Rational Limitation of Deflections in Reinforced Concrete Members

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Rational Limitation of Deflections in Reinforced Concrete Members

Mehmel, A.

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PREFACE

S a t i s f a c t o r y c r i t e r i a f o r p e r m i s s i b l e d e f l e c t i o n s I n r e i n f o r c e d c o n c r e t e members has n o t y e t been c l e a r l y

e s t a b l i s h e d . New forms o f c o n s t r u c t i o n and h i g h e r w o r k h g s t r e s s e s a r e making t h e need f o r such c r i t e r i a nrore a p p a r e n t . T h i s p a p e r b y D r . A . Mehnlel was t r a n s l a t e d a t t h e r e q u e e t of t h e B u i l d i n g S t r u c t u r e s S e c t i o n of t h e D i v i s i o n of B u i l d i n g Research because i t c o n t a i n s an i n t e r e s t 1 : l g approach t o t h e s u b j e c t of d e f l e c t i o n s which should be o f i n t e r e s t t o

N a t i o n a l B u i l d i n g Code and Canadian S t a n d a r d s A s s o c i a t i o n Committees on Reinforced Concrete and t o d e s i g n e r s i n g e n e r a l .

The D i v i s i o n i s g r a t e f u l t o M r . D.A. S l n c l a i r of t h e

N.R.C. T r a n s l a t i o n s S e c t i o n f o r p r e p a r i n g t h e t r a n s l a t i o n .

Ottawa R.F. Legget

Title :

NATIONAL FESEARCH COUNC IL OF CANADA

Technical Translatlon 1069

A

rational limitation of deflections in reinforced concrete

members

( h e r eine sinnvolle ~ e s c h r g n k u n ~ der Durchbiegungen von Stahlbe tonbauteilen )

Author:

A.

Mehmel

Re.ference: Bauingenieur,

36

(8):

293-300, 1961

A RATIONAL LIlclITATION OF DEFLECT IOl\TS I31 _{REJXPORCED CONCRETE bEFTBE1IS}

Every s t r u c t u r a l member. s u b J e c t t o bending s t r e s s has a c e r t a i n d c f l e c - t i o n . This depends n o t o n l y on t h e load and t h e e x l s t i n ~ s t a t i c system, but a l s o on t h e " s t i f f n e s s t ' E J and hence a l s o on t h e ditnensions and t h e m a t e r i a l s enlployed ( c f . e.g. r e f . 1 and 2 ) . According t o t h e former r u l e l a i d down i n

DIN 10145

-

" ~ e i n f o r c e d c o n c r e t e rnenibers"

-

f o r c e r t a i n s t r u c t u r a l members t h e r e were minirnurn requirements with r e s p e c t t o t h e e f f e c t i v e depth h , which could n o t be l e s s t h a n a c e r t a i n f r a c t i o n of t h e span 1. This f r a c t i o n d i d n o t depend on t h e c o n c r e t e and s t e e l s t r e s s e s o c c u r r i n g i n t h e member. There was a b a s i c d i f f e r e n c e I n t h e value of t h i s f r a c t i o n f o r one-way and two-way s l a b s . I n p a r t i c u l a r , DIN 1045 l a y s dovm t h e following:

"Sec. 22. S l a b s with main reinforcement i n one d i r e c t i o n

1.

...

2.

...

The e f f e c t i v e depth h of t h e s l a b must a l s o be a t l e a s t : f o r simple s u p p o r t b o t h s i d e s , 1/35 of t h e span,

f o r continuous o r f i x e d s l a b s , 1/35 of t h e maximum d i s t a n c e between p o i n t s of i n f l e c t i o n . I f t h i s d i s t a n c e i s n o t c a l c u l a t e d , i t can be assumed a t 4/5 of t h e span.

For s l a b s t h a t a r e walked on o n l y d u r i n g r e p a i r s and c l e a n i n g o p e r a t i o n s , e t c . , t h e corresponding v a l u e s a r e 1/40 of t h e span and 1/40 of t h e maximum d i s t a n c e between p o i n t s of i n f l e c t i o n .

"Sec

.

23. Two-way r e i n f o r c e d s l a b s

The e f f e c t i v e depth hU r e f e r r e d t o t h e bottom s t e e l reinforcement must be a t l e a s t :

f o r f r e e l y supported, s i n g l e - s p a n s l a b s

1/50,

and f o r continuous o r f i x e d

slabs 1/60 of t h e s m a l l e r span.

I f t h e r a t i o of l a r g e r t o s m a l l e r span i s g r e a t e r than 1.5, then Sec. 22, 2 , holds f o r the e f f e c t i v e depth hu.

2.

..."

Despite observance of t h e s e minimun~ e f f e c t i v e depths, a c c o r d i n g t o DIN

1045 i n c r e a s i n g darnage of widely v a r y i n g k i n d s has occurred a s

a

r e s u l t o f e x c e s s i v e d e f l e c t i o n s . Deserving of s p e c i a l mention i n t h i s connection a r e craclcs a p p e a r i n g i n non-bearing i n s i d e walls supported on f l o o r s . Such damage

was f o r r i ~ c r l y l e s s pi-cvalcnt, and i t i s an obvious i n f e r e n c e t h a t whercafi t h e niininl~un e f f c c t i v c d c p t h s rcquil,ctd 1 1 1 D I N 10115 vrerc for:rierly a d e q u a t e , urider

p r e s e n t c o n d i t i o r l s t h c y no l o n g c r r e s u l t i n s u f f i c i e n t l l - m i t a t i o n o f t h e d e f l e c t i o n . The d i f f e r e n c e between former and p r e s e n t - d a y c o n d i t i o n s a r e sulu~led up i n t h e f o l l o v ~ i n g two p o i n t s :

( a ) The use of h i g h q u a l i t y m a t e r i a l s and hence t h e u t i l i z a t i o n o f h i g h a d ~ l l i s s i b l e s t r e s s e s now p e r m i t s g r e a t e r s p a n s t h a n f o r m e r l y f o r a g i v e n s t i f f - n e s s v a l u e , o r s m a l l e r d i n ~ e n s i o n s f o r a g i v e n s p a n .

( b ) It was n o t f o r n l e r l y customary t o e r e c t i n s i d e p a r t i t i o n i n g w a l l s on f l o o r s , 1 . e . w i t h o u t s u p p o r t i n g then1 on t h e ground.

There i s a l s o a l o n g l i s t o f o t h e r t y p e s of damage which c a l l f o r t h e s t u d y of d e f o m i a t i o n s , e . g . t h e b r e a k i n g o f window panes i n l a r g e - a r e a fa$ade p a n e l s , t h e jamming of d o o r s , t h e damaging of i n s t a l l a t i o n s and many o t h e r s . Alvrays, however, i t i s a q u e s t i o n o f d e f i c i e n c i e s i n t h i n g s which a r e a d j o i n e d t o t h e deforming lneinbers, n o t i n t h e s e members themselves. T h i s i s obvious, s i n c e I n r e g u l a r d e s i p p t h e d e f o r m a t i o n o c c u r r i n g i n any g i v e n c a s e , w i t h i n t h e l i m i t s imposed by v a r i a t i o n s i n t h e m a t e r i a l , i s c l e a r l y a f u n c t i o n o f t h e c a l c u l a t e d s t a t e o f s t r e s s . The s t r u c t u r e s a f e t y o f t h e deforming member

i t s e l f , t h e r e f o r e , i s n o t a s u b j e c t o f d i s c u s s i o n i n t h e p n e s e n t c o n s i d e r a - t i o n s . T h i s must be emphasized i f t h e p o i n t s which a r e t o be d i s c u s s e d l a t e r i n c o n n e c t i o n w i t h t h e s t r u c t u r a l d e s i g n and t h e s t a t i c c a l c u l a t i o n a r e t o be k e p t i n p r o p e r p e r s p e c t i v e . These a r e c e r t a i n l y o f i n t e r e s t , b u t t h e i r i r p o r - t a n c e f a l l s c o n s i d e r a b l y s h o r t o f s t r e n g t h a s such.

It i s sonieti~nes p o i n t e d o u t t h a t e x c e s s i v e d e f o r m a t i o n s would a l s o endanger t h e s t r u c t u r a l s a f e t y of t h e deforrlled lilembers themselves o r t h e members on which t h e y a r e s u p p o r t e d . The f i r s t o f t h e s e c a s e s can a r i s e when l e v e r arms a r e c r e a t e d by t h e d e f o r m a t i o n which no l o n g e r p e r m i t t r a n s m i s s i o n of t h e f o r c e s i n t h e undeforrned s t r u c t u r e . We t h e n have a s t r u c t u r e which r e q u i r e s a s t a t i c c a l c u l a t i o n o f t h e deformed system. Such s p e c i a l c a s e s , however, do n o t come w i t h i n t h e scope of t h e d i s c u s s i o n h e r e . They were n o t , n o r a r e , a s u b j e c t of DD? 1045, n o r can t h e y i n any c a s e be d e a l t w i t h by

" l i ~ n i t a t i o n o f d e f l e c t i o n " . The second c a s e could o c c u r , f o r example, i f e x c c s s i v c d e f o r m a t i o n o f a c e i l i n g r e s u l t e d i n u n f o r e s e e n bending moments i n t h e s u p p o r t e d members. I i i t h o u t s p e c i a l v e r i f i c a t i o n , t h e r e f o r e , i t i s

c e r t a i n l y n o t p o s s i b l e , from t h e p o i n t of view of s t a b i l i t y , t o s u p p o r t a s i n g l e - s p a n r e i n f o r c e d c o n c r e t e s o l i d - s l a b c e i l i n g w i t h a 1 0 m span on masonry w a l l s 24 cm t h i c k and t o t r e a t t h e b e a r i n g p r e s s u r e a s uniform. Once a g a i n , t h i s i s n o t a q u e s t i o n o f a d e f l e c t i o n l i r i l i t a t i o n f o r t h e c e i l i n g , b u t r a t h e r one o f c o n t i n u i t y between t h e deforrnations o f c e i l i n g and w a l l . I n such a

c e i l i n g and w a l l and v e r i f y t h e r e s u l . t i n g s t r e s c e s . Unless he makes t h i s v e r i f i c a t i o n hirllself, h i s c a l c u l . a t l o n s a r e Incolnpletc and t h e i n s p e c t i o n d c p a r t n c n t w i l l demand t h a t i t be niadc. I call t h e r e f o r e s e e no p o s s i b i l i t y of endangering t h e s t r e n g t h of t h e s u p p o r t i n g member because of e x c e s s i v e deformations of t h e supported one u n l e s s t h e ~ t a t i c c a l c u l a t i o n i s i n c o r r e c t , o r a t l e a s t i n c o ~ n p l e t e

.

It i s c l e a r from what has been s a i d , t h a t I n o r d e r t o a s s e s s a d m i s s i b l e d e f o r ~ ~ i a t i o n s i t i s n e c e s s a r y t o d i s t i n g u i s h between:

1. Reinforced c o n c r e t e members t h a t cannot do any damage by r e a s o n o f t h e i r d e f l e c t i o n , and

2 . Reinforced c o n c r e t e members whose d e f l e c t i o n s can cause damage. By damage h e r e we mean harmful e f f e c t s which d e t r a c t from t h e s e r v i c e - a b i l i t y of t h e s t r u c t u r e . If l a r g e d e f l e c t i o n s do r e s u l t i n e f f e c t s which reduce o r endanger t h e s t r u c t u r a l s a f e t y , t h i s Is due t o t h e exceeding o f t h e p e r m i s s i b l e m a t e r i a l stresses, which can o n l y happen i f t h e s t r e n g t h c a l c u l a - t i o n s have been i n c o r r e c t l y o r i n c o m p l e t e l y executed. Such e x c e s s i v e s t r e s s e s cannot be avoided by s e t t i n g minimum requirements f o r t h e stiffness of t h e f l o o r .

In group 1 we can d o u b t l e s s i n c l u d e a l l f l o o r s which do n o t s u p p o r t w a l l s beyond t h e main s t r u c t u r e ( ~ u s b a u w 8 n d e ) and which themselves r e s t on b e a r i n g w a l l s o r r i g i d crossbeams. T h i s t a k e s i n t h e m a j o r i t y of f l o o r s i n m u l t i - s t o r e y i n d u s t r i a l b u i l d i n g s , a s w e l l a s t h e f l o o r s of apartment houses i n which t h e i n s i d e p a r t i t i o n s a r e borne by t h e s o i l o r on f l o o r s of s m a l l span.

In n g o p i n i o n t h i s a p p l i e s t o a l a r g e p r o p o r t i o n of modern apartment houses.

( 3

1

Recent s t a t i s t i c a l i n v e s t i g a t i o n s a l s o b e a r t h i s o u t

.

Group 2 I n c l u d e s , e . g . f l o o r s of wide span w i t h non-bearing w a l l s , f l o o r s o r s p a n d r e l beams ( ~ a n d t r : ~ e r ? ) with web ri~er~ibers mounted on t h e s e unsupported edges ( g l a s s panes!), e t c . In c e r t a i n c a s e s a e s t h e t i c f a u l t s can be included among t h e harrnful e f f e c t s , f o r example i f i n s t e a d o f a s t r a i g h t l i n e a t t h e edge o f a t h i n continuous f l o o r a d i s t i n c t c a t e n a r y shape can be recognized. The harmfulness o f a d e f l e c t i o n must a l s o be judged from t h e s t a n d p o i n t of t h e uses t o wiilch t h e roolcs a r e t o be p u t . It Is c e r t a i n l y n o t d e s i r a b l e , f o r example, f o r t h e wlde-span f l o o r o f a schoolroom t o s a g v i s i b l y . Even from t h e s e few examples it w i l l be e v i d e n t t h a t a given s t r u c t u r e can be a s s i g n e d e i t h e r t o group 1 o r 2, even when i n t e n d e d f o r t h e same purpose, depending on t h e i n d i v i d u a l p o i n t o f view. However, s i n c e q u e s t i o n s of s t r e n g t h a r e n o t involved, a s a l r e a d y explained i n d e t a i l , i t a p p e a r s a p p r o p r i a t e t o me t o make c e r t a i n minimurn requirements o b l i g a t o r y f o r a l l r e i n f o r c e d c o n c r e t e members

( 1 . e . f o r both groups 1 and 2), a s h a s a l r e a d y been done i n D I N 1045 f o r s l a b s and ribbed f l o o r s , and i n a d d i t i o n t o recommend t h e a p p l i c a t i o n o f more

stringent rcquirerllcnts f o r group 2. T l ~ c conscqucnces of l a r g e d e f l e c t i o n s w i l l t h c n be c l c a r l y indicated, s o t h a t t h e c o n s c i e n t i o u s e n g i n e e r can t a k e hced. It would then be up t o him and h i s c l i e n t whether he should merely s a t i s f y t h c o b l i g a t o r y requirements, o r should improve t h e q u a l i t y of t h e s t r u c t u r e by o b s e r v i n g f u r t h e r requirements which w i l l p r e v e n t o r a t l e a s t m i t i g a t e t h e harl~iful e f f e c t s . \ h e n a p u b l i c a u t h o r i t y i s t h e c l i e n t , e . g . i n t h e c a s e of a school b u i l d i n g , i t w i l l then be i n a p o s i t i o n t o demand observ- ance of t h e recommendations f o r group 2.

The German Committee f o r Reinforced Concrete has l o n g been concerned

with t h e problem of l i m i t i n g d e f l e c t i o n s . S i n c e t h e r e v i s i o n of DIN 1045 w i l l

s t i l l t a k e some time, a temporary s e t of i n s t r u c t i o n s should b e i s s u e d drawing t h e a t t e n t i o n of t h e d e s i g n i n g e n g i n e e r s t o t h e importance of t h e problem of l a r g e d e f l e c t i o n s and t h e v a r i o u s k i n d s of damage t h a t may r e s u l t from them. With t h i s i n view t h e German Committee f o r Reinforced Concrete has d r a f t e d

a

" ~ e n i p o r a r y r e g u l a t i o n on t h e l i m i t a t i o n of d e f l e c t i o n of r e i n f o r c e d c o n c r e t e members a c c o r d i n g t o DIN 1 0 4 5 " ~ which has a l r e a d y been i n t r o d u c e d i n some s t a t e s of t h e F e d e r a l Republic of Germany ( c f . e . g . r e f . 4 ) . The a p p l i c a t i o n of t h i s r e o o l u t i o n has l e d i n some c a s e s t o c o n s i d e r a b l e i n c r e a s e s i n t h e dimensions even f o r members i n r e s p e c t t o which v e r y l i t t l e damage, i f any, h a s h i t h e r t o been e x p e r i e n c e d . We s h a l l l e a v e a s i d e f o r t h e moment t h e ques- t i o n of whether t h i s was t h e f a u l t of t h e f o r m u l a t i o n o r t h e i n t e r p r e t a t i o n of t h e r e g u l a t i o n . A l a r g e number of p u b l i c a t i o n s p r e s e n t i n g c h a r t s and t a b l e s f o r t h e d e t e r m i n a t i o n of s l e n d e r n e s s have appeared (5-8)

,

some of which comment on t h e r e g u l a t i o n a p p a r e n t l y without knowledge even of t h e p u b l i s h e d p r i n c i - p l e s on which it was based. A t t h e working s e s s i o n of t h e Betonverein

( c o n c r e t e A s s o c i a t i o n ) on March 24, 1961, i n B e r l i n , a number of p a p e r s

r e f e r r e d t o inadequacies of t h e new d e f l c c t i o n T h e r e f o r e it

appeared n e c e s s a r y t o r e v i s e t h e s e r e g u l a t i o n s . The s t a t e of IIessen, c i t e d a s an example f o r p u b l i c a t i o n of t h e decree('), took accourit of t h i s n e c e s s i t y by suspending i t s compulsory a p p l i c a t i o n f o r t h e time b e i n g (lo, ' I ) . The a u t h d r of t h e p r e s e n t r e p o r t , himself a member of t h e subcommittee on " l i m i t a t i o n of d e f l . e c t i o n s " of t h e German Committee f o r Reinforced Concrete, w i l l p r e s e n t i n what f o l l o w s h i s own i d e a s on t h e s e m a t t e r s , and a t t h e end of t h e r e p o r t w i l l

o f f e r a s u g g e s t i o n f o r a r e v i s i o n of d e f l e c t i o n l i m i t a t i o n s . The i n t e n t i o n i s n o t , and t h i s must bc emphasized, t o s e c u r e g r e a t e r "accuracy", b u t r a t h e r t o i n d i c a t e by c l e a r d e f i n i t i o n s t h e p o s s i b i l i t y of a p p l i c a t i o n even i n c a s e s n o t n u m e r i c a l l y conprchcnded i n any d e c r e e , and a t t h e sanie time t o d i s t i n g u i s h c l e a r l y between t h e t r e a t m e n t of members f o r which t h e d e f l e c t i o n s can o r cannot have harmful e f f e c t s .

For rcinforcccl c o i ~ c r c t c mclrtbcrs trhcrc the _{d e f l e c t i o n s can have no harmful}

e f f e c t s t h c p r e v i o u s ret;ul.ations of DDJ 1045, Sec. 22, 2, can be r e t a i n e d , r c q u l r - l n g n mini~~irus e f f e c t i v e h e i g h t of t i = 1 _I

/35.

The r e d u c t i o n t o h = 11/40

f o r s l a b s vrlllch a r c walked on o n l y f o r r e p a i r s and c l e a n i n g work, should be dropped, because owing t o t h e slriall l i v e l o a d s on such rnerrtbers t h e supplernen- t a r y d e f l e c t i o n s r e p r c s e n t a l a r g e r p r o p o r t i o n o f t h e t o t a l d e f l e c t i o n . The i d e a l l e n g t h t i was d c f l n e d i n D I N 1045 f o r onc-vray s l a b s a s t h e d i s t a n c e between p o i n t s o f i n f l e c t i o n ; f o r t h i s a more meaningful d e f i n i t i o n should be chosen, which w i l l a t t h e same time cover two-way s l a b s . Such a d e f i n i t i o n i s d e r i v e d from t h e problem of spanning a space of g i v e n dimensions f o r an

e f f e c t i v e load of q = g -I- p w i t h a c e i l i n g supported i n any manner. T h i s can be r e s o l v e d e i t h e r with a one-way c e i l i n g s l a b hinged a t b o t h ends o r , depend- i n g on t h e a d j o i n i n g s p a c e s , w i t h a c e i l i n g system t h a t i s continuous i n one o r both d i r e c t i o n s . I have a l l u d e d above t o t h e f a c t t h a t f o r m e r l y t h e i n s t a l l a - t i o n of i n s i d e p a r t i t i o n s r e s t i n g on f l o o r s b u t n o t supported on t h e s o i l was unusual, 1 . e . members were g e n e r a l l y designed i n such a way t h a t t h e i r d e f l e c - t i o n s could have no d e l e t e r i o u s e f f e c t s . In such members, t h e r e f o r e , we can r e l y on t h e e s t a b l i s h e d f a c t t h a t t h e minimum s t i f f n e s s o b t a i n e d w i t h min h = 1/35 i s adequate. However, o n l y a few decades ago continuous f l o o r systems were n o t n e a r l y a s common a s t h e y a r e today, s o t h a t t h e e x p e r i e n c e i s d e r i v e d e s s e n t i a l l y from s i n g l e - s p a n f l o o r s . It i s r e a s o n a b l e , t h e r e f o r e , t o allow s m a l l e r e f f e c t i v e nlinimum h e i g h t s f o r continuous f l o o r s t h a n f o r s i n g l e - s p a n f l o o r s , such t h a t f o r e q u a l l e n g t h t h e d e f l e c t i o n corresponds t o a s i n g l e - s p a n s l a b w i t h h = 1/35. Ile t h e r e f o r e I n t r o d u c e a n " i d e a l l e n g t h "

such t h a t f o r t h e s l a b on two s u p p o r t s and hinged a t both ends t h e system- dependent c o e f f i c i e n t i s k = 1, and f o r o t h e r boundary c o n d i t i o n s k i s s o deterrnincd t h a t t h e sarne load q f o r t h e same span 1 c a u s e s t h e same d e f l e c t i o n f a s i n t h e two-way hinged s l a b . For one-way and two-way members t h e d e f l e c - t i o n a t mid-span o r f o r c a n t i l e v e r c o n s t r u c t i o n s a t t h e end of t h e c a n t i l e v e r and f o r continuous mectbers, f u l l load i s always a p p l i e d . Regardless of t h e k i n d s of load a c t u a l l y o c c u r r i n g i n an i n d i v i d u a l c a s e , t h e uniform load q i s

chosen h e r e , s i n c e i t i s n o t a q u e s t i o n of e x a c t l y p r e d i c t i n g a d e f l e c t i o n but n e r c l y a r e a s o n a b l e vray o f d e t e r m i n i n g an a p p r o p r i a t e minimum s t i f f n c s s

.

For t h e sane r e a s o n t h e v a r i o u s load d i s t r i b u t i o n s i n continuous systems were excluded. The f o l l o w i n g c o n s i d c r a t i o ~ ~ s a r e f o r s o l i d r e i n f o r c e d c o n c r e t e s l a b s ; however, t h e rninilllurn c f f e c t i v e d e p t h s c a l c u l a t e d from them must hold a l s o f o r o t i i e r t y p e s of ~ r ~ c n ~ b e r s such a3 r i b b e d f l o o r s , beams, T-beams, e t c .

This i s p o s s i b l e ~ ~ i l ; l l i n t h c limits imposed by t h e r e q u i r e d accuracy,

especially as t h e dimensioning i s d e c i s i v c l - y de t c m ~ i n c d by t h e minimum cf f e c - t i v c depth, p r a c t i c a l l y speaking, o n l y i n t h c c a s e of s o l i d s l a b s .

Tlle deflection of t h e two-way hinged s l a b , r e f e r r e d t o t h e span 1, a c c o r d i n g t o F i g . 1 ( s u b s c r i p t o ) nlay be w r i t t e n

For any s l a b , r e g a r d l e s s of how supported, e.g. a c c o r d i n g t o F i g . 2

( s u b s c r i p t k ) :

From t h e requirement of e q u a l i t y of d e f l e c t i o n s f o = f k , w i t h e q u a t i o n ( 1 ) it f o l l o w s t h a t

Now, .A = 5/384, and e .g. f o r a one-way r e i n f o r c e d s i n g l e - s p a n s l a b f i x e d both ends (assuming p = 0 ) A k = 1/384, and t h e r e f o r e , f o r t h i s s u p p o r t c a s e k =

3

40.2

= 0.58. For a square s l a b hinged on f o u r s i d e s

$

= l/246, hence k =

3 7 . -

4 384/(5 246) =

0.63.

The k v a l u e s c a l c u l a t e d f o r one-way, s i n g l e - s p a n _- s l a b s a r e shovrn i n F i g .

3

and f o r two-way s l a b s i n F i g . 4. For any span o f a one-way continuous system a c c o r d i n g t o F i g . 5 we o b t a i n with

ml = -Mdq12 and mr = - 1 ~ l d ~ 1 2 , ( 5 4

M1 and Mr a r e rnornents a t t h e s u p p o r t s t h a t g e n e r a t e t e n s i l e s t r e s s e s on t o p and a r e t h e r e f o r e p u t n e g a t i v e i n e q u a t i o n ( 5 a ) , s o t h a t f o r n e g a t i v e

moments nil and IT', a r e p o s i t i v e . Thc k v a l ~ c s under f u l l load f o r a number of

s l a b s of two t o f i v e spans a r e c o l l e c t e d i n Plg.

6

-

8. For an e l a s t i c a l l y f i x e d overhangrlng s l a b acco13ding t o F i g . 9 w e o b t a i n with

The 1c v a l u e s f o r an overhanging s l a b w i t h one s p a w a n d w i t h two ad j o i n i n g spans a r e shown i n F i g . 10; t h e f u l l load has been p l o t t e d f o r t h e e n t i r e s y s tem.

With t h e d . e f l n i t i o n introduced above two-way s l a b s a r e a l s o covered a d e q u a t e l y and i n a way analagous t o t h e former DIN 1045, which h a s proved i t s e l f . The c o n d i t i o n h

2

li/35 f o r a s q u a r e s l a b hinged on f o u r s i d e s means t h a t h

2

0.681/35 = 1/51.5, s i n c e I i = 0.681. It i s t h u s almost i d e n t i c a l w i t h t h e former

DIN

_{c o n d i t i o n h}

2

_1/50

.

_{For a s l a b w i t h b/l = 1 . 5 a c c o r d i n g}

t o F i g . 4, hinged on f o u r s i d e s , L i = 0.64; t h e r e f o r e h L _{li/35 means h} =

0.841/35 = 1/42.

T h i s r e v i s i o n of t h e previous h

2

1/50 a p p e a r s q u i t e a p p r o p r i a t e , because w i t h i n c r e a s i n g r a t i o b/l t h e b e a r i n g e f f e c t of a two-way s l a b c o n t i n u o u s l y approaches t h a t of a one-way s l a b .

DIN

_{1045 p r e v i o u s l y took account of t h i s}

f a c t o n l y inasmuch a s t h e v a l u e s f o r one-way s l a b s were s u b s t i t u t e d from b/l = 1.5 on. A s l a b w i t h b/l = 1.4, f i x e d on f o u r s i d e s , h a s l i = 0.551, s o t h a t t h e c o n d i t i o n h li/j5

is

e q u i v a l e n t t o t h e c o n d i t i o n h

2

1/64. A s l a b t h a t

is r i g i d l y f l x e d on a l l f o u r s l d e s can t h u s be made more s l e n d e r t h a n b e f o r e . Since t h e above c o n s i d e r a t i o n s f o r d e f l e c t i o n s of r e i n f o r c e d c o n c r e t e members t h a t can have l i t t l e o r no damaging e f f e c t lead t o r e s u l t s t h a t

approximate t h e former

DIN

_{1045 r e g u l a t i o n , i t seems a p p r o p r i a t e t o a s k whether}

one should r e v i s e t h i s proven r e g u l a t i o n a t a l l a s f a r a s t h e s e members a r e concerned. The a u t h o r b e l i e v e s t h a t we should do so, because t h e d e f i n i t i o n given h e r e o f f e r s t h e p o s s i b i l i t y of a s s e s s i n g every p o s s i b l e b e a r i n g s i t u a - t i o n . T h i s i s most c l e a r l y dernonstratcd on t h e overhanging beam ( c f . F i g . l o ) , f o r which i n f u t u r e , g r e a t e r e f f e c t i v e d e p t h s w i l l be n e c e s s a r y i n some c a s e s t h a n were f r e q u e n t l y p r o j e c t e d i n t h e p a s t . O f course t h i s does n o t seem n e c e s s a r y f o r a l l overhanging c o n s t r u c t i o n , e s p e c i a l l y i n t h e c a s e of f i l l i n g s t a t i o n r o o f s , canopies o v e r e n t r a n c e s , e t c . In such c a s e s t h e diminished c o n d i t i o n h = li/50 i s suggested, a f t e r t h e p a t t e r n of t h e s t r u c t u r a l s t e e l r e g u l a t i o n s .

F o r r e i n f o r c e d c o n c r e t e members whose d e f l e c t i o n s may have harmful e f f e c t s we s t a r t from t h e f a c t t h a t f o r f o r m e r l y customary a d m i s s i b l e s t r e s s e s up t o

about tsb

-

65

bg/cm2 and o = 1800 1ce/cri12 and t h e former c o n d i t i o n h = 1/35

e

f o r t h e s l a b hinged on two s i d e s no harmful e f f e c t s of d e f l e c t i o n s were

experienced. With _-

-

l i = lc

-

1 _{( 7}

1

we s h a l l i n t r o d u c e , f o r s t r e s s c s g r e a t e r than 65/1800 ke/cm2, a s t r e s s -

dependent value ~ ( o ) and a s u i t a b l e niininiun s l e n d e r n e s s h L

TI/@.

Accordingly

w e e s t a b l i s h p =

Po

=

35

and

TI

= 1 f o r a one-way s l a b hinged a t b o t h s i d e s and s u b j e c t e d t o a uniformly d i s t r i b u t e d load q, i f i t s s t r e s s e s a t mid-span come t o ab = 65 kg/cm2 and oe = 1800 ke/cm2. F o r o t h e r s t r e s s e s a t mld-span t h e s t r e s s - d e p e n d e n t value @ i s determined such t h a t f o r e q u a l s u p p o r t and e q u a l b e a r i n g c o n d i t i o n s t h e same d e f l e c t i o n f o c c u r s . The e l a s t i c d e f l e c t i o n of t h e beam of span 1 supported a t two p o i n t s i s

f o , e l /1 = A.

-$$-

w i t h hO = 5/j84.

With t h e mid-span moment No = q12/8 and mo = ~ d =

1/8,

~ i t 1 f o l l o w s t h a t ~

and f i n a l l y w i t h s u b s t i t u t i n g t h e c o n c r e t e and s t e e l s t r e s s e s t h e n t h e e l a s t i c d e f l e c t i o n i s given by In o r d e r t o t a k e i n t o account t h e i n f l u e n c e of c r e e p and s h r i n k a g e , t h e e l a s t i c d e f l e c t i o n f a c c o r d i n g t o r e f . 1 Is m u l t i p l i e d by a c o r r e c t i o n o,e

f a c t o r u which a l s o depends on t h e s t r e s s . F o r ob = 65 kg/cm2 and o e =

1800 kg/cm2 t h i s i s denoted by o o . The t o t a l d e f l e c t i o n r e f e r r e d t o t h e span

For a r b i t r a r y s t r e s s e s 0 _{b,k and oc, k~} b u t t h e same a t a t i c system i t f o l l o w s t h a t

We now wish t o know t h e e f f e c t i v e d e p t h

%

= 1/J3 which f o r t h e stresses o b

,

k

and o g i v e t h e same d c f l e c t i o n and t h e e f f e c t i v e d e p t h ho = f o r t h e e , k

s t r e s s e s a = 65 kdcn12 and a = 1800 kg/cm2. Equating ( 8 a ) and ( 8 b ) we

b , o e , 0

g e t

The v a l u e s n o and xk a r e o b t a i n e d from r e f . 1, assuming S = 3 0

With xo = 1.32, n =

15,

a b J o =

65

kg/cm2, a = 1800 kg/cm2, and

Po

=

35

e , o we g e t These

B

v a l u e s a r e p l o t t e d i n F i g . 11 f o r 0

2

a b J k

6

_{120 kg/cm2 and 1400}

_I

u _e.k

_-

1

2800 kp/cm2. For p r a c t i c a l u s e i t i s a d v i s a b l e t o combine t h e v e r i f i c a t i o n w i t h t h e u s u a l d e s i g n procedure. Accordingly

%

=

dm,

where h i s i n om, M i n t m and b i n m. If we wish t o r e s t r i c t d e f l e c t i o n t o t h e amount recommended f o r members which may cause damage, t h e n we must t a k e h =

Ti/B,

i . e . hrequired

-

-

Ti/& From t h e two e q u a t i o n s

h r e q u i r e d [cm] = kh

.

d n ~ b hobtained [cm] = l 0 0 i i / ~ w i t h

ii

i n m i t f o l l o w s t h a t With t h e a b b r e v i a t i o n s kf =

Ti/

./

Ti

i n ~ n ,

M

_{i n t m , b i n m} +

In

r e f . 1 t h i s v a l u e i s denoted ~y x,; i n F i g . 3b o f t h a t a r t i c l e t h e r i s a t y p o g r a p h i c a l e r r o r ; t h e s o l i d c u r v e s h o l d f o r s s =

15

.

lo",

t h e broken l i n e c u r v e s f o r e S = 3 0

lom5

we o b t a i n

The recommended l i m i t a t i o n of d e f l e c t i o n i s t h u s observed i f

The kh values and corresponding k? v a l u e s a r e c o l l e c t e d i n Fig. 12. For double reinforcement t h e k? v a l u e s which correspond t o t h e v a l u e s which determine t h e c o n c r e t e s t r e s s e s . I would recommend b a s i n g t h e assessment of d e f l e c t i o n s f o r bending and compression s t r e s s e s on t h e c o n c r e t e s t r e s s e s t h a t would be e f f e c t i v e I n s t a t e

_-

I1 without normal f o r c e .

The i d e a l l e n g t h

TI

= k 1 f o r members whose d e f l e c t l o n s can have harm- f u l e f f e c t s should be d e f i n e d , u n l i k e l i f o r members whose d e f l e c t i o n s can produce no damage, by e q u a t i n g t h e s t r e s s e s i n s t e a d of t h e l o a d s ( o n l y t h e f3

values, n o t t h e

'E

values, i n f l u e n c e t h e s t r e s s e s ! ) . For t h e s l a b hinged on both s i d e s we have a l r e a d e s t a b l i s h e d k = 1. For o t h e r boundary c o n d i t i o n s k i s determined such t h a t f o r e q u a l span 1 and e q u a l s t r e s s e s

ab

and ae i n mid- span, or, i n t h e case of overhanging beams a t t h e f i x i n g p o i n t , t h e same d e f l e c t i o n s f a r e obtained a s f o r hinged s u p p o r t on both s i d e s . For one-way

and two-way members t h e d e f l e c t i o n i n mid-span i s a p p l i e d ; f o r overhanging beams a t the overhang end and f o r continuous members f u l l load I s a p p l i e d . For two-way members t h e s h o r t e r span i s always t h e d e c i s i v e one. The maximum s t r e s s e s ab and oe i n mid-span can occur, of course, i n t h e d i r e c t i o n of t h e longer span and I n such c a s e s they must be introduced i n t o t h e c a l c u l a t i o n . For a s i n g l e - s p a n s l a b hinged on two s i d e s ( s u b s c r i p t o ) t h e d e f l e c t i o n

r e f e r r e d t o t h e span, according t o e q u a t i o n (Ba), f o r given s t r e s s e s ob and oe

For a s l a b supported i n any manner (index k ) of e q u a l span 1 with t h e same s t r e s s e s ob and oe, u s i n g e q u a t i o n ( 7 ) and i i / h = f3, is

where

q

i s t h e bending moii~ent a t mld-span d i v i d e d by q12 ( i n t h e c a s e of

overhanging s l a b s a t t h e p o i n t of f i x i n g ) and f o r two-way s l a b s i s t h e g r e a t e r of t h e moments a c t i n g i n both d i r e c t i o n s . A s an example we a g a i n choose a one-way r e i n f o r c e d s l a b f i x e d a t both ends. Here = 5/304, mo =

1/8,

and f o r f i x a t i o n on two s i d e s

\

= 1/384 and

5

= 1/24, hence f o r t h i s case k =

24/40 = 0.6. This value d i f f e r s b u t l i t t l e from t h e v a l u e k = 0.58 o c c u r r i n g f o r t h e same c a s e . The main d i f f e r e n c e s a r c found f o r two-way s t r e s s e d s l a b s . The v a l u e s computed a c c o r d i n g t o e q u a t i o n ( 1 1 ) f o r one-way s i n g l e - s p a n s l a b s

are shown in F i g . 13 and f o r two-way s i n g l e - s p a n s1.abs i n F i g . 14. In a l l c a s e s p = 0 was employed. In two-way s l a b s v a l u e s of

%

> 1 a r e o b t a i n e d , owing t o t h e f a c t t h a t f o r t h e same load q and t h e same span 1 t h e d e f l e c t i o n of a two-way s l a b d e c r e a s e s t o a l e s s e r degree in r e l a t i o n t o t h e one-way s l a b t h a n t h e bending moment. For t h e c a s e of a one-way continuous system accord- i n g t o F i g .

5

_{we o b t a i n , w i t h}

_rill

_{and mr a i c o r d i n g t o ( 5 a )}

The

i?

v a l u e s f o r a number of two t o f i v e - s p a n s l a b s under f u l l load a r e given i n F i g .

15

-

17. For an e l a s t i c a l l y f i x e d s l a b a c c o r d i n g t o F i g .

9,

w i t h rnr

a c c o r d i n g t o ( 6 a ) we g e t

I n F i g . 18

E

v a l u e s a r e given f o r overhanging s l a b s with one span and with two a d j o i n i n g spans. F u l l load on t h e e n t i r e system i s a g a i n assumed.

Wlth t h e d e f l n l t i o n i n t r o d u c e d h e r e we g e t

Ti

v a l u e s f o r two-way s l a b s s i m i l a r t o t h o s e o b t a i n e d f o r one-way s l a b s , t h e s u p p o r t c o n d i t i o n s of t h e two s l d e s p a r a l l e l t o t h e s m a l l e r span d i r e c t i o n I n t h l s c a s e d e t e r m i n i n g t h e v a l u e s f o r t h e two-way s l a b s . For members whose d e f l e c t i o n s can have harmful e f f e c t s t h i s i s a l t o g e t h e r a p p r o p r i a t e , because f o r e q u a l s t r e s s e s and e q u a l span t h e two-way s l a b can t r a n s m i t a much l a r g e r load t h a n a one-way s l a b .

According t o F i g . 11 v a l u e s of f3 >

35

a l s o o c c u r , depending on t h e rnagni- tude of t h e s t r e s s e s ab and a e . It would n o t be c o r r e c t t o r e p l a c e t h e s e v a l u e s by B = 35, a s provided i n t h e former temporary r e g u l a t i o n , because Ti

#

l . Rather, t h e a p p l i c a b l e v a l u e s of l3 a r e l i m i t e d by t h e c o n d i t i o n

In

_{members i n t h e open a i r o r i n unheated s p a c e s s m a l l e r permanent}

d e f l e c t i o n s occur t h a n i n heated rooms. It i s r e a s o n a b l e t h e r e f o r e , i n compli- ance with t h e a l r e a d y e s t a b l i s h e d r u l e , t o allow f o r 15% g r e a t e r v a l u e s of

B

t h a n t h o s e a c c o r d i n g t o F i g . 11 and 12, where t h e l i m i t 11/35 a g a i n a p p l i e s . In o t h e r words we nlust have

min h = maxPi/(l.15B); 1 ~ / 3 5 ] .

Experience shows t h a t i n c o n v e n t i o n a l l y designed apartment b u i l d i n g s where t h e spans a r e determined by average room dimensions, harmful e f f e c t s due

t o l a r g e d e f l e c t i o n s h a r d l y e v e r occur, ever1 when e x t r a w a l l s ( ~ u s b a u w h d e ) n o t supported on t h e ground a r e e r e c t e d on t h e f l o o r s . F o r t h e s e f l o o r s ,

t h e r e f o r e i t i s o n l y n e c e s s a r y t o observe t h e minimum c o n d i t i o n h = 11/35.

A s a l i m i t beyond which t h i s c o n d i t i o n no l o n g e r s u f f i c e s I would s u g g e s t

Ti

=

5.0 m. I n one-way systems -under s t a n d a r d s u p p o r t c o n d i t i o n s t h e f o l l o w i n g t r u e spans correspond t o t h i s v a l u e :

Single-span s l a b hinged b o t h s i d e s 1 =

5.0

m End span of a continuous s l a b hinged

a t t h e o u t s i d e l

=

5.5

m Single-span s l a b hinged one s i d e ,

f i x e d t h e o t h e r a i d e 1 =

6.2

m I n s i d e span of a continuous s l a b 1

6.6

m Single-span s l a b f i x e d b o t h s i d e s l =

8.3

m

T h i s t a b l e shows t h a t t h e d e s i g n s of t h e c o n v e n t i o n a l apartment house a r e s c a r c e l y a f f e c t e d by t h e s u g g e s t i o n s made h e r e .

F i g u r e 19 g i v e s a swnmary of k and v a l u e s . These a r e v a l u e s t h a t can be c o n s i d e r e d a d m i s s i b l e without v e r i f i c a t i o n . For t h i s r e a s o n upper l i m i t i n g v a l u e s have been given f o r one-way continuous systems f o r overhanging o u t s i d e and i n s i d e spans. The same h o l d s f o r t h e k v a l u e s of two-way s l a b s where t h e curves i n F i g . 4 have been r e p l a c e d by s t r a i g h t l i n e s s i t u a t e d above them, which a t b/l = 2 a t t a i n t h e k v a l u e s f o r one-way nlembers of span 1, s o t h a t f o r 1 b/1 5 2 a l i n e a r i n t e r p o l a t i o n i s p o s s i b l e . A v a l u e s f o r two-way s l a b s t h o s e of one-way s l a b s with t h e s u p p o r t c o n d i t i o n s of t h e s h o r t e r span d i r e c t i o n were i n t r o d u c e d , a l t h o u g h t h e v a l u e s g i v e n i n F i g . 14 a r e a l i t t l e h i g h e r . In view of t h e r e l a t i v e l y low accuracy requirements t h i s s i n i p l i f i c a - t i o n seemed j u s t i f i a b l e .

I n l i n e w i t h t h e above c o n s i d e r a t i o n s I recommend r e p l a c i n g t h e former " ~ e n i p o r a r y r e g u l a t i o n on t h e l i r n l t a t i o n of d e f l e c t i o n of r e i n f o r c e d c o n c r e t e members a c c o r d i n g t o DIN 1045" by t h e f o l l o w i n g r l e g u l a t i o n :

1. In e s t a b l i s h i n g p e r m i s s i b l e minimum s l e n d e r n e s s v a l u e s f o r r e i n f o r c e d c o n c r e t e members t h e l a t t e r a r e d i v i d e d i n t o two c a t e g o r i e s :

1.1 Reinforced concre t c lrrelilbcrs whose dcf l e c t i o n s can have no harmful e f f e c t s .

1 . 2 Reinforcetl c o n c r e t e menlbers whose d e f l e c t i o n s can have harmful e f f e c t s .

The term "harmful e f f e c t s " niearis h e r e consequences which may i m p a i r t h e s e r v i c e a b i l i t y of t h e s t r u c t u r e . E f f e c t s which i m p a i r t h e s t r u c t u r a l s a f e t y a r e t h e consequence of a n i n c o r r e c t o r incorr~plete s t r e n g t h c a l c u l a t i o n and a r e n o t a f f e c t e d by t h e f o l l o w i n g l i m i t a t i o n s .

Group 1.1 i n c l u d e s e . g . a l l f l o o r s which do n o t s u p p o r t w a l l s o u t s i d e t h e main s t r u c t u r e ( ~ u s b a u w 8 n d e ) and a r e s u p p o r t e d a t t h e i r b o u n d a r i e s by b e a r i n g w a l l s o r r l g i d crossbeams, a s

_-

w e l l a s a l l members w i t h s p a n s I and s u p p o r t c o n d i t i o n s such t h a t

Ti

= k I = 5 . 0 m. For t h e d e f l n i t i o n o f s e e no.

3 .

Group 1 . 2 I n c l u d e s e . g . wide-span f l o o r s

(Ti

> 5 . 0 m) w i t h n o n - b e a r i n g w a l l s , f l o o r s w i t h o u t crossbeams a t t h e b o u n d a r i e s and w i t h n o n - b e a r i n g p a n e l s

( g l a s s p a n e s ) mounted on t h e s e b o u n d a r i e s , e t c .

2 . Where members o f group 1.1 a r e s u b J e c t t o b e n d i n g s t r e s s i t i s

r e q u i r e d t h a t h 2 11/35. T h i s c o n d i t i o n a p p l i e s t o one-way s t r e s s e d and two- way s l a b s , r i b b e d f l o o r s , beams, T-beams e t c . , b o t h as s e p a r a t e s t r u c t u r a l members and a s p a r t s of c o n t l n u o u s o r framing composite s t r u c t u r e s . F o r over- hanging members o f minor importance, e . g . r o o f s o f f i l l i n g s t a t i o n s and w a i t - i n g sheds, e n t r a n c e c a o p i e s e t c . , i t i s p e r m i s s i b l e t o p u t h & 11/50.

Here h i s t h e s t a t i c e f f e c t i v e d e p t h and l i t h e a v a i l a b l e span m u l t i p l i e d by a c o e f f i c i e n t k, i . e . li = k

.

1 . F o r s i n g l e - s p a n members hinged on b o t h

s i d e s t h e system-dependent c o e f f i c i e n t k = 1. F o r o t h e r l i m i t i n g c o n d i t i o n s k

Ps _{determined such t h a t f o r g i v e n l o a d s q and g i v e n a v a i l a b l e s p a n 1 t h e same}

d e f l e c t i o n s f a r e o b t a i n e d a s f o r hinged. s u p p o r t on b o t h s i d e s . F o r one-way and two-way members t h i s r e f e r s t o t h e d e f l e c t i o n a t mid-span, o r f o r overhang- i n g beams t o t h e d e f l e c t i o n a t t h e end of t h e overhang, and f o r c o n t i n u o u s members f u l l l o a d i s assumed.

3 . _{I n t h e c a s e of inembers of group 1 . 2 s u b j e c t t o bendlng s t r e s s h}

Ti/

l3 i s recomniended. T h i s a p p l i e s a l s o f o r t h e same members as u n d e r 2.

The s t r e s s - d e p e n d e n t i3 v a l u e s a r e c o l l e c t e d . i n t h e t a b l e ( F i g . 11). F o r s u p p o r t hinged on b o t h s i d e s and s t r e s s e s ob =

65

kg/cm2,0e = 1800 kg/cm2,

B

-

3 5 . F o r o t h e r s t r e s s e s ob and oe,

8

i s d e t e r m i n e d s u c h t h a t f o r e q u a l spans 1 and e q u a l boundary c o n d i t i o n s e q u a l d e f l e c t i o n s f a r e o b t a i n e d . For one-way and two-way members t h i s r e f e r s t o d e f l e c t l o n s and s t r e s s e s a t m l d - span, o r f o r overhanging beams t h e d e f l e c t i o n s a t t h e end o f t h e overhang and t h e s t r e s s e s q t t h e p o i n t of f i x a t i o n u n d e r f u l l l o a d .

-

In t h i s h 1.9 t h e s t a t i c e f f c c t i v c depth, l l t h e a v a i l a b l e span m u l t i p l i e d by a c o e f f i c i e n t ( c f . No. 2 ) . For s i n g l e - s p a n members hinged on both s i d e s t h e system-dependent c o e f f i c i e n t i? = 1. For o t h e r boundary c o n d i t i o n s

i s made such t h a t f o r equal spans 1 and e q u a l s t r e s s e s a b and a,, t h e same d e f l e c t i o n s f a r e obtained a s f o r h i n g i n g on both s i d e s . For t h e l o c a t i o n of t h e determining d e f l e c t i o n s t h e same a p p l i e s a s has a l r e a d y been s a i d I n con- n e c t i o n w i t h

B.

For two-way members t h e maximum s t r e s s i s d e c i s i v e even when i t does n o t occur i n t h e d i r e c t i o n of t h e s h o r t e r span.

The m i n i m u m e f f e c t i v e h e i g h t h computed h e r e must n o t be l e s s t h a n t h a t computed a c c o r d i n g t o No. 2. I f a lower value i s obtained then t a k e t h e minimum depth a c c o r d i n g t o No. 2, 1 . e . min h = max

fli/B;

_li/35].

4.

For members i n t h e open a i r o r i n unheated s p a c e s t h e c o e f f i c i e n t s f3

a c c o r d i n g t o t h e t a b l e ( F i g . 11) can be i n c r e a s e d by 15s. The r e s u l t i n g minimum d e p t h s h must n o t be l e s s t h a n those computed a c c o r d i n g t o No. 2,

1 . e . m i n h = max[Ti/(1.15~); 1,/35].

5. I n s t e a d of a n i n d i v i d u a l v e r i f i c a t i o n t h e k and

TI

v a l u e s a c c o r d i n g t o t h e t a b l e ( ~ i g . 1 9 ) can be used. In t h e case of two-way s l a b s t h e k v a l u e s f o r 1.0

&

b/l

5

2.0 can be l i n e a r l y i n t e r p o l a t e d . For b/l > 2 . 0 t h e v a l u e s f o r one-way s l a b s with t h e s u p p o r t c o n d i t i o n s of t h e s h o r t e r span d i r e c t i o n aPP l y

I n c l o s i n g i t must be remarked t h a t j u s t t h e observance o f c e r t a i n minimum e f f e c t i v e d e p t h s w i l l n o t by i t s e l f p r e v e n t harmful consequences due t o

d e f l e c t i o n s . Much more can be c o n t r i b u t e d t o t h i s by e x p e r t d e s i g n i n g and by t h e p a r t i c i p a t i o n of t h e e n g i n e e r i n t h e p l a n n i n g and development of d e t a i l . Even when high a d m i s s i b l e s t r e s s e s a r e e x p l o i t e d i t i s d o u b t l e s s p o s s i b l e t o execute e n t i r e l y s a t i s f a c t o r y c o n s t r u c t i o n s i n modern r e i n f o r c e d c o n c r e t e . Of course, r e s p o n s i b l e e n g i n e e r s must do t h e d e s i g n i n g and e q u a l l y r e s p o n s i b l e c o n t r a c t o r s must execute t h e work.

References

1. 11. h l c l ~ n ~ ~ - I : I:in ncitrnf 711r n(-rcrlrnc~ric (lrr c,l.rstizrlicn und p I n s t i ~ ~ 1 1 ~ 1 1 D ~ I ~ C . I ~ I J I S ~ < I I I I ~ L , ~ .sclrI:~lf l~(wrlrr~t!r S t ~ ~ l ~ l l ~ t ~ l o i ~ l ~ : ~ l k c n , Drr

~ ~ ~ I I I ~ I I ~ I ~ I I ~ ~ I I ~ 3.1 (19:!1), S , 9,

2 . 1:. 1 , c n n h n r d t : hnfiinglidrc ~rncl r ~ n d l t r k ~ l i d i c D t ~ r ~ l r h i r g ~ r n - gc.li v1,11 S ~ : r h l l > r ~ ( ~ ~ i l ~ : ~ l k ( ~ ~ ~ im Zustnnd 11. Bcton- ~ r n d Sti~lill~eton-

I ) A I I 5.1 (Inan). s.2.10.

3. 11'. E r n 3 t : 1st dil: nrrtc SJilnnkI~citsvorsclrriIt Ilrrrd~tiyt?

R : ~ ~ ~ r t . ~ l i l ~ c ~ ~ ~ ~ ~ l i r - N n ~ l ~ r i r l ~ ~ O (lDOl), 11. 5.

.I. I?rl;tG clcs Ilcss. hli~rihtcr.s clrs Inncrn \,om 2. 11. 1080. (Aklz. Vll -- (; 1.1 : ,if03 1/60).

5 11. S c . l ~ r i i c l c ~ r : K o l ~ i r n r ~ ~ t n r 7.n D I S 10.13. (Ziff, 11. hIi11.131. XI\!\' S r . ! ) I ; vo111 26. 8 . 1980.) Ilnlr- 11nd n n ~ ~ i n d u s t r i c 7 (lROI\.

S. 262.

ti. K. O l > l : ~ d c n 11. E. R i c h t o r : Ililfsmittcl zrlr \,creit~f;~dilcn Rerrclr~ru~~g dcr I>~rrdrl,irs~rng von Stal~ll)ctonl~nutviIrn. B;rn ulrd H . ~ ~ ~ - i n d ~ ~ \ t r i t , 9 (1961). S. 363.

I . K . O p l ; i d e i ~ : nvt~rcc<~rnjir- uncl IIilfstnfeln zllr l3urii~kqidlti-

g~rllji clrl. z~~lii*si;cn Sd~l;~irLilcit fiir Stal~lbctonteile. 1X.r 1><.11tiAc I?nuinrirl(~r 3 (19131), S. 157.

S. ;\I. S t i l l c r : IIills~nlcln fiir die Ilenicssung v n n Stnlilbt.ton- tr:lgu~,rk~*n 11vi 13escirriinkt11ig der ,D11rclil)ic~1tn~ rinvli dchr Ergiitlzr~nz

Z I I ]>I9 Il)4.i. Dcto~i- u ~ ~ d S t n l ~ l l ~ c t o n l ~ m i 58 (1961). S. 00. 9. 11. n e c k : netontag 1001. Der Bnuingenieur 36 (1961). S. 321. 10. Erl:lR des IIcss. htinislors des Innern vom 4. 1. 1961. (Ak1.r. \'I> - 64;) 26/03 - 1/81.)

11. Erlnl3 (les I-lrss. Ministers dcs Innorn vom 4. 4. 1061. (Akls. \'I> - 8411 16119 - 20161.)

F i g . 1

-

1A

-

8 -8

I

-

- 1 -

-

! A A - A

F i g . 2

Rectanwlar s l a b supported on four s i d e s ,

one o f the 1onge.r s i d e s being f i x e d , the others hinged

k values f o r single-span s l a b s under d i f f e r e n t boundary conditions

F i g .

4

k v a l u e s f o r r e c t a n l a r s l a b s supported on f o u r s i d e s ; s i d e r a t i o s 1 . 0

L

b,%

6.

2.5;

v a r i o u s boundary c o n d i t i o n s

F i g . 6

k v a l u e s f o r two-span s l a b s with span r a t i o s

0.5 _1,/1, 5 2.0 k v a l u e s f o r t h r e e - s p a n s l a b s with v a r i o u s span r a t i o s F i g .

8

k v a l u e s f o r t h r e e - , f o u r - , and f i v e - s p a n s l a b s w i t h v a r i o u s span r a t i o s Feld = F i e l d

4 / 1 ,

-

F i g . 1 0

k v a l u e s f o r overhanging s l a b s f o r v a r i o u s r a t i o s o f overhanging span l e n g t h t o a d j o i n i n g span l e n g t h

F i g . 1 2

Table of kh and k; values

F i g .

13

k v a l u e s f o r s i n g l e - s p a n s l a b s under v a r i o u s boundary c o n d i t i o n s

Fig. 14

k values f o r rectangular s l a b s supported on f o u r s i d e s with s i d e r a t i o s 1.0 Si b/l S 2 . 5

F i g . 15

k values for two-span slabs wlth span ratios

0.5

& 1,/1, & 2 . 0

Fig. 16

k values for three-span slabs wlth various span ratios

Fig. 17

k values for three-, four- and five-span slabs with various span ratios

Fig.

IS

k values f o r overhanging slabs for various r a t i o s o f overhanging span l e n g t h t o a d j o i n i n g span l e n g t h

Fig. 19