<p>To address the stability challenges of non-circular openings in deep underground engineering, this study focuses on four common opening types in underground rock engineering-three-centered arches, squares, circles, and isosceles trapezoids-establishes a stress solution model and a discrete element numerical calculation model based on complex function theory, investigates the stress distribution and failure characteristics around the openings under different lateral pressure coefficients and inclination angles. The relationships among shape parameters, stress states, and failure modes are analyzed. The results indicate that the stress distribution is primarily governed by the shape and inclination angle of the opening, whereas the lateral pressure coefficient mainly influences the magnitude of stress concentration. Within the scope of this study, isosceles trapezoidal openings are most sensitive to the lateral pressure coefficient, followed by square and three-centered arch openings, with circular openings being the least sensitive; compressive stress concentration zones form at the endpoints of the openings or locations with abrupt curvature changes (such as the corners of squares and trapezoids, and the shoulders/corners of arches), and the compressive stress concentration is most pronounced when the opening inclination angle is approximately 45°. The minimum circumferential stress increases monotonically with the lateral pressure coefficient, transitioning from tensile to compressive and eventually stabilizing. DEM simulations further reveal that the initial failure zone corresponds closely with the stress concentration zone: tensile cracks dominate at low lateral pressure coefficients, while shear-dominated failures emerge as the coefficient increases. Among the studied shapes, circular openings show the greatest capacity to suppress tensile stress concentration, whereas trapezoidal openings are the most unfavorable in this regard.</p>