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White Paper

an introduction to non-linearfinite element analysis

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White Paper - An Introduction to Non Linear Finite Element Analysis (1)

NIALL SMYTHNiall Smyth is a qualified mechanical engineer and has over 10 years’ experience working in petrochemical, automotive, aeronautical and defence industries.

He has specialisation certification in Simulation and the Product and Factory Design Suites. Niall’s key skills and areas of expertise include data management and using Simulation in digital prototyping to reduce development time and drive product development.

Technical Consultant, A2K Technologies



Introduction



1.

As Finite Element Analysis (FEA) within CAD environments has become increasingly commonplace in today’s engineering office, there is a greater demand to understand the types of analysis that can be conducted.

Having an understanding of non-linear analysis and when to use it helps designers simulate product performance and optimise designs.

FEA is a computational technique for verifying the structural integrity of components. It provides a greater understanding of how a product works, optimisation of current designs, and validation of ‘what if’ scenarios, which in turn lead to greater innovation and development.

Failure to account for non-linear behaviour, in large deformations and/or inelastic material behaviour, can lead to product failures, safety issues, and unnecessary cost to product manufacturers.

This white paper highlights the differences between linear and non-linear analysis from the point of view of a practical application, and can benefit designers who are looking to further their knowledge in FEA.

We look at non-linear analysis performed using Autodesk Nastran and Simulation Mechanical FEA software, and explore when to use non-linear versus linear analysis. This paper also covers the differences between these analyses, the importance of using the correct analysis type, and the risk posed from running an incorrect type of analysis.

what is linear behaviour in fea?



1.

Designers are typically introduced to linear static stress analysis during initial training on FEA software. This type of tress analysis assumes a linear relationship between stress and strain, as shown in the diagram below. From this, one can assume that if the force doubles, the displacement (and stresses) will also double in a linear analysis.

As stiffness is assumed to be constant, essentially only linear behaviour is allowed. In many real-world situations, however, this small-displacement theory may not be valid, as the stiffness of a part varies with the force applied. Consider an example of pulling an elastic band, where the more it displaces, the greater the force required to stretch it. In such situations, non-linear analysis may be required.

K

F

X

A linear analysis using Hooke’s Law which states:

Where [K] represents the stiffnessand represents displacement. { } x

[ ]{ } { }FxK =

Standard assumptions when conducting a linear analysisAll the materials in the model comply with Hooke's law: that is stress is directly proportional to strain. This means material properties of the component remain linear after the yield limit has been reached.

The deflections of components are small compared to overall component size, so that the change in the stiffness caused by loading can be ignored.

The components are rigid and ductile like metal

Boundary conditions do not vary throughout the analysis, i.e. a drop test would be considered non-linear, as the stiffness of a structure can change due to impact of the drop.

The components deform equally in all three directions, that is, the material properties are isotropic.

A typical example of non-linear behaviour is that when a paperclip is flexed, permanent deformation is achieved. This means that to predict the material’s behaviour, the FEA software needs to know what happens beyond the yield limit - unlike a linear stress analysis. An example of how a material behaves beyond its yield limit is shown below.

what is non-linear behaviour in fea?

Strength to weight optimisation.

Use of new materials that do not have a linear relationship between stress and strain before the yield strength is reached.

Addressing safety-related issues of structures thoroughly i.e. earthquake analysis.

Collapse or buckling of structures due to sudden overloading or geometric instability.

Progressive damage due to long-lasting severe loads.

Certain structures like cables or pressure vessels where membrane stresses are critical to the design.

Several common everyday applications exhibit either large deformations and/or inelastic material behaviour. Failure to account for non-linear behaviour can lead to product failures, safety issues, and unnecessary cost to product manufacturers.

The need for non-linear analysis has increased in recent years due to requirements for:

Several important engineering phenomena can only be assessed on the basis of a non-linear analysis. These include:



As can be seen, the relationship between stress and strain is non-linear after the yield

strength of the material is reached.

Yield Strength(Elastic Limit) Strain

HardeningNecking

Ultimate Strength

Failure

Change in Length

Original LengthStrain =

Stre

ss =

Forc

e

Area

sources of non-linearities



1.

There are three main sources of non-linearities: geometric non-linearities, material non-linearities, and contact.

In a non-linear static analysis, the stiffness [K] is dependent on the displacement {x}:

A large deformation is considered to be one that is greater than 1/20 of the part’s largest dimension. If a structure experiences large deformations, its changing geometric configuration can cause non-linear behaviour.

A typical example of this is of a fishing rod bending as shown below.

In this case, there is a non-linear relationship between the force applied at the tip of the fishing rod and the deflection it is undergoing.

When large deformations exist, the changing shape of a model causes non-linear changes in the component’s stiffness. It is very important to choose a non-linear solver in this case, as the change in deflection could cause different load and reaction orientations, cross sections, stresses, and/or displacements in the model.

1. Geometric non-linearities

( )[ ]{ } { }FxxK =

The resulting force versus displacement curve may be non-linear, so doubling the force does not necessarily result in doubling of the displacements and stresses, as shown below

In a design where there is large deformation, the direction of the load needs to be considered, i.e. does the direction of the load change due to the changing shape of the part? This is illustrated below.

Displacement

Force



2. Material non-linearities

A non-linear stress-strain relationship, such as metal plasticity (shown on the sample stress strain curves below), is another source of non-linearities. Plastics and soft rubber materials are further examples of where material non-linearities can be observed.



Change in Length

Original LengthStrain =

Stre

ss =

Forc

e

Area

Hard and Brittle

Hard and Strong

Hard and Tough

Soft and Tough

Differing material models allow software to correctly simulate the behaviour of a variety of material types.

As an example, below is a list of the material models available within some of the Autodesk Nastran and Simulation Mechanical FEA software .

Isotropic

Hardening

Mooney-Rivlin

Ogden

Drucker-Prager Variable Tangent

Yield

Custom stress-strain curve

Hyperelastic

Anisotropic Orthotropic

Non-linear materials

2D & 3D 2D & 3D

Non-linear elasticElasto-plastic

Plastic

Neo-Hookean

IsotropicKinematicCombined

For rubber-likematerials

For higher order rubber-like materials

For soil-likematerials

For rock-likematerials

Von MisesTresca

Mohr-CoulombDrucker-Prager

YeohGeneralised polynomialExperimental data function Simple tension/compression Equibiaxial tension Simple shear Pure shear Pure volumetric compression



Viscoelastic

Arruda-Boyce

Support for NEi Explicit materials

Duncan-Chang

Fatigue

Hydrodynamic Hyperfoam

Blatz-Ko

Failure TheoryVariable Tangent

PPFA StiffnessReduction Factors Temperature

Dependant Property Support

For creepFor fluid properties

and stresscalcuations

For hyperelasticcompressible

materials


materialsFor soil


materials

For rock-likematerials

Von Mises stressPrincipal stress

Rigid (isotropic, orthotropic 2D)Brittle

ConcreteCrushable foam

S-N dataE-N data Isotropic orthotropic

2D and 3D



3. contact

When evaluating the effects of contact, there can be a type of ‘changing status’ non-linearity; this is a change in stiffness that occurs when bodies come into or out of contact with each other.

Boundary conditions that involve components in contact with one another often produce disproportionate changes in deformation.

When boundary conditions change, the analysis would require a non-linear solver, for example, a drop test for a part or assembly. Here the stiffness matrix of the part being dropped changes abruptly due the sudden impact of the drop.



Many structures require an evaluation of their structural stability. Thin columns, compression members, and vacuum tanks are all examples of structures where stability considerations are important.

Consider an example of an empty soft drink can. If you press down on the can with your hand, not much appears to happen.

If you put the can on the floor and gradually increase the force by stepping down on it with your foot, at some point it will suddenly collapse. This sudden collapse is known as ‘buckling’.

At the onset of instability (buckling), a structure will have a very large change in displacement under essentially no change in the load, as illustrated graphically below.

F

Stable Unstable

F



Examples of Non-Linear Analysis

In the normal use of most products, buckling can be catastrophic if it occurs. The failure may not be caused by excess stress, but rather by geometric instability. An example of this is shown below.

Linear Solver Non-linear Solver

The linear analysis shows a maximum compressive stress that is below the compressive yield strength of the structure. This is the reason designers need to check for buckling strength as well as compressive strength.

The non-linear model shows that the onset of buckling makes the structure unstable, and illustrates stresses due to buckling in the model.



Once a part begins to deform due to buckling, it can no longer support even a fraction of the force initially applied, and becomes unstable.

The most concerning part about elastic buckling for designers is that it can occur at relatively low compressive stress values, compared to what the material can withstand. A separate check must be performed to see if a product or section is acceptable with respect to buckling criteria. In other words, the stresses, as determined by a linear static analysis, may mislead the engineer into believing that the structure is safe, when it is actually geometrically unstable.

Another typical example of where a non-linear analysis is required is where strain hardening of a material needs to be considered. The strain hardening modulus is the slope of the stress versus strain curve after the point of yield of a material.

Ultimate strength

Yield strength

Yield strain

Strain hardeningmodulus

Elongationat 2 in.

ε

σ



A simple analysis of a bar with a hole in it acting as a stress concentration illustrates the need for a non-linear analysis and the effect of strain hardening.

Linear Solver

Non-linear SolverThe non-linear analysis maps the behaviour of the material in a more accurate manner, due to the strain hardening modulus that is being used. The designer can see whether the ultimate tensile strength of the part has been reached, and if the stress is local to one area, or across the entire load bearing surface.

The linear solver shows a high stress singularity near the hole that is above the yield strength of the material. However, the question is: will the part fail or plastically deform?



A non-linear analysis shows a more accurate picture of how the part is behaving after the yield strength of the material is reached. It could also show how much residual stress is held in the part due to strain hardening once the load is removed.

A final example of where non-linear analysis is required in application is the design of pressure vessels where membrane stresses are generated.

The reason a designer needs to use non-linear analysis for pressure vessel design is that as the deformation of the structure increases, the membrane stiffness also increases, which changes the stiffness of the structure. A linear analysis assumes a constant stiffness, so would not take into account the changes as the structure deforms.



Conclusion



1.

As designers and engineers increasingly use FEA, the need for greater understanding of the capabilities of different methods is required. Understanding the difference between linear and non-linear FEA analysis is the key to using FEA correctly.

As shown in this white paper, the results from the analysis can be very different depending on which approach is taken.

Using FEA in design leads to a better understanding of how a product works, optimisation of current designs and valida-tion of ‘what if’ scenarios, which in turn, can lead to greater innovation and development.

References



1.

Autodesk, 2013, Autodesk® Simulation Mechanical 2014 – Part 2 – Seminar Notes, San Rafael, CA.

Autodesk, 2014, ‘Autodesk Simulation Mechanical Online Help Portal’, accessed online 16/10/2014 http://help.au-todesk.com/view/ASMECH/2014/ENU/

Bathe, Klaus-Jürgen, 2007, Finite Element Procedures for Solids and Structures, Cambridge, MA

Younis, W., 2013, Up and Running with Autodesk Inventor Professional 2014 Part 1-Stress and Frame Analysis, CreateSpace Independent Publishing Platform.

Zienkiewicz O. C., Taylor, R. L. Zhu, J.Z. 2005, The Finite Element Method: Its Basis and Fundamentals, Sixth Edition, Butterworth-Heinemann, New Delhi.

1800 223 562 (AU) or 0508 232797 (NZ)[email protected]

62 Brandl StreetEight Mile Plains QLD 4113 Australia

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