Tessler 1993 Qualitative Structural Analysis Diagrammic Reasoning

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

1


3/10

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

4


4/10

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

243


5/10

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


6/10

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


7/10

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

Tessler 1993 Qualitative Structural Analysis Diagrammic Reasoning

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