Fill in a narrow stope is held up by shear on the walls, so the vertical stress saturates with depth (Janssen arching): sv(z) = gamma B / (2 K tan d) * (1 - exp(-2 K tan d z / B)), with K = 1 - sin(phi) (at rest) and wall friction d = 2/3 phi. For z large, sv tends to gamma B / (2 K tan d), much less than gamma z.

A cemented face stands freely when its unconfined strength is enough: Rankine/Bell vertical cut h_max = 2 UCS / gamma, so UCS_req = gamma h / 2 (times a safety factor). The cement-strength link UCS = 90 * c^1.1 kPa (c in weight %) is an illustrative empirical curve; real mixes need laboratory tests. Simplifications: 2D plane strain, uniform fill, no pore pressure or consolidation. Educational model, not for design.

Numeric checks (shown live): closed-form Janssen against a numerical integration of d(sv)/dz = gamma - 2 K tan d sv / B, and the Rankine height recomputed from the required UCS.