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
.