Deduction of a Formulation for the Minimum Reinforcement for Crack

Disclaimer: The following models are published for study and discussion purposes only and not thought and neither corrected to be finally used to solve design tasks, yet. There is no guarantee given for any results obtained by using that formulae. Potential users need to verify the deduction process on their own to form their conclusion. Despite the fact that the models shown are derived from a normative starting point, the kind of fitting them together to form a simplified design approach is not normative in the current stage and is not more or less than an author’s current opinion. Limitation: The following deductions assume the application of high-bond-bars. Achtung: (Deutsch) Die folgenden Modellformulierungen sind nur zu Studien- und Diskussionszwecken veröffentlicht worden. Sie sind letztendlich weder fertig korrigiert noch dazu gedacht, reale Designaufgaben z. Zt. zu lösen. Es wird keine Garantie für jegliche mit diesen Formeln erzeugten Ergebnisse gegeben. Mögliche Nutzer müssen versuchen den Herleitungsprozess nachzuvollziehen und zu verstehen, um ihre eigenen Schlüsse daraus zu ziehen. Trotz des Fakts, dass am Anfang der Herleitung normative Modelle stehen, ist die Art, diese für die Bemessungsaufgabe zu einer einfacher zu nutzenden Formulierung umzuformen definitiv nicht als normativ zu bezeichnen und nicht mehr und nicht weniger als eine Autorenmeinung. Abgrenzung: In den Ableitungen wird von der Verwendung üblicher, gerippter Betonstähle ausgegangen, (keine glatten Rundstähle, keine Litzen, keine tief gerippten Stähle). Deduction of a formulation for the minimum reinforcement for crack control during primary cracking Eckfeldt, L.11 Research assistant at the Institute of Concrete Structures, TU Dresden, Dresden, Germany 1. Scope of minimum reinforcement to control crack widths in primary cracking - definitions In principle, the tension force that is set free during primary cracking should be overtaken by the minimum reinforcement. The need for a limiting value develops from following:

s,min s,min sct ,min eff e

c ct c,eff c

ct ct

A A E(1), (2), (3)

k k A A E (t)A h b;

ρ = ρ = α =⋅ ⋅

= ⋅where: is the part of the section under tension and verification, in case two faces of a structural member are in tension, both ca ct

brutto concrete areact ct ,c e s1

c,eff c,ef

h h / 2

A A ( 1) A ,

A h

= − −

=

= + α − ⋅

=

nd

ct

n be treated separately with Precisely is but the 2 part can be switched off because of the uncertainty of f (x) in later calculations.

f c,eff 1 1b; h m d dm h

m)

⋅ = ⋅ −

−

being distance bargroup centroid to tension face usually 2,5, but should be chosen depending on

(see [König/ Tue, Bergner] for

1 1

1 1

1

for : 0 h / d 5, m h /(2d ) 2,5for : 5 h / d 35, m (3,33 h / d 0,33) / 2for : h / d 35, m 15 / 2 7,5

≤ ≤ = ≤≤ ≤ = + ⋅

> = =

c,eff

acc. to [Bergner], it gives:

(the "/2" in m results, because m is thought for A on one tension face)

(4)

Eckfeldt, Deduction of minimum reinforcement

1

( )

c ct ct

s,minc ct cts,cr,I e

s,min c ct

k k A f (t) (5)

Ak k A f (t)1 1

A k k A

≈ ⋅ ⋅ ⋅

⎡ ⎤⋅ ⋅ ⋅σ = ⋅ + α − ⋅⎢ ⋅ ⋅⎣ ⎦

cr,I

Cracking Force to be set free during primary cracking:FThe stress induced into the reinforcement in the primary crack is:

s,minct ,min

c ct

(6)

Ak k A

⎥

ρ = σ⋅ ⋅ s,cr-1

The extension [1+...] ensures the strain compatibility in full force transfer between steel and concretesurrounding.Because

gives very small values , can be taken as

c ct cts,cr,I

s,min

c,eff ct

k k A f (t)(7)

A

A f (t) (8)

⋅ ⋅ ⋅σ ≈

≈ ⋅cr,II

:

The necessary cracking force for the activated tension chord in order to produce secondary cracking is:F

( )c,eff ct s,mins,cr,II e

s,min c,eff

c,eff cts,cr,II

s,min

A f (t) A1 1 (9),

A A

A f (t)(10)

A

⎡ ⎤⋅σ = ⋅ + α − ⋅⎢ ⎥

⎢ ⎥⎣ ⎦⋅

σ ≈

stress induced into the reinforcement by secondary cracking is:

to be simplified as:

As a rule, the stresses

s,cr,I s,cr,II

s,cr,I s,cr,II1, 2...1,5σ > σ

σ > ⋅σ

compare to each other as follows, depending on the specific geometricalconditions:

In case can be proved, primary cracking automatically induces secondary cracks in the neighbouring zones . This shortens transfer lengths and eases restraints. In conseqence and over a long time period, in most situations secondary cracking can be expected. The minimum reinforcement can be derived from the equation to calculate the characteristic crack width in a structural situation. 2. Deduction of close solutions for the minimum reinforcement in DIN 1045-1, Kap. 11.2. a) In cases where , the crack width may be obtained from the following: s,cr,I s,cr,II1, 2...1,5 σ > ⋅σ

k rk m

m s m c m

s,cr,II s t s,cr,IIk t

bk s

s s,cr,I bk ctm

s,cr,II s,crk

ct

0,40,6

w s (11)

(12)

kw ; (k )

2 E1,8 f (t) (14)

w3,6 f (t)

= →= →

= ⋅ε

ε = ε − ε

σ ⋅φ σ − ⋅σ= ⋅

⋅ τσ = σ τ = ⋅

σ ⋅φ σ= ⋅

⋅

long time loadingshort time loading

where is:

DIN, MC90:

using

(13)

,I t s,cr,II

s

k(15)

E− ⋅σ

For primary cracking the crack distance srk is mainly influenced by the possibility of occurrence of secondary cracks and the steel stress that should be expected is very much related to Fcr,I. If wk is substituted by wlim and the simplified terms for the steel stresses are taken over into calculation, the above equation changes to:


2

ctlim

f (t)w = c,eff

s,min ct ( t )

A

A 3,6 f

⋅ ⋅ φ

⋅ ⋅ct c ct t ct c,eff

s,min s

ct c,eff c ct t c,effs,min

s lim

ct c,eff c ct t c,effct ,min

s lim c ct

f (t) k k A k f (t) AA E

f (t) A (k k A k A )A (16)

3,6 E w

f (t) A (k k A k A )(17)

3,6 E w (k k A )²

⋅ ⋅ ⋅ − ⋅ ⋅⋅

⋅

⋅ ⋅ φ ⋅ ⋅ ⋅ − ⋅=

⋅ ⋅

⋅ ⋅φ ⋅ ⋅ ⋅ − ⋅ρ =

⋅ ⋅ ⋅ ⋅ ⋅

b) If it remains uncertain whether those fortunate secondary cracks develop or not, (mostly in cases where the steel stress is found to fit the following inequality:

s,cr,I s,cr ,II1,3σ < ⋅σ , The approach to wk should change slightly to:

k tkw 2l= ⋅Δεm (18) In ltk it is thought that a full transfer of the steel force within the primary cracking is enabled. A position x within sr can be reached for which strain compatibility can be stated: εs(x) = εc(x).

s,cr,I t s,cr ,Ilim

ct s

ctlim

(1 k )w (19)

3,6 f (t) E

f (t)w

σ ⋅φ − ⋅σ= ⋅

⋅

= c ct

ct

k k A

3,6 f (t)

⋅ ⋅ ⋅ ⋅φ

⋅t ct c ct

s s,mins,min

c ct t ct c ctlim

s,min s s,min

t cts,min c ct

lim s

t ctct ,min

lim s

(1 k ) f (t) k k AE AA

k k A (1 k ) f (t) k k Aw

3,6 A E A

(1 k ) f (t)A k k A (20)

w 3,6 E

(1 k ) f (t)w 3,6 E

− ⋅ ⋅ ⋅ ⋅⋅

⋅⋅

⋅ ⋅ ⋅ φ − ⋅ ⋅ ⋅ ⋅= ⋅

⋅ ⋅

− ⋅φ ⋅= ⋅ ⋅ ⋅

⋅ ⋅

− ⋅φ ⋅ρ =

⋅ ⋅tlate restraints,(1-k ) = ct

lim s

f (t)(21)

w 6 Eφ⋅

⎯⎯⎯⎯⎯⎯⎯⎯→=⋅ ⋅

0,6

The steel stress is limited to 0,8 fy, allowing 25% more than the calculated load for any uncovered restraints. In that way, it is an additional measure for more reliability. The simplicity of the formulae does not require any further adjustments. Verification tools can be produced in the style of diagrams related to φ and ct ,minρ .


3

Figure 2: Illustration of the influence of secondary cracking to the problem of controlled cracking MC 90 and DIN 1045-1 are similarly structured for serviceability. In some circumstances it is easier to refer to MC 90 because it is covered by far more background information. Explanations of some differences in MC 90 DIN 1045-1 and the explanation of [Tue] et.al relies very much on the assumption that, however, stabilised cracking or at least secondary cracks in the surrounding of a primary crack are always likely to occur. It also applies in restraint situations, over a longer period of time. On the other hand, pure single cracking in MC 90, meaning s,cr,I s,cr,II1, 2...1,5σ < ⋅σ ,is defined as follows:

s,cr,I s,cr,I t s,cr,Ik

bk s2,25fctm

bk ctm bm

bk ct,0.05 ctm

t

s,cr,Ik

kw (22)

2 E

where : 1,35 f (t) /1,7Another possible explanation 2, 25f /1, 25 would have lead to 1,26 f (t)

k 0,6

leading to:

w2 1,35

=

σ ⋅φ σ − ⋅σ= ⋅

⋅ τ

τ = ⋅ ←= ττ = ⋅

=

σ ⋅φ=

⋅ ⋅s,cr,I s,cr,I s,cr,I

ctm s ct,min s ct,min s

s,cr,I s,cr,I

ct,min s ct,min s

0, 4 0,4 0,6(23)

f (t) E 2,7 E 3,6 E

as above suggested for DIN 1045-10,6 0,6

(24,05 E 3,6 E

⋅σ ⋅σ ⋅σφ φ⋅ = ⋅ < ⋅

⋅ρ ⋅ρ

⋅σ ⋅σφ φ⋅ < ⋅

⋅ρ ⋅ρ4)

This means that, compared to the original approach of MC 90 for single cracking, the suggested approach for DIN 1045-1 stays on the safe side. In any case, the definition of τbm is not free of contradictions. Eligehausen expected a mean bond strength to be dependent on c/φ.


4

This expression may be transformed in a quadratic equation:

2c,efflim s u ss,min2

s,min s,min

2lim s s,min c,eff

u s s,min

2 21,2

c,effu ss,min

s lim s l

Aw E 2 min[c ;c ]A

1,7 A ks A

w E A A2 min[c ;c ] A

1,7 ks

p p0 x px q; x ( ) q (32 2A 1,71,7 2 min[c ;c ]

0 A ²E w E w

1)

φ⋅ ⋅β⋅ ⋅ ⋅β= + ⋅

⋅

⋅ ⋅ φ ⋅ ⋅β= ⋅ ⋅β ⋅ +

→ = + + = − ± −

φ⋅ ⋅β ⋅⋅ ⋅ ⋅β→ = − −

⋅ ⋅ im

s,min,1

s,min,2

ks

A 1,7 2A

⋅

⋅= + u smin[c ;c ]

2⋅ ⋅β

s lim

1,7 2E w

⋅± −

⋅ ⋅u smin[c ;c ]

2⋅ ⋅β

2c,eff

s lims lim

2c,effu s u s

s,mins lim s lim s lim

u s

s lim

ct ,min

5( )10

A 1,7E w ksE w

A 1,71,7 min[c ;c ] 1,7 min[c ;c ]A (32)

E w E w E w ks

1,7 min[c ;c ] 1,7 miE w

==

⎛ ⎞ φ ⋅ ⋅β ⋅+⎜ ⎟⎜ ⎟ ⋅ ⋅⋅ ⋅⎝ ⎠

φ ⋅ ⋅β ⋅⎛ ⎞⋅ ⋅β ⋅ ⋅β= + + +⎜ ⎟⋅ ⋅ ⋅ ⋅⎝ ⎠

⋅ ⋅β ⋅+ +

⋅ρ =

pure tension(bending)

2c,effu s

s lim s lim

c ct

5( )10

A 1,7n[c ;c ]E w E w ks

(33)k k A

==

φ ⋅ ⋅β ⋅⎛ ⎞⋅β+⎜ ⎟⋅ ⋅ ⋅⎝ ⎠

⋅ ⋅


b) In case primary cracking occurs but the steel stress is too small to activate secondary cracking (that will be most likely if ,) it is: s,cr,I s,cr ,II1,3σ < ⋅σ

t s,cr,Ik u s 5( )

s10 ct ,min

c ct t ct c ctlim u s

s,min s s,min

lim su s t

(1 k )w 1,7 2 min[c ;c ] (

Eks

k k A (1 k ) f (t) k k Aw 1,7 2 min[c ;c ]

ks A E A

w E2 min[c ;c ] (1 k )

1,7

==

⎛ ⎞ − ⋅σφ= ⋅ ⋅ + ⋅⎜ ⎟⎜ ⎟⋅ρ⎝ ⎠

⎛ ⎞φ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅= ⋅ ⋅ + ⋅⎜ ⎟⎜ ⎟⋅ ⋅⎝ ⎠

⋅= ⋅ ⋅ − ⋅


34)

2

c ct t ct c ctct

s,min s,min

2

2lim s t ctu s t ct ct,min

ct ,min ct ,min

2lim s ct ,min

u s t c

k k A (1 k ) f (t) k k Af (t) (35)

A ks A

w E (1 k ) f (t)1 12 min[c ;c ] (1 k ) f (t)1,7 ks

w E2 min[c ;c ] (1 k ) f

1,7

⎛ ⎞⋅ ⋅ φ ⋅ − ⋅ ⋅ ⋅⋅ + ⋅⎜ ⎟⎜ ⎟

⎝ ⎠

⎛ ⎞⋅ φ ⋅ − ⋅= ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ρ⎜ ⎟⎜ ⎟ρ ρ⎝ ⎠

⋅ ⋅ρ= ⋅ ⋅ − ⋅ t ct

t ct ,min(1 k ) f (t)

(t)ks

φ⋅ − ⋅⋅ρ +

2lim s ct ,min t ct

u s t ct ct ,min

2 u s t ct t ctct ,min ct ,min

lim s lim s

ct ,min,1 u

ct ,min,2

w E (1 k ) f (t)0 2 min[c ;c ] (1 k ) f (t)

1,7 ks1,7 2 min[c ;c ] (1 k ) f (t) 1,7 (1 k ) f (t)

0w E w E ks

1,7 2 min[c ;c

⋅ ⋅ρ φ ⋅ − ⋅→ = − + ⋅ ⋅ − ⋅ ⋅ρ +

⋅ ⋅ ⋅ − ⋅ ⋅φ ⋅ − ⋅→ = ρ − ⋅ρ −

⋅ ⋅ ⋅

ρ ⋅ ⋅=

ρ

(36)

2

s t ct u s t ct t ct

lim s lim s lim s

] (1 k ) f (t) 1,7 2 min[c ;c ] (1 k ) f (t) 1,7 (1 k ) f (t)(37)

2 w E 2 w E w E ks⎛ ⎞⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅φ ⋅ − ⋅

± +⎜ ⎟⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎝ ⎠

2

u s t ct u s t ct t ctct ,min

lim s lim s lim s

(1 kt) 0,6

2

u s ct u s ctct ,min

lim s lim s

1,7 min[c ;c ] (1 k ) f (t) 1,7 min[c ;c ] (1 k ) f (t) 1,7 (1 k ) f (t)(38)

w E w E w E ks

min[c ;c ] f (t) min[c ;c ] f (t)w E w E

− =

⎛ ⎞⋅ ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅φ ⋅ − ⋅ρ = + +⎜ ⎟⋅ ⋅ ⋅ ⋅⎝ ⎠

⎯⎯⎯⎯→

⎛ ⎞⋅ ⋅ρ = + +⎜ ⎟⋅ ⋅⎝ ⎠

ct

lim s

f (t)(39)

w E ksφ⋅⋅ ⋅


6

2

u s ct ct

lim s lim s

ct

lim su s ct ctct ,min

lim s lim s ct

lim s

ct

min[c ;c ] f (t) f (t)(40)

w E w E ks

f (t)1,4

w E ksmin[c ;c ] f (t) f (t)~ (

w E w E ks f (t)1,1...1, 2

w E ks

⎛ ⎞⋅ φ ⋅⎜ ⎟⋅ ⋅ ⋅⎝ ⎠

φ ⋅→

⋅ ⋅⋅ φ ⋅ρ ≈ +

⋅ ⋅ ⋅ φ ⋅→ <

⋅ ⋅

ρ

because of:

Bending: <

Pure tension: 41)

ct

lim s,min

ct

lim s

ctct ,min,EN1992 1 1

lim s

cts,min c ct

lim s

f (t)0,44

w E(42)

f (t)0, 49...0,53

w E

f (t)~ 0,5 (43)

w E

f (t)A 0,5 k k A (44)

w E

− −

φ ⋅→

⋅≈

φ⋅→ <

⋅

φ⋅⎯⎯⎯⎯→ρ

⋅

φ⋅≈ ⋅ ⋅ ⋅

⋅

simplified

Bending: <

Pure tension:

That can be easily compared to the outcome of DIN 1045-1 or MC 90:

ct ct ctct ,min,MC90

lim s lim s lim s

f (t) f (t) f (t)1~ 0, 4w 6 E 6 w E w Eφ⋅ φ ⋅ φ ⋅

ρ = ⋅ =⋅ ⋅ ⋅ ⋅

(45)

4. The influence of factor k on the difference between minimum reinforcement acc. to EN 1992-1-1 and DIN 1045-1 Because k is smaller in the DIN 1045-1 definition, one should really study its influence before applying this important reduction, especially if EN 1992-1-1 defines stronger values:

Figure 4: Differences in the definition of k-factors between EN 1992-1-1 and DIN 1045-1


7

ctc ct

s,min,EN lim s EN1992 1 1

s,min,DIN DIN1045 1ctc ct

lim s

f (t)0,5 k AA w E k

1, 25 1,25...1,3 1,56...1,62 (46)A kf (t)0,4 k A

w E

− −

−

φ ⋅⋅ ⋅

⋅= ⋅ = ⋅ =

φ⋅⋅ ⋅

⋅

probably upper limit

The own recommendation is to choose As,min acc. to the DIN/MC 90 model with k-factors as described in EN 1992-1-1.

Figure 5: The simplified and pure DIN (MC 90) model

Figure 6: The simplified DIN (MC 90) model with k from EN 1992-1-1

Scientific reading to k offers [Bergner]: There is k = kE = σbz/fctm. [Bergner] warns about using the offered reduction by the height depending factor kE with a view on Figure 7, delivering scary test results. This also means that the more conservative approach is the better.

Figure 7: Differences between the DIN-definition of k and test results acc. to [Bergner]

Bergner’s own replacement (kz,t · λBew) uses a different and more time dependent approach focussing on a time-dependent cement reduction for early cracking and the influence of the reinforcement ratio. [Röhling] gives a very detailed view on temperature and restraint related problems that helps to cover specialised and known influences on the inner stress distribution.


8

5. Notes to the factor kc: Factor kc is similar defined in both codes. In that paper we use for rectangular sections or webs

( )( ,

c( ,c *

1* ct

*

*

Ed 1**

Ed 1*

k 0,4 1 (47k h / h f (t)

h 1,0 h 1,0h 1,0 h hN 0(comp) k 1,5

2hN 0(tens) k3h

+−

⎡ ⎤σ⎢ ⎥= ⋅ −

⋅ ⋅⎢ ⎥⎣ ⎦> → =

≤ → =< → =

< → =

Ed

Ed

wenn N <0)wenn N >0)

where: m m m

)

kc =1, in case of pure tension Factor kc for flanges, subjected to tension

crc

ct ct

Fk 0,9 0,5 (48

A f (t)= ⋅ ≥

⋅)

kc is a kind of distribution factor for tensile stresses with the boundary values, multiplied by the change within the inner liver arm between state I and II (zI/zII) kc = 0,4 (Bending) theoretically: 0,5 for a triangular distribution kc = 1 (Tension theoretically: 1,0 for a rectangular distribution If real situations need calculation, mind that the distribution of tensile stresses of very high sections due to bending might differ and be less than triangular. It can be explained with fracture mechanics and size effect in background. For example, a concave parable leads to a distribution of kc = 0,33. 6. Literature EN 1992-1-1, Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings. CEN, Mai 2004 CEB-FIP Model Code 1990. CEB Bulletin d’information No. 213, Lausanne, 1993. DIN 1045-1: Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1: Bemessung und Konstruktion. DIN, Juli 2001 König, G.; Tue, N. V.: Grundlagen und Bemessungshilfen für die Rissbreitenbeschränkung im Stahl- und Spannbetonbau. DAfStb, Heft 466. Beuth-Verlag, Berlin 1996 Curbach, M.; Tue, N.; Eckfeldt, L.; Speck, K.: Beitrag: Zum Nachweis der Rissbreitenbeschränkung gemäß DIN 1045-1. In: Erläuterungen zu DIN 1045-1. Teil 2. Heft 525 des DAfStb, Beuth Verlag, Berlin 2003, S. 190 ff. Bergner, H.: Rissbreitenbeschränkung zwangbeanspruchter Bauteile aus hochfestem Normalbeton. Heft 482 des DAfStb, Beuth-Verlag 1997 Helmus, M.: Mindestbewehrung zwangbeanspruchter dicker Stahlbetonbauteile. Heft 412 des DAfStb, Beuth-Verlag 1990 Röhling, S.: Zwangspannungen infolge Hydratationswärme. Verlag Bau + Technik, Düsseldorf, 2005 Eligehausen, R., Popow, E. P. und Bertero, V. V.: Local Bond Stress-Slip Relation-ships of Deformed Bars under Generalized Excitations, Report No. UCB/EERC-83/23 Earthquake Engineering Research Center, University of California, Berkeley, Oktober 1983


9

Documents

Deduction of a Formulation for the Minimum Reinforcement for Crack