Contribution on the Structural Behaviour of Skew Slabs, Using a Model

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Contribution on the Structural Behaviour of Skew Slabs, Using a Model

Mehmel, A.; Weise, H.

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A s t h e a u t h o r of t h i s paper p o i n t 8 o u t t h e r e i s l i t t l e a v a i l - a b l e i n f o r m a t i o n on t h e s t r u c t u r a l behaviour of continuous skewed

s l a b s . A t t h e s u g g e s t i o n of P r o f . M.W. Huggins, Chairman of t h e J o i n t CSA/NBC Committee on Reinforced Concrete Design, t h e Building 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 r e q u e s t e d t h i n t r a n s l a t i o n i n o r d e r t o make it a v a i l a b l e t o Canadian e n g i n e e r s . The D i v i e i o n i s g r a t e f u l t o M r . D . A . S i n c l a i r of t h e T r a n s l a t i o n 8 S e c t i o n of t h e N a t i o n a l Research Council L i b r a r y 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 May

1964

R.F. Legget D i r e c t o r

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TI tle:

NATIONAL RESEARCH COUNC IL OF CANADA

Technical Translation 1128

A contribution on the atruotural behaviour of skew slabs, using

a

model

in

modellatatiecher Beltrag

zum

Tragverhalten schiefwlnkliger

platten)

Authors : A. Mehmel and H. Welae

Reference: Beton- und Stahlbetonbau,

57

(10) : 233-239, 1962

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A CONTRIBUTION ON THE STRUCTURAL BEHAVIOUR OF SKEW SLABS, USING A MODEL

1. I n t r o d u c t i o n

Although almple span skew s l a b s have been d e a l t w i t h t h o r o u g h l y i n t h e l i t e r a t u r e , very few r e s u l t s a r e a v a i l a b l e on c o n t i n u o u s skew s l a b s . Refer- ence 1, f o r example, d e s c r i b e s t h e behaviour of a two-span s l a b skewed 40° w i t h spans 1, : 1, = 41.20 : 34.60 and width b = 16.60 m . Because t h e method of c u r v a t u r e measurement i n 1954/55 was s t i l l very crude and i n a c c u r a t e , t h e p r i n c i p a l moments ( l o a d p e r span) g i v e n i n t h i s work a r e n o t very r e l i a b l e . I n r e f . 2 r e s u l t s o b t a i n e d by t h e o p t i c a l r e f l e c t i o n method on a 38.5O skewed two-span s l a b w i t h 1, = 1, = 20.80, b = 8.00 m a r e r e p o r t e d f o r v a r i o u s e q u a l l o a d c a s e s and v a r i o u s p o s i t i o n s of u n i t l o a d . A p a r a l l e l t e s t shows t h e l n - f l u e n c e of a haunch on t h e p r i n c i p a l moments. I n r e f . 3 some g e n e r a l consid- e r a t i o n s r e p r e s e n t e d on t h e b e a r i n g p r o p e r t i e s of t h e c o n t i n u o u s skew s l a b , and moment boundary l i n e s a r e g i v e n q u a l i t a t i v e l y a s a f u n c t i o n of t h e a n g l e of skew, d i f f e r e n t i a t i n g between end and c e n t r e s p a n s . I n f l u e n c e a r e a s f o r t h e moments o f a space p o i n t i n t h e end span of a curved skew s l a b a r e

p r e s e n t e d i n r e f . 4, w h i l e r e f .

5

g i v e s t h e i n f l u e n c e a r e a s f o r t h e boundary e l o n g a t i o n s of t h e d e s i g n p o i n t s of a s i m i l a r s l a b . Reference 6 c a l c u l a t e s

a 50° skewed, narrow box-type b r i d g e w i t h t h r e e s p a n s 1 , : 1, : 1, = 72.00 : 96.00 : 72.00 a c c o r d i n g t o r e f . 7 a s a skewed s t r i p , a f t e r f i r s t d e m o n s t r a t i n g beam-like t r a n s v e r s e d i s t r i b u t i o n and r i g i d i t y c o n d i t i o n s by p l a n e b e a r i n g

s t r u c t u r e c a l c u l a t i o n .

N e i t h e r i n number n o r i n scope, however, do t h e above-mentioned examples s u f f i c e f o r a n a d e q u a t e u n d e r s t a n d i n g of t h e b e a r i n g p r o p e r t i e s of t h e con- t i n u o u s skew s l a b . The t e s t r e s u l t s r e p o r t e d below r e p r e s e n t a f u r t h e r c o n t r i b u t i o n t o t h i s s p e c i a l f i e l d . To c l a r i f y t h e b e a r i n g p r o p e r t i e s of t h e c o n t i n u o u s skew s l a b , a t h r e e - s p a n c o n t i n u o u s skew s l a b (11) i s compared w i t h a c o n t i n u o u s s t r a i g h t s l a b ( I ) of t h e same span i n t e r v a l and width, and w i t h a simple span skew s l a b of e q u a l span and a n g l e of skew (111).

e ere

t h e r e a d e r i s r e f e r r e d t o t h e comparison of t h e moments of a simple span skew s l a b

(40°) w i t h a s q u a r e s l a b ( I V ) i n r e f . 7 and 8 . ) 2 . The Messerich Bridge

The t h r e e - s p a n skew s l a b was s t u d i e d i n t h e c o u r s e of a model t e s t f o r a c o m p a r a t i v e l y s m a l l , n o n - p r e s t r e s s e d r e i n f o r c e d b r i d g e s t r u c t u r e a t Messerich

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and t h e a n g l e of skew s l i g h t l y l e s s t h a n 40°. Five e l a s t i c b e a r i n g s s u p p o r t t h e s l a b a l o n g each b e a r i n g l i n e . Some of t h e s e s t a n d on abutment w a l l s and some on c o n c r e t e columns. A l o n g i t u d i n a l s e c t i o n through t h e a x i s of t h e executed s u p e r s t r u c t u r e i s shown i n F i g . 1. F i g u r e 2a shows a c r o s s - s e c t i o n of t h e b r i d g e , F i g . 2b t h e corresponding model c r o s s - s e c t i o n . I n o r d e r t o r e t a i n t h e s t i f f n e s s of t h e m o n o l i t h i c a l l y - j o i n e d c a n t i l e v e r arms t h e s e a r e added t o t h e s o l i d s l a b .

h he

i d e a l i z e d width of t h e model, t h e p l a n (11) of which i s seen I n F i g . 3, i s c a l c u l a t e d from t h e c o n d i t i o n t h a t t h e edge bending s t i f f n e s s e s of t h e s o l i d s l a b and of t h e c a n t i l e v e r e d s u p e r s t r u c t u r e a r e assumed equal .)

3 . Comparison of P r i n c i p a l Moments under Equal Load

S l a b s I , I1 and 111 i n F i g . 3 a r e d i r e c t l y comparable because a l l s p a n s have equal l e n g t h and w i d t h and hence equal p l a n a r e a . The t o t a l l o a d s p e r span a r e t h e r e f o r e c o n s t a n t :

P = p

.

b

.

1 = 1

-

14.00

-

16.00 = c o n s t .

A l l p o i n t s i n v e s t i g a t e d a r e s i t u a t e d a t t h e c e n t r e of t h e span i n each c a s e , over t h e s u p p o r t s on t h e c e n t r e a x i s of t h e s l a b and over t h e l i n e j o i n i n g t h e edge columns, w i t h o u t n e c e s s a r i l y i n c l u d i n g t h e p o s i t i o n s of

maximum moment. I n t h e skew s l a b s , f o r example, t h e maximum p o s i t i v e moments of t h e edge zone a r e d e f i n i t e l y d i s p l a c e d towards t h e o b t u s e c o r n e r . The p r i n c i p a l moments f o r t h e g i v e n dimensions and p = 1 a r e shown i n F i g . 3 .

The u n i t v a l u e s m u l t i p l i e d by p12 a r e g i v e n i n Table I . The comparison b r i n g s o u t t h e f o l l o w i n g p o i n t s :

The b e a r i n g p r o p e r t i e s of t h e skew s l a b s a r e b e t t e r t h a n t h o s e of t h e r e c t a n g u l a r ones. The maximum moments a t t h e span c e n t r e s and o v e r t h e i n t e r - mediate columns a r e g r e a t e r f o r I t h a n f o r 11. Even t h e simple span skew s l a b h a s a more f a v o u r a b l e p a t t e r n of moments t h a n t h e s t r a i g h t continuous s l a b , a t l e a s t i n t h e end spans of t h e l a t t e r . T h i s i s due primarj-ly t o t h e f a v o u r a b l e b e a r i n g d i r e c t i o n of t h e skew s l a b . It d i s t r i b u t e s i t s l o a d a s f a r as p o s s i b l e over t h e s h o r t e s t d i s t a n c e t o t h e s u p p o r t i n g l i n e s 1 s i n 9. T h i s circumstance can be recognized most c l e a r l y from t h e d i r e c t i o n of t h e p r i n c i p a l moments a l o n g t h e c e n t r e a x e s . Moreover, t h e e l a s t i c r e s t r a i n i n g of t h e open edges a t t h e o b t u s e c o r n e r s

-

it i s absorbed by bending and t o r s i o n a l moments i n t h e s l a b s t r i p over t h e s u p p o r t s

-

r e n d e r s t h e moment s t r e s s i n g of t h e skew s l a b more f a v o u r a b l e , because i t i s more uniform.

The c o n t i n u i t y of o r t h o g o n a l b e a r i n g systems i s l e s s t h a n t h a t of t h e skewed systems. I n t h e c a s e of I t h e boundary and c e n t r e span moments a r e i n a r a t i o , on t h e average, of t h r e e t o one, whereas f o r t h e a n g l e of skew of s l a b I1 t h e r a t i o i s o n l y two t o one. The moments of t h e i n d i v i d u a l s p a n s

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a r c of t h e same o r d c r of m a ~ n i t u d e , and hence t h e support moments a r e s m a l l e r f o r t h e skew s l a b t h a n f o r t h e r e c t a n g u l a r s l a b . I t i s t o be expected t h a t t h e d i f f e r e n c e w i l l l a r g e l y v a n i s h a s t h e a n g l e becomes s m a l l e r . The r e a s o n s may a g a i n be s t a t e d a s f o l l o w s : t h e load d i s t r i b u t i o n a t r i g h t a n g l e s t o t h e l i n e s of support

-

a s a consequence t h e boundary zones a r e p a r t i a l l y excluded from t h e c o n t i n u i t y

-,

t h e e l a s t i c r e s t r a i n i n g of t h e boundary zones a t t h e o b t u s e c o r n e r s , which i s observed i n a l l spans; a s a r e s u l t of t h i s boundary r e s t r a i n i n g an unfavourable e f f e c t on t h e span moments of t h e boundary zones due t o t h e p a r t i a l e x c l u s i o n from t h e c o n t i n u i t y i s compensated f o r .

F r m t h e comparison of skew s l a b s 11 and I11 t h e f o l l o w i n g f a c t a come t o l i g h t : The c o n t i n u i t y e f f e c t t o a neighbouring span b r i n g s l i t t l e improve- ment i n t h e moment d i s t r i b u t i o n of a skew s l a b , s i n c e t h e o b t u s e c o r n e r s

c o n t r i b u t e p r i m a r i l y t o a r e d u c t i o n of t h e span moments. The c o n t i n u i t y e f f e c t i s t o t a l l y a b s e n t f o r t h e boundary zones of t h e o u t s i d e spans which t e r m i n a t e a t t h e s h a r p c o r n e r s ; t h e moments a t t h e edge c e n t r e a r e p r a c t i c a l l y of t h e same magnitude and d i r e c t i o n a s t h o s e of t h e simple span. The span moment o f t h e edge s t r i p o p p o s i t e , however, h a s been d e c r e a s e d . These zones a r e a l s o r e s t r a i n e d a t t h e i r s h a r p c o r n e r s .

Comparison of s l a b s I1 and I11 a l s o shows t h a t t h e d i r e c t i o n of t h e p r i n c i p a l moments i s a l s o changed by t h e a d d i t i o n of s p a n s . A t a m a j o r i t y of p o i n t s i n v e s t i g a t e d t h e a n g l e between t h e p r i n c i p a l b e a r i n g d i r e c t i o n and t h e l o n g i t u d i n a l a x i s of t h e s l a b d e c r e a s e s . The span i n t h e p r i n c i p a l b e a r i n g d i r e c t i o n i s t h u s i n c r e a s e d by t h e c o n t i n u i t y . Reinforcement of t h e c r o s s - s e c t i o n i n t h e d i r e c t i o n of t h e i n t e r m e d i a t e s u p p o r t s (haunch) i n c r e a s e s t h i s e f f e c t ( 2 ) , a s does an i n c r e a s e i n t h e c o e f f i c i e n t of expansion, a s c a l c u l a t e d f o r t h e simple span skew s l a b i n r e f . 9 .

A comparison of t h e b e a r i n g r e a c t i o n s of s l a b s I1 and I11

-

f o r a t o t a l of f i v e columns on each support l i n e

-

shows that t h e c o n t i n u i t y e f f e c t

smooths o u t t h e d i f f e r e n c e s between t h e s e p a r a t e column ~ ~ e a c t i o n s

a able

11). The curve of column r e a c t i o n s f o r I11 i s d i s t i n g u i s h e d by a very h i g h peak v a l u e a t t h e o b t u s e c o r n e r and a s t e e p drop towards t h e neighbouring second column, w h i l e t h e 3 r d , 4 t h and 5 t h columns v a r y about t h e mean v a l u e ~ / 1 0 .

For t h e c o n t i n u o u s s l a b I1 t h e b e a r i n g r e a c t i o n curve i s f l a t t e r , e s p e c i a l l y between t h e s p a n s .

I n t h i s connection n o t e t h a t t h e d i f f e r e n c e s between t h e column r e a c t i o n s depend on t h e p l a n shape o f t h e s l a b , t h e number of columns p e r b e a r i n g

1 i n e ( l o ) and t h e e l a s t i c c o n s t a n t s of t h e columns. I n t h e p r e s e n t I n s t a n c e t h e l a t t e r two parameters were k e p t c o n s t a n t . The model s u p p o r t was v i r t u a l l y r i g i d . With e l a s t i c support t h e colwnn r e a c t i o n s can be more o r l e s s

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p r e s e n t time i n t h e I n s t i t u t e f o r Concrete C o n s t r u c t i o n of t h e Technische Hochschule Darmstadt.

4 .

Comparison of I n f l u e n c e Areas

4 . 1 Loading p o i n t 1 a t t h e c e n t r e of t h e s l a b ( ~ i g . 4 )

The i n f l u e n c e s u r f a c e s confirm what was i n d i c a t e d i n F i g .

3

w i t h r e s p e c t t o b e a r i n g d i r e c t i o n and c o n t i n u i t y e f f e c t of t h e m u l t i p l e span skew s l a b . For example, i f we compare t h e a r e a s enclosed by t h e contour l i n e s 0.05, t h e d i f f e r e n t b e a r i n g e f f e c t becomes c l e a r . I n s l a b s I t h e a r e a s mx and m d i f f e r Y g r e a t l y . m remains s m a l l f o r a l l p o s i t i o n s of t h e l i v e l o a d , 1 . e . l e s s t h a n xy 0.05. On account of t a n 2 a = 2

m d ( m x

-

"Y

,

t h e d i r e c t i o n s of t h e p r i n -

c i p a l moments a t p o i n t 1 a r e l a r g e l y independent of t h e load p o s i t i o n . They c o i n c i d e w i t h t h e d i r e c t i o n of t h e a x i s . Hence t h e measured a x i a l moments a r e a l s o t h e p r i n c i p a l moments. I n I1 t h e a r e a s mx and m a r e approximately

Y

equal; m

xy

'

however, a t t a i n s c o n s i d e r a b l y g r e a t e r v a l u e s t h a n i n I . The main

d i r e c t i o n s , however, do not c o i n c i d e w i t h t h e x and y d i r e c t i o n s . The skew s l a b t e n d s t o t r a n s m i t t h e load a t r i g h t a n g l e s t o t h e s u p p o r t s . I n I11 mx

and m i n t h e r e g i o n of t h e l o a d i n g p o i n t a r e s i m i l a r t o t h o s e i n 11, b u t t h e

Y

t o r s i o n a l moment i s s t i l l g r e a t e r . N e v e r t h e l e s s t h e d i r e c t i o n s of t h e p r i n - c i p a l m o m e n t s of I11 do not depend any more s t r o n g l y on t h e load t h a n t h o s e of 11, because t h e d i f f e r e n c e s mx

-

m have i n c r e a s e d a l o n g w i t h m

Y XY I n t h e

c a s e of 11, f o r example, l i n e a r load a l o n g one of t h e open edges, r e s u l t s i n a change of

l o 0

i n t h e p r i n c i p a l moment d i r e c t i o n s a t p o i n t 1, b u t o n l y of 5 O

i n t h e c a s e of s l a b 111. It i s found, however, t h a t t h e d i r e c t i o n s of t h e p r i n c i p a l moments of t h e skew s l a b , e s p e c i a l l y a l o n g t h e open edge, depend more on t h e load t h a n do t h o s e of t h e s t r a i g h t s l a b .

The d i f f e r e n c e s i n c o n t i n u i t y e f f e c t a r e expressed most c l e a r l y i n t h e zones of i n f l u e n c e , f o r which t h e s i g n s a r e r e v e r s e d . S l a b I (mx f o r 1)

p o s s e s s e s marked n e g a t i v e c o l l e c t i n g zones which, a s i n t h e c a s e of t h e con- t i n u o u s beam, extend over t h e e n t i r e a r e a of t h e neighbouring s p a n s . I n t h e c a s e o f s l a b I1 we f i n d t h e corresponding spans d i v i d e d by l i n e s of equal

( z e r o ) l o a d . Only t h e zones a l o n g t h e e x t e n s i o n s of t h e normals from t h e l o a d i n g p o i n t t o t h e row of columns make any a p p r e c i a b l e n e g a t i v e moment

c o n t r i b u t i o n . These zones, however, a r e much s m a l l e r t h a n f o r I . V a r i a t i o n s of load from span t o span, a s may be expected from beam s t a t i c s , d e c r e a s e i n importance f o r continuous s l a b s of l a r g e skew a n g l e . I f we determine t h e p r i n c i p a l moments r e s u l t i n g from t h e uniform l o a d i n g of t h e o u t s i d e spans a l o n e , o r t h e i n s i d e span a l o n e , and r e l a t e them t o t h e p r i n c i p a l moments f o r f u l l l o a d i n g , we f i n d

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f o r I 1 -1.8/1 (end spans l o a d e d / f u l l l o a d ) +2.8/1 ( i n s i d e apan l o a d e d / f u l l l o a d ) f o r I1 1 -0.9/1 (end spans l o a d e d / f u l l l o a d )

+ l . g / l ( i n s i d e span l o a d e d / f u l l l o a d ) . 4.2 Loading p o i n t 2, edge c e n t r e , i n s i d e span ( ~ i g .

5 )

Here a t f i r s t we a r e s t r u c k by t h e b e t t e r load d i s t r i b u t i o n o v e r t h e width of s l a b I . A u n i t load on t h e c e n t r e of t h e s l a b ( l o a d i n g p o i n t 1)

produces p r i n c i p a l moments a t t h e edge of t h e o r d e r m , = +0.14 and

m , = f 0.00 m/m. S l a b 11, under t h e same c o n d i t i o n s , g e t s o n l y m , = 0.02,

m , = -0.01 tm/m. D e s p i t e t h e g r e a t width p a r t i c i p a t i n g i n t h e l o a d , however, t h e loading-point moments a r e g r e a t e r f o r I . The r e s t r a i n t of t h e edge s t r i p a t t h e o b t u s e c o r n e r (11, 111) exceeds t h e r e s t r a i n t on both s i d e s due t o c o n t i n u i t y ( I ) . Consequently t h e p o s i t i v e moment under a s i n g l e load becomes s m a l l e r f o r t h e skew s l a b t h a n f o r t h e s t r a i g h t o n e . The c o n t i n u i t y e f f e c t f o r t h e boundary zones i s a l s o found t o be small f o r t h e c a s e of s l a b 11.

Again t h e c o l l e c t i n g r e g i o n i s s i t u a t e d e s s e n t i a l l y on t h e e x t e n s i o n s t o t h e support normals. The r a t i o of p r i n c i p a l moments a t p o i n t 2 ( g i v e n s e p a r a t e l y f o r end-span and centre-span l o a d i n g ) t o f u l l load moment i s a s f o l l o w s f o r t h e two s l a b s :

f o r I 2 -2.5/1 (end spans l o a d e d / f u l l l o a d ) +3.4/1 ( i n s i d e span l o a d e d / f u l l l o a d ) f o r I1 2

-1.5/1

(end spans l o a d e d / f u l l l o a d )

+2.3/l ( l n s i d e span l o a d e d / f u l l l o a d ) . Space p o i n t s

3

and

4,

i n t e r m e d i a t e columns ( ~ i g .

6

and

7 )

The b e a r i n g d i r e c t i o n s of t h e s l a b s a r e a g a i n e v i d e n t from t h e l i n e s of c o n t o u r f o r p o i n t 3. The c e n t r e s of g r a v i t y of t h e moment p a t t e r n s a r e d i s -

placed i n t h e skew s l a b s away from t h e normal d i r e c t i o n t o t h e s u p p o r t s , f o r

mx towards t h e x a x i s , f o r m towards t h e y a x i s and f o r m towards t h e 45*

Y XY

a x e s . The moment p a t t e r n s of I1 a r e even s m a l l e r t h a n t h o s e of I . The con- t i n u i t y t o t h e end span i s of no importance f o r p o i n t 3 . However, s i n c e t h e deformations of t h e end span can a c t on t h e end column p o i n t 4 a t r i g h t a n g l e s t o t h e l i n e of s u p p o r t , t h e r e i s a p o s i t i v e moment p a t t e r n (mx and m of 11).

Y

The shape of t h e z e r o l i n e s i n t h e b e a r i n g d i r e c t i o n which d i v i d e t h e i n d i v i d - u a l spans of t h e skew s l a b i n t o p o s i t i v e and n e g a t i v e zones i s a g a i n s t r i k i n g . The f a c t t h a t no i n d i v i d u a l columns a r e s i t u a t e d on t h e z e r o l i n e s of t h e a r e a s of i n f l u e n c e i s due t o t h e e l a s t i c i t y of t h e columns.

(9)

5 . Conclusions and R e s e r v a t i o n s

The t e s t s have shown t h a t t h e c o n t i n u i t y e f f e c t h a s l e s s e f f e c t on t h e d i s t r i b u t i o n of moments i n skew s l a b s than i n s t r a i g h t o n e s . The m u l t i p l e span skew s l a b b e a r s a c l o s e r resemblance, a s f a r a s i t s s t r u c t u r a l behaviour i s concerned, t o t h e simple span skew s l a b t h a n t o t h e continuous s t r a i g h t s l a b . The main moment-resisting d i r e c t i o n of t h e skew s l a b , a s known from t h e simple span s l a b , namely a t r i g h t a n g l e s t o t h e s u p p o r t l i n e s o r , i n t h e c a s e of s l a b s comparatively elongated i n t h e x - d i r e c t i o n , t r a n s v e r s e t o t h e l i n e j o i n i n g t h e o b t u s e c o r n e r s , i s l a r g e l y r e t a i n e d i n t h e continuous s l a b .

The g e n e r a l a p p l i c a b i l i t y of t h e o b s e r v a t i o n s and c o n c l u s i o n s can be e s t a b l i s h e d o n l y by f u r t h e r t e s t s . (some of t h e o b s e r v a t i o n s a g r e e w i t h t h e r e s u l t s d e s c r i b e d i n r e f . 2 and

3.)

The r e s u l t s do n o t s u f f i c e f o r t h e d e v i a t i o n of conversion f a c t o r s f o r t h e p r i n c i p a l moments. It i s c l e a r , however, t h a t t h e moments o f continuous skew s l a b s cannot be determined i n advance by a p p l y i n g t h e conversion f a c t o r s f o r simple bay r e c t a n g u l a r t o skew s l a b s . Such a c a l c u l a t i o n would r e s u l t i n o v e r - e s t i m a t i o n of t h e con- t i n u i t y e f f e c t and t o over-dimensioning e s p e c i a l l y a l o n g t h e i n t e r m e d i a t e s u p p o r t s

.

6 .

Models and Model Procedures

A l l t h r e e s l a b s were prepared from 5 rnm c r y s t a l l i n e p l a t e g l a s s . T h i s m a t e r i a l , because of i t s high e l a s t i c i t y and low t r a n s v e r s e c o e f f i c i e n t of expansion ( 0 . 2 2 ) , i s p a r t i c u l a r l y w e l l s u i t e d f o r t h e model t e s t i n g o f r e i n - f o r c e d c o n c r e t e s l a b s . The s c a l e was 1 : 50, making t h e l e n g t h of t h e simple span 32 _{cm and t h e width 28 cm.} _{The models were mounted on socketed s u p p o r t s}

I n each c a s e one r i g i d and one hinge-type s u p p o r t a s s u r e d t h e s t a b i l i t y of t h e s t r u c t u r e . Aluminium p l a t e s

6

rnm wide and 2 mrn t h i c k between t h e s l a b and t h e b a l l - b e a r i n g mounted hinge a s s u r e d adequate load d i s t r i b u t i o n without a p p r e c i a b l y s t i f f e n i n g t h e s l a b . A p r e l i m i n a r y load of about 70 kg p r e s s e d t h e model o n t o i t s s u p p o r t ( ~ i g .

8 ) .

The i n s t r u m e n t s used f o r measuring t h e bending and t o r s i o n a l moments, based on t h e p r i n c i p l e of mechanical c u r v a t u r e t r a n s m i s s i o n through c r o s s b r i d g e s , have a l r e a d y been d e s c r i b e d i n d e t a i l ( 1 3 ) . The t e s t base was 30 nun

f o r t h e bending and 40 mm f o r t h e t o r s i o n a l moments. The pick-up o p e r a t e d i n d u c t i v e l y . The moments were obtained i n t h e form of i n f l u e n c e s u r f a c e , and t h e s u r f a c e s of s t a t e ( ~ i g . 3 ) were determined s u b s e q u e n t l y by i n t e g r a t i o n

( ~ i m p s o n ~ s r u l e , modified f o r unequal s t e p w i d t h s ) . I n r e c o r d i n g t h e span moments a s u i t a b l e a p p a r a t u s p r e s s e d t h e measuring i n s t r u m e n t from below a g a i n s t t h e s l a b , and s i m i l a r l y f o r t h e r e c o r d i n g of t h e t o r s i o n a l moment

(10)

over t h e s ~ l p y o i ~ t . I n t h c 1nt;tci7 c a s e t h e Instrument p i e r c e d t h e support a t t h e c e t l t r c ( F i g . 9 ) . T h i s solut.lon was n o t p o s s i b l e f o r t h e bending moment instrument because of t h e c e n t r e f o o t . The support i n q u e s t i o n t h u s had t o be forked (prongs 1 cm a p a r t ) , 30 t h a t i t supported t h e s l a b a t two neigh-

bouring p o i n t s . Owing t o t h i s d i s a d v a n t a g e t h e i n f l u e n c e a r e a s of t h e support moments were recorded w i t h t h e inotrument on t o p and t h e l o a d i n g p o i n t zone of about 4 x

6

cm was superimposed a f t e r w a r d s w i t h t h e i n s t r u m e n t underneath.

F i n a l l y , we s h a l l a l s o d e s c r i b e a semi-automatic scanning i n s t r u m e n t which q u i c k l y r e c o r d s and reduces c o n t o u r l i n e p l a n s ( ~ i g . 1 0 ) . T h i s com- p r i s e s a system of r o d s s i m i l a r t o t h e c a r t o g r a p h e r l s pantograph w i t h a

w r i t i n g magnet a t one end and a f r e e l y d i s p l a c e a b l e moving load a t t h e o t h e r . During t h e scanning t h e l o a d i n g r o l l r o t a t e s about i t s v e r t i c a l a x i s , but can b e stopped i f d e s i r e d and removed by r e v e r s i n g t h e e l e c t r i c motor. When t h e load i s removed ( P = 0 ) t h e d e f l e c t i o n of t h e g i v e n contour l i n e i s shown on t h e i n d i c a t o r w i t h s i g n r e v e r s e d . When loaded ( P = 1) t h e p o i n t e r o s c i l l a t e s about z e r o a s soon a s t h e load h a s reached t h e d e s i r e d l i n e . The o b s e r v e r

s i t s a t t h e desk w i t h t h e reduced p l a n p r o j e c t i o n o f t h e model i n f r o n t of him and a t t e m p t s t o f o l l o w l i n e s of c o n s t a n t p o i n t e r p o s i t i o n ( l i n e s of con- t o u r o f t h e p l a n e s of i n f l u e n c e ) . I n doing s o he does n o t have t o watch t h e p o i n t e r , because a s p e c i a l z e r o p o i n t s w i t c h and a supplementary a m p l i f i e r which c l o s e s a c i r c u i t whenever t h e p o i n t e r i s a t z e r o p e r m i t s t h e magnet t o w r i t e o n l y when t h e c o n t o u r l i n e i s c r o o s e d . The a l r e a d y smoothed image of t h e c o n t o u r l i n e t h u s a p p e a r s on t h e p l a n a s a d o t t e d l i n e . A low-pass f i l t e r p r e v e n t s t h e o s c i l l a t i o n s of t h e system from i n t e r f e r i n g w i t h t h e p r e s s u r e p r o c e s s d u r i n g t h e scanning.

References

111 S e h ~ i l x . E . : Spmnnbelr,mbriirkc ilber dl. Eiaenbmhn i m 7.~6. drr Rundrm.trnO. 4 l s r i Uarlleohsittrl. Ilntnterlrnik 1955. 11e11 3. S. 7 3 .

( 2 ) Sehm.rl,er. I,.. nraodca. G . . Sclaembrek. 11.: A u d ( L h r u n 8 m r h i ~ : l w i n k l i ~ e r k n m t i n n i r r - l i c h r r 1'l.ttrnl~rilcksn. n a u i n g r n i e u r 1 9 i l l . H r l i 5. 5. 1711.

1 1 1 V o ~ t . 11.: Dar ~ t a t i n e h s B r h a n d l u a p von n r h i r l r m I ' l a t t r n h r l i r k r a t i h r m e h r v r r 1:a:ldrr. 11. 1 8 . St. 1958. H e l l 8. 9. 209.

1.11 Andra. W.. L r o n h n r d t . F.. K r i q r r . R.: V r r e i n l n e h t r r V r r l s h r r n rur M c m u n g -,on MomenlenlluOll.rhca h r i I ' l a t t r n . Ha~rinertnnrur 19Sfl. l l r l i 11. S. 407.

151 I l c i n i * r h , K . : F : i n l l u O l l i r h r n l o r cine a ~ h i r l w i n k l i ~ r . g e k r h m l e D r ~ i l r l d ~ l m l t r m i l vcrdnslcrlichrr I l a t l r n d i c k r . 0. u. St. 10611, l l e l t 3. S . 58.

161 Schmniz. 11.: Kon.irtaktisn und Avzrlill~rung d r r Autnhsl~nbrliekr. b r i \\rmtholcmn. R.ainpmirllr 1961. l l r l l 111. S. 3 6 5 .

1,') l l o ~ n l ~ c r g , 11.. Mar'. W. TI.: S r l > i r l r S l l h r u u r l P l a t t m . P r r n r r - V r r l a y 1.rnbll. UU.""l,l".f I l S f l .

[RI I l ~ , m b ~ r g . 11.. Ma-. U . n.: H m u p l m o r n r u t . u n d deren U i r h t u a g e n e i n r r 111.. s r l u r l - u YImttr i t n V c r p l ~ i r h m i l d c n r n c i n r r q u a d r a l i r r l ~ e n I ' l a t l r . n. 8 , . St. 19x8. ll r l t 2. 5. 1.1.

(91 nnlna. J . . Ilsn!alkm, A , : Drr F:n#tlluO rlrr Q ~ n r r t l r h a ~ r n p r a h l .\#I r l c . ~ ~ 9 1 , a n n u n p ~ - 7,~umtand e i m r 15- a c l t i e l ~ n l ' l m t t ~ , L l a ~ f i i ~ ~ ~ e n i ~ ~ ~ r 1'461. l l e f l 3 , l<ItB.

[ l o ] Andre, UJ.. I,.~nllnr,lt. Y.: R l n l l w l l dcs I,nrrr.b.t.ndr. a u l I l b r ~ r n m l , m r n t r vnrl AIII. I ~ ~ c r k r a l t c m r l ~ i r l v i n k l i ~ c r b i n l s l d p l o l ~ m . I t . u. St. l'lhtl, H r l t 7. S. 151.

111) Unniel. I[., I \ i c l ~ t ? r , U'.: M ~ ~ l ~ l l v ~ c ~ ~ ~ ~ l ~ ~ m i l ~ l m ~ l t w h g e l q e r l e m w l t < c l e n I 1 l a t t ? n . U m u i n p r ~ ~ i c ~ u r 105'1. l l r l t 10. 5. I l l b .

(1.71 l l n ~ n l ~ ~ r ~ . 11.. J i i r k l r . 11.. hlarx, U'. n . . l i n l l u f l clcr cla.lisrltrn I.ngrrunl mul Dirge. mnrnrol, ~ l n d h n l l l ~ l ( r r h r r ~ l t r .r l ~ ~ ? l v ~ n k l i p r r I:.inlrl~ll,l.tttrn 13mllingmlr8rr I q b l . I l ? I l I, s. l<l,

1131 \Vri#ler. If.. U'rirr. I f . : Y n ~ l ~ ~ l l . t a t i ~ e l ~ r ~ V e r l n h r r n zur Atalmnhnnr ,,,n E i n l l u O - II~<.I,F~ v,Dm ~*IUIIF,,. n. ,,. 51. I ~ I ~ V , ~I~.II 5 , s. 1z.1.

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Table

I

P r i n c i p a l moments under uniform load l/p12

Angle

a

between m, and i n t e r s e c t i o n of

x

and

y

axes, load

transmission according t o Fig. 3.

Slab

I:

Continuous rectangular s l a b over 3 equal spans,

l/b =

1/0.875#

p =

0.22. Slab

11:

Continuous s l a b skewed 37.8' over

3 equal spans,

l/b =

1/0.875,

=

0.22.

I11

(

IV

Simple span

Slab

111:

Simple span s l a b skewed 37.8",

l/b =

1/0.875,

p =

0.22.

1 I

Outside

span

+0.072

+0.008

0"

+0.069

+0.008

l o

-0.005

-0.021

24.6"

s l a b s

+O .045

-0 -008

39.3"

+0.051

+0.001

21.5'

+0.003

-0.024

15.2'

4 . 0 2 7

-0.075

38.5"

I

Inside

span

Slab

IV:

_{Square s l a b}

f r m Ref.

7,

l/b = 1/1, y =

0.31. I n s i d e

span

+0.019

+0.001

43'

+0.014

+0.001

2.5'

-0.034

-0.055

39.6"

-0.037

-0.059

15.4'

+O

.I22

+O ,028

0"

+ 0 . 1 3

+0.003

0"

Centre of span

m, (1) in, a

Edge centre

m,

(2

m, a

Centre column

m ,

(3)

m 2 a

End column

m ,

( 4 )

m, a

+0.026

+0.004

0"

+0.021

+0.006

0

O

-0.034

-0.104

o

O

-0.035

-0.111

1 "

L I Out

s i d e

span

+0.032

-0.003

42.5"

+0.023

+0.008

3.2"

+0.050

+0.000

24.6"

t

Contin. o v e r t

obtuse

sharp

1 corner

1 -0.014

-0.043

38.2'

-0.054

-0.087

10.8"

(12)

Table I1

Support r e a c t i o n s under uniform load ( r e l a t i v e v a l u e s )

The v a l u e s g i v e n a r e t o be m u l t i p l i e d by t h e t o t a l load of a span, i n c l u d i n g edge overhang.

S l a b 11: Continuous s l a b skewed 37.8', w i t h

3

e q u a l spans, l/b = 1/0.875, p, = 0.22. I11 0 .219 0 .007 0.110 0.088 0.076 0.500

S l a b 111: Simple span s l a b skewed 37.8O, l/b = 1/0.875,

p, = 0 . 2 2 . Column I F i g . 1 L o n g i t u d i n a l s e c t i o n through t h e Messerich b r i d g e , E i f e l I1 Outside span g f j j & c 6 - c & 0

2"

rl Q) Q m

B B

O D U 0 I n s i d e row 0 .296 0.144 0.216 0.186 0.198 1. 2 .

3.

4.

5 . 2 Outside row 0.142 0.054 0.112 0.083 0.069 1.500

(13)

Fig. 2

(a) Cross-section of superstructure (section parallel to the line of columns, viewed from the direction

parallel to the bridge axis)

(b) Model cross-section (section parallel to the line

of supports, viewed from the direction parallel to the bridge axis)

(14)

(15)

n, fir t

my fui. r

mr# fur f

Fig.

4

Influence surfaces of the moments at point 1 (centre of slab):

x and y axes parallel to

and normal to longitudinal slab axis respectively

Influence surfaces of the moments

at point 2 (edge centre, lnalde

span): x and y axes parallel

to and normal to longitudinal slab axis respectively

(16)

m,, ld- 3

Fig.

6

Influence surfaces of the moments at point

3

(centre column):

Influence surfaces of the moments at point 4 (boundary column):

(17)

Fig.

3

(18)

Apparatus f o r measuring t o r s i o n a l m o m e n t s , s e t up t o determine t h e i n f l u e n c e s u r f a c e s of a

column m o m e n t m

XY

F i g . 10

Scanning a p p a r a t u s f o r r e c o r d i n g and reducing l i n e s of contour ( i n f l u e n c e s u r f a c e s )