A04 Introduction to FEM

8/12/2019 A04 Introduction to FEM

1/75

A04 - Introduction to the Finite Element Method 1

Introduction to the Finite Element Method

Background Material AERO 306 notes andIntroduction to Aerospace Structural Analysis,Allen and Haisler

http!!ceaspu"#eas#asu#edu!structures!FiniteElementAnal$sis#htm http!!%%%#m$"&o#net!m$"&ous!Anal$sis!Features!10'4(#htm

http!!%%%#m$"&o#net!m$"&ous!Anal$sis!)ools!*rocess!1040(#htm http!!larcpu"s#larc#nasa#+o!randt!13!Rand)!.ection/!/11#htmlAssumptionsIt is assumedthat $ou are amiliar %ith "asic FEM theor$ AERO 3062

and %ith applications to truss or "eam elements and structures, and

no% FEM theor$ "ased on an ener+$ or ariational ormulation, no% %hat a stiness matri is, no% ho% to assem"le element stiness and orce matrices into

+lo"al structural2 stiness and orce matrices,

no% ho% to sole the resultin+ e5uili"rium e5uations 78 9 8 9K q Q= or displacements, and


2/75

A04 - Introduction to the Finite Element Method &

no% ho% to determine resultin+ strains and stresses#


3/75


.tructures are oten anal$:ed usin+ comple inite element anal$sis

methods# )hese tools hae eoled oer the past decades since

earl$ 160;s2 to "e the "asis o most structural desi+n tass# Acandidate structure is anal$:ed su"


4/75


Moral of the story as presented by the chart above= >on;t "eliee that


5/75

A04 - Introduction to the Finite Element Method '

Aircrat hae man$ main structural components in the %in+s, usela+e,

tail section, landin+ +ear, etc# as sho%n "elo%


6/75


.tructural mem"ers sho%n a"oe ma$ "e ariousl$ modeled as

"eams, thin plates, mem"ranes, shells, etc# In some cases, it ma$

"e necessar$ to perorm a ull 3-> stress anal$sis an elasticit$

t$pe anal$sis as opposed to an approimate one lie plate theor$2#


7/75

A04 - Introduction to the Finite Element Method (

)he websin %in+ ri"s and loor "eams, and %in+ and usela+e

skinsare t$picall$ thin mem"ers that mi+ht "e considered as "ein+

in a state oplane stress#

*lane stress descri"es a three-dimensional +eometr$ %herein the

non-:ero stresses all occur in a sin+le plane# For a thin plate, %e"

or sin l$in+ in the -$ plane, the onl$ non-:ero stress components

are , ,xx yy xy #


8/75

A04 - Introduction to the Finite Element Method ?

@e no% consider the deelopment o a plane stress inite element#

.uppose %e hae a +eometr$ lie that sho%n "elo% %here the

thicness is small compared to the other t%o dimensions

)he onl$ ma


9/75

A04 - Introduction to the Finite Element Method

Cracet /eometr$ D FEM Mesh, oadin+ and C

t = 0.5 in.


10/75


otice that the +eometr$ has "een diided up into a num"er o

rectan+ular re+ions elements2 - these are called Guad elements#

@e could also use trian+ular elements# @e %ill demonstrate the

deelopment o the stiness matri and load ector or a trian+ular

element as sho%n

"elo%# @e assume that

plane stress occurs in the

-$ plane and deine

displaceent

coponents , 2u x y and , 2v x y # @e deine

nodal displaceentsat

the three corners as

ode 1 1u and 1v

ode & &u and &v

ode 3 3u and 3v


11/75


)he process o deelopin+ the equilibriu equationsor a +ien

element re5uires that %e utili:e an ener+$ or ariational principle

or eample, the principle o minimum potential ener+$ +ien "$

2 0! " + =

%here is the internal strain ener+$ and J is the eternal potential

ener+$#

For a plane stress state, the internal potential2 ener+$ is +ien "$

1&

8 9 8 9#"

! d" =

%here

8 9 8 9

xx xx

yy yy

xy xy

and

= =


12/75

A04 - Introduction to the Finite Element Method 1&

Assumin+ a linear elastic material, the constitutie e5uation ma$

"e %ritten as8 9 78 9$ =

%here >7 or plane stress is +ien "$

&

1 0

7 1 010 0 1 2 ! &

%

$

= ote that >7 is s$mmetric#

)he ininitesimal strains are +ien

, ,xx yy xyu v u v

x y y x

= = = +

%here , 2u x y and , 2v x y are the displacement ields#


13/75


.u"stitutin+ into +ies the strain ener+$ as

1

& 8 9 78 9

#

"! $ d" = %here use has "een made o the s$mmetr$ o >7# ote that i %e

su"stitute into , is no% in terms o the displacement ields , 2u x y and , 2v x y #

)he eternal potential J can "e ealuated once the eternal

tractions and "od$ orces are speciied# In +eneral, J %ill hae the

orm o

2 2x y x yS "" p u p v ds & u & v d" = + +

%here xp and yp are "oundar$ tractions, . is the element

"oundar$ surace, x& and y& are "od$ orces# ote that J is in

terms o displacements , 2u x y and , 2v x y #


14/75


15/75

A04 - Introduction to the Finite Element Method 1'

Assume the corners o

the trian+le nodes2 are

num"ered @, and

hae coordinates

1 1 , 2x y , etc# as sho%n#

At each node iK1,&,32,

assume the nodal

displacements are +ien

"$ , 2i iu v # @e can no%

%rite 6 B"oundar$

conditionsB as ollo%s

For u,$2

At node 1 1 1 1 1 & 1 3 1 , 2u u x y x y = = + +At node & & & & 1 & & 3 & , 2u u x y x y = = + +At node 3 3 3 3 1 & 3 3 3 , 2u u x y x y = = + +


16/75


For ,$2

At node 1 1 1 1 1 & 1 3 1 , 2v v x y x y = = + +

At node & & & & 1 & & 3 & , 2v v x y x y = = + +At node 3 3 3 3 1 & 3 3 3 , 2v v x y x y = = + +

@e can no% sole or the constants in terms o nodal

displacements# E5s# can "e %ritten in matri orm as

1 1 1 1

& & & &

3 3 3 3

1

1

1

x y u

x y u

x y u

=

.olution is1 1 1 & & 3 3

& 1 1 & & 3 3

3 1 1 & & 3 3

2 !& 2

2 !& 2

2 !& 2

a u a u a u A

b u b u b u A

c u c u c u A

= + += + += + +


17/75

A04 - Introduction to the Finite Element Method 1(

%here

1 & 3 3 & & 3 1 1 3 3 1 & & 1

1 & 3 & 3 1 3 1 &

1 3 & & 1 3 3 & 1

, ,

, ,, c , c

a x y x y a x y x y a x y x y

b y y b y y b y yc x x x x x x

= = =

= = = = = =

and

1 1

& &

3 3

1

& 1 & 2

1

x y

A x y area of trian'le

x y= =

.u"stitutin+ into and rearran+in+, u,$2 can "e %ritten

1 1 1 1 & & & &

3 3 3 3

1 , 2 2 2

&

2 7

u x y a b x c y u a b x c y uA

a b x c y u

= + + + + ++ + +

ote that the a;s, ";s and c;s are constants and depend onl$ upon the

nodal coordinates ,$2 o the 3 corner nodes#


18/75

A04 - Introduction to the Finite Element Method 1?

>einin+ the coeicients o iu as iN , e5uation "ecomes3

1

, 2 i ii

u x y N u

=

=

%here1

, 2 2&

i i i iN x y a b x c yA

= + +

A similar result is o"tained or ,$2

3

1

, 2 i ii

v x y N v=

=

)he 5uantities , 2iN x y are calledshape functions# ote that the

same shape unctions appl$ or "oth , 2u x y and , 2v x y #

@e can no% o"tain the strains "$ su"stitutin+ displacement

unctions and into strain epressions to o"tain


19/75


3 3

1 1

3 3

1 1

3 3 3 3

1 1 1 1

&

&

& &

i ixx i i

i i

i iyy i i

i i

i i i ixy i i i i

i i i i

N buu u

x x A

N cvv v

y y A

N N c bu vu v u v

y x y x A A

= =

= =

= = = =

= = =

= = =

= + = + = +

)he last 3 e5uations or strains can "e put into matri notation as

1

1

1 & 3&

1 & 3&

1 1 & & 3 33

3

0 0 010 0 0

&

xx

yy

xy

u

v

b b b uc c c

vAc b c b c b

u

v

=


20/75

A04 - Introduction to the Finite Element Method &0

Or, more compactl$ as or an$ element BeB2

8 9 78 9e e e( q =

%here

1 & 3

1 & 3

1 1 & & 3 3

0 0 01

7 0 0 0&

e

b b b

( c c cA

c b c b c b

=

and

1

1

&

&

3

3

8 9e

u

v

uq

v

u

v

=

.ince the terms in 7e( are constant or an element, the strains

8 9e are constant %ithin an element hence the name )constant

strain trian'le) or *S##


21/75


@e can no% ealuate the internal strain ener+$ # .u"stitutin+

into +ies

1&

1&

8 9 7 7 78 9

K 8 9 7 7 7 8 9

e e # e # e e e

"

e # e # e e e

"

! q ( $ ( q d"

q ( $ ( d" q

=

)he 5uantit$ in parentheses can "e identiied as the eleentstiffness atrix 7ek and can "e %ritten as

1&

8 9 78 9e e # e e! q k q=

%here the eleent stiffness atrix 7ek is deined "$

7 7 7 7e e # e e"

k ( $ ( d" =


22/75

A04 - Introduction to the Finite Element Method &&

I the element has a constant thicness te, then dJKtdA# Assumin+

that E is constant oer the element and notin+ that the terms in C

are constants, then 7 7 7 7e e e e # e ek t A ( $ (=

ote that the element stiness matri 7ek is a 66 matri, i#e#, %e

hae a 6 de+ree-o-reedom do2 element#

ote that the +eneral orm or the strain ener+$ can "e%rittenininde notation also

6 61 1

& &1 18 9 78 9

e e # e e e e e

i+ i +i +! q k q k q q= == =

Cecause 7e$ is s$mmetric, the stiness matri 7ek deined "$

either or is a s$mmetric matri al%a$s the case2#


23/75


24/75


)o deine the eternal potential

ener+$ J, %e hae to deine the

eternal load# .uppose %e hae a

uniorm traction pressure2 p applied

on the element ed+e deined "$

nodes 1 and )he eternal

potential then "ecomes

0 2 cos 2 sin 7

,e" u s p v s p tds = +ote that cosp is the component o p in the direction#>isplacements u and on "oundar$ 1-& must "e %ritten asunctions o position s on the "oundar$

1 & 1 &

1 & 1 &

2 1 ! 2 ! 2

2 1 ! 2 ! 2

u s s , u s , u

v s s , v s , v

= +

= +


25/75

A04 - Introduction to the Finite Element Method &'

.u"stitutin+ us2 and s2 into J, and inte+ratin+ oer the

"oundar$, +ies

( ) ( ) ( ) ( )1 1 1 11 1 & && & & &cos sin cos sin" pt, u pt, v pt, u pt, v = + + + )he last result can "e %ritten in matri notation as

6

1

8 9 8 9e e # e e ei ii

" q & & q=

= =

%here

1&

1&

1

&1&

cos

sin

cos8 9

sin

0

0

e

e

ee

e

pt ,

pt ,

pt ,&

pt ,

=


26/75


)he matri 8F9 represents the equivalent 'enerali-ed nodal force

vectordue to pressure load on "oundar$ 1-&, i#e#, %e hae replaced

the pressure p on "oundar$ 1-& "$ the nodal orces 8F9 at nodes 1

and

ote that the total orce due to p

on "oundar$ 1-& is pt2 and diides e5uall$ "et%een nodes 1 and

=


27/75

A04 - Introduction to the Finite Element Method &(

Another set o orces eists on the "oundar$ o an$ element#

)hese are due to surroundin+ elements that appl$ orces due to

contact %ith the element in 5uestion, i#e#, surroundin+ elements are

"ein+ deormed and hence the$ tr$ to deorm the element in

5uestion and there"$ put orces on this element# Additionall$,

%here a node is at a support or Bied,B there %ill "e a reaction

orce on the element node# all these reaction orces 8 9S #

1S1

&

3

&S3S

4S

'S6

S

iS reactions fro ad+acent eleents=

)he et# pot# ener+$ due to reactions is 8 9 8 9e e # e" q S=


28/75

A04 - Introduction to the Finite Element Method &?

@e can determine the e5uations o e5uili"rium or the element#

sin+ and notin+ that e! and e" are unctions o nodal

displacements , 1,###,6

e

iq i= , %e hae6

1

2 2 0

e ee e e

iei i

! "! " q

q

=

++ = =

.ince 0iq , then 2

0 1,&,###,6e e

ei

! "for i

q

+ = =

.u"stitutin+ and J and 7 into +ies the equilibriu equation

for any eleent#

78 9 8 9 8 9e e e ek q & S = +

ote that 7eK is 662 and 8 9e& D 8 9eS are 612 matrices#


29/75

A04 - Introduction to the Finite Element Method &

E5uations - proide the e5uili"rium e5uation or a sin+le element#

.uppose %e loo at a collection o elements i#e#, a complete

structure2# )hen the total ener+$ o the structure is +ien "$ the

sum o internal and potential ener+$ o all the elements elN 2

( )6 6

1 1& &

1 1 1 1 1

8 9 78 9el el el N N N

e e # e e e e estr i+ i +

e e e i +

! ! q k q k q q= = = = =

= = =

and

( ) ( )1 1 1

6 6

1 1 1 1

8 9 8 9 8 9 8 9

el el el

el el

N N Ne e # e e # e

str

e e e

N Ne e e e

i i i ie i e i

" " q & q S

& q S q

= = =

= = = =

= =

=

)he principle o minimum potential ener+$ or the structure

re5uires that


30/75


1

2 2 0

Mstr str

str str i

i i

! "! " q

q

=

++ = =

%here 8 9q contains the M de+rees o reedom or the structure

O) do or each element2# For 0iq , the last e5uationre5uires that

20 1,&,###,str str

i

! "for i M

q

+= =

.u"stitutin+ str! and str" into +ies

( ) ( ) ( )1&1 1 1

8 9 78 9 8 9 8 9 8 9 8 9

0

el el el N N Ne # e e e # e e # e

e e e

i

q k q q & q S

q= = =

=

1,&,###,for i M=

*ro"lem )he ener+$ terms or each element are in terms o the

element do, "ut in order to o"tain the e5uations o e5uili"rium or


31/75


the structure a"oe e5uation2, %e hae to tae the partial

deriaties %ith respect to the +lo"al structural do# In order to

complete the a"oe, the element de+rees o reedom8 9

e

qmust "e

%ritten in terms o the M +lo"al structural de+rees o reedom 8 9q #

For an$ element, %e can %rite a transormation "et%een element

local and +lo"al do called the local-+lo"al transormation2

6 26 12 12

8 9 7 8 9e e

xMx Mx

q # q=

)he transormation %ill "e nothin+ more then 1;s and 0;s# As an

eample, suppose %e hae the ollo%in+ element and structural

node num"erin+

1 2 3 4

5 6 7 8

1211109x

y

1q&q

3q4q

'q6q

(q?q

&3q&4q

-q10q

1(q1?q

1 2 3 4

5 6 7 8

1211109

12

34

56

78

910

1112

x

y

p


32/75


onsider element (# .uppose %e place element node 1 at +lo"al

node 6#

1

&

31eq&eq

3eq

4eq

'eq6

eq

(

Element nodesand local dofs

(

Structural nodesand global dofs

611q

1&q

13q14q

(

11&1q

&&q

@e see that or element (, there is a correspondence "et%een the 6

element local dos at element nodes 1, & and 3, and the 6 structural

+lo"al dos at nodes 6, 11and (# @e see that local element2 node1 corresponds to +lo"al node 6, local element2 node & corresponds

to +lo"al node 11, and local node 3 corresponds to +lo"al node (#

@e can %rite this local to +lo"al transormation 8 9 78 9e eq # q= as


33/75


1,&

3,4

',6

(,?

( ,1,&

(3,4

(',6

1 007 07 07 07 07 07 07 07 07 07 07

0 1

1 007 07 07 07 07 07 07 07 07 07 07

0 1

1 0

07 07 07 07 07 07 07 07 07 07 070 1

e

q

q

q

q

qq

q

q

=

=

10

11,1&

13,14

1',16

1(,1?

1,&0

&1,&&

&3,&4

q

q

q

q

q

q

q

Each 07 is a &&2# )he a"oe sa$s that or element (, local

element2 node 1 corresponds to +lo"al node 6, i#e#, local dos 1,&correspond to +lo"al dos 11,1& local node & corresponds to +lo"al

node 11, i#e#, local dos 3,4 correspond to +lo"al dos &1,&&, etc#

+lo"al node L

1 & 3 4 ' 6 ( ? 10 11 1&


34/75


o% transorm e! and e" rom local to +lo"al do "$ su"stitutin+

into and to o"tain

1 1 1& & &8 9 78 9 8 9 7 7 78 9 8 9 78 9

8 9 7 8 9 8 9 7 8 9 8 9 8 9 8 9 8 9

e e # e e # e # e e # e'

e # e # e # e # e # e # e' '

! q k q q # k # q q K q

" q # & q # S q & q S

= =

=

o% %e can deine the ollo%in+ element matrices in +lo"al do

instead o local element do2

62 6 62 6 2 2

62 6 12 12

7 7 7 7

8 9 7 8 9

8 9 7 8 9

e e # e e'

Mx x xMMxM

e e # e'

Mx xMx

e e # e'

K # k #

& # &

S # S

)o see %hat an element stiness and orce matri %ritten in +lo"al

do loos lie, consider element ( a+ain# @e o"tain or( 7'K and

(

8 9'&

6 11 7


35/75


Element #

Each "loc is a &&2 su"-matri

1 & 3 4 ' 6 ( ? 10 11 1&

1

&

3

4'

6 (11k

(13k

(1&k

( (31k

(33k

(3&k

(3&

?

10

11 (&1k

(&3k

(&&k

1&

o% the internal and eternal potential ener+$ is +ien "$

( ( (11 1& 13

( ( (&1 && &3

( ( (31 3& 33

k k k

k k k

k k k

6 11 7

6

11

7

(1

(&

(3

&

&

&

( 7'K = (8 9'& =


36/75


( )1 1& &1 1 1 1 1

8 9 78 9el el el N N N M M

e # e estr ' 'i+ i +

e e e i +

! ! q k q k q q= = = = =

= = =

( ) ( )1 1 1

1 1 1 1

8 9 8 9 8 9 8 9

el el el

el el

N N Ne # e # e

str ' '

e e e

N NM Me e

'i i 'i i

e i e i

" " q & q S

& q S q

= = =

= = = =

= =

=

o% %e can su"stitute and into to o"tain

( ) ( ) ( )1

&1 1 1

8 9 78 9 8 9 8 9 8 9 8 9

0

el el el N N N# e # e # e

' ' 'e e e

i

q k q q & q S

q

= = =

=

1,&,###,for i M=


37/75


%hich +ies a s$stem o M e5uations in terms o the structural

displacements

( )1 1 1

78 9 8 9 8 9 809el el el

N N Ne e e

' ' '

e e e

k q & S = = =

= or

1 1 1

7 8 9 8 9 8 9el el el N N N

e e e' ' '

e e e

k q & S

= = =

= +

@hen all the element contri"utions hae "een summed, %e simpl$

%rite 78 9 8 9 8 9K q Q S= +

ote that %hen the element stiness and orce matrices are %ritten

in terms o structural displacements usin+ local to +lo"al

transormation2, the$ "ecome additie see e5# 7 i#e#, to +et the

structural stiness matri 7, %e sum the contri"utions or all

elements#


38/75


Assem$lage o% Elements

A sin+le element "$ itsel is useless# @e ust deterine the

equilibriu equations for an assebla'e of eleents that coprise

the entire structures#

onsider the ollo%in+ structure onl$ a e% elements are taen to

simpli$ the discussion2 %ith a uniorml$ pressure p on the ri+ht

"oundar$ and ied on the let "oundar$ assume a constantthicness t2#

@e num"er the structural nodes

rom 1 to 1& as sho%n# @e also

num"er the elements rom 1 to1& as sho%n in an$ order2#

For each +lo"al node o the structure, %e can speci$ the ,$2

coordinates ix , iy , iK1, &, N, 1

1 2 3 4

5 6 7 8

1211109

12

34

56

78

910

1112

x

y

p


39/75


Each node o the structure %ill

hae t%o de+rees o reedom

do2# @e la"el these

structural .'lobal/ de'rees of

freedoin order as sho%n to

the ri+ht# ote that the

structural nodal displacements

are %ritten %ithout the superscript Be#B )he nodal displacement

ector is %ritten as 8 9q and is &412 or this pro"lem#

@e note that the let side is ied nodes 1, ' and 2# Hence,

displaceent boundary conditions%ill re5uire that

1 & 10 1( 1? 0q q q q q q= = = = = = #

ote %e do not hae to num"er the do consistentl$ and inse5uence %ith the structural nodes# Ho%eer, this maes the

"ooeepin+ much, much simpler

For each element, %e can construct a ta"le called the eleent

connectivitythat speciies %hich structural +lo"al2 nodes are

1 2 3 4

5 6 78

1211109x

y

1q&q

3q4q

'q6q

(q?q

&3q&4q

-q

10q

1(q1?q


40/75


connected "$ an element# Hence, or the pro"lem a"oe, %e hae

the ollo%in+ eleent connectivity table

Element &o. Element &ode ' Element &ode ( Element &ode )1 1 ' &

& ' 6 &

3 ' 10 6

4 ' 10

' & 6 36 6 ( 3

( 6 11 (

? 6 10 11

3 ( 4

10 ( ? 411 ( 1& ?

1& ( 11 1&

ote that or the .), element nodes M.) "e +ien as @#

Element node 1 can "e attached %ith an$ +lo"al node o the

element#


41/75


ote that i %e are careul in num"erin+ the nodes and choosin+

the element connectiit$ in a Bs$stematicB manner, there %ill "e a

pattern to the element connectiit$ ta"le see a"oe2# An

autoatic esh 'enerator, lie the one in FEMA*, tries to ollo%

this pattern#

ote that the +lo"al node num"ers or the structure are some%hat

ar"itrar$, i#e#, %e could num"er them in an$ order# Ho%eer, it

%ill turn out that there are optimum %a$s to num"er nodes or a+ien structure and mesh2 in order to reduce the "and%idth o the

structural stiness matri 7 - this saes time solin+ the

e5uations# For the mesh a"oe, it %ould "e optimum to num"er

do%n%ard and let-to-ri+ht, as opposed to let-to-ri+ht and

do%n%ard# @e;ll discuss that later# ie%ise, the elementnum"erin+ is ar"itrar$, "ut a+ain there ma$ "e optimum

approaches# An automatic mesh +enerator tries to do the

num"erin+ in an optimum ashion#


42/75


ote that or this structure, %e hae 1& +lo"al nodes# )here are &

de+rees o reedom do2 at each node u and 2# Hence, the

structure has &4 do and the structural stiness matri 7 %ill "e

&4&42# )he structural e5uili"rium e5uations can "e %ritten as

&4 &42 &4 12 &4 12

7 8 9 8 9 8 9x x x

K q Q S= +

%here 7Kstructural stiness matri,

8G9Kstructural orces matri due to applied tractionsand "od$ orces2

8.9Kstructure reaction orces due to "oundar$ conditions

ets see ho% each element contri"utes to +lo"al matrices# )ae

element 1 to start %ith# ote that %e can use su"-matri notationto diide the element matrices as ollo%in+# se a superscript o 1

on the terms to indicate element 1#


43/75


1 1 1 111 1& 13 1

& &2 & 12

1 1 1 1 1 1&1 && &3 &

6 621 1 1 131 3& 33 3

7 , 8 9

x x

x

k k k &

k k k k & &

k k k &

= =

@e no% loo at element 1 and note that element node num"ers 1, &, 3

correspond to +lo"al node num"ers 1, ', & rom the dra%in+ o themesh, or rom the element connectiit$ ta"le2# @e can indicate this

inormation on the stiness and orce matrices as ollo%s

1 ' &1 1 1 111 1& 13 1

& &2 & 12

1 1 1 1 11&1 && &3 &

6 62 6 121 1 1 131 3& 33 3

7 , 8 9

x x

x x

k k k &

k k k k & &

k k k &

= =

1

'

&

1

'

&


44/75


Hence, %e see that element 1 contri"utes stiness and orces to

+lo"al nodes 1, ' and *lacin+ these contri"utions into the +lo"al

stiness matri +iesElement ' only

1 & 3 4 ' 6 ( ? 10 11 1& 859 G

1 111k113k

11&k 1,&

q 11&

& 131k133k

13&k 3,4

q 13&

3 ',6q4 (,?q

' 1&1k1&3k

1&&k ,10

q 1&&

6 =

(

?

10

11

1& &3,&4q

PP remem"er, each "loc is a &&2 su"-matri

6 11 7


45/75


o% tae element (#

Element # only

1 & 3 4 ' 6 ( ? 10 11 1& 859 G

1 1,&q

& 3,4q

3 ',6q

4 (,?q

' ,10q

6 =

(

?

10

11

1& &3,&4q


( ( (11 1& 13

( ( (&1 && &3

( ( (31 3& 33

k k k

k k k

k k k

6 11 7

6

11

7

(1

(&

(3

&

&

&


46/75


ote that the distri"uted pressure load p is applied onl$ to the ri+ht

"oundar$ o elements 10 and 11# Hence 8F9 or all elements

ecept 10 and 11 %ill "e :ero# For elements 10 and 11, %e %ill

hae

14 ?&10

14 ?&

0

0

8 9 0

0

pt,

&

pt,

=

8

4

7

1? 1&&11

1? 1&&

0

0

8 9 0

0

pt,

&

pt,

=

7

12

8

%here 4 ?, is the len+th "et%een +lo"al nodes 4 and ?, etc#

I %e assem"le all element stiness matrices 7 and orces

matrices 8F9 to the +lo"al e5uili"rium e5uations, %e hae the

ollo%in+ result

4(


47/75


!tructural E*uations o% E*uili$rium

1 & 3 4 ' 6 ( ? 10 11 1& 859 G

1 Q Q Q 1,&q

& Q Q Q Q Q 3,4q

3 Q Q Q Q Q ',6q

4 Q Q Q Q (,?q Q

' Q Q Q Q Q Q ,10q

6 Q Q Q Q Q Q Q =

( Q Q Q Q Q Q Q

? Q Q Q Q Q

Q Q Q

10 Q Q Q Q Q

11 Q Q Q Q Q

1& Q Q Q Q &3,&4q Q

Q means that one or more elements hae contri"uted here


ote that 7 issyetric also it is bandedsemi-"and%idthK1&2#

4?


48/75


In the preious pa+e, each Q

means that one or more elements

hae contri"uted to that &&2

su"-matri# For eample, %e notethat node & %ill hae stiness

rom elements 1, & and '# Hence,

the &,& position o the +lo"al

stiness matri %ill "e e5ual to

note $ou hae to reer to the element connectiit$ to see %hichelement node or each element corresponds to +lo"al node &2

1 & '&& 33 33 11 7 7 7 7K k k k= + + each su"-matri is &&2

)he +lo"al node &-6 couplin+ term&6

7K %ill hae contri"utions

rom elements & and ' since onl$ these elements share the

"oundar$ "et%een nodes & and 6 & '&6 3& 1& 7 7 7K k k= + #/lo"al node 6 %ill hae stiness contri"utions rom elements &, 3,

', 6, (, and ? & 3 ' 6 ( ?66 && 33 && 11 11 11 7 7 7 7 7 7 7K k k k k k k= + + + + + #

1 2 3

56

7

11109

1

23

4

5

67

8

1

2

3 1

1 22

33

1111

22

2

22

3

3 3

3

4


49/75


Guestion= @hat happened to the reactions 8.9 or each element=

@h$ don;t the$ sho% up in the structural stiness matri=

.imple# It is e5uili"rium# Recall that %hen %e mae a ree-"od$,in this case tae a sin+le inite element as the ree-"od$, %e %ill

hae e5ual and opposite reactions %here the cut is made thou+h the

"od$# onsider elements 1 and & "elo%

1

&

3

1 1

22

11S

1&S

13S

1

4S

1

'

S

16S

1

&

3

&6S

&'S

&4S

&3S

&&S &

1S

At the "oundar$ "et%een elements 1 and &, the reactions are e5ual

and opposite# Hence, %e add them up %e hae 1 &1 3 0S S+ = ,1 &

& 4

0S S+ = , 1 &' '

0S S+ = , and 1 &6 6

0S S+ = # Hence, all the reactions

'0


50/75

A04 - Introduction to the Finite Element Method '0

"et%een elements sum to :ero and do not hae to "e put into the

structural e5uili"rium e5uations#

O, "ut %hat a"out the "oundar$ %here there are supports= @hathappens to the reactions there= For eample, the cantileer plate

eample a"oe

)he$ don;t disappear and should

"e included in the structuralstiness matri#

@e no% that there %ill "e

unno%n reactions at +lo"al

nodes 1, ' and # @e could call

these reactions 10 , &0 , 0 , 100 ,

1(0 and 1?0 consistent %ith

+lo"al displacements2# .o %e

hae the ree "od$ o the

structure

1 2 3 4

5 6 7 8

1211109

12

34

56

78

910

1112

x

y

p

1 2 3 4

5 6 7 8

1211109

12

34

56

78

910

1112

x

y

p

10

&0

-0100

1(0

1?0

'1


51/75


!tructural E*uations o% E*uili$rium +ith !upport ,eactions

1 & 3 4 ' 6 ( ? 10 11 1& 859 G

1 Q Q Q 1,&q 1,&0

& Q Q Q Q Q 3,4q

3 Q Q Q Q Q ',6q

4 Q Q Q Q (,?q Q

' Q Q Q Q Q Q,10q

,100

6 Q Q Q Q Q Q Q =

( Q Q Q Q Q Q Q

? Q Q Q Q Q

Q Q Q 1(,1?0

10 Q Q Q Q Q11 Q Q Q Q Q

1& Q Q Q Q &3,&4q Q



A04 I d i h Fi i El M h d '&


52/75

A04 - Introduction to the Finite Element Method '&

O, no% one last step# @e hae to appl$ displaceent boundary

conditions# )he structure is ied at nodes 1, ' and thus,

1 & 10 1( 1? 0q q q q q q= = = = = = # )he easiest %a$ to appl$"oundar$ conditions to an$ s$stem o e5uations is as ollo%s

1# ero out the ro% and column on the let side matri the 7

matri2 correspondin+ to each C##, and :ero out the ro% o the

ri+ht side the 8G9 matri2 correspondin+ to each C##

*lace a 1 on the dia+onal o the let side matri the 7 matri2

correspondin+ to each C##

Sou %ill notice that eer$ do that has a C## also corresponds toa do %here a support reaction R2 occurs# Appl$in+ C## as

descri"ed a"oe %ill thus eliminate the reactions rom the

e5uili"rium e5uations# A theoretical reason %h$ %e donTt hae to %orr$ a"out reactionsin structural e5uations o e5uili"rium= Cecause these support

reactions R do no %or displacement is :ero at support2 and

hence do not aect e5uili"rium o the structure

A04 I t d ti t th Fi it El t M th d '3


53/75


!tructural E*uations o% E*uili$rium +ith B.. Applied

1 & 3 4 ' 6 ( ? 10 11 1& 859 G

1

1

1

0 0

0 0

0 0

0 0

1,&q 0

0

&0 0

0 0

Q Q

0 0

0 0

Q 3,4q

3 Q Q Q Q Q ',6q

4 Q Q Q Q (,?q Q

'0 0

0 0

0 0

0 0

1

1

0 0

0 0

0 0

0 0

0 0

0 0

,10

q 00

6 Q Q 0 00 0 Q Q Q Q =( Q Q Q Q Q Q Q

? Q Q Q Q Q

0 0

0 0

1

1

0 0

0 0

0

0

100 0

0 0

Q

0 0

0 0

Q Q

11 Q Q Q Q Q

1& Q Q Q Q &3,&4q Q



A04 I t d ti t th Fi it El t M th d '4


54/75


)he structural e5uations %ith C## ma$ no% "e soled or the

unno%n displacements# ote that %hen %e sole the s$stem o

e5uations, the solution %ill +ie 1 & 10 1( 1? 0q q q q q q= = = = = = ,i#e, the 1ste5uation simpl$ sa$s 112 0q = , etc#

Element !trains and !tresses

o% %e are read$ to sole or the element strains and stresses# For

each element, %e can su"stitute the 6 +lo"al displacements

correspondin+ to that element into

3 623 12 6 12

8 9 78 9e e e

xx x

( q = eK1, &, N, no# o elements

)he stresses or each element can then "e o"tained "$ su"stitutin+the strains or that element into

3 323 12 3 12

8 9 78 9e e e

xx x

$ = eK1, &, N, no# o elements

A04 Introduction to the Finite Element Method ''


55/75

A04 - Introduction to the Finite Element Method ''

Ealuation o stress results "ased on stress components in the

artesian coordinates directions , , , #xx yy xy etc 2 leaes

somethin+ to "e desired# @h$= .tresses in these directions ma$

not necessaril$ represent the lar+est stresses and %e need these in

order to consider $ieldin+ or ailure# Sou alread$ no% that $ou

can calculate principal stresses and maimum shear stress usin+

stress transormation e5uations or Mohr;s ircle# Hence, stress

results stress components2 are oten represented in t%o additional

%a$s

*rincipal stresses and maimum shear stress, and on Mises stress#

*rincipal stresses can, as noted a"oe, "e o"tained "$ either stresstransormation e5uations or throu+h the use o Mohr;s ircle# An

alternate approach to deine principal stresses is to %rite

A04 Introduction to the Finite Element Method '6


56/75


0

xx p xy xy

yx yy p y-

-x -y -- p

=

Epansion o the determinant proides a cu"ic e5uation that can "e

soled or the three principal stresses p # omparin+ principal

stresses to a tensile $ield stress proides some measure oealuation ho%eer, one has to eep in mind that comparin+ the

principal stress o"tained rom a three-dimensional stress state2 to

a $ield stress o"tained rom a uniaial tension test is ris$ at "est#

)he on Mises stress proides a means to etrapolate uniaialtensile test data or $ield stress2 to a three-dimensional stress state#

In eect, the on Mises stress proides an Be5uialentB uniaial

stress approimation to the three-dimensional stress state in a "od$

throu+h the ollo%in+ e5uation

A04 Introduction to the Finite Element Method '(


57/75

A04 - Introduction to the Finite Element Method '(

1& & & &

1&

2 2 2

6 6 6

xx yy yy -- -- xx"M

xy y- -x

+ + =

+ + + or

1& & & &1

1 & & 3 3 1& 2 2 2"M p p p p p p = + +

%here 1 & 3 , , 2p p p are the principal stresses# /ien the stresscomponents , , , #xx yy xy etc 2 or principal stresses, one can

compute the on Mises stress#

)his representation has "een used 5uite successull$ to model the

onset o $ieldin+ in ductile metals and colla"orates %ell %itheperiment# It is %idel$ used in industr$# For a material to remain

elastic,

"M y < or no $ieldin+2

A04 - Introduction to the Finite Element Method '?


58/75

A04 - Introduction to the Finite Element Method '?

E5uation orms an ellipsoid in 3-> ellipse in &->2 %hen the

stresses are plotted in principal stress space# As lon+ as the stress

state represented "$ the principal stresses is inside the ellipse the

$ield surace2, the material is elastic#



59/75


Element -i$raries

.or1 choose the ri'ht eleent for a structural coponent and

loadin'1 in order to axii-e potential for correct results with the

least aount of coputation/

Man$, man$ inite elements hae "een deeloped or use in

modern FEM sot%are# hoosin+ the correct element or a

particular structural is paramount# For eample,

i a structural mem"er "ehaes lie a "eam in "endin+, %eshould choose a "eam element to model it,

i a structural mem"er "ehaes lie a thin plate in plane stress,%e should choose an appropriate element to model it,

i a structural mem"er loos lie a shell o reolution, %e

should use a thin shell o reolution element,

i a structural mem"er %ill eperience a three-dimensionalstress state, %e hae to choose an element that models that

"ehaior,

etc#



60/75

A04 Introduction to the Finite Element Method 60

Here are some eamples o the t$pes o elements aaila"le

)russ element &-> and 3->2 Ceam "endin+ element &-> and 3-> strai+ht and cured2

Mem"raneelement no "endin+ lat and cured2

)rian+ular, Guad "oth strai+ht and cured sides2

dof at eac node



61/75

A04 Introduction to the Finite Element Method 61

*lanes .tress and *lane .train elements )rian+ular and Guadrilateral shapes "oth strai+ht and cured

sides2#

*lane stress re5uires that the onl$ non-:ero stresses occur in

the plane o the element ho%eer, strain does occur normal

to plane2# /enerall$ applica"le to thin +eometries# )%o

displacement do per node O rotational do2#

*lane strain re5uires that the onl$ non-:ero strains occur in

the plane o the element strain is :ero normal to plane, "ut

stress is not :ero2# on+ constrained +eometries or

eample, a lon+ pipe, a dam2# Elements %ith cured

"oundaries %ill al%a$s hae 3 or more nodes per ed+e#



62/75

0 t oduct o to t e te e e t et od 6

*late and shell "endin+ elements "endin+ and in-planestresses lat and cured elements2

)rian+ular, Guad "oth strai+ht and cured sides2

*late and shell "endin+

elements are characteri:ed

as "ein+ thincompared to

other dimensions, and

hain+ no stress normal to

the plate similar to plane

stress2#



63/75

*late and shell "endin+ elements %ill hae in-plane and normal

displacements , ,u v w2 and rotations ,x y 2 a"out the t%o aes

in the plane o the plate!shell# o stiness a"out the normal

aes#



64/75

Ais$mmetric shell "endin+ element or shell o reolution2

)he shell o reolution can "e

descri"ed +eometricall$ "$ a curedor strai+ht2 line that is reoled

a"out the ais o s$mmetr$# )op

t%o i+ures are eamples o thin

shells o reolution# >e+rees o

reedom and stress state or a shello reolution are similar to plate

and shell "endin+ elements, i#e#,

displacements parallel and

perpendicular to the shell surace,

and rotations a"out the t%o aesthat lie in the plane o the shell# o

stress normal to surace#



65/75

o%er i+ure multicell tu"e2 is O) a shell o reolution, "ut

%ould re5uire plate or shell elements#

Ais$mmetric "od$ o reolution element 3-> stress anal$sis2

Cod$ is solid and ais$mmetric a"out some ais o reolution#

.tress state is ull$ three-dimensional includes all stress

components2# ote that the

element orms a trian+ular-shaped

rin+, i#e#, the element itsel is a"od$ o reolution#



66/75

/eneral solid element or ull 3-> stress anal$sis2

)etrahedron, Cric shapes "oth strai+ht and cured sides2

.tress state is ull$ three-dimensional includes all stress

components2# )hree-dimensional solid elements hae 3

displacements per node O rotational dos2# Elements %ith



67/75

cured "oundaries %ill al%a$s hae 3 or more nodes per

ed+e#



68/75

Mesh eneration/ re and ost1processing/ !ol2ers

In order to +enerate the structural stiness matri, %e must speci$

the nodal coordinates ,$,:2 o each +lo"al node and elementconnectiit$ or each element# For a lar+e pro"lem %ith comple

+eometr$, this is a monumental tas i done "$ hand# Mesh

+enerators are sot%are tools that automate this process#

Mesh eneration and reprocessingIn +eneral, mesh +eneration starts "$ speci$in+ the coordinates o

a e% e$ locations that %ill suicientl$ deine the outer "oundar$

t = 0.5 in.



69/75

o a structural component and other distin+uishin+ eatures such as

holes, illets, etc# onsider the "racet sho%n a"oe# @e could

irst deine the corners o a 10B &'B rectan+le# )o speci$ the

rounded ed+es on the ri+ht "oundar$, %e could speci$ that a 4Bradius illet is to "e placed at each ri+ht corner# Finall$, to speci$

the location o the hole, %e %ould deine a &B radius circle %hose

center is located 'B rom the ri+ht "oundar$#

)o speci$ the mesh ho% man$ elements are to "e used2, %e%ould speci$ so man$ elements in the direction and so man$

elements in the $ direction#

.ince the structure is &-> and in plane stress, %e %ould speci$ the

t$pe o element trian+ular, 5uad, etc#2 to "e used in deinin+ themesh#

A04 - Introduction to the Finite Element Method (0


70/75

)he mesh +enerator %ould then automaticall$ determine the

coordinates o all +lo"al nodes and determine the element

connectiit$ or all elements#

As a part o the mesh +eneration, %e %ould also speci$ the

thicness to "e used or each element# /enerall$, the mesh

+enerator %ould allo% or a linear thicness ariation %ithin the

re+ion "ein+ meshed# For a aria"le thicness case, one %ould

hae to speci$ the thicness at seeral e$ locations and mesh+enerator %ould determine the thicness or all elements in the

re+ion#

Also, the material properties %ould "e speciied or each element

in the re+ion# Most mesh +enerators onl$ proide or one materialset to "e used %ithin a re+ion i#e#, same material or all elements2#

et, the mesh +enerator also called thepreprocessor, "ecause it

processes all the re5uired input2 %ould re5uire the t$pe o loadin+



71/75

that is applied to elements or element "oundaries point orces,

"od$ orces, surace tractions2# For a thermal stress anal$sis, the

preprocessor %ould also re5uire inormation on thermal loadin+#

For a d$namic transient2 anal$sis, %e %ould hae to speci$ mass

properties or each element and the time histor$ o the loadin+#

astl$, the displacement "oundar$ conditions must "e speciied#

/ien all this inormation, the mesh +enerator!preprocessor %illhae +enerated a model somethin+ lie that sho%n "elo%

A04 - Introduction to the Finite Element Method (&


72/75

@ith this inormation, the preprocessor %ould create a data set an

output ile2 in a ormat suita"le to "e read and processed "$ the

inite element anal$sis pro+ram also called thesolver2#

!ol2er

Once thesolverhas created the structural stiness and orce

matrices, soled or +lo"al nodal displacements, and soled or

stresses and strains plus a e% other thin+s appropriate to each

element t$pe2, it +enerates an output ile and %e are no% read$ toeamine and interpret the results#

ost1processing

learl$, or a complicated structure %ith man$ nodes and

elements, the eamination and ealuation o all results is anenormous tas "ecause o the shear olume o data displacements,

strains, stresses, etc2# )hepost2processorno% taes oer# Its

purpose is to proide output such as the deormed +eometr$,

contour plots o principal stress components, contour plots o on



73/75

Mises stress, contour plots o strain components, etc# %hat eer

youdecide is important2# @hat, and ho%, the postprocessor can

displa$ inormation depends on the t$pe o elements "ein+ used#

Celo% are outputs o the deormed +eometr$ and onMises stress#



74/75

Oten the pre and post-processor are com"ined into one sot%are

paca+e lie FEMA* or *A)RA2# Most preprocessor pro+rams

%ill create output iles in ormats accepta"le to arious solers

A04 - Introduction to the Finite Element Method ('


75/75

lie AEFEM, A.)RA, A.S., ACA., etc#2# ie%ise,

most postprocessor pro+rams %ill accept the output ile o most

solers and displa$ a ariet$ o data results +enerated "$ the

soler#

ast )hou+hts on Mesh /enerators, *re!*ost-*rocessors D .olers

)he$ %ill do a lot the tedious %or or $ou#

)he$ are not a su"stitute or +ood en+ineerin+ no%-ho%,

Documents

A04 Introduction to FEM