Ground engineering · Tunnels & shafts

Circular shaft lining (ring compression) calculator

Lateral earth and water pressure at the design depth, ring thrust and an ovalisation moment from non-uniform pressure, and the lining stresses.

Standard: Thin-ring thrust N = p·r · ovalisation moment M = Δp·r²/6 (p varying as cos 2θ) · plain/lightly reinforced concrete

Inputs in the worked example

The live calculator starts with these values; change any of them and every result, check and diagram updates instantly.

Input	Value	Unit
Internal radius	3	m
Lining thickness	0.3	m
Design depth	15	m
Soil unit weight	19	kN/m³
Lateral pressure coefficient (K0, or reduced for arching)	0.5
Water table depth	3	m
Surface surcharge	10	kPa
Pressure non-uniformity (pmax − pmin)/p	0.1
Concrete f′c	40	MPa

Diagram for the worked example

Pore pressure = 9.81*MAX(0,15-3) = 117.72 kPa
σ′v = 19*15-117.72 = 167.28 kPa
Lateral pressure K(σ′v + q) + u = 0.5*(167.28+10)+117.72 = 206.36 kPa
External radius = 3+0.3 = 3.3 m
Ring thrust p·re = 206.36*3.3 = 680.99 kN/m
Ovalisation moment η·p·rm²/6 = 0.1*206.36*(3+0.3/2)^2/6 = 34.127 kNm/m
Min stress N/t − 6M/t² (− = tension) = (680.99/0.3-6*34.127/0.3^2)/1000 = -0.005159 MPa
Limit: compression 0.6 × 0.85f′c = 0.6*0.85*40 = 20.4 MPa
Limit: tension −0.6 × 0.36√f′c = -0.6*0.36*SQRT(40) = -1.3661 MPa
Max compression N/t + 6M/t² = (680.99/0.3+6*34.127/0.3^2)/1000 = 4.5451 MPa
Circular shafts in soil carry much of the load by arching; K may be reduced below K0 (Berezantzev) where justified. Check base heave and hydraulic uplift at the base separately.

Tunnel lining & rock load — Overburden, lining hoop thrust and stress, ovalisation moment, and rock support pressure.
Convergence–confinement (circular tunnel) — Critical support pressure, plastic zone radius and wall displacement for a support pressure pi.
Tunnel face stability (Broms & Bennermark) — Stability ratio N = (σv − σT)/su and the support pressure needed to keep N at a target.
Tunnelling settlement trough (Peck) — Maximum surface settlement, trough width i = K z0 and settlement at an offset for a volume loss.
Consolidation settlement — Primary settlement (NC or OC clay) and time to a degree of consolidation.
Prefabricated vertical drains — Radial consolidation with smear, time to target degree, triangular or square spacing.

For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.