Experimental study on concrete beams without web reinforcement based on fractal theory

Based on the development and distribution of cracks, we explored the shear performance of concrete beams without web reinforcement under different shear span ratios and longitudinal reinforcement ratios. Nine groups of concrete beams without web reinforcement with shear-span ratios of 1.5, 2, 2.5 and longitudinal reinforcement ratios of 1.28%, 1.62%, and 1.99% were used for four-point loading shear tests. The cracks on the surface of the test beam were analyzed by applying fractal geometry theory, and the box counting method was used to calculate the fractal dimension of the cracks on the surface of the beam under the effect of the graded load and the ultimate load. The relationship among the fractal dimension of the beam surface, the ultimate load, the graded load and the span was discussed. The results show that the shear-span ratio is inversely proportional to the ultimate load and cracking load, while the longitudinal reinforcement ratio is directly proportional to the ultimate load and exhibit a small influence on the cracking load. Concrete beams without web reinforcement have obvious fractal characteristics under the effect of graded loading or ultimate load. The fractal dimension under the effect of graded load is 0.964–1.449, and the fractal dimension under the effect of ultimate load is around 1.33. The graded load, mid-span deflection and fractal dimension show a good logarithmic relationship. The change curve of graded load and fractal dimension is affected by the shear-span ratio and the beam longitudinal reinforcement ratio. The intermediate deflection is less affected by the shear-span ratio. Under the effect of the longitudinal reinforcement ratio, the curvature of the curve shows a trend of first increasing and then decreasing, but the relationship between the ultimate load and the fractal dimension has certain differences. The ultimate load first increases and then decreases with the increase of the shear span ratio, and the difference is greater with the increase of the longitudinal reinforcement ratio.