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.
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Settlement Trough Width (Peck)
However well a tunnel is excavated, some ground loss always occurs at the face and around the shield, and that loss propagates to the surface as a depression — the settlement trough. Ralph Peck observed in 1969 that the transverse profile of this trough follows the shape of a Gaussian curve (an inverted bell), and that its width is characterised by the parameter i = K·z₀, where z₀ is the depth of the tunnel axis and K is a trough width parameter that depends on the soil type: around 0.5 for clays (wide, gentle troughs) and 0.25 to 0.35 for sands (narrow, deep troughs). The parameter i is the standard deviation of the Gaussian curve — the horizontal distance from the tunnel axis to the point of inflection of the trough, where the curvature changes sign. It is fundamental in the design of urban tunnels (metros, sewer interceptors, TBM drives beneath cities): from i and the maximum settlement over the axis, the whole trough can be plotted and the angular distortion and horizontal strain that neighbouring building foundations will undergo can be computed — the basis for assessing damage risk and specifying protective measures. A wide trough (large i) spreads the settlement smoothly and causes less damage; a narrow trough concentrates the deformation. Enter the parameter K and the tunnel depth.
Related Tools
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.
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.
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.
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.
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