Theoretical H-M Capacity Envelope for Large-Diameter Rigid Piles Embedded in Sandy Slopes

Large-diameter rigid piles often serve as foundations on sloping ground, where combined lateral force and overturning moment constitute important design considerations. While bearing-capacity envelope approaches have been developed for rigid piles in clay, corresponding theoretical methods for piles in sand remain relatively limited, despite the strong influence of overburden pressure and slope effects on their lateral resistance. This study extends the previous six-mode methodology to develop a theoretical framework for the H-M capacity envelope of large-diameter rigid piles embedded in sandy slopes. The proposed method adopts the previously established failure wedge model, incorporating a variable wedge base angle and equivalent-depth correction, and extends its unidirectional formulation to account for uphill and downhill resistance through the sign of the slope angle. Additional resistance components, including distributed vertical shear-induced moment, base shear, and base moment, are incorporated to account for the size effects of large-diameter piles. The complete H-M envelope is constructed from parametric equations corresponding to six soil-pile interaction models. Comparisons with finite element limit analyses and independent experimental and numerical results show reasonable agreement within the investigated conditions. Parametric analyses indicate that, within the parameter ranges considered, the fan angle has a more pronounced influence on the envelope than the interface friction angle, while the additional resistance components become increasingly relevant with increasing pile diameter and decreasing length-to-diameter ratio. The slope angle progressively increases the asymmetry of the H-M envelope; in the investigated case with a slope angle of 20°, the Q2 and Q4 regions account for 89.9% and 6.3% of the total envelope area, respectively. The proposed framework provides a relatively simple theoretical approach for estimating the ultimate H-M capacity of rigid piles on sandy slopes and may serve as a useful reference for preliminary assessment and design, where available analytical guidance remains limited.