

where

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3.1 Incremental methodThe load is divided into a set of small increments DFi .

Increments of displacements are calculated from the set of linear simultaneous equations

iiiT FdK DD )1(

iii ddd D 1

Nodal displacements after force increment of DFi are

where KT(i-1) is tangent stiffness matrix computed form displacements d(i-1) obtained in previous incremental step.

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Incremental method

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3.2 Iterative methodsNewton-Raphson methodConsider that di is estimation of nodal displacement. As it is only an estimation, the condition of equilibrium would not be satisfied

This means that conditions of equilibrium of internal and external nodal forces are not satisfied and in nodes are unbalanced forces

FdR )( i

FdRr )( ii

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Correction of nodal displacements can be then obtained from the set of linear algebraic equations

iiiT rdK D)(

and mew, corrected estimation of nodal displacements is

iii ddd D1

The procedure is repeated until the sufficiently accurate solution is obtained.

The first estimation is obtained from linear analysis

FKd 1

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Standard Newton-Raphson (NR) method

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Modified Newton-Raphson (MNR) method - the same stiffness matrix is used in all iterations

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Combination of Newton-Raphson and incremental methods

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4. Material nonlinearities 4.1 Nonlinear elasticity models

For any nonlinear elastic material model, it is possible to define relation between stress and strain increments as

εDσ dd T

Matrix DT is function of strains . Consequently, a set of equilibrium equations we receive in FEM is nonlinear and must be solved by use of any method described above

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4.1 Elastoplastic material models

The total strains are decomposed into elastic and plastic parts

pe εεε

σε

Qp dd

The yield criterion says whether plastic deformation will occur.The plastic behaviour of a material after onset of plastic deformations is defined by so-called flow rule in which is the rate and the direction of plastic strains is related to the stress state and the stress rate. This relation can be expressed as

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Constitutive equation can be formulated as

εDσ dd TThe tangential material matrix DT is used to form a tangential stiffness matrix KT. When the tangential stiffness matrix is defined, the displacement increment is obtained for a known load increment

FdK DDT

As load and displacement increments are final, not infinitesimal, displacements obtained by solution of this set of linear algebraic equation will be approximate only. That means, conditions of equilibrium of internal and external nodal forces will not be satisfied and iterative process is necessary.

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The problem - not only equilibrium equations but also constitutive equations of material must be satisfied. That means that within the each equilibrium iteration step check of stress state and iterations to find elastic and plastic part of strains at every integration point must be included. The iteration process continues until both, equilibrium conditions and constitutive equations are satisfied simultaneously. The converged solution at the end of load increment is then used at the start of new load increment.

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Example of non-linear static analysis – bending of the beam,

considering elastoplastic material

bilinear material model

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Detail of finite element mesh – SHELL4T elements are colored according to their thickness

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Beginning of plastic deformations

Normal stress distribution in the cross-section at mid of the beam span.

Maximal stress x approaches value of the yield stress

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Normal stress distribution after increase of the load.

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COLLAPSE – inability of the beam to resist further load increase

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Deflection versus load

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Example of nonlinear dynamic analysis - drop test of container for radioactive waste.Simulation of a drop from 9 m at an angle to steel target

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Reduced stresses at time 0,00187 s

Time courses of reduced stresses at selected nodes

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Drop on side of the lid – check of screwed bolts

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0,00165 s.

0,0027 s

43Maximal displacements

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0

50

100

150

200

250

300

350

400

450

0 1000 2000 3000 4000 5000 6000 7000

time step

[ M

Pa]

Bolt 1

Bolt 2

Bolt 3

Bolt 4

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Drop along the top on the mandrel

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0

50

100

150

200

250

300

350

400

450

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

t [ms]

[M

Pa]

Drop along the top on the mandrel - time course of maximal stress in the lid

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Drop aside on the mandrel

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Reduced stresses at time 0,00235 s

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0

50

100

150

200

250

300

350

400

450

500

0,0 0,5 1,0 1,5 2,0 2,5 3,0

t [ms]

[M

Pa]

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Study of influence of residual stresses due to arc welding on load-bearing capacity of a thin-walled beam.

Example:

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Coupled thermal and stress analysis in following steps:

1.Nonlinear transient thermal analysis

• temperature dependent thermal material properties c, k and density

• temperature dependent convective heat transfer coefficient

2.Nonlinear stress analysis

• plastic deformations

• large displacements

• temperature dependent material mechanical properties

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Temperature field at time t = 5 s, 1st phase of welding

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Temperature field at time t = 10 s, 1st phase

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Temperature field at time t = 5 s, 2nd phase

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Temperature field at time t = 10 s, 2nd phase

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Temperature field after end of welding

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Reduction coefficients for yield stress and modulus of elasticity

yfy fTkTy

)()(

20)()( ETkTE E

fy – yield stress and E20 modulus of elasticity at 20 oC

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temperature field

stress field

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60

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-2

-1

0

1

2

3

4

5

0 500 1000 1500 2000 2500 3000

t (s)

v (m

m)

Deflection of the beam during welding

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0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 5 10 15 20 25

F (kN)

v (m

m)

bez reziduálnych napätí(žíhaný)nežíhanýunannealedannealed

Maximum deflection versus load

F