An Mathematic Model For Fracture Process Zone Of Concrete Beam

Fracture mechanics of concrete is different from the metal material. A large micro crack zone exists in the tip of crack in which the stress singularity disappears. So it is obviously inappropriate to analyze the crack problems of the concrete structure with the linear elastic fracture mechanics. The analytical solution of simply supported beam with the fracture process zone was deduced which was based on the “Duan-Nakagawama” model and the superposition of several solutions of elastic mechanics problems. In addition, the influence of crack length and the fracture process zone length on the natural vibration frequency of structure is analyzed. The main work is as follows:Based on the virtual crack model and Westergaard stress function, the singularity of stress function in the crack tip disappears. A series of elastic solutions of infinite or semi infinite plates under different loads are superposed and the method of selected points was chosen to satisfy the boundary conditions. So the stress function of fracture process zone of simply supported beam is obtained and the tensile strain softening curve of corresponding fracture process zone is depicted. The softening curve is closed which slowly shrinks to zero in the crack tip with the tendency of concave at the whole. It coincides with the bilinear softening curve which is adopted in the research of concrete fracture mechanics.According to the displacement function of the problems mentioned above, the deflection curve of the simply supported beam is obtained. And then the energy method is used to get the first-order natural vibration frequency of the concrete beam with the fracture process zone. At the same time, by analyzing the impact of ligament width and the fracture process zone length on the frequency, the corresponding relation curve is depicted. Ligament width has larger influence on the frequency of the beam. With the reduction of the ligament width, the frequency decreases fast which is obviously nonlinear. However, the effect of fracture process zone length on the natural frequency of the structure is smaller.The fracture process zone of beam is represented by the nonlinear spring and Ansys is used to simulate the simply supported beam of concrete with the fracture process zone. By comparison, the calculation results are in good agreement with the theoretical calculation results which include the stress distribution of the typical cross section, the vertical displacement of beam and the opening displacement of crack tip and so on.By superposing weighted integral stress functions with different crack lengths, the method of selected points is chosen to get the stress function solution of the simply supported beam under uniform loads. The solution meets the condition of fracture process zone length and the bond stress distribution.