An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams

An analytical model to predict the effective fracture toughness K IC s of concrete was proposed based on the fictitious crack model. Firstly, the equilibrium equations of forces in the section were formed in combination with the plane section assumption. Then a Lagrange function was presented through the equilibrium equations and the relationship formula between the effective crack length and crack tip opening displacement. Taking into account Lagrange Multiplier Method, the maximum load Pmax was obtained, as well as the critical effective crack length ac. Furthermore, K IC s was gained in an analytical manner. Subsequently, some material and structural parameters from other literatures were adopted into the proposed model for the calculation. Compared with the experimental results, most of the calculated values show a good agreement for Pmax and ac. In order to study the influence of the softening curve in the fictitious crack on the calculated fracture parameters, three series of constants determining the shape of the softening curve were chosen in the calculation. The results show that the calculated fracture parameters are not sensitive to the shape of the softening curve. Therefore, only if the elastic modulus Ec and flexural tensile strength fr were measured, Pmax, ac and K IC s can be predicted accurately using the proposed model. Finally, the variations of the calculated fracture parameters with the specimen size and a0/h (i.e., the ratio of the initial crack length to the depth of the specimen) were studied. It was found that both K IC s and the pre-critical crack propagation length Δac increase with the specimen size. However, the two parameters increase to the maximums and then decrease gradually with a0/h. Moreover, the theories of free surface effect were utilized to explain the observed size effects.