.,. C'.......L- 1. ~...,

T..

-' 0: ....

....:'..r. r.r: S.

Ii •

eele

1

.,

T e ~eport gives a sumrr.a~ized p_e entation of +ne four

theories deve~oped for t e c . utaticl of t"e deflect~ons or

beams. the application of the heories to va~ious beam and

connection tes s is disc ssed, an the theo~et1cal resu~ts

a 4 e co~pared witn the test results~ l~O+ all studies are COl

plete because of the limite ti e available in ~. ~. 113.,

and it wil oe necess ry to do some f rther worK on the co

putatio: of theoretics: curves •

•

.-

,"'....1 u:-~ <...

....,:: c .. e

u .... l_ A...Y

.... _0 ,y

•

• • •

• •

•

...1

2

..

.......:......

r.. e e 0..; nJ.J" Ie ~al

L i~ures and

f' e5! S, continuou~ beams

L

UOU 1'r.

T e ptlrp)se

.. ir COFPO:lt:l:ts·'.

t, is re rt ~s to s l:')8.d defle,:t ion

relations 1p tween the t~st result a d the ~heorles t~8t ...have been developed by va~ious researcners :n this field.

Sc::)pe

This stul1y was conducted on the basis of a three crea.it

hour course. ~he tiree as li ited, tlerefore, and so~ sub-

e~!s could o~ly be touched po~ because o~ the lack of time.

In the cont L uous De m se es pract ically a ... l vea I.S ..'ere in­

veEtig8ted ara. the res~lts plottea. uP. rom the connections,/'

ts. everal nJ~ arisons were

only "fe ~ere sleeted or s

uf the simply s ported bea~

dy an~ presenI . ~~,

in t his report.

made and discussed in the repo t.

.t .....l..- .I. 1u... oJ

.- vU Lloi t ~a 0:

"_!'e~~ r "il :ar.l .; lJ••-~

..... L:m: ;:, . ' '"4 ......~- .l._

::I"TJ

" . j uer

-2-

To_ first sec:ion of the report ~ e~ent t.e r ur de-

flec~ion theo~i~s and the ad .t ges ar: disad ontaGes of

each t.lec:-y. "=>everal fi res are i elude' we i h explain

certa'n aspects )f each theo~y.

I!i he second part of the report. tr. e ,9.. icati.ll of the

1 rt omin s of t.e theories

S disCll"

I is very v: ~nt,'" ... rf r~nt testv5r~o s theories to

frol a 1 10+ ... :Lat:n t:.e hi uer :'ast'c tln'e I.d .?ar 10-

. larl i.r .... e s .. i'l- ar

at :: is 01 t i very .i h f. Le

e e ::'e_+ f tIeJJ. ....

• I.. .;: _= U6.:. hortc _:.. .... n •

.-... : t . eul . )

c . p the . r... 0 fshe'i. tt1

fo

. nd i~..; sed •

wubjec , to B:, to find

teel remuer-.

-...-

e 11 it +ions o~ d~fle~.1 ns f ~

J'he oe .... 1\" e uati . for fincins 1-: e deflectL.. n... ­= - - \r'i .. i!.J~icates) o~ ~o ..:..·~e, the

is,

_ rota-

tio. of tV! f.L r.e .• lent;..t: apa ... t. ....n t .. e el t:.c

" ge, t h i =e1 t 1..

¢ : ~l. It .as oe fo~

~m 11 stic ef... ectio .. r ~ "I.l. • tha tb rela.' r~s .. ·p 1s

':

n longer linear.

'or detern inate struc ures, he ..,eflec!.<: 0... c . ue fo lnd

e~s 'I.' be.... ause t' .." mo ~. t i_ 1. de.. c dent of" tn,:; -e.lectlon

aLd t e X~¢ curve can e in e rated by numerical or nalyt1cal

me~hods.

III statically inoeter ir.a te s true 'ures, howt;'ver, this is

n ~ possible and a new idea has to be introduced t et re-::;ults. 'his is done y considering tne cont n ity at t e

boundaries. n the following, four .e~hods are discussed

ri;3fly by 'uich deflections cati be s .... lvcu.

~imple Plastic ~heor

~fue ... a otruc ure is loadwd be.ond the elastic imit it

goes int w e plastic ra get ~he stru ture ~ill car'Y load

'nti 't laS Oecor....e C lurle el plastic•.~w ~ •• ·s taint, a

.. U

-4-

hinge is deve'oped at the support and continued loading

, ill produce more deflect1 .• T 1 Ii o::aent at this point is•

called the plastic hinge 0 ent. '1 he 18[,raI_s will explain

this:

Zlast1c Ir.i tiel1 ... tic

Plas . cF nr-e

N.A

c.

'i'his tht::ory ~ mp· e ly net.;:ec~s whe straL -hardening

.. effe~~ 8Ld ¥'ll, tlerefore i Te 0.1.:" approx'mate res It •

H wever, since eflectio. ~S usually li~1te by spe~ificat10~s,

the strai h r eni:l c' effect ill be of· little influence on

t ,Le result and fa ir a ..$wers can oe expec ted in t !'J.e early part

I t e pIa-tic range.

,e iskopf' s Letbod

l'his ap oach is also called the rr.ethod of mathelhatical

in egration. It h s different e ua ons for t.e thr6e parts

of the '.:- cirve elastic, nlastic, st ....aln-!1 den1 • l~'h

c'lo' i. ai I arr..'7il_ s ow t ese . ,",le t ions:

..

-~- ---------------------

-~-

A'- - -, --.;------=-=--=-=-=-::::--:-=--=-=--=----=~---B.-

Mo i----r

Mp

M

o ¢

Loment-~ atlon I,,; rve

iel pint of tile aterial

: ~I for >1 stic portio

6'.' V Crt: ~ ~.~ fo ... plastic portion..

-'

.. (f,

ondi equat ons are:F

l:p = plastic linge mo nt

¢d2 M- !..Zdx.... • -:BT""

A a d l:S are consta 1ts of the material i- t e trein har en1ng

!

range •

. he cor:putation car.. e s 'mplified ne lecti~ t:e

.rain harden"Ii;, froll.rl or to IJ. 'h s

an e

ne elastic

el1min tea a d lef¢ i

las~ 1c range in t ".e t .rl;:~ ,leis opf equ t ion •

f=o~ 0 to al.an" the pl~..... 0 ge " i 1 t ..e exte d...

f~om1 to an +~o

A -, ....... \wt

~he second eq tion 0

¢2"

•v-...

-c-

2¢ & 1.3

In symet4ic - " loaded struct .... es, e_e e 1 a lor.s c n..."

..e int ~a",e ve y sir. n::'y ..0 11 \ ark sn rter.

,.l he 8S-•

v tion aoe t ~ -ia t ~ fa .... fr eisl f' ... o_y and

e'1vE:o ace ra:e ~es It • ... or t: i tethod the tier ectiOI c

will e of the sar, e s 8. e, as ",U?VB .. J'"Len~ncies b~"': eel th's t etho c:h 0 ~ ••e or! ' ...1&- i",1 kc f

n..eth d, bL.t ••ley do not a ..o r.: tu ~ :ilffel'e (.~ foe than

InOl 1.t •

Fe.......... ter: t "' ~..1 L. d easi b~' • ! er.s io•. s 0 t e ...-'/; i . ~m.

.. 5 ~eq 1-e5, of ~_u~s~, .. 0 t t e 'tre s-~t ain I lation of

1. inceterl 1.. ate ::.tr c-, ,o

.. res olu t ions a. be~ 0:' a serie;:;. of r lfJ.xat· o__~...

"he str .. u e is e::"st'c aLj t.e aef e : on c

u e b~~o='in~ to t~e e1 oJ C theori~s. 1f, 1. wev r, t 11

.Lor:.el.t ets ::'arge- ;;:an thE; yie::'d .orr.~I.t, tae shear at t:.e

t.e struot' .....e, :r .:. t ':'s c 1 tiL \1<:, 1',11 not e

aro; a Q :.e ax~s as to 1.1 e .. , r:':'s ie ne :r..ro t.: ..

t ... tion i tn Dear is z

'..ls D.e .0 ~ ~ duces t. e ost acc": e" re5 '1- • it Co so r",-

ui as rea t 81'.0 t If i e:;:-ef - ,no ale •. 9 OIly ryl ce wn.~~ : .~S ~e 4- .... r:ot... 1 the e r: st 1"" .... 8 .. .e..:aus~

1 '" r S5"" . ne ._...oe" 8.

o .. ,.e

...

Or ly the ear y plasti0 ra.nge ".,as c ~p te f r these

bea:r.s nec' se not enough ti e was aVliilucle to fig re th

t£leoretical curves for the whole ran r;e. .l;i.S far as toe

theoretical curves were con.,uted, ho.,./ev:> r t ley snO'l1 GX-

cellent a 'reerrent uetween test result. 8Ld theory Lhe~ ,

iscre,a c' for the as de~ivereu wears y8s less tlar. t.at for~ ~

t e ~nnealeQ onev. ~ pare.tly t ,e ar eb'in ~oes not in-,

crease toe stren ~th of t!ie M~' ers a: preciaul '.

.-nly test ... o. 3 s' .O,ie ..... a lllore than 1 IV • ifference be-

t~een theor and test. Po epnarent an_ er can De give

tho gh for this, because all toe test ~un iti ns ~ere ex-

aetly the s me

!..os t of tie studies 'fer made o. the c ntinuous eanlS

.shd co.. pl~te ~urves were plotted for almo t all C3U!lS. .in

the COT... t inuous t:::a " series, lar~e - se t;;T) • ...; 1_ .. l"eCl:ime ap arent

and "') i:1t 0Ut t.1_ sh rteo 11 g" of the tieo les.•

In al_ ses, tr.e .;J eld int tL 1 test is -0-&

pro:.i..ate y 1 elO'1 t: ~ "•.eore 1: ice.l ~o P' te . v lue. ',1e] -e ',od lY g) n-9.S tee bi€..l1est y e a • + "Ih eh is-~ .0_I:

iu. uti ir. ...et-:. i . t,e - n ry. .1.1 s

, :i t a •.

ex t,U] .... o' = •10_ 0:3 • J

.p~

~ 7 .ic !.~ ." r..,.-t,e r.- •• .J .. ~ .1 . ~, - ..... ..L..'"t ;:.. -

O· e tc 4. :;

s in~plif ied

c "' e

't,

n

v~. i ... 0.

.if

.4. ........... _

L s

C:O i.-C. tr

e.

:oad':' ...

u"" o",t inea.

.0

1'., _ rve

~l d 1 ...\. .. '"' ~ I... . v ~ .)_....

o

be s tnoe ty iCC1

tnst ll_ ... ..... r...e t.e 51 e r

..e 1;;> •c ~e.r re.

te \,; r. t L ucu.~ \lea. f .t~L~ .. .. I., ~

\;.p' ard e l enc./

also t Co .CO • C. f~or.. '"'

L.t:J <2"C.. - ur. I i:.. ir.f:l te 4- 1e a ave....

..

-

.'

y

-1,-, -

...... . .... ,'re~s

ht•

.ieiskopfl{t:in -e II ,

Yor '.

[al ~, ~hin

Pl'ogre~

..

Yar.g, _ h., " ....i.::>serta 01 :!'e~ enive .ity, l~ ... l.

-1 -

ITIhe tes ts re rted. OIl here:) ... are a

le o. 1 c i''F 40 s del·verea. 14'-v" ...... 6.n

It -.o. 2 8 ,F At' - .. !Laled "_v

If 1"0. 3 8 ,7 ~ e~i~leredTt It

n '0. 4 8 .rF 67 b. ..nealed " tt

C

....eat. D2 ./F 40 ::>' t.ila -:ed. (,; ntinuotl.s e m

If 8 .•F 40 :::>illlulate f an'e3'

" D4 8 'F 40 imulated Jontinuous ueaniS •.~

It B 8 ;rF 40 t1 tt n

B7 14 .J' 30 tt It " -Strai""h

'ormection A 8 b 13 >.Ilee conr........ c ~ion

Tf K 8 B 13 t1 1f

tt 1. 8 B 1~,. It

" 1: 8 .d 13 " t1

- 2-

pFor an exan.ple a cuil t-in oean is used as sho r~ on

P2

Po

P p

fe- .!- b

.~P.

A - B c ..

Load- efle c .. i '"'n vur-v~

In order to f'n the . flec~iont Ge values of Pl , Pa

a d P2 have to oe found which is possible TIit a hbrd~ook.

n oL2

"'1 ab2

l~",L

Pa = ..,9-

PO) - .c.. 0- -....

1 - ~ (f _[~t...::_:.) l )11~ u+,",\f-1A~

• II - f J" en:

d :

d :d.d •• (P -

P2 - "[) )"al~ b a}

10 curves of ~e siIil.le 1 stic hod were sho n ec se

the I ere presented in an e rlier re ort.

--'his . thod i very time consQ~inc a d to rrive at a

satisfuct,ry solution, a nu~ er of trials have to e rna e.

line ffiathematics involv d are very simp] e.lin that the arddS

of e ..-1; curves Cire compllted and tneir eoout

the ee ter line 0 tne ue~:w.

~his ~etnod 's rathe 8 "e'" 8 •• 1 .} •..,e. son:e bee se

-che to tl:il aflec tior. 1... bro} r. p i.t 0 ... 1.' P l·ts aLe. ~ c

part 1'e ui es a great a ount or cc :u:st'on .wor. It i

posst to w rk th's ,et d tle Q ~e ~ ·s n~drieal inte-

'ration mort::: or less f;raphical.,

elovl are the e({ua ti ::m .mien tire nece..,sl:i1'Y for an an-

alytic!::l solu ion. ...·he eflectiOl: is fi"'ure for "he elasti ,

.1aEtic an strail -hBruenin o tiOlL~.

y ='l'he total efl c t i . is '<;' a to:

"111 + (,)1

2+ (y) 13 + di} 12 L-l2-l3)

1s cor p03ed f:

(y)ll di ~. )-,.

( il2 : 4~ V (Fr t a x

(y) +~

'"

C :. + = t,).:) J...

-l'-t-

= 2-6' 7

V"Oj t e :.r.~

.....u....

,L• ...

I d • a " + :x..~ ::... I

3.....

a

b

1. n.a n t

.1 .,'1 . ". -} ~ i .

_....operty of s. ~:' .. _ J.. ... S~ .. J. -

.. 1 i. ..e 1a

of ::ot ....

cf e':&2tic' y

, ex r. Ie o~: t.

t' e r·~'. tor the ot~pr t"eo 'es this .. ',~~d oe very rd •

to solve ....; Vley ..... _·, 'it,. YCJ.~.. t J th c'y, t1e .. 1"::>' r- 0eCCl....es

• c~ctal ef.ecti .,.. is "':'81. - .'" ..... ..v.

0= , \d_ .,. d.J)...'''no "" da

:. d f.!.e :'on 0: t ,- ca!l~'::'evert ..... '-" ~ .

0' = Q':.flec l.~ 0

"

Oa - y + ]2 +',-re' [ ~ 'j, G' - . ~~~= -'- AX --=. _

:r :; ..>Li

- V(L'Y2 - X - X

(I -X.)Y =....

he ... e:

- rr.X = - ( , -

y;­•

1 [ .,2 J]L. .... - Z 'l .

~= - ! - .... . l··2 J.oJ ... "a

d - ¢3 L- X0

N :re n< i 9'1'<7-....

ea fI L oit .-1' C lr e

M a -l--...---,,-=-=.--------=====---r-­"

M

'*' X AL

f • 0 < •.

for 0 < •

..

•.. ...'

"G)

~~r--

s,

~t:---~

8 2

~,

/ --C. E. 113.

P '.

DE F LE CTIONS OF BEAMS

(s imple be ams . B I and B 2)

I~

- actual-

C!> Weiskopf,

I Il:. num. iIlte"gr.

II/

f.-- .

0.5 1.0 1.5 2.0- .

~ .

10

o

60

40

50

(/)

Cl.. 30~

z

0 20<{

0"-.J

DEFLECTION IN INCHES

•.. .,

60, .

~ ::::==B~

50

~~ B...

/"

40 I(I / C.E.1I3.

Deflect ion of Beams

CflQ.. 30 (simple beams B3 and B 4)- v-~ - actu al

Jf' .'

z. (9 Weisk·opf,

0 208. n u m. int e gr.

<{ /0..J

10 /.

I

of

0.5. \.0

D"EFLECTION' IN INCHES

1.5 2.0

. ' . •.. .'.

2.01.51.0

DE F L ~G T ION IN INCHE S

0.5 '

" -.<

-

~~"

~ L--- B4-

~v---- --------

,-'

/~

82

C.E.113."

~

Deflections of 8eams

(continuous beams 82,4,5).

II'/- act u 01

.~:~:~.;.~; }" 'eo num. inte gr.II', .

H. Yang8

«:> Wp.i!=;konf

,

..

In" k;)

40

10

o

20

50

30

60

z

o<{

o...J

(J)

CL,~

;T

40

o

,l

:

.'y--~

\

1/

I ~ Ba

I~. , .

;)." ..

.. " ~

C.E.113.-. ,.1' f.~i-1;: ,... , ..' .. -,.,'

Deflectio ns of Beams

(simula te d frame 83')

- actualCl nU,m. integr.

/I

,

/

,

. j

(f)

CL

~

za«a...J

i;' ,

60

50

20

10

0.5

. ..... \ l~.

1.0

o E F LEe T ION ' IN INC H E S

1.5 2.0."

G>

~

•... ..

70

60

50

(J)

Q.. 40~

z

~ 30o.-J

20

10

o

~ ,

) V

.~

B,

V "

/17 G. E. 113..

Deflections of. Beams

/ ( con tin 'i: Q.lJ"S be am B 7)".'

- ci ctual

~ num. integr.

8 H. Yang

f 0 We isk'opf

. .

..

.«'

/.

,

O. 5 I. 0 1.5 2. 0

DEFLECTION IN INCHE S

..", '".. . ."

DEFLECTION IN INCHESv

0 /,

~~'7'-

,,/~

l0

v- ~~

~n

~ A -.....0::::::

~0 /

~

/C. E. 113.

Detlection of Knees::T8

C A,K ,L, M ).

II~( connections

_ actual

.. H. Yang

;:Ic

/I~~

O. I O. 2 0. 3 0 4 O. 5 O. 6 O. 7 '0 8 ~

o

20

70

100

60

300

z

. - 50ena..~

I

r0 40z