Prediction of Shear Strength for Large Anchors Considering the Prying Effect and Size Effect

The failure cracking area of concrete under tensile loading as shown in Fig. 5a can be expressed as a function of the embedment depth of anchor \( h_{ef} \). However, the cracking area of the concrete under shear loading in Fig. 5b is affected by the edge distance \( c_{1} \), diameter \( d_{o} \), and depth of failure surface \( l_{f} \). The cracking area is not uniform through the edge distance. Figure 6 shows the failure shape roughly. The shape of the failure surface is actually round and curved, but the area of the failure surface \( A_{f} \) can be defined as Eq. (2) by assuming a curved surface to a flat one as shown in Fig. 6c.where \( A_{b} \) and \( A_{s} \) are the area of the bottom and side surface, respectively, \( m_{1} \) and \( m_{2} \) are empirical constants.

The bottom area of failure surface is proportional to the square of edge distance \( c_{1} \), and the proportional constant \( m_{1} \) is a function of crack angles \( \theta_{1} \) and \( \theta_{2} \). The side area is proportional to the product of the edge distance and the depth of failure \( l_{f} \), and the constant \( m_{2} \) depends on the flexural stiffness of the anchor, which is a function of the depth of failure \( l_{f} \) and diameter \( d_{o} \) as well as the angles \( \theta_{1} \) and \( \theta_{2} \). Equation (2) can be rewritten as Eq. (3) by considering the characteristic of constants \( m_{1} \) and \( m_{2} \).where \( m_{3} \), \( m_{4} \) and \( m_{5} \) are empirical constants.

The failure surface area of concrete is actually a function of the edge distance \( c_{1} \) and depth of failure \( l_{f} \). However, Eq. (3) can be simplified as Eq. (4) which is the same form as suggested in ACI 349-13 (2015) without much error.where \( m_{6} \), \( m_{7} \), \( m_{8} \), and \( m_{9} \) are empirical constants.

The size effect can be defined as follows: The resisting force per unit area of concrete decreases while the area increases. In other words, the strength of the section is not proportional to the section area. According to fracture mechanics, the size effect is relatively small when the member size is small compared to the fracture process zone (Bazant and Planas 1997). When the member size becomes infinite, the strength decreases in proportion to the square root of the size based on linear fracture mechanics as defined in Eq. (5a), and in case of dissimilar cracking, the size effect converges toward a certain value as shown in Eq. (5b). where \( s_{e} \) is the relative strength, \( \alpha_{1} \) and \( \alpha_{2} \) are the empirical constants, and \( d \) is the characteristic length. Design codes such as fib2010 (Federation Internationale du Beton (fib) 2013) simply use Eq. (6) instead of Eq. (5) because the size of the structural members is ordinarily within a specific range.where \( \alpha_{3} \) is an empirical constant (\( \alpha_{3} \ge 2 \)).

A comparison between Eq. (5a) and Eq. (6) is shown in Fig. 7. As shown in Fig. 7, if the sizes of the specimens or members are within some range, the size effect can be simply expressed by Eq. (6) rather than Eq. (5). However, Eq. (5) more precisely predicts the size effect than Eq. (6), especially when the sizes of the members are quite different.

The failure area of the anchor for tension and shear is increased as the diameter and its effective embedment depth are increased. In case of the anchor for shear force, the failure area is more complicated than the tension because the failure does not occur through the entire embedment depth \( h_{ef} \). The failure area of both anchors is increased for large anchors, but the resisting strength of the concrete is not proportionally increased due to the size effect. The characteristic length \( d \) for the anchor under shear load can be roughly replaced by the edge distance \( c_{1} \).

As shown in Fig. 8, the reaction force \( P_{2} \) is created near the end of the anchor when the shear force is applied to the anchor. Equation (7) establishes an equilibrium of the force. where \( P_{1} \) is the force applied to the anchor, \( P_{2} \) is the reaction force, \( P_{3} \) is the resultant force of \( P_{1} \) and \( P_{2} \), i.e., the shear force to the concrete, \( h \) is the distance between \( P_{1} \) and \( P_{2} \), \( h_{1} \) is the distance between \( P_{1} \) and \( P_{3} \), \( h_{2} \) is the distance between \( P_{2} \) and \( P_{3} \), and \( m_{f} \) is load-magnification factor.

The resultant force \( P_{3} \) induces the breakout failure, which is greater than the externally applied force \( P_{1} \) as shown in Eq. (7b). The prying effect represents the generation of a reaction force \( P_{3} \) greater than the externally applied force \( P_{1} \), and \( h/h_{2} \) in Eq. (7b) is called the load-magnification factor \( m_{f} \). This load-magnification factor is related to the equilibrium distance of the reaction forces and these reactions are related to the stress distribution. The stress distribution is affected by the displacement of the anchor, a crack in the concrete and so on. Displacement of the anchor is a function of its flexural rigidity, which is proportional to \( h_{ef}^{3}{/}EI \) for a concentrated load and \( h_{ef}^{4}{/}EI \) for a uniformly distributed load. Therefore, the load-magnification factor \( m_{f} \), depending on the stress distribution due to the prying effect, can be expressed as Eq. (8).where \( m_{10} \), \( m_{11} \), and \( m_{12} \) are empirical constants.

Elastic analysis for a simple beam with spring supports is conducted to support the prying effect even though concrete is not elastic. In other words, the prying effect in concrete can be predicted through the elastic analysis. A beam used for the analysis has elastic modulus of 200,000 MPa, and the spring constant of the spring support is 20,000 N/mm. A point load is applied at the end of the beam, and reaction force at each support is obtained. One of the analytical results, the reaction force, is as shown in Fig. 8b, c. From this results, \( P_{1} \), \( P_{2} \), \( P_{3} \), and \( m_{f} \) can be found. The resultant \( P_{3} \) and the distances between the forces are different depending on the size of the anchor, as previously explained.

Load-magnification factor \( m_{f} \) depending on the embedment depth \( h_{ef} \) and diameter \( d_{o} \) is presented in Fig. 9a. As shown in Fig. 9a, the load-magnification factor is related with the embedment depth and the diameter which is closely related to flexural rigidity, and it can be explained by Eq. (8). It can be also found that the load-magnification factor has little effect on the load-magnification factor whereas the diameter does when \( h_{ef}{/}d_{o} \ge 8 \). In this case, Eq. (8) can be simplified as Eq. (9).where \( m_{13} \) and \( m_{14} \) are empirical constants.

Figure 9b shows the depth of failure \( l_{f} \) depending on the embedment depth \( h_{ef} \) and diameter \( d_{o} \) based on the elastic analysis results. Depth of failure \( l_{f} \) increases with an increase in the embedment depth \( h_{ef} \). Therefore, the depth of failure \( l_{f} \) can be written as Eq. (10) with some empirical constants.where \( m_{15} \) and \( m_{16} \) are empirical constants.

The failure depth becomes almost constant about 3.8 for \( h_{ef}{/}d_{o} \ge 8 \) in this elastic analysis as shown in Fig. 9b. This result is similar to the experimental result in Fig. 4. The front view of failed large anchors with the condition \( h_{ef}{/}d_{o} \ge 8 \) shows the depth of failure surface as \( l_{f} = 4d_{o} \) in Fig. 4.

As shown in Table 3, the existing prediction equations in ACI 349 (2015) do largely overestimate the shear strength for large size anchors, while the equations predict with a safety margin the shear strength for small size anchors. Therefore, it is necessary to develop a new prediction equation by considering the size effect and the prying effect, and also including the test results for large size anchors.

The concrete strength for the breakout failure of an anchor under shear load can be defined as Eq. (11) based on the failure area, size effect and prying effect. In other words, Eq. (11) is composed of Eqs. (3) or (4) for the failure area, concrete strength \( f_{ck} \), Eqs. (5a) or (6) for size effect, and the load-magnification factor in Eqs. (8) or (9) for the prying effect.where \( m_{17} \) and \( m_{18} \) are empirical constants.

This equation, Eq. (11) can be divided into four types depending on the decision of the failure area \( A_{f} \) and the relative strength \( s_{e} \) as in the following Eq. (12). where \( n_{1} \), \( n_{2} \), \( n_{3} \), \( n_{4} \), \( n_{5} \), \( n_{6} \), and \( n_{7} \) are empirical constants that can be defined by a regression analysis of the experimental results and the results from the literature.

For load-magnification factor \( m_{f} \) in Eq. (9) is used for the final prediction equation in order to make the equation simple and most of the experimental results are within the range of \( h_{ef}{/}d_{o} \ge 8 \). Although Eq. (10) is suggested through the elastic analyses, the actual depth of failure of the anchor is not exactly the same as in the analysis. Furthermore, the calculation of the depth of failure for every anchor is not practically appropriate. Therefore, the depth of failure \( l_{f} \) is substituted for the model equations as the same as \( l_{e} \) in the ACI model equations in which \( l_{e} = h_{ef} \) for anchors with a constant stiffness over the full length of an embedded section, and \( l_{e} \) will never exceed \( 8d_{o} \) (ACI Committee 349 2015), and empirical constants are found within this condition.

Regression analysis for identification of Eq. (12) is conducted by using not only the experimental results for large anchors but also the results for small anchors from the literature (Bailey and Burdett 1977; Klingner and Mendonca 1982; Hallowell 1996; McMackin et al. 1973; Klingner et al. 1982, 1999; Swirsky et al. 1977; Ollgaard et al. 1971; Korea Concrete Institute (KCI) 2012). For the regression analysis, however, it is necessary to accommodate some limitation because some values must have some limitations for physical phenomena. In addition, the exponent of compressive strength by regression analysis is fixed to be the same as the current value 0.5 (\( n_{6} = 0.5 \)) to maintain the form of the current equation as much as possible and to represent the tensile strength of the concrete. The exponent \( n_{8} \) for considering the load-magnification factor is fixed as 0.034 (\( n_{8} = 0.034 \)), which is from the regression of \( m_{f} \) by Eq. (9). This fixation of parameters does not considerably change the coefficient of determination (\( R^{2} \)) of the regression analysis. The final empirical constants by the regression analysis are shown in Table 4.

In this paper, the most sophisticated and simplest equation among all the types of Eq. (12), Eqs. (12a) and (12d), are finally suggested to predict the breakout shear strength of anchors because it is practically hard to use all the types of Eq. (12). Regression analysis is conducted again to reduce the significant figure of constants in Table 4 by fixing the exponents \( n_{i} \) (\( i \) = natural number) rounded off and finding the coefficients \( n_{i} \) (\( i \) = natural number) one by one. The final model equations are the following types of Eq. (13).

Equation (13b) is directly comparable with the ACI model because both have the same format. The exponent of diameter \( d_{o} \) is decreased in Eq. (13b) compared to the ACI model because it includes the experimental results of large anchors that have a prying effect that is larger than that in small anchors. The exponent of edge distance \( c_{1} \) in Eq. (13b) is similar to the ACI model even though the ACI model does not explicitly include the size effect. However, the effect of the load-bearing length is disregarded in Eq. (13) by regression because load-bearing lengths of large anchors by experiment become identical by the upper limit 8 \( d_{o} \) because the embedment depths of large anchors are deep enough. In addition, every embedment depth \( h_{ef} \) of data used for the regression is over \( 3.5 \times d_{o} \), and the load-bearing length does not change significantly on this condition as shown in Fig. 9b. This fact also leads to the neglect of the load-bearing length in Eq. (13).

A comparison between the prediction by each type of Eq. (13) and the test results is included in Fig. 10. Improvement of the suggested Eq. (13) can be seen in Fig. 11, which represents the ratio of the experimental results to the prediction by the equations according to the diameter of the anchor. The ACI model overestimates the shear strength of large size anchors as shown in Fig. 11a. The average ratio between the test and the prediction by the ACI model for anchors with small diameters of less than 50 mm is about 1.14 and it reveals that the safety margin of ACI is about 14 %. Overestimated cases are located under the dashed line in Figs. 11b and c in which the safety margin is fixed as the same as the ACI model. The suggested models reduce the portion of the overestimation, which is dangerous for the structures. In conclusion, both proposed equations are more reliable and improve the prediction for large anchors compared to the current ACI model.