12.

Load

ing

& S

olut

ion

Load

ing

& S

olut

ion

Ove

rvie

w

�So

far,

we

have

see

n ho

w to

app

ly th

e fo

llow

ing

type

s of

lo

ads:

�D

ispl

acem

ent (

DO

F co

nstr

aint

s)�

Pres

sure

and

con

vect

ion

(sur

face

load

s)�

Gra

vity

(ine

rtia

load

s)�

�Str

uctu

ral�

tem

pera

ture

s (b

ody

load

s)

Thes

e lo

ads

acco

unt f

or fo

ur o

f the

five

mai

n ca

tego

ries.

In

this

cha

pter

, we

will

acc

ount

for t

he re

mai

ning

cat

egor

y �

conc

entr

ated

load

s, s

uch

as n

odal

forc

es in

a s

tres

s an

alys

is.

Load

ing

& S

olut

ion

...O

verv

iew

�W

e w

ill c

over

the

follo

win

g to

pics

in th

is c

hapt

er:

A.

Forc

e Lo

ads

B.

Nod

al C

oord

inat

e Sy

stem

C.

Solv

ers

D.

Mul

tiple

Loa

d St

eps

E. W

orks

hop

Load

ing

& S

olut

ion

A.

Forc

e Lo

ads

�A

forc

e is

a c

once

ntra

ted

load

(or

�poi

nt lo

ad�)

that

you

can

app

ly a

t a

node

ork

eypo

int.

�Po

int l

oads

suc

h as

forc

es a

re

appr

opria

te fo

r lin

e el

emen

t mod

els

such

as

beam

s, s

pars

, and

spr

ings

.

In s

olid

and

she

ll m

odel

s, p

oint

load

s us

ually

cau

se a

str

ess

sing

ular

ity,

but a

re a

ccep

tabl

e if

you

igno

re

stre

sses

in th

e vi

cini

ty.

Rem

embe

r, yo

u ca

n us

e se

lect

logi

c to

�ig

nore

� th

e el

emen

ts in

the

vici

nity

of t

he

poin

t loa

d.

Load

ing

& S

olut

ion

...Fo

rce

Load

s

�In

the

2-D

sol

id m

odel

sho

wn

at b

otto

m le

ft, n

otic

e th

at

max

imum

str

ess

SMA

X (=

1206

4) is

repo

rted

at t

he lo

catio

n of

th

e fo

rce.

Whe

n th

e no

des

and

elem

ents

in th

e vi

cini

ty o

f the

forc

e ar

e un

sele

cted

, SM

AX

(=61

41) m

oves

to th

e bo

ttom

cor

ner,

whi

ch

is a

noth

er s

ingu

larit

y (b

ecau

se o

f the

dis

plac

emen

t co

nstr

aint

at t

he c

orne

r).

Load

ing

& S

olut

ion

...Fo

rce

Load

s

By

unse

lect

ing

node

s an

d el

emen

ts n

ear t

he b

otto

m c

orne

r, yo

u ge

t the

exp

ecte

d st

ress

dis

trib

utio

n w

ith S

MA

X (=

3946

) ne

ar th

e to

p ho

le.

Load

ing

& S

olut

ion

...Fo

rce

Load

s

Not

e th

at fo

raxi

sym

met

ricm

odel

s:

�In

put v

alue

s of

forc

es a

re b

ased

on

the

full

360°

.

�O

utpu

t val

ues

(rea

ctio

n fo

rces

) are

als

o ba

sed

on th

e fu

ll 36

0°.

�Fo

r exa

mpl

e, s

uppo

se a

cyl

indr

ical

she

ll of

radi

us r

has

an e

dge

load

of P

lb/in

. To

app

ly th

is lo

ad o

n a

2-D

axis

ymm

etric

shel

l mod

el

(SH

ELL5

1 el

emen

ts, f

or e

xam

ple)

, you

wou

ld s

peci

fy a

forc

e of

2πr

P.

r

P lb/in

2πrP

lb

Load

ing

& S

olut

ion

...Fo

rce

Load

s

�To

app

ly a

forc

e, th

e fo

llow

ing

info

rmat

ion

is n

eede

d:�

node

ork

eypo

intn

umbe

r (w

hich

you

can

iden

tify

by p

icki

ng)

�fo

rce

mag

nitu

de (w

hich

sho

uld

be c

onsi

sten

t with

the

syst

em o

f un

its y

ou a

re u

sing

)�

dire

ctio

n of

the

forc

e �

FX, F

Y, o

r FZ

Use

:�

Solu

tion

> -L

oads

-App

ly >

For

ce/M

omen

t�

Or t

he c

omm

ands

FK

or F

�Q

uest

ion:

In

whi

ch c

oord

inat

e sy

stem

are

FX,

FY,

and

FZ

inte

rpre

ted?

Load

ing

& S

olut

ion

B.

Nod

al C

oord

inat

e Sy

stem

�A

ll fo

rces

, dis

plac

emen

ts, a

nd o

ther

dire

ctio

n-de

pend

ent

noda

l qua

ntiti

es a

re in

terp

rete

d in

the

noda

l coo

rdin

ate

syst

em.

�In

put q

uant

ities

:�

Forc

es a

nd m

omen

ts F

X, F

Y, F

Z, M

X, M

Y, M

Z�

Dis

plac

emen

t con

stra

ints

UX,

UY,

UZ,

RO

TX, R

OTY

, RO

TZ�

Cou

plin

g an

d co

nstr

aint

equ

atio

ns�

Etc.

�O

utpu

t qua

ntiti

es:

�C

alcu

late

d di

spla

cem

ents

UX,

UY,

UZ,

RO

TX, R

OTY

, RO

TZ�

Rea

ctio

n fo

rces

FX,

FY,

FZ,

MX,

MY,

MZ

�Et

c.

Load

ing

& S

olut

ion

...N

odal

Coo

rdin

ate

Syst

em

�A

nod

al c

oord

inat

e sy

stem

is a

ttach

ed to

eve

ry n

ode

in th

e m

odel

.

�B

y de

faul

t, th

e no

dal C

S is

par

alle

l to

glob

al C

arte

sian

, i.e

, all

appl

ied

forc

es a

nd d

ispl

acem

ent c

onst

rain

ts a

re in

terp

rete

d in

glo

bal C

arte

sian

by

defa

ult.

XY

X nY nX n

Y n

X nY n

X nY n

Load

ing

& S

olut

ion

...N

odal

Coo

rdin

ate

Syst

em

�If

nece

ssar

y, y

ou c

an ro

tate

the

noda

l CS

to a

diff

eren

t or

ient

atio

n.

For e

xam

ple:

�To

sim

ulat

e an

incl

ined

ro

ller s

uppo

rt.

�To

app

ly ra

dial

forc

es.

�To

app

ly ra

dial

con

stra

ints

(p

erha

ps to

sim

ulat

e a

rigid

, pre

ss-fi

tted

pin)

.

Load

ing

& S

olut

ion

...N

odal

Coo

rdin

ate

Syst

em

�To

�ro

tate

nod

es,�

use

this

four

-ste

p pr

oced

ure:

1.Se

lect

the

desi

red

node

s.2.

Act

ivat

e th

e co

ordi

nate

sys

tem

(or c

reat

e a

loca

l CS)

in

to w

hich

you

wan

t to

rota

te th

e no

des,

e.g

, C

SYS,

1.3.

Cho

ose

Prep

roce

ssor

> M

ove/

Mod

ify >

-Rot

ate

Nod

e C

S-To

Act

ive

CS,

then

pre

ss [P

ick

All]

in th

e pi

cker

.O

r iss

ue N

RO

TAT,

ALL.

4.R

eact

ivat

e al

l nod

es.

�N

ote:

Whe

n yo

u ap

ply

sym

met

ry o

n an

ti-sy

mm

etry

bo

unda

ry c

ondi

tions

, AN

SYS

auto

mat

ical

ly ro

tate

s al

l nod

es o

n th

at b

ound

ary.

Load

ing

& S

olut

ion

...N

odal

Coo

rdin

ate

Syst

em

�D

emo:

�R

esum

e rib

.db.

�O

ffset

wor

king

pla

ne to

cen

ter o

f bot

tom

circ

le (u

sing

ave

rage

keyp

oint

loca

tion)

.�

Cre

ate

loca

l cyl

indr

ical

CS

at w

orki

ng p

lane

orig

in.

�Se

lect

nod

es a

t rad

ius

= 0.

35 a

nd p

lot t

hem

.�

Rot

ate

all s

elec

ted

node

s in

to a

ctiv

e sy

stem

.�

App

ly a

UX

disp

lace

men

t con

stra

int (

or a

n FX

forc

e) a

t all

sele

cted

no

des.

Not

e th

e ra

dial

dire

ctio

n.�

Now

act

ivat

e gl

obal

Car

tesi

an (C

SYS,

0).

�R

otat

e al

l sel

ecte

d no

des

into

act

ive

syst

em.

�R

eplo

t, an

d no

te th

e ne

w d

irect

ion

of th

e lo

ads.

Load

ing

& S

olut

ion

C.

Solv

ers

�Th

e fu

nctio

n of

the

solv

eris

to s

olve

the

syst

em o

f lin

ear

sim

ulta

neou

s eq

uatio

ns re

pres

entin

g th

e st

ruct

ure�

s de

gree

s of

free

dom

.

�Th

e so

lutio

n co

uld

take

any

whe

re fr

om a

few

sec

onds

to

seve

ral h

ours

dep

endi

ng p

rimar

ily o

n th

e si

ze o

f the

mod

el

and

the

spee

d of

you

r com

pute

r.

�A

line

ar s

tatic

ana

lysi

s w

ith o

ne lo

ad s

tep

requ

ires

only

one

su

ch s

olut

ion,

but

a n

onlin

ear o

r tra

nsie

nt a

naly

sis

may

re

quire

tens

, hun

dred

s, o

r eve

n th

ousa

nds

of s

uch

solu

tions

.

Ther

efor

e, th

e ty

pe o

f sol

ver y

ou c

hoos

e fo

r sol

utio

n co

uld

be im

port

ant.

Load

ing

& S

olut

ion

...So

lver

s

�Th

e so

lver

s av

aila

ble

in A

NSY

S ca

n be

cat

egor

ized

into

two

type

s:�

Dire

ct e

limin

atio

nso

lver

s�

Fron

tal

�Sp

arse

�Ite

rativ

eso

lver

s�

PCG

(Pre

-con

ditio

ned

Con

juga

te G

radi

ent)

�IC

CG

(Inc

ompl

ete

Cho

lesk

yC

onju

gate

Gra

dien

t)�

JCG

(Jac

obiC

onju

gate

Gra

dien

t)

Load

ing

& S

olut

ion

...So

lver

s

�D

irect

elim

inat

ion

solv

ers

calc

ulat

e th

e so

lutio

n as

follo

ws:

1.Fo

rmul

ate

indi

vidu

al e

lem

ent m

atric

es.

2.R

ead

in d

egre

es o

f fre

edom

(DO

F) fo

r the

fir

st e

lem

ent.

3.El

imin

ate

any

DO

F th

at h

as a

kno

wn

valu

e or

can

be

expr

esse

d in

term

s of

ot

herD

OFs

, the

n w

rite

an e

quat

ion

to th

e .tr

i file

. Th

e re

mai

ning

DO

Fsco

nstit

ute

the

wav

efro

nt.

4.R

epea

t ste

ps 2

& 3

for a

ll el

emen

ts u

ntil

allD

OFs

have

bee

n el

imin

ated

. Th

e .tr

i fil

e no

w c

onta

ins

atr

iang

ular

ized

mat

rix.

5.C

alcu

late

the

DO

F so

lutio

n by

bac

k su

bstit

utio

n, th

en u

se e

lem

ent m

atric

es

to c

alcu

late

the

elem

ent s

olut

ion.

Form

ulat

e el

emen

tm

atric

es

Ass

embl

ean

dtr

iang

ular

ize

glob

al m

atrix

Bac

k-su

bstit

ute

for s

olut

ion

.em

atfil

e

.tri

file

resu

ltsfil

e

Load

ing

& S

olut

ion

...So

lver

s

�Th

ew

avef

ront

is th

e nu

mbe

r of D

OF

reta

ined

by

the

solv

er

durin

gtr

iang

ular

izat

ion

beca

use

they

can

not y

et b

e el

imin

ated

. It

swel

ls a

nd s

hrin

ks a

s th

e so

lutio

n pr

ogre

sses

, an

d fin

ally

bec

omes

zer

o w

hen

all D

OF

have

bee

n el

imin

ated

.

�Th

e va

lue

ofw

avef

ront

dire

ctly

affe

cts

solu

tion

time:

the

hi

gher

the

wav

efro

nt, t

he lo

nger

the

solu

tion

time.

�R

eord

erin

g th

e el

emen

ts �

choo

sing

a p

rope

r ord

er in

whi

ch

elem

ents

are

pro

cess

ed b

y th

e so

lver

�ca

n re

duce

the

wav

efro

nt.

AN

SYS

does

aut

omat

ic re

orde

ring

at th

e be

ginn

ing

of a

sol

utio

n.

Load

ing

& S

olut

ion

...So

lver

s

�Ite

rativ

e so

lver

sca

lcul

ate

the

solu

tion

as fo

llow

s:1.

Form

ulat

e in

divi

dual

ele

men

t mat

rices

.2.

Ass

embl

e th

e gl

obal

stif

fnes

s m

atrix

.3.

Star

t with

an

assu

med

zer

o va

lue

for a

ll D

OF

and

itera

te to

con

verg

ence

(bas

ed

on a

n in

put t

oler

ance

on

resi

dual

forc

e).

4.U

se e

lem

ent m

atric

es to

cal

cula

te th

e el

emen

t sol

utio

n.

�Th

e m

ain

diffe

renc

e be

twee

n th

e ite

rativ

e so

lver

s in

AN

SYS

�PC

G,

JCG

, IC

CG

�is

the

type

of p

re-

cond

ition

erus

ed.

Form

ulat

e el

emen

tm

atric

es

Ass

embl

egl

obal

mat

rix

Itera

teto

sol

utio

n

.em

atfil

e

.full

file

resu

ltsfil

e

Load

ing

& S

olut

ion

...So

lver

s So

lver

W

hen

to U

se

Mod

el S

ize

(DO

Fs)

Mem

ory

Use

D

isk

Use

Fron

tal W

hen

robu

stne

ss is

requ

ired

(non

linea

r ana

lysi

s) o

r w

hen

mem

ory

is li

mite

d.

< 50

k Lo

w

Hig

h

Spar

se W

hen

robu

stne

ss a

nd s

olut

ion

spee

d ar

e re

quire

d (n

onlin

ear a

naly

sis)

; for

line

ar a

naly

sis

whe

re it

erat

ive

solv

ers

are

slow

to c

onve

rge

(esp

ecia

lly fo

r ill-

cond

ition

ed m

atric

es, s

uch

as p

oorly

sha

ped

elem

ents

).

10k

- 500

k (m

ore

for

shel

l &

beam

m

odel

s)

Med

ium

Hig

h

PCG

W

hen

solu

tion

spee

d is

cru

cial

(lin

ear a

naly

sis

of la

rge

mod

els,

esp

ecia

lly th

ose

with

sol

id e

lem

ents

). 50

k -

1000

k+

Hig

h Lo

w

ICC

G W

hen

solu

tion

spee

d is

cru

cial

in m

ultip

hysi

cs

appl

icat

ions

. Han

dles

mod

els

that

hav

e co

nver

genc

e di

fficu

lties

with

oth

er it

erat

ive

solv

ers

(nea

rly in

defin

ite

mat

rices

).

50k

- 10

00k+

H

igh

Low

JCG

W

hen

solu

tion

spee

d is

cru

cial

in "s

ingl

e-fie

ld" p

robl

ems

(ther

mal

, mag

netic

s, a

cous

tics,

and

mul

tiphy

sics

). 50

k -

1000

k+

Med

ium

Low

Load

ing

& S

olut

ion

...So

lver

s

�To

cho

ose

a so

lver

:�

Solu

tion

> -A

naly

sis

Type

-Sol

�nC

ontro

l, th

en c

hoos

eSo

l�nO

ptio

nsta

b�

Or u

se E

QSL

Vco

mm

and

The

defa

ult i

s to

use

a �

prog

ram

cho

sen�

sol

ver [

eqsl

v,-1

], w

hich

is u

sual

ly th

e sp

arse

dire

ct s

olve

r.

Load

ing

& S

olut

ion

D.

Mul

tiple

Loa

d St

eps

�So

far,

we

have

see

n ho

w to

sol

ve fo

r one

set

of l

oadi

ng

cond

ition

s, i.

e, o

ne lo

ad s

tep.

�Im

port

or c

reat

e th

e m

odel

�M

esh

it�

App

ly lo

ads

�So

lve

(one

load

ste

p)�

Rev

iew

resu

lts

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

�If

you

have

mul

tiple

load

ing

cond

ition

s,

you

can

choo

se o

ne o

f tw

o w

ays:

�So

lve

for a

ll lo

ads

toge

ther

in a

sin

gle

load

ste

p�

Or a

pply

eac

h lo

adin

g co

nditi

on

sepa

rate

ly a

nd s

olve

mul

tiple

load

ste

ps.

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

�A

load

ste

pca

n be

def

ined

as

one

set o

f loa

ding

con

ditio

ns

for w

hich

you

obt

ain

a so

lutio

n.

�B

y us

ing

mul

tiple

load

ste

ps, y

ou c

an:

��i

sola

te�

the

stru

ctur

e�s

resp

onse

to e

ach

load

ing

cond

ition

.�

com

bine

thes

e re

spon

ses

in a

ny d

esire

d fa

shio

n du

ring

post

proc

essi

ng, a

llow

ing

you

to s

tudy

diff

eren

t �w

hat-i

f�

scen

ario

s. (

This

is c

alle

d lo

ad c

ase

com

bina

tion

and

is v

alid

for

linea

r ana

lyse

s on

ly.

It is

cov

ered

in C

hapt

er 1

4.)

�Th

ere

are

two

way

s to

def

ine

and

solv

e m

ultip

le lo

ad s

teps

:�

Mul

tiple

sol

ve m

etho

d�

Load

ste

p fil

e m

etho

d

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

Mul

tiple

Sol

ve M

etho

d

�A

n ex

tens

ion

of th

e si

ngle

-load

-st

ep s

olut

ion,

whe

re y

ou s

olve

ea

ch lo

ad s

tep

sequ

entia

lly w

ithou

t le

avin

g th

e So

lutio

n pr

oces

sor.

�B

est s

uite

d fo

r bat

ch m

ode.

�W

hen

used

in in

tera

ctiv

e m

ode,

th

is m

etho

d is

use

ful o

nly

for

mod

els

that

sol

ve q

uick

ly.

�Im

port

or c

reat

e th

e m

odel

�M

esh

it�

App

ly lo

ads

�So

lve

(load

ste

p 1)

�A

pply

diff

eren

t loa

ds�

Solv

e (lo

ad s

tep

2)�

App

ly d

iffer

ent l

oads

�So

lve

(load

ste

p 3)

�Et

c.�

Rev

iew

resu

lts

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

Load

Ste

p Fi

le M

etho

d

�In

this

cas

e, in

stea

d of

sol

ving

each

lo

ad s

tep,

you

writ

eth

e lo

ad s

tep

info

rmat

ion

to a

file

, cal

led

the

load

st

ep fi

le:

�So

lutio

n >

-Loa

d St

ep O

pts-

Writ

e LS

Fi

le�

Or u

se L

SWR

ITE

com

man

d.

�Th

e lo

ad s

tep

file

is n

amed

jobname.

s01,

.s02

, .s0

3, e

tc.

�A

fter a

ll lo

ad s

teps

hav

e be

en w

ritte

n ou

t, yo

u ca

n us

e on

e co

mm

and

�LS

SOLV

Eor

Sol

utio

n >

-Sol

ve-F

rom

LS

File

s�

to re

ad in

eac

h fil

e se

quen

tially

and

sol

ve it

.

�Im

port

or c

reat

e th

e m

odel

�M

esh

it�

App

ly lo

ads

�W

rite

to L

S fil

e (.s

01)

�A

pply

diff

eren

t loa

ds�

Writ

e to

LS

file

(.s02

)�

App

ly d

iffer

ent l

oads

�W

rite

to L

S fil

e (.s

03)

�Et

c.�

Solv

e fr

om L

S fil

es�

Rev

iew

resu

lts

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

�Th

e ad

vant

age

of th

e lo

ad s

tep

file

met

hod

is th

at y

ou c

an

inte

ract

ivel

yse

t up

all l

oad

step

s ev

en fo

r a la

rge

mod

el a

nd

then

sol

ve th

em w

hile

you

are

aw

ay fr

om th

e co

mpu

ter.

�N

ote:

The

load

ing

com

man

ds o

n th

e lo

ad s

tep

file

are

alw

ays

in te

rms

of n

odes

and

ele

men

ts, e

ven

if yo

u ap

ply

load

s on

th

e so

lid m

odel

.

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

�Fo

r bot

h m

etho

ds:

�Lo

ads

appl

ied

in a

pre

viou

s lo

ad s

tep

will

sta

yin

the

data

base

un

less

they

are

del

eted

. So

be

sure

to d

elet

e an

y lo

ads

that

are

no

t par

t of t

he c

urre

nt lo

ad s

tep.

�R

esul

ts fo

r eac

h lo

ad s

tep

are

appe

nded

to th

e re

sults

file

and

id

entif

ied

as lo

ad s

tep

1, lo

ad s

tep

2, e

tc.

�In

pos

tpro

cess

ing,

you

firs

t �re

ad in

� th

e de

sire

d se

t of r

esul

ts

and

then

revi

ew th

em.

�Th

e da

taba

se c

onta

ins

the

load

s an

d re

sults

for t

he la

st lo

ad

step

that

was

sol

ved.

Load

ing

& S

olut

ion

...M

ultip

le L

oad

Step

s

�D

emo:

�R

esum

e rib

.db

�Fi

x le

ft lin

e in

UX

and

botto

m li

ne in

UY

�A

pply

pre

ssur

e =

100

on to

p lin

e�

Writ

e LS

file

1, t

hen

list i

t and

sho

w F

.E. l

oad

com

man

ds�

App

ly p

ress

ure

= 50

to 1

00 (t

aper

ed) o

n rig

ht li

ne�

Del

ete

the

top

pres

sure

load

�W

rite

LS fi

le 2

�LS

SOLV

E,1,

2�

Rev

iew

resu

lts fo

r eac

h lo

ad s

tep

sepa

rate

ly

Load

ing

& S

olut

ion

E. W

orks

hop

�Th

is w

orks

hop

cons

ists

of t

hree

exe

rcis

es:

W11

A.

3-D

Bra

cket

W11

B.

Con

nect

ing

Rod

W11

C.

Whe

el

Ref

er to

you

r Wor

ksho

p Su

pple

men

tfor

inst

ruct

ions

.