For the derivations below, the following assumptions are made: (1) a critical planar crack exists across the entire section of the tensile member (Fig. 3.13), (2) the crack is normal to the tensile stress field, (3) the contribution of the matrix along the crack is negligible, and (4) the fibers crossing the crack are in a general state of pull-out. This is typically the case when steel fibers are used in concrete.

The detailed derivation of the equation to predict the post-cracking strength of the composite is time consuming. Some background is given in [13,15,38]. Here, it suffices to say that the equation can be most generally put in the following form:

For circular fibers:

where X is the product of several coefficients:

X = X^X 2X3X5 X2 = 4^2X4

Similarly to the approach followed in Eq. (3.4), the use of a single coefficient, X, in Eq. (3.21) allows for future expansion of the equation, should new research justify the case. For instance, in the past [13, 15], the author used only X^2X3and later added X5 on the basis of new research.

The coefficients in Eqs. (3.21) and (3.22) are defined as follows:

X1 = average or expected value of the ratio of fiber shorter embedded distance from a forming crack to the length of the fiber. Its value is % as derived from probability theory considerations. It is illustrated in Fig. 3.14. The shorter embedded length is assumed to be the one that will pull-out under load.

X2 = 4X4^2; coefficient that takes into consideration orientation effect on pull-out resistance. It can be considered the efficiency factor of orientation in the cracked state.

«2 = efficiency factor of fiber orientation in the uncracked state of the composite; it is equal 1 for unidirectional fibers; 2/n = 0.636 for

L

Average pull-out length: L / 4

Fig. 3.14 Range of length of shorter pull-out segment, and average fiber pull-out length.

fibers randomly oriented in planes; and 0.5 for fibers randomly oriented in space. This factor directly influences the number of fibers intersecting a unit area of composite, be it cracked or uncracked (Eq. (3.3)).

Л3 = group reduction coefficient for bond, to simulate the fact that the bond strength resistance per fiber decreases when the number of fibers pulling out from the same area increases [41,42].

Л4 = expected value of ratio of maximum pull-out load for a fiber oriented at angle в to maximum pull-out load of same fiber aligned with the direction of pull-out. The angle в is the angle between the longitudinal axis of the fiber and the pull-out direction and varies between 0 and 90 degrees; thus (п/2 — в) represents the angle between the longitudinal axis of the fiber and the cracking plane. Л4 = 1 for fibers with longitudinal axis oriented in the direction of pull-out loading.

Л5 = reduction coefficient to account for the fact that fibers inclined

at an angle of more that about 60 degrees with the pull-out load direction contribute very little, due to spalling of the wedge of matrix at the cracking plane (Fig. 3.15). This is particularly true for stiff fibers such as steel. For aligned fibers, Л5 = 1.

Note that the two coefficients «2 and Л4 are placed together since Л4 and a2 are correlated. For instance, if a2 equals one, then Л4 also

Fig. 3.15 Wedge of matrix likely to fail under fiber pull-out at large angles of fiber inclination to the loading direction. |

equals one; that is the case of unidirectional fibers. Thus, the two coefficients are related and are both associated with orientation effects; «2 affects the number of fibers at an angle to a plane, and Л4 affects the pull-out force associated with an inclined fiber. Integration over the angle of orientation from zero to 90 degrees leads to the expected values of these coefficients.

In previous publications [13,15,29,], the author had not considered the coefficient Л5, and the coefficient, Л2, was simply termed the efficiency factor of orientation in the cracked state.

The coefficient, Л4, was termed “snubbing coefficient” by Li and Wu [11]; using analogy with a pulley, they showed that Л4 varies

theoretically between 1 and 2.32. In analyzing the experimental results of Visalvanich and Naaman [43], Li and Wu back-calculated the values of Л4 and obtained a value close to 2 for steel fibers.

The above form of Eq. (3.21) is convenient in allowing the experimental determination of a single coefficient Л from a direct tensile test [28]. Indeed, because of the many uncertainties associated with most of the above coefficients (Л = 4Л1Л3(Л4а2)Л5), one can undertake a tensile test and, assuming the bond strength г known with relative accuracy, find out the coefficient Л by comparing the test value of <7pc with the value predicted from Eq. (3.21). It is best, in such test, to

use a notched tensile prism of sufficient cross-section to represent the

real tensile member. The average bond strength, т, used in the equation could be determined from separate pull-out tests.

Note that for fibers aligned in the loading direction, Л1 = 1/4, «2 = 1, Л-2 = 4, A4 = 1, A5 = 1, A3 = A3. Thus for aligned circular fibers Eq. (3.21) becomes:

(3.23)

3.10.2.1 Non-dimensional form

Equation (3.21) can be put in a non-dimensional form as follows:

= A—— — Vf < Eq. (3.25) (3.24)

Gmu Gmu d