Critical Observations on the German Specifications for the Calculation of the Supporting Beams of Two-Way Slabs and Proposals for a New Method of Calculation

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Critical Observations on the German Specifications for the Calculation

of the Supporting Beams of Two-Way Slabs and Proposals for a New

Method of Calculation

Dischinger, F.

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PREFACE

The D i v i s i o n of B u i l d i n g Research of t h e N a t i o n a l Research Council has t h e p r i v i l e g e of a s s i s t i n g t h e A s s o c i a t e Committee on t h e N a t i o n a l B u i l d i n g Code w i t h i t s s e c r e t a r i a l and t e c h n i c a l work i n r e l a t i o n t o t h e N a t i o n a l B u i l d i n g Code of Canada. A new e d i - t i o n of t h e Code was published d u r i n g 1960 and t h i s c o n t a i n s a n e n t i r e l y new S e c t i o n on r e i n f o r c e d c o n c r e t e .

During t h e p r e p a r a t i o n of t h i s S e c t i o n ( 4 . 5 ) on P l a i n ,

Reinforced and P r e - s t r e s s e d Concrete, t h e "luIarcusM method f o r t h e d e s i g n of two-way. s l a b s was adopted because i t was considered t o be one of t h e b e s t a v a i l a b l e . It was p o i n t e d o u t t o t h e Revision Committee t h a t a p a p e r i n German by F. Dischinger showed t h a t a l t h o u g h t h i s method r e s u l t s i n s a t i s f a c t o r y s l a b d e s i g n s , i t does n o t lead t o t h e p r o p e r d e s i g n of s u p p o r t i n g beams.

Accordingly t h e assumptions f o r beam l o a d i n g c o n t a i n e d i n t h e 1953 e d i t i o n of t h e Code were r e t a i n e d i n t h e 1960 e d i t i o n because t h e y were known t o lead t o more a c c u r a t e r e s u l t s . The Revision Committee, under t h e chairmanship of P r o f e s s o r M . W . Huggins, recommended t o t h e D i v i s i o n t h a t M r . D i s c h i n g e r ' s p a p e r be t r a n s l a t e d

.

T h i s h a s now been done by M r . E.M. Rensaa of Rensaa and

Minsos, Edmonton, A l b e r t a . The D i v i s i o n r e c o r d s i t s a p p r e c i a t i o n t o M r . Rensaa f o r t h i s t r a n s l a t i o n . The p u b l i c a t i o n of t h i s t r a n s l a t i o n i s i n d i c a t i v e of t h e way i n which DBR/NRC i s a b l e t o assist t h e A s s o c i a t e Committee on t h e N a t i o n a l B u i l d i n g Code w i t h i t s important work.

Ottawa R.F. Legget

Title :

NATIONAL RESEARCH COUNCIL OF

CANADA

Technical Translation 1048

Critical observations on the German specifications for the calculation of the supporting beams of two-way slabs and proposals for a new method of calculation

(~ritische Betrachtungen zu den deutschen Bestlmmungen far die

Berechnvg der ~r8~erroste kreuzweis gespannter Platten und

~orschlh~e fur eine neue ~erechnungsweise)

Author:

F.

Dischinger

Reference : Beton und Eisen, 41: 86-91, 103-108, 127-130, 1942

CRITICAL OBSERVATIONS ON

THE

GERMAN SPECIFICATIONS FOR THE CALCULATION OF THE SUPPORTING BEAMS OF TWO-WAY SLABS AND

PROPOSALS FOR A NEW METHOD OF CALCULATION Foreword

With t h e i n t r o d u c t i o n of equation (10) of A r t i c l e 23 ( t h e Marcus Method) of t h e German s p e c i f i c a t i o n s f o r r e i n f o r c e d concrete dated 1932 a very simple method of a n a l y s i s of two-way s l a b s was introduced. This method is s u f f i c - i e n t l y a c c u r a t e f o r p r a c t i c a l purposes and has proved i t s e f f i c a c y in every r e s p e c t . Paragraph

5

of A r t i c l e 23 a l s o e s t a b l i s h e d t h e s p e c i f i c a t i o n s f o r c a l c u l a t i n g t h e s u p p o r t i n g beams. According t o t h i s paragraph, t h e load t r a n s f e r r e d from t h e uniformly loaded r e c t a n g u l a r s l a b t o t h e beams o r

p e r i p h e r a l w a l l s could be assumed t o be uniformly d i s t r i b u t e d a l o n g t h e beam o r w a l l . This l a t t e r assumption, however, does n o t a t a l l correspond t o a c t u a l c o n d i t i o n s and i s c o n t r a r y t o t h e o r e t i c a l p r i n c i p l e s . Even with

square, f r e e l y supported s l a b s t h e a c t u a l bending moments in t h e beams, a l s o f r e e l y supported, a r e 61% g r e a t e r than c a l c u l a t e d a c c o r d i n g t o t h e 1932 s p e c i f i c a t i o n , and t h e e r r o r increased by a m u l t i p l e f o r t h e s h o r t marginal beams in elongated s l a b s . Thus, in t h e case of a f r e e l y supported s l a b of

1

a span r a t i o E = 2 = 2, e x a c t c a l c u l a t i o n s g i v e s bending moments about

8.3

x

times g r e a t e r f o r t h e beams of t h e s h o r t e r span than a r e obtained by t h e s p e c i f i e d method of c a l c u l a t i o n .

This conventional method of c a l c u l a t i n g t h e s u p p o r t i n g beams i s n o t o n l y extremely i n a c c u r a t e , but i s a t t h e same time cumbersome because f o r continuous beams d i f f e r e n t dead and l i v e load e f f e c t s a r e obtained from each type of a d j a c e n t s l a b s .

A s a r e s u l t of my r e p r e s e n t a t i o n s , t h e new German r e i n f o r c e d c o n c r e t e s p e c i f i c a t i o n (1943) w i l l , t h e r e f o r e , b e arnmended. In t h e p r e s e n t paper t h e b a s i s f o r t h e s e changes a r e discussed and proposals a r e made f o r more cor- r e c t c a l c u l a t i o n s adapted t o a c t u a l c o n d i t i o n s .

A t t h e same time, new t a b l e s of bending moments f o r two-way s l a b s a r e published. It i s w e l l known t h a t i t is p o s s i b l e t o reduce t h e support con- d i t i o n s t o s i x and U s e r ' s book c o n t a i n s t a b l e s f o r t h e e e c o n d i t i o n s . Every d e s i g n e r probably knows how d i f f i c u l t i t i s t o use t h e s e t a b l e s in p r a c t i c e both f o r o r i g i n a l c a l c u l a t ' i o n s and f o r checking of work done by o t h e r s . It i s e a s y t o g e t conmsed because of t h e n e c e s s i t y of changing

x

and y c o o r d i n a t e s which depend on t h e boundary c o n d i t i o n s of t h e s l a b . G s e r B. Bemessungsverfahren, 7. Auflage (30. u. 31. ~ a u s e n d ) . Wilheim E r n s t & Sohn. B e r l i n , 1941.

E r r o r s from t h i s source can be e l i m i n a t e d by always u s i n g

I x

f o r t h e s h o r t span of s l a b and t o make a l l bending moments i n t h e s l a b a f u n c t i o n of t h i a l e n g t h . T h i s r e s u l t s , as w i l l be shown in S e c t i o n B, i n n i n e s l a b t y p e s i n - s t e a d of six. Only t h e range of s l a b r a t i o s from 6 = 1 . 0 0 t o & = 2.00 needs

t o be taken i n t o account, s o t h a t t h e t o t a l number of f i g u r e s r e q u i r e d i s l e s s .

In S e c t i o n A we s h a l l f i r s t show t h a t t h e e r r o r s involved in t h e con- v e n t i o n a l method of c a l c u l a t i o n a r e i n a d m i s s i b l e , while in S e c t i o n B new t a b l e s f o r c a l c u l a t i n g bending moments i n two-way s l a b s w i l l be i n t r o d u c e d . S e c t i o n C w i l l show how simple b u t v e r y a c c u r a t e c a l c u l a t i o n s of t h e sup- p o r t i n g beams can be c a r r i e d o u t .

A . I n v e s t i g a t i o n s of Bending Moments i n S u p p o r t i n g Beams

1. The s u p p o r t i n g beams f o r a f r e e l y supported two-way s l a b a e shown on F i g . 1

Let u s assume t h a t t h e p e r i p h e r a l beams a r e supported in such a manner a s t o r o t a t e f r e e l y (and i n d e p e n d e n t l y ) a t t h e f o u r c o r n e r s and t h u s have no t o r s i o n a l r e s i s t a n c e t o s l a b r o t a t i o n . We nay t h e n d i s r e g a r d t h e dead load- i n g a s t h i s does n o t i n f l u e n c e o u r i n v e s t i g a t i o n s .

Unlike t h e s p e c i f i c a t i o n s , we denote t h e bending moments i n t h e s l a b w i t h t h e small l e t t e r s (mx and

%)

because we wish t o u s e t h e c a p i t a l l e t t e r s f o r t h e beam moments. In what f o l l o w s , t h e bending moments of two p a r a l l e l beams w i l l be taken t o g e t h e r as one member.

( a ) Beam moments by t h e e a r l i e r method of c a l c u l a t i o n . The bending moments of two p a r a l l e l beams a r e c a l c u l a t e d on basis o f t h e uniformly d i s - t r i b u t e d l o a d s q, and qy by means of t h e f o l l o w i n g e q u a t i o n s :

1

From t h i s , f o r v a r i o u s values of span r a t i o s & =

2

we o b t a i n :

l x

A l l beam moments, both in t h e x and t h e y d i r e c t i o n r e f e r t o t h e u n i t t o t a l load on t h e width of t h e s t r u c t u r e . In t h e y d i r e c t i o n we t h e n have

=

ox.

( b ) Considerably more a c c u r a t e r e s u l t s a r e o b t a i n e d by d i s t r i b u t i n g t h e loads t o marginal Foams in t h e manner i n d i c a t e d i n F i g . 2. The bending moment in t h e s h o r t beam span by t h i s load d i s t r i b u t i o n w i l l n o t depend on

l Y The equations f o r t h e beam moments w i l l then be:

The bendlng moments ( f o r two p a r a l l e l beams taken t o g e t h e r ) f o r d i f f e r -

rent

:wan

r a t i o s w i l l then be:

F o r a square s l a b t h e bending moment f r o m ( b ) i s

3356

g r e a t e r t h a n t h a t found by t h e conventional method o f c a l c u l a t i o n ( a ) For s = 2 t h e moments

M

a r e about s i x times g r e a t e r than f r o m ( a ) b u t even t h e v a l u e s from ( b ) a r e -x

l e s s t h a n t h e a c t u a l ones. I n t h e conventional method of c a l c u l a t i o n t h e bending moments

Ex

d e c r e a s e s very r a p i d l y with i n c r e a s i n g spans, whereas in method(b) t h e y remain c o n s t a n t . S t r i c t l y speaking, according t o t h e r i g o r o u s s l a b equation, t h e y must even i n c r e a s e somewhat with i n c r e a s i n g c; f u r t h e r - more, t h e t r u e bending moments a r e a l l g r e a t e r t h a n t h o s e developed above.

( c ) The c o r r e c t bending moments in t h e s u p p o r t i n g beams a r e o b t a i n e d

r 2 L

rhen from t h e t o t a l bending moments i n t h e p a n e l

Ex

= ( q l y )

-8

a n d H =

1 -Y

( q I X ) we deduct, t h e bending moment t a k e n by t h e s l a b i t s e l f i n t h a t d i r e c t i o n ( ~ 1 ~ .

3 ) .

I n o r d e r t o d i s t i n g u i s h t h e bending moments o f t h e whole system from t h o s e of t h e beams we w r i t e t h e i r symbols w i t h a dash on

t o p . Wen t h e complete system becomes s t a t i c a l l y indeterminate, t h e c o r r e - sponding bending moments

%

and

H

r e p l a c e t h e s t a t i c a l l y determinate

Y

moments

Ex

and and t h e s t a t i c a l l y i n d e t e r m i n a t e beam moments Mx and M

-Y'

Y

r e p l a c e t h e s t a t i c a l l y determinate beam moments

Ex

and M The e q u a t i o n s

-Y

Here m(x) and m ( y ) a r e t h e s l a b moments which v a r y a l o n g t h e y and x

a x e s r e s p e c t i v e l y , mx a n d m a r e t h e maximum s l a b moments on rrhich t h e Y

calculation i s based i n accordance with e q u a t i o n ( 1 0 ) o f t h e German r e i n f o r c - ed c o n c r e t e s p e c i f i c a t i o n s of 1932. F u r t h e r QX and $ a r e f a c t o r s which a r e

Y

t o be determined on t h e b a s i s of a r i g o r o u s s o l u t i o n of t h e s l a b e q u a t i o n . The bendlmg moments mx a l o n g y and t h e bending moments

%

a l o n g x can, w i t h s u f f i c i e n t accuracy, be assumed t o v a r y . p a r a b o l i c a l l y f o r a l l t y p e s of s l a b s s o t h a t we may p u t Jrx = $Y

-

-

7 -

S i n c e i n a l l c a s e s by f a r t h e g r e a t e r p a r t

-

o f t h e t o t a l moments a r e c a r r i e d by t h e beams, I& and

W

a s c a l c u l a t e d from

-Y

2

e q u a t i o n s ( 3 ) a r e r e l a t i v e l y u n a f f e c t e d by t h e assunlptio~l of

-

f o r t h e v a l u e 3

o f

qX

and $ The r i g o m u s s o l u t i o n s of t h e , s l a b e q u a t i o n given i n S e c t i o n

Y ' n L

( d ) show t h a t t h e f a c t o r Jr = 7 i s q u i t e a c c u r a t e .

J

The above e q u a t i o n s f o r a s l a b f r e e l y supported on a l l s i d e s w i l l now be e v a l u a t e d . The valuqs mx and m can be o b t a i n e d from g s e r f s t a b l e s o r

Y

from t h e numerical t a b l e s of s e c t i o n B. For moments in two p a r a l l e l beams we o b t a i n :

The beam moments ( c ) o b t a i n e d by means of e q u a t i o n

( 3 )

a r e s t i l l c o n s i d e r a b l y g r e a t e r than t h o s e determined from t h e t r i a n g u l a r and t r a p e z o i d a l l o a d s . The reason f o r t h i s l i e s i n t h e t o r s i o n a l e f f e c t s which have n o t been taken

i n t o account i n ( b ) . The t o r s i o n a l moments c r e a t e n e g a t i v e f o r c e s Z ( t e n s i l e f o r c e s ) a t t h e c o r n e r s of t h e f r e e l y supported s l a b and corresponding e q u a l

p o s i t i v e r e a c t i o n s ( o u t on t h e beam) and t h e l o a d s on t h e p e r i p h e r a l beams a r e i n c r e a s e d by t h e v a l u e s of t h e s e . A s shown on F i g . 4, we o b t a i n f o r a s q u a r e s l a b and w i t h P o i s s o n l s r a t i o v = 0 . 3 ( s t e e l ) , Z = -0.2375 qa2 f o r t h e s e t e n s i l e f o r c e s . The t o t a l load on a p e r i p h e r a l beam w i l l t h e r e f o r e be I n c r e a s e d from A = q a 2 t o A = 1.2375 q a 2 . The s l a b moments c o r r e s p o n d i n g t o

2 "

t h i s value of P o i s s o n l s r a t i o w i l l be mx = m = q

.

-

Y 21.2 ' A c t u a l l y , however

t h e bending moments i n o u r s p e c i f i c a t i o n s a r e based on t h e v e r y f a v o u r a b l e c o n d i t i o n of P o i s s o n l s r a t i o e q u a l zero, whereby t h e t o r s i o n a l e f f e c t i s

i n c r e a s e d and t h e s l a b moments a r e reduced t o mx = m _{= q}

.

1'

Y 27,iJ3' T h i s in-

c r e a s e d t o r s i o n a l e f f e c t corresponds t o a g r e a t e r t e n s i l e f o r c e of t h e o r d e r of Z = -0.34 qa2, whe,reby t h e l o a d on each p e r i p h e r a l beam i s t h e i n c r e a s e d t o A = 1 . 3 4 q a 2 .

F i g u r e 4a shows t h e d i s t r i b u t i o n of t h e beam l o a d i n g f o r t h e t h r e e c a s e s ( a ) , ( b ) and ( c ) . I n c a s e ( a ) t h e load on a p e r i p h e r a l beam i s con- s t a n t o v e r t h e e n t i r e beam l e n g t h and i s 0 . 5 qa per u n i t l e n g t h : t h e c o r r e - sponding beam moment M = ?$ 0.0625 q 1 2 . I n c a s e ( b ) , which i s Inore I n

conformity with f a c t s , t h e beam l o a d i n g h a s a t r i a n g u l a r shape w i t h o u t any change of t h e t o t a l load A = q a 2 . T h i s load d i s t r i b u t i o n g i v e s a c o n s i d e r - a b l y l a r g e r beam moment M =

$

*

0.0833

q 1 2 .

In

r e a l i t y , however, t h e beam l o a d i s d i s t r i b u t e d approximately a c c o r d i n g t o t h e c u b i c p a r a b o l a w i t h a maximum o r d i n a t e of 0.903 qa t o which corresponds a t o t a l load A = 1 . 3 4 q a 2 . The r e a s o n f o r t h i s i n c r e a s e i s b t h e a d d i t i o n a l l o a d of 0.34 q a 2 caused by t h e c o r n e r f o r c e . ( I t should be noted t h a t a t e n s i l e f o r c e a c t i n g a t t h e end of t h e beam would n e c e s s i t a t e

a

corresponding compressive f o r c e f a r t h e r o u t on t h e beam because t h e a l g e b r a i c sum of such f o r c e s must be z e r o . The c o r n e r f o r c e Z i t s e l f does n o t cause any bending in t h e beam b u t t h e oppo- s i t e l y d i r e c t e d f o r c e f a r t h e r o u t on t h e beam d o e s . ) A s

a

r e s u l t of t h i s c o n d i t i o n t h e bending moments In t h e p e r i p h e r a l bearns a r e f u r t h e r i n c r e a s e d t o M =

5

0.1007 q12. T h i s c o n d i t i o n a l s o g i v e s a v e r y simple e x p l a n a t i o n

1" 12 of why t h e s l a b moments a r e n o t merely reduced from q

-

t o q

8 due t o

two way d i s t r i b u t i o n of t h e load b u t r a t h e r t o a value of q

&

owing t o t h e supplementary t o r s i o n e f f e c t . The u n i f o r m l y d i s t r i b u t e d l o a d i n d i c a t e d on F i g .

5

would have a moment l e v e r arm of d

= $ = a

and

a

s l a b moment f o r

a

a

t h e vridth a of m = qa2

.

3 = q o r f o r a u n i t l e n g t h of s t r i p a moment o f

a2 1

m = q a 7 = q - z .

On account of t h e p a r a b o l i c shape of t h e load d i s t r i b u t i o n , i n combina- t i o n w i t h t h e n e g a t i v e c o r n e r f o r c e 2, t h e l e v e r a r m d of t h e i n t e r n a l f o r c e s in F i g . 5a h a s a l e n g t h of o n l y

58s

of t h e l e v e r a r m i n F i g .

5,

t h u s r e d u c i n g t h e s l a b moment t o q

.

0 . 5 8 = q

m.

12 T h i s r e d u c t i o n o f t h e s l a b moments, however, must l e a d t o a c o r r e s p o n d i n g i n c r e a s e i n t h e beam moments

from t h o s e o b t a i n e d from c a s e ( a ) t o t h o s e o f c a s e ( c ) .

Going from t h e f r e e l y supported s l a b t o a s l a b f i x e d on a l l s i d e s , t h e t o r s i o n moments a t t h e p e r i m e t e r o f t h e s l a b v a n i s h and hence a l s o t h e t e n - s i l e f o r c e s a t t h e c o r n e r s . The r e l i e f of t h e s l a b r~ioments from t o r s i o n i s t h e r e f o r e much s m a l l e r i n a continuous s l a b t h a n i n a s l a b f r e e l y s u p p o r t e d .

q 1"

The s l a b moments d e c r e a s e o n l y from

2

= q _T1 _{C ~ ~ O} I" _{~ P .We should} t h e r e f o r e e x p e c t t h a t w i t h a s l a b f i x e d a t t h e s u p p o r t s in accordance w i t h S e c t i o n 2, t h e l a r g e d i f f e r e n c e between t h e c o r r e c t beam moments ( c ) and t h o s e o f ( b ) w i l l be s m a l l e r . ( d ) Comparison of t h e v a l u e s o b t a i n e d w i t h t h e r e s u l t s of t h e r i g o r o u s s o l u t i o n by means of t h e ( m a t h e m a t i c a l l y c o r r e c t ) s l a b e q u a t i o n

%

=

PIN.

( a ) F o r a s q u a r e s l a b w i t h I = l y = 1 we o b t a i n e d a c c o r d i n g t o ( c ) t h e x- beam moments

Ex

= M =

M

= 0.1007 q 1 2 . On t h e b a s i s of t h e ( e x a c t ) s l a b -Y

-

e q u a t i o n w i t h v = 0, we o b t a i n

M

= 0.1012 c12, which a g r e e s v e r y c l o s e l y w i t h t h e v a l u e found from ( c ) ( d i f f e r e n c e

%).

7 L ( P ) I n t h e c a s e of a l o n g r e c t a n g u l a r s l a b w i t h E = = 2 we g e t X

a c c o r d i n g t o ( c ) t h e beam mornents x! = 0.1238 81x2, and M = 0.121.0

ql

2 .

-Y Y On t h e b a s i s o f t h e ( e x a c t ) s l a b e q u a t i o n and v = 0 we o b t a i n

Ex

= 0.1233 ?Ix* and M = 0.1222 5 1 2. -Y Y I n t h e f o l l o w i n g i n v e s t i g a t i o n s i t i s assumed t h a t t h e p e r i p h e r a l beams a r e a l s o f i x e d a t t h e i r ends. We s h a l l a g a i n denote t h e s t a t i c a l l y d e t e r - minate beam moments by

-

M and t h e s t a t i c a l l y i n d e t e r m i n a t e p o s i t i v e and nega- t i v e moments by M.

( a ) Beam moments a c c o r d i n g t o c o n v e n t i o n a l c a l c u l a t i o n . The l o a d i n g qx and qy a r e t h e same f o r a f i x e d s l a b a s f o r a simply supported one. The bending moment

M

f o r a simply supported beam a c c o r d i n g t o 1 ( a ) i s i n t h e

7

c a s e o f a bean1 w i t h f i x e d s u p p o r t s d i s t r i b u t e d w i t h as p o s i t i v e moments i n 2

t h e span and

-

₃ a s n e g a t i v e moments a t t h e s u p p o r t s .

7

( b ) Detemnination o f t h e beam moments by t r a p e z o i d a l and t r i a n g u l a r l o a d i n g . The s t a t i c a l l y d e t e r m i n a t e moments from t r i a n g u l a r and t r a p e z o i d a l

Mx = -0.0417 -0.0325 -0.0241 -0.0177 -0.0131 -0.0099 ( I 1 [M = - 0 . 1 -0.0563 -0.0662 -0.0723 -0.0760 -0.0785 Y <Ix 2 ELI $- o a o 8lY2]4$& 03

l o a d i n g w i l l now be d i s t r i b u t e d i n t o p o s i t i v e and n e g a t i v e moments ( a c t i n g a t mid-span and a t s u p p o r t s ) .

I

Mid-span

I

Supports

-

T r i a n g u l a r load: _-x M = q

. -

_{1 2} T r a p e z o i d a l l o a d . 1

₊

')

4E

=

E 2 The s t a t i c a l l y d e t e r m i n a t e momen~s of t h e t o t a l s y s t e ~ c o n s i s t i n g of 1 L -

-

-

X

-

s l a b and beams ( F i g . 3 ) :

M

= q

-

* -x 12 e and -Y = q

-

8

(1

-

$)

a r e d i s t r i b u t e d Fn accordance with t h e boundary c o n d i t i o n s ( f u l l f i x i t y ) between t h e p o s i t i v e and n e g a t i v e zones I n t h e manner I n d i c a t e d . I n o r d e r t o save .space w e s h a l l omit t h i s simple procedure.

The v a l u e s o b t a i n e d by means o f t h e s e e q u a t i o n s a r e shown i n t h e f o l l o w i n g t a b l e :

( c ) Determination of bending moments i n t h e edge beams from t h o s e of t h e whole system. Bending moments of t h e s t a t i c a l l y i n d e t e r m i n a t e t o t a l system a r e denoted by i n p l a c e of i n e q u a t i o n ( 3 ) . For t h e beam moments we now o b t a i n t h e f o l l o w i n g e q u a t i o n s which a p p l y f o r mid-span and a t t h e s u p p o r t s . Mid-span The f a c t o r s

Ix

and

ly

o b t a i n e d on b a s i s of t h e r i g o r o u s s o l u t i o n s f o r t h e s l a b f i x e d a t a l l edges a r e $,

-

2

-

JIY

= $ =

-5

and t h i s h o l d s b o t h f o r t h e A t s u p p o r t s =

W

-

m 1 $ h e r e i s X _{x Y Y'} =

W

-

m I

p

h e r e i s Y y X x' Ix2

-

nx

= ( q l y ) = q

.

E 1 = ( q l x )

&

= q

.

Y 2

-

-

_{IX . E} Mx = -q

.

-

1 2

-

-

1 2 ( 5 ) M = - q . + Y

p o s i t i v e and n e g a t i v e moments, because t h e bending moments mx a l o n g t h e y a x i s and a l o n g t h e x a x i s can a g a i n w i t h s u f f i c i e n t a c c u r a c y be considered

'%

t o vary p a r a b o l i c a l l y . N e v e r t h e l e s s , t h e f a c t o r s

Jr

w i l l need a c o r r e c t i o n because i n o u r r e i n f o r c e d c o n c r e t e s p e c i f i c a t i o n s we do n o t f i g u r e with t h e maximum n e g a t i v e bending moment o b t a i n e d from t h e e x a c t s l a b t h e o r y . T h i s n e g a t i v e f i x i n g moments on t h e l o n g s i d e o f a r e c t a n g u l a r s l a b a r e , of c o u r s e , 1_L 2 X c a l c u l a t e d by t h e e q u a t i o n mx = -qx

-

-12

while t h o s e a l o n g t h e s h o r t s i d e s 1 2 a r e given by m = -q

.

.x Y

x.

For t h e s q u a r e s l a b ( I x _l2 = L _Y = I , qx =

4

q )

and t h e same value mx = m = -q

.

3

i s o b t a i n e d from both e q u a t i o n s .

Y

According t o t h e r i g o r o u s - t h e o r y , however, t h e maximum f i x i n g moment f o r P o i s s o n ' s r a t i o v = 0 i s mx = m = -0.0474 qL2r 1 . e . 14% g r e a t e r t h a n t h e

Y

f i x i n g moment we a c t u a l l y s p e c i f y . In F i g .

6,

t h e a c t u a l f i x i n g moment mx

a l o n g y i s shown a s a curve w h i l e t h e t r a p e z o i d l i n e r e p r e s e n t s t h e moment which we a c t u a l l y u s e i n d e s i g n . A s i m i l a r d i f f e r e n c e i s encountered w i t h

t h e l o n g s l a b shown i n F i g . 7 which a l s o shows t h e d i f f e r e n c e between e x a c t n e g a t i v e f i x i n g moments and t h o s e we use a c c o r d i n g t o o u r s p e c i f i c a t i o n s . The t r u e maximum n e g a t i v e f i x i n g moment a c c o r d i n g t o t h e r i g o r o u s s l a b equa- t i o n i s mx = -0.0905 qLx2, whereas we u s e i n o u r c a l c u l a t i o r ~ s a maximum

I x '

. - -

x2

value of mx = -qx

-

-0.941 q

-

=

1 2 -0.0784 q L x 2 The t r u e f i x i n g moment i s a c c o r d i n g l y 15% g r e a t e r . A s i m i l a r differ.ence i s a l s o found f o r . t h e maximum f i x i n g moment m on t h e s h o r t e r s i d e of t h e r e c t a n g l e .

Y

These d i f f e r e n c e s between t h e a c t u a l n e g a t i v e moments and t h e average moments from t h e s i m p l i f i e d e q u a t i o n s a r e t a k e n i n t o c o n s i d e r a t i o n by u s i n g

2

1

=

7

f o r t h e p o s i t i v e moments in t h e s u p p o r t i n g beams as b e f o r e , b u t f i g u r e

3

with

Jr

= f o r t h e n e g a t i v e moments in t h e beams.

1,

=

qy

=

_'5

( f o r p o s i t i v e moments)

3

I

qX = qy = ( f o r n e g a t i v e moments).

We s h a l l now e v a l u a t e e q u a t i o n s

( 5 )

and

( 5 a )

f o r s l a b s of d i f f e r e n t span r a t i o s f i x e d a t t h e s u p p o r t s . The maximum p o s i t i v e bending moments can be o b t a i n e d from G s e r l s d e s i g n manual o r from t a b l e s in S e c t i o n B of t h i s paper, while t h e n e g a t i v e bending moments i n t h e s l a b a t t h e s u p p o r t s a r e

1 2 A c a l c u l a t e d a c c o r d i n g t o t h e above-mentioned e q u a t i o n s mx = -qx

12

and 1 2 m = - q . Y

*.

We t h e n o b t a i n t h e f o l l o w i n g v a l u e s :

A s might be expected, t h e d i f f e r e n c e s i n t h e beam moments between t h o s e corresponding t o t h e e x a c t s l a b t h e o r y ( c ) and t h o s e obtained by means of t r i a n g u l a r and t r a p e z o i d a l l o a d i n g s a r e o n l y s l i g h t . Contrary t o t h i s we f i n d t h a t t h e conventional method of c a l c u l a t i o n a c c o r d i n g t o ( a ) ,

as

allowed by t h e r e i n f o r c e d c o n c r e t e s p e c i f i c a t i o n s , i s q u i t e a s unusable as w i t h

f r e e l y supported s l a b s s i n c e i n t h e l i m i t i n g c a s e s t h e y r e s u l t i n v a l u e s about seven times t o o small.

3 . S l a b s w i t h a r b i t r a r y boundary c o n d i t i o n s

Here method ( c ) , i n which t h e beam moments were determined from t h e whole system, always g i v e s very a c c u r a t e v a l u e s . Method ( b ) of c a l c u l a t i n g

t h e beam moments from t h e t r i a n g u l a r and t r a p e z o i d a l loads, on t h e o t h e r hand, becomes r a t h e r u n c e r t a i n because d i f f e r e n t boundary c o n d i t i o n s w i l l change t h e load d i s t r i b u t i o n i n d i c a t e d in F i g . 2 . A t f r e e l y supported s l a b c o r n e r s t h e r e w i l l be

a

t e n s i l e f o r c e ( w i t h a corresponding i n c r e a s e d load on t h e beam i t s e l f ) . It i s shown i n S e c t i o n C, however, t h a t d e s p i t e t h i s r e d i s t r i - b u t i o n of t h e s l a b r e a c t i o n s on t h e beanls

a

c a l c u l a t i o n by means of t r i a n g u -

l a r and t r a p e z o i d a l loads i s s u f f i c i e n t l y a c c u r a t e .

The r e d i s t r i b u t i o n of t h e s l a b r e a c t i o n s , because of d i f f e r e n t boundary c o n d i t i o n s and t h e l a c k of a s u i t a b l e method of c a l c u l a t i o n which could t a k e c a r e of such v a r i a t i o n s , were probably t h e reason why t h e p r e s e n t day (1942) c o n v e n t i o n a l method ( a ) of c a l c u l a t i n g bean moments by use of uniformly d i s t r i b u t e d l o a d s was adopted i n t h e s p e c i f i c a t i o n s of 1932. This, however, a s shown i n t h e foregoing, l e a d s t o i n a d m i s s i b l y i n a c c u r a t e beam moments. Despite t h e s e s e r i o u s e r r o r s i n d e s i g n c a l c u l a t i o n s f o r s u p p o r t i n g beams f o r

t h e s h o r t e r spans o f t h e s l a b s no a c c i d e n t s have occured o r have become known. This i s probably because t h e bending moments d e r i v e d from t h e dead

load of t h e s e beams and t h e l i v e l o a d s coming d i r e c t l y upon them a r e added t o t h e l o a d s from t h e s l a b , and a l s o because i n most c a s e s d e s i g n e r s h3ve employed l a r g e r s t e e l p e r c e n t a g e s i n view of t h e small amount of r e i n f o r c e - ment d e r i v e d from t h e c a l c u l a t i o n s .

A more complete p r e s e n t a t i o n of t h e method of c a l c u l a t i o n f o r c o n t l n u - ous s u p p o r t i n g beams in s l a b systems, b o t h f o r t h e c a s e where a l l and t h a t where o n l y some o f t h e p o i n t s of beam i n t e r s e c t i o n s a r e supported by columns,

i s given f o r b o t h methods o f c a l c u l a t i o n i n S e c t i o n C . F i r s t , however, a s was mentioned i n t h e Foreword, t h e n e x t s e c t i o n w i l l c o n t a i n t a b l e s f o r

c a l c u l a t i o n of bending nloinents i n two-way s l a b s i n a simple and c l e a r manner. B. Tables of Bending Moments f o r Two-rray S l a b s

1. Development of e q u a t i o n s f o r d i f f e r e n t t y p e s o f s l a b s

The r e l e v a n t e q u a t i o n s from t h e s p e c i f i c a t i o n of s l a b bending moments a r e :

. .

-

[ m y = m v Y Y' h e r e i s v Y =

Contrary t o t h e specifications, s m a l l l e t t e r s a r e used f o r t h e s l a b moments t o d i f f e r e n t i a t e them from t h e beam moments, i n accordance with what

i s done i n t h e g e n e r a l t h e o r y of e l a s t i c i t y . mx and m a r e t h e maximum

Y

moments, mox and m mean t h e moments i n a simply supported s l a b i n which t h e O Y

t o t a l uniform load i s c a r r i e d i n one d i r e c t i o n o n l y ( m o x = q

'$

1 2 ,

-

T

J

OY

m

=

q

%)

,

mx and

iii

a r e t h e bending moments i n t h e two-way s l a b t a k i n g i n t o

Y

account t h e kind of s u p p o r t s p r e s e n t a s shown i n F i g . 8 b u t without t a k i n g i n t o account t h e e f f e c t of t o r s i o n a l moments. The t h r e e p o s s i b l e k i n d s o f s u p p o r t c o n d i t i o n s , with t h e corresponding bending moments and d e f l e c t i o n s , a r e shown on F i g .

8.

qx and qy a r e t h e load components of t h e t o t a l l o a d q

c a r r i e d i n t h e x d i r e c t i o n and y d i r e c t i o n , r e s p e c t i v e l y . In o r d e r t o

determine t h e s e components, t h e t o t a l uniformly d i s t r i b u t e d load q

i s

d i v i d - ed i n such a way t h a t t h e c e n t r e s t r i p s shown i n F i g .

9

_{have e q u a l d e f l e c t i o n}

a t c e n t r e f o r t h e two component l o a d i n g s . In c a l c u l a t i n g d e f l e c t i o n s t h e kind o f s u p p o r t c o n d i t i o n s a s shovm on F i g .

8

must be c o n s i d e r e d . In t h e g e n e r a l form t h e s e d e f l e c t i o n s may be w r i t t e n :

In view of t h e e q u a l s l a b t h i c k n e s s i n b o t h t h e x and y d i r e c t i o n s , and d i s r e g a r d i n g t h e d i f f e r e n c e i n amount of r e i n f o r c e m e n t , t h e moments of i n e r t i a a r e Ix = I ( p e r u n i t w i d t h ) . Since qx + q~ = q and 6, = 6 we ob- Y Y t a l n t h e f o l l o w i n g v a l u e s f o r load components:

The bending moments

lx

and

iii

a r e t h u s

Y

The c o n s t a n t s

Bx

and

P

which c h a r a c t e r i z e t h e magnitude of t h e bending

Y 1

moments f o r t h e t h r e e k i n d s of s u p p o r t c o n d i t i o n s of F i g . 6 a r e

a,

-&

and The s l a b rnornents

Ex

and

ii

a r e reduced due t o t h e t o r s i o n a l r e s i s t a n c e

FT-

Y

t o t h e v a l u e s mx and m The r e d u c t i o n i s c h a r a c t e r i z e d by t h e c o e f f i c i e n t s Y '

vx and v Y '

I n F i g . 10 a r e shown t h e c l a s s i f l c a t i o n of s l a b s i n t o s i x k i n d s now used and F i g . 10a shovls t h e proposed new c l a s s i f i c a t i o n i n t o n i n e t y p e s . This l a t t e r c l a s s i f i c a t i o n enable3 u s t o r e s t r i c t t h e span r a t i o s between

s = 1 t o E = 2 and t o e x p r e s s a l l moments only h terms of t h e span 'Lx where

1, always i s t h e s h o r t e r span. Confusion between spans i s no l o n g e r p o s s i b l e w i t h t h i s method. (This c l a s s i f i c a t i o n i s a l s o t h e b a s i s f o r t h e t a b l e s of vx and v i n t h e c o n c r e t e c a l e n d e r * which d i d n o t , however, make _Y use of t h e p o s s i b i l i t y of r e s t r i c t i n g t h e E v a l u e s t o t h e range from E = 1 t o s = 2 . )

We s h a l l now e x p r e s s a l l v a l u e s given by e q u a t i o n

( 6 )

i n terms of t h e parameter E and t h e r e b y a r r i v e a t d e f i n i t e e q u a t i o n s f o r t h e n i n e k i n d s of

s l a b .

In Table I t h e v a l u e s f o r mx, m and vx, v a r e developed a s a f u n c t i o n Y Y of e , and we t h u s o b t a i n t h e f o l l o w i n g e q u a t i o n s f o r t h e n i n e k i n d s of s l a b : 6 1 tS . q I 2

(

I - - .

-

my

-

my lVY = 8 '

-

5 6 l + r 4 ₁

+

t' . q 1 , 2 '

(

1-

-. -

ti 1 + t 4 t I 45 r ? my .= Ilr 1 9 -

-

.

-

'-128 5 + 2 r 4 .(I/,?

(

I--.- 32 5 + 2 C 6 p4 1 5 = ,.* --.

- .

-

.

8 5 + r 4 1

The above e q u a t i o n s a r e e v a l u a t e d i n Table I1 f o r a l l v a l u e s from e = 1 . 0 0 t o E = 2.00, w i t h t h e s m a l l i n t e r v a l s of Ae = 0.02, s o t h a t i n t e r p o l a - t i o n can e i t h e r be dispensed w i t h o r e a s i l y e s t i m a t e d .

According t o t h e Reinforced Concrete S p e c i f i c a t i o n s A r t i c l e 23,

paragraph 2, t h e corresponding t o r s i o n moments need n o t be checked and i t i s n o t n e c e s s a r y t o i n c l u d e any a d d i t i o n a l t o r s i o n a l reinforcement a c c o r d i n g t o F i g . 12 of t h e s p e c i f i c a t i o n s i f t h e s l a b i s m o n o l i t h i c w i t h t h e p e r i p h e r a l beams o r t h e a d j a c e n t s l a b . If t h e r e i s no such j o i n i n g t h e n a d d i t i o n a l t o r s i o n reinforcement can o n l y be omitted i f t h e r e d u c t i o n c o e f f i c i e n t v i n t h e above e q u a t i o n s ( 6 ) i s r e p l a c e d by t h e c o e f f i c i e n t

-

+ whereby l a r g e r

2

s l a b moments a r e o b t a i n e d . This a p p l i e s , a s t h e f i g u r e s and t a b l e s show, o n l y t o s l a b t y p e s 1 t o 4, where two f r e e l y r o t a t a b l e edges meet, t h u s r e s u l t i n g i n c o n s i d e r a b l e t e n s i l e f o r c e s due t o t h e t o r s i o n moment f o r t h e f r e e c o r n e r s in q u e s t i o n , t h e non-absorption of t h e a e f o r c e s producir,g a n i n c r e a s e i n t h e s l a b moments. I n o r d e r t o f a c i l i t a t e t h e work of c a l c u l a t i o n f o r t h i s c a s e a s well, t h e s e i n c r e a s e d bending moments a r e given i n Table 111 f o r t h e s l a b t y p e s 1 t o 4 in q u e s t i o n .

A l l bending moments mx and

$

of T a b l e s I1 and I11 a r e given as func-

1 2 ' x t i o n s of t h e s h o r t e r span

Ix

= I , e . g . i n t h e form o f mx = q

- -

n l m y

-

2 3

Ix _{where t h e i n d i c e s 3 x and 3y show t h a t t h e moments mx and m of s l a b}

q.,

3Y Y

2. The c a l c u l a t i o n o f c o n t i n u o u s s l a b s

The problems a r i s i n g f o r c o n t i n u o u s s l a b s and c o r r e s p o n d i n g s u p p o r t i n g beams a r e e x t r e m e l y d i f f i c u l t because a d j a c e n t s l a b s and beams i n f l u e n c e each o t h e r a s a r e s u l t o f t h e c o n t i n u i t y c o n d i t i o n s . The bending s t i f f n e s s of t h e beam, however, i s c o n s i d e r a b l y g r e a t e r t h a n t h a t o f t h e s l a b s , s o t h a t f o r l o a d i n g on a l l s l a b s we can s t a r t from t h e assumption, i n c l o s e approximation t o t h e r i g o r o u s t h e o r y , t h a t t h e i n d i v i d u a l s l a b s do n o t i n f l u - ence one a n o t h e r . T h i s w i l l be s t i l l c l o s e r t o r e a l i t y i f t h e t o r s i o n a l s t i f f n e s s o f t h e beams i s a l s o t a k e n i n t o a c c o u n t . Accordingly a t e v e r y t r a n s i t i o n from one s l a b t o a n o t h e r complete f i x i t y can be assumed s o t h a t

-

g i v e n uniform l o a d i n g o f a l l s l a b s

-

t h e new e q u a t i o n s ( 6 a ) , which were d e r i v e d w i t h o u t t a k i n g i n t o a c c o u n t t h e c o n t i n u i t y , a r e v a l i d f o r t h e i n d i - v i d u a l s l a b s .

In t h e c a s e of a l i v e l o a d v a r y i n g from s l a b t o s l a b , however, a r e c i - p r o c a l i n f l u e n c e i s p r e s e n t . I n o r d e r t o t a k e t h i s i n t o a c c o u n t we d i s t r i - b u t e t h e l i v e l o a d on t h e s l a b system i n a checker-board f a s h i o n as shown on F i g . 11. In t h i s way t h e maximum and minimum p o s i t i v e moments a r e determined f o r a uniform c o n t i n u o u s load o f

+

$

and a n a n t i m e t r i c l o a d o f f

5.

Thus from dead l o a d and l i v e load we o b t a i n f i r s t a c o n t i n u o u s l o a d and t h e n an

a n t i m e t r i c a l t e r n a t i n g one from s l a b t o s l a b w i t h v a l u e s o f

F o r t h e uniform load q t t h e s l a b s may be c o n s i d e r e d a s c o m p l e t e l y f i x e d a t

t h e s u p p o r t s i n accordance w i t h t h e above c o n s i d e r a t i o n s , s o t h a t f o r d e t e r - mining t h e maximum p o s i t i v e s l a b moments from q t t h e s l a b t y p e s o f F i g . 11

can be used. In computing t h e a n t i m e t r i c l o a d , on t h e o t h e r hand, s l a b t y p e

1 i s always a p p l i c a b l e , because i n t h i s l o a d i n g c a s e t h e i n d i v i d u a l s l a b s do n o t i n f l u e n c e one a n o t h e r , a s i s e v i d e n t from t h e c o r r e s p o n d i n g moment and bending c u r v e s . The i n d i v i d u a l s l a b s may t h e r e f o r e be regarded a s f r e e l y s u p p o r t e d f o r t h e l o a d 2

$?.

S t r i c t l y s p e a k i n g t h i s i s o n l y c o r r e c t on t h e assumption t h a t a l l s l a b s a r e o f e q u a l s i z e and t h a t t h e t o r s i o n a l s t i f f n e s s of t h e beams i s n e g l e c t e d . However, s i n c e t h e p o s i t i v e moments of t h e s l a b a r e reduced f o r t h e a n t i m e t r i c l o a d k

5

because o f t h e t o r s i o n a l s t i f f n e s s o f t h e beams, we a r e on t h e s a f e s i d e i n t h i s assumption and can t h e r e f o r e a c c e p t i n a c c u r a c i e s t h a t may r e s u l t from t h e p r e s e n c e o f s l a b s o f d i f f e r e n t s i z e s .

The c a l c u l a t i o n of t h e n e g a t i v e moments a l o n g t h e l o n g s i d e s of t h e r e c t a n g u l a r c o n t i n u o u s s l a b s i s c a r r i e d o u t in t h e same way as f o r one-way s l a b s e x c e p t t h a t t h e l o a d component qx i s p u t i n p l a c e o f load q . The

following equations a r e t h u s obtained f o r t h e s u p p o r t moments of t h e s l a b s .

,

( a ) mx = -qx

.

12

IX2 = _{-*xq '}

12

IX2

a t t h e common support o f two s l a b s f i x e d on both long s i d e s according t o F i g . 12a;

( b mx a -qx '

-

'x2

8

= -Xx9 •

7

IX2

a t t h e common support of two s l a b s f i x e d on one s i d e accord-

i n g t o F i g . 12b; ( 8 )

( c m = -qx

.

1

Ix2

0

= -xXq

. -

I&

X 1 0

a t t h e common support between a s l a b f i x e d one a i d e and a s l a b f i x e d on two s i d e s according t o Fig. 12c.

Equation ( 8 b ) i s obtained a s a mean value o f e q u a t i o n ( 8 ) and e q u a t i o n ( 8 a ) s o t h a t t h e x x value i s a l s o taken a s

a

mean value of t h e xx f i g u r e s of e q u a t i o n s ( 8 ) and ( 8 a ) . The load component qx o r t h e a s s o c i a t e d xx v a l u e s can be taken from Table

IV.

However, t h e above e q u a t i o n s cannot be used t o c a l c u l a t e t h e support moments on t h e s h o r t s i d e s , a s t h e r i g o r o u s t h e o r y shows. In t h e i r p l a c e we use t h e following e q u a t i o n s ( 8 ) , i n which t h e support moments

s e n t e d a s a f u n c t i o n of t h e span 1, and load q = g

+

p: are repre-

a t t h e i n t e r i o r support of two s l a b s f i x e d on both s h o r t s i d e s according t o Fig. 12d;

1 2

a t t h e i n t e r i o r s u p p o r t of two s l a b s f i x e d one s i d e accord- i n g t o F i g . 12e;

r 2 (8)

x

2 0

% = - q ' -

a t t h e i n t e r i o r support of one s l a b fFxed one s i d e and one s l a b f i x e d two s i d e s according t o F i g . 12f a s a mean of e q u a t i o n s ( 8 d ) and ( 8 e ) .

I n t h e c a s e of square s l a b s w i t h s i n i i l a r boundary c o n d i t i o n s ( s l a b types 1,4,9) e q u a t i o n s ( 8 d ) t o ( 8 f ) change t o t h e corresponding e q u a t i o n s ( 8 a ) t o ( 8 c ) .

Accordingly, we d i s c e r n two groups of t h r e e f i x i n g moments each, t h e

f i r s t group r e l a t i n g t o t h e long s i d e s and t h e second t o t h e s h o r t s i d e s of t h e r e c t a n g l e . Each group i s f u r t h e r . s u b d i v i d e d i n t o t h r e e I n d i v i d u a l c a s e s , depending on t h e boundary c o n d i t i o n s of t h e edges s i t u a t e d o p p o s i t e t h e s i d e s i n q u e s t i o n , while t h e boundar'y c o n d i t i o n s of t h e edges p e r p e n d i c u l a r t o t h e s i d e s in q u e s t i o n a r e d i s r e g a r d e d . To t h i s e x t e n t e q u a t i o n s

(8)

a r e o n l y a n

approxlmatlon, and moreover t h e moments mx run a l o n g y and t h e moments m Y a l o n g x approximately p a r a b o l i c a l l y i n accordance w i t h F i g .

6

and 7, whereas we based t h e i r dimensioning on t h e trapezium a s shown i n t h e s e f i g u r e s . D e s p i t e t h i s 15$ underdimensioning, however, a reinforcement based on t h e s e

t r a p e z i a a f f o r d s adequate s a f e t y , because t h e overdimensioned s l a b s t r i p s compensate f o r t h e ur~dcrdimensioned s u p p o r t ends of t h e s t r i p and t h e r e b y e s t a b l i s h f u l l s a f e t y . F i n a l l y , we a g a i n s t a t e t h a t t h e (maximum) p o s i t i v e moments of t h e n i n e s l a b t y p e s 1 t o 9 a r e t o be c a l c u l a t e d by means o f t h e continuous load q 1 = g

+

5

and t h e a n t i m e t r i c checker-board l i v e load q" =

f

2

while t h e n e g a t i v e moments of t h e s i x d i f f e r e n t s l a b edges

a

t o f a r e 2

c a l c u l a t e d from t h e t o t a l load q = g

+

p .

C . The C a l c u l a t i o n of t h e Supporting Beams of Two-way S l a b s

The s u p p o r t i n g beams can be c a l c u l a t e d b o t h from t h e d i f f e r e n c e between t h e moments of t h e whole system and t h e s l a b moments ( c a s e c ) , o r by means of t h e t r i a n g u l a r and t r a p e z o i d a l l o a d s ( c a s e b ) . The method of c a s e ( c ) is

of course t h e more a c c u r a t e one, because i t i s based on t h e r i g o r o u s s o l u - t i o n of t h e s l a b problem. D i f f e r e n c e s i n t h e r e s u l t s a r e found especially

when s l a b t y p e s 1 t o 4 a r e used w i t h t o r s i o n e f f e c t , i n which c a s e as shown I n S e c t i o n A, t h e c a l c u l a t i o n by means o f t r i a n g u l a r and t r a p e z o i d a l l o a d s

is based on s l a b r e a c t i o n s t h a t a r e t o o small. However, even when s l a b t y p e s 5 t o

8

a r e employed, and where no a d d i t i o n a l t o r s i o n reinforcement is pro- vided, beam moments c a l c u l a t e d from t r a p e z o i d a l and t r i a n g u l a r l o a d s are t o o

small. Only w i t h s l a b type 9 which i s f i x e d on a l l s i d e s a r e beam moments a c c o r d i n g t o c a s e ( b ) and ( c ) s u f f i c i e n t l y a c c u r a t e . The d i f f e r e n c e o b t a i n e d with t h e two methods of c a l c u l a t i o n w i l l be shown w i t h an example.

I. C a l c u l a t i o n of t h e bend in^ moments i n t h e s u p p o r t i n g beams from t h e t o t a l moments The e q u a t i o n s f o r t h i s a r e t h e same a s f o r t h e f i x e d s l a b , e q u a t i o n ( 5 ) . f o r t h e p o s i t i v e momenta

-

2

tx

-

Py

=

5.

I

f o r t h e s u p p o r t rnoriients

-

However, now Mx and

F

l

a r e t h e bending moments of t h e continuous t o t a l system Y

i n c l u d i n g beanis and s l a b s .

From t h e bending moments of t h e beams we g e t t h e corresponding s h e a r loads from equations

Here t h e v a r i a t i o n of t h e moments

Mx

and

M

can be taken a6 p a r a b o l i c

Y

with s u f f i c i e n t accuracy, whereby a s t r a i g t h l i n e v a r i a t i o n i s obtained f o r t h e t r a n s v e r s e s h e a r a s a d i f f e r e n t i a x curve of t h e moment l i n e .

According t o A r t i c l e 18, paragraph 3 , of t h e s p e c i f i c a t i o n s t h e support r e a c t i o n s t r a n s f e r r e d t o t h e beam may be t r a n s f e r r e d without t a k i n g i n t o account any c o n t i n u i t y t h a t may e x i s t , so t h a t i n c a l c u l a t i n g t h e t o t a l moments in t h e beam in t h e x d i r e c t i o n we would have t o use a loading width

I and I n c a l c u l a t i n g t h e beam In t h e y d i r e c t i o n a loading width

Ix.

Y'

In

t h e following example we s h a l l t r y t o determine whether t h i s s p e c i f i c a t i o n , which o f f e r s a considerable f a c i l i t a t i o n of t h e c a l c u l a t i o n work f o r s l a b s of equal s i z e , i s i n f a c t p e r m i s s i b l e .

1. The s l a b system

In

two rows a s i n Fig. 13 a s a numerical example

We have chosen t h i s s l a b system of two rows because t h e maximum loading width w i l l r e s u l t f o r t h e c e n t r a l beam

In

t h e x d i r e c t i o n because of t h e c o n t i n u i t y of t h e two-span s l a b I n t h e y d i r e c t i o n .

We use t h e following values f o r spans and loading:

q 1 I x 2 =

6 2

0.625 = 22.5 t m ; q1'Ix2 = 24.5 t m .

The dead weight of t h e beams

i s

neglected, s i n c e t h i s i s of no h p o r t a n c e i n o u r i n v e s t i g a t i o n s .

( a ) Calculation of continuous load q l . This Is determined from s l a b types 4 and 7. The s l a b moments a r e found t o be

P o s i t l v e moments : 1

S l a b type 4 m, =

m-

q ' I, = _37.2 = +0.605 t m

1

F i x i n g moments: I,' Edge a _{mx = -qx}

.

₁₂

_{= -%Xqt}

.

_-

IX2 _{1 2 =} -0.667

a

= -1.25 t r n L L X . - = X Edge c mx = -qx _-%xqt

_-

_{10 =}

_-

0.667

+

₂ 0.500

22.5

_{1 0} & x Edge e m = - q t

-

= - q t 1 0.0625 ( d a l x = Y 16 Y I y )

From

%

we g e t t h e l o a d width of t h e c e n t r e beam of x d i r e c t i o n as I

Y

(1

+

2 0.0625) = 1.125 I while f o r t h e o r d i n a r y continuous beam w i t h two Y

spans t h e r e a c t i o n a t t h e c e n t r e s u p p o r t i s 1.25 p l

.

We now c a l c u l a t e t h e t o t a l moment f o r t h e row two s l a b s wide, t h e load- i n g of which p e r l i n e a l metre i s

-

q = q t 21y = 0.625

.

12 = 7 . 5 t,

-

qIx2 = 7.5

36

= 270 t m ,

-

M = = 33.75. t m .

-

According t o t h e t a b l e s f o r continuous beams of f i v e spans t h e s u p p o r t moments a r e

R,

= -0.105

.

270 = -28.35 t m

-

M 2 = -0.079 270 = -21.25 t m .

The moment of t h e f r e e l y supported span was

Pf

= 33.75 t m . Thereby we o b t a i n t h e bending moments a t t h e c e n t r e of t h e span as

From t h i s it i s n e c e s s a r y t o deduct t h e s l a b moments of t h e two s l a b s

a c c o r d i n g t o t h e f o l l o w i n g t a b l e , where f o r t h e p o s i t i v e moment I)

-

-

,

f o r t h e s u p p o r t moment I) y - 3

I n the second l a s t l i n e of the above t a b l e the moments f o r a l l the

supporting beams of x d i r e c t i o n a r e taken together, while the l a s t l i n e gives only the moments of t h e c e n t r e beam, and from the moments of a l l t h r e e beams, according t o the load w i d t h of 1.125 1 found above,

Y 100 = 56.25% i e

c a r r i e d by the c e n t r e beam. The moments i n the c e n t r e beam a r e given I n t h e l a s t l i n e .

( b ) Calculation f o r t h e a n t i m e t r i c load q = f0.125 t / m 2 according t o Fig. 13a.

,In

the f i g u r e the s l a b s have superimposed loads of +0.125 t / m and -0.125 t / m

,

r e s p e c t i v e l y . The determining kind of s l a b i s t h e r e f o r e type 3. 1 Slab type 3 mx = q " l x 2

-

=

3

= 0.122 t m . 36.8 36.8 The t o t a l moment i s I X 2 - I t =

-

- C "

. -

x 8 ' 12

.

0.125 = 1.5 t/m, M, = 26.75 t m .

Accordingly, the moment f o r a l l t h r e e p a r a l l e l beams t o g e t h e r is Mx = 6.75

-

1 2 0.122 = f5.77 t m .

7

The load w i d t h i s the same a s f o r ( a ) , so t h a t on t h e c e n t r e beam t h e r e devolves

( c ) The bending moments of the c e n t r e beam a t mid-span from dead weight and l i v e load a r e found from load cases ( a ) and ( b ) a s

( a ) :

+

8.30 +2.73 +4.69 t m

( b ) :

+

3.24 i-3.24 i-3.24 t m Mx = ( a )

+

( b ) : +11.54 +5.97 +7.93 t m

2 . For comparison we now c a l c u l a t e a s l a b s t r i p of t h e same type, but assume t h a t It i s fixed a t the upper and lower boundaries

Here the load w i d t h of a c e n t r e beam i n the x direction i s only 1

Y instead of 1.125 1 ( F i g . 1 4 ) .

Y

( a ) Calculation f o r the continuous load q l . In place of s l a b type 4

P o s i t i v e moments:

Slab type 8

mx =

+s6

=

+0.444

t m

50

Slab type 9

m =

+

X

3

55.7

=

+o.

403

t m ;

Fixing moments

:

E d g e a

m = - _{q x . r =}

Ix2

Ix2

x

-%s1

12

=

-0.5

22.5

12 =

-0.937

t m lX2

Edge c

mx =

-

qx

1

0

= -"xq'

₁

₀

0.5

+

0.333

=

-

2 '

a

10 =

-0.938

t m .

The t o t a l moments which a c t on a c e n t r e beam

in

the

x

d i r e c t i o n a r e t h e r e f o r e

h a l f a s g r e a t a s f o r ( l a ) , since we a r e now required t o load only a

s t r i p

of

w i d t h ly.

According t o the subsequent t a b l e , we t h e r e f o r e obtain the

following moments

:

Mx

=

+8.01

-

9.95

+2.84

-

6.45

+4.59

trn

( b )

Calculation f o r t h e a n t i m e t r i c load'

q" =

f0.125

t/m2

( ~ i g .

14a).

For

t h i s

the p e r t i n e n t s l a b type i s No.

6.

+

4

.

5

=

20.081

t m . mx =

-

55.7

The t o t a l moment f o r the load width

1 is

h a l f a s g r e a t a s f o r ( l b ) , there-

Y

fore

From t h i s t h e s l a b moment

m 1 $ =

f0.081

-

6

.

J =

t o .

32

t m i s

deducted,

s o

X Y Y

t h a t f o r the beam the moment

Mx =

3.37

-

0.32

=

3.05

t m

remains.

( c ) The a d d i t i o n of load cases

(a)

and ( b ) r e s u l t s In:

0 -1 1-2 2-3

(a) : -!- 8.01 $ 2,W ?- 4 3 9 tm

3.

The c o n c l u s i o n s t o be drawn frorn numerical examples 1 and 2

We s h a l l now compare t h e bending moments i n t h e c e n t r e beam f o r c a s e 1 and 2 , f o r t h e d e t e r m i n i n g l o a d i n g c a s e s

( a )

p l u s ( b ) made up o f dead weight and l i v e load.

0- 1 1 1-2 2 2-3

Case 1

Mx

= +11.54 -9.32 +5.97 -5.68 +7.93 t m Case 2 Mx = +11.06 -9.95 +5.89 -6.45 +7.64 t m

D e s p l t e t h e g r e a t e r load width of t h e c e n t r e beam o f example 1, 1.125 Ly a s compared with 1.00 1 o f example 2, t h e bending moments a c c o r d i n g t o 2 a r e

Y

on t h e average s l i g h t l y g r e a t e r t h a n a c c o r d i n g t o 1. We can conclude from t h i s t h a t we do n o t have t o t a k e i n t o account t h e c o n t i n u i t y in t h e y d i r e c - t i o n , 1.e. we may c a l c u l a t e t h e beams i n t h e x d i r e c t i o n a s though t h e s t r i p of s l a b in t h e y d i r e c t i o n were completely f i x e d . The same h o l d s c o n v e r s e l y t r u e f o r t h e beams of y d i r e c t i o n .

The same r e s u l t i s a l s o o b t a i n e d from t h e l i m i t i n g c a s e E = 2, f o r

example where ( l x = 6 . 0 m, 1 = 1 2 . 0 m ) . For t h e sake o f s a v i n g space o n l y

Y

t h e f o l l o w i n g f i n a l r e s u l t s a r e c i t e d f o r s = 2:

Case 1

M~ = +12.94 -9.40 +5.82 -5.10 +9.43 t m Case 2 I/Ix = +13.37

-3.85

+6.1'7 -5.50 +9.67 t m

Here t h e rnoments a c c o r d i n g t o 2 a r e p r e f e r a b l e t o t h o s e of 1 because o f t h e s i m p l e r c a l c u l a t i o n . Case 2 on t h e average g i v e s h i g h e r v a l u e s t h a n 1,

which t a k e s i n t o account a c t u a l c o n d i t i o n a t t h e upper and lower boundaries. D e s p i t e t h e small l o a d i n g width, t h e bending moments i n t h e s u p p o r t i n g beams a r e somewhat h i g h e r on t h e average f o r c a s e 2. The r e a s o n i s t h a t

vrhere t h e upper and lower edges o f t h e s l a b a r e f i x e d , a s i n c a s e 2, t h e s l a b moments mx become s m a l l e r t h a n f o r t h e f r e e l y supported edges o f c a s e 1, and

t h u s t h e i n f l u e n c e of t h e g r e a t e r l o a d i n g width i s removed. I n c a l c u l a t i n g t h e beams of x d i r e c t i o n , t h e r e f o r e , t h e c o n t i n u i t y o f t h e s l a b I n t h e y d i r e c t i o n can be d i s r e g a r d e d . I n a manner s i m i l a r t o t h a t f o r t h e o r d i n a r y continuous beam we t h u s o b t a i n t h e maximum p o s i t i v e moments of t h e beam i n t h e x d l r e c t i o n due t o l i v e load by a l t e r n a t i n g , a c c o r d i n g t o F i g . 15 and 15a, t h e l i v e load on t h e slab^ i n t h e y d i r e c t i o n i n r e l a t i o n t o t h e x d l r e c t i o n .

The niaximun~ s u p p o r t moments, on t h e o t h e r hand, a r e o b t a i n e d by l o a d i n g a c c o r d i n g t o F i g . 15b and 15c. The corresponding bending moments o v e r t h e s u p p o r t s of t h e e n t i r e system f o r t h e l o a d i n g width can be taken from t h e a v a i l a b l e t a b l e s f o r continuous beams with e q u a l spans. From t h i s vre deduct t h e s l a b moments f o r t h e continuous load q f and t h e a l t e r n a t i n g load q" I n t h e manner i n d i c a t e d above.

A s f a r a s t h e a l t e r n a t i n g load q" i s concerned, t h e c a l c u l a t i o n , of course, Is now i n a c c u r a t e , because a t t h e s u p p o r t i n q u e s t i o n two s l a b s load- ed with q" a r e continuous w i t h one a n o t h e r . The r e s u l t i n g inaccuracy, how- e v e r , l s o f no inlportance, e s p e c i a l l y a s t h e r e i n f o r c e m e n t s of t h e s l a b In x d i r e c t i o n i n t h e v i c i n i t y of t h e beam a r e n o t f u l l y u t i l i z e d and can t h e r e - f o r e be counted on t o c o o p e r a t e . This i s i n d i c a t e d i n F i g . 6 and 7 , accord- i n g t o which t h e moments mx a l o n g t h e y a x i s a r e p a r a b o l i c . Every d e s i g n e r , however, w i l l i n c l u d e r e i n f o r c e n ~ e n t s i n t h e s l a b n e a r t h e s u p p o r t i n g beams s o t h a t t h e s e can s e r v e a s a r e s e r v e f o r t h e beams.

In

S e c t i o n 111 t h e c a l c u l a t i o n method d e r i v i n g from t h e above conclu- s i o n s are i l l u s t r a t e d w i t h a numerical example.

In

t h e course of t h i s i t Is

a g a i n shown how t h e s h e a r f o r c e s a r e o b t a i n e d i n a simple manner from t h e moments.

In S e c t i o n I1 below, t h e c a l c u l a t i o n f o r t h e s l a b of F i g .

13

i s c a r r i e d o u t by means of t r i a n g u l a r and t r a p e z o i d a l loads, i n o r d e r t o check i f such

a c a l c u l a t i o n i s s u f f i c i e n t l y a c c u r a t e e s p e c i a l l y f o r t h e beams a t t h e boundaries of t h e s l a b .

11. The c a l c u l a t i o n of t h e s l a b of F i g .

31

by t r a p e z o i d a l and t r i a n g u l a r l o a d s ( ~ i g . 1 6 )

The three-moment e q u a t i o n of t h e continuous g i r d e r with c o n s t a n t moment o f i n e r t i a i s :

-

6

r - + 4Mr + Mr+ 4 -

-

( T I + P ~ ~ ' ) .

Here 0' and q" a r e t h e a n g l e s of r o t a t i o n of t h e s t a t i c a l l y d e t e r m i n a t e beam a t t h e r - t h support ( F i g . 1 7 ) . For a c o n s t a n t load q t h e moment and t h e corresponding angle of r o t a t i o n a r e found from