025210 SIM Nonlinear Analysis WP ENG

8/9/2019 025210 SIM Nonlinear Analysis WP ENG

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Unann Nnna Ana

SUMMARY

In this pape you wi ean about the dieences between inea and

noninea anaysis and eaize thee ae optimum times to use one typ

o anaysis vesus the othe. We wi discove that negecting noninea

eects can ead to seious design eos. Ate eviewing the exampe

taken om eveyday design pactice, you wi see how noninea anay

can hep you avoid ovedesign and buid bette poducts.

w h i t e p A p

insight


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inroducon

Ove the ast decade, nite eement anaysis (FEA) stopped being egaded o

as an anaysts too and enteed the pactica wod o design engineeing. CA

sotwae now comes with buit-in FEA capabiities and design enginees use

FEA as an eveyday design too in suppot o the poduct design pocess.

Howeve, unti ecenty, most FEA appications undetaken by design enginee

wee imited to inea anaysis. Such inea anaysis povides an acceptabe

appoximation o ea-ie chaacteistics o most pobems design enginees

encounte. Nevetheess, occasionay moe chaenging pobems aise, pob

ems that ca o a noninea appoach.

Histoicay, enginees wee euctant to use noninea anaysis, because o its

compex pobem omuation and ong soution time. Thats changing now, as

noninea FEA sotwae inteaces with CAD and has become much easie to

use. In addition, impoved soution agoithms and poweu desktop compute

have shotened soution times. A decade ago, enginees ecognized FEA as a

vauabe design too. Now they ae stating to eaize the benets and geateundestanding that noninea FEA bings to the design pocess.

Drncs bn lnar and nonlnar analyss

The tem stiness denes the undamenta dieence between inea and

noninea anaysis. Stiness is a popety o a pat o assemby that chaacte

izes its esponse to the appied oad. A numbe o actos aect stiness:

1. Sa: An I-beam has dieent stiness om a channe beam.

2. Maral: An ion beam is ess sti than the same size stee beam.

U N D E r S T A N D I N g N O N l I N E A r A N A ly S I S

The tem stiness deines the un-

damenta dieence between ineaand noninea anaysis. Stiness is apopety o a pat o assemby that

chaacteizes its esponse to theappied oad. Thee pimay actos

aect stiness: Shape, Mateia, andPat Suppot.


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3. par Suor: A beam with a simpe suppot is ess sti and wi defect

moe than the same beam with buit-in suppots as shown in Figue 1.

igURe 1: CANtilever beAm (top) hAs lower stiffNess thAN the sAme

beAm sUpported oN both eNds (bottom).

When a stuctue deoms unde a oad its stiness changes, due to one o

moe o the actos isted above. I it deoms a geat dea, its shape can chan

I the mateia eaches its aiue imit, the mateia popeties wi change.

On the othe hand, i the change in stiness is sma enough, it makes sense

to assume that neithe the shape no mateia popeties change at a duing

the deomation pocess. This assumption is the undamenta pincipe o

inea anaysis.

That means that thoughout the entie pocess o deomation, the anayzed

mode etained whateve stiness it possessed in its undeomed shape pio

to oading. regadess o how much the mode deoms, whethe the oad geappied in one step o gaduay, and no matte how high the stesses that

deveop in esponse to that oad may be, the mode etains its initia stiness

This assumption geaty simpies pobem omuation and soution. reca th

undamenta FEA equation:

[] = [K] * [d]

whee: [F] is the known vecto o noda oads

[K] is the known stiness matix

[d] is the unknown vecto o noda dispacements

This matix equation descibes the behavio o FEA modes. It contains a vey

age numbe o inea agebaic equations, vaying om sevea thousand tosevea miion depending on the mode size. The stiness matix [K] depends

on the geomety, mateia popeties, and estaints. Unde the inea anaysis

assumption that the mode stiness neve changes, those equations ae asse

bed and soved just once, with no need to update anything whie the mode i

deoming. Thus inea anaysis oows a staight path om pobem omuat

to competion. It poduces esuts in a matte o seconds o minutes, even o

vey age modes.

Eveything changes upon enteing the wod o noninea anaysis, because

noninea anaysis equies enginees to abandon the assumption o constant


I the change in stiness is sma

enough, it makes sense to assumethat neithe the shape no mateiapopeties change at a duing the

deomation pocess. This assump-tion is the undamenta pincipe o

inea anaysis.


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stiness. Instead, stiness changes duing the deomation pocess and the

stiness matix [K] must be updated as the noninea sove pogesses thou

an iteative soution pocess. Those iteations incease the amount

o time it takes to obtain accuate esuts.

Undrsandn drn ys o nonlnar bavorAthough the pocess o changing stiness is common to a types o nonin-

ea anayses, the oigin o noninea behavio can be dieent, making it ogi-

ca to cassiy noninea anayses based on the pincipa oigin o nonineaity

Because it isnt possibe to point out a singe cause o noninea behavio in

many pobems, some anayses may have to account o moe than one type

o nonineaity.

Nonlnar omry

As aeady discussed, noninea anaysis becomes necessay when the

stiness o the pat changes unde its opeating conditions. I changes instiness come ony om changes in shape, noninea behavio is dened

as geometic nonineaity.

Such shape-caused changes in stiness can happen when a pat has age

deomations that ae visibe to the naked eye. A geneay accepted ue o

thumb suggests conducting a noninea geomety anaysis i the deoma-

tions ae age than 1/20th o the pats agest dimension. Anothe impota

acto to ecognize is that in cases o age deomations, the oad diection

can change as the mode deoms. Most FEA pogams oe two choices to

account o this diection change: oowing and nonoowing oad.

A oowing oad etains its diection in eation to the deomed mode as shoin Figue 3. A nonoowing oad etains its initia diection.

igURe 3: followiNg (or NoNCoNservAtive) loAd ChANges its direC-

tioN dUriNg the proCess of deformAtioN ANd remAiNs NormAl to th

deformed beAm (left). NoNfollowiNg, or CoNservAtive, loAd retAiNs

its origiNAl direCtioN (right).

A pessue vesse subjected to vey high pessue that undegoes a dastic

change o shape povides anothe good exampe o the atte situation. The

pessue oad aways acts noma to the was o the pessue vesse. Whie

inea anaysis o this scenaio assumes that the shape o the vesse does no

change, eaistic anaysis o the pessue vesse equies anayzing geometic

nonineaity with nonconsevative (o oowe) oading.

A geneay accepted ue o thumb

suggests conducting a nonineageomety anaysis i the deoma-tions ae age than 1/20th o the

pats agest dimension. Anotheimpotant acto to ecognize is that

in cases o age deomations, theoad diection can change as themode deoms.



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Changes in stiness due to shape can aso occu when the deomations ae

sma. A typica exampe is an initiay fat membane defecting unde pessu

(see Figue 4).

igURe 4: ANAlysis of A flAt membrANe UNder pressUre loAd reqUire

NoNliNeAr geometry ANAlysis eveN thoUgh the mAgNitUde of defor

tioN mAy be very smAll.

Initiay, the membane esists the pessue oad ony with bending stiness.

Ate the pessue oad has caused some cuvatue, the deomed mem-

bane exhibits stiness additiona to the oigina bending stiness (Figue 5)

Deomation changes the membane stiness so that the deomed memba

is much stie than the fat membane.

igURe 5: A flAt membrANe respoNds to loAd oNly with beNdiNg stif

Ness. beCAUse of deformAtioN it Also ACqUires membrANe stiffNess

therefore, it is mUCh stiffer thAN prediCted by liNeAr ANAlysis.

Some FEA pogams use conusing teminoogy, caing a anaysis o geome

ic nonineaities age deomation anaysis. This ignoes the necessity to p

om noninea anaysis o smae deomation.

Nonlnar maral

I changes o stiness occu due ony to changes in mateia popeties unde

opeating conditions, the pobem is one o mateia nonineaity. A inea mat

ia mode assumes stess to be popotiona to stain (Figue 6, beow). That

means it assumes that the highe the oad appied, the highe the stesses an

deomation wi be, popotiona to the changes in the oad. It aso assumes

that no pemanent deomations wi esut, and that once the oad has been

emoved the mode wi aways etun to its oigina shape.

Changes in stiness due to shape

can aso occu when the deoma-tions ae sma. A typica exampe isan initiay at membane deect-

ing unde pessue. Initiay, themembane esists the pessue oad

ony with bending stiness. Atethe pessue oad has caused somecuvatue, the deomed membane

exhibits stiness additiona to theoigina bending stiness.



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figUre 6

Athough this simpication is acceptabe, i the oads ae high enough to cau

some pemanent deomations, as is the case with most pastics, o i the

stains ae vey high (sometimes> 50 pecent), as occus with ubbes and

eastomes, then a noninea mateia mode must be used.

Due to the vast dieences in how vaious types o mateias behave unde

thei opeating conditions, FEA pogams have deveoped speciaized tech-

niques and mateia modes to simuate these behavios. The tabe beow oe

a shot eview o what mateia modes wok best o which pobem.

Maral Classfcaon Modl Commns

Eastopastic Von Mises o Tesca These modes wok we o mateia

o which a stain-stess cuve shows

a pateau beoe eaching the utimate

stess. Most engineeing metas and som

pastics ae we-chaacteized by this

mateia mode.

Ducke-Page This mode woks o sois and ganua

mateias.

Hypeeastic Mooney-rivin Wok we o incompessibe eastomes

and Ogden such as ubbe.

Batz-Ko This mode woks o compessibe poyu

than oam ubbes.

Viscoeastic Sevea (optiona with This mode woks o had ubbe o gas

othe modes)

Ceep Sevea (optiona with Ceep is a time-dependent stain poduc

othe modes) unde a state o constant stess. Ceep isobseved in most engineeing mateias,

especiay metas at eevated tempeatu

high poyme pastics, concete, and soid

popeant in ocket motos.

Supeeastic Nitino Shape-memoy-aoys (SMA) such as

(Shape memoy aoys) Nitino pesent the supeeastic eect. Th

mateia undegoes age deomations in

oading-unoading cyces without showin

pemanent deomations.

i the oads ae high enough to

cause some pemanent deoma-tions, as is the case with mostpastics, o i the stains ae vey

high (sometimes >50 pecent), asoccus with ubbes and eastomes,

then a noninea mateia modemust be used.



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igURe 7: stress-strAiN CUrve of AN elAstiC-perfeCtly plAstiC mAte-

riAl model. with this mAteriAl model the mAximUm stress mAgNitUde

CANt exCeed the limit of plAstiC stress (yield stress).

In deaing with anaysis o an eastic-peecty pastic mateia modethat is,

mateia that has ost a abiity to etun to its oigina shape ate deomation

the stess emains constant above a cetain stain vaue. It descibes the cast

ion mateia (this mode is one o the simpest o noninea mateia modes,

and its stain-stess cuve is shown in Figue 8.) o a bukhead hed in pace

vey we by eight bots.

linea anaysis eveas a maximum von Mises stess o 614 MPa (89,600 psas compaed to a mateia yied o 206 MPa (30,000 psi). The esuts o that

inea anaysis ae shown in Figue 8.

igURe 8: liNeAr stress solUtioN of A bUlkheAd shows very high AN

loCAlized stress CoNCeNtrAtioNs.

I the stesses exceed yied, wi the bukhead a apat? To nd out, an easto

pastic mateia mode needs to be used o examining how much mateia wi

go pastic. Figue 9 shows the noninea soution whee the maximum stess

equas the yied stess. Pastic zones ae sti oca, indicating that the bukhea

wi not a apat. O couse, caeu engineeing judgment is equied to decidi this design is acceptabe.

In deaing with anaysis o an

eastic-peecty pastic mateiamodethat is, a mateia that hasost a abiity to etun to its oigina

shape ate deomationthe stessemains constant above a cetain

stain vaue.



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igURe 9: NoNliNeAr stress solUtioN obtAiNed with AN elAstiC-

perfeCtly plAstiC mAteriAl model. red zoNes iNdiCAte mAteriAl

goiNg plAstiC. the exteNt of plAstiC zoNes is loCAl.

Figue 10 shows the inea stess soution o an auminum backet. The maxi

mum stess eads 44 MPa (6,400 psi) and ignoes the act that the mateia

yieds at 28 MPa (4,100 psi).

igURe 10: liNeAr stress solUtioN of A hollow brACket reports

stress wAy Above the limit of mAteriAl yield stress.

Noninea mateia anaysis can account o these esuts, whee the mateia

yieds when the maximum stess hods at 28 MPa (4,100 psi) (Figue 11, nex

page). The noninea stess esuts indicate that the backet is vey cose to c

apsing. Pastic zones occupy amost the entie coss-section o the cantieve

and a sight incease in the oad magnitude woud cause the coss-section to

become competey pastic and deveop a pastic hinge that causes the back

to coapse.


I the stesses exceed yied, wi the

bukhead a apat? To ind out, aneastopastic mateia mode needsto be used o examining how much

mateia wi go pastic. Pastic zonesae sti oca, indicating that the

bukhead wi not a apat.


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igURe 11: NoNliNeAr stress solUtioN shows mAximUm stress Not

higher thAN yield stress. the exteNt of the plAstiC zoNes iNdiCAtes

thAt the brACket is very Close to formiNg A plAstiC hiNge. it is At

the limit of loAd-beAriNg CApACity.

Modeing the simpe act o taking an odinay stee pape cip, unbending it,

and then bending it back, equies consideation o both noninea mateia

and noninea geomety. Figue 12 shows the deomed shape o the pape

cip using an eastic-peecty pastic mateia mode. Figue 13 shows esidua

stesses ate the cip has been bent back to the oigina shape.

igURe 12: ANAlysis of pAper Clip beNdiNg reqUires NoNliNeAr

mAteriAl ANd NoNliNeAr geometry ANAlysis. pAper Clip iN UNbeNt

positioN shows plAstiC stresses

igURe 13: pAper Clip beNt bACk iNto the origiNAl shApe shows

residUAl stresses..



Loss o lasc sably (buckln)

Stiness in a pat aso changes due to appied oads. Sometimes, oads

depending on how they ae appiedcan eithe incease the stiness (tensio

oads) o decease it (compessive oads). Fo exampe, a tight ope can take

an acobats weight. A oose one, howeve, wi make him a. In cases o com

pessive oad, i the changes in stiness ae sucient to cause the stuctuestiness to dop to zeo, bucking occus and the stuctue expeiences a ap

deomation. It then eithe as apat o acquies a new stiness in its postbu

ing state.

linea bucking anaysis can be used to cacuate the oad unde which a stu

tue wi bucke (Eue oad). Howeve, the esuts o inea bucking anaysis a

not consevative. In addition, ideaizations in the FEA mode may esut in the

pedicted bucking oad being much highe o the FEA mode than o the e

pat. Thus the esuts o inea bucking anaysis shoud be used caeuy.

Bucking does not necessaiy equa catastophic aiue and the stuctue ma

sti be abe to suppot the oad ate bucking has taken pace. Noninea anasis wi expain postbucking behavio.

figUre 13

Figues 13 and 14 show a snap-though eect. The pat etains its oad-

beaing capabiities even ate bucking has taken pace.

igURe 14: ANAlysis of sNAp-throUgh effeCt reqUires NoNliNeAr

ANAlysis.



I suppot conditions, incuding

contacts, change duing the appica-tion o opeating oads, nonineaanaysis is needed.

Conac srsss and nonlnar suors

I suppot conditions, incuding contacts, change duing the appication o

opeating oads, noninea anaysis is needed.

Contact stesses deveop between two contacting suaces. Theeoe, the

contact aea and the stiness o the contact zone ae unknown pio to sou-

tion. Figue 15 shows a stess soution o a typica contact pobem. Even

though the contact stess aea is vey sma compaed to the ovea mode

size, the changing stiness o the contact zone cas o noninea anaysis.

igURe 15: CoNtACt stress ANAlysis thAt models stresses developiN

betweeN two spheres (oNly oNe of the two CoNtACtiNg pArts

is showN) beloNgs to the CAtegory of ANAlysis with NoNliNeAr

sUpports.

Figue 16 shows an exampe o noninea suppots. The eective beam engt

and its consequent stiness depend on the amount o beam deomation.

When the beam contacts the suppot, its stiness inceases because o the

dop in its active ength.

igURe 16: this sUpport (wheN ACtivAted) ChANges the effeCtive

leNgth of beAm. CoNseqUeNtly, beAm stiffNess ChANges ANd the

problem reqUires NoNliNeAr ANAlysis.



In many cases, howeve, inea

assumptions die too much omeaity and povide cude o misead-ing inomation. Using the esuts o

inea anaysis to decide i a pat wiai unde its opeating oads may

ead to ove-design.

Nonlnar dynamc analyss

Dynamic anaysis accounts o inetia eects, damping, and time-dependent

oads. A dop test, vibations o an engine mount, aibag depoyment, o

cash simuation a equie dynamic anaysis. But is dynamic anaysis inea

o noninea? The quaiying ues ae exacty the same as in static anaysis.

I mode stiness does not change signicanty unde the appied oad, then

inea dynamic anaysis suces. A vibating engine mount o a tuning ok bo

expeience sma deomations about the point o equiibium and so can be

anayzed with inea dynamic anaysis.

Pobems such as cash simuation, anaysis o aibag depoyment, o mode

ing a meta stamping pocess a equie noninea dynamic anaysis becaus

both age deomations (noninea geomety) and age stains (noninea

mateia) occu.

ho can nonlnar analyss l us buld br roducs?

Natue is noninea. That means inea anaysis can ony appoximate the ea

noninea behavio o pats and assembies. Most o the time, such an appox

mation is acceptabe, and inea anaysis can povide vauabe insight into po

uct chaacteistics. In many cases, howeve, inea assumptions die too muc

om eaity and povide cude o miseading inomation.

Using the esuts o inea anaysis to decide i a pat wi ai unde its opeat

ing oads may ead to ovedesign. Fo exampe, a backet design anayzed on

with inea anaysis equies the designe to stick with a equiement that ste

must not exceed the yied. But noninea anaysis may show that some yiedin

is acceptabe. In that event, it becomes possibe to save on the amount

o mateia used o to choose a ess expensive mateia without compomisinstuctua integity. An enginee may be concened about too age a defectio

o a fat pane as tested with inea anaysis, o anothe exampe, and ove-

design it to compensate o that defection without eve knowing that inea

anaysis exaggeated the deomations, and it was ne as oiginay designed

Nonlnar analyss n vryday dsn racc

Once an enginee gains enough expeience to ecognize noninea pobems

it becomes obvious that appication o this technoogy isnt conned to exotic

situations. Designs that equie o may benet om noninea anaysis aboun

in evey industy and in eveyday design pactice.Hee ae sevea exampes o poducts whee the coect design decision

equies noninea anaysis. Many o these pobems invove moe than one

type o noninea behavio.



idlr ully (ur 17)

This stamped stee puey may bucke unde bet oad beoe it deveops

excessive stesses. Athough a inea bucking anaysis may be enough

to detemine the bucking oad, noninea anaysis is equied to study its po

bucking behavio.

figUre 17

Daram srn (ur 18)

The noninea sping chaacteistic equies a noninea geomety anaysis

to account o membane eects.

figUre 18

Rollovr rocv srucur (ur 19)

In the case o a oove, the stuctue deoms past its yied, and absobs

oove enegy. Duing this pocess it expeiences age deomation

Undestanding the eects o oove equies combining noninea mateia

and noninea geomety anaysis.

figUre 19



So obsrc orcs (ur 20)

Sot obstetic oceps ae designed to mod aound a babys head duing

oceps-assisted deivey. I too high taction and/o compession ae appied

the oceps ae designed to sip o the babys head to pevent injuies. Anay

o such oceps must combine noninea mateia and noninea geomety to

account o age deomations and noninea eastic mateia.

figUre 20

an uard (ur 21)

This pat equies noninea geomety anaysis because o the membane

stesses that deveop duing the deomation pocess. A noninea mateia

anaysis may be equied as we.

figUre 21

Sna rn (ur 22)

A noninea geomety anaysis is equied because o age deomations.

This ing may aso be a candidate o a noninea mateia anaysis.

figUre 22



Arln lua conanr (ur 23)

This aiine uggage containe equies a noninea geomety anaysis becaus

o membane eects in the bue lexan panes. In addition, the ame equi

a bucking o postbucking anaysis.

figUre 23

Oc car (ur 24)In this exampe, age deomations o the ame may necessitate a noninea

geomety anaysis. The seat and backest equie noninea geomety and no

inea mateia anaysis.

figUre 24

Alln rnc (ur 25)

The contact between the wench and the socket scew necessitates a conta

stess anaysis.

figUre 25


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w w w . s o l i d w o r k s . c

daau s sw C.300 ba AnuCnc, mA 01742 UsApn: 1 800 693 9000ou Us: +1 978 371 5011ea: [email protected]

Concluson

The natue o equenty encounteed anaysis pobems shoud be the yad

stick by which to justiy a decision to add noninea anaysis capabiities to

enginees FEA sotwae. I day-to-day wok equies noninea anaysis on

occasionay, it may be bette to ask o the hep o a dedicated anayst o to

hie a consutant. I, because o the natue o the designed poducts, desiganaysis pobems outiney invove age deomations, membane eects,

noninea mateia, contact stesses, bucking, o noninea suppots, then

noninea anaysis capabiities shoud be added to in-house FEA sotwae

intended o use by design enginees.

The past ten yeas have conditioned enginees to the use o FEA as a design

too. Now FEA sotwae and compute hadwae have matued enough so th

noninea anaysis can be added to thei tooboxes.

Documents

025210 SIM Nonlinear Analysis WP ENG