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Presented at Nordic Mini-seminar: “Fibre reinforced concrete”, Trondheim, November 15 th

2007.

1(12)

Calculation of crack width and crack spacing

Ingemar Löfgren

Thomas Concrete Group

E-mail: ingemar.lofgren@tcg.nu

ABSTRACT

The present paper discusses crack propagation and special attention is given

to how the combined effect of reinforcement and fibre bridging influences

the crack spacing and width in the serviceability limit state. Two analytical

approaches, for calculating the crack spacing and crack width, are

presented. The first model is a modification of the conventional crack

spacing model presented in Eurocode 2 and is valid for the case when

cracking is caused by an external load. The second model, which is based

on a bond-slip relationship and a compatibility requirement, is valid for

cracking caused by restraint stresses. Moreover, in the paper some

examples are provided of how the models can be used.

Key words: Fibre-reinforced concrete, Cracking, Restraint, Serviceability, Shrinkage.

1. INTRODUCTION

Concrete has a low tensile strength and tensile strain capacity and cracking is initiated at a

tensile strain of about 0.1 mm/m which can be compared to the drying shrinkage of concrete of

about 0.6 to 0.8 mm/m. Hence, cracks are almost unavoidable and reinforcement is needed to

control the behaviour after cracking and to limit crack widths. Large crack widths are not

aesthetic but may also lead to accelerated reinforcement corrosion in severe environments,

leakage in water-retaining/resisting structures, insanitary conditions, or obstructions and

interruptions in production processes. Cracking may be caused by external applied forces,

imposed deformations, by shrinkage or thermal strains which are externally and/or internally

restrained, or by a combination of these. When cracking is caused by an external applied force

the crack width, if sufficient amount of reinforcement is added, will depend on the applied force.

However, if cracking is caused by an imposed deformation the force in the member depends on

the actual stiffness and the crack width on the number of cracked formed. However, most codes

do not distinguish between these two cases. Furthermore, for structures having both fibre- and

bar reinforcement there exist almost no guidelines exists for structural engineers.

2. THE CRACKING PROCESS

The cracking process differs depending on whether it is caused by an external load, imposed

deformation or restrained shrinkage, see Figure 1. When cracking is caused by an external load

the reinforcement is usually designed such that it is able to transfer the load after cracking

without yielding. For this case the load will cause an immediate cracking process where several

cracks are formed and which are relatively uniformly distributed. For this type of situation the

standard method in Eurocode 2 can be used to determine the minimum reinforcement and for

estimating the crack spacing and crack width. For a member with combined reinforcement

mailto:ingemar.lofgren@tcg.nu

Presented at Nordic Mini-seminar: “Fibre reinforced concrete”, Trondheim, November 15 th

2007.

2(12)

(fibre- and bar reinforcement) this approach has to be modified. When the cracking is caused by

an imposed deformation a different behaviour can be observed. When a crack is formed this is

accompanied by a sudden drop in the force N and the stiffness of the element also decreases. For

a new crack to be formed the deformation has to be increased so that the force N again reach the

critical value (N > Ncr). However, the force depends on the stiffness of the member and if this is

low a large deformation may be required before a new crack can be formed, compare (b-1) and

(b-2) in Figure 1, and this results in fewer but larger cracks. For this type of cracking process the

standard approach for determining crack spacing and crack width cannot be used.

N N

u Crack

u u

N N

(a) (b-1)

Stadium II

(neglecting tension

stiffening)

Tension

stiffening Ncr Ncr

u

N

(b-2)

Ncr

Small reinforcement

ratio

Force Imposed deformation

Large reinforcement

ratio

Imposed deformation

Figure 1. A reinforced concrete member subjected to: (a) axial force; (b) imposed

deformation, (b-1) with a large reinforcement ratio and (b-2) with a small

reinforcement ratio. Based on Ghali et al [1].

Compared to plain concrete (i.e. without fibres) fibre-reinforced concrete exhibits the ability to

transfer tensile stresses also after cracking, see Figure 2. This material property is referred to as

the residual tensile strength or, for describing the whole curve, the stress-crack opening

relationship (-w relationship). The residual tensile strength increases with increased fibre

dosage but is also influenced by the type of fibre (e.g. slenderness, geometry, material, etc.)

fct

l

w FRC

Concrete

w l

l

wc 0.3 mm wc = lf / 2 w 0.05 mm

Fibre

contribution Residual tensile

stress

Figure 2. Schematic description of the fracture behaviour of fibre-reinforced concrete (FRC).

Presented at Nordic Mini-seminar: “Fibre reinforced concrete”, Trondheim, November 15 th

2007.

3(12)

3. FORCE INDUCED CRACKING

The crack spacing in reinforced concrete structures (without fibres) can be calculated using the

following expression presented in Eurocode 2:

effs

r kkkcks ,

4213max.

[mm] (1)

where:

c is the concrete cover

is the bar diameter

s,ff is the effective reinforcement ratio, effcseffs AA ,, and Ac,eff is the effective area of

concrete in tension surrounding the reinforcement

k1 = 0.8 for high bond bars and 1.6 for bars with an effectively plain surface

k2 = 0.5 for bending, 1.0 for pure tension or 121 2 for eccentric tension k3 = 3.4

k4 = 0.425

For a section with combined reinforcement a similar expression, which takes into account the

contribution from the fibre reinforcement, can be derived. Consider a reinforced tension rod

loaded with the crack load, Ncr, according to Figure 3. The rod is reinforced with a centrally

placed reinforcement bar, with an area of As, and fibres. The force equilibrium in the region

between two cracks with the maximum crack distance sr,max = 2lt,max is analysed, see Figure 3.

Ncr Ncr

Stress acting on the concrete

Stress introduced to concrete

through bond, c (x)

Residual tensile strength, fft.res(w)

Total concrete tensile stress, ct (x,w)

lt,max

ct fct

Possible location of new crack

New crack

Crack Crack

fft.res bm

ct fctm

Ac

As

lt,max

sr,max

lt,max

0.5 sr,max

Figure 3. Equilibrium of forces for a tension rod.

At the crack the fibre reinforced concrete transfers a stress fft.res. At the midpoint between the

two cracks the concrete is about to crack and the stress is thus ct fctm. The increase of stress is

a result of stresses being transferred from the reinforcement to the concrete through bond. The

Presented at Nordic Mini-seminar: “Fibre reinforced concrete”, Trondheim, November 15 th

2007.

4(12)

bond stress b varies along the transmission length and has an average value of bm which can be

calculated as:

maxt,

l

b

bm l

dxx maxt

,

0 )(

(2)

If the tension rod is cut in the middle between the two cracks and along the interface between

the reinforcement and concrete the following equilibrium condition can be formulated:

cctmcresftmaxr,bm AfAfs .)5.0( (3)

The concrete gross cross-sectional area can be formulated as:

s

s

s

c sc

A

A

A AA

(4)

with s = reinforcement ratio

Inserted in (3) gives

resftctm s

maxr,bm ffs .

2

4 )5.0(

(6)

sbm

resftctm

maxr,

ff s

.

2

1 (7)

The minimum crack spacing is equal to half the maximum crack spacing. Accordingly, the

minimum crack spacing can be calculated as:

sbm

resftctm

minr,

ff s

.

4

1 (8)

The average crack spacing during the crack formation can be estimated as the average value of

(7) and (8) which gives (in Eurocode 2 it is assumed that sr,max = 1.7×s