Maximum Surface Settlement (Tunnel)
Calculate the maximum surface settlement, over the tunnel axis, S_max = Vs ÷ (i·√(2π)), from the settlement trough volume per metre of tunnel Vs (m³/m, the lost soil volume surfacing) and the trough-width parameter i (m, Peck's method). Since the trough is Gaussian, integrating the curve gives Vs = √(2π)·i·S_max, isolating the maximum settlement, which occurs right over the axis. This is the critical value for damage assessment: compared to allowable limits (typically 10-25 mm for sensitive structures), it decides whether the excavation is safe or needs mitigation. Enter the trough volume and the width parameter.
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Maximum surface settlement of a tunnel
The maximum settlement at the ground surface, which occurs directly above the tunnel axis, is S_max = Vs ÷ (i·√(2π)), where Vs is the volume of the settlement trough per metre of tunnel (the volume of ground loss that surfaces, in m³/m) and i is the Peck trough width parameter. The formula follows from the geometry of the Gaussian trough: the area under the Gauss curve equals √(2π)·i·S_max, and that area is precisely the lost volume Vs per metre, so isolating the maximum settlement is all it takes. This is the critical value of any urban tunnelling assessment, because it marks the point where the surface sinks the most. S_max gets compared with allowable limits — typically 10 to 25 mm for sensitive buildings, and less still for historic structures or railways — to decide whether the excavation is safe or calls for mitigation measures: compensation grouting, which injects grout into the ground to lift the surface back up, higher TBM face pressure, or even artificial ground freezing. Maximum settlement, together with volume loss, is the indicator that links the machine operation underground to the potential damage to buildings at the surface — the heart of the geotechnical challenge of tunnelling beneath dense cities without cracking buildings. Enter the trough volume and the width parameter.
Related Tools
Peck Settlement Trough Width
Calculate the trough-width parameter of the surface settlement induced by tunnelling, i = K·z₀, by Peck's method, from the trough-width parameter K (~0.5 for clays, ~0.25-0.35 for sands) and the tunnel axis depth z₀. The surface settlement from ground loss follows a Gaussian (inverted bell) curve, and i is its standard deviation — the horizontal distance from the tunnel axis to the inflection point, defining the trough width. Larger i means a wider, gentler trough (clays); smaller means narrower and deeper (sands). This parameter is essential to predict damage to nearby buildings in urban tunnels. Enter the K parameter and the tunnel depth.
Tunnel Volume Loss
Calculate the volume loss of a tunnel excavation, VL = Vs ÷ (π·D²/4)·100, the percentage ratio between the settlement trough volume per metre Vs (m³/m) and the excavated cross-section area (from diameter D). Volume loss quantifies how much soil 'disappeared' relative to the theoretical tunnel volume — caused by face relaxation, overexcavation, tail-gap closure behind the TBM shield and consolidation. It is the key control parameter for urban excavation: well-run EPB/slurry TBMs achieve 0.5-1.5% in soils; values above 2-3% indicate problems and excessive settlement. Enter the trough volume and the tunnel diameter.
Tunnel Convergence
Calculate a tunnel's convergence — the relative radial deformation of the excavation, ε = (u ÷ r)·100 — from the radial displacement u (the inward movement of the walls toward the center, measured by extensometers or total station) and the excavation radius r, in the same unit. Convergence is the primary monitoring indicator in NATM (New Austrian Tunnelling Method): it measures how much the rock mass deforms after excavation, reflecting stress mobilization and support effectiveness. Low, stabilized convergence indicates a stable mass; high, growing or accelerating convergence signals squeezing, instability or insufficient support, requiring immediate reinforcement. Enter the radial displacement and the tunnel radius.
Tunnel Support Pressure
Calculate the support pressure a tunnel lining must resist, pv = γ·Hp, from the rock mass unit weight γ (kN/m³) and the rock load height Hp (m) — typically from Terzaghi's method or geomechanical classifications (RMR, Q-system). Support pressure is the vertical stress the loosened rock zone exerts on the support (shotcrete, steel sets, final lining), and it drives the structural design of the lining. In shallow tunnels the load may be the full overburden; in deep tunnels, arching reduces it to a fraction. Estimating it correctly is decisive: underestimating leads to collapse, overestimating raises cost. Enter the unit weight and the rock load height.
Tunnel Face Pressure (EPB/Slurry)
Estimate the face support pressure needed to stabilize the excavation front of a mechanized tunnel, p = K·γ·H, from the earth pressure coefficient K (at rest K₀ ≈ 1−sinφ, or active), the soil unit weight γ (kN/m³) and the axis depth H (m). In closed-face TBMs (EPB or slurry), the pressurized chamber must balance the earth and water pressure at the front, avoiding both collapse (insufficient pressure) and blow-out (excessive pressure). Face pressure is the most critical operational parameter of a TBM, adjusted in real time per cover, water table and soil type. This gives the earth component; total pressure adds hydrostatic water pressure and a safety margin. Enter the earth pressure coefficient, unit weight and depth.
Terzaghi Rock Load Height
Estimate the rock load height over a tunnel crown by Terzaghi's classic method, Hp = Cf·(B + Ht), from the rock load factor Cf (depending on mass quality — ~0 for intact rock to >2 for heavily fractured or swelling rock), the width B and the height Ht of the excavation. Hp represents the loosened rock zone above the tunnel that effectively loads the support — Terzaghi proposed that, due to arching in the mass, only a fraction of the total overburden acts on the lining. This loosening-load model is the historic basis for rock tunnel support design. Multiplying Hp by the unit weight gives the support pressure. Enter the load factor, width and height.
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