Effects of Scale Parameters and Counting Origins on Box-Counting Fractal Dimension and Engineering Application in Concrete Beam Crack Analysis

Fractal theory provides a powerful tool for quantifying complex geometric patterns such as concrete cracks. The box-counting method is widely employed for fractal dimension (FD) calculation due to its intuitive principles and compatibility with image data. However, two critical limitations persist in existing studies: (1) the selection of scale parameters (including minimum measurement scale and cutoff scale) lacks systematization and exhibits significant arbitrariness; (2) insufficient attention to the sensitivity of counting origins compromises the stability and comparability of FDs, severely limiting reliable engineering application. To address these limitations, this study first employs classical fractal images and crack samples to systematically analyze the impact of four minimum measurement scales (2, $\sqrt{2}$, 3, $\sqrt{3}$) and three cutoff scale coefficients (cutoff-to-minimum image side ratios: 1, 1/2, 1/3) on computational accuracy. Subsequently, the farthest point sampling (FPS) method is adopted to select counting origins, comparing two optimization strategies—Count-FD-Mean (mean of fits from multiple origins) and Count-Min-FD (fit using minimal box counts across scales). Finally, the optimized approach is validated through static loading tests on concrete beams. Key findings demonstrate that: the optimal scale combination (minimum scale: $\sqrt{2}$; cutoff coefficient: 1) yields a mere 0.5% average error from theoretical FDs; the Count-Min-FD strategy delivers the highest stability and closest alignment with theoretical values; FDs of beam cracks increase continuously with loading, exhibiting an exponential correlation with midspan deflection that effectively captures crack evolution; uncalibrated scale parameters and counting strategies may induce >40% errors in inferred mechanical parameters; results stabilize with 40–45 counting origins across three tested fractal patterns. This work advances standardization in fractal analysis, enhances reliability in concrete crack assessment, and provides critical support for the practical application of fractal theory in structural health monitoring and damage evaluation.