Principles of the "N" - and "N-Free" Methods of Calculating Reinforced Concrete Sections Subject to Bending

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Principles of the "N" - and "N-Free" Methods of Calculating Reinforced Concrete Sections Subject to Bending

Mehmel, A.

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T i t l e :

Author: Reference:

Translator:

NATIONAL RESEARCH COUNCIL OF CANADA

Tecllnical T r a a s l a t

i o n

TT-664

The p r i n c i p l e s of the

"ntt-

and ttn-free't n~ethods of'

c a l c u l a t i n g reinforced concrete s e c t i o n s s u b j e c t t o bending.

(

zu

den GrunQlngen d e ~ , , ' ~ n " - und des wn-fr-eien't Benessungsverf zhrens fur b i e ~ ~ m g s b e n n s p r u c h t e ~ t + 3 h l b e t o n q u e r s c h n i t t e ) .

AlTred IAelmlel.

Beton- und Stzhlbetonbau, 50 ( 2 ) : 51-54, 1955.

D.A, S i n c l z i r , Trr:l?slations Section,

::,Re

C , Library,

P r e f a c e

I n r e i n f o r c e d concrete d e s i g n a t t e n t i o : - - h-i.s, i n

r e c e n t y e w s , been d i r e c t e d towards a b e t t e r understanding- of t h e behaviour of s e c t i o n s s u b j e c t e d t o bending. The conventional assur.rption of a f i x e d r n t i o "n" bet7:<een t h e e l s s t i c riloduli of s t e e l :ind c o n c r e t e h a s been rlucstioncc'l and "n-free" ( o r ultir?slte l o a d ) methods of calcu3.:ition :-:.re b e i n g advcinced. This development i s ? a r t 1 7 due

to

t h e

i n t r o d u c t i o n of p r e s t r e s s e d c o n c r e t e nhere t h e ~ ~ ~ j c c t i s of considerxble iriiportnnce.

I t i s b e l i e v e d thtit the t r a n s l a t i o n of t h i s ixiyer

which colnpares "nqt and ltn-freel' methods 7 : ~ i l l 5 e a ~ ~ o r t h -

while c o n t r i b u t i o n t o t h e information on tile s u - ' ~ j e c t :~ v ~ ; i l - a b l e t o engineers i n t h i s c o u ~ t r y .

The D i v i s i o n of Building Research i s g r a t e f u l t o

M r , S i n c l a i r f o r p r e p a r i n g t h i s t r a n s l a t i o n T;~2nic:l has beell done with t h e permission of t h e author.

Ottawa

I n reinfor.ccd colzcrete b~:.,:.i c r o s s - s e c t i o n s ::re cos:~-.:only claxsil^iec? 11s o ~ e r - i ~ ~ ~ i ~ ~ ~ o r c e d . , cnder--re ii ~ r 1 0 ~ c c d O P 5: .l;,nced

s e c t i o n s .

fl beail i s s a i d 'tio be over-pein;'orccL ii' 'Lile co;i$ression

, "

zone f a i l s (er:treme f i b p z co~iipressio~!. s t r e s s t l . l ; . i ; . . - ; ,,,nc- , -I;-;c value 01'

tile c o n c r e t e s t r e n g t h K,), ?,:?ith the s t e e l s t r e s s peii;ii~iiliiij n i t h i n t h e e l a s t i c range (below t h e y i e l d p o i n t o _S ),

A n under-reinforced beax reaches t h e l i r . ! i t of i t s b e a r i n g c a p s c i t y nhen t h e rein2'orceiiierlt peaciies t h e y i e l d p o i n t ;

t h e co;i:pression zone then r a i l s clue t o j::'urther y i e l d i;Jl~lich r e s u l t s i n a r e d u c t i o n i n t h e h e i ~ h l ; of t h e co.:.~)iaesc;ion zose clnd an i x - c r e s s e i n t h e extrerrie f i b p e s t r e s s , I n b o t h c a s e s , t h e r e f o r e , tlie b e a r i n g c c p z c i t y o r t h e c r o s s - s e c t i o n i s a t t a i n e d when tile conerelie

coi.gression zone f a i l s d i r e c t l y ( o v e r - r e i n f o r c e d s e c t i o n ) o r in- d i r e c t l y ( u n d e r - r e i n f o r c e d s e c t i o n ) ,

11 b e m ~ i s c l a s s i f i e d a s b a d l y under'-rcini'orecd :-:hen t h e rixixi~num t e n s i l e f o r c e i n t h e reinf'orcernent i s l e s s l;:.!::n t!n::t of t n e t e n s i o n f o r c e t h a t could b e cieveloped by t h e unreinl^o:-ced c r o s s - s e c t i o n , i. e., : . ; s s u x i n ~ a ITavier s t r e s s d i s t l > i l : ~ c t i o n

As f a r ns t h e i r b e z r i w c - ~ ~ p a c i t y i s concerned :;re::tlg under-reirdorced c r o s s - s e c t i o n s :;ye t o b e tre::..ted t h e f;s.l;;e ; I ~ E cn- r e i n f o r c e d c r o s s - s e c t i o n s . They a r e o r no d i r e c t in.Lel-est i n t h e

problems of' r e i n f o r c e d c o n c r e t e and do n o t iieed t o b e considered f u r t h e r here.

Both t h e n- and t h e 11-free ~ i ~ e t h o i l s ace bc-~ued on the assumption t h a t t h e c r o s s - s c c t i o ~ l s rer,:ain plane: ~ ( v ) = v * s , . , I n a d d i t i o n , t h e nr method assumes a c o n s t a n t moc~ulus o f e l a s t i - c i t y of c o n c r e t e o r , i l l o't!lrlh : ~ O ~ Y L S + a s t r a i g h t - l i n e a-s -curve

( 0 = constp. . o )

.

whereas t h e n-f r e e rllethod i n t r o d u c e s t h e a c t u a l f u n c t i o n a l r e l a t i o n a = f ( s ) i n t o t h e c a l c u l a t i o n and t h u s claims t o achieve a c a l c u l a t i o n which corresponds more c l o s e l y t o t h e a c t u a l c o n d i t i o n s ,

1.1 The S t r e s s D i s t r i b u t i o n a s C a l c u l a t e d by t h e n-Tqlethod.

The s t r e s s d i s t r i b u t i o n i n r e i n f o r c e d c o n c r e t e c r o s s - s e c t i o n s s u b j e c t t o bending a s c a l c u l a t e d by t h e n-method ( b r i e f l y c a l l e d the "nt' c r o s s - s e c t i o n ) i s , on reaching t h e u l t i m a t e l o a d , l i n e a r , i n accordance with Fig. 1 and 2, f o r b o t h over- and under- r e i n f o r c e d beans.

Fig. 2a shows t h e s t r e s s curve of t h e under-reinforced c r o s s - s e c t i o n a t t h e y i e l d p o i n t of -the reinf'orcemcnt,

Fig. 2b shows

i t

a t t h e moment of f r a c t u r e , Ee

" = T

i n F i g , 2a i s

r e p l a c e d by t h e s m a l l e r v a l u e

fi

=

3 e ( ~ )

where E e ( € ) i s t h e s e c a n t Eb

modulus of t h e e n t i r e s t r a i n of t h e s t e k l beyond t h e y i e l d zone ( ~ i g . 3 ) .

When f a i l u r e occurs t h e i n t e r n a l f o r c e s a r e approximately e q u a l t o those a t t h e beginning of y i e l d (Fig. 2a). The r e s i s t i n g moment i s i n c r e a s e d only s l i g h t l y due t o t h e i n c r e a s e i n t h e l e v e r arm ( s e e Sec. 2).

The s a f e t y f a c t o r s a r e chosen l a r g e r i n t h e German S t a n d a r d s f o r c o n c r e t e under compression than f o r t h e s t e e l i n

o r d e r t o t a k e i n t o account t h e v a r i a t i o n s of q u a l i t y of c o n c r e t e m a t e r i a l s . For t h e p r e s e n t i t w i l l s t i l l b e n e c e s s a r y t o observe

t h i s p r i n c i p l e f o r t h e g r e a t m a j o r i t y of c o n c r e t e s mixed a t t h e b u i l d i n g s i t e , If v t b e t h e supplementary s a f e t y f a c t o r f o r t h e

concrete the following working safety coefficient may be estab- lished:

If

the allowable stress values are used in the stress diagram in place of the critical values, as is usually done, then instead of the stress diagram according to Fig, 1 and 2a we obtain those seen in Fig,

4

and

5,

Here we have

"0

wb a

u

-

- - - -

= - . S

b adm,

_v

_v

_{V t} ; e' adm, _V

The stress diagram of the ty-pe of Fig. 2b is not used for practical n-calculation,

The dividing line between over- and under-reinforcement, the so-called balanced reinforcenent, is obtained when G

b adm, and a _{e adm,} are reached simultaneously (~i.g.

6 ) .

'!/hen this is the case, with a constcnt concrete cross-section, the bearing capacity can

only be increased ir, besides increasing the tensile reinforcement, a corresponding compression reinforcement is also inserted. For the balanced reinforcement of the r e c t ~ n g u l z ~ bending cross-section the follo~ving equation is obtained:

u b adm,

u

_{e a h a F e , g r} = a _badma b

x/2;

x

= h = s o h a _{b adm,}

+

0 _{e adm,}

_/n

gr

1 a (1)

Fe, l3-r

1,2 The n-Free Method.

The n-free method is not based on allowable stresses but on the ultimate load capacity, i. e,

,

substantially corresponding to Figs. 1 and 2b, except that in the compression zone an assu!ied

In the n-free method present practice is characterized by two assumptions:

(a) the 0-c -curve is concave relative to the &-axis, do

i.e.,

-

_d& decreases with increased stress;

(b) all concretes show, under similar test conditions (including the loading rate, etc,) the same strain '

b max, at failure.

Since at failure, both for over- and under-reidorced cross-sections, the compressive strength of the concrete is ex- ceeded in all cases, the stress-strain diagram for the state at failure has a characteristic appearance shown in Fig,

7

and

8

with v

-

_1.5

_{for concrete in compression.}

t

-

The curve in Fig,

8

corresponds to the n-method according to Fig. 2b, Under-reinf'orced cross-sections according to Fig,

8

are in the great majority, so that in the following attention will be concentrated on them. There can be no objection to this because considerations are basically the same for the over-reinf'orced sections,

The u (v)-curve is afrined to the o(& )-curve ( ~ i g , 91,

The affinity factor by which the &-coordinates must be multiplied in order to be transformed into the v-coordinates is

X

_-

_-

-

1 &

b rnax. ' 1

According to Fig.

8

the resultant compressive force at the point of failure is given by

X X

Db

_rnax. = b J o

(v)

d v = b 0 [ o

1

= a o

% *

x o b 9 (2)

0 F1o

where a is termed the area ratio, With a =

0.5

we get the stress distribution which is linear over v, i. e,

,

the triangular form; with a = 1.0 we obtain the stress distribution which is constant over v, i.e., the rectangular distribution ( ~ i g , 10a, lob).

I n d e t a i l s , of course, t h e r e i s d i v e r s i o n of opinion, e.g, i n connection w i t h the ayes r a t i o a of t h e 0-E-cwve and on

the numerical value of Eb maxm

.

(DIN 4227 assumes a = Om75. Eb maxe

= 2

The balanced reinforcement B e

*

i s obtained, a s i n the

P gr

n-method, when cb maxe and E s a r e reached simultaneously. &

0 * = a . K O m x * b ; x = h e b m x * = s * e h ;

S Fe,gr s _{b max.}

+

E _s g r

The influence of the a-s-curves o f the concrete, p a r t i c u l a r l y

through i t s a r e a r a t i o and the f a i l u r e s t r a i n , on t h e reiflorcement and u l t i m a t e load w i l l be shorm i n what f o l l o ~ v s , and then the

q u e s t i o n w i l l be discussed of whether p r e s e n t l y a v a i l a b l e t e s t re-, s u l t s a r e s u f f i c i e n t t o introduce t h e appropriate c o e f f i c i e n t s i n t o the c a l c u l a t i o n ,

The f o l l o ~ v i n g symbols w i l l be used: a: a r e a r a t i o of the coi7~d)ression zone,

@: d i s t a n c e of t h e c e n t r e of g r a v i t y of the compression zone from the extreme f i b r e i n r e l a t i o n t o the height of the cortrpression zone.

( The " r e l a t i v e u l t i m a t e bendiru momentst1

m

a r e r e l a t e d % , g r

*

= r e l a t i v e moment of t h e n-free beam with t h e

balanced reinforcement Fe

*

= @gr*

-

b h P

m'

%*

= r e l a t i v e moment of t h e n-free beam, Fe* < F e9@'

*

% , g r = r e l a t i v e rnoriient of the n-bean with the balanced

ri\ = r e l a t i v e nornent of t h e n-beam Fe < Fe ? g r

aFe and F i = a d d i t i o n a l t e n s i o n o r cornpression reinforcement of t h e n-bean i n o r d e r t o accommodate a r e l a t i v e moment beyond

X

s * , s

*,

s , sgr: r e l a t i v e h e i g h t s of t h e p r e s s u r e zone

r;

.

g r

2.1 The I n t e r n a l Lever A r m a s a Function of t h e Area R a t i o

J e

= r ( a ) l . Let t h e r e l a t i v e s t r e s s - s t r a i n curve f o r t h e r r e n r a t i o of 0.25

2

a 0.75 be r e p r e s e n t e d by t h e 3rd degree parabola (pig. 11) where EL 4, I t then f o l l o w s t h a t 1 ₁

a

=

d

q d& =

4

(3.

-

lc) and

2.2 Comparison of the n- and n-Free Cross S e c t i o n s w i t h Respect t o Their Bendin[< Reinforcement and Their Ultimate Load.

2.21 Both c r o s s - s e c t i o n s a r e siniply r e i n f o r c e d . _- The balanced r e i n f o r c e r ~ e n t s a r e determined by e q u a t i o n s ( 1 ) :lnd

( 3 )

; t h e

S i n c e t h e d i s c u s s i o n concerns t h e i n t ~ ~ o c l u c t i o n of t h e n-free nethod i'or o r d i n a r y r e i n f o r c e d ( n o t p ~ e s t r e s s e d ) c o n c r e t e

s e c t i o n s , t h e fo1lo;i'in;; numerical exarrrples w i l l be based on t h e r r ~ a t e r i a l s B 225 and S t , I , x i t h a d i s t i n c t y i e l d p o i n t 0- = 2400

LJ

~;gi:;, /c;.ii. f r e q u c n t l y used i n r e i n f o r c e d c o n c r e t e c o n s t r u c t ion. Tlie ~ 2 . f e t y f a c t o r v _{= 1*75,}

v t

_{= 1.5 i s used i n the c ~ l c u l a t i o n ,} lvhich corresponds t o the mezn v a l u e s given i n D I N 1045,

4225

and 4227.

I n o r d e r t o shon _I.. t h e e f f e c t of t h e v:>lue c b mnx. t h e

,.-

Fe, p;r ri\ %k

c u r v e s

,

( a ) and ( a ) a c c o r d i w t o eq.u&tions (1) 2nd I1*a

,

C;r

( 3 )

m ; ~ ~ ~ r z n c

(6).

r e s p e c t i v e l y , a r e p l o t t e d Tor b o t h v a l u e s of'

E = 2

low3

and

4

lom3,

r c s p c c t i v e l y , :IS pa;"r~r,:eters i n

b

m

a

:

:

,

With i n c r e a s i n g

o

t h e r a t i o s of t h e balanced reinf'orce- -I? i t

l-eilts

+Rr'

i n c r e a s e 2nd f o r

a

= 0.75, o r

a

= 1 ? 0 an6 E b rl.,.-

-

-

i 'Ad,..

!?9D"

2

lom3

rescll v z l u e s of 2,10 and 2.77, r e s p e c t i v e l y . The '~ei'iC~2r~g ~;~or.:eilts which c:in be s u s t a i n e d do not of course v a r y l i i i e a r l y w i t h F

*

b u t i n c r e a s e 1;':oye slowly, s i n c e t h e l e v e r arms 3ls0 v a r y

e , g r

,

p;p

*

( c f . 2 , l ) ; i t s r a t i o h a s the v a l u e s 1,63 and 2.24 f o r a =

r'o

cr

0.75 and 1 - 0 , r e s p e c t i v e l y .

The n-r~ethod does n o t g i v e a s high a v a l u e f o r t h e

r e s u l t a n t c o r p r e s s i v e f o r c e f o r equal K.,, a s does t h e n-free lnethod,

n f a c t v11ich becomes apparent i n t h e a r e s r a t i o a, and t k i s

disi'erence becomes i n c r c n s i ~ ; l y i l o t i c e a b l e a s t h e ~ e i n f o ~ c e - i i e n t approaches t h e balanced value p, ::

.

b

.a'

h

.

On t h e :::hole :.re a r r i v e a t c r o s s - s e c t i o n s vhich t ; . e p ~ . ~ t coi-~siderz21ly fro12 present-day exper.ience i n r e i n f o r c e 6 c o n c r e t e c o i i s t r u c t i o n , t h e r e s u l t being t h a t over a c ~ n s i d e ~ a i ~ l e range beztcms a r e obtained, 1?1hich b y t h e n-method must c o n t a i n co~iqr-ession s t e e l , ;-~lzerecs, by t h e n-free method, they r e q u i r e only a t e n s i o n reil.;i"olace-8 ment ( ~ i g , 1 2 ) ,

2 . 2 2 Tlie n-cross s e c t i o n i s unubly r-cinforced vrhereas

--

t h e n - f r e e s e c t i o n i s o n l y sir..plj~ r e i n f o r c s . 'i'he t o t a l r ~ i r ~ f o r c e - rnent of' both c r o s s s e c t i o ~ l s i s t71c same. 1Iov.i do t h e p c r : . . i s s i b l e

-

~norients behave a s a f u > t t i o n of t h e r a t i o

IT1or e q u a l s a f e t y w i t h r e g a r d t o t h e y i e l u p o i n t of' tlle t e n s i o n s t e e l of t h e n- anti. t h e n - f r e e beai?.s, i . e . , w i t h

.s. 1 "- F = F and F =

P

e e , g r e e,Ep

+

AJ',

+

b" e we o b t a i n

-

-

--. I,--

-

-' Db ( n - f r e e ) and .I, I n s e r t i n g t h e v a l u e of s." from e q u a t i o n ( 8 ) i n t o t h e expr4ession we t h e n o b t a i n P

-

z

Sgr (1 + A,

7

+

-

""I

---

--

-

-.- e , g r

Equation ( 9 ) g i v e s t h e r e l a t i o n s h i p of t h e moment of a simply r e i n f o r c e d s e c t i o n determined 'by t h e n - f r e e r:~ethod t o t h e nion~en t

m,,

g r a s a f u n c t i o n of

The r a t i o moment of a doubly r e i n f o r c e d c r o s s - s e c t i o n d e t e r m i n e d by t h e n method t o

m,

i s g i v e n by 9 g r t o g e t h e r w i t h 1 - s F ' = AE'

ar

e e c + s - 1 . gr S u b s t i t u t i n g c from e q u a t i o n ( 1 1 ) i n e q u a t i o n ( 1 0 ) we o b t a i n

The d e s i r e d r a t i o i s o b t a i n e d by combining e q u a t i o n s ( 1 0 a ) and

( 9 ) :

F i g .

14

shows t h e r e l a t i o n s h i p e x p r e s s e d i n e q u a t i o n ( 1 2 ) f o r v a r i o u s v a l u e s of a w i t h c = 0.93 h. Froni t h i s we a r r i v e a t t h e i m p o r t a n t r e s u l t t h a t t h e n and n - f r e e rnethods g i v e approximately t h e same u l t i m a t e l o a d c a p a c i t y f o r beams w i t h b a l a n c e d r e i n f o r c e m e n t F of t h e n-beam. The d i f f eren.ce

between t h e t w o nic'ihods i s r e v e a l e d only when t h e simple r e i n f o r c e : ; ~ e n t i s i n c r e a s e d beyond t h i s v a l u e , all i n c r e a s e which i s p e r m i t t e e without conipression s t e c l or-ly by t h e n-

f r e e method. The r a t i o of t h e l o a d c a p a c i t i e s shows t h e

s u p e r i o r i t y of t h e n - f r e e bean over t h e n-bear,). For a, = ~j.75 t o 1.00, f o r t h e saxe t o t a l r e i n f o r c e m e n t , t h e n - f r e e beam achieves a load c a p a c i t y 1.20 t o 1.25 tilrles a s g r e a t a s t h e n-beam. This e s t a b l i s h e s t h e econoi i c s u p e r i o r i t y of t h e n-free rr-ethod.

2.3 On Yhat *&-Curve ( a - v a l u e ) Shoulci a n n-E'ree --- Lethod

-

be Based?

I n t h e i n v e s t i g a t i o n s r e p o r t e d i n S e c t i o n s 2.21 an2 2.22, a r e a r a t i o s w i t h i n t h e range of 0.25 < a < 1.00 have been considered. A t f i r s t c o n s i d e r a t i o n t h e r e docs n o t appear t o be any good reason f o r t a k i n g such a l a r g e range of v a l u e s i n t o a c c o u n t , s i n c e t h e g e n e r a l l y known anu used +&-curves of con- c r e t e a r e curveu i n such a way t h a t t h e i r concave s i u e i s to- wards t h e & - a x i s . However, a s f a r a s t h e p r e s e n t a u t h o r has been a b l e to ciiscover, t h e s e curves have a l l been determined from

s h o r t term t e s t s . I n r e p o r t s i s s u e d by t h e P r o b s t I n s t i t u t e i n Karlsrukle ( l p 2 ) , t h e r e s u l t s of r e p e a t e d load t e s t s on corlcrete a r e d e s c r i b e d i r l which t h e 0-&-curve r e a u c e s i t s c u r v a t u r ~ c , a t f i r s t concave towards t h e & - a x i s , ~ u r i n g the repented l o a d i n g s , e v e n t u a l l y b e c o ~ i ~ e s a s t r a i g h t l i n e and even changes t o a convex c u r v a t u r e , whereby t h e l o a d i n g anG unloading curves e i t h e r form a h y s t e r e s i s loop o r c o i n c i d e ( F ~ Q . 1 5 ) . These phenomena becorrLe a l l t h e more marked t h e g r e a t e r t h e range of l o a c i n g anci i t s maximu? s t r e s s , i . e .

,

f o r one and t h e salLe c o n c r e t e t h e value of a depends on t h e l o a d i n g c a t e g o r y , and hence i s n o t a

m a t e r i a l c o n s t a n t . For example, a, would be sl;laller f o r a r a i l - road b r i d g e beau than f o r a high b u i l d i n g beam b e a r i n g a s t a t l c o r only s l i g h t l y v a r i a b l e l o a d .

Bach ( 3 ) i n t e s t s w i t h I-narble and Graf

( 4 )

i n t e s t s w i t h g r a n i t e and m o t t l e d s a n d s t o n e s , a l s o observed o-s-curves w i t h

t h e concave s i d e s toward t h e a-axis.

A a-s-curve w i t h t h e concave s i d e towards t h e 0-axis i n d i c a t e s t h a t t h e m a t e r i a l i n c r e a s e s i n stiffness w i t h i n c r e a s - i n g s t r e s s . h~ehniel

(I)

i n t e r p r e t e d t h e r e v e r s a l of t h e curve a s an e f f e c t of f a t i g u e . The a u t h o r i s indebted t o I r o f . Graf of S t u t t g a r t f o r having r e c e n t l y drawn t o h i s a t t e n t i o n t h e f a c t t h a t t h e j u s t mentioned a- &-curves i n riiarble ( 3 ) have been i n t e r p r e t e 6 t o mean t h a t i n i t s g e o l o g i c a l p a s t t h e rock may

have been s t r e s s e d by rnovenlents and t h e l i k e , wl~ereby i t s u f f e r e d f i n e i n t e r n a l f r a c t u r i n g . This might p o s s i b l y e x p l a i n t h e f a c t t h a t t h e Bach and Graf experiments

( 3 , 4 )

show t h e curve r e v e r s a l a t coi:~p::rativcly siiiall s t r e s s e s . Berg ( 5 ) r e p o r t s o b s e r v i n g

t h a t c o n c r e t e specimens f o r which t h e !<nown 0-s-curves were

o b t a i n e d i n short-teri:. t e s t s , show d i s t u r b a n c e s i n t 7 - ~ c i r i n t e r n a l s t r u c t u r e due t o f i n e c r a c k s when t h e s t r e s s e s i n t h ~ silort-term t e s t r e a c h 50 t o

60%

of t h e b r e a k i n r s t r e q - t h . ;Ir,nr.g ' o r r n

r e p o r t s

(6)

t h a t D r . R. Jones i n t h e Road Research I a l ~ o r a t o r y made s i m i l a r o b s e r v a t i o n s t o those of Berg. T h i s would s u p p o r t

t h e i n t e r p r e t a t i o n s of t h e r e v e r s a l of t h e a-s-curve j u s t given. Berg and J o n e s ' o b s e r v a t i o n s appear t o b e a r o u t t h e f a c t t h a t t h e s t r e n g t h i s reduced by continuous l o a d i n g , f o r i t i s reason- a b l e t o suppose t h a t i n t h e course of tirrie t h e f i n e c r a c k s under continuous s t r e s s w i l l sooner o r l a t e r develop i n t o c r a c k s lead- i n g t o f a i l u r e .

T h i s 'brings us t o s a f e t y c o n s i d e r a t i o n s . D I N kc227 l e a d s t o the permissible r e s u l t a n t cor;~pressive fo r c e by t h e

w i t h v = 1.75, L whereby, a s a l r e a d y n o t e d ,

\

=

-

₃ i i ~accounts i ' o r t h e f a c t t h a t t h e s t r e n g t h v a l u e s of t h e c o n c r e t e show a c e r t a i n s c a t t e r an6 c o n c r e t e t h e r e f o r e r e q u i r e s a g r e z t e r s a f e t y f a c t o r than s t e e l , The 1.75 s a f e t y f a c t o r i s based on t h e

%-

v a l u e o b t a i n e d i n

the short-tern test. It is well known, however, that prolonged loading, especially when it is in the forrn of a repetitive load, has a strength reducing effect.

The desired safety factor

1.75 (assuming

that the concrete strength in the structure attained is only

%

= 2/3 dB) is thus smaller by the amount by which I$, is reduced to the long- term strength, which corresponds to repeated load with a range of fluctuation peculiar to the structural part considered.

Despite this consideration I have no objection to the evaluation of the resultant compression force in DIK 4227. The prestressed constructions are at present produced either in

stationary plants in the form of prefabricated parts or on the building site for specially irriportant constructions, generally bridges. In both cases it may be assumed that the scatter l/v

t for tflc concrete strength, which is taken as 2/3 in accordance witii the suggestion of DII'3 4227, can be taken as equal at least to unity.

In

my experience the concretes of prestressed bridges and of plant manufacture are made so exactly that one niay assuriie that their rated strengths are not only reached but as a rule even exceeded.

'fie may then use the safety factor

vt

=

1.5

to coiupensate for the fact that long-terln loading is not taken into account in

V =

1.75

and for the too favourable assumptions of the n-free

method (too favourable a-values, see above). This possibility of compensation is however invalid if the n-free rr:etllod is extended to cover all reinforced concrete constructions with the calculation values deterained in DIE 4227, and the physical coefficients must then be determined more accurately.

3.

Summary

3.1

The area ratio a is of great importance in assessing the load capacity. According to the presently available long- term tests, which of course do not nearly cover the whole range of the problen:, area ratios are to be expected which can fall

considerably below

0.75.

Possibly different values of a will be obtained for predominantly static and frequently changing loads, depending

on the loading range and the kind of aggregate. It nay also be expected that long-term tests will affect not only the for~~i of the a-E-curves (a-value), but also the strength of the concrete, on which the safety factor must be based.

3.2 The assumption

%

max.

= constant for all concretes (for given test conditions) is at first surprising. A11 the other physical coefficients of concrete depend on a great 1r;any factors such as weight (light weight or heavy concrete), grain corr-posi- tion, water

-

cernent ratio, curing, etc.

3.3

If a -values between

0.75

and 1.0 were employed in the present-day n-free method, balanced reinforcement values of up to twice those of the n-bearn would result. Apart frorn the con- siderations of 3.1 and 3.2, it should be borne in mind that practical application of this to reinforced constructions would abruptly take us fsr beyond the scope of existing construction experience,

3.4

The n-free method has so far been deemed applicable only to prestressed structures in which the conditions are

especially favourable for it. Before it can be applied generally to ordinary reinforced constructions, i.e., to the great ~ a J o r i t y of designs, more thorough investigations are needed along the

lines indicated in 3.1 and

3.2,

and possibly also a step-by-step extension of experience in accordance with

3.3.

References

1. Mehrnel. Untersucl-iungen Uber den Einf'luss hiiufig wiederholter Druckbeanspruchungen auf Druckelastizitat und Durck-

festigkeit von Beton (~nvestigations into the effect of frequently repeated colnpressive strains on the compres- sive elasticity and compressive strength of concrcte), Berlin, Springer-Verlag, 1926,

2. Treiber. Das Verhalten von Eisenbeton-T-Balken unter dern Einfluss dauernd ruhender und hgufig wiederholter

belastung

b he

behaviour of reinforced concrete T-beams under continuously resting and frequently repeated

loading), Bauing,

1934,

p. 131.

3.

Bach. ElastizitHt und Festigkeit (~lasticity and strength), 8th ed. Berlin, Springer-Verlag, 1920.

4.

Graf. Versuche Uber die DruckelastizitHt von Basalt, Gneis, Muschelkalk, Quarzit, Granit, Buntsandstein sowie

HochofenstUckschlacke ( ~ e s t s on the cor:ipressive elasticity of basalt, gneiss, shell limestone, quartzite, granite, bunter and broken blast-furnace slag). B. u.

X.

1926, P.

399.

5 .

Berg, 0. Y. Road Research Library Comrriunication No.

165.

London,

1951.

6.

Cowan, Henry J. Inelastic deformation of reinforced concrete in relation to ultimate strength. Engng.

174(4518),

1952.

S t r o n g l y r e i l l r o r c e d c r o s s - s e c t i o n Fig. 1 z u l = adm. Fig.

4

S t r o n g l y r e i n f o r c e d c r o s s - s e c t i o n Fig.

7

,.leakly r e i n f ' o r c e d c r o s s - s e c t i o n Fig. 2a

-

2b F i g .

5

Fig.

6

I - - c - + - ~ ~ i

4 J e a k l y r e i n f o r c e d c r o s s - s e c t i o n F i g . 8 Fig. 9

Fig. IOa

Fig. 11

Fig. l o b

- E

-

"nu cross-section

Fig, 13a

- b

-

:In-f ree" cross-section

Fig, 13% I - . - -Curves f o r para-meter values E b m a r c 2 - - - C u r v e s f o r 2 4 &

< 4

.

- - - * -- - Fig.

14

Fig. I 5