Journal of the American Concrete Institute, Vol. 16, No. 6

■i « ' i i 'n i ' n i c a m m d e s h ip in fin d in g the b e s t w a y s to d o

'+¿¡1 hi %,'u' iJtn Ifi <P/K I kin d s a n d in s p re a d in g th at k n o w le d g e

J O U R N A L

P-95I^H}H5~ the

AMERICAN CONCRETE

( A C I P R O C E E D IN G S ¥ \ T O H H T T T T ' l ? (C o n te n ts on

V ol. 41) U N O 1 1 1 U 1 £ j B a ck C o v e r)

V o l. 16 June 1945 N o . 6

TO THE .AMERICAN PEOPLE:

Your s o n s, husbands and b r o th e r s who are stand ­ in g tod ay upon th e b a t t l e f r o n t s a re f ig h t in g f o r more than v ic t o r y in war. They a re f i g h t ­ in g f o r a new w orld o f freedom and p ea ce.

We, upon whom h as been p la c e d th e r e s p o n s ib il­

i t y o f le a d in g th e American f o r c e s ,, appeal to you w ith a l l p o s s i b l e e a r n e s tn e s s to in v e s t in War Bonds to th e f u l l e s t e x te n t o f your

c a p a c ity .

Give u s n ot o n ly th e needed implements o f war, but th e a ssu ra n ce and backing o f a u n ited p eo p le so n e c e ssa r y to h a ste n th e v ic t o r y and speed th e r etu rn o f your f ig h t in g men.

¿ 0

$ 7 .5 0 b y th e y e a r

$ 1 .5 0 p e r c o p y Extra copIm to m em ber. $ 1 .0 0

DISCUSSION

( P U B L IC A T I O N O F A L L D IS C U S S IO N T E N T A T I V E L Y S C H E D U L E D F O R J U N E 1 9 4 5 IS P O S T P O N E D T O N O V E M B E R 1 9 4 5 S U P P LE M E N T )

Discussion closed

P re p a ra tio n o f T e ch n ica l Papers— W . D. B ig le r

The Effectiveness o f V a rio u s Treatm ents a n d C o a tin g s fo r C oncrete in R e d u cin g th e P ene­

tra tio n o f K erosene— F. B. H o rn ib ro o k T w o -W a y R e in fo rce d C oncrete S labs— Paul Rogers

R e a c tiv ity o f A g g r e g a t e C onstituents in A lk a li n e S o lu tio n s — L e o n a rd B ean a n d J . J- T re g o n in g

A d m ix tu re s fo r C oncrete— R epo rt o f C o m m itte e 2 1 2 , F. B. H o rn ib ro o k , C hairm an

The E ffe ct o f A lk a lie s in P o rtla n d C em ent on th e D u r a b ility o f C oncrete— B a ile y T re m p e r The E ffe ct o f C uring C o n d itio n s on C om pressive, T e nsile a n d F le x u ra l S trength o f C o n cre te

C o n ta in in g H a y d ite A g g r e g a te — E. B. H a n s o n , J r. a n d W . T. N e e la n d s

A L im ite d In v e s tig a tio n o f C a p p in g M a te ria ls fo r C oncrete Test Specim ens— Thomas B.

K e n n e d y

C oncrete O p e ra tio n s in th e C oncrete S hip P ro g ra m — L e w is FT. T u th ill F u lly a n d P a rtly Prestressed R e in fo rc e d C o n cre te — Paul W illia m A b e le s

Discussion closes J u ly 1, 1945

A n Instrum ent a n d a T echnic fo r F ie ld D e te rm in a tio n o f th e M o d u lu s o f E la s tic ity , a n d F le x u ra l S tren gth, o f C oncrete (P avem ents)— B a rtle tt G . L o n g , H e n r y J. K urtz, a n d Thomas A . S a n d e n a w

A W o rk in g H y p o th e s is fo r Further S tudies o f Frost R esistance o f C o n c re te — T. C . P o w e rs P rop osed Test P ro ce d u re to D e te rm in e R e la tiv e B o n d V a lu e o f R e in fo rc in g Bars

R e p o rt o f A C I C o m m itte e 2 0 8 , H . J . G IL K E Y , C h a irm a n

E ffe ct o f T y p e o f B a r on W id th o f C racks in R e in fo rc e d C o n cre te S u b je c te d to Tension

— D a vid W a ts te in a n d N o rm a n A . S eese , J r.

C ra c k in g a n d T e m pe ratu re C o n tro l o f M a s s C o n cre te — C la re n c e R a w h o u s e r C o n cre te C u rin g M e th o d s — A S T M S ta n d a rd s

A p r . J l. ’4 5 P re s id e n tia l A d d re s s to A m e r ic a n C o n cre te In s titu te — R. W . Crum

Precast C oncrete P it S h e e tin g — J a c o b F eld

A P ra c tic a l P ro ce d u re fo r R ig id Fram e D e s ig n — D. R. C e rv in D y n a m ic Testing o f Pavem ents— G e r a ld P ickett

E stim ating 2 8 -D a y S trength o f C o n cre te fro m E a rlie r S trengths— In c lu d in g th e P ro b a b le E rror o f th e E stim ate— J a c o b J . C re s k o ff

Discussion closes A u g u s t 15, 1945

J u n e J l. '4 5 S labs S u p p o rte d on Four S ides— R. L. B e rlin , J o se p h Di S tasio, a n d M . P. V a n B uren

Ju n e 1 9 4 5 (P ro c e e d in g s V o l. 4 1 )

: v \ MBUfllEIA 2«

J O U R N A L

of t

AMERICAN C O N C R E T E INSTITUTE

P u b lish e d b y th e A m e ric a n C o n c r e te In s titu te . T h e In s titu te w a s fo u n d e d 1 9 0 5 ; in c o rp o ra te d in th e D is tric t o f C o lu m b ia in 1 9 0 6 as T h e N a t io n a l A s s o c ia tio n o f C e m e n t Users; th e nam e c h a n g e d 1 9 1 3 b y c h a r te r a m e n d ­ m e n t. T h e J o u r n a l is issued six tim es y e a r ly in th e months o f J a n u a r y , F e b ­ ru a ry , A p r il , J u n e , S e p te m b e r a n d N o v e m b e r u n d e r th e a u th o r ity o f th e

BOARD O F DIRECTION, A M E R IC A N CONCRETE INSTITUTE

President D O U G L A S E. P A R S O N S

V ic e Presidents

H A R R IS O N F. G O N N E R M A N , S T A N T O N W A LK E R

Secretary-Treasurer H A R V E Y W H IPP LE

R e g io n a l D irectors P A U L W . N O R T O N

M Y R O N A . S W A Y Z E A L E X A N D E R FOSTER, JR.

D ire c to rs -a t-L a rg e

H A R R Y F. T H O M S O N ROBERT F. B LA N K S H E N R Y L. K E N N E D Y

Past Presidents

RODERICK B. Y O U N G R A Y M O N D E. D A V IS

BEN M O R E E LL M O R T O N O . W IT H E Y

R O Y W . C R U M

F R A N K H . J A C K S O N C H A R L E S S. W H IT N E Y

HERBERT J. G IL K E Y

P ap e rs a n d o th e r c o n trib u tio n s p re p a re d w it h a v ie w t o J o u r n a l p u b lic a tio n s h o u ld b e s u b m itte d in tr ip lic a te , a d d re s s e d : S e c re ta ry , P u b lic a tio n s C o m m itte e , A m e r ­ ic a n C o n c r e te In s titu te , 7 4 0 0 S e co n d B o u le v a rd , D e ­ t r o it 2 , M ic h ig a n . P u b lic a tio n o f a c o n trib u tio n d o e s n o t im p ly th e a c q u ie s c e n c e o f c ritic s (w h o s e a d v ic e is s o u g h t b y th e C o m m itte e , p r io r to a c c e p ta n c e ) o r o f th e In s titu te in th e o p in io n s w h ic h it expresses n o r th e a p p r o v a l o f d a ta o r p ra c tic e w h ic h th e c o n tr i­

b u tio n re c o rd s . In s titu te a u th o r ity a tta c h e s o n ly to S ta n d a rd s fo r m a lly a d o p te d as p ro v id e d in th e By- L a w s . A co m m itte e r e p o r t im p lie s m e re ly th e jo in t c o n tr ib u tio n o f a n a p p o in te d g ro u p .

S u b s c rip tio n p ric e $ 7 .5 0 p e r y e a r p a y a b le in a d v a n c e . T o m em bers, $ 7 .5 0 p e r y e a r , in c lu d e d in th e a n n u a l

d u e s . ( A s p e c ia l dues ra te o f $ 3 .0 0 p e r y e a r a p p lie s fo r " a s tu d e n t in re s id e n c e a t a re c o g n iz e a te c h n ic a l o r e n g in e e r in g s c h o o l” a n d in c lu d e s J o u r n a l s u b s c rip tio n . B ou n d vo lu m e s 1 to 4 0 o f P R O C E E D IN G S O F T H E A M E R IC A N C O N C R E T E IN S T IT U T E (1 9 0 5 to 1 9 4 4 ) a r e fo r sa le as f a r as a v a ila b le , a t p ric e s to b e h a d on in q u ir y o f th e S e c re ta ry -T re a s u re r. S p e c ia l p ric e s a p p ly fo r members o r d e r in g b o u n d volum es in a d d itio n to th e m o n th ly J o u r n a l.

P u b lic a tio n address: 7 4 0 0 S eco n d B o u le v a rd , D e tr o it 2 , M ic h ig a n . C o p y r ig h t, 1 9 4 5 , A m e ric a n C o n c r e te In ­ s titu te , P rin te d in U . S. A . E n te re d a t th e Post O f f ic e a t D e tro it, M ic h ig a n , as m ail o f th e se con d class u n d e r p ro visio n s o f th e A c t o f M a r c h 3 ,1 8 7 9 .

the A C I J o u r n a l

is e d ite d b y th e S e c re ta ry o f th e P u b lic a tio n s C o m m itte e u n d e r th e d ire c tio n o f th e C om m ittee

ROBERT F. B L A N K S C h a irm a n

D O U G L A S E. P A R S O N S ( e x - o ffic io )

R. D. B RAD B U RY

HERBERT J. G IL K E Y

A . T. G O L D B E C K

H A R R IS O N F. G O N N E R M A N

F R A N K H . J A C K S O N

W . H . KLE IN

S T A N T O N W A L K E R

RODERICK B. Y O U N G

H A R V E Y W H IP P LE S e cre ta ry

ft Is th e p o lic y o f th e A m e r ic a n C o n cre te In s titu te to e n c o u ra g e p a r tic ip a tio n b y its m e m bers a n d o thers in th e w o rk o f e x te n d in g th e k n o w le d g e o f c o n c re te a n d re in fo rc e d c o n c re te as a basis fo r im p ro v e d p ro d u cts a n d structures.

To this e n d th e B o a rd o f D ire c tio n has a ssig n e d to th e P u b lic a tio n s C o m m itte e th e re s p o n s ib ility o f s e le c tin g fo r p u b lic a tio n such p a p e rs , c o m m itte e re p o rts , d iscussions a n d o th e r c o n trib u tio n s o r p a rts o f such c o n trib u ­ tio n s , as in th e ju d g m e n t o f th e C o m m itte e , seem to o ffe r m ost o f v a lu e in a tta in in g In sti­

tu te o b je c tiv e s w ith in sp a c e re q u ire m e n ts co n siste n t w ith b u d g e t lim ita tio n s .

A M E R I C A N C O N C R E T E INSTITU TE N E W C E N TE R B U IL D IN G D E T R O IT 2 , M I C H I G A N

X PLEASE NOTE

I

ra tio n s ta n d a rd s p ro c e d u re under the S ta n d a rd s C o m m itte e in 1 9 3 7 . They have b e en a n d w ill c o n tin u e to be a v a ila b le s in g ly in separate prints b u t w ill s h o rtly b e a v a ila b le also in on e " A C I B o o k o f Standards

$ 1 .5 0 ( $ 1 .0 0 to A C I M embers).

2

a w a re o f th e a v a ila b ilit y of separate prints o f e a ch p a p e r and report.

M a n y w h o a re a w a re of their a v a il­

a b ilit y h a ve n o t been aware that the In s titu te , o rg a n iz e d a n d financed as it is, is n o t in a p o s itio n to make free d is trib u tio n o f its literature. See th e new a n n o u n c e m e n t which tops th e first p a g e o f e a ch p a p e r and re p o rt—

se p a ra te prints are usually a v a ila b le a t 25 or 5 0 cents. In q u a n titie s th e prices are lo w e r— for large q u a n titie s m uch lo w e r.

3

" A m e r ic a n C oncrete In s titu te P u b li­

c a tio n s P o lic y ” (a n 8 - p a g e re p rin t from the S e p tem be r 1 9 4 1 J o u rn a l).

It w ill be sent w ith o u t c h a rg e , on request.

ii

P

I I

[“ T o f a c ilit a t e s e le c tiv e d is tr ib u tio n , se p a ra te p rin ts o f this t i t l e ( 4 1 - 2 2 ) a re c u r r e n tly “ ] a v a ila b le fro m A C l a t 2 5 c e n ts e a c h — q u a n tity q u o ta tio n s on re q u e s t. Discussion L o f this p a p e r (c o p ie s in t r ip lic a t e ) s h o u ld re a c h th e In s titu te n o t la te r th a n A u g . 1 5 , 1 9 4 5 J

T itle 4 1 - 2 2 a p a rt o f P R O C E E D IN G S , A M E R IC A N C O N C R E T E IN STITU TE V o l . 41

J O U R N A L

o f the

A M E R I C A N C O N C R E T E I N S T I T U T E

( c o p y r ig h te d )

V o l . 1 6 N o . 6 7 4 0 0 S E C O N D B O U L E V A R D , D ETROIT 2 , M I C H I G A N June 1 9 4 5

Slab s Sup p orted on Fo u r Sid es*

Suggested Changes in A C l Building Regulations

By R. L. BERTIN, JOSEPH Dl STASIO, and M . P. V A N BURENf

M e m b e rs A m e ric a n C o n c r e te In s titu te

S Y N O P S I S

The A C l Building Regulations for Reinforced Concrete provide, w ith respect to slabs supported on four sides, a m ethod of analysis which reflects a clear picture of the elastic action of the structure, and, through the use of equivalent uniform load factors, perm its the direct solution for bending m om ents and shears in the slabs and beams with th e same coefficients as prescribed for one-way construction. To clarify the m anner of presentation, the authors have prepared a suggested change of the entire C hapter 7 of the Code. While retaining all the original basic features, notation has been simplified, non-essential formulas and extraneous theory eliminated, and the regulations con­

densed to the fundam entals requisite for direct design. Final results are unchanged from those obtained through the use of the present 1941 regulations.

In this paper, the proposed changes are stated and reasons for them given. Suggested regulations are presented in new form. Comparisons are shown to indicate conformity w ith theory. Finally, an analysis of a typical series of floor panels is given to illustrate the facility w ith which com putations could be made under suggested changes. I t is believed th a t engineers would find the suggested modification of this section of th e Code simple and easy to apply.

IN T R O D U C T IO N

T h e d ev elo p m en t of em pirical m eth o d s of design for inclusion !in bu ild in g codes is gen erally m o tiv a te d b y tw o basic co ncepts s ta te d ’ in th e o rd er of th e ir im p o rta n c e to th e designers :

^ R e ceiv e d b y th e I n s tit u te J a n . 25, 1945.

f R . L . B e rtrn , C h ie f E n g r. W h ite C o n s tru c tio n C o ., Jo se p h D i S tasio , a n d M . P . V a n B u ren J T)i

S ta sio & C o ., C o n s u ltin g E n g rs ., N . Y . C ity . ’

(5 3 7 )

5 3 8 J O U R N A L O F TH E A M E R C A N C O N C R E T E INSTITU TE Ju n e 1945

(a) S im p licity of ap p lic a tio n ,

(b) A d herence to th e o re tic a l correctness.

T h is is p a rtic u la rly tru e of tw o -w a y slab design b ecau se of th e com­

p le x ity of th e th e o re tic a l analysis.

I t is believed t h a t th e A C I re g u la tio n s yield re su lts th ro u g h o u t the ra n g e of tw o -w a y slabs co nform ing m ore n e a rly th a n a n y o th e r to the th e o re tic a l an aly sis, a n d in th e new sug g ested form i t is n o t only simple of ap p lic a tio n , b u t co m p a tib le w ith th e fram e an aly sis of continuous s tru c tu re s in w hich tw o -w a y slab s occur.

I t is th e p u rp o se of th is p a p e r to p re se n t c o m p a ra tiv e d a ta from which th ese beliefs are deduced. Sub-divisions of th e p re se n t paper are:

P a r t 1— Suggested m odified re g u la tio n s; P a r t 2— C o m p a ra tiv e analysis:

P a r t 3— G en eral a p p lic a tio n .

P A R T 1 S uggested changes

T h e re g u la tio n s first developed for th e N ew Y o rk C ity Building C ode w ere reco m m en d ed to th e A C I b y C o m m itte e 501 in its Proposed B uild­

ing R eg u la tio n s in 1935, a n d th e basic th e o ry explained by a paper in 1936*.

T h e original 1935 tr a n s c r ip t w as re -a rran g e d a n d am plified by ta b le s in th e A C I B u ild in g R eg u latio n s for R ein fo rced C oncrete, adopted in 1941. T h e re -e d itin g of th is section as su g g ested b y th e authors, includes th e follow ing p ro p o se d m odifications:

1— E lim in a tio n of ta b le s a n d fo rm u las in v o lv in g d eterm in atio n of lines of inflection. E la stic an aly sis is a defin itely established th e o ry an d req u ire s no special exposition u n d e r th is h eading. L im itatio n s are re ta in e d w ith in w hich p rescrib ed v alu es of th e d istan ce betw een in ­ flection lines m a y be used.

2— E lim in a tio n of special tr e a tm e n t of u n u su a l cases by d e fin ite ly specifying t h a t all slabs be securely a tta c h e d to supports. S tru c tu re s in w hich p a n e l edges are free to u p lift are of lim ite d p ractical v alu e a n d sh o u ld be considered as in d iv id u a l problem s.

3— S im p lificatio n of fo rm u las for m o m e n t a n d shear. T erm s in v o lv in g w id th of slab s trip a n d tr ib u ta r y w id th c a rrie d by beam , are e lim in a te d b y expressing th e lo ad in te rm s of IT, th e to ta l slab load on e ith e r sla b s trip or beam . All fa c to rs, b y w hich m o m e n t an d sh ear a t a n y sectio n in o ne-w ay c o n stru c tio n in e ith e r d ire c tio n are o b tain ed , a re p re s e n te d in tw o sh o rt tab le s. C o rresp o n d in g c h a rts are also show n as g rap h ic illu s tra tio n s of th e tab le s.

*“ S lab s S u p p o rte d on F o u r S id e s ,” b y D i S tasio a n d V an B u ren , A C I J o u r n a l J a n .- F e b . 1936; P r o ­ ceedings V. 32, p . 350.

S LA BS SUPPORTED O N FO U R SIDES 5 3 9

4— S im plification of n o ta tio n . Special sym bols are elim in ated an d rep laced w ith a condensed a n d m ore fa m iliar n o m en clatu re. Of these, som e, such as L a n d W m a y be used w ith o u t change in definition for o th e r sections of th e Code.

5— E lim in a tio n of all fo o tn o te form ulas. T hese are no longer re q u ire d as all in fo rm a tio n fo r th e design of slabs m a y be o b ta in e d from th e ta b le s in th e ir sug g ested form . F o r general in fo rm atio n , th e fo rm u las on w hich th e ta b le s are b ased are given h ere as derived from th e 1941 reg u latio n s b y d ire c t s u b s titu tio n in th e new n o ta tio n .

r >

•5 C “ ( i T ? ) ( ï 7= t ) ^{( 1 )}

r < -5-c = i q h ... ...

C 8 = ( ,

5

6x c )

3

S uggested re g u la tio n s

700— N o tation (For Slabs Supported On Four Sides) L = S p an le n g th

L \ = S p an le n g th in th e d irectio n n o rm al to L in floors s u p p o rte d on fo u r sides.

m — R a tio of sp a n betw een lines of inflection to L in th e d irectio n of sp a n L , w hen sp a n L only is loaded.

TOi = R a tio of sp a n betw een lines of inflection to L i in th e directio n of sp a n L i, w hen sp a n L x o n ly is loaded.

r = D egree of re c ta n g u la rity b etw een lines of inflection of a panel m L

su p p o rte d on fo u r sides = ^ "

w = U niform ly d is trib u te d to ta l load p er u n it a re a of slab.

W = T o ta l u niform load for one w ay c o n stru c tio n betw een opposite su p p o rts on slab strip of a n y w id th or on b eam in th e directio n of L.

X = R a tio of d istan c e fro m su p p o rt to a n y section of slab or beam , to sp an L or L x.

C = F a c to r m o difying b en d in g m o m en ts p rescrib ed for one-w ay co n stru c tio n for use in p ro p o rtio n in g th e slabs a n d beam s in th e d irec tio n of L of slabs su p p o rte d on fo u r sides.

Cs = R a tio of th e sh ear a t a n y section of a slab strip d is ta n t x L from th e su p p o rt to th e to ta l lo ad W on th e strip in d irectio n of L.

C or

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Fig. 1 (a t to p ) a n d 2

S LA BS SUPPORTED O N F O U R SIDES 541

Cb = R a tio of th e S h ear a t a n y sectio n of a B eam d is ta n t x L from th e s u p p o rt to th e to ta l lo ad W on th e B eam in d irec tio n of L . B .M .C .= B en d in g M o m e n t C oefficient.

W i, Ci, Csi, Cm, a re c o rresponding values of W , C, C3, Cb, for slab strip or b eam in d ire c tio n of L X.

7 0 9 — Floors su p p o rte d on fo u r sides

(a) T h is co n stru c tio n , con sistin g of floors reinforced in tw o d irectio n s a n d su p p o rte d on fo u r sides, includes solid reinforced con crete slabs;

con crete jo ists w ith b u r n t clay or con crete tile fillers, w ith or w ith o u t con crete to p slab s; a n d con crete jo ists w ith to p slabs p laced m o n o lith i- cally w ith th e joists. T h e slab shall be su p p o rte d a ro u n d its p e rip h e ry along th e sp an s L a n d L \, a n d shall be securely a tta c h e d to said su p p o rts b y m o n o lith ic co n stru c tio n or o th e r a d e q u a te m eans.

(b) M in im u m slab thickness

T h e slab th ick n ess shall sa tisfy p rescrib ed w orking stresses a n d shall n o t be less th a n 4" nor less th a n th e sum of th e clear le n g th of all su p ­ p o rts a t w hich th e slab is co n tin u o u s w ith th e a d ja c e n t panels d ivided b y 180 plus th e clear le n g th of all o th e r s u p p o rts d iv id ed b y 144.

(c) Lanes of inflection fo r determ ination of r

T h e lines of inflection shall be d e te rm in e d b y elastic an aly sis of th e c o n tin u o u s s tru c tu r e in each d irectio n , w hen th e sp a n u n d e r co n sid era­

tio n only is loaded.

W h en th e sp an L or L x is a t le a st 2 /3 an d a t m o st 3 /2 of th e a d ja c e n t co n tin u o u s sp a n or spans, th e values of m or m x m ay be ta k e n as 0.87 for ex te rio r sp an s a n d 0.76 for in te rio r spans. (See Fig. 1).

F o r freely su p p o rte d sp an s m or m x shall be ta k e n as u n ity . (d) B ending moments and shear

B en d in g m om ents shall be d e term in ed in each d irectio n w ith th e coefficients prescribed for one-w ay c o n stru ctio n in section 701 a n d 702 a n d m odified b y fa c to r C or Ci from T ab les 1 or 2 or from c h a rt. (See F ig. 2).

I n L Direction I n L x Direction B . M . for slab strip = M = C W L (B .M .C .) M i = C iW xL x (j3.M .C .) B . M . for beam = M = (1— C) W L { B .M .C .) M i= { \— Cf) W XL X

{B .M .C .) W hen th e coefficients p rescrib ed in 701(c) are used, th e av erag e value of Cw or CiW for th e tw o sp an s a d ja c e n t to a su p p o rt shall be used in d e te rm in in g th e n eg a tiv e b e n d in g m o m en t a t th e face of th e su p p o rt.

T h e sh ear a t a n y section d ista n ce x L or x L , from su p p o rts shall be d e te rm in e d b y m odifying th e to ta l load on th e slab strip or b eam b y th e fa c to rs Cs, Csl, Cb or Chi ta k e n from T ab le 1 or 2 or from ch arts.

(See Fig. 3 or 4).

T A B L E 1— S L A B S

5 4 2 J O U R N A L O F T H E A M E R IC A N C O N C R E T E IN S TIT U TE J u n e 1 9 4 5

Upper Figure

Lower Figure c.

c>i

CST

r 1_

r ^{X 0 0} . 1 .2 .3 .4

0.00 C£ .50

.00 .40

.00 .30

.00 .20

.00 .10

.00 1.00 .00

.50 2.00 .44

.06 .36

.03 .27

.02 .18

.00 .09

.00 .89 .06

.55 1.82 .43

.07 .33

.04 .23

.02 .15

.01 .07

.00 .79 .08

.60 1.67 .41

.09 .30

.05 .20

.03 .12

.01 .05

.00 .70 .10

.65 1.54 .39

.11 .28

.06 .18

.03 .10

.01 .04

.00 .64 .13

.70 1.43 .37

.13 .26

.08 .16

.04 .09

.01 .03

.00 .58 .15

.80 1.25 .33

.17 .22

.10 .13

.06 .07

.02 .02

.00 .48 .21

.90 1.11 .29

.21 .19

.13 .11

.07 .05

.03 .01

.01 .40 .27

1.00 1.00 .25

.25 .16

.16 .09

.09 .04

.04 .01

.01 .33 .33

1.10 .91 .21

.29 .13

.19 .07

.11 .03

.05 .01

.01 .28 .39

1.20 .83 .18

.32 .11

.21 .06

.13 .02

.06 .00

.02 .23 .45

1.30 .77 .16

.34 .10

.23 .05

.14 .02

.07 .00

.03 .19 .51

1.40 .71 .13

.37 .08

.25 .04

.16 .02

.09 .00

.03 .16 .57

1.50 .67 .11

.39 .07

.27 .04

.17 .01

.10 .00

.04 .14 \

.61

1.60 .63 .10

.40 .06

.29 .03

.19 .01

.11 .00

.05 .12 .66

1.80 .55 .07

43 .04

33 .02

23 .01

15 .00

.07 .08 .79

2.00 .50 .06

44 .03

36 .02

27 .00

18 .00

09 .06 .89 oc

0.00 .00

50 .00

40 .00

30 .00

20 .00

10 .00 1.00

S L A B S S U P P O R T E D O N F O U R S I D E S

T A B L E 2 — B E A M S

5 4 3

U p p e r F i g u r e L o w e r F i g u r e

c „ c bl

1 - C l - c x

r 1

T

X 0 . 0 . 1 . 2 . 3 . 4

0 . 0 0 cc

. 0 0 . 5 0

. 0 0 . 4 0

. 0 0 . 3 0

. 0 0 . 2 0

. 0 0 . 1 0

. 0 0 1 . 0 0

. 5 0

2 . 0 0

. 0 6 . 4 4

. 0 4 . 3 7

. 0 3 . 2 8

. 0 2 . 2 0

. 0 1 . 1 0

. 1 1 . 9 4

. 5 5

1 . 8 2

. 0 7 . 4 3

. 0 7 . 3 6

. 0 7 . 2 8

. 0 5 . 1 9

. 0 3 . 1 0

. 2 1 . 9 2

. 6 0

1 . 6 7

. 0 9 . 4 1

. 1 0 . 3 5

. 1 0 . 2 7

. 0 8 . 1 9

. 0 5 . 1 0

. 3 0 . 9 0

. 6 5

1 . 5 4

. 1 1 . 3 9

. 1 2 . 3 4

. 1 2 . 2 7

. 1 0 . 1 9

. 0 6 . 1 0

. 3 6 . 8 7

. 7 0

1 . 4 3

. 1 3 . 3 7

. 1 4 . 3 2

. 1 4 . 2 6

. 1 1 . 1 9

. 0 7 . 1 0

. 4 2 . 8 5

. 8 0

1 . 2 5

. 1 7 . 3 3

. 1 8 . 3 0

. 1 7 . 2 4

. 1 3 . 1 8

. 0 8 . 1 0

. 5 2 . 7 9

. 9 0

1 . 1 1

. 2 1 . 2 9

. 2 1 . 2 7

. 1 9 . 2 3

. 1 5 . 1 7

. 0 9 . 0 9

. 6 0 . 7 3

1 . 0 0

1 . 0 0

. 2 5 . 2 5

. 2 4 . 2 4

. 2 1 . 2 1

. 1 6 . 1 6

. 0 9 . 0 9

. 6 7 . 6 7

1 . 1 0

. 9 1

. 2 9 . 2 1

. 2 7 . 2 1

. 2 3 . 1 9

. 1 7 . 1 5

. 0 9 . 0 9

. 7 2 . 6 1

1 . 2 0

. 8 3

. 3 2 . 1 8

. 2 9 . 1 9

. 2 4 . 1 7

. 1 8 . 1 4

. 1 0 . 0 8

. 7 7 . 5 5

1 . 3 0

. 7 7

. 3 4 . 1 6

. 3 0 . 1 7

. 2 5 . 1 6

. 1 8 . 1 3

. 1 0 . 0 7

. 8 1 . 4 9

1 . 4 0

. 7 1

. 3 7 . 1 3

. 3 2 . 1 5

. 2 6 . 1 4

. 1 8 . 1 1

. 1 0 . 0 7

. 8 4 . 4 3

1 . 5 0

. 6 7

. 3 9 . 1 1

. 3 3 . 1 3

. 2 6 . 1 3

. 1 9 . 1 0

. 1 0 . 0 6

. 8 6 . 3 9

1 . 6 0

. 6 3

. 4 0 . 1 0

. 3 4 . 1 1

. 2 7 . 1 1

. 1 9 . 0 9

. 1 0 . 0 5

. 8 8 . 3 4

1 . 8 0

. 5 5

. 4 3 . 0 7

. 3 6 . 0 7

. 2 8 . 0 7

. 1 9 . 0 5

. 1 0 . 0 3

. 9 2 . 2 1

2 . 0 0

. 5 0

. 4 4 . 0 6

. 3 7 . 0 4

. 2 8 . 0 3

. 2 0 . 0 2

. 1 0 . 0 1

. 9 4 . 1 1

cc

0 . 0 0

. 5 0 . 0 0

. 4 0 . 0 0

. 3 0 . 0 0

. 2 0 . 0 0

. 1 0 . 0 0

1 . 0 0 . 0 0

X i ( w i t h C b | )

5 4 4 J O U R N A L O F TH E A M E R IC A N C O N C R E T E IN S TITU TE J u n e 1 9 4 5

X (w ith C b )

Fig. 3

I n L Direction I n L x D irection

\ {Shear for Slab S trip . . . . V = CSW V x = CsXW X S h ear for B e a m ...V = CbW V x = CblW x

Foi sp an s w here th e end m o m en ts are u n b a la n c e d , sh e a r v a lu es a t a n y section shall be a d ju s te d in a cco rd an ce w ith S ectio n s 701 a n d 702.

(e) A rrangem ent of reinforcement '

1. I n a n y panel, th e a re a of rein fo rc e m en t p e r u n it w id th in th e long d ire ctio n shall be a t le a st o n e -th ird t h a t p ro v id e d in th e s h o rt d ire c tio n .

2. t h e a re a of p o sitiv e m o m e n t re in fo rc e m en t a d ja c e n t to a c o n tin u ­ ous edge only a n d for a w id th n o t exceeding o n e -fo u rth of th e s h o rte r dim ension of th e p an el m a y be red u ced 25 p e r cent.

3. A t a n o n -co n tin u o u s edge th e a re a of n e g a tiv e m o m e n t rein fo rce­

m e n t p er u n it w id th shall be a t le a st-o n e-h alf of t h a t re q u ire d fo r m axi-

X , ( w i t h C s i )

S LA BS SUPPORTED O N F O U R SIDES 5 4 5

.2 .3 .4-

X (w ith C s )

to u

F ig. 4

m u m p o sitiv e m o m en t for th e ce n te r one-half of th e panel a n d shall b e p ro v id ed across th e e n tire w id th of th e ex te rio r su p p o rt.

4. T h e spacing of th e b a rs sh all be a t m o st th re e tim es th e slab th ic k ­ ness an d th e ra tio of re in fo rce m en t a t le a st 0.0025.

P A R T 2 C o n fo rm ity w ith th e o ry

T h is sectio n is d e v o te d to a co m p ariso n of th ese reg u latio n s w ith th e re su lts of o th e r a u th o r ita tiv e in v e stig a tio n s an d codes, n am ely , th e fo rm u las of D r. M arcu s* , D r. W e s te rg a a rd t, th e 1940 J o in t C o m m ittee, a n d th e B o sto n C ode. I n o rd er to elim in a te a n y difference in p rim a ry a ssu m p tio n s as to p ro p o rtio n or a rra n g e m e n t of th e live load, pan els

* '‘D esig n of R ein fo rced C o n c re te S la b s ,” Jo se p h A . W ise, A C I Proceedings, V. 25, 1929.

+“ F o rm u la s for th e D esig n of R e c ta n g u la r F lo o r S lab s a n d th e S u p p o rtin g G ird ers, ’ H . M . W e ste r- g a a rd , A C I Proceedings V. 22, 1926.

5 4 6 J O U R N A L O F T H E A M E R IC A N C O N C R E T E IN S TITU TE J u n e 1 9 4 5

a = S h o rt span t^/¿¡ b= L o n q s p a n

.20

IQ is

I 7 16 .15 .14

13 12 II 10

Kz = K + Kb /

in K OJbc*2 + KbUJ ba2 _/'¿>4y o r K (jJatD2 +. KbLJ a b 2 s #

cila b beams V /

i

+P>

- J C Sp<an 3 0 - Spc3n b

4 4 C I -¡nar s / *

- a n d - - V_ ioc: L — -

- -...x

0

160 0

140 cQ

120T>

IQo

100 To

?

00

10 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.Ô 1.9 2.0 ^

a - S h o rt span | b = L o n q s p a n

F ig. 5 (to p )— S la b m om ent c o e ffic ie n ts in sim p le p a n e l

Fig. 6 Sum o f m om ent c o e ffic ie n ts fo r p a ra lle l be am a n d s la b in sim p le p a n e l

S LA B S SUPPORTED O N F O U R SIDES 5 4 7

w ith v a rio u s c o m b in a tio n s of free a n d fu lly fixed edges h a v e been selected as re p re se n tin g th e e x tre m e ran g e of c o n tin u ity for one definite loading co n d itio n . As in d ic a te d on th e g ra p h s of Fig. 5 to 8, th e A C I C ode m ore n e a rly conform s to th e th e o re tic a l analy ses of tw o -w ay slabs th a n a n y o th e r code, as follow s:

(a) Load D istribution. The equivalent uniform load causing slab bending (8K,w) in either direction of a sim ply supported panel agrees closely w ith the results of D r.

M arcus, which, in the opinion of th e authors, present the most direct theoretical ap ­ proach to this problem. (See Fig. 5).

(b) T otal Panel M om ent. The sum of th e beam and slab mom ents in either direction (K 2WL) equal the statical m om ents (1/8W L). (See Fig. 6).

(c) T otal Slab Bending. The sum of the equivalent uniform loads causing slab bending in both directions (8K3w or 8K 4w) for extrem e variations in edge restrain t are in close agreem ent w ith D r. W estergaard’s 1926 formulas. In determ ining D r. W ester- gaard’s to ta l slab m om ent for one direction, the m ethod of averaging simple and con­

tinuous panels was followed as described in his paper. For th e Jo in t Comm ittee, this m om ent was taken as l b j tim es the negative m om ent in accordance w ith Section (812(b) of th a t Code.

These criteria are presented as typical illustrations of the consistency by which the A CI regulations properly distribute and provide for the effect of the to tal load, both on the slab and on the supporting beams. Similar consistency prevails for other conditions of loading and continuity. Comparison of Fig. 7 and 8 dem onstrates the wide variation in slab bending caused by changes in edge restraint, and establishes th e A CI m ethod of using lines of inflection as a satisfactory measure of true rectangu- larity. In consequence, it follows th a t the panel will be affected by the loading and stiffness of adjacent panels in a continuous structure, as prescribed in the Code. Thus, fundam entally, the A CI formulas and m ethod are in substantial accord w ith accepted theory.

P A R T 3 G e n e ra l A p p lic a tio n

A u n iq u e fe a tu re of th e A C I fo rm u la is its flexibility an d general a p p lic a b ility to a w ide ran g e of d ifferen t conditions. T h ro u g h th e sim ple process of m odifying th e load b y th e ta b u la te d facto rs, th e re re su lts a n e q u iv a le n t u n ifo rm lo ad to be used as in one-w ay co n stru ctio n . A ny c o m b in a tio n of live a n d dead load can be tr e a te d in th is m an n er w ith re s u lta n t econom y of design. W here w a rra n te d , accep ted m eth o d s of a d ju s tm e n t fo r th e tra n sv e rse to rsio n al resistan ce of th e girders can be in clu d ed in th e d e te rm in a tio n of th e one-w ay coefficients, as such r e s tra in t is in no w a y p e cu liar to tw o -w ay slabs. In th e tre a tm e n t of u n e q u a l panels, th e A C I C ode surpasses all o th e rs in directness an d fa c ility , an d , th ro u g h th e use of eq u iv a le n t u n ifo rm loads, p e rm its an a c c u ra te a n d easy so lu tio n of th e com plete s tru c tu re as a rigid fram e.

A lte rn a te ly , in sim ple s tru c tu re s, w here a r b itr a r y coefficients are p e r­

m issible, th e se m a y be used as p rescrib ed fo r one-w ay spans. T herefore, c o n siste n t fa c to rs of sa fe ty are m a in ta in e d th ro u g h o u t w ith o th e r ty p e s of floor.

5 4 8 J O U R N A L O F TH E A M E R IC A N C O N C R E T E IN S TITU TE J u n e 1 9 4 5

a = S h o rt span t p á b = L o n q s p a n

oo

*-l 1 2 1.3 1 .4 1.5 1.6 1.7 1 .8 1.9 2 .0

a = S h o rt span t ^ / â b = L o n q s p a n

Fig. 7 (to p ) a n d 8 Sum o f to ta l m om ent c o e ffic ie n ts o f s la b fo r tw o d ire c tio n s .1 4

13 .12 .11

10

09 .00 .07 0 6 0 5

K4 = K 3+- K 1b ./

in K aCUa2 4- KibCo b 1 ( Mⁿ + Mp)

£ -

S ' .__

~--- --- A

V es' erqc ar3 ^\tyii \J .O.

Spar a, both en< f re e Spar b, both e n ds f ixed

100 XSt- CO

6 0 f\3

M—

4 0 O

S LA BS SUPPORTED O N FO U R SIDES 5 4 9

S pans L ^{19 ' - s ’ 1 6 - 0}

b 2

c<TJ

Cl,

CO

b-13

3

I

"T cdi

i

2

5-24-

N o te . C o lu m n s a nd I'-O " W id e

Fig. 9 — T y p ic a l e x a m p le

beam s

T h u s, th e A C I C ode pro v id es a m e th o d of an aly zin g slabs su p p o rte d on four sides w hich is a cc u ra te, flexible, a n d also as now proposed, v ery sim ple to a p p ly . A n exam ple will be in stru c tiv e in illu stra tin g th e design p ro ced u re in a ty p ic a l building.

T y p ic a l e x a m p le

L e t Fig. 9 re p re se n t fo u r floor panels a t th e corner of a b u ild in g in w hich th e slabs a n d b e am s are n u m b e re d as show n. F irs t, d eterm in e th e m in im u m perm issible slab thicknesses in accordance w ith Sec.

709(b). U sing th e clear spans, th e m in im u m th ick n ess for P an el 1 becom es

'18 + 1 5 \ , / 1 8 + 15s ( 1 8 + .15 ) 12 +

V 144 / \ 180 / 12 = 4.95 in.

144 / \ 180

Sim ilarly, in pan els 2, 3 a n d 4, th e re su lts are 4.65 in., 4.70 in. an d 4.40 in., resp ectiv ely . A 5-in. slab could th ere fo re be used th ro u g h o u t, pro v id ed

5 5 0 J O U R N A L O F T H E A M E R IC A N C O N C R E T E IN S TIT U TE J u n e 1 9 4 5

T A B L E 3— R E C T A N G U L A R IT Y

Panel r in L D irection = VILl

m \L\

1 .87 X 18 _

.87 X 15

2 .76 X 15 _ g7

.87 X 15

3 .87 X 18 _ , K

.76 X 18

4 .76 X 15 _

.76 X 18

T A B L E 4 — T W O - W A Y S L A B F A C T O R S

Panel r

X = O

C c, 1—C 1— Ci

O *« O a = c sl

1 1.20 .18 .32 .23 .45 .77 .55

2 .87 .30 .20 .42 .26 .58 .74

3 1.15 .20 .30 .26 .42 .74 .58

4 .83 .32 .18 .45 .23 .55 .77

“ C ” v a lu e s ta k e n f ro m T a b le s 1, 2 o r g r a p h F ig .2.

w orking stresses d e te rm in e d fro m m o m e n ts a n d sh ears fall w ith in th e p rescrib ed values.

As all sp a n s a re “ a t le a st 2 /3 a n d a t m o st 3 /2 of th e a d ja c e n t con­

tin u o u s sp a n or sp a n s ” , lines of inflection, w hen th e sp a n u n d e r co n sid er­

a tio n only is lo aded, m a y be d e te rm in e d w ith o u t reco u rse to elastic an aly sis from th e ra tio s m or m x = .87 for e x te rio r sp a n s a n d .76 for in te rio r spans. T h e degree of r e c ta n g u la rity r of th e v a rio u s p a n e ls follows d ire c tly fro m F ig. 1, a n d are given in T a b le 3. I t is o n ly neces­

sa ry to co m p u te r fo r one d ire c tio n in each p an el, as w ith th e se v alu es fa cto rs for d e te rm in in g all m o m e n ts a n d sh ea rs m a y be selected fro m th e tab les. G en erally , only th e v alu es of C, 1 — C. a n d C s a n d Cb for x = 0, are re q u ire d as listed in T a b le 4. Id e n tic a l re su lts c a n be o b ta in e d b y using 1 /r w ith L a n d L x rev ersed . I t is to be n o te d t h a t for recip ro cal valu es of r w ith X e q u al to zero, Cs a n d Cb are in te rc h a n g e d in a m o u n t, a n d t h a t C s plus Cb is a c o n s ta n t, in th is case .50. T h is m u s t b e tr u e to ac c o u n t for th e to ta l p a n el load. T h e a p p a r e n t c o n fo rm ity of p an els 3 a n d 4 w ith 1 a n d 2 is co in cid e n tal to th e sh a p e of th e s tru c tu re .

G iven a u n ifo rm ly d is tr ib u te d to ta l d e a d a n d live slab lo ad , w, 100 lb. p er sq. ft., an a d d itio n a l in te rio r b eam lo ad of 100 lb. p e r lin. f t., a n d an e x terio r w all load of 1000 lb. p e r lin. ft., sh e ars a n d b e n d in g m o m e n ts

S LA BS SUPPORTED O N FO U R SIDES 551

m a y b e d e te rm in e d as in one-w ay co n stru c tio n m odified b y th e a p p ro p ri­

a te fa cto rs. As, in th e exam ple chosen, th e la rg er of tw o a d ja c e n t sp a n s does n o t exceed th e sh o rte r b y m ore th a n 2 0 p er cent, th e one-w ay coefficients, p rescrib ed u n d e r S ection 701(c), m a y be used in th e fo rm u las of 709 (d ):

V = CeW (coefficient), M = C W L (B .M .C .)

T h e re su lts for slabs m a y be ta b u la te d as in d ic a te d in T a b le 5: W ith th e se m o m en ts, m in im u m slab th ick n esses should be checked for s tru c ­ tu r a l re q u ire m e n ts a n d rein fo rcem en t d e term in ed in th e u su a l w ay. D u e re g a rd should be given to th e difference in effective d e p th in th e tw o d irections. I t will u su a lly b e fo u n d b e st p ractice to place th e steel in th e d irectio n of th e h e a v ier b e n d in g m o m e n t closest to th e surface, T h e reg u latio n s p ro v id e t h a t p o sitiv e rein fo rce m en t a d ja c e n t to a c o n tin u ­ ous edge, a n d for a w id th n o t exceeding o n e -fo u rth of th e sh o rte r d im en ­ sion of th e p anel, m a y be red u ced 25 p er cent. A ccordingly, in th is case, b a r sp acin g w ith in a w id th of 3 ft. 9 in. from B12, B13, B 24 a n d B34, m a y be in creased one th ird . N eg ativ e rein fo rce m en t n o t less th a n h alf t h a t re q u ire d fo r th e u n re d u ced p o sitiv e m o m en t should be p ro v id ed across B O , B l, B 2 n d B3. T h e a m o u n t of steel a t a n y section is also lim ite d b y m in im u m p ercen tag e, m ax im u m b a r spacing, an d general d e ta ils as in one-w ay co n stru ctio n .

B eam s m ay be an aly zed in a sim ilar m a n n e r w ith th e ad d itio n of th e effects of special loads. B eam B l is se le c te d as ty p ic a l of th e m eth o d , a n d shears a t in te rm e d ia te p o in ts d istan ce X L from th e s u p p o rt will be in v e stig a te d . F ro m T a b le 2 th e fa c to rs listed in T a b le 6 a re found.

C0D. U sin g th e coefficients prescribed in S ectio n 701(c), th e r e s u lta n t shears

^ a n d b en d in g m o m en ts are as given in T a b le s 7 a n d 8 resp ectiv ely . W here necessary , sh e a r a t in te rm e d ia te sections of th e slab m a y be in v e stig a te d

■0... in a sim ilar m a n n er, using th e fa c to rs C s or C81.

panels I n th e design of s tru c tu re s , it is o ften sufficient to d e te rm in e th e end neces- sh e a rs a n d m ax im u m m o m e n ts only, as sh ears a t o th e r sections can be

e s tim a te d w ith s a tisfa c to ry a c c u ra cy for th e sp acin g of th e stirru p s. In th is in sta n c e , i t m a y be c o n v en ien t to w ork w ith e q u iv a le n t u n ifo rm ly

= 0, d is trib u te d loads. T h e fa c to rs to be u se d are (1— C) for bending, an d tw ice th e v a lu e of Cb w hen X — O, for shear. T h e e q u iv a le n t loads ,r0Cal re q u ired in th e design of all b eam s of th e exam ple are given in T ab le 9.

W ith th e se loads, end sh ears a n d b en d in g m o m e n ts m a y be calcu lated in e x a c tly th e sam e m a n n e r a s in one-w ay co n stru c tio n . T h is is of anels p a rtic u la r a d v a n ta g e in th e an aly sis of rigid fram es. W here th e o re tic a l

re fin em en t is desired, th e e q u iv a le n t loads m a y be a d ju ste d for th e jflO elastic re a c tio n s of th e slab sp a n n o rm a l to th e b eam b y th e algebraic afl(j a d d itio n of th e difference in slab end m o m e n ts d iv id ed b y its span.

ats

TABLE5—SLABSHEARSANDMOMENTS

5 5 2 J O U R N A L O F TH E A M E R IC A N C O N C R E T E IN S TIT U TE J u n e 1 9 4 5

§ ^

° o

<D sII n

tO00

00 1184 885 667 965 1198 086 ¿99 S«

V 1/10 1/10 O

i—l 1/10 Î&0>bO u oj

&

*»Ö tb COT—1

SoCO So COi-H

SoCO X

3 Xsic

*O

100 100

'ÔîÖ

oo oo1—1 Ö COoÖ

CM rä3fi

a.a .325 .435 Ph

<d

<D

CQ .245 .355 Ph

a?CD m

Ph

CDCD m

Ph

CDCD m S

II o SPQ X

530 723 590 419 oCO to00 633 465

I s 1/14 1/14 1/16 1/14 1/14 1/16 1/16 1/16

<D .

*43

<N

001—1 tO1—4 So Soi—H 00 00 to 00CJ o 0)

Pl, a d ai ShP

3 X o

100 100 100 oo1—4 100 ooi-H o o 1—1 oo1-H

*0 '

s.s

■ .42 .45 COCM

< wX L X Coef. = V 372 554 450 345 415 540 480 324

.5 1.15 1.15 1.15 1.15

?-<

cj ^—

■st:GO g

00i-H tOi—( ^{I Q}rH ^»Qi-H 00 00 i—1 ^{t o} 00i—l w « Ö

l ag S Sh4

100 OO

\

oo_1—( 100 oo1-H 100 ooi-H 100

o 00i—l CMCO .30 .20 .20 OCO .32 00_T-H

Span in Ft. L or L, L = 18 ^{I QtH}

II

I Q

II

t o i-H

II

»4 L = 18 00

T-H

II

L = 15 00

II

►5

Panel

i-H CN CO

♦Average values ofspans either side ofsupport.

T A B L E 6 — F A C T O R S F O R B E A M B1

S LA B S SUPPORTED O N F O U R SIDES' 55 3

c„ 11 Ü

X

r \ 0.00 .1 .2 .3 .4

1.20 .32 .29 .24 .18 .10 .77

T A B L E 7— S H E A R S IN B E A M 1

a t

X [ Î C b X w X

1) + S x ( j - x ) J

.0 .32 100 7 .5 1000 .5 18' 1.15 15350

.1 .29 It Cl a .4 11 a 12790

.2 .24 It it ii .3 11 ii 9900

.3 .18 It '11 a .2 It a 6930

.4 .10 It It ii .1 11 it 3620

T A B L E 8— M O M E N T S IN B E A M 1

MaxAVIoment j j^(l-C )X w X + Load | X X B ' M 'C - = Max. Pos. M o m en t. . ^{. 7 7} 100 ^{7 . 5} 1000 182 1/14 36400

M ax. Neg. M o m en t. . ^{. 7 7} It a It 182 1/10 51000

F o r in te rio r beam s in T ab le 9 u n d e r B eam load, th e first colum n gives th e floor lo ad d ire c tly over th e beam , a n d th e second colum n th e w eight of th e b eam itself. If p referred, th ese tw o te rm s could be o m itte d w ith ­ o u t m a te ria l error,- p ro v id e d th e beam w eights are d is trib u te d in th e u n it floor load, an d calcu latio n s are b ased on c e n te r to c e n te r dim ensions.

A com parison of th e e q u iv a le n t loads carried b y th e v ario u s sp an s in th is p ro b lem illu s tra te s th e te n d e n c y of th e lo ad to be d istrib u te d in d ire c t p ro p o rtio n to th e stiffness of th e resp ec tiv e slab spans. In te rio r sp an s are re la tiv e ly m ore rigid th a n end sp an s, an d in w all panels p a r ti­

cu larly , a g re a te r p ro p o rtio n of th e lo ad is a ttr a c te d to th e sp an in th e d ire ctio n of th e sm aller m o m e n t coefficient. A n a tu r a l econom y of m a te ria ls is th e resu lt.

C O N C L U S IO N

T h e suggested changes of th e 1941 reg u latio n s for Slabs S u p p o rted on F o u r Sides are believed to p re se n t th is su b je c t in th e sim plest form so far