For anchor bars provided with ribs, indentations or threads a tensile force applied to the bar is transferred along the steel/concrete interface to the surrounding concrete by bond. This is in general a favourable way to anchor connection details, since the tensile force is transferred successively along the anchorage length and high stress concentrations can be avoided.
No anchorage is perfectly rigid, but the bond transfer results in a certain slip between the anchor bar and the surrounding concrete. It should be noted that the bond stresses along the steel/concrete interface are not normally uniformly distributed and, accordingly, the slip varies along the anchorage length. This means that the slip at the loaded end of the bar exceeds the slip at the passive end. It can even be the case that the active end has a slip, while the passive end is still firmly fixed without any slip at all. Accordingly, the anchor bar should not be regarded as a rigid body.
For an anchor bar subjected to a tensile force N that increases step by step, typical anchorage
behaviour is illustrated in Fig. 7-12. At the loaded end of the anchor bar, the whole tensile force is resisted by the bar only. Along the steel/concrete interface the tensile force is transferred to the surrounding concrete by bond stresses #b that decreases along the bar. For a small tensile force, see Fig. 7-12 a, bond stresses appear only within a limited length that is shorter than the anchor bar. This ‘contributing’ length is referred to as the ‘transmission length’. The bond stresses are distributed along the transmission length with the maximum value near the loaded end and approach zero at the end of the transmission length. In every section within the transmission length the steel strain exceeds the strain of the surrounding concrete, which means that the bar must elongate and slip in relation to the concrete. The total elongation of the steel bar in relation to the concrete can be recognised as the slip at the loaded end, the so-called ‘end-slip’. In the actual case the active end of the bar has a certain slip, but the passive end of the bar has no slip at all.
#b(x) Fs2 Fs1 $b(x) #b(x) $b(x) #b(x) a) b)
Fig. 7-12: Typical anchorage behaviour of an anchor bar loaded in tension and typical distributions of steel stress and bond stress for a) a small tensile force, b) an intermediate tensile force. Dotted lines indicate possible effect of local concrete failure near the free edge
In Fig. 7-12 b the tensile force has increased and bond stresses appear along the entire anchor bar. This means that the whole anchor bar moves in the concrete, but the slip is greater at the loaded end. It should be noted that while the steel stress always becomes zero at the passive end of the bar, there could still be a considerable bond stress at this end. Its actual value depends on the slip that has occurred at the passive end. When the tensile force increases even more, it results in an increase of the average bond stress and of the overall slip. The bond stress distribution also becomes more uniform.
In design for the ultimate limit state it is generally assumed that the bond stress in each section along the anchorage length reaches the bond strength. However, this assumption of a uniform bond
stress distribution is a simplification of the real behaviour. A uniform bond stress distribution means that the tensile force and the steel stress increase linearly along the transmission length. Various failure modes are possible depending on the actual detailing and material properties.
The failure can take place in the concrete or in the steel. In case of small concrete covers, the anchorage can fail due to splitting of the concrete, see Fig. 7-9 a. If the concrete cover is sufficient to prevent a splitting failure, the anchor bar can loose its grip in the concrete by a shear failure that develops along the interface starting from the loaded end. This failure mode is referred to as a pullout failure, see Fig. 7-9 b. It is generally assumed that a concrete cover greater than about 3 times the bar dimension is sufficient to prevent a splitting failure. In CEB-FIP Model Code 1990 %CEB-FIP (1992) it is stated that the anchorage condition can be considered as ‘well confined’, when the concrete cover is 5 times the bar dimension or greater.
In case of a short anchorage length, a pullout failure can occur before yielding of the steel is reached. Ordinary design rules prevent that this would be the case. However, a pullout failure can also occur in the post-yield stage before the steel reaches its ultimate strength due to rupture of the steel. In ordinary design the steel strength is based on the yield strength, but in cases where ductility is important the connection detail should be designed so that the tensile capacity at rupture can be safely anchored. In that case the plastic behaviour of the bar can be fully utilized.
Near the free edge, inclined cracks starting from the ribs of the anchor bar develop towards the edge and may cause a local concrete cone failure as indicated in Fig. 7-13. Such local cone failures have been observed in tests, see Fig. 7-15. The depth of the cone have been about 2!, where ! = the anchored bar diameter. When these cracks appear, the bond will be reduced as shown in Fig. 7-13 a. At larger end-slips, when the concrete cone is separated from the concrete element, the bond will be totally lost near the edge, Fig. 7-13 b.
reduced bond
stresses ' 2!&r
b=
##b=00
Fig. 7-13: Local bond failure near the free edge because of inclined cracks, a) reduced bond at an early stage of cracking, b) loss of bond due to local concrete cone failure
Fig. 7-14 shows results from pullout tests on anchor bars of various lengths anchored in well- confined concrete (concrete cover not less than 12 times the bar diameter) %Engström et al. (1998) . The anchor bars were !16 mm ribbed hot-rolled bars with characteristic yield strength of 500 MPa. According to tensile tests on samples the average yield capacity was 114 kN and the average ultimate tensile capacity (at steel rupture) was 130 kN.
Influence of embedment length 0 20 40 60 80 100 120 140 0 100 200 300 400 500 Position x [mm] Tensile force [kN] N220 N290a N500 H90b H170 H210 H250 tensile capacity yield capacity
Fig. 7-14: Results from pullout tests on anchor bars anchored in well-confined concrete, according to Engström et al. (1998). Tensile force variation along anchor bars of various lengths in two concrete types just before failure
The figure shows how the tensile force varied along the anchor bar at the maximum load just before failure. Three bars were anchored in normal strength concrete (mean compressive strength about 25 – 30 MPa) and four bars were anchored in high strength concrete (mean compressive strength about 100 – 105 MPa). It appears from the figure that in normal strength concrete an anchorage length of either 220 mm or 290 mm was insufficient to prevent pullout failure before yielding of the bar, while an anchorage length of 500 mm was sufficient to anchor not only the yield capacity but also the ultimate tensile capacity. When this bar ruptured yielding had been reached within a length of about 100 mm from the loaded end. This was the final extension of the plastic zone (yield penetration). For bars anchored in high strength concrete an anchorage length of 90 mm was insufficient to avoid pullout failure. When the anchorage length was 170 mm or 210 mm, pullout failure occurred in the post-yield stage. An anchorage length of 250 mm resulted in rupture of the steel bar. The yield penetration was in that case about 70 mm.
In all of these pullout tests a local concrete cone failure, see Fig. 7-13 b, occurred near the loaded end of the bar. Of a total of seven tests in normal strength concrete the depth of the failure cone varied between 18 and 35 mm with an average value of 26,4 mm, which corresponds to 1,65 times the bar diameter. For seven tests in high-strength concrete specimens the depth of the failure cone varied between 16 and 30 mm with an average value of 27, 0 mm, or 1,68 times the bar diameter.
The response of an anchor bar can be characterised by the relation between tensile force and end- slip. By combining models for the material behaviours, equilibrium and deformation conditions a differential equation that governs the anchorage behaviour can be established, see for instance the CEB-FIP Model Code 1990. If the local relationship between bond stress and slip is known and can be mathematically formulated, it is possible to solve this differential equation analytically and the response can be determined. Bond stress-slip relations are presented in Section 7.2.2 and prediction of the response is shown in Section 7.2.3.
Fig. 7-15: Typical example of a local concrete cone failure at the loaded end, from pullout tests on anchor bars anchored in well-confined concrete %Engström et al. (1998)