Analytical Model for Fictitious Crack Propagation in Concrete Beams
Journal of engineering mechanics
An analytical model for load-dispiacement curves of unreinforced notehed and un-notched concrete beams is presented. The load displacement-curve is obtained by combining two simple models. The fracture is modelled by a fictitious crack in an elastic layer around the rnid-section of the beam. Outside the elastic layer the deformations are modelled by the Timoshenk o beam theory. The state o f stress in the elastic layer is assumed to depend bi-linearly on local elongation corresponding to a
... r softening relation for the fictitious crack. For different beam size results from the analytical model are compared with results from a more accurate model based on numerical methods. The analytical model is shown to be in good agreement with the numerical results i f the thiclcness of the elastic layer is taken as half the beam depth. Several general results are obtained. It is shown that the point on the load-dispiacement curve where tbe fictitious crack starts to develop, and the point where the real crack starts to grow will always correspond to the same bending moment. Closed form solutions for the maximuro size of the fraelure zone and the minimum slope on the load-dispiacement curve is given. Tbe latter result is used for derivation o f a general snap-back eriterion depending only on beam geometry. Introduetion Since Kaplan (Kaplan 1961) performed his linear elastic fracture mechanical (LEPM) investigation of notehed concrete beams subjected to three and four point bending much attention has been paid to fracture of concrete and rock. In thi s pioneering work and in three subsequent discussions (Blakey and Beresford (1962), Gliicklich (1962), Irwin (1962)) the applicability of LEPM was discussed and the views given in these contributions are still popular (e.g. slow crack growth). Today it is realized that LEPM is only applicable to large scale structures and ultra brittie concrete, Pianas and Elices (1989), and that it is necessary to apply nonlinear fracture mechanics for description of fracture in ordinary concrete structures. Different models based on nonlinear fracture mechanical ideas describe the softening behaviour of concrete e. g. the Pictitious Crack Model (PC-model) by Hillerborg, Modeer and Peterson (1976), the Crack Band model by Bazant (1983) and the Two Parameter Model by Jenq and Shah (1985) . In this paper the PC-model will be used to describe fracture in concrete. Few researchers have considered analytical methods based on non-linear elastic fracture mechanical models to describe fracture in concrete structures. A model has been developed by T . Chuang and Y. W. Mai (1989) basedon the Crack Band Model. Also, a model based on the fictitious crack model has been developed by Llorca and Elices (1990). The idea of modeiling the bending failure of concrete beams by development of a fictitious crack in an elastic layer with a thickness proportional to the beam height was introduced by 2 Ulfk:jaer, Brincker and Krenk (1990). In the present paper this model is presented using a linear softening relation and the model is validated by comparing with results from a numerical model. Several general results are obtained. It is shown that the point on the load-dispiacement curve where the fictitious crack starts to develop, and the point where the real crack starts to grow always will correspond to the same bending moment, the points lying on each side of the peak point. Closed form solutions for the maximum size of the fracture zone and the minimum slope on the load-dispiacement curve are given. The last result is used for derivation of a general snap-back eriterion depending only on beam geometry. Basic Assumptions The failure of a simply supported beam loaded in three point bending is modelled by assuming development of a single fictitious crack in the midsection of the beam. In the FC-model material points on the crack extension path are assumed to be in one of three possible states: A) a linear elastic state, B) a fracture state where the material is softened, caused by cohesive forces in the fracture process zone and finall y, C) a state of no stress transmission. In the fracture state the cracking process is deseribed by a softening relation which relates stress normal to the cracked surface a to the crack opening displacement, w (distance between the cracked surfaces) a = f(w) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . (l) where f (·) is a material function determined by uniaxial tensile tests, see Fig. l . The area under the material function is termed the specific fracture energy, G p, which is assumed to be a material constant, Elfgren (1989) . Dsually the FC-model is combined with numerical methods like the finite element method Hillerborg et al. (1976) or a boundary element method like the substructure method introduced by Petersson (1981), and therefore no simple method of analysis is directly available. Therefore, in thispaper the model is further simplified by two additional assumptions: l) the camplex stress field around the crack is modelled by simple spring-action in an elastic layer around the crack, and outside the layer the deformations are modelled by beam theory 3) the softening relation is assumed to be linear. The first assumption is characteristic of the model concept and eannot be changed without changing the whole idea of the model. The second assumption however is not inherent with the model and the linear softening relation might be changed to a Dugdale relation or another softening relation. Using the assumption of a linear softening relation however, the fracture en erg y is given by G p= ~a u w c where a u the ultimate tensile stress and w c is the critical crack opening displacement, see Fig. l . In the elastic layer only bending stresses are assumed to be present and the stress is assumed to depend linear ly on the local elongation o f the layer. Assuming a linear softening relation, the constitutive relation of the layer becomes a bi-linear relation between the axial stress a and the elongation v, Fig. 2 . On the ascending branch the elongation is linear elastic v=ve and no crack opening is present. The linearresponseis given by ve =ah/E where h is the thickness of the layer, and E is Young's modulus. On the descending branch, however, the total deformation v consists of two contributions v = ve + w, where w is the crack opening dispiacement The peak point corresponds to the deformation v = vu, and total fracture corresponds to v=vc. Therefore, the critical crack opening 3 dispiacement wc correspond to wc=vc.