Chapter 7 — Structurally Controlled Instability in Tunnels¶

7.1 Kinematic Release of Polyhedral Wedges¶

In shallow to intermediate depth tunnels driven through blocky, jointed rock masses, failure is rarely driven by crushing of intact rock. Instead, gravity-induced fallouts of polyhedral rock wedges bounded by intersecting discontinuities represent the dominant hazard.

For a wedge to detach and fall or slide into a tunnel void, two conditions must be met: 1. Kinematic Feasibility: The geometry of intersecting joint planes must form a closed tetrahedral or pentahedral wedge whose apex points away from the excavation void, and whose sliding direction daylighting into the opening is physically unobstructed. 2. Limit Equilibrium Instability: Driving forces (wedge gravity weight \(W\) and joint water cleft pressures \(U\)) must exceed resisting forces (joint cohesion \(c_j\), friction \(\sigma_n' \tan\phi_j\), and installed rockbolt capacities \(T\)).

7.2 Stereographic Projection & Kinematic Analysis¶

Stereonets (equal-angle Wulff or equal-area Schmidt projections) plot 3D orientation data (Dip \(\psi\) and Dip Direction \(\alpha\)) onto a 2D plane:

Great Circles: Represent planes of discontinuities and tunnel excavation faces.
Poles: Normal vectors perpendicular to discontinuity planes.
Intersections (\(\vec{I}_{12}\)): The plunge (\(\beta\)) and trend (\(\theta\)) of the line of intersection formed by two joint planes:

\[\vec{I}_{12} = \vec{n}_1 \times \vec{n}_2\]

Sliding Modes:¶

Direct Falling (Roof Dropouts): The wedge apex lies vertically above the crown; all bounding planes dip steeper than their friction angles.
Single-Plane Sliding: The intersection line daylights, but the wedge slides along the steeper of the two joint planes while detaching from the second.
Double-Plane Sliding: The wedge slides along the line of intersection \(\vec{I}_{12}\), with contact maintained on both joint faces simultaneously.

       Crown of Tunnel
     +-----------------+
      \     WEDGE     /
       \   (Weight W)/
        \    /\     /
         \  /  \   /   Joint Plane 2
          \/    \ /   (Dip psi_2)
 Joint Plane 1   v
 (Dip psi_1)   SLIDE VECTOR (Line of Intersection)

7.3 Limit Equilibrium Calculation: The UNWEDGE Method¶

For a wedge sliding along the line of intersection of Plane 1 and Plane 2 with plunge \(\beta\):

Driving Force (\(F_{\text{drive}}\)):¶

\[F_{\text{drive}} = W \sin\beta\]

Normal Forces on Joint Faces (\(N_1, N_2\)):¶

Resolved by wedge equilibrium perpendicular to the intersection line:

\[N_1 + N_2 = W \cos\beta \cdot f(\text{wedge wedge geometry})\]

Resisting Force (\(F_{\text{resist}}\)):¶

\[F_{\text{resist}} = (c_1 A_1 + N_1' \tan\phi_1) + (c_2 A_2 + N_2' \tan\phi_2) + \sum T_b \cos\theta_b\]

Where \(A_1, A_2\) are face areas, \(N'\) are effective normal forces reduced by pore cleft water pressures, and \(T_b\) is the tensile capacity of intersecting rockbolts oriented at angle \(\theta_b\) to the sliding vector.

\[FS = \frac{F_{\text{resist}}}{F_{\text{drive}}}\]

7.4 Support Design for Roof and Sidewall Wedges¶

Dead-Weight Support Rule: Pattern rockbolts must anchor the maximum anticipated wedge volume back into stable rock beyond the failure envelope with an anchor embedment length \(\ge 1.5 - 2.0\text{ m}\).
Bolt Spacing: Bolt spacing \(s\) must be smaller than the minimum dimensions of the expected wedge face to prevent small blocks falling out between bolts:

\[s \le \frac{1}{2} \text{Wedge Dimension}\]

Shotcrete Integration: Mesh-reinforced or steel fiber reinforced shotcrete (SFRS) provides immediate surface containment, preventing small keyblocks from ravelling and unravelling larger wedges.

7.5 Canonical Terminology¶

English Term	Canonical Vietnamese Translation	Definition
Kinematic analysis	Phân tích động học	Geometric assessment determining whether a rock wedge can physically detach
Stereographic projection	Phép chiếu lập thể cực	2D graphical representation of 3D plane and line orientations
Line of intersection	Đường giao tuyến	Vector formed by the intersection of two discontinuity planes
Wedge failure	Phá hoại trượt nêm	Detachment of a polyhedral rock block bounded by intersecting joints
Keyblock	Khối nêm khóa	Critical surface block whose removal destabilizes adjacent rock masses