The current ACI Code provisions for the pullout failures of bars and anchors are based on plastic limit analysis (ACI Committee 408, 1979; ACI Committee 349, 1989). However, the size effect is very strong in this kind of failure, and considerable work has been done in the last decade to increase the understanding of anchor pullout in terms of fracture mechanics, a topic of considerable interest because it is at the base of the design of anchors and of the recently introduced nondestructive test method for concrete strength based on pullout of a headed stud (the Swedish “lok” test).
The ACI Code provision (Sec. 15.8.3) requires that the “anchor bolts and mechanical connections shall be designed to reach their design strength prior to anchorage failure or failure of surrounding concrete". This means the anchor bar must yield before fracture occurs, but this can be ensured only if the load causing the fracture is correctly predicted. ACI Committee 349 (1989) recommends that “the design pullout strength of concrete, Pd, for any anchorage shall be based on a uniform tensile stress of 4ff’c acting on an effective stress area which is defined by the projected area of stress cones radiating toward the attachment from the bearing edge of the anchors”. This gives the pullout force Pu and nominal strength
Г7Т p Г7Т
Pu — ~^txD2 , ct/vu a: —— ka\ — , u = 1 psi — 6.895kPa (10.5.3)
у о 1 7ГD – V of
in which к і = empirical constant, /’ = standard compression strength of concrete, and D = the embedment depth of the anchor bolt. This expression obviously corresponds to plastic limit analysis, the size effect being ignored.
A clear confirmation of a strong size effect in the pullout failure of reinforced concrete bars without anchors was provided by the tests of Bazant and §ener (1988); see Fig. 1.5.6 (series II). They tested microconcrete cubes of size ratio 1:2:4. The bar diameter and the embedment length were scaled so as to maintain geometric similarity. As is seen from Fig. 1.5.6, in the logarithmic size effect plot, the test results lie very close to the LEFM asymptote. This reveals an extremely high brittleness number for this type of failure. Eligehatisen and Ozbolt (1990) and Bazant, Ozbolt and Eligehausen (1994) further showed that these test results agreed closely with nonlocal finite element solutions using a realistic material model for concrete (the microplane model).
Eligehausen and Sawade (1989) proposed a L. EFM-based formula for pullout strength. This formula was written as
Pu – 2.1 s/ECh-D7’12 or су и u – 0.67 ft]pjy (10.5.4)
in which the fracture parameters appear explicitly. To avoid the explicit use of fracture parameters — which have not been measured in most of the available test series in the literature, Eligehausen et al. (1991) proposed the following formula based only on the cube compression strength:
Pu = at fjcc – D3/2 (10.5.5)
They evaluated the results of 209 pullout tests of headed anchors carried out at different laboratories. In all tests, the failure occurred by a conical crack surface. The tests were done on concretes of various strengths, and, therefore, the measured maximum loads were normalized to the cube compression strength fcc — 25 MPa, by multiplying them with the factor /25 jfcc. The normalized failure loads are plotted in Fig. 10.5.1 as a function of the effective embedment depth D, together with the fit of Eq. (10.5.5). From this fit it turns out that ci ~ 15.5 for Pu in N, fcc in MPa, and D in mm. The formula is seen to closely describe the experimental results, which means that the nominal strength almost follows LEP’M. This was confirmed numerically by Eligehausen and Ozbolt (1990) using a microplane nonlocal model: the l. EFM formula and Bazant’s size effect law differed less than 6% up to embedment length of 400 mm.
Apart from the nonlocal microplane model just mentioned, many different approaches have been used in the last decade to analyze this interesting problem. A two-dimensional LEE, VI analysis with a mixedmode crack was used by Ballarini, Shah and Keer(1985) to study the pullout of rigid anchor bolts. They used the Green’s function for a concentrated force in an elastic half space, represented the crack opening by means of dislocations and tlius reduced the problem to a system of singular integral equations, whose numerical solution yielded the mixed-mode stress intensity factor. The calculations provided the crack profdes and crack growth. Stability checks were made and the results were compared with anchor pullout experiments. An interesting point was that if the support reactions are sufficiently removed from the axis of the anchor, crack propagation becomes unstable and the load capacity is reduced (this is obviously due to the higher stored energy when the support reactions act farther away). On the other hand, for support reactions close to the anchor axis, as well as for sufficiently deep embedments, the crack propagation was found to be stable.
The pullout of circular disc-shaped anchors was studied by Elfgren, Ohlsson, and Gylltoft (1989). They used the finite element discrete crack approach, in which the tensile and shear softening were taken into account according to the formulation of Gylltoft (1984). They studied straight cracks inclined by 45° and 67° from the pullout axis as well as a crack starting at angles 73° and curving according to the principal tensile stress direction. They found the lowest pullout strength to occur for the 45° straight crack. They did not study the size effect, nor the effect of geometry.
The plane stress and axisymmetric problems of anchor pullout were analyzed by numerous researchers in a recent round-robin contest (Elfgren 1990) using various numerical procedures based on various fracture mechanics models, from LEFM to lattice models. Elfgren and Swartz (1992) published summaries of the contributions and a state-of-the-art report is in preparation by Elfgren, Eligehausen, and Rots.
Some of the results can be found in the proceedings of a special seminar on anchorage engineering held at Vienna Technical University in 1992 (Rossmanith, ed., 1993).