Real-time plotting of the ground reaction curve (GRC) and support characteristic curve (SCC) using Mohr-Coulomb elasto-plastic theory. Instantly computes plastic zone radius, convergence displacement, and support utilization ratio.
Ground Parameters
Support Parameters
GRC / SCC Curves (Ground Reaction vs Wall Displacement)
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Plastic Zone Radius rp (m)
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Convergence u (mm)
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Critical Support Pressure (MPa)
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Support Utilization (%)
GRC / SCC
Theory & Key Formulas
Plastic zone radius:
\(r_p = r\left[\frac{2(p_0\sin\phi + c\cos\phi)}{(1-\sin\phi)(p_i + c\cot\phi)}\right]^{\frac{1}{2k}}\)
What exactly is a "Ground Reaction Curve" in tunnel design? It sounds abstract.
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Basically, it's a graph that shows how the rock or soil around a tunnel pushes back as it deforms. Think of it like a spring: when you dig a tunnel, the ground moves inward, and this movement creates a resisting pressure. The GRC plots this internal support pressure (\(p_i\)) against the tunnel wall's radial displacement (\(u_r\)). Try moving the "Support Stiffness" slider above—you'll see the curve shift, showing how different ground conditions change the required support.
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Wait, really? So the ground itself is providing support? How does that work with the plastic zone formula shown?
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Exactly! The ground isn't just a load; it's a structural partner. The key is the plastic zone—the ring of material around the tunnel that yields. Its radius \(r_p\) tells us how far this yielding extends. In the simulator, as you increase the internal support pressure \(p_i\), watch how the calculated plastic zone radius shrinks. A smaller \(r_p\) means less ground yielding and more stability. The GRC is derived from this elasto-plastic behavior.
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So where does the actual tunnel lining, like shotcrete, come in? How do we know if it's strong enough?
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Great question! That's where the Support Characteristic Curve (SCC) comes in. It's the other line on the graph. The SCC shows how much pressure a support system (like shotcrete or steel ribs) can provide as it deforms. The magic happens at the intersection point. Try changing the "Support Type" parameter—you'll see the SCC jump. The pressure at that intersection is what your design achieves. If it's too high, the support might fail.
Physical Model & Key Equations
The core of the analysis is determining the extent of the yielded ground around the tunnel, defined by the plastic zone radius. This is calculated using the Mohr-Coulomb failure criterion for a thick-walled cylinder under initial in-situ stress.
$$r_p = r \left[\frac{2(p_0\sin\phi + c\cos\phi)}{(1-\sin\phi)(p_i + c\cot\phi)}\right]^{\frac{1}{2k}}$$
Where: \(r_p\) = Radius of the plastic zone [m] \(r\) = Tunnel excavation radius [m] \(p_0\) = Initial in-situ stress [kPa] \(p_i\) = Internal support pressure at the tunnel wall [kPa] \(c\) = Cohesion of the ground [kPa] \(\phi\) = Friction angle of the ground [°] \(k\) = A constant, \(k = \frac{\sin\phi}{1-\sin\phi}\)
The equation shows that increasing the support pressure \(p_i\) directly reduces the plastic zone size \(r_p\).
The Ground Reaction Curve (GRC) defines the relationship between the support pressure \(p_i\) and the radial displacement of the tunnel wall \(u_r\). It has two parts: an elastic portion (before yielding) and a plastic portion (after yielding begins).
This is the simplified form of the GRC equation in the plastic region. The wall pressure \(p_{wall}\) is equal to the internal support pressure \(p_i\) at equilibrium. The term \((r/r_p)^{2k}\) captures the influence of the plastic zone size. As displacement increases (\(r_p\) grows), the ground's resisting pressure \(p_{wall}\) decreases along the curve, which is why timely support installation is crucial in NATM.
Real-World Applications
Urban Metro Tunneling: In cities, tunnels must be built close to existing foundations with minimal surface settlement. Engineers use this GRC-SCC analysis to determine the optimal stiffness and timing for shotcrete lining. For instance, a stiffer, early-installed lining (moved earlier on the GRC) minimizes plastic zone growth and protects nearby buildings.
Mine Access and Haulage Drifts: In mining, tunnels are often in weak, fractured rock. The analysis helps decide between light, yielding supports (like split-sets) or heavy, rigid supports (like concrete liners). A common case is designing for a specific "yield" displacement to activate ground arching without causing collapse.
High-Speed Rail Tunnels in Mountains: These long tunnels encounter varying rock masses. The interactive model helps engineers design a flexible support system that can be adapted zone-by-zone. For example, in a fault zone with low cohesion, the graph shows a much larger plastic zone, requiring a stronger, more ductile SCC.
Hydropower Pressure Tunnels: Tunnels carrying high-pressure water must resist internal bursting forces and external ground loads. The GRC-SCC intersection analysis ensures the final concrete lining is designed for the long-term ground pressure that develops after the initial rock bolts and shotcrete have deformed.
Common Misunderstandings and Points to Note
When you start using this tool, there are a few common pitfalls to watch out for. The first one is whether you are simply setting the initial in-situ stress p0 as the overburden pressure. While it's common to calculate p0 = γH (γ: unit weight of soil, H: overburden), in actual mountain tunnels, the lateral pressure coefficient (the ratio of horizontal to vertical stress) is rarely 1. For example, in folded zones, horizontal stress can be 1.5 times or more the vertical stress. The tool requires you to input p0 as a single value, but that value needs to be an appropriate estimate that considers the influence of such geological structures.
The second point is using the tool without understanding parameter sensitivity. Particularly, the "cohesion c" and "friction angle φ" are sensitive parameters where small changes can significantly alter the results. For instance, ground with c=100 kPa, φ=30 degrees and ground with c=80 kPa, φ=28 degrees might appear similar, but the final convergence displacement without support can differ by more than a factor of two. Since survey data has variability, you should always perform a "sensitivity analysis" by gradually changing parameters to see how the curves shift. This is the first step in risk assessment.
Finally, don't forget that this analysis is based on the idealizations of a "circular cross-section" and "isotropic, homogeneous ground." Actual cross-sections are horseshoe-shaped, and ground is often layered or full of fractures. Therefore, the plastic zone radius and wall support pressure calculated here are "guidelines," not "absolute values." By comparing them with field measurement data and empirically accumulating correction factors—like "for this ground condition, displacement tends to be 1.2 times the calculated value"—you build true design capability.
Enter tunnel radius (r) in meters—typical values 4–8 m for urban NATM applications
Input initial ground stress (p0) in kPa—use γh where γ=20 kN/m³ for overburden depth h
Set cohesion (c) in kPa—soft clay ~15–30 kPa, weathered rock ~50–150 kPa
Click Calculate to generate Ground Reaction Curve (GRC) and support pressure curves simultaneously
Adjust support stiffness (shotcrete thickness, bolt spacing) to intersect GRC at stable convergence point
Worked Example
6 m diameter tunnel in silty clay: r=6 m, p0=180 kPa (9 m depth), c=25 kPa, φ=28°. Mohr-Coulomb analysis yields GRC peak support demand ~120 kPa at 2.5% wall convergence. 200 mm shotcrete (25 GPa modulus) with 2×2 m bolt grid provides support curve intersecting GRC at 1.8% convergence (radial displacement δ ≈ 108 mm). Equilibrium achieved without face instability or excessive deformation.
Practical Notes
GRC intersection below support curve indicates insufficient support—increase shotcrete thickness or reduce bolt spacing
Adjust p0 for pore pressure: effective stress p0'=p0−u; ignore u only in dry rock
Cohesion loss occurs during excavation; use residual c for conservative design in fissured/jointed ground
Convergence >3% signals plastic yield; deploy forepoling or compressed air support in weak clay