Predicting the pullout response of inclined straight steel fibers

Abstract

Fiber pullout tests have been used for decades to characterize and optimize bond strength on fiber reinforced concretes. However most of the investigations focus on the behavior of fibers aligned with load direction whose pullout mechanisms are not representative of the ones existing in real applications, where random orientation of fibers is likely to occur. In this paper a new predictive model for the pullout response of steel fibers embedded in cement matrices with any inclination respect to loading direction is provided. Comparisons with experimental results highlight the capacity of the model on describing appropriately the entire load–crack width behavior. The procedure differentiates itself from previous works by introducing clear and comprehensive concepts within a straightforward approach.

Appendix 1: Definition of the matrix spalled length (L SP1)

Spalling of the matrix is a consequence of an extremely complex phenomenon in which failure of the matrix occurs due to local curvature and stretching of the fiber at the matrix cracked surface. However, depending upon a variety of parameters (fiber diameter and embedded length, fiber elastic modulus and tensile strength, magnitude and rate of external loading, etc.) the action imposed over the matrix wedge might change considerably. Moreover, regarding the magnitude of L _SP1 in steel fiber reinforced cementitious matrices (up to few mm) uncertainties associated to the microstructure of the matrix might play a major effect.

To quantify the average matrix spalled length a simplified failure criterion is herein proposed taking into account the resisting mechanism provided by the matrix at the cracked surface (R _SP1) and the spalling force imposed by fiber curvature (F _SP1). Hence L _SP1 represents the minimum length along fiber main axis at which the matrix wedge stabilizes, such as defined by Eq. 23.

$$ R_{{{\text{SP}}1}} \ge F_{{{\text{SP}}1}} $$

(23)

The resisting mechanism provided by the matrix (R _SP1) is based on the assumption that the tensile strength of the matrix (f _ctm) is the major parameter controlling resistance against spalling. Therefore R _SP1 becomes defined as following:

$$ R_{{{\text{SP}}1}} = A_{{{\text{SP}}1}} f_{\text{ctm}} $$

(24)

where A _SP1 is the thorough surface failure of the matrix wedge (Fig. 17) which is defined in Eqs. 25–27.

Fig. 17

Geometry of the matrix wedge spalled off

$$ A_{{{\text{SP}}1}} = A_{{{\text{SP}}1,1}} + A_{{{\text{SP}}1,2}} $$

(25)

$$ A_{{{\text{SP}}1,1}} = L_{{{\text{SP}}1}} {\frac{\cos \theta }{\sin \theta }}\left( {d + L_{{{\text{SP}}1}} {\frac{\cos \theta }{\sin \theta }}} \right) $$

(26)

$$ A_{{{\text{SP}}1,2}} = L_{{{\text{SP}}1}}^{2} \sqrt 2 \,{\frac{\cos \theta }{\sin \theta }} $$

(27)

The spalling force (F _SP1) is taken as the component of the deviation force (D _F1) parallel to the failure surface A _SP1, which according with experimental observations from [19] is also perpendicular to the fiber main axis (Fig. 18).

Fig. 18

Schematic equilibrium of equivalent forces at fiber exit point

Although the component of the deviation force parallel to the fiber main axis (F _R1) might introduce a sort of stabilization effect, due to the uncertainties on appropriately defining the stress field along the thorough matrix wedge it will be disregarded. Thereby the equivalent force which induces spalling on the matrix (F _SP1) becomes defined as following:

$$ F_{{{\text{SP}}1}} = P_{{{\text{S}}01}} \sin \theta \cos \theta $$

(28)

Applying the failure criterion defined in Eq. 23 L _SP1 becomes defined by simply solving a quadratic function a L ²_SP1  + b L _SP1 + c = 0 with the parameters defined in Eqs. 29–31.

$$ a_{1} = {\frac{\sqrt 2 }{\sin \theta }} + {\frac{\cos \theta }{{\sin^{2} \theta }}} $$

(29)

$$ b_{1} = {\frac{d}{\sin \theta }} $$

(30)

$$ c_{1} = - {\frac{{P_{{{\text{S}}01}} \sin \theta }}{{f_{\text{ctm}} }}} $$

(31)