A Geometry-Dependent Dimensionless Reinforcement Placement Index for Planar Geosynthetic-Reinforced Shallow Foundations: Boussinesq-Based Formulation and Multi-Source Experimental Assessment

Reinforcement depth, vertical spacing, and layer number are commonly interpreted as separate variables, although their effectiveness also depends on footing size and geometry. This study reformulates the dimensional reinforcement efficiency factor uz as a dimensionless, geometry-dependent reinforcement placement index, uz*, for planar-reinforced shallow foundations. The Boussinesq point-load solution was integrated over circular, square, finite rectangular, and ideal strip loaded areas, and the reinforcement influence depth was defined at a normalized vertical stress threshold of η=0.150. The resulting influence-depth coefficients were 1.478, 1.663, and 4.205 for circular, square, and ideal strip loading, respectively, while the rectangular coefficient varied continuously with L/B. Continuity was assessed using the original circular-loading dataset. The external database comprised 15 independent sources and provided 149, 123, and 86 reinforced observations in 32, 25, and 18 source/condition-specific groups at s/Bc=0.050, s/Bc=0.100, and s/Bc=0.150, respectively. All 75 study-specific bearing pressure–uz* relationships had positive slopes, and 74 yielded R2 ≥ 0.80. Group fixed-effects models explained 60.6%, 62.6%, and 68.8% of the within-system variation in lnBCR, respectively, with positive source-specific slopes in every available source. Varying η from 0.100 to 0.200 changed Rwithin2 by less than 0.012. Compared with conventional single-parameter descriptors, uz* provided the most balanced performance at the principal and larger settlement levels. The proposed index is therefore a physically interpretable coordinate for comparing vertical reinforcement placement across footing sizes and geometries, rather than a stand-alone predictor of absolute bearing-capacity improvement.