Nl Analysis

- Nonlinear

Nonlinear AnalNonlinear AnalNonlinear Anal

This chapter includes material from the book Practical This chapter includes material from the book Practical Documentation. Additional material was also added by Patrick Documentation. Additional material was also added by Patrick

IntroductionIntroduction

IIn this chapter, we discuss the practical aspects of nonlinear static In this chapter, we discuss the practical aspects of nonlinear static But how do we know that our problem is nonlinear? The best wBut how do we know that our problem is nonlinear? The best w

characteristic load introduction points. As discussed in an earliercharacteristic load introduction points. As discussed in an earlier(deformation, stress and strain) is linearly proportional to the magnitude(deformation, stress and strain) is linearly proportional to the magnitudeetc.), then the analysis of such a structure is known as linearetc.), then the analysis of such a structure is known as linearproportional, then the analysis falls under nonlinear analysis (seeproportional, then the analysis falls under nonlinear analysis (seestiff metal is subjected to a load relatively lower in magnitude asstiff metal is subjected to a load relatively lower in magnitude asstructure will be linearly proportional to the load and the structurestructure will be linearly proportional to the load and the structureBut most of the time either the material behavior is not linearBut most of the time either the material behavior is not linearitself keeps it from responding linearly. Due to the cost or weightitself keeps it from responding linearly. Due to the cost or weightover metals, nonmetals are replacing metals for a variety of applicationsover metals, nonmetals are replacing metals for a variety of applicationscharacteristics, even under mild loading conditions. Also the structurescharacteristics, even under mild loading conditions. Also the structuresload level so close to the strength of the material, that it starts behload level so close to the strength of the material, that it starts behthe structures in these circumstances, it is necessary to performthe structures in these circumstances, it is necessary to perform

Linear response: Linear response: P1/d1 = P2/d2P1/d1 = P2/d2

P2

P2 P2

P1 P1

Load

P1

Load

Load

Load

Load

Load

d1 d2

Comparison of linear and

d1 d2

Comparison of linear and

As discussed in the earlier chapter, the stiffness matrix relatingAs discussed in the earlier chapter, the stiffness matrix relatinganalysis; however, all the real world structures behave nonlinearlanalysis; however, all the real world structures behave nonlinearlas length, cross sectional area and moment of inertia, etc. and matas length, cross sectional area and moment of inertia, etc. and matThe static analysis assumes that these parameters do not changeThe static analysis assumes that these parameters do not changestatic analysis takes into account the changes in these paramestatic analysis takes into account the changes in these parameaccommodated in the analysis by rebuilding the stiffness matrix withaccommodated in the analysis by rebuilding the stiffness matrix witheach incremental load application. Although, the world is nonlineaeach incremental load application. Although, the world is nonlineaof linearity is valid and in that way a linear analysis can be doneof linearity is valid and in that way a linear analysis can be doneexpensive approach.expensive approach.

Comparison Of Linear And Nonlinear FEAComparison Of Linear And Nonlinear FEAComparison Of Linear And Nonlinear FEA

The table below provides a short summary overview about the multipleThe table below provides a short summary overview about the multipleis not meant to explain the differences in detail. Some of the mainis not meant to explain the differences in detail. Some of the mainrelation, stress-strain measures are explained in some more detailrelation, stress-strain measures are explained in some more detail

11

Nonlinear Analysis -

AnalysisAnalysisAnalysis

Practical Finite Element Analysis and the HyperWorks Help Practical Finite Element Analysis and the HyperWorks Help atrick Zerbe, Kristian Holm and Matthias Goelke.atrick Zerbe, Kristian Holm and Matthias Goelke.

nonlinear static Finite Element Analysis.nonlinear static Finite Element Analysis.

way is to look at the load-displacement response of one or moreway is to look at the load-displacement response of one or more

earlier chapter on Types of Analysis, when the structural responseearlier chapter on Types of Analysis, when the structural responsemagnitude of the load (force, pressure, moment, torque, temperaturemagnitude of the load (force, pressure, moment, torque, temperature

analysis. When the load to response relationship is not linearlyanalysis. When the load to response relationship is not linearly(see figure below). For example, when a compact structure made of(see figure below). For example, when a compact structure made of

compared to the strength of the material, the deformation in thecompared to the strength of the material, the deformation in thestructure is known to have been subjected to linear static deformation.structure is known to have been subjected to linear static deformation.

linear in the operating conditions or the geometry of the structurelinear in the operating conditions or the geometry of the structureeight advantage of nonmetals (polymers, woods, composites etc.)eight advantage of nonmetals (polymers, woods, composites etc.)applications. These applications have nonlinear load to responseapplications. These applications have nonlinear load to response

structures are optimized to make most of its strength, pushing thestructures are optimized to make most of its strength, pushing thebehaving nonlinearly. In order to accurately predict the strength ofbehaving nonlinearly. In order to accurately predict the strength ofa nonlinear analysis.a nonlinear analysis.

Nonlinear response:Nonlinear response:P1/d1 P2/d2P2 P1/d1 P2/d2P2

P1P1

d1 d2d1 d2

and nonlinear responseand nonlinear response

relating to the load and response is assumed to be constant for staticrelating to the load and response is assumed to be constant for staticnonlinearly. The stiffness matrix consists of geometric parameters suchnonlinearly. The stiffness matrix consists of geometric parameters such

material properties such as elastic modulus, rigidity modulus etc.material properties such as elastic modulus, rigidity modulus etc.change when the structure is loaded. On the other hand, nonlinearchange when the structure is loaded. On the other hand, nonlinear

eters as the load is applied to the structure. These changes areeters as the load is applied to the structure. These changes arewith respect to the deformed structure (i.e. altered properties) afterwith respect to the deformed structure (i.e. altered properties) after

nonlinear, it should be mentioned that in many cases the assumptionnonlinear, it should be mentioned that in many cases the assumptioninstead. Also, from a computation point of view, it is a much lessinstead. Also, from a computation point of view, it is a much less

FEAFEAFEA

multiple differences between a linear and nonlinear FE analysis andmultiple differences between a linear and nonlinear FE analysis andmain differences such as load-displacement relation, stress-strainmain differences such as load-displacement relation, stress-strain

etail in the remainder of this chapter.etail in the remainder of this chapter.

- Nonlinear

Characteristic Feature Linear ProblemsCharacteristic Feature Linear Problems

Load - Displacement Displacements vary linearly withLoad - Displacement Displacements vary linearly with

Relation applied loads. Thus StiRelation applied loads. Thus Sticonstant. Changes in geomconstant. Changes in geomdue to displacement are assumeddue to displacement are assumedto be small and hence are ignored.Original or undeformed staOriginal or undeformed staalways used as a reference staalways used as a reference sta

Stress-Strain Relation Linear up toStress-Strain Relation Linear up toproportionalproportional/ elastic limit. Prope/ elastic limit. Propesuch as Youngs Modulussuch as Youngs Moduluseasily available.

Scalability Applicable. That is if a 1Scalability Applicable. That is if a 1

causes x units of displacement ,causes x units of displacement ,then 10 N magnitude of then 10 N magnitude of cause 10x displacementcause 10x displacement

Superposition Applicable. That is a combinationSuperposition Applicable. That is a combination

of load cases is possible.of load cases is possible.

Reversibility The behaviour of the structure is Reversibility The behaviour of the structure is fully reversible upon the remfully reversible upon the remof external loads. Thisof external loads. Thismeans that loading senot important and the final stanot important and the final stais unaffected by the load hisis unaffected by the load his

Solution scheme The load is applied inwith no iterations .with no iterations .

Computational Time Small

Users Interaction with the Least requiredUsers Interaction with the Least required

softwaresoftware

Types Of NonlinearityTypes Of Nonlinearity

There are three basic source of nonlinearity: Geometric NonlinearitThere are three basic source of nonlinearity: Geometric Nonlinearit

Recall the definition of linear elastic small displacements and stress Recall the definition of linear elastic small displacements and stress

Linear ElasticLinear Elastic

1) Geometric Nonlinearities1) Geometric Nonlinearities

Geometric nonlinearity may be related to 1) Large strain 2) Large roGeometric nonlinearity may be related to 1) Large strain 2) Large ro

Geometric nonlinearity takes into account that the geometry crossGeometric nonlinearity takes into account that the geometry crossstatic analysis the cross section is considered to be constant). Largestatic analysis the cross section is considered to be constant). Large

22


Nonlinear ProblemsNonlinear Problems

y linearly with It is nonlinear. Thus stiffnessy linearly with It is nonlinear. Thus stiffness

applied loads. Thus Stiffness is varies as a function of load.applied loads. Thus Stiffness is varies as a function of load.geometry Displacements can be very largegeometry Displacements can be very large

o displacement are assumed and changes in geometry cannot beo displacement are assumed and changes in geometry cannot beo be small and hence are ignored. ignored. Thus stiffness varies as a

ormed state is function of load.ormed state is function of load.erence state.erence state.

the It is nonlinear function of stress-the It is nonlinear function of stress-strain and time. These are

opertiesstrain and time. These are difficult to obtain and requires lot operties

Modulus aredifficult to obtain and requires lot of additional experimental Modulus are of additional experimental material testing. Mind the differences between true stress differences between true stress and engineering stressand engineering stress

a 1 N force Not applicable.a 1 N force Not applicable.

causes x units of displacement ,causes x units of displacement ,magnitude of force willmagnitude of force willdisplacementdisplacement

a combination Not applicable .a combination Not applicable .

possible.possible.

viour of the structure is The final state after removal of viour of the structure is ersible upon the removal

The final state after removal of loads is different from the ersible upon the removal

also loads is different from the initial state. Due to this also

means that loading sequence is tant and the final state

initial state. Due to this superpositionof load cases is not possible. tant and the final state

y the load history.of load cases is not possible. Load history is very important.y the load history. Load history is very important.

one stepLoad is split into smallLoad is split into smallincrements with iterationsincrements with iterationsperformed to ensure thatperformed to ensure thatequilibrium is satisfied at everyequilibrium is satisfied at everyload increment.load increment.Large

Requires lot of monitoring as theRequires lot of monitoring as the

software may fail to convergesoftware may fail to convergesometimes.

tric Nonlinearity, Material Nonlinearity, and Contact Nonlinearity.tric Nonlinearity, Material Nonlinearity, and Contact Nonlinearity.

displacements and stress is proportional to straindisplacements and stress is proportional to strain

rotation, and 3) Large deformationrotation, and 3) Large deformation

oss section may change as a result of large deformation (in lineaross section may change as a result of large deformation (in linearLarge displacements may also be introduced by geometric buckling.Large displacements may also be introduced by geometric buckling.

- Nonlinear

A phenomena which is characterized by a sudden failure of a structuralA phenomena which is characterized by a sudden failure of a structuralthe actual compressive stress at the point of failure is less thanthe actual compressive stress at the point of failure is less thanof withstanding. Thus, the equilibrium equations must be writtenof withstanding. Thus, the equilibrium equations must be writtenapplied loads may change their direction as they increase, as whenapplied loads may change their direction as they increase, as whenof finite element analysis; R.D. Cook et. al, p. 595).of finite element analysis; R.D. Cook et. al, p. 595).

Large Displacements And Rotations (small strain; linear orLarge Displacements And Rotations (small strain; linear or

Large Displacements, Rotations, And Large Strain (linear orLarge Displacements, Rotations, And Large Strain (linear or

Images after K.J. Bathe, FiniImages after K.J. Bathe, Fini

In linear FEA, strain, for example in the x-direction, is expressed asIn linear FEA, strain, for example in the x-direction, is expressed asu/x + ...[(u/x) + (v/x) + (w/x) ] is considered. In the caseu/x + ...[(u/x) + (v/x) + (w/x) ] is considered. In the case

z z z

also considered. Additionally, that the stress-strain (material) relationship also considered. Additionally, that the stress-strain (material) relationship

33


structural member subjected to high compressive stress, wherestructural member subjected to high compressive stress, wherethan the ultimate compressive stresses that the material is capablethan the ultimate compressive stresses that the material is capable

en with respect to the deformed structural geometry. Additionally,en with respect to the deformed structural geometry. Additionally,when pressure inflates a membrane (Ref. Concepts and applicationswhen pressure inflates a membrane (Ref. Concepts and applications

or nonlinear material)or nonlinear material)

or nonlinear material)or nonlinear material)

Finite Elemente MethodenFinite Elemente Methoden

as = u/x. Thus, in other words, only the first order term of =as = u/x. Thus, in other words, only the first order term of =x x

case of large displacements (nonlinear), the second order terms arecase of large displacements (nonlinear), the second order terms arerelationship may or may not be linear.relationship may or may not be linear.

- Nonlinear

2) Material Nonlinearities2) Material Nonlinearities

Stress

StrainStrain

HyperelasticNonlinear HyperelasticNonlinearElastic

Stress

Elastic

Stress

20 %20 %

Features of MaFeatures of Ma

Nonlinear Material (small displacements)Nonlinear Material (small displacements)

Images after K.J. Bathe, Fini

All engineering materials are inherently nonlinear as it is not feasibleAll engineering materials are inherently nonlinear as it is not feasiblelaw for the entire range of environmental conditions such as loading,law for the entire range of environmental conditions such as loading,simplify the material behavior to account for only certain effects whichsimplify the material behavior to account for only certain effects whichHookean) material assumption is the simplest of all. The materialHookean) material assumption is the simplest of all. The materialif it is irrecoverable. If the temperature effects on the material propeif it is irrecoverable. If the temperature effects on the material propeand thermal behavior should be properly taken into considerationand thermal behavior should be properly taken into considerationrate has significant effects on the material, we have to consider therate has significant effects on the material, we have to consider thematerial nonlinear behavior has been given in the above figures.material nonlinear behavior has been given in the above figures.

Abrief classification can be given as follows:Abrief classification can be given as follows:

1. Nonlinear elastic1. Nonlinear elastic

2. Hyperelastic2. Hyperelastic

3. Elastic-perfectly plastic3. Elastic-perfectly plastic

4. Elastic-time independent plastic4. Elastic-time independent plastic

44


Elastic plasticElastic plastic

Elastic-Perfectly plastic

StrainStrain

HyperelasticHyperelastic

Strain Strain

of Material Nonlinearityof Material Nonlinearity

Finite Elemente Methoden

easible to characterize a nonlinear material by a single constitutiveeasible to characterize a nonlinear material by a single constitutiveloading, temperature and rate of deformation. We can idealize orloading, temperature and rate of deformation. We can idealize orwhich are important for the analysis. The linear elastic (also calledwhich are important for the analysis. The linear elastic (also called

rial is nonlinear elastic if the deformation is recoverable and plasticrial is nonlinear elastic if the deformation is recoverable and plasticoperties are important, then the coupling between the mechanicaloperties are important, then the coupling between the mechanical

consideration through thermo-elasticity and thermo-plasticity. If the strainconsideration through thermo-elasticity and thermo-plasticity. If the strainthe theories of visco-elasticity and visco-plasticity. An example ofthe theories of visco-elasticity and visco-plasticity. An example of

- Nonlinear

5. Time dependent plastic (Creep)5. Time dependent plastic (Creep)

6. Strain rate dependent elasticity plasticity6. Strain rate dependent elasticity plasticity

7. Temperature dependent elasticity and plasticity7. Temperature dependent elasticity and plasticity

If we observe the stress-strain curves in the above figures, then matIf we observe the stress-strain curves in the above figures, then mat

1. Linear elastic-perfectly plastic1. Linear elastic-perfectly plastic

2. Linear elastic-plastic. The plastic part in the stress-strain curve 2. Linear elastic-plastic. The plastic part in the stress-strain curve

a. Elastic Piecewise linear plastic (as shown in following figures)a. Elastic Piecewise linear plastic (as shown in following figures)

b. Elastic Actual stress strain curve information.b. Elastic Actual stress strain curve information.

Str

ess

Str

ess

Str

ess

E1

Strain(a) Elastic Material(a) Elastic Material

Stress Strain Curve for Material with

3. Nonlinear elastic model characterizing materials with no fixed3. Nonlinear elastic model characterizing materials with no fixedlimiting well below say 20 %.limiting well below say 20 %.

4. Hyperelastic materials such as rubber undergoing very large displacements. 4. Hyperelastic materials such as rubber undergoing very large displacements.

3) Boundary Nonlinearities/Contact Nonlinearity3) Boundary Nonlinearities/Contact Nonlinearity

After displacements in the order of have been reached (occurred),After displacements in the order of have been reached (occurred),degree of freedom.degree of freedom.

ContactContact

Images after K.J. Bathe, FiniImages after K.J. Bathe, Fini

Boundary nonlinearity arises when boundary conditions in a FEBoundary nonlinearity arises when boundary conditions in a FEconditions could be added to or removed from the model due toconditions could be added to or removed from the model due tononlinearity typically involves contact sets in the model which couldnonlinearity typically involves contact sets in the model which couldThe load transfer mechanism via a contact pair is a complicatedThe load transfer mechanism via a contact pair is a complicated

55


material nonlinearity can be classified further as follows:material nonlinearity can be classified further as follows:

e is time independent and can be analysed into two main types:e is time independent and can be analysed into two main types:

wing figures)wing figures)

E32

Str

ess

E32

Str

ess

E2

Str

ess

E1E1

Strain(b) Elastoplastic Material(b) Elastoplastic Material

with Linear and Nonlinear Properties

ed definition of yield point such as say plastic but the strain stilled definition of yield point such as say plastic but the strain still

displacements. A typical application is a gasket material.displacements. A typical application is a gasket material.

(occurred), the boundary condition is changing, i.e. changes in/of the (occurred), the boundary condition is changing, i.e. changes in/of the

Finite Elemente MethodenFinite Elemente Methoden

model change during the course of the analysis. The boundarymodel change during the course of the analysis. The boundaryo boundary nonlinearity as the analysis progresses. This kind ofo boundary nonlinearity as the analysis progresses. This kind of

could get engaged or disengaged as a response to applied loads.could get engaged or disengaged as a response to applied loads.ed phenomenon. Researchers have developed several theories toed phenomenon. Researchers have developed several theories to

- Nonlinear

describe the load transfer using contact sets. Atypical contact algorithmdescribe the load transfer using contact sets. Atypical contact algorithm

First, check for either open or closed contact surfaces. This is doneFirst, check for either open or closed contact surfaces. This is doneforming the contact pairs. If a contact is closed, then apply reactionforming the contact pairs. If a contact is closed, then apply reactionthrough each other. If a contact is open, then no load is transferredthrough each other. If a contact is open, then no load is transferredare calculated based on stiffness of the materials of each of the suare calculated based on stiffness of the materials of each of the suset is not constant and it requires continuous check to determineset is not constant and it requires continuous check to determinebe handled by linear static analysis codes. More accurate resultsnodes in the contact set more frequently.nodes in the contact set more frequently.

Contact TreatmentContact Treatment

There are two approaches to deal with contact:There are two approaches to deal with contact:

Penalty method Penalty method

Lagrange Multiplier method Lagrange Multiplier method

The Penalty method is the most commonly used in explicit codesThe Penalty method is the most commonly used in explicit codesMultiplier method is used in special case studied.Multiplier method is used in special case studied.

Interfaces using the Penalty method are based on master / slaveInterfaces using the Penalty method are based on master / slaveand a set of master segments. Master segments are defined dependingand a set of master segments. Master segments are defined depending4-node shell, the segment is the surface of the element. If it is a4-node shell, the segment is the surface of the element. If it is a2D solid element (quad), the segment is a side.2D solid element (quad), the segment is a side.

Contrary to the penalty method, the Lagrange Multiplier methodContrary to the penalty method, the Lagrange Multiplier method(springs) to model contact. A nonlinear system of equations is sol(springs) to model contact. A nonlinear system of equations is soltime step collapse due to high interface stiffness, but it takes moretime step collapse due to high interface stiffness, but it takes moreby nonlinear solver. The method has the advantage to stop slavby nonlinear solver. The method has the advantage to stop slavprecisely satisfied); however the friction cannot be computed.precisely satisfied); however the friction cannot be computed.

Additional information about this topic is available with ChapterAdditional information about this topic is available with ChapterProblems and Chances written by Professor rolf Steinbuch (ReutlingenProblems and Chances written by Professor rolf Steinbuch (Reutlingen

Stress-Strain Measures For Nonlinear AnalStress-Strain Measures For Nonlinear AnalStress-Strain Measures For Nonlinear Anal

This is a major distinguishing factor between linear static and nonlinearThis is a major distinguishing factor between linear static and nonlinearstress strain definitions, whereas in nonlinear analysis typicallystress strain definitions, whereas in nonlinear analysis typicallystress/strain relationships is also available at http://dolbow.cee.dustress/strain relationships is also available at http://dolbow.cee.du

We will examine these various stress and strain definitions throughWe will examine these various stress and strain definitions through

ll

ll0

A one dimensional rod undergoing

1) Engineering Strain And Engineering Stress1) Engineering Strain And Engineering Stress

Engineering strain is a small strain measure, which is computedEngineering strain is a small strain measure, which is computedlinear measure since it depends on the original geometry, (i.e. length)linear measure since it depends on the original geometry, (i.e. length)of the material because a moderate rigid body rotation will resultof the material because a moderate rigid body rotation will result

= x

= x

66


algorithm in FE codes works as follows:algorithm in FE codes works as follows:

done by measuring the distances between the nodes on the surfacesdone by measuring the distances between the nodes on the surfacesreaction forces to the nodes in contact to keep them from penetratingreaction forces to the nodes in contact to keep them from penetrating

erred between those nodes. The reaction forces applied to the nodeserred between those nodes. The reaction forces applied to the nodessurfaces in the contact set. Since load transfer through a contactsurfaces in the contact set. Since load transfer through a contact

ermine the open or closed status of the contact, this analysis can notermine the open or closed status of the contact, this analysis can notresults can be obtained by checking the open or closed status of the

and can be found in most of OptiStruct interfaces. The Lagrangeand can be found in most of OptiStruct interfaces. The Lagrange

e treatment. Contact can only occur between a set of slave nodese treatment. Contact can only occur between a set of slave nodesdepending on the type of element they lie on. If it is a 3-node or adepending on the type of element they lie on. If it is a 3-node or a

a solid element, the segment is defined as a face, Finally, if it is aa solid element, the segment is defined as a face, Finally, if it is a

thod is purely mathematic and does not require physical elementsthod is purely mathematic and does not require physical elementssolved to account for contact conditions. Therefore, there is nosolved to account for contact conditions. Therefore, there is no

more CPU time to do 1 cycle as new equations need to be solvedmore CPU time to do 1 cycle as new equations need to be solvedve nodes at the contact surface (that is, the contact condition isve nodes at the contact surface (that is, the contact condition is

17.9 Contact Modelling in Structural Simulation Approaches,17.9 Contact Modelling in Structural Simulation Approaches,eutlingen University, Germany).eutlingen University, Germany).

AnalysisAnalysisAnalysis

nonlinear FEA. In linear analysis we always use the engineeringnonlinear FEA. In linear analysis we always use the engineeringtypically the true stress and train is used. An excellent summary abouttypically the true stress and train is used. An excellent summary about

duke.edu/TENSILE/tutorial/node3.html.duke.edu/TENSILE/tutorial/node3.html.

ough a simple one-dimensional example.ough a simple one-dimensional example.

ll

FF

undergoing axial deformation

using the original geometry. The engineering strain measure is ausing the original geometry. The engineering strain measure is alength) which is known beforehand. It is limited to small rotationslength) which is known beforehand. It is limited to small rotations

in non-zero strains.in non-zero strains.

ll

ll0

- Nonlinear

Engineering stress (), is the conjugate stress measure to engineeringEngineering stress (), is the conjugate stress measure to engineeringA in its computation.

0A in its computation.

0

= =

2) Logarithmic Strain And True Stress.2) Logarithmic Strain And True Stress.

Logarithmic strain / natural strain / true strain is a large strain measure, Logarithmic strain / natural strain / true strain is a large strain measure,

l dll

= =dl

= =l l

ll0l

0

This measure is a nonlinear strain measure since it is a nonlinearThis measure is a nonlinear strain measure since it is a nonlinearlog strain (or true strain) and represents an additive strain measurelog strain (or true strain) and represents an additive strain measureinitial length of 1m undergoing deformation in 3 steps as follows:initial length of 1m undergoing deformation in 3 steps as follows:

Step 1 : deformation from 1 m to 1.2 mStep 1 : deformation from 1 m to 1.2 m

Step 2 : deformation from 1.2 to 1.5 mStep 2 : deformation from 1.2 to 1.5 m

Step 3 : deformation from 1.5 to 2 mStep 3 : deformation from 1.5 to 2 m

In the following table we compare the engineering strains againstIn the following table we compare the engineering strains againstretained in only true strain and hence this one should be used in nonlinearretained in only true strain and hence this one should be used in nonlinear

Step Engineering Strain

1 0.21 0.2

2 0.3/1.2=0.252 0.3/1.2=0.25

3 0.5/1.5=0.333 0.5/1.5=0.33

Addition of strains in 0.78Addition of strains in 0.78

all the stepsall the steps

Total : From 1 to 3 (2.0 1.0 )/1.0 = 1.0Total : From 1 to 3 (2.0 1.0 )/1.0 = 1.0

Conclusion Engg. Strain is not additive and

preserve the history of depreserve the history of decorrectly .

True stress (s) is the conjugate 1-D stress measure to the log straincorrectly .

True stress (s) is the conjugate 1-D stress measure to the log strain(or deformed) area A. This measure is also commonly referred to as (or deformed) area A. This measure is also commonly referred to as

s =

3) Green-Lagrange Strain And 2Nd Piola-Kirchoff Stress3) Green-Lagrange Strain And 2 Piola-Kirchoff Stress

Green-Lagrange strain is another large strain measure, which is coGreen-Lagrange strain is another large strain measure, which is co

1 =

G 2

1 =

G 2

This measure is nonlinear because it depends on the square ofThis measure is nonlinear because it depends on the square ofadvantage of this strain measure, over the log or Hencky strain,advantage of this strain measure, over the log or Hencky strain,in large strain problems. The conjugate stress measure for thein large strain problems. The conjugate stress measure for thecomputed in 1-D bycomputed in 1-D by

ls =

0l

s = 0

l l

It should be noted that this stress has little physical meaning.It should be noted that this stress has little physical meaning.

77


engineering strain (). It uses the current force F and the original areaengineering strain (). It uses the current force F and the original area

FFAA

00

strain measure, which is computed asstrain measure, which is computed as

l = Ln

l = Ln l l 0

nonlinear function of the unknown final length. It is also referred to as thenonlinear function of the unknown final length. It is also referred to as themeasure as compared to linear strain. Let us consider a bar with saymeasure as compared to linear strain. Let us consider a bar with say

::

against true strains and one can clearly see that the additivity isagainst true strains and one can clearly see that the additivity isnonlinear analysis.nonlinear analysis.

True StrainTrue StrainLog (1.2/1) = 0.1823Log (1.2/1) = 0.1823

Log (1.5/1.2) = 0.2231Log (1.5/1.2) = 0.2231

Log (2/1.5) = 0.2876Log (2/1.5) = 0.2876

0.69310.6931

Log(2/1) = 0.6931Log(2/1) = 0.6931

and doesnt True Strain is and thusadditive

deformation preserves the history of deformationdeformation preserves the history of deformationcorrectly.

strain ( ), which is computed by dividing the force F by the currentcorrectly.

strain ( ), which is computed by dividing the force F by the currentlas the Cauchy stress.as the Cauchy stress.

FA

F

A

computed in 1-D ascomputed in 1-D as

l2 - l 2

l 2

l2 - l 2 0

l 2

00

of the updated length l, which is an unknown. A computationalof the updated length l, which is an unknown. A computationalis that it automatically accommodates arbitrarily large rotationsis that it automatically accommodates arbitrarily large rotations

the Green-Lagrange strain, is the 2nd Piola-Kirchoff stress (s). It isthe Green-Lagrange strain, is the 2nd Piola-Kirchoff stress (s). It is

l F0l F

0

l A

0l A

0

- Nonlinear

Essential Steps To Start With Nonlinear FEAEssential Steps To Start With Nonlinear FEAEssential Steps To Start With Nonlinear FEA

Learn first how the software works on a simple model before Learn first how the software works on a simple model beforehow your structural component will behave, i.e. check for available how your structural component will behave, i.e. check for available

Try to understand the softwares supporting documentation, its Try to understand the softwares supporting documentation, its

Know what you are looking for. Prepare a list of questions Know what you are looking for. Prepare a list of questionsanalysis, including the model, material model, and boundaryanalysis, including the model, material model, and boundary

Keep the final model as simple as possible. A linear analysis Keep the final model as simple as possible. A linear analysisthe high stresses in the model, where the initial contact mthe high stresses in the model, where the initial contact mmodel. The results of the linear analysis may even point out thatmodel. The results of the linear analysis may even point out thatsituation include the yield limit not being reached, there is nosituation include the yield limit not being reached, there is no

Verify and validate the results of the nonlinear FEA solution. V Verify and validate the results of the nonlinear FEA solution. Vnumerical point of view. Wrong discretization with respect tonumerical point of view. Wrong discretization with respect toasks the questions if the correct model is computed e.gasks the questions if the correct model is computed e.gcoincide with the one acting in reality.coincide with the one acting in reality.

Try to look into the assumptions made with respect to the structural Try to look into the assumptions made with respect to the structuralstrain (On/Off), look into different material models if the earlierstrain (On/Off), look into different material models if the earliersoftware only make some models compatible with commonlysoftware only make some models compatible with commonlyof changing the element formulation).of changing the element formulation).

General Procedure For Nonlinear Static AnalGeneral Procedure For Nonlinear Static Anal

Atypical nonlinear static analysis project requires following steps:Atypical nonlinear static analysis project requires following steps:

Meshing: FE model building is a very important step in FE analysis,Meshing: FE model building is a very important step in FE analysis,The selection of the appropriate element type for certain applicationsThe selection of the appropriate element type for certain applicationsis provided with a part with surface data, which is required to beis provided with a part with surface data, which is required to bethe parts in the assembly are meshed, they are all connected togthe parts in the assembly are meshed, they are all connected togCBUSH. In general, quad and hex elements should be preferred oCBUSH. In general, quad and hex elements should be preferred oholes, and cutouts should be captured in the model appropriatelholes, and cutouts should be captured in the model appropriatelattempts should be made to create a similar mesh on both surfacesattempts should be made to create a similar mesh on both surfacesnormal to the surfaces without distorting the shell elements. Honormal to the surfaces without distorting the shell elements. Hois based on the tied contact concept. This allows the FEA user tois based on the tied contact concept. This allows the FEA user towelded. Its recommended to mesh the complex portion of a parwelded. Its recommended to mesh the complex portion of a para good quality mesh in the model. Correct representation of fastenea good quality mesh in the model. Correct representation of fastenetransfer within the structure. The stiffness and preloading shouldtransfer within the structure. The stiffness and preloading shouldIf load transfer is supposed to take place from one surface in aIf load transfer is supposed to take place from one surface in athem. Each FEA code has its own format to input contact paramethem. Each FEA code has its own format to input contact parameor elements, coefficient of friction, and an offset (gap) distance betor elements, coefficient of friction, and an offset (gap) distance bet

Apply loads and boundary conditions: No matter how good theApply loads and boundary conditions: No matter how good themodel is not constrained appropriately or the load applied is notmodel is not constrained appropriately or the load applied is notlocation sometimes puts constraints on how and where loads andlocation sometimes puts constraints on how and where loads andto re-mesh the FE model locally at the locations where the loadsto re-mesh the FE model locally at the locations where the loadsrepresented in the best possible way.represented in the best possible way.

If the FE analysis is being done to virtually validate any test done inIf the FE analysis is being done to virtually validate any test done inimportant measurements on test fixtures and loading devices. Theseimportant measurements on test fixtures and loading devices. Thesein the FE model in the same way as the part or test specimen is subjecin the FE model in the same way as the part or test specimen is subjeca loading device would help when placing the loads on certain nodesa loading device would help when placing the loads on certain nodesconstraint locations (nodes) and its degrees of freedom.constraint locations (nodes) and its degrees of freedom.

Assign material properties: Material nonlinearity is defined inAssign material properties: Material nonlinearity is defined instructure depends on the properties supplied to the FE model. Thestructure depends on the properties supplied to the FE model. Theformat of the material data card, as different software codes mformat of the material data card, as different software codes mdisplacement is measured. This information is then expressed asdisplacement is measured. This information is then expressed as

88


FEAFEAFEA

you use a nonlinear feature which you havent used. Also guessyou use a nonlinear feature which you havent used. Also guessailable studies, reports and benchmarks .ailable studies, reports and benchmarks .

its output and warnings.its output and warnings.

you think your analysis should be able to answer. Design theyou think your analysis should be able to answer. Design theconditions, in order to answer the questions you have in mind.conditions, in order to answer the questions you have in mind.

analysis done first can provide a lot of information such as where areanalysis done first can provide a lot of information such as where aremay occur, and what level of load will introduce plasticity in themay occur, and what level of load will introduce plasticity in thethat there is no need for a nonlinear analysis. Examples of such athat there is no need for a nonlinear analysis. Examples of such a

no contact, and the displacements are small.no contact, and the displacements are small.

Verification means that the model is computed correctly from theVerification means that the model is computed correctly from theo the mesh size and time stepping are common errors. Validationo the mesh size and time stepping are common errors. Validationg. the geometry, material, boundary conditions, interactions etcg. the geometry, material, boundary conditions, interactions etc

structural component, its geometry behavior with respect to largestructural component, its geometry behavior with respect to largeearlier model is unable to give you a result you expect (sometimesearlier model is unable to give you a result you expect (sometimes

commonly used elements and in this case you might look into a possibilitycommonly used elements and in this case you might look into a possibility

Analysis ProjectAnalysis Project

::

analysis, irrespective of what kind of analysis is going to be performed.analysis, irrespective of what kind of analysis is going to be performed.applications has already been discussed in Chapters 4-7. The FEA groupapplications has already been discussed in Chapters 4-7. The FEA group

be meshed with elements to get the component mesh. When allbe meshed with elements to get the component mesh. When allogether using appropriate fastening elements such as CWELD orogether using appropriate fastening elements such as CWELD orover tria, penta and tetra elements. Important features like fillets,over tria, penta and tetra elements. Important features like fillets,ely. If there are fasteners or welds between two parallel surfaces,ely. If there are fasteners or welds between two parallel surfaces,faces. This will facilitate the placement of weld or rigid elementsfaces. This will facilitate the placement of weld or rigid elementsowever, many FEA codes support node independent welds, whichowever, many FEA codes support node independent welds, which

place weld elements independent of the nodes in the parts to beplace weld elements independent of the nodes in the parts to bert first, then proceed towards the simple or plane areas to ensurert first, then proceed towards the simple or plane areas to ensure

eners, joints and welds are necessary in the model for correct loadeners, joints and welds are necessary in the model for correct loadshould be defined for these elements as applicable for better accuracy.should be defined for these elements as applicable for better accuracy.

structure to the other, a contact set should be defined betweenstructure to the other, a contact set should be defined betweeneters. A typical contact definition requires master and slave nodeseters. A typical contact definition requires master and slave nodesetween surfaces and contact algorithm.etween surfaces and contact algorithm.

the FE mesh is, the results are not going to be accurate if the FEthe FE mesh is, the results are not going to be accurate if the FErepresentative of the intended loading. The mesh size and noderepresentative of the intended loading. The mesh size and node

and boundary conditions are applied to the model. It would not hurtand boundary conditions are applied to the model. It would not hurtare applied to make sure the loads and boundary conditions areare applied to make sure the loads and boundary conditions are

in a lab, then it is a good practice to visit the testing facility and takein a lab, then it is a good practice to visit the testing facility and takeThese measurements will help apply loads and boundary conditionsThese measurements will help apply loads and boundary conditions

subjected to at the time of testing. For example, measurements ofsubjected to at the time of testing. For example, measurements ofnodes or elements. Similarly, the fixture dimensions would dictatenodes or elements. Similarly, the fixture dimensions would dictate

in the FE model in this very important step. The response of thein the FE model in this very important step. The response of theThe software manual should be referred to understand the inputThe software manual should be referred to understand the inputmay have different formats. Typically in the lab, loading versusmay have different formats. Typically in the lab, loading versus

as engineering stress and strain. If the software expects the trueas engineering stress and strain. If the software expects the true

- Nonlinear

stress-strain data then the engineering data needs to be convestress-strain data then the engineering data needs to be convemodel. Sufficient stress-strain data points should be included tomodel. Sufficient stress-strain data points should be included tomaterial suppliers to provide certified properties for the exact samematerial suppliers to provide certified properties for the exact sameproperties for commonly used materials can either be pulled fromproperties for commonly used materials can either be pulled fromcom.com.

Specify nonlinear analysis control parameters: The basicSpecify nonlinear analysis control parameters: The basicincrement, minimum and maximum increment, maximum numberand convergence criteria for iterations (acceptable residual load, fand convergence criteria for iterations (acceptable residual load, f

Run the analysis: The FE model is now ready to be run. TheRun the analysis: The FE model is now ready to be run. Theversion, memory size, and number of CPUs to better control executionversion, memory size, and number of CPUs to better control execution

Review and interpret results: It is highly recommended that theReview and interpret results: It is highly recommended that theaccuracy before making any conclusions based on the simulationsaccuracy before making any conclusions based on the simulationsof them are:of them are:

- Observe for unexpected movements in the animation.- Observe for unexpected movements in the animation.

- Compare the reaction forces against applied forces.- Compare the reaction forces against applied forces.

- Check if stresses and strains are as per material properties - Check if stresses and strains are as per material properties

- Check interacting surfaces in the contact set for any malfunction.- Check interacting surfaces in the contact set for any malfunction.

- Make quick hand calculations by simplifying the problem and - Make quick hand calculations by simplifying the problem and

While reporting the FE results, the analysts should always discloseWhile reporting the FE results, the analysts should always discloseindustrial environment, there is always pressure to complete the FEAindustrial environment, there is always pressure to complete the FEAmust check the model thoroughly and review the results carefullymust check the model thoroughly and review the results carefullyalways be verified using engineering judgment and past results withalways be verified using engineering judgment and past results withtrigger further reviewing of the model. It is highly recommended ttrigger further reviewing of the model. It is highly recommended tresults, especially when the analysis is being performed to evaluaresults, especially when the analysis is being performed to evalua

Nonlinear Analysis With OptiStruct & RADIOSSNonlinear Analysis With OptiStruct & RADIOSS

General RemarksGeneral Remarks

With the 12.0.210 release (October 2013), a renaming of our structuralWith the 12.0.210 release (October 2013), a renaming of our structuralthe growing trend of applying optimization technologies more intensithe growing trend of applying optimization technologies more intensirun analysis. Moving forward, our structural and thermal analysisrun analysis. Moving forward, our structural and thermal analysiscalled OptiStruct and will continue to leverage the widely usedcalled OptiStruct and will continue to leverage the widely usedsuch as:such as:

Linear static analysis Linear static analysis

Nonlinear implicit quasi-static analysis Nonlinear implicit quasi-static analysis

Linear buckling analysis Linear buckling analysis

Normal modes analysis Normal modes analysis

Complex eigenvalue analysis Complex eigenvalue analysis

Frequency response analysis Frequency response analysis

Random response analysis Random response analysis

Linear transient response analysis Linear transient response analysis

Geometric nonlinear explicit and implicit analysis Geometric nonlinear explicit and implicit analysis

Linear fluid-structure coupled (acoustic) analysis Linear fluid-structure coupled (acoustic) analysis

360360


erted into true stress-strain data before feeding them to the FEerted into true stress-strain data before feeding them to the FEcapture the nonlinearity of the material. Analysts should requestcapture the nonlinearity of the material. Analysts should request

same material which is going to be used to build the parts. Thesame material which is going to be used to build the parts. Theom the suppliers website or it could be available at www.matweb.om the suppliers website or it could be available at www.matweb.

basic controlling parameters for nonlinear static analysis are initialbasic controlling parameters for nonlinear static analysis are initialnumbers of iterations, the interval at which results file are to be output

force, displacement).force, displacement).

analysis run command may have options to specify the solveranalysis run command may have options to specify the solverecution.ecution.

the analysis results should be carefully reviewed and checked forthe analysis results should be carefully reviewed and checked forsimulations. There are many ways the FEA results can be checked, somesimulations. There are many ways the FEA results can be checked, some

ties supplied to the FE model.ties supplied to the FE model.

malfunction.malfunction.

and compare it with the FE results.and compare it with the FE results.

disclose all the assumptions made while building the FE model. In thedisclose all the assumptions made while building the FE model. In theFEA sooner to meet tight launch schedules. Still the FEA analystsFEA sooner to meet tight launch schedules. Still the FEA analystsbefore releasing the results to the clients. The FEA results shouldbefore releasing the results to the clients. The FEA results shouldwith similar FE models. Unrealistically high or low results shouldwith similar FE models. Unrealistically high or low results shouldto maintain consistency in the model to increase accuracy of theto maintain consistency in the model to increase accuracy of the

aluate the effect of changing certain parameters in the model.aluate the effect of changing certain parameters in the model.

RADIOSSRADIOSS

structural solver products has taken place in order to accommodatestructural solver products has taken place in order to accommodateensively in the product development process, instead of just singleensively in the product development process, instead of just single

analysis and optimization product for linear and nonlinear events will beanalysis and optimization product for linear and nonlinear events will bebulk data input format. OptiStructs includes analysis disciplinesbulk data input format. OptiStructs includes analysis disciplines

360360

- Nonlinear

Linear steady-state heat transfer analysis Linear steady-state heat transfer analysis

Coupled thermo-mechanical analysis Coupled thermo-mechanical analysis

Inertia relief analysis with static, nonlinear contact, modal fre Inertia relief analysis with static, nonlinear contact, modal fre

Component Mode Synthesis (CMS) for the generation of flexible Component Mode Synthesis (CMS) for the generation of flexible

Reduced matrix generation Reduced matrix generation

One-step (inverse) sheet metal stamping analysis One-step (inverse) sheet metal stamping analysis

Fatigue analysis Fatigue analysis

Typical problems in OptiStruct (implicit) generally consist of low-speedTypical problems in OptiStruct (implicit) generally consist of low-speedare not limited to:are not limited to:

Material Snap-through Material Snap-through

Forming springback Forming springback

Static dummy positioning Static dummy positioning

Buckling mode analyses Buckling mode analyses

Working with rubberized materials Working with rubberized materials

The design and optimization capabilities of OptiStruct allowThe design and optimization capabilities of OptiStruct allowimprovement of existing designs based on finite element analysesimprovement of existing designs based on finite element analysessize optimization. Sizing, shape and free-shape optimization are asize optimization. Sizing, shape and free-shape optimization are a

Altair RADIOSS is a leading structural analysis solver forAltair RADIOSS is a leading structural analysis solver fordifferentiated for Scalability, Quality and Robustness, and consistsdifferentiated for Scalability, Quality and Robustness, and consistssuch as composites. RADIOSS will continue to use the well suitedsuch as composites. RADIOSS will continue to use the well suited

Finite element solutions via Altair RADIOSS include:Finite element solutions via Altair RADIOSS include:

Explicit dynamic analysis Explicit dynamic analysis

Nonlinear implicit static analysis Nonlinear implicit static analysis

Transient heat transfer and thermo-mechanical coupling Transient heat transfer and thermo-mechanical coupling

Explicit Arbitrary Euler-Lagrangian (ALE) formulation Explicit Arbitrary Euler-Lagrangian (ALE) formulation

Explicit Computational Fluid Dynamics (CFD) Explicit Computational Fluid Dynamics (CFD)

Smooth Particle Hydrodynamics (SPH) Smooth Particle Hydrodynamics (SPH)

Incremental sheet metal stamping analysis with mesh adaptivity

Linear static analysis Linear static analysis

Normal modes analysis Normal modes analysis

Linear and nonlinear buckling analysis Linear and nonlinear buckling analysis

1010


frequency response, and modal transient response analysesfrequency response, and modal transient response analyses

xible bodies for multi-body dynamics analysisxible bodies for multi-body dynamics analysis

speed transient, nonlinear, or quasi-static events, including, but speed transient, nonlinear, or quasi-static events, including, but

for the development of preliminary design concepts and for thefor the development of preliminary design concepts and for theanalyses. Structural design tools include topology, topography, and free-analyses. Structural design tools include topology, topography, and free-

available for structural optimization.available for structural optimization.

highly nonlinear problems under dynamic loadings. It is highlyhighly nonlinear problems under dynamic loadings. It is highlyconsists of features for multi-physics simulation and advanced materialsconsists of features for multi-physics simulation and advanced materials

block format.block format.

adaptivity

1010

- Nonlinear

Although there are a range of problems which can be solved withAlthough there are a range of problems which can be solved withvelocity of the model components, the more suitably it can be modeledvelocity of the model components, the more suitably it can be modeledmaterial and geometric nonlinearity. The figure below generally illustramaterial and geometric nonlinearity. The figure below generally illustra

A comparison of explicit and A comparison of explicit and

Nonlinear Quasi Static Analysis (Analysis Nonlinear Quasi Static Analysis (Analysis

Quite obviously, nonlinear effects related to the following do not hQuite obviously, nonlinear effects related to the following do not h

Material (i.e. plasticity) Material (i.e. plasticity)

Geometry (i.e. large displacements, large strain, large rotation) Geometry (i.e. large displacements, large strain, large rotation)

Contact (i.e. changing boundary conditions) For instance, Contact (i.e. changing boundary conditions) For instance, nonlinearities may be present due to contact or material propenonlinearities may be present due to contact or material prope

Naturally, the user must always understand the dominating nonlinear Naturally, the user must always understand the dominating nonlinear

In case mild nonlinearities are present (i.e. nonlinear statics), matIn case mild nonlinearities are present (i.e. nonlinear statics), matdisplacements and rotations are assumed, making the geometry displacements and rotations are assumed, making the geometry

This scenario can then be investigated in the context of a NonlinearThis scenario can then be investigated in the context of a Nonlinearsolution method.solution method.

Please note that the loads stay in the undeformed coordinates and Please note that the loads stay in the undeformed coordinates and

1111


with either implicit or explicit solutions, generally, the higher thewith either implicit or explicit solutions, generally, the higher themodeled with explicit equations. The same is true for the amount ofmodeled with explicit equations. The same is true for the amount ofillustrates this concept.illustrates this concept.

and implicit solution domainsand implicit solution domains

sis Type = NLSTAT)sis Type = NLSTAT)

have to occur concurrently.have to occur concurrently.

tation)tation)

deformations and rotations may be small (infinitesimal), butdeformations and rotations may be small (infinitesimal), butoperties.operties.

nonlinear mechanism in the structure of interest.nonlinear mechanism in the structure of interest.

material plasticity and small-sliding contact may occur while smallmaterial plasticity and small-sliding contact may occur while smally linear.y linear.

Nonlinear Statics (NLSTAT) Analysis which invokes the nonlinear implicitNonlinear Statics (NLSTAT) Analysis which invokes the nonlinear implicit

and simply move along the axis they are defined in.and simply move along the axis they are defined in.

1111

- Nonlinear

NLSTAT - Stress-Dependent Material Definition (Plasticity)NLSTAT - Stress-Dependent Material Definition (Plasticity)

Implicit analysis in OptiStruct allows the user to define stress-dependentImplicit analysis in OptiStruct allows the user to define stress-dependentinformation.information.

A simply modeled plasticity region defined A simply modeled plasticity region defined

E is the slope of the uniaxial stress-strain curve in the plastic region E is the slope of the uniaxial stress-strain curve in the plastic region T

is possible to calculate H (i.e. plasticity hardening) by the equation:is possible to calculate H (i.e. plasticity hardening) by the equation:

The MATS1 card is used within a OptiStruct deck to extend the functionalityThe MATS1 card is used within a OptiStruct deck to extend the functionalitycard is associated with the proper MAT1 card by using the samecard is associated with the proper MAT1 card by using the samebelow).below).

FormatFormat

(1) (2) (3) (4) (5) (6)(1) (2) (3) (4) (5) (6)

MATS1 MID TID TYPE H YFMATS1 MID TID TYPE H YF

TYPSTRNTYPSTRN

General parameters:General parameters:

MID: ID of respective MAT1 MID: ID of respective MAT1

TYPE: Currently only PLASTIC available for NLSTAT TYPE: Currently only PLASTIC available for NLSTAT

YF: Yield function criterion = von Mises (only option right now) YF: Yield function criterion = von Mises (only option right now)

HR: Hardening rule = isotropic and kinematic HR: Hardening rule = isotropic and kinematic

Reminder: since deformation/rotation are assumed to be small, engineering Reminder: since deformation/rotation are assumed to be small, engineering

1212


dependent material settings in addition to the basic linear staticdependent material settings in addition to the basic linear static

region defined by E and Y .region defined by E and Y .T 1

region as a continuation of E past the yield point. Using E and E, itregion as a continuation of E past the yield point. Using E and E, itT

uation:uation:

functionality of MAT1 cards to allow for plasticity effects. The MATS1functionality of MAT1 cards to allow for plasticity effects. The MATS1MID for the MAT1 and MATS1 corresponding entries (see imageMID for the MAT1 and MATS1 corresponding entries (see image

(7) (8) (9) (10)(7) (8) (9) (10)

HR LIMIT1HR LIMIT1

w)w)

engineering stress/strain are approximately true stress/strain.engineering stress/strain are approximately true stress/strain.

1212

- Nonlinear

There are two ways of defining the elastoplastic material behavior:There are two ways of defining the elastoplastic material behavior:

1. One parameter model with a constant slope in the plastic region (H defined):1. One parameter model with a constant slope in the plastic region (H defined):

FormatFormat

(1) (2) (3) (4) (5) (6) (7)(1) (2) (3) (4) (5) (6) (7)

MATS1 MID TID TYPE H YF HRMATS1 MID TID TYPE H YF HR

TYPSTRNTYPSTRN

In the above card image, TID is not required, while H and LIMIT1In the above card image, TID is not required, while H and LIMIT1elastic-plastic steel which essentially is defined by its yield stress (limit) elastic-plastic steel which essentially is defined by its yield stress (limit)

The respective working steps are:The respective working steps are:

Create material collector (Card Image MAT1) Create material collector (Card Image MAT1)

Extend/activate the MATS1 option and specify the TYPE (Plastic Extend/activate the MATS1 option and specify the TYPE (Plasticslope H (=0) and insert the yield stress e.g. 230 MPa.slope H (=0) and insert the yield stress e.g. 230 MPa.

Full stress strain curve for the plastic region Full stress strain curve for the plastic region

Alternatively, if a full stress-strain relationship is desired, users canAlternatively, if a full stress-strain relationship is desired, users canwith the full curve data and associate the TID for that table to the with the full curve data and associate the TID for that table to the

In the MATS1 card image, TID, LIMIT1, and TYPSTRN are required In the MATS1 card image, TID, LIMIT1, and TYPSTRN are required

1313


vior:vior:

region (H defined):region (H defined):

(7) (8) (9) (10)(7) (8) (9) (10)

HR LIMIT1HR LIMIT1

LIMIT1 are. For instance, you may need to create the following ideallyLIMIT1 are. For instance, you may need to create the following ideallyyield stress (limit) and the slope of the plasticity hardening (H=0):yield stress (limit) and the slope of the plasticity hardening (H=0):

(Plastic is the only option usable in NLSTAT), define the work hardening(Plastic is the only option usable in NLSTAT), define the work hardening

can create a table (via load collector using the Card Image TABLES1)can create a table (via load collector using the Card Image TABLES1)the MATS1 card.the MATS1 card.

uired while H is not required.uired while H is not required.

1313

- Nonlinear

FormatFormat

(1) (2) (3) (4) (5) (6)(1) (2) (3) (4) (5) (6)

MATS1 MID TID TYPE H YFMATS1 MID TID TYPE H YF

TYPSTRNTYPSTRN

TID: ID of a TABLES1 card TID: ID of a TABLES1 card

LIMIT1: Yield stress LIMIT1: Yield stress

TYPSTRN: TYPSTRN:

0 = total strain - First point must be at the origin ( 0 = total strain - First point must be at the origin ((y2=LIMIT1) - The slope of the line joining the origin t(y2=LIMIT1) - The slope of the line joining the origin tstrain, rotation and deformation are assumed, engineeringstrain, rotation and deformation are assumed, engineering

1 = plastic strain - First point at x1=0 must have y1=LIMIT1 1 = plastic strain - First point at x1=0 must have y1=LIMIT1

All data points must be in ascending order for both options All data points must be in ascending order for both options

Definition of stress strain curve of the plastic region defined by discrDefinition of stress strain curve of the plastic region defined by discr

FormatFormat

(1) (2) (3) (4) (5)(1) (2) (3) (4) (5)

TABLES1 TIDTABLES1 TID

x1 y1 x2 y2x1 y1 x2 y2

x5 y5 ... ...x5 y5 ... ...

This table (its ID) is then referenced in the MATS1 card in column 3This table (its ID) is then referenced in the MATS1 card in column 3

Example - Definition of elastic-plastic material via curve/tExample - Definition of elastic-plastic material via curve/t

Total stress vs. plastic strain plTotal stress vs. plastic strain pl

The data from the total stress vs. plastic strain plot can be defined The data from the total stress vs. plastic strain plot can be defined

Note: If the table is defined in terms of total strain (TYPSTRN = 0Note: If the table is defined in terms of total strain (TYPSTRN = 0second point (X2, Y2) must be at the initial yield point (Y1) specifiedsecond point (X2, Y2) must be at the initial yield point (Y1) specifiedthe yield stress must be equal to the value of E. If the table is definedthe yield stress must be equal to the value of E. If the table is definedY1), corresponding to yield point (Y1), must be at X1=0.Y1), corresponding to yield point (Y1), must be at X1=0.

1414


(7) (8) (9) (10)(7) (8) (9) (10)

HR LIMIT1HR LIMIT1

( x1=0, y1=0) - Second point must be at the initial yield point( x1=0, y1=0) - Second point must be at the initial yield pointto the yield stress must be equal to the value of E. Note, as smallto the yield stress must be equal to the value of E. Note, as small

engineering stress/strain ~ true stress/strain.engineering stress/strain ~ true stress/strain.

e y1=LIMIT1e y1=LIMIT1

th optionsth options

discrete data values in TABLES1:discrete data values in TABLES1:

(6) (7) (8) (9) (10)(6) (7) (8) (9) (10)

x3 y3 x4 y4x3 y3 x4 y4

... ... ... ...... ... ... ...

column 3 (TID).column 3 (TID).

ve/tableve/table

plastic strain plotplastic strain plot

defined as a TABLES1 (load collector with card image TABLES1).defined as a TABLES1 (load collector with card image TABLES1).

0), the first point must be at the origin (X1 = 0, Y2 = 0) and the0), the first point must be at the origin (X1 = 0, Y2 = 0) and thespecified on the MATS1 entry. The slope of the line joining the origin tospecified on the MATS1 entry. The slope of the line joining the origin to

defined in terms of plastic strain (TYPSTRN = 1), the first point (X1,defined in terms of plastic strain (TYPSTRN = 1), the first point (X1,

1414

- Nonlinear

For analyses where small deformations are assumed, there shouldFor analyses where small deformations are assumed, there shouldand the engineering stress-strain curve, so either of them mayand the engineering stress-strain curve, so either of them maydeformations are not assumed, the engineering stress-strain curvdeformations are not assumed, the engineering stress-strain curv

If the deformations go past the values defined in the table, the cuIf the deformations go past the values defined in the table, the cu

The table from above (with ID3) is then referenced in the MATS1 caThe table from above (with ID3) is then referenced in the MATS1 ca

TID: select the TABLES1 collector previously created (total stressTID: select the TABLES1 collector previously created (total stress1 = plastic strain; LIMIT1: yield stress, matching the value from the 1 = plastic strain; LIMIT1: yield stress, matching the value from the

NLSTAT - Contact (small sliding)NLSTAT - Contact (small sliding)

For contacts in OptiStruct, interface definitions are created as For contacts in OptiStruct, interface definitions are created as algorithm should not exist in both groups except, in special cases algorithm should not exist in both groups except, in special cases

..

Figure: A graphic representing the operation Figure: A graphic representing the operation

Generally, master entities are treated as surfaces and the slaveGenerally, master entities are treated as surfaces and the slavepenetration against the master surface.penetration against the master surface.

The CONTACT, PCONT, and CONTPRM cards are used within OptiStructThe CONTACT, PCONT, and CONTPRM cards are used within OptiStructentities or sets.entities or sets.

The working steps regarding contact definition are:The working steps regarding contact definition are:

Specify the contact surface (i.e. define master and slave elem Specify the contact surface (i.e. define master and slave elemrespect to solid elements).

Define a contact interface (in the example below, it is named Define a contact interface (in the example below, it is named

Figure: InterFigure: Inter

Inside the same panel, select the add subpanel and define Inside the same panel, select the add subpanel and definebefore).before).

1515


should be little or no difference between the true stress-strain curveshould be little or no difference between the true stress-strain curvey be used in the TABLES1 definition. For analyses where smally be used in the TABLES1 definition. For analyses where smallve should be used.ve should be used.

curve is extrapolated linearly.curve is extrapolated linearly.

card as shown below:card as shown below:

stress-plastic strain); TYPSTRN: defines how the curve is given, here:stress-plastic strain); TYPSTRN: defines how the curve is given, here:the curve.the curve.

master-slave groups. Entities to be considered in the contactmaster-slave groups. Entities to be considered in the contactspecial cases of self-contact.special cases of self-contact.

representing the operation of contacts in OptiStructrepresenting the operation of contacts in OptiStruct

e set is reduced to nodes. These nodes are then checked fore set is reduced to nodes. These nodes are then checked for

OptiStruct to create and define the relationships between contactingOptiStruct to create and define the relationships between contacting

ments; in the example shown below, the contact is defined withments; in the example shown below, the contact is defined with

named Contact_1 and is stored as group in HyperMesh).named Contact_1 and is stored as group in HyperMesh).

rfaces panelrfaces panel

define the master and slave elements (defined as contact surfacesdefine the master and slave elements (defined as contact surfaces

1515

- Nonlinear

Click on the card image subpanel in order to view and edit the Click on the card image subpanel in order to view and edit the

FormatFormat

(1) (2) (3) (4) (5)(1) (2) (3) (4) (5)

CONTACT ID Contact Slave Set Master SCONTACT ID Contact Property for

Slave Set Master SProperty for Predefined Predefined TYPETYPE

In column 3 SLIDE, STICK and FREEZE can be referenced as predefined In column 3 SLIDE, STICK and FREEZE can be referenced as predefined

The SLIDE type creates a sliding contact between the master The SLIDE type creates a sliding contact between the master

The STICK type creates contact where the sticking property applies The STICK type creates contact where the sticking property applies

The FREEZE contact type enforces zero relative displacement The FREEZE contact type enforces zero relative displacementelements).elements).

The following example is thought to illustrate the differences: The following example is thought to illustrate the differences:

Two blocks with small gapTwo blocks with small gap

Fixed at one end Fixed at one end

Load on edge of top block only Load on edge of top block only

Contact interface between blocks Contact interface between blocks

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the interface definitionthe interface definition

(6) (7)(6) (7)

er Set Push-out Search er Set Push-out direction

Search distancedirection distance

predefined TYPEs:predefined TYPEs:

er and slave.er and slave.

applies to all closed contacts.applies to all closed contacts.

e displacements between both open and closed contacts (tied contact for solid e displacements between both open and closed contacts (tied contact for solid

1616

- Nonlinear

The left image was set up with the STICK condition and showsThe left image was set up with the STICK condition and showsdisplacements to each other at the contact interface and that thedisplacements to each other at the contact interface and that theapplied. By contrast, the right image illustrates a SLIDE condition.applied. By contrast, the right image illustrates a SLIDE condition.slave nodes are not restricted with respect to the displacement ofslave nodes are not restricted with respect to the displacement of

In both the downward loading and upward loading cases, the FREEZEIn both the downward loading and upward loading cases, the FREEZEor closed contact). In both cases, the plates act as if they are oneor closed contact). In both cases, the plates act as if they are onethe interface.the interface.

Figure: Downward loading (left) and upward loading Figure: Downward loading (left) and upward loading

Alternatively, the interface contact may be defined by referencing Alternatively, the interface contact may be defined by referencing

(1) (2) (3)(1) (2) (3)

PCONT PID Padding (account PCONT PID Padding (account shell thickness)shell thickness)

The PCONT card can be used to change the default parametersThe PCONT card can be used to change the default parametersto account for shell thickness, the gap stiffness coefficient, the coefficient to account for shell thickness, the gap stiffness coefficient, the coefficient

Note that the selection for Property Option in the lower part of the caNote that the selection for Property Option in the lower part of the ca

1717


ws that the contacting faces of the plates have identical relativews that the contacting faces of the plates have identical relativethe ends of both plates have remained coplanar as the load wasthe ends of both plates have remained coplanar as the load was. When SLIDE is activated, penetration is checked but the surface. When SLIDE is activated, penetration is checked but the surfacethe master surface, but instead move along the master surface.the master surface, but instead move along the master surface.

FREEZE condition ties the relative displacements for all nodes (openFREEZE condition ties the relative displacements for all nodes (openone part with a common enforced displacement at every point ofone part with a common enforced displacement at every point of

loading (right) with FREEZE contact, respectivelyloading (right) with FREEZE contact, respectively

erencing a property collector with card image PCONT:erencing a property collector with card image PCONT:

(4) (5) (6)(4) (5) (6)

Gap Coefficient of Coefficient ofGap stiffness

Coefficient ofstatic friction

Coefficient ofkinetic frictionstiffness static friction kinetic friction

for contact definition and includes the padding through whichfor contact definition and includes the padding through whichcoefficient of static friction, and the coefficient of kinetic friction.coefficient of static friction, and the coefficient of kinetic friction.

the card image now set to Property Id.the card image now set to Property Id.

1717

- Nonlinear

The PID references the property collecThe PID references the property collec

Solver Setup NLPARMSolver Setup NLPARM

The parameters for Nonlinear Static Analysis Control (i.e. iteration The parameters for Nonlinear Static Analysis Control (i.e. iteration created using a load collector with the card image NLPARM:created using a load collector with the card image NLPARM:

FormatFormat

(1) (2) (3) (4) (5) (6) (7)(1) (2) (3) (4) (5) (6) (7)

NLPARM ID NINC KSTEP MAXITERNLPARM ID NINC KSTEP MAXITER

EPSU EPSP EPSW MAXLSEPSU EPSP EPSW MAXLS

NINC: NINC > 0 represents the number of equal subdivisions thatNINC: NINC > 0 represents the number of equal subdivisions thatblank (default =1), the entire load for a given subcase/loadstep isblank (default =1), the entire load for a given subcase/loadstep isanalysis (ANALYSIS = NLSTAT) is full Newton. The stiffness matrixanalysis (ANALYSIS = NLSTAT) is full Newton. The stiffness matrix

MAXITER: This is the limit on the number of implicit iterations forMAXITER: This is the limit on the number of implicit iterations for(ANALYSIS = NLSTAT only). The default value is 25 (Integer > 0)(ANALYSIS = NLSTAT only). The default value is 25 (Integer > 0)

CONV: Flags to select implicit convergence criteria. Default = UPWCONV: Flags to select implicit convergence criteria. Default = UPW

U: Displacement P: Load W: WorkU: Displacement P: Load W: Work

EPSU, EPSP, EPSW: Error tolerances (see help menu NLPARM forEPSU, EPSP, EPSW: Error tolerances (see help menu NLPARM for

Finally, the NLPARM card (load collector) must be referenced in theFinally, the NLPARM card (load collector) must be referenced in the

1818


collector with Card Image PCONTcollector with Card Image PCONT

eration strategy) are defined through the NLPARM card which is eration strategy) are defined through the NLPARM card which is

(8) (9) (10)(8) (9) (10)

MAXITER CONVMAXITER CONV

LSTOLLSTOL

that the total load in a given subcase will be divided into. If NINC isthat the total load in a given subcase will be divided into. If NINC isis applied at once. The solution method for quasi-static nonlinearis applied at once. The solution method for quasi-static nonlinearis updated at each iteration.is updated at each iteration.

each load increment. If this is reached, the solution is terminatedeach load increment. If this is reached, the solution is terminated

UPW for ANALYSIS = NLSTAT UPW for ANALYSIS = NLSTAT

or details)or details)

the subcase/loadstep definition:the subcase/loadstep definition:

1818

- Nonlinear

Edit the load case to view and verify that the analysis type is NLSTEdit the load case to view and verify that the analysis type is NLST

Subcase Continuation (CNTNLSUB)Subcase Continuation (CNTNLSUB)

Subcase continuation in OptiStruct allows users to apply loadingSubcase continuation in OptiStruct allows users to apply loadingThis feature is critical for creating complex loading paths for pathThis feature is critical for creating complex loading paths for pathanalysis with friction.analysis with friction.

Subcase continuation is available through the use of CNTNLSUB andSubcase continuation is available through the use of CNTNLSUB andedit the respective subcase).edit the respective subcase).

The CNTNLSUB subcase entry can be used within subcases to continueThe CNTNLSUB subcase entry can be used within subcases to continueof the same type. Each subcase can only contain one CNTNLSUBof the same type. Each subcase can only contain one CNTNLSUB

Yes: This option continues the nonlinear solution from the Yes: This option continues the nonlinear solution from thePlease note that the order in which OptiStruct continues thePlease note that the order in which OptiStruct continues thenumbering order.numbering order.

No: This option returns to the free unloaded and stress-free model No: This option returns to the free unloaded and stress-free model

SID: When CNTNLSUB is called with a specific subcase id, the SID: When CNTNLSUB is called with a specific subcase id, theexecution type to be valid, the subcase listing and informationexecution type to be valid, the subcase listing and informationthe CNTNLSUB = entry in the deck and must be of thethe CNTNLSUB = entry in the deck and must be of the

When solving NLSTAT analysis, OptiStruct breaks the loadingWhen solving NLSTAT analysis, OptiStruct breaks the loadingapplied and solved with a full Newton method, with each iterationapplied and solved with a full Newton method, with each iterationsolver. User-defined convergence criteria are available in additionsolver. User-defined convergence criteria are available in addition

1919


TAT :TAT :

ing successively to a model that goes through different end-states.ing successively to a model that goes through different end-states.path-dependent problems such as materials with plasticity or contactpath-dependent problems such as materials with plasticity or contact

and can be activated within the loadstep/subcase definition (i.e.and can be activated within the loadstep/subcase definition (i.e.

continue nonlinear subcases from a preceding nonlinear subcasecontinue nonlinear subcases from a preceding nonlinear subcasesubcase information entry. The options for this entry are:subcase information entry. The options for this entry are:

nonlinear subcase which immediately preceds it in the deck.nonlinear subcase which immediately preceds it in the deck.the nonlinear runs is dependent upon the deck order, not on thethe nonlinear runs is dependent upon the deck order, not on the

model state and executes the solution sequence from that point.model state and executes the solution sequence from that point.

the nonlinear run continues from that subcase. In order for thisthe nonlinear run continues from that subcase. In order for thisormation for subcase number must come before the subcase withormation for subcase number must come before the subcase with

same type (NLGEOM, NLSTAT) as the current subcase.same type (NLGEOM, NLSTAT) as the current subcase.

sequence up into progressive steps. Each step is sequentiallysequence up into progressive steps. Each step is sequentiallyof the Newton method handled through a direct or iterative linearof the Newton method handled through a direct or iterative linearto OptiStructs default settings for NLSTAT runs.to OptiStructs default settings for NLSTAT runs.

1919

- Nonlinear

Nonlinear Geometry - NLGEOM (quasi-staticNonlinear Geometry - NLGEOM (quasi-staticNonlinear Geometry - NLGEOM (quasi-static

Of course, you can also run a nonlinear implicit analysis with OptiStructOf course, you can also run a nonlinear implicit analysis with OptiStructrequire large displacement and rotations, contact with friction andrequire large displacement and rotations, contact with friction andThis kind of analysis will be enabled within the loadstep (subcase)This kind of analysis will be enabled within the loadstep (subcase)nonlinear (impl. dynamic) analysis.nonlinear (impl. dynamic) analysis.

Subcase/loadstep panel >Subcase/loadstep panel >

Within this section, we will consider OptiStruct specific methods for seWithin this section, we will consider OptiStruct specific methods for se

MaterialsMaterials

There are a large number of material types available within OptiStruct There are a large number of material types available within OptiStruct

MATS1 Elastic-plastic material (see discussion before NLST MATS1 Elastic-plastic material (see discussion before NLST

MATX0 - Void MATX0 - Void

MATX02 Johnson-Cooke elastic-plastic material MATX02 Johnson-Cooke elastic-plastic material

MATX13 Rigid material MATX13 Rigid material

MATX21 - Rock-Concrete material MATX21 - Rock-Concrete material

MATX25 - Composite shell material, TSAI-WU and CRASURV formulations MATX25 - Composite shell material, TSAI-WU and CRASURV formulations

MATX27 Elastic-Plastic brittle material MATX27 Elastic-Plastic brittle material

MATX28 - Honeycomb material MATX28 - Honeycomb material

MATX33 Visco-Elastic foam material MATX33 Visco-Elastic foam material

MATX36 Piece-wise linear elastic-plastic material MATX36 Piece-wise linear elastic-plastic material

2020


atic nonlinear analysis)atic nonlinear analysis)atic nonlinear analysis)

OptiStruct including nonlinear geometry. These include models thatOptiStruct including nonlinear geometry. These include models thatand large sliding, and post-buckling and snap-through problems.and large sliding, and post-buckling and snap-through problems.

(subcase) by defining a geometry nonlinear (impl. static) or geometry(subcase) by defining a geometry nonlinear (impl. static) or geometry

panel > analysis typepanel > analysis type

or setting up and applying each of these types of problems.or setting up and applying each of these types of problems.

OptiStruct for implicit NLGEOM analysis. Of these, the main types are:OptiStruct for implicit NLGEOM analysis. Of these, the main types are:

TAT)TAT)

ormulationsormulations

2020

- Nonlinear

MATX42 Ogden-Mooney Rivlin material MATX42 Ogden-Mooney Rivlin material

MATX43 - Hill orthotropic material MATX43 - Hill orthotropic material

MATX44 Cowper-Symonds elastic-plastic material MATX44 Cowper-Symonds elastic-plastic material

MATX60 - Piece-wise nonlinear elastic-plastic material MATX60 - Piece-wise nonlinear elastic-plastic material

MATX62 Hyper-visco-elastic material MATX62 Hyper-visco-elastic material

MATX65 Tabulated Strain Rate Dependent Elastic-Plastic ma MATX65 Tabulated Strain Rate Dependent Elastic-Plastic ma

MATX68 - Honeycomb material MATX68 - Honeycomb material

MATX70 Tabulated visco-elastic foam material MATX70 Tabulated visco-elastic foam material

MATX82 Ogden material MATX82 Ogden material

The definition of MATS1 (simple elastic plastic material) was discussedThe definition of MATS1 (simple elastic plastic material) was discussedmodels are defined in the material collector (card image) by actimodels are defined in the material collector (card image) by actiunderstand the material law, such as what kind of stresses are used,understand the material law, such as what kind of stresses are used,

NLGEOM - Nonlinear MaterialNLGEOM - Nonlinear Material

In most and many cases LAW2 (=MATX02), which correspondsIn most and many cases LAW2 (=MATX02), which co

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Nl Analysis