Formula sheet

Mine Slope and Excavation Formula Sheet

Mine slope formulas for unit weight, stress, pore pressure, Mohr-Coulomb strength, safety factor, planar sliding, infinite slopes, earth pressure, monitoring, and risk.

Branch: Mining and Georesources Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.8 · reviewed

This formula sheet collects first-pass relationships used in mine slope, quarry face, excavation, and geotechnical risk checks. These equations support screening and review. They do not replace site-specific geological mapping, groundwater interpretation, laboratory testing, numerical modelling, monitoring, or competent geotechnical judgement.

State the failure mode, scale, drainage condition, unit system, groundwater assumption, material parameters, consequence category, and monitoring basis before using any result.

Unit weight and vertical stress

Unit weight:

\gamma=\rho g

where $\rho$ is density and $g$ is gravitational acceleration.

Vertical total stress at depth $z$ for uniform unit weight:

\sigma_v=\gamma z

Layered vertical stress:

\sigma_v=\sum_i \gamma_i h_i

where $h_i$ is the thickness of layer $i$ .

Use representative unit weights for the material state being analysed: intact rock, fractured rock mass, saturated soil, waste rock, backfill, or ore.

Hydrostatic pressure and effective stress

Hydrostatic pore pressure:

u=\gamma_w h_w

where $\gamma_w$ is unit weight of water and $h_w$ is pressure head.

Effective normal stress:

\sigma'=\sigma-u

Effective vertical stress:

\sigma'_v=\sigma_v-u

Water pressure in a tension crack of depth $h_w$ :

\displaystyle P_w=\frac{1}{2}\gamma_w h_w^2

This is the hydrostatic resultant per unit out-of-plane width. Multiply by the represented width when converting it to total force.

Pore pressure can reduce shear resistance even when total stress and slope geometry are unchanged.

Mohr-Coulomb shear strength

Effective-stress shear strength:

\tau_f=c'+\sigma'_n\tan\phi'

where $c'$ is effective cohesion, $\sigma'_n$ is effective normal stress, and $\phi'$ is effective friction angle.

For a discontinuity with no effective cohesion:

\tau_f=\sigma'_n\tan\phi'

Peak and residual parameters can be very different. Use parameters that match the expected mechanism, scale, roughness, infill, weathering, and displacement level.

Factor of safety

General factor of safety:

\displaystyle FS=\frac{\text{available resistance}}{\text{driving demand}}

For shear along a defined surface:

\displaystyle FS=\frac{\tau_f}{\tau_{mobilized}}

For force equilibrium:

\displaystyle FS=\frac{\sum R}{\sum D}

where $R$ represents resisting forces and $D$ represents driving forces.

Required factors of safety depend on consequence, uncertainty, design standard, monitoring, and operating state. A single $FS$ value is not enough unless the failure mechanism and assumptions are clear.

Planar sliding check

For a block sliding on a plane with dip angle $\alpha$ :

Driving force:

T=W\sin\alpha

Normal force without external loads:

N=W\cos\alpha

Effective normal force with water pressure resultant $U$ :

N'=N-U

Shear resistance:

R=c'A+N'\tan\phi'

Factor of safety:

\displaystyle FS=\frac{c'A+(W\cos\alpha-U)\tan\phi'}{W\sin\alpha}

where $A$ is sliding-plane area. Add surcharge, seismic, anchor, and support forces only when their direction and reliability are defined.

Infinite slope approximation

For a dry cohesion-friction slope with soil or weathered material thickness $z$ normal to slope:

\displaystyle FS=\frac{c'+\gamma z\cos^2\beta\tan\phi'}{\gamma z\sin\beta\cos\beta}

With pore pressure $u$ :

\displaystyle FS=\frac{c'+(\gamma z\cos^2\beta-u)\tan\phi'}{\gamma z\sin\beta\cos\beta}

where $\beta$ is slope angle.

The infinite slope model is a simplification for shallow translational failure. It is not appropriate for deep-seated, discontinuity-controlled, wedge, toppling, or circular failures without justification.

Overall slope geometry

Overall slope angle:

\beta=\operatorname{atan2}(H,L)

where $H$ is vertical height and $L$ is horizontal projection.

Bench face angle:

\theta_b=\operatorname{atan2}(H_b,L_b)

Inter-ramp slope angle depends on bench height, berm width, ramp interruption, and face angle. The built geometry should be checked against the design geometry because overbreak and poor scaling can reduce catch capacity.

Earth pressure checks

Rankine active earth pressure coefficient:

\displaystyle K_a=\frac{1-\sin\phi'}{1+\sin\phi'}

Rankine passive earth pressure coefficient:

\displaystyle K_p=\frac{1+\sin\phi'}{1-\sin\phi'}

Active lateral pressure at depth $z$ :

\sigma_h=K_a\gamma z

Resultant active force for dry level backfill:

\displaystyle P_a=\frac{1}{2}K_a\gamma H^2

Hydrostatic pressure must be added if drainage is not assured:

\displaystyle P_{water}=\frac{1}{2}\gamma_w H_w^2

Earth pressure models require assumptions about wall movement, drainage, surface slope, surcharge, and soil strength.

Surcharge and design loads

Uniform surcharge contribution to lateral pressure:

\Delta\sigma_h=Kq

where $q$ is surcharge and $K$ is the selected earth pressure coefficient.

Total surcharge lateral force over wall height $H$ :

P_q=KqH

Equipment, haul trucks, stockpiles, blast vibration, nearby structures, and temporary works can all act as surcharge or dynamic loading.

Support demand and resistance

Required support force for a simplified sliding block:

T_{support}\ge D-R

where $D$ is driving demand and $R$ is existing resistance without support.

Bolt or anchor load utilization:

\displaystyle U_L=\frac{T_{applied}}{T_{allowable}}

Support reliability depends on installation quality, bond length, corrosion protection, load direction, inspection access, and ground compatibility.

Monitoring rates

Displacement increment:

\Delta s=s_2-s_1

Average displacement velocity:

\displaystyle v=\frac{\Delta s}{\Delta t}

Average acceleration:

\displaystyle a=\frac{v_2-v_1}{\Delta t}

Rainfall intensity:

\displaystyle I=\frac{P}{\Delta t}

where $P$ is precipitation depth.

Trigger action plans may use displacement, velocity, acceleration, rainfall, pore pressure, rockfall frequency, or observed cracking. The trigger must specify the action, not only the measurement.

Numerical model checks

Residual norm:

\|r\|_2=\sqrt{\sum_i r_i^2}

Mesh refinement change in quantity of interest:

\displaystyle \Delta Q=\frac{|Q_{fine}-Q_{coarse}|}{|Q_{fine}|}

Finite element equilibrium form:

K u=f

where $K$ is stiffness matrix, $u$ is displacement vector, and $f$ is load vector.

Numerical convergence is not physical validation. Check boundary conditions, constitutive model, groundwater coupling, excavation stages, and mesh sensitivity.

Probabilistic stability

Estimated probability of instability from simulation:

\displaystyle \hat{P}(FS<1)=\frac{N_{FS<1}}{N}

Sample mean factor of safety:

\displaystyle \overline{FS}=\frac{1}{N}\sum_{i=1}^{N}FS_i

Sample standard deviation:

\displaystyle s_{FS}=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(FS_i-\overline{FS})^2}

Reliability index approximation:

\displaystyle \beta_R=\frac{\overline{FS}-1}{s_{FS}}

Probabilistic results depend strongly on input distributions, correlations, and data quality.

Risk ranking

Traditional risk priority number:

RPN=SOD

where $S$ is severity, $O$ is occurrence, and $D$ is detection ranking.

Simple risk expression:

Risk=P_f C

where $P_f$ is probability of failure and $C$ is consequence.

RPN is a screening tool, not a physical risk measure. High-consequence geotechnical hazards may require action even when estimated occurrence is low.

Validation checklist

Minimum calculation review should confirm:

Failure mode matches geology and geometry.
Strength parameters match drainage, scale, and displacement condition.
Groundwater assumptions are measured or conservatively bounded.
Surcharge, blast effects, and excavation sequence are represented.
Monitoring triggers are tied to defined actions.
Sensitivity checks identify controlling assumptions.
Field observations are used to update the geotechnical model.

The equations are useful only when the engineering model describes the slope that actually exists.

REF

Disciplines