Steel – Concrete Bond

1.4 Behaviour of Reinforced Concrete Sections

1.4.3 Steel – Concrete Bond

To ensure the full contribution of the steel reinforcement in a given section of reinforced concrete, before that cross section the steel has to be anchored in the concrete for a length equal to at least a given multiple of its diameter. Taking into account different values of partial safety factors, withcC/cS= 1.5/1.15 ≅1.3, for a ribbed steel bar, withf_v/sbr≅ 400/4.5 = 90, one has

1offi901:3=4ffi30/:

Such value is approximately valid for a proper coupling of the qualities of the two materials, for which higher steel strength shall be associated with a higher bond capacity. And this, as it will be shown later on, depends upon both the nature of the contact surface and the tensile strength of concrete.

The presence of hooks at the ends of the bars gives a different anchorage mechanism (see Fig.1.34) and reduces the minimum required lengthl_o. Given the developed length of the hook needed for bending the bar itself, such reduction is significant for a smooth bar, little for a ribbed one. For the latter, a straight end of equal length is equally effective.

Types of Bond

Bond between steel and concrete is due to several phenomena of a different nature.

Thefirst one ismolecular chemical adhesionthat ensures a union without slippage, but that is limited to small strength values. There is then the geometrical pene- trationdue to the roughness of the contact surfaces (see Fig. 1.35a). When forces increase effective contacts activate in a non-uniform way, thanks to small slippages that lead the surface irregularities to push one against the other. In order to enhance this phenomenon, actual interlocks can be obtained with appropriate ribs pro- truding from the reinforcing bars (see Fig.1.35b).

Friction contributions due to possibletransverse compressions(see Fig. 1.35c) can affect bond. These compressions occur to a small extent because of the concrete shrinkage. More significant is the self-anchoring phenomenon of pre-tensioned strands in pre-stressed concrete that, when released, tend to shorten exhibiting at the same time a transverse expansion. Furthermore, in certain zones direct actions can be applied, such as the flux of compressions that goes through the beams of a multi-storey frame at the columns location. There is eventually the contribution of

Fig. 1.34 End anchorage mechanism of a bar

transverse confinementwhich also has the properties of friction and is provided by transverse reinforcement or hoops with a truss behaviour (see Fig.1.35d).

In the two main bonding mechanisms described in Fig.1.35a, b, the bar pull-out occurs with pure tension failure of the surrounding concrete. One can therefore set

1b¼ /fy

4b_bfct¼/fy

4fb;

where f_b=b_bf_ct is the equivalent strength of bond and where b_b is the effective contact ratio. The values of this ratio for smooth bars are largely lower than 1, because of the limited extent of the effective contact zones with respect to the total surface. The ribs of the deformed bars increase the size of the concrete sleeve geometrically interlocked to the steel and this increases the equivalent bond strength.

Transverse compressions extend the effective contacts and at the same time they reduce, for the same longitudinal shear force, the principal tensile stress in concrete, increasing its resistance. Greater values of the ratiobbare therefore observed, even greater than 1 for ribbed bars.

Theconfinementprovided by transverse reinforcement leads to a different bond mechanism, establishing a resisting truss that, leaving the tensions to the steel reinforcement, stresses the concrete mainly with an inclinedflux of compressions.

When the transverse reinforcement is adequately proportioned and diffused, this leads to a much higher resistance, not related anymore to the pure tensile strength of concrete.

Several appropriate measures should be adopted in the detailing of reinforcement to ensure bond: First of all, an adequate limitation of bars diameters to avoid excessive anchorage lengths. As mentioned before, a consistent combination of Fig. 1.35 Types of bond of steel bars

materials qualitiesalso has to be ensured. One also has to take into account the negative effect of cracking, which causes detachments and damages of the surface of effective contact. It is preferable to anchor the bars in compression zones whenever possible. The proximity of reinforcing bars to the external concrete surface also reduces the bond strength, because of the reduced or null effectiveness of the surface layer. Therefore, bars normally have to be anchored bending their ends inwardsor with appropriate shapes. It is eventually to be noted how the rebar lapping, that is their junction by simple superimposition, implies the transfer of stressflow through concrete. Such stresses are therefore to be accurately verified and appropriate staggered laps are required, not to concentrate the disturbance causing the possible excessive weakening of the concerned section.

The values ofequivalent bond strength f_brequired for the design are deduced from specific tests. The easiest one is thepull-out test, which consists of measuring the force required to extract the reinforcing bar from a cubic concrete specimen as shown in the scheme of Fig.1.36a. More significant results, as they are more similar to the actual structural situations, are obtained from thebeam testwhere the pull-out force is measured indirectly through the bending action of a beam as shown on the scheme in Fig.1.36b.

From the tensile bond tests, with the appropriate measurement of the slippaged, diagrams similar to the one described in Fig.1.37 can be obtained. They are characterized by:

• stage OA without significant slippage up to the failure of the chemical adhesion;

• stage AB with progressive activation, thanks to initial slips, of the effective contacts and initiation of microcracking at the concrete interlocks;

• stage BC with progressive failure of the concrete interlocks up to failure limit of bond;

• stage CD measurable only with tests under displacement control, decreasing up to complete detachment of the steel bar.

Fig. 1.36 Pull-out (a) and beam (b) tests for bond measurement

For ribbed reinforcing bars the obtained values, expressed as a function of the characteristic strength of concrete, are given by

f_bk¼2:25f_ctk;

valid for diameters/ 32 mm. The design strength value isfinally obtained from f_bd=f_bk/cC.

RIBBED BARS

SMOOTH BARS Fig. 1.37 Bond–slip

experimental diagrams

Appendix: Characteristics of Materials Table 1.1: Hardening Curves of Concrete

The following table shows the values of the ratiosf_cj/f_cbetween the strength at time tfrom casting and the strength at 28 days, where values deduced from the following formula:

fcj

fc ¼e^b11=pffiffi_s

ð Þ;

and the values of the analogous ratioE_cj/E_cbetween elastic moduli, values deduced from the following formula:

E_cj

Ec ¼ e^b11=pffiffi_s

ð Þ

h i0:3

withs¼t=28;

wheretis expressed in days (t= 0.58 corresponds to about 14 h of ageing, time of possible demoulding of precast elements).

Age Strengths Moduli

Accelerated curing (indicative values)

Concrete Accelerated

curing (indicative values)

Concrete Fast

setting

Normal setting

Slow setting

Fast setting

Normal setting

Slow setting Days b= 0.08 b= 0.20 b= 0.25 b= 0.38 b= 0.08 b= 0.20 b= 0.25 b= 0.38

0.58 0.62 0.30 0.23 0.10 0.87 0.70 0.64 0.51

1 0.71 0.42 0.34 0.20 0.90 0.77 0.72 0.61

2 0.80 0.58 0.50 0.35 0.94 0.85 0.81 0.73

3 0.85 0.66 0.60 0.46 0.95 0.88 0.86 0.79

4 0.88 0.72 0.66 0.54 0.96 0.91 0.88 0.83

5 0.90 0.76 0.71 0.59 0.97 0.92 0.90 0.86

6 0.91 0.79 0.75 0.64 0.97 0.93 0.92 0.88

7 0.92 0.82 0.78 0.68 0.98 0.94 0.93 0.89

10 0.95 0.87 0.85 0.77 0.98 0.96 0.95 0.93

14 0.97 0.92 0.90 0.85 0.99 0.98 0.97 0.95

21 0.99 0.97 0.96 0.94 1.00 0.99 0.99 0.98

28 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

60 1.03 1.07 1.08 1.13 1.01 1.02 1.02 1.04

90 1.04 1.09 1.12 1.18 1.01 1.03 1.03 1.05

180 1.05 1.13 1.16 1.26 1.01 1.04 1.05 1.07

365 1.06 1.16 1.20 1.32 1.02 1.04 1.06 1.09

∞ 1.08 1.22 1.28 1.46 1.02 1.06 1.08 1.12

Table 1.2: Strength Classes of Concrete

The following tables show the strength and deformation parameters for different codified classes of concrete, of ordinary and controlled classes. Classes are char- acterized by characteristic values of cylinder and cubic strengths. Cylinder strength f_c, cubic strengthR_c, tensile strengthf_ctand elastic modulusE_care reported in the consecutive columns, indicating the mean values with subscript mand the char- acteristic values with subscriptk. Data are expressed in MPa and are calculated with the following formulas:

Rcm¼fcm=0:83 fctm¼0:27 ffiffiffiffiffi f_m² p3

forfcm58 fctm¼2:12 ln 1½ þðfcm=10Þ forfcm[58 Ecm¼22;000½fcm=10^0:3 E_cm ¼Ecm=1000

:

In design previsions it is assumedfcm¼fkþDf, withDf ¼8 MPa for ordinary production (common construction sites) and withDf ¼5 MPa for controlled pro- duction (prefabrication plants). For the two types of production, it is assumed respectivelyfctk¼0:7fctm and fctk¼0:8fctm.Table 1.2a

Class Ordinary production—Df ¼8 MPa

Class f_ck f_cm R_cm f_ctm f_ctk E_cm

Low C16/20 16 24 29 2.2 1.6 29

C20/25 20 28 34 2.5 1.7 30

C25/30 25 33 40 2.8 1.9 31

Medium C30/37 30 38 46 3.1 2.1 33

C35/43 35 43 52 3.3 2.3 34

C40/50 40 48 58 3.6 2.5 35

C45/55 45 53 64 3.8 2.7 36

Table 1.2b

Class Controlled production—Df ¼5 MPa

Class f_ck f_cm R_cm f_ctm f_ctk E_cm

Medium C30/37 30 35 42 2.9 2.3 32

C35/43 35 40 48 3.2 2.5 33

C40/50 40 45 54 3.4 2.7 35

C45/55 45 50 60 3.7 2.9 36

High C50/60 50 55 66 3.9 3.1 37

C55/67 55 60 72 4.1 3.3 38

C60/75 60 65 78 4.3 3.4 39

C70/85 70 75 90 4.5 3.6 40

Table 1.3: Deformation Parameters of Concretes

The following table shows the values of the main mechanical characteristics of concrete calculated as a function of the compressive strength with the formulas specified below:

Ec¼22000½fc=10^0:3 E_c¼Ec=1000 j¼1:05Ecec1=fc

ec1¼0:7f_c^0:3110³2:810³ e_c1¼1000ec1

ecu¼n2:8þ27 98½ð f_cÞ=100⁴o

10³3:510³ e_cu¼1000ecu

f_ct¼0:27 ffiffiffiffi f_c² p3

forf_c58 f_ct¼2:12 ln 1½ þðf_c=10Þ forf_c[58 at¼f_ct=f_c

jt¼1:05E_cect1=f_ct ðect1¼0:00015Þ:

Such values are to be used in the constitutive modelsr–eof concrete in com- pression and tension, respectively, expressed in the following form:

r¼ jgg²

1þ ðj2Þgfc ðg¼e=ec1Þ

r¼ jtg_t ð2jt3Þg²t þ ðjt2Þg³t

atf_c ðg_t¼e=ect1Þ:

Stresses and elastic moduli are expressed in MPa.

The other deformation characteristics are

• Poisson’s raiov= 0.20

• coefficient of thermal expansion aT¼1:010⁵C¹:

f_c E_c j e_c1 e_cu f_ct at jt

24 28.6 2.35 1.90 3.50 2.25 0.094 2.01

28 30.0 2.21 2.00 3.50 2.49 0.089 1.90

33 31.5 2.07 2.10 3.50 2.78 0.084 1.78

38 32.8 1.96 2.20 3.50 3.05 0.080 1.69

43 34.1 1.87 2.20 3.50 3.31 0.077 1.62

48 35.2 1.79 2.30 3.50 3.57 0.074 1.56

53 36.3 1.72 2.40 3.50 3.81 0.072 1.50

35 32.0 2.03 2.10 3.50 2.89 0.083 1.75

40 33.3 1.92 2.20 3.50 3.16 0.079 1.66

45 34.5 1.84 2.30 3.50 3.42 0.076 1.59

(continued)

(continued)

fc E_c j e_c1 e_cu fct at jt

50 35.7 1.76 2.40 3.50 3.66 0.073 1.53

55 36.7 1.70 2.40 3.50 3.90 0.071 1.48

60 37.7 1.64 2.50 3.36 4.13 0.069 1.44

65 38.6 1.59 2.60 3.12 4.27 0.066 1.42

75 40.3 1.50 2.70 2.88 4.54 0.060 1.40

Table 1.4: Drying Shrinkage of Concrete

Drying shrinkage is given by

ecdðt⁰Þ ¼ecd1gsðt⁰Þ;

where t′ is time expressed in days and measured starting from the onset of the phenomenon.

The following tables show the final value of the drying shrinkage e_cd∞ for different relative humidities h of the ageing environment, for different strength classescof concrete and for different equivalent thicknessess. Values are deduced from the following formula:

ecd1 ¼k_secdo; with

ks¼0:7þ0:0094ð5sÞ^2:5 fors\5

ks¼0:7 fors\5

edo¼870 1ð h³Þe^0:12c10⁶; where

h¼RH=100 relative humidity ratio;

c¼fcm=10 mean strength in kN=cm²; s¼2Ac=u

100

equivalent thickness in dm;

(Ac= cross-sectional area in mm²;u = perimeter of the section in mm).

The ones reported in the tables are mean values, for cement of classNand for water/cement ratio 0.55, with coefficient of variation of about 0.30. For higher water/cement ratios, shrinkage is greater. For underwater ageinge_cd∞= 0.00 can be assumed.

Table 1.4a: Values ofe_cd1¼1000e_cd1for RH = 50%

Class f_cm(MPa) Equivalent thicknesses in mm

50 100 150 300 500

C16/20 24 0.63 0.57 0.52 0.43 0.40

C20/25 28 0.60 0.54 0.50 0.41 0.38

C25/30 33 0.57 0.51 0.47 0.39 0.36

C30/37 38 0.53 0.48 0.44 0.36 0.34

C35/43 43 0.50 0.45 0.42 0.34 0.32

C40/50 48 0.47 0.43 0.39 0.32 0.30

C45/55 53 0.44 0.40 0.37 0.30 0.28

C30/37 35 0.55 0.50 0.46 0.38 0.35

C35/43 40 0.52 0.47 0.43 0.35 0.33

C40/50 45 0.49 0.44 0.41 0.33 0.31

C45/55 50 0.46 0.42 0.38 0.31 0.29

C50/60 55 0.43 0.39 0.36 0.30 0.28

C55/67 60 0.41 0.37 0.34 0.28 0.26

C60/75 65 0.39 0.35 0.32 0.26 0.24

C70/85 75 0.34 0.31 0.28 0.23 0.22

Table 1.4b: Values ofe_cd1¼1000ecd1for RH = 60%

Class f_cm(MPa) Equivalent thicknesses in mm

50 100 150 300 500

C16/20 24 0.56 0.51 0.47 0.39 0.36

C20/25 28 0.54 0.49 0.45 0.37 0.34

C25/30 33 0.51 0.46 0.42 0.35 0.32

C30/37 38 0.48 0.43 0.40 0.33 0.30

C35/43 43 0.45 0.41 0.37 0.31 0.28

C40/50 48 0.42 0.38 0.35 0.29 0.27

C45/55 53 0.40 0.36 0.33 0.27 0.25

C30/37 35 0.49 0.45 0.41 0.34 0.31

C35/43 40 0.47 0.42 0.39 0.32 0.30

C40/50 45 0.44 0.40 0.36 0.30 0.28

C45/55 50 0.41 0.37 0.34 0.28 0.26

C50/60 55 0.39 0.35 0.32 0.27 0.25

C55/67 60 0.37 0.33 0.30 0.25 0.23

C60/75 65 0.35 0.31 0.29 0.24 0.22

C70/85 75 0.31 0.28 0.25 0.21 0.19

Table 1.4c: Values ofe_cd1¼1000ecd1for RH = 70%

Class f_cm(MPa) Equivalent thicknesses in mm

50 100 150 300 500

C16/20 24 0.47 0.43 0.39 0.32 0.30

C20/25 28 0.45 0.41 0.37 0.31 0.29

C25/30 33 0.42 0.38 0.35 0.29 0.27

C30/37 38 0.40 0.36 0.33 0.27 0.25

C35/43 43 0.38 0.34 0.31 0.26 0.24

C40/50 48 0.35 0.32 0.29 0.24 0.22

C45/55 53 0.33 0.30 0.28 0.23 0.21

C30/37 35 0.41 0.38 0.34 0.28 0.26

C35/43 40 0.39 0.35 0.32 0.27 0.25

C40/50 45 0.37 0.33 0.30 0.25 0.23

C45/55 50 0.35 0.31 0.29 0.24 0.22

C50/60 55 0.33 0.30 0.27 0.22 0.21

C55/67 60 0.31 0.28 0.25 0.21 0.19

C60/75 65 0.29 0.26 0.24 0.20 0.18

C70/85 75 0.26 0.23 0.21 0.18 0.16

Table 1.4d: Values ofe_cd1¼1000e_cd1for RH = 80%

Class f_cm(MPa) Equivalent thicknesses in mm

50 100 150 300 500

C16/20 24 0.35 0.32 0.29 0.24 0.22

C20/25 28 0.33 0.30 0.28 0.23 0.21

C25/30 33 0.32 0.29 0.26 0.22 0.20

C30/37 38 0.30 0.27 0.25 0.20 0.19

C35/43 43 0.28 0.25 0.23 0.19 0.18

C40/50 48 0.26 0.24 0.22 0.18 0.17

C45/55 53 0.25 0.22 0.21 0.17 0.16

C30/37 35 0.31 0.28 0.26 0.21 0.20

C35/43 40 0.29 0.26 0.24 0.20 0.18

C40/50 45 0.27 0.25 0.23 0.19 0.17

C45/55 50 0.26 0.23 0.21 0.18 0.16

C50/60 55 0.24 0.22 0.20 0.17 0.15

C55/67 60 0.23 0.21 0.19 0.16 0.14

C60/75 65 0.21 0.19 0.18 0.15 0.14

C70/85 75 0.19 0.17 0.16 0.13 0.12

Table 1.5: Drying Shrinkage Curves of Concrete

The following table shows the values of the functiongs(t′) which expresses the time law of drying shrinkage for different values of the equivalent thickness 2Ac/ u(A_c= cross-sectional area of concrete;u = its perimeter).

Age 2A_c/u(mm) Small thickness

Medium– small

Medium thickness

Medium– large

Large thickness

Days 50 100 150 300 600

0.58 0.00 0.00 0.00 0.00 0.00

1 0.23 0.10 0.05 0.02 0.01

2 0.50 0.26 0.16 0.06 0.02

3 0.63 0.38 0.25 0.10 0.04

4 0.71 0.46 0.32 0.14 0.05

5 0.76 0.52 0.38 0.18 0.07

6 0.79 0.58 0.42 0.21 0.08

7 0.82 0.62 0.47 0.24 0.10

10 0.87 0.70 0.56 0.31 0.14

14 0.90 0.77 0.65 0.39 0.19

21 0.94 0.84 0.74 0.50 0.26

28 0.95 0.87 0.79 0.57 0.32

60 0.98 0.94 0.89 0.74 0.50

90 0.98 0.96 0.92 0.81 0.60

180 0.99 0.98 0.96 0.90 0.75

365 1.00 0.99 0.98 0.95 0.86

∞ 1.00 1.00 1.00 1.00 1.00

The onset of the phenomenon is assumed at 14 h from casting (t′ =t–0.58).

The values are calculated with the following formula:

g_s¼ t⁰ t⁰þ4 ffiffiffiffi

s³

p withs¼2A_c=u 100 : For the calculation of shrinkage at timetit can be set as

ecd¼ecd1g_s; whereecd1 is deduced from Table1.4.

Table 1.6: Autogenous Shrinkage of Concrete

Autogenous shrinkage is given by

ecaðtÞ ¼eca1gaðtÞ;

wheretis the concrete age expressed in days.

The following table shows the final value of autogenous shrinkage e_ca∞ for different mean strengthsf_cmof concrete. The values are deduced from the following formula:

eca1¼2:5ðfcm18Þ 10⁶ (in tablee_ca1¼1000eca1).

Ordinary f_cm(MPa) e_ca1

Class

C16/20 24 0.02

C20/25 28 0.03

C25/30 33 0.04

C30/37 38 0.05

C35/43 43 0.06

C40/50 48 0.08

C45/55 53 0.09

Controlled f_cm(MPa) e_ca1

Class

C30/37 35 0.04

C35/43 40 0.06

C40/50 45 0.07

C45/55 50 0.08

C50/60 55 0.09

C55/67 60 0.11

C60/75 65 0.12

C70/85 75 0.14

Table 1.7: Autogenous Shrinkage Curves of Concrete

The following table shows the value of the functionga(t) that expresses the time law of autogenous shrinkage. The values are calculated with the following formula:

ga¼1e^0:2pffi_t

;

wheretis the concrete age expressed in days starting from casting.

Age g_a

0.58 0.14

1 0.18

2 0.25

3 0.29

4 0.33

5 0.36

6 0.39

7 0.41

10 0.47

14 0.53

21 0.60

28 0.65

60 0.79

90 0.85

180 0.93

365 0.98

∞ 1.00