Vignan Chapter 7 Common Errors in FEA

8/18/2019 Vignan Chapter 7 Common Errors in FEA

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Common Errors in Finite Element Analysis

Idealization error Discretization error Idealization Error

•Posing the problem

•Establishing boundary conditions

•Stress-strain assumption

•Geometric simpliication

•Speciying simpliication•Speciying material beha!iour

•"oading assumptions

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error - 1


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Discretization Error

•Imposing boundary conditions

•Displacement assumption

•Poor strain appro#imation due to element distortion

•Feature representation

•$umerical integration

•%atri# ill-conditioning

•Degradation o accuracy during Gaussian elimination

•"ac& o inter-element displacement compatibility

•Slope discontinuity bet'een elements



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Idealization Error

(ne must be able to understand the physical nature o an

analysis problem 'ell enough to concei!e a proper

idealization) Engineering assumptions are al'ays

re*uired in the process o idealization)

Example:

•Establishing boundary conditions

•Speciying %aterial beha!iour



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Establishing +oundary Conditions

,here are innumerable e#amples o ho' the speciication o

improper boundary conditions can lead to either no results or

poor results)

It is impractical to re!ie' e!ery possible manner in 'hich one can

error 'hen establishing boundary conditions) ,he inite element

analyst must gain a suicient theoretical understanding o

mechanical idealization principles so that he can understand

'hat boundary conditions are applicable in any particular case)

Consider a brac&et loaded by the 'eight o a ser!o motor as

illustrated in Figure . /



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Fig 1. Actual Motor Bracket

Fig. 2 Simplifed



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Idealization Error 0 Speciying %aterial +eha!iour

Elasto-Plastic state 1Incompressible in plastic region2

Elastomeric material 1ϑ 3 4)5 Constituti!e relations undeined2

Discretization Error

Discretization is the process 'here the idealization ha!ing

an ininite number o D(F6s is replaced 'ith a model

ha!ing inite number o D(F6s) ,he ollo'ing are errors

associated 'ith discretization)



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$o o Points 3 7do per point 3 /

,otal E*uations 3 .8



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Element Errors•Imposing essential boundary conditions

•Displacement assumption

•Poor strain appro#imation due to element distortion

•Feature representation

Global Errors•$umerical integration

•%atri# ill-conditioning

•Degradation o accuracy during Gaussian elimination

•"ac& o inter-element displacement compatibility

•Slope discontinuity bet'een elements

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error -


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Element Error 0 Imposing Essential +oundary

Conditions

9igid body motion is displacement o a body in such a mannerthat no strain energy is induced) :hen one considers inite

elements in three-dimensional space 'ith nodes ha!ing three

translations D(F6s restraining rigid body motion can be morecomplicated) Consider the 7-node bric& element in three-

dimensional space depicted in Figure ;) :hat are the

minimum nodal displacement restraints re*uired to pre!entrigid body motion o this element<

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error - !


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Fig 4. -"ode Brick Eleme"t i" 3-

# SpaceIn =D space there e#ists potential or si# rigid body modes

1 = displacements 1# y z2 and = rotations ># >y and >z 2

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error -1$


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Fig 5. %e&trai"i"g '"e (ode

9estraining one node 'ill allo' the body to rigidly rotate)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error -11


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Fig 8) $odal 9estraint as a +all and Soc&et ?oint

,ranslational motion arrested but the body tend to rotate



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Fig 7. )"*i+iti"g %igid Bod, Motio"Displacement is restrained at 5 and 7) ,his pre!ents rotation

about z a#is)



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%ost inite element sot'are programs 'ill issue a 'arning

@singular system encountered or @$on-positi!e deinite

system encountered) ,he latter 'arning reers to the act

that the strain energy is not greater than zero using the

gi!en boundary conditions suggesting that the structure is

either unstable as in the case o structural collapse or not

properly restrained such that rigid body modes o

displacement are possible)



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Element Error 0 Displacement Assumption

,he h-method o inite element analysis endea!ours to

minimize this error by using lo'er order displacement

assumptions 1typically linear or *uadratic2 then reining the

model using more smaller elements) Bsing more smaller

elements to gain increased accuracy is &no'n as h-

con!ergence)



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Element Error 0 Poor Strain Appro#imation Due to

Element Distortion

Distorted elements inluence the accuracy o the inite

element appro#imation or strain) For instance some

bending elements 1beams plated shells2 compute

trans!erse shear stain) ,his type o elements may ha!e

diiculty computing shear strain 'hen the element becomes

!ery thin)



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Figure Discretization Error



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Global Errors

Global errors are those associated 'ith the assembled inite

element model) E!en i each element e#actly represents the

displacement 'ithin the boundary o a particular element

the assembled model may not represent the displacement

'ithin the entire structure due to global errors)

Global Error 0 $umerical Integration

,he use o numerical integration instead o closed-orm

integration introduces error)



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Global Error 0 %atri# Ill-Conditioning,he inite elements solution can render poor solutions or

displacement due to round o error) Ill-conditioning errors

typically maniest themsel!es during the solution phase o

an analysis)

Figure Structure 'ith an Ill-conditioned Stiness %atri#

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error -1!


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A stepped shat characterized by t'o dierent cross sections

is depicted in Figure) ,'o rod elements is used to model the

structure 'ith the e#panded e*uilibrium e*uations or

Element I gi!en as

=

−

−

3

2

1

3

2

1

1

11

000011

011

F

F

F

U

U

U

L

A E

1.2

,he e*uilibrium e*uations or Elements / are

=

−

−

3

2

1

3

2

1

2

22

110

110

000

F

F

F

U

U

U

L

A E 2

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error -2$


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Su+&tituti"g t*e k"o/" 0aria+le& i"to t*e euatio"&a+o0e t*e &ti"e&& matrice& are

−

−

=

−−

−

=

000

00100.00100.0

00100.00100.0

000

111

011

00.5

)0100.0(00.51 K

−

−=

−

−=

00.100.1000.100.10

000

110110

000

00.5

)00.5(00.12 K

3

4

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Error-21


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,he element stiness matrices are added together to

represent the global stiness

=

−

−−

−

P

F

U

U

U

00.0

00.100.100.0

00.101.10100.

00.00100.0100. 1

3

2

1

1.2

+oundary conditions are no' imposed upon the global system

o E*uations 1.2) Since B. 3 4 and F. is un&no'n the = =

system in 1.2 is replaced by a / / systems

=

−

− P U

U 00.000.100.100.101.1

3

2

1/2



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Bsing Gaussian elimination E*uations 1/2 is manipulated to

yield

P U P U

U 10100.000990.00.0

00.101.13

3

2=∴

=

− 1=2

Consider a small error in computation o &.. gi!en by

E*uation1/2

=

−

−

P U

U 00.0

00.100.1

00.102.1

3

2

4



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Bsing Gaussian elimination to sol!e the system containing

the small error

2

3

3

1.02 1.00 0.0051.0

0.00 .0196

U U P

U P

− = ∴ =

152

It may be surprising to note that a . error in one o the

entries o the stiness matri# is responsible or a 54 error

in displacement)



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Con!ergence

,o ensure monotonic con!ergence o the inite element

solution both the indi!idual elements and the assemblageo elements 1@the mesh2 must meet certain re*uirements)

%onotonic Con!ergence Bsing the Displacement +ased

h-%ethod

:ith mesh reinement the inite element solution is

e#pected to con!ergence monotonically to the e#act

solution)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras o"0er - 1


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Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras o"0er -


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9e*uirements or %onotonic Con!ergence

(i) Requirements for Each Element’s Displacement Assumption

.) 9igid +ody 9epresentation assumption must be able to

account or all rigid body displacement modes o the

element)

/) Bniorm Strain 9epresentation Constant strain states or

all strain components speciied in the constituti!e

e*uations o a particular idealization must be represented

'ithin the element as the largest dimension o the

element approaches zero)


(ii) R i t f th M h


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(ii) Requirements for the Mesh

1. Compatibility Beteen Elements! ,he dependent

!ariable1s2 and p-. deri!ati!es o the dependent !ariable

must be continuous at the nodes and across the inter-

element boundaries o adacent elements)

". Mesh Refinement! Each successi!e mesh reinement must

contain all o the pre!ious nodes and elements in their

original location)

#. $niform %train Representation! ,he mesh must be able to

represent uniorm strain 'hen boundary conditions that are

consistent 'ith a uniorm strain condition are imposed)



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Bnderstanding Con!ergence 9e*uirements

.) Con!ergence and 9igid +ody 9epresentation



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=) Con!ergence and Inter-Element Compatibility

Continuity must be maintained across element boundaries

as 'ell as at the nodes)

A) Incompatibility due to Elements $ot Properly Connected



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+) Incompatibility due to Diering (rder Displacement

Assumptions

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras o"0er -1$


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In igure that the mid-side node o Element . is connected to

corner nodes o Element / and =) Higher order elements must

be matched such that the mid-side node o one element is

connected to the mid-side node o the other)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras o"0er -12


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Incompatibility also occurs 'hen element ha!ing diering types o nodal

D(F are oined) Figure sho's a /-node beam element attached to one

node o a 7-node element 'hich ha!e translational D(F only 'hile

structural elements ha!e both translational and rotational D(F6s)

:hen oining continuum and structural elements a special constraint must

be imposed upon the structural element6s rotational D(F the least 7-node

bric& element sho'n in Figure is 'ell restrained) Its nodes do not ha!e

rotational D(F6s thereore at the node 'here the beam is attached there

e#its no D(F rom the bric& to couple 'ith the rotational D(F o the beam) As sho'n in igure the beam element 'ill e#perience rigid body rotation)



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D) Incompatibility due to incompatible elements

,pe o i"compati+ilit, to +e me"tio"ed occur& /*e"

eleme"t& i" me&* are o t*e 8i"compati+le t,pe9



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Consider the t'o elements in Figure both are identically

ormulated ; node *uadrilateral surace elements designed

'ith incompatible modes o displacement) Assume that t'o

nodes o Element / are gi!en a displacement o Jy as

depicted in igure 1a2) :ith no other displacement the

elements 'ould appear as sho'n) Ho'e!er displacement

or Element / 'ould be computed as sho'n in igure 1.72)


; Con!ergence and the Discretization Process


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;) Con!ergence and the Discretization Process

Each successi!e mesh reinement must contain all o thepre!ious nodes and elements in their original location)



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5) Con!ergence and Bniorm Strain 9epresentation 'ithin %esh

,he mesh must be able to represent a uniorm strain state

'hen suitable boundary conditions are imposed) A &atch

'est has been de!ised to test the uniorm strain condition)Either displacements or loads can be applied depending

upon 'hat characteristics o the mesh are to be in!estigated)



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Elementary +eam Plate and Shell Elements

Euler-+eronoulli +eams

+eams are slender structural members 'ith one dimension

signiicantly greater that the other t'o)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Beam - 1

Euler +eronoulli +eam Assumptions


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Euler-+eronoulli +eam Assumptions

I Euler- +ernoulli 1@elementary2 beam theory is to be used

some restrictions must be imposed to ensure suitable

accuracy) Si# items related to the Euler- +eronulli beam

bending assumptions are considered in brie

.) Slenderness 1,rans!erse shear neglected "Kh L .4

(rthogonal planes remain plane and orthogonal2

/) Straight narro' and uniorm beams 1Arch structure 0

membrane orces)2

=) Small deormations 1Second order terms and dropped

rom the Green-"agrange Strain mertic2


; +ending loads only 1"oads that cause t'ist or cause # or


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;) +ending loads only 1"oads that cause t'ist or cause #- or

y- displacement o neutral a#is are ignored)2

5) "inear isotropic homogeneous materials response

8) (nly normal stress in the #-coordinate direction 1M##2

signiicant 1M## !ary linearly in the z direction and Mzz 3 4


Nirchho Plate +ending Assumptions


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Nirchho Plate +ending Assumptions

Se!eral restrictions must be imposed 'hen using the

Nirchho plate bending ormulation i suitable accuracy is

to be obtainedO si# related items are considered briely

.) ,hinness

/) Flatness and uniormity

=) Small deormation

;) $o membrane deormation

5) "inear isotropic homogeneous materials

8) M## Myy M#y the only stress components o signiicance


. ,hin Plates


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.),hin PlatesI the ratio o the in-plane dimensions to thic&ness is greater than ten a

plate may be considered thinO in other 'ords thin plates are such that

.10// ≥≥ t bt l

(i) 'ranserse %hear %tress Does ot Affect 'ranserse

Displacement!

Although trans!erse displacement o a plat loaded normal to its neutral

surace is aected by both normal stress and trans!erse shear stress it is

assumed that the shear stress does not ha!e a signiicant impact on the

magnitude o trans!erse displacement) It can be sho'n that the magnitude

o trans!erse displacement due to shear stress is small in thin plates)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Beam -5


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(ii) *rtho+onal &lanes Remain &lane an, *rtho+onal!

In a thin plate any cross sectional plane originally orthogonal

to the neutral surace is assumed to remain plane and

orthogonal 'hen the plate is loaded) As 'ith deep beams

cross sections in thic& plates originally orthogonal and planar

'ill be neither under trans!erse load due to shearing eects)



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=) Small Deormation

.) ,he e#pression or stain actually contains second order

terms and these terms are truncated i the strains are

presumed small)

/) "i&e the Euler-+ernoulli beam plate cur!ature can be

e#pressed in terms o second order deri!ati!es i the

s*uare o the slop is negligible)

=) As a plate delects its trans!erse stiness changes)

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Beam -

:hy does the trans!erse stiness change 'ith delection< As


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a plate deorms into a cur!ed 1or a doubly cur!ed2 surace

trans!erse loads are resisted by both bending and membrane

deormation) ,he addition o membrane deormation aects the

trans!erse stiness o the plate) Consider igure 'here a ball is

placed on a !ery thin @plate)

$otice that as the amount o plate delection increases more

load is carried by membrane tension and less by bending)

Shell theory is re*uired to account or structure that carry loads

through simultaneous bending and membrane deormation

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Beam - !

;) $o %embrane Deormation


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Nirchho plates do not account or membrane deormation)

,his means that at the neutral surace there can be no

displacement in the #- or y-coordinate directions 'hich

'ould suggest the e#istence o membrane deormation

5) Signiication Stress Components are M##

Myy

M#y

As in shallo' beams it is assumed that trans!erse shear

stress in thin plates is relati!ely insigniicant 'hen compared

to the normal stress in the longitudinal direction o the beam

hence

Myz

3 Mz#

3 4 and Mzz

3 4

Prof .N. Siva Prasad,Prof .N. Siva Prasad, Indian Institute of Technology MadrasIndian Institute of Technology Madras Beam -1$

Shear Stress Aects the Edge 9eaction Forces


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Shear Stress Aects the Edge 9eaction ForcesIn plate bending stress aects trans!erse displacement) In

addition shear stress aects reaction orces on restrained

edges o plates and shells)

,he restrained edges o shell and plate structures de!elop

reaction orces) ,he magnitude o the reaction is dependent

upon the beha!iour in the !icinity !ery near the restrained

edge) ,his boundary layer eect cannot be captured 'ithout

accounting or trans!erse shear stress) In addition e!en in

models that do account or shear eects the boundary layer

eect is diicult to model 'ithout special care)

Documents

Vignan Chapter 7 Common Errors in FEA