tessler 1993 qualitative structural analysis diagrammic reasoning
TRANSCRIPT
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7/25/2019 Tessler 1993 Qualitative Structural Analysis Diagrammic Reasoning
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Qual i t at i ve Structural
Anal ysi s
Usi ng Dagrammati c
Reasoni ng
Shi r l ey Tessl er Yum
I wasaki
and
Ki ncho
Law
Knowedge Syst ems
Labor at or y
Stanf or d Uni versi t y
7 1Wel chRoad Bl dgC
Stanfor d Cal i f orni a
943 5
i wasaki @ksl
. stanf ord
ed u
1
I nt r oduct i on
Ci vi l
Engi neer i ngDepartment
Stanf or d
Uni versi t y
Terman
Engi neer i ng
Cent er
Stanf or d Cal i forni a
943 5
l aw@i ve. stanf ord
. ed
u
Abst r act
Di agrammat i c r easoni ng i s
a
type of
r easoni ng i n
whi ch t he
pr i mary means of i nf erence
i s
t he d i r e c t
mani pul at i on
and
i nspecti on of a
di agr am
Di agrammat i c r easoni ng i s
preval ent
i n
human
probl em
sol vi ng behavi or
especi al l y
f or
pr obl ems
i nvol vi ng spat i al
rel at i onshi ps
among physi cal
obj ect s
Our
research examnes t he
rel at i onshi p
bet ween
di agrammat i c
r easoni ng
and
symbol i c
r easoni ng
i n a
comput at i onal
f r amework
Wehave
b u i l t
a
system cal l edREDRAW t h a t
emul at es t he human capabi l i t y
f or reasoni ng w t h
pi ctures i n
c i v i l
engi neer i ng
The
c l a s s
of
s t r u c t u r a l
anal ysi s
pr obl ems
chosen
provi des a
r e a l i s t i c
domai n
whose sol ut i on
process
requi res
domai n- speci f i c
know edge
as
wel l
as
pi c t or i a l r easoni ng s k i l l s Wehypot hesi ze
t h a t
di agrammat i c
r epresent ati ons such as
t hose used by s t r u c t u r a l engi neer s
provi de
an envi r onment
wher e
i nf er ences about t he
physi cal r e s u l t s of pr oposed s t r u c t u r a l
conf i gur at i ons
can t ake
pl ace
i n a more i n t u i t i v e
manner than t h a t possi bl e
t hrough
pur el y symbol i c
r epresent ati ons
Humans often use
di agr ams
t o f a c i l i t a t e pr obl em
sol vi ng I n many t ypes of
pr obl ems
i ncl udi ng
but not l i m t ed t o
pr obl ems
i nvol vi ng behavi or s of physi cal
o bj e ct s dr aw ng a
di agram
i s a cruci al st ep i n t he
sol ut i on
process
Drawng
can
reveal i mpor t ant
i nf ormati on that may not
be
e x p l i c i t i n a wri t t en descri pt i on
and can
hel p one gai n
i nsi ghts i nto t he nature of the
pr obl em Though
such use
of
di agrams
i s
an
i n t e gr a l
part
of human
pr obl em
sol vi ng
behavi or
has not recei ved nearl y
as much
at t ent i on
i n
I as
symbol i c r easoni ng has
One
i mpor t ant
advant age of di agr ammat i c
r epr esent at i on
i n some
t ypes
of
pr obl ems
i s
that
makes
expl i c i t t he s pa t i a l
rel at i onshi ps
t h a t
mght requi re ext ensi ve
search and
numerous
i nf er ence steps t o determne usi ng a
symbol i c
r epr esent at i on
Lar ki n and
Simon
have
shown t h a t even when t he
i nf ormati on cont ent s
of
symbol i c
and
di agrammat i c r epr esent at i ons ar e equi val ent a
di agr ammat i c
r epr esent at i on
can offer
computat i onal
advant age
i n pr obl ems
where s pa t i a l
rel at i onshi ps pl ay
a pr omnent
rol e
[ Lark i n
Simon
1987]
Si nce
humans
r eason
w t h so much
apparent
ease
i n
some
pr obl ems
a program
ha t coul d
reason d i r e c t l y w t h a
di agrammat i c
r epresent ati on woul d
be
moreunderst andabl e t o
t he
user than a
program
t h a t
r easons
excl usi vel y w t h
a
pur el y
symbol i c
r epr esent at i on of
t he
same
i nf ormati on
I n
ad di t i o n
a
di agrammat i c
r easoni ng
program
shoul d
offer i nsi ght
into
t he
rel ati onshi p
bet ween di agr ammat i c
r easoni ng
and
symbol i c
r easoni ng
Such
a
pr ogr am
may
al so
be usef ul
i n
i mpar t i ng vi sual i zati on
s k i l l s
t o
st udent s of
di sci pl i nes wher e such a f a c i l i t y i s c r uc i a l
such
as i n c i v i l or
mechani cal
engi neer i ng
and
desi gn
I n t h i s paper we present our work
ai med
t owar ds
unders t andi ng
t he rol e of
di agr ammat i c
r easoni ng i n
probl emsol vi ng
The
pr obl em
we
chose
f or
st udyi ng
di agr ammat i c
r easoni ng i s t h a t
of
det ermni ng t he
def l ecti on
shape
of a bui l di ng
f r ame
s t r u c t u r e under
240
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l oad
W
have
const r uct ed a
computer
programcal l ed
REDR W
Reasoni ng
w t h
Daw
ngs
that
sol ves t h i s
probl emqual i tati vel y
usi ng a
di agrami n
away s im lar
to
humanengi neers
1
Rol es of
di agram
i n
Probl em
Sol vi ng
Some research has been
done on
t he
rol es
t h a t diagrammati c
r easoni ng pl ay
i n human
probl em
sol vi ng
Novak
and
Bul ko,
[Novak
Bul ko
1992] , f or
exampl e,
have
asserted
t hat
a di agram
and
i t s annotat i ons
serve
as a short-termmemory devi ce i n t he probl em
sol vi ng
process
Such a
devi ce al l ows t empor ar i l y- needed i nf ormat i on to be retr i eved
l ater i n t he
same manner that wri t i ng
down i nt ermedi at e
resul t s i n
mul ti pl i cati on
probl em
f rees
t he per son to performf urt her cal cul ati ons
They
al so post ul ate that a
di agrammayact
as
a substrate
or
concept
anchor t hat al l ows t he
new
part of a
probl em o
be descr i bed rel ati ve to
wel l - under st ood probl em
base
Larki n and S mon di scuss
extensi vel y t he
advantages
of di agram f or
f aci l i tat i ng i nf erence about t opol ogi cal
or
geometri c r el at i onshi ps [Lark i nmon
98 ]
Chandrasekaran and
Narayanan
[Chandrasekaran
Narayanan
1992] , Novak
and
Bul ko
[Novak
ul ko 1992] , Borni ng
[Borni ng
1979]
and
others have al so poi nted out the usef ul ness of di agram to human
probl em
sol vers
as a devi ce
t o
ai d i n vi sual i zati on, gedanken
exper i ment s
or predi ct i on
Fi nal l y,
Novak
and
Bul ko
[Novak
ul ko 1992] ,
Koedi nger
[Koedi nger 1992]
and
others
have
expl ored
t he
i dea
that
di agram
maysometi mes
be
used
not pri mari l y
f or
maki ng base- l evel
i nf erence, but rather
to
hel p i n
t he
sel ecti on
of an
appropr i ate method
t o sol ve a probl em t hat i s as an
ai d
i n
t he
organi zat i on
of
cogni t i ve
acti vi ty
[Chandrasekaran et
al
1993]
sal i ent
f eat ure
of
di agrammati c
r easoni ng
i n many si tuat i ons
i s
i t s
qual i t at i veness
Peopl e r eason wth
di agram
to
get r ough, qual i tati ve answers
I f a
more preci se,
quant i t at i ve answer
i s
needed,
t hey
must
resort to
more
f ormal ,
mathemat i cal
t echni ques
However ,
qual i tati ve
t echni ques
are ext r emel y
useful
i n gai ni ng
val uabl e
i nsi ght i nto the
range
of possi bl e sol ut i ons ni n i t i a l
qual i tati ve underst andi ng
t hus
obt ai ned
can gui de
t he l ater anal ysi s
f or more detai l ed answers I n t he cont ext of
structural
anal ysi s,
knowng t he
qual i tati ve
def l ected
shape al l ows
one
to i denti f y c r i t i c a l f eatures of t he
shape
One
can
t hen
set
up
rel evant equati ons
i n
order
to
obtai n
more
preci se
i nf ormat i on
such as actual
magni tudes
of f orces
and
di spl acement s
at
speci f i c
poi nt s of
i nterest
How
do
di agram actual l y hel p c i v i l engi neer s
t o
make qual i tati ve i nf erences? From
studyi ng t ext books on
el ement ary
structural
anal ysi s, such
as
[Brohn 1984] , that ai mto
develop a i ntui t i ve
underst andi ng
of
t he r esponse of t he structure under a l oad,
we f ind
that di agram f u l f i l l
many of
t he
same rol es as t hose art i cul ated by researchers i n
other
f i el ds
First,
di agram
are used as a vi sual l anguage of structural behavi or that
can
be
understoodw t h t hemnimumof
textual
comments [Brohn
1984]
The l anguage al l ows
t he engi neer
to express
expl i ci t l y
t he const rai nt or physi cal
l aw
that
i s
rel evant at each
part of t he
proposed
structure,
i n
such a way that t he const rai nt s
and
some of t he
consequences
are i mmedi ately
apparent to t he r eader wthout f ur t her
reasoni ng
Secondl y, t he di agramserves as a
pl ace
hol der or
short-term
memorydevi ce by
al l ow ng
t he desi gner
to
sketch
out
t he
resul t
of
one
def ormat i on
and
t hen
go
back
to
see
i f
there
i s
a
f urt her ef f ect
or
i nt eracti on
that needs to be addressed
Fi nal l y, vi sual
i nspecti on of
di agram seem to gui de
t he
engi neer i n
choosi ng t he next
st ep,
resul t i ng
i n a
more
ef f i ci ent
probl emsol vi ng process
t han
i t woul dbe
ot herw se
Havi ng studi ed t he use of di agram i n al l t hese capaci t i es i n t he context of
det erm ni ng
deformati onshape of f ramestructures,
we
have
constructed
REDR W
o
use di agrams i n
al l
those capaci t i es i nways
s im lar
to humans
W
w l l f i r s t
expl ai n
t he
def l ecti on shape
24
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probl em n Secti on The ar chi tecture of RE R W l l be descr i bed
i n det ai l
i n
Secti on
Def l ecti on ShapeProbl em
Determni ng the qual i tat i ve def l ected shape of a f rame structure
under a l oad i s a
cr uci al
step i n anal yzi ng t he
behavi or
of a structure Structural engi neer s
f i r s t make
asi mpl e, 2-
drawng
of
t he
shape of
t he
gi ven
f rame
structure
Gven
a
l oad
on
t he
structure,
they
modi fy
t he shape of t he structural member under t he
l oad
They
i nspect
t he modi f i ed
shapeto i denti f y t he pl aces where
constr ai nts
f or
equi l i br i umof t he
structure
ar e
vi ol ated
Those
const rai nt vi ol at i ons are
cor r ectedby
modi fyi ng t he shape of connected structural
members
pr opagat i ng
def l ecti on t o other parts of t he structure Thi s pr ocess
i s
r epeat ed
unti l al l t he const r ai nts ar e sati sf i ed
The
drawng thus produced
shows t he
f i nal
def l ected shapeof t he f rameunder the gi ven l oad
G ven a
di agram
of a f rame structure
and
a l oad,
t he
programproduces an under l yi ng
symbol i c r epr esentat i on i n order to f i l i t t e
r easoni ng about engi neer i ng concept s
Then
t heprogramw l l use i t s structural
engi neer i ng
knowedge t o pr opagat e
constr ai nts
on t he
di agramof t he structure and
w l l
i nspect
andmodi fy t h i s
pi cture
unti l a f i nal shape i s
produced
that r epr esents a stabl e
def l ected structure under the gi ven l oad
As w th t he qual i tat i ve nature of
human vi sual r easoni ng, the r easoni ng
carr i ed
out by
RE R W
s
al so
qual i tat i ve
The
answer produces
i s
a
pi cture
of a
def l ected
shape
A though
t he resul t i ng pi cture i s
qual i tat i vel y consonant wi th t he probl emsol uti on,
i t
i s
not
nor
does
i t
need t o
be
mathemat i cal l y
accurate
or to scal e
J 2
Fi gur e
1
Steps i n det erm ni ng t he def l ected shape
RE R Wsol ves t h i s
t ype
of def l ected shape probl em
by
di rect l y
mani pul ati ng a
r epr esentat i on
of
t he shape i n t he
manner
shown above A though t heprobl em
coul d
be
sol ved by sett i ng
up
equati ons,
vi sual i zat i on
i s a i ndi spensabl e f i r s t step that pr ovi des
an
engi neer wi th an i ntui t i ve
under st andi ng
of
t he
behavi or
of t he
structure and
enabl es
her
to r ecogni ze agood
strategy
f or further anal ysi s
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Before descri bi ng howRE R W
nal yzes
structures, we expl ai n br ie f ly
t he
r easons
f or
our
choi ce of t h i s def l ect ed shape probl em n
advantage of t hi s c i v i l engi neeri ng
probl em
domai n
f or st udyi ng t he r ol e of vi sual r easoni ng i n
probl em
sol vi ng
i s
t he
f act
t hat i t
i s r i ch
w t h
domai n- speci f i c
knowedge t hat has si gni f i cant i mpl i cat i ons
on
how
t he di agram s mani pul ated
and
i nt erpret ed
One possi bl e
domai n
i n wh ch t o st udy
pi ctori al r easoni ng
i s
geomet r y, where pi ctures are abst ract di agram w thout bei ng
a
r epr esent at i on of
anyt hi ng
i n
t he
worl d
I n
geomet r y,
t he
onl y
pr oper t y
one
r easons
about
i s
t he
geometri c
pr oper t y There
are
no ot her
t ypes of
i nf ormat i on,
apart
f rom
t hat
r epr esent ed i n t he di agr am
t hat
one must
t ake i n t o account when mani pul at i ng
and
i nspect i ng t he di agram
I n cont r ast ,
pi ct ures
used f or
r easoni ng i n
engi neer i ng desi gn are not si mpl y abst ract
geometri c shapes but
act ual l y
r epr esent t hi ngs
i n
t he
real worl d
Fur t hermore, how
a
pi ct ure
i s
i nt erpret ed
and
mani pul ated
depends
si gni f i cant l y
onwhat
r epr esent s
For
exampl e a l i ne i n
our
domai n
r epr esent s abeamor a
co umn
hangi ng
t he
l engt h
of
t he
l i ne woul d change
t he
i nf or mat i on
r epr esented
by
t he
di agram
I n
a
ci rcui t
di agr am on
t he ot her
hand one
coul d change t he l engt h or curvature of t he l i ne
r epr esent i ng
an
el ectr i cal
connect i on
w t hout
changi ng t he i nf ormat i onal cont ent of t he di agram
For
t he
goal of bet t er
unders t andi ng
t he
rol e of vi sual reasoni ng i n probl emsol vi ng and
i t s
rel at i on
t o
symbol i c
r easoni ng,
i t
i s
i mpor t ant
f or
us
t o work
w t h
a
probl em
requi r i ng
a
weal t h
of
domai n knowedge
t hat
has si gni f i cant
i nf l uence
on t he waydi agram are
used
and
i nt erpret ed
3 Archi tecture
of thesystem
Fromexamni ng
t he
way def l ecti on shape probl em are sol ved by
humans
i s
appar ent
t hat sol v i ng
t h i s
t ype
of probl em
r equi r es
not
onl y
an abi l i t y t o mani pul at e
and i nspect
di agram
but
al so
subst ant i al structural
engi neer i ng
knowedge St ruct ural
engi neer i ng
knowedge
about t he
propert i es of vari ous
t ypes
of j oi nts
and
support s
i s
necessary
t o
i dent i f y
const rai nt s on t he shape f or
t he
structure
t o be i n
equi l i br i um Suchknowedge
i s best r epr esent ed
and
mani pul ated
symbol i cal l y
On t he
ot her
hand,
i nf ormat i on
about
shapes
i s
best
r epr esented
as
a
pi ct ure
Many
t ypes
of
modi f i cat i on
and
i nspect i on
of
t he
shape
are al somore easi l y carr i ed out w t h api cture
The requi rement f or bot h pi ctori al
and
non- pi ctori al represent at i on and reasoni ng
suggest s
a
l ayer ed archi t ecture
Thus,
RE R W
ncl udes
both symbol i c r easoni ng and
di agrammati c
reasoni ng components
The
former
cont ai ns t he
knowedge base of
structural engi neer i ng knowedge
about var i ous
t ypes
of
structural
members j oi nt s,
support s,
and
t he
const r ai nt s
t hey
i mpose
on t he structure
t
al so
i ncl udes a
const rai nt -
based i nf er ence mechani sm
o make use of t he knowedge
The l a t t e r
diagrammati c
r easoni ng component i ncl udes
an i nt ernal
r epr esent at i on
of t he
t wo- di mensi onal
shapeof
t he f rame structure as wel l
as
a
set of
operat ors t o mani pul at e
and
i nspect t he shape
These operat ors, some of wh ch are shown i n Fi gure
2,
corr espond t o
t he
mani pul at i on
and
i nspect i on operat i ons
peopl e
perform
f r equent l y
and
easi l y w t h di agram wh l e
sol vi ng def l ect ed- shape
probl em
TheStructure
Layer
cont ai ns
a symbol i c
r epresent at i on of domai n- speci f i c
knowedge t
r epr esent s
non- vi sual
i nf ormat i on such as
hi nged
j oi nt rotat i on ,
var i ous
t ypes
of
structural
members equi l i bri umcondi t i ons, as wel l as heur i s t i c knowedge
f or
cont rol l i ng
t he
structural
anal ysi s
pr ocess
TheDagram
Layer r epr esent s t he two dimensi onal shape of a structure There are
several operat or s t hat di rect l y act on t h i s
r epr esent at i on
t o
al l ow i nspect i on
as
wel l
as
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t r ansf ormat i on of t he
shape These operators
correspond
to the operat i ons people
perform
easi l y
w t h
di agram The
i nternal
represent at i on of a shape i s a combi nat i onof
a
bi tmap
whose el ements
corr espond
t o each
poi nt
i n a pi cture, and a
more
symbol i c
representat i on where
each
l i ne
i s r epresent edby a set of
coordi nate poi nt s
The D agramLayer i s i ndependent of t he structural
engi neer i ng domai n
i n t he sense t h a t
i t does not cont ai n
any
structural engi neer i ng concepts
However
t he types of
both
mani pul at i on
and
i nspecti on operators
provi ded
f or the
l ayer
refl ects
t he
r equi r ement s
of
t he
domai n
For exampl e t he assumpti on that t he f rames consi st of i ncompressi bl e
members
made apart i cul ar set of operators necessary e. g t he program equi res a
bend
operator
but not a stretch or compress
operator) , and
al so
by
the speci f i c f uncti oni ng of
t hose
requi r ed
operators
f or exampl e, t he
bend
operator creates amoderate curve rather
t han
a
compl ete
bend
t hat
woul d
cause t he
l i ne
endpoi nt s
to
touchor cross
; or,
t he i nspect
operator
may
l ook at components connected to
t he component
i n
questi on, but w l l not
compare
that
component
to any other, as
mght i n
some
other domai n
Structure
Layer
Obj ect s :
beams,
col umns, connect i ons,
supports,
l o ad e t c
Operators :
generate- f orce-equi l i bri umcondi t i ons,
generate- moment - equi l i br i umcondi t i ons,
et c
Dagram
Layer
Obj ect s :
Operators :
l i n e s s pl i n es c i r c l e s
Mani pul at i on r o t a t e bend, t r a n s l a t e smoot h,
e t c
I nspecti on
get- angul ar - di spl acement , get- di spl acement ,
symmet r i c
a l p,
et c
Fi gure 2 Types of
obj ects
andoperat or s i nREDR Wrogram
There i s a
cl ose
l i nk between
t he i nf ormat i on i n t he two l ayers The systemrel ates t he
represent at i on of
a part i cul ar beami n t he
Structure
Layer to a spl i ne i n t he
D agram
Layer,
and
t he concept of
def l ecti on of a
beam
to an operat i on on aspl i ne t o transformi t s
shape Li kewse, t he system
i s
abl e to i denti f y f eat ures
of a shape e g
di recti on
of
bendi ng, exi st ence of an
i nf l ecti on
poi nt )
and
to communi cate
them
to
t he
Structure
Layer
Communi cat i on between
t he two l ayers
takes
pl ace
by sendi ng commands and posti ng
constr ai nt s
by
t he Structure Layer,
whi ch
i s carr i ed
out
or checked
by
t heD agram
Layer
Fi gure
5 shows
t he t wo- l ayer ed
archi tectur e schemat i cal l y There i s a
t ransl ator
between
t he two
l ayers
t o
medi ate t he communi cat i onbetween t he
two
l ayers
Went he
Structure
Layer
posts
a constrai nt or a
command
t he Transl at or
t ransl ates i t
i nto a cal l to a
D agram
Layer operator
t hat
can di rectl y act on t he r epresent at i on of t he shape to
mani pul at e
or
i nspect The
resul t
i s agai n
t ransl ated back to concepts that the Structure
Layer underst ands
STRU TURE
L YER
Def l ect
B1 Ai r down
00
D GR ML YER
BendB pi c : y
D agram
Representati on
Fi gure
3 Two l ayered archi t ecture of t he
REDR Wrogram
24
4
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The
RE R W
rogramhas
been i mpl ement ed and has
successful l y
anal yzed
s i x
of t he
3 basi c def l ected
shape
probl ems
descri bed
by Al l en [A l len
1978]
An i nf ormal
eval uat i on by a c i v i l
engi neer shows t h a t t he
pr ogram
r e f l e c t s t he
q u a l i t a t i v e
reasoni ng
process used
i n anal yzi ng
f rame s t r u c t u r e s , and
t h a t i t woul d be usef ul i n hel pi ng
students
and
novi ce
engi neers
l earn
t o
sol ve
t h i s
type of
probl em
3. 1 Exampl e
I n t h i s s e c t i o n,
we
i l l u s t r a t e
t he probl emsol vi ng
process byRE R W
i t h t he exampl e
present ed ear l i e r i n Fi gure
3
Wi l l u s t r a t e the type of
communi cat i on t h a t
takes pl ace
between t he l ayers G ven t he
f rame
s t r u c t u r e of
Fi gure
4 a ,
wi th a l o ad ,
Load3,
pl aced on
i t ,
t he Structure
Layer, S
sends a command,
Def l ect
Beam3 i n
t he
same di recti on as the l oad, whi ch
t he
Transl ator, T
t r a ns l a t es
into
an operat i on
Bend
Beam3 p i c i n t he negat i ve
di recti on
of
t he y- coordi nat e . Carryi ng
out
t h i s
operat i on
wi l l
r es ul t i n t he shape shown i n
Fi gure
4 b)
S i n f e r s t h a t si nce J oi nt3
i s
a
r i gi d
j o i n t , Beam3
and Col umn3 must
r emai n
perpendi cul ar
t o each other at J oi nt3
S
i s s u es a query t o t es t t hi s
const rai nt The
query
i s tr ansl ated into
get t he angl e
between
Beam3. p i c
and
Col umn3. p i c
at t he ends
connect ed
by
J oi nt3
. p i c
f o r
t heD agram a y e r , The
answer , t he actual angl e between
t he
two
l i n e s ,
i s
communi cated
t o
S
as
t he
answer
t h a t
t he
const rai nt
i s
not
s a t i s f i e d
S
now
i ssues a command t o
s a t i s f y
t h i s constrai nt whi l e
keepi ng
Beam3 f i x ed, whi ch i s
t ransl ated
i n t o make the angl e
between
Beam3. p i c
and
Col umn3
. p i c at J oi nt3 . p i c be 9
degrees
wi t hout
modi f yi ngBeam3
p i c
f o r
Carr yi ng
out the
operati on wi l l r es ul t i n t he
shape
shown i n
Fi gure 4 c)
Communi cat i on
wi l l cont i nue i n
t h i s
manner
unt i l
al l t he
const rai nt s
a r e s a t i s f i e d
Fi gure
5
shows REDRAWs
symbol i c
r easoni ng a c t i v i t y f or the
same
exampl e
a
I
b
d )
e )
Fi gure
4
RE R W
ol uti on t o f rame
s t r u c t u r e probl emsketched
i n
Fi gure 1
24 5
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Def l ect
BEA.0
i n t he
same
di rect i on as
l oad
i o t r r r 3 i s
of support
type f i xed at
90
.
Const r ai nt
A Angl e
at
coL3 must be
90
t o BEAMS
Get angl e
bet ween
COL
and
BEAMSat J o t r r r 3
Sat i s fy
Constr ai nt
A
Make
angl ebetweencoL3
andBEAMS
be 90 whi l e keepi ngBEAMS f i x ed
( S im l a r l y ,
f or
angl e
between coL4 and
BEAMS )
suPPoRT3 i s
of support
type
f i x ed at
90
.
Get angl e
of
COL3 at
suppoRT3
S a t i s f y
Constr ai nt B Make
angl e
o f c oL3
t o
suPPoRT3 be
90
( S im l a r l y , f o r angl e of coL4to suppoRT4.
Constr ai nt C
Moment
ar ound
i our r 3must
be
zero
Get
moment
ar ound i o m r 3
Sat i s fy
Constr ai nt
C
Establ i sh
moment
equi l i br i um
ar ound i o i N - r 3
S i m l a r l y
f o r momnt ar ound j o i a
C 1
3 . 2
Di scussi on
Constr ai nt B Angl e at
suppoRT3
must be 90
t o
COL3
6
BendBEAMPic i n t he negati ve
d i r e c t i o n
of t he
y-coordi nate .
Get angl e
bet ween
coL3 . P i c andBEAM3 PIcat
i o L NT 3
. P I G
Angl e
bet ween
coL3
. P i c andBEAM3 Pl c i s