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July 2000Page4 BENCHmark

How to Model & Interpret Results

Part 1: An Illustration of Idealisation & Modelling Strategies

Editors IntroductionThe text of this article has been put

together by a series of selective

extracts from two books written for

NAFEMS by D Baguley & D R Hose.

The first deals with How To Model

with Finite Elements and the second

How to Interpret Finite Element

Results. The intention is, after setting

the scene, particularly for the

relatively new analyst, to follow an

example using the methodology

advocated by the authors. In the nextissue of BENCHmark we will then

look at the selection of element types,

consideration of mesh density and

other checks. The final article will look

at the interpretation of the results.

Hopefully interested readers will find

the time to prepare similar models and

study them ahead of the subsequent

issues to maximise the benefit from

the series of articles.

IdealisationThere is a strong tendency for newusers of FE to focus on the generation

of an accurate geometric model of a

structure to be analysed, and to see

this as the substance of FE modelling.

The facilities to import geometry from

CAD packages might reinforce the

tendency to concentrate on geometric

modelling.

The purpose of a finite element analysis

is to model the behaviour of a structure

under a system of loads, not just itsgeometry. How much the behaviour is

influenced by the geometric details

varies greatly from case to case.

Although computer resources now

permit large complex models to be

processed, in most analyses there

remains a requirement to simplify the

problem to be solved, and most often

the greatest economies can be made

by idealising the geometry.

The degree of simplification admissible

depends on the accuracy required, and

the degree of simplification required

depends on the resources available(man-time, elapsed time, computer

time, computer capacity, and software

capabilities). If there is not some area

in which the two overlap there is no

point in proceeding with an analysis.

The booklet How to plan a Finite

Element Analysis deals with the

setting of realistic targets for accuracy

and the allocation of resources.

Idealisation is not peculiar to FE. When

classical methods or handbookformulae are used for the calculation

of stresses, it is rare that the structure,

its supports and loading will be

identical to one for which a solution is

available. The major advantage of the

FE method is that, for complex

structures, less idealisation is

required.

It can be argued that the fewer the

simplifications made, the easier the

analysis will be for the inexperienced

user, since it is necessary to makefewer decisions based on theoretical

knowledge and practical experience.

This may be another explanation for

the novices affinity for geometrically

accurate three-dimensional models. It

will not take long to realise that if these

decisions are not made, the areas of

simplification achieved and

simplification required will very rarely

overlap.

EmpathyThe analyst will find many of the

decisions necessary in preparing a

finite element model simpler if he

develops empathy with the structure

to be analysed. The ability to do this

will depend on the analysts experience

of structural analysis and knowledge

of the particular structure. The latter

may be gained from many sources

such as hand calculations, previous

analyses, physical tests and in-service

failures.

Let us take as an example a

simplification to the geometry. Quite

irrespective of the method of solution,the analyst can compare the expected

behaviour of the structure in question

with an imaginary structure

corresponding to the FE idealisation.

Although the stresses in either may

not be predicted accurately, it may be

possible to assess within a few percent

the difference in stress between the

real and the idealised structure. If not,

a more important judgement can be

made about whether the idealisation

increases or decreases the stress. Ifthe expected difference is acceptable

within the accuracy requirements, then

the simplification is acceptable. If it is

not possible to make this judgement,

the simplification should not be made

without further steps to assess its

effect. The use of sensitivity tests to

assess the validity of assumptions is

described in the booklet How to plan

a Finite Element Analysis.

If the idealised structure is analysed

by the FE method, some differencesfrom the real structure will arise from

the assumptions embedded in the

technique. Comparison of the expected

behaviour and the real structure

therefore requires some empathy with

the FE model, based on an

understanding of the theory and

practical experience. Although the

theory can be learned in advance, it is

not possible to be instantly

experienced. Fortunately it is possible

to mitigate a lack of experience bycareful appraisal of FE results, since

these give great insight into the

behaviour of the model. They may

show that the idealisation was

unsatisfactory, and indicate that an

iteration through the analysis process

is necessary. Experience will increase

the chances of obtaining satisfactory

results first time, but will not remove

the requirement for qualifying the model

by examination of the results, as

described in the booklet How to

Interpret Finite Element Results.

Education & Training


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Balancing ApproximationsThe behaviour of a structure is

influenced by many factors including

the geometry, the distribution of the

loads, the rate at which loads areapplied, interaction with adjoining

structures, the material and the

temperature. In simulating the

behaviour of the structure, it may be

necessary to simplify all of these to

some ideal form. The overall accuracy

of the results depends on all of the

approximations made, and it is

necessary to balance the extent to

which each is idealised.

Stages of Model CreationThere are several major stages in the

creation of a finite element model and

for most types these are:

a) Selection of Analysis Type

b) Ideal isat ion of Mater ia l

Properties

c) Creat ion of the Model

Geometry

d) Application of Supports orConstraints

e) Application of Loads

f) Solution Optimisation

Chronologically the preparation of data

usually follows the order shown above,

and these stages are addressed in this

order in the example chosen for this

article. Generally the data forcto e

can only be created if the data for the

previous stage exists. The material

data for a model consisting of a single

isotropic material is not necessarily

required for stage c, but the decisionto idealise it as such should have been

made.

The data for stage c is also greatly

influenced by the requirements for the

stages following, as well as those

preceding. As well as fitting or creating

the geometry, the definition of the

element mesh must allow for non-

geometric features, such as changes

in material, directions of orthotropic

material properties, and the location

of supports. Because the data is sostrongly interrelated, it is necessary

to consider all the stages before any

data is prepared. This may appear

obvious, but experienced practitioners

have often discovered at stage e that

the mesh created incprevents efficient

application of the loads.

Modelling Summary

s Develop a feel for thebehaviour of the structure.

s Assess the sensitivity of the

results to approximations of

the various types of data.

s Develop an overall strategy

for the creation of the model

s Compare the expected

behaviour of the idealised

structure with the expected

behaviour of the real structure.

Axi-symmetric Example

This example demonstrates the use

of an axi-symmetric model to

investigate possible design changes to

remedy the failure of a bolted flange

joint in a cylindrical shell. Figure 1

shows a cross-section through one of

a pair of identical flanges at a bolt hole.

There are sixteen equi-spaced bolts

around the circumference. Testing of

a prototype has resulted in failure by

rupture through the section where the

fillet blends into the outer diameter of

the adjoining cylinder.

The stages of idealisation and model

creation are dealt with in the order on

the check list as described in How to

Plan a Finite Element Analysis and

shown in Figure 4 . Those that are

irrelevant to this example have been

omitted. Some knowledge of the

behaviour of bolted flange joints is

useful. In this case there is metal to

metal contact at the flange faces.Pressure in the cylinder causes a

tensile force across the joint equal to

the piston force (pressure times bore

area). The offset of the bolt forces from

the tensile force in the cylinder wall

results in a moment on the flange

causing it to rotate as shown in figure

3. If there is a compressive force

between the flange faces near the outer

circumference as shown in figure 2, it

appears from a purely two dimensional

viewpoint that it is possible to obtain

static balance. The forces as shownresult in radial bending of the flanges

so that they separate at the inner

circumference. As the pressure force

increases, the gap between the flanges

extends outwards. If the bolts were

sufficiently flexible compared to the

flanges, it would be possible to

increase the pressure until the faces

were no longer in contact (assuming

the seal remained effective). In this

condition it is not possible to obtain

moment balance without consideringthe hoop stresses in the flange and

adjoining cylinder.

Reference to Roark & Young (Article

10.9) shows that the rotation of a ring

subjected to a distributed overturning

moment is inversely proportional to the

second moment of area about the

Figure 1 Section through Bolted Flange



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July 2000Page6 BENCHmark

radial neutral axis, as shown in figure

3. The hoop stresses at the face AA

are compressive because the rotation

causes the radius to decrease.

Conversely the hoop stresses on faceBB are tensile. The magnitude of the

hoop stress at any location is

proportional to the distance from the

neutral axis. The calculation of hoop

stress is directly analogous to the

calculation of fibre stresses in a beam

using simple bending theory. In reality

it is unlikely that the flanges will

separate completely and the

overturning moment will be resisted by

a combination of radial bending stress

and hoop stress.

Tightening of the bolts during assembly

of the joint results in an initial tensile

strain in the bolts and a compressive

pressure between the mating faces of

the flanges. Hence the effects of the

pressure loading are superposed on

this set of balanced initial forces. As

the pressure is increased the interface

pressure decreases and hence the

increase in bolt tension is less than

the applied pressure force.

Consequently the bolt forces are notstatically determinate.

Analysis TypeIn this case the material has low

ductility and nonlinear material

behaviour is believed to be

unimportant. Unless it is known

beforehand that nonlinear behaviour is

important, it is usually prudent to carry

out a linear analysis first. Even if the

material were ductile it might be

possible to use the results from a linear

analysis as described in the booklet

How to interpret Finite Element

Results. Large deflections are unlikely

to be significant since gross distortion

would result in leaking, and the

deflections will not make any

significant difference to the way in

which loads are applied. Since the bolt

tightening and the pressure force may

result in separation of the flanges to

an unknown extent, it is necessary to

adopt an iterative procedure to

determine how much of the faces are

in contact. Gap elements could beemployed to automate the iterative

process as described in section 12,

Figure 4 Finite element Analysis Input Check List

Figure 2 Flange Forces Figure 3 Neutral Axis of Ring


FINITE ELEMENT ANALYSIS INPUT CHECK LIST

job reference job number section index sheet number approved by date

analysis title:

file name/date/time:

C OMME NT S S EC TI ON A UT HO R C HE CK ER

analysis type

units

extent of model

material data

co-ordinate systems

major dimensions

element types & options

real constants

mesh density

element plots

element shapes

internal edges

elements missing

elements duplicated

consistent normals

constraint equations

symmetry constraints

supports

rbm & mechanisms

load cases

summed mass

master freedoms

front/band width

output options


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July 2000BENCHmark Page 7

but in this case a manual process is

used to demonstrate the method.

There is no rapid variation of the loads

with time. Hence for each iteration the

analysis type is static linear elastic.

UnitsDimensions on the drawing are mm

and the stresses are required in MPa

(N/mm2). Hence the modulus of

elasticity and pressures are in MPa,

and forces are in N.

Extent of ModelSince the geometry and loading on

each side of the joint are identical, the

use of mirror symmetry at the plane of

the flange interface is an obvious way

of halving the size of the model. Mirror

symmetry about radial planes through

a bolt and midway between two bolts

could be used to reduce the model to

one thirty-second of the circumference.

However, the bolts are quite closely

spaced, and except for locations close

to bolts, the effect of smearing the bolts

and holes around the circumference is

expected to be acceptable, particularly

for a parametric design study. This

smearing permits a 2D axi-symmetricmodel to be used. Compared with a

3D solid model, even one reduced to a

thirty-second of the circumference, the

assumption of axi-symmetry results in

great savings in man-time for model

preparation and results processing,

and machine time for solution. The

latter is more significant in this case

than usual because an iterative

procedure is to be used.

The rate at which the effects of an axi-symmetric shear or moment at the end

of a thin cylinder die away along the

length depends on the thickness and

the radius, and to a small degree on

the Poissons ratio for the material.

Formulae for the decay use the

parameter where

At a distance of 6/ the effects are

negligible and cylinders with this lengthare considered long. Thus the cylinder

can be discontinued at this distance

from the joint without any significant

effect on the stresses at the joint.

Since the failure was not initiated at

the seal groove, it is assumed that the

stresses in that region are small. Theresults can be examined to confirm the

validity of the assumption. The groove

makes little difference to the overall

stiffness of the structure and it is

neglected in the model.

Material DataFor most of the structure the material

data is simple and consists of the

modulus of elasticity and Poissons

ratio for the flange and cylinder, and

just the modulus of elasticity for the

bolts. The modulus of elasticity isadjusted for the elements in the flange

corresponding to the location of the bolt

hole. Since the stresses in this region

are not of particular interest, the main

objective is to make an adjustment to

reflect the change in stiffness.

Obtaining an accurate value for the

effective stiffness would require further

analysis. In this case the elastic

modulus is multiplied by the ligament

efficiency at the bolt PCD. Tables of

equivalent properties for perforatedplates given in references 3 and 4

indicate that this gives a fairly good

approximation, erring slightly on the

flexible side. A qualitative assessment

of how the stresses in the region of

failure leads to the belief that it is

conservative for the flexibility to err on

the high side. The effect on the hoop

stiffness is believed to be similar, and

assuming this avoids the use of

orthotropic material properties. As the

area over which the modified modulus

is used is relatively small, imprecision

in the value will have only a small

effect on the effective stiffness of the

flange in resisting overturning by hoop

stresses.

Co-ordinate SystemsIn common with many others, the

analysis program used requires a

particular orientation of the global axes

for axi-symmetric analyses. In this

case the X axis is radial and the Y axis

is along the axis of symmetry. Forconvenience Y=O is chosen to be the

face of the flange.

Major DimensionsThe inside diameter is fixed. Other

dimensions are to be varied to

investigate the effect on the stresses.

The possible variations considered are:

s increase the shell thickness

local to the flange

s increase the thickness of the

flange

s increase the outside diameter

of the flange

s change the fillet radius

sincrease the number of bolts

Some of the dimensions are

interrelated. For example there is a

minimum distance between the fillet

and the bolt hole required to clear the

washer under the bolt head. This

means that if the shell thickness is

increased while the outside diameter

of the flange is retained, it may be

necessary to reduce the fillet radius.

Many programs permit models to be

defined in parametric form so that the

required changes to geometry can be

made by modifying just a few data

items. A similar effect can be achieved

in associative solid modellers by

moving key points.

It is important that the model is created

in such a way that there is always a

smooth blend at the ends of the fillet,

as an artificial sharp corner will result

in spurious stresses at an important

location.

It is also necessary to make some

provision for the application of the

pressure loading on the flange face by

locating a key point or node at the seal

radius.

The dimensions of the model can be

checked by sampling the co-ordinates

of a few key points, either printed in

the output from a data check or

extracted from the model database. For

example, the co-ordinates of the top

outer corner of the flange will indicatethe correctness of the flange thickness

and outside radius.

(To be Continued .......)


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