Engineering Theory and Assumptions

Orientation Convention

Plane orientations use dip and dip direction in degrees. Trends are azimuths in degrees. Plunge is positive downward and may range from -90 to 90 degrees for force vectors.

The internal coordinate system treats horizontal azimuths as x = sin(azimuth), y = cos(azimuth), with z positive upward. Plane normals are built from the entered orientations and forced upward where needed.

Wedge Formation

For planar analysis, the app constructs the two joint planes, upper face, and slope face from their orientations and a slope height. A valid tetrahedral wedge requires the planes to intersect in a finite, usable geometry.

For profile analysis, wedge formation, geometry, and force balance use the entered multi-bench profile directly. The toe-to-crest equivalent planar slope is used only for the overall daylighting classification.

Daylighting and Kinematics

The intersection line of Joint 1 and Joint 2 is evaluated against the slope face or profile proxy. The stereonet panel plots the active great circles, friction circle, daylighting/sliding zone, and intersection point.

A wedge may form geometrically but be marked non-daylighting if the intersection line does not exit through the slope in the current orientation context.

Tunnel Intersection Blocks

The single three-joint input is searched across every junction region and all eight three-joint plus twelve two-joint half-space families. Two-joint families use a dedicated active-set evaluation for free fall, either single joint, the true joint-intersection line, and stable states. A candidate translation may not penetrate any connected excavation-contact patch and must exit through at least one.

Search diagnostics are emitted even when no block is recovered. Joint planes are searched by independent signed offsets using CAD-feature seeds, deterministic NLopt DIRECT-L global exploration, and Sbplx local polishing. converged means the global and local numerical convergence criteria were satisfied. boolean-budget-limited, optimizer-limited, and optimizer-failed results are provisional and remain viewable. The Boolean budget applies independently to every combination, junction region, and block family. None of these numerical statuses proves a global maximum.

Force Balance

The force balance is evaluated using vector equilibrium on the current wedge geometry. The active vector includes weight, cohesion contribution, external force, water force, and seismic force. Passive face pressures are handled as a resisting/passive contribution.

The app reports falling, unsupported, and supported factors of safety where applicable. Sliding mode can occur on Joint 1, Joint 2, both joints, lifting, falling, or be stable.

Underground rock bolts are passive forces and do not influence the active-force sliding direction. A row contributes only when its collar lies on the wedge excavation face and the bolt exits a joint face before reaching its tip. Available capacity is the tensile strength, limited when enabled by bond strength multiplied by stable-rock embedment beyond the wedge. Cosine efficiency is clamp(-bolt direction · sliding direction, 0, 1) when selected. The resultant bolt vector is the vector sum of all engaged rows.

Unsupported joint reactions use the active vector A; supported reactions use A + P, where P contains the bolt vector. Reported factor of safety is max(FSf, FSu, FSs). Moment equilibrium, bolt shear capacity, plate/stripping failure, pretension, axial bolt angle, and row offset are not modelled.

Strength Model

Joint shear resistance is based on cohesion plus normal stress times tan(friction angle). Cohesion and pressure values are stored and computed directly in kPa, unit weights in kN/m3, and resulting force terms in kN.

Runout

Runout uses Gibson-style conical and pyramidal spill-berm calculations where the current geometry provides usable volume, length, and local slope angle inputs. The reported runout distance is the governing lesser value when both modes are valid.

In profile mode, the steepest exposed profile face is used as the local slope-angle approximation for runout notes.

Implemented Load Terms