Notation and glossary — OGR Slip2D documentation

The documentation uses a single notation from the first chapter to the last. Below, the symbols are grouped by subject with their meanings and units, and a glossary defines the slope-stability and program terms used in the other pages. Where a symbol has two meanings in different chapters, both are given.

5.6.1Conventions#

Units. SI throughout: lengths in m, forces in kN per metre of slope (the analysis is two-dimensional, in plane strain), stresses and pressures in kPa, unit weights in kN/m³, angles in degrees. The program stores and computes in these units whatever unit system its window displays.
Axes. $x$ grows to the right and $y$ upwards; a slope fails from left to right when the ground falls to the right.
Effective quantities carry a prime: $c'$ , $\phi'$ , $\sigma'_n$ .
Slices are numbered $i = 1 \dots n$ ; a symbol with the index $i$ belongs to slice $i$ , and vectors are set in bold ( $\mathbf{W}_i$ ).
A dash (—) in the unit column marks a dimensionless quantity.

5.6.2Symbols#

Slices, forces and the factor of safety#

Table 5.6.1 Symbols of the method of slices.

Symbol	Meaning	Unit
$F$	factor of safety	—
$F_f,\ F_m$	factor of safety from force equilibrium and from moment equilibrium	—
$F_0$	factor of safety of Janbu's simplified method, before the correction	—
$f_0$	Janbu's correction factor, $f_0 = 1 + b_1\left[d/L - 1.4\,(d/L)^2\right]$	—
$b_1$	coefficient of $f_0$ for the type of soil: 0.69 cohesion only, 0.31 friction only, 0.50 both	—
$d,\ L$	largest perpendicular distance from the chord to the slip surface, and length of the chord joining the two ends of the surface	m
$n,\ i$	number of slices; index of a slice	—
$W$	weight of the slice (vertical load on it)	kN/m
$b$	width of the slice	m
$l$	length of the base of the slice, $l = b\sec\alpha$	m
$\alpha$	inclination of the base of the slice	°
$\beta$	inclination of the ground surface above the slice (for the reliability index, see Table 5.6.4)	°
$R$	radius of a circular slip surface	m
$N,\ N'$	total and effective normal force on the base	kN/m
$S,\ S_m$	available and mobilised shear force on the base, $S_m = S/F$	kN/m
$U$	pore-water force on the base, $U = u\,l$	kN/m
$E,\ X$	interslice normal and shear forces	kN/m
$Z,\ \theta$	interslice resultant and its inclination (for the volumetric water content, see Table 5.6.2)	kN/m, °
$\lambda$	scale factor of the interslice force function, $X = \lambda f(x)\,E$	—
$f(x)$	interslice force function ( $f \equiv 1$ in Spencer's method)	—
$t$	position along the slip surface, normalised from 0 at one end to 1 at the other, at which $f$ is evaluated	—
$m_\alpha$	denominator of the base normal force in Bishop's method, $m_\alpha = \cos\alpha + \sin\alpha\tan\phi'/F$	—
$n_\alpha$	the same in Janbu's methods, $n_\alpha = m_\alpha\cos\alpha$	—
$H$	net external horizontal force on a slice, positive in the sense of sliding (for the total head, see Table 5.6.2)	kN/m
$T$	support force	kN/m
$k_h,\ k_v$	horizontal and vertical seismic coefficients; $k_v$ positive downwards	—
$\varepsilon$	convergence tolerance	—
$\sigma_n,\ \sigma'_n$	total and effective normal stress on the base	kPa
$\tau,\ \tau_f,\ \tau_m$	shear stress; shear strength; mobilised shear stress, $\tau_m = \tau_f/F$	kPa
$\sigma_t$	tensile strength allowed at a slice base	kPa

Water, pore pressure and seepage#

Table 5.6.2 Symbols of pore pressure, groundwater and unsaturated soil.

Symbol	Meaning	Unit
$u$	pore pressure (at the middle of the base of a slice)	kPa
$\gamma_w$	unit weight of water (9.81 by default)	kN/m³
$h$	vertical distance from a point up to the water table or piezometric line of its material	m
$H_u$	pore-pressure coefficient of a water surface, $u = \gamma_w H_u h$ ; 1 by default, $\cos^2$ of the inclination of the water surface with Auto $H_u$	—
$r_u$	pore-pressure ratio, $u = r_u\,\sigma_v$ , with $\sigma_v$ the vertical stress over the point: its depth below the ground surface times the unit weight of its material, plus any water ponded above; external loads are not included	—
$\bar{B}$	Skempton's coefficient for the pore pressure of an undrained change of vertical stress, $\Delta u = \bar{B}\,\Delta\sigma_v$	—
$H$	total head, $H = y + P$	m
$P$	pressure head, $P = u/\gamma_w$	m
$\psi$	suction, positive where the pore pressure is negative: suction head or matric suction, according to the permeability model	m or kPa
$k_s$	saturated hydraulic conductivity (permeability)	m/s
$k_r$	relative hydraulic conductivity of the unsaturated soil, $k = k_r k_s$	—
$\theta$	volumetric water content, in the seepage equations	—
$\phi^b$	angle of the increase of shear strength with matric suction (`phi_b` in the program)	°
$\mathrm{AEV}$	air-entry value: matric suction up to which the soil is treated as saturated, the suction acting through $\phi'$ ; the suction above it adds a cohesion $(\psi - \mathrm{AEV})\tan\phi^b$	kPa

Strength models#

Table 5.6.3 Symbols of the strength models.

Symbol	Meaning	Unit
$c',\ \phi'$	effective cohesion and effective friction angle	kPa, °
$c_u\ (s_u)$	undrained shear strength	kPa
$\gamma,\ \gamma_{sat}$	unit weight, and saturated unit weight used below the water surface when the material asks for it	kN/m³
$\sigma'_v$	effective vertical stress	kPa
$\sigma_{ci}$	uniaxial compressive strength of the intact rock	kPa
$\mathrm{GSI}$	Geological Strength Index of the rock mass	—
$m_i$	Hoek–Brown constant of the intact rock	—
$D$	disturbance factor of the rock mass	—
$m_b,\ s,\ a$	Hoek–Brown constants of the rock mass: $m_b = m_i\,e^{(\mathrm{GSI}-100)/(28-14D)}$ , $s = e^{(\mathrm{GSI}-100)/(9-3D)}$ , $a = \tfrac12 + \tfrac16\left(e^{-\mathrm{GSI}/15} - e^{-20/3}\right)$	—
$m$	constant of the classic Hoek–Brown criterion (with $s$ )	—
$\mathrm{JRC}$	joint roughness coefficient	—
$\mathrm{JCS}$	joint wall compressive strength	kPa
$\phi_r$	residual friction angle of a joint	°
$A,\ S,\ m,\ \mathrm{OCR}$	SHANSEP constant term, strength ratio, exponent and overconsolidation ratio: $s_u = A + \sigma'_v\,S\,\mathrm{OCR}^m$ , with a lower bound	—
$K$	vertical stress ratio: $\tau_f = K\,\sigma'_v$ , with a minimum strength	—

Seismic, probabilistic and design symbols#

Table 5.6.4 Symbols of seismic, probabilistic and partial-factor analysis.

Symbol	Meaning	Unit
$K_y$	critical (yield) seismic coefficient: the $k_h$ that brings $F$ down to its target value, 1 by default	—
$\mathrm{PF}$	probability of failure: fraction of the samples with $F < 1$	—
$\beta$	reliability index, $\beta = (\mu_F - 1)/\sigma_F$ for a normal distribution of $F$ , or computed from $\ln F$ for a lognormal one	—
$\mu_F,\ \sigma_F$	mean and standard deviation of the sampled factors of safety	—
$\gamma_G,\ \gamma_Q$	partial factors on permanent and on variable actions	—
$\gamma_{c'},\ \gamma_{\phi'}$	partial factors on the effective cohesion and on $\tan\phi'$	—
$\gamma_{cu}$	partial factor on the undrained shear strength	—
$\gamma_{\gamma}$	partial factor on the unit weight	—
$\gamma_{R;e}$	partial resistance factor of a slope (earth resistance)	—
$\gamma_{\tau}$	factor that OGR Slip2D applies to the strength of models that cannot be split into $c'$ and $\tan\phi'$ ; EN 1997-1 does not define it	—

5.6.3Glossary#

Table 5.6.5 Terms used in this documentation.

Term	Definition
Active support	A support whose force reduces the driving force; it is not divided by the factor of safety.
Admissibility check	A check made after the factor of safety of a surface has converged (tensile stress, m-alpha) that can reject the surface.
Agent bridge	The local connection through which an MCP server works on the model open in the program window.
Back analysis	The calculation of the horizontal support force needed to reach a target factor of safety.
Block Search object	A window, line, point or polyline from which the Block Search takes the vertices of non-circular trial surfaces.
Critical surface	The admissible slip surface with the lowest factor of safety among those analysed (with the lowest $K_y$ in a seismic coefficient analysis).
Design approach	A combination of partial factors of Eurocode 7: DA1 combinations 1 and 2, DA2 and DA3.
Drawdown line	The lowered water level of a rapid-drawdown analysis.
Effective stress	Total stress minus pore pressure; the stress that governs the drained strength of a soil.
External boundary	The closed polygon that outlines the whole soil or rock mass of a model.
Factor of safety	The number by which the shear strength must be divided for the sliding mass to be just in equilibrium.
Failure direction	The direction in which the mass slides: left to right when the ground falls to the right.
Focus object	A window, line, point or tangent that restricts the trial surfaces a search considers, before they are evaluated.
Grid search	The circular search over a grid of slip-circle centres, with several radii per centre.
Hydraulic conductivity	The ratio between the flow per unit area and the hydraulic gradient (Darcy); also called permeability in this documentation.
Interslice force	The force exerted between two neighbouring slices, with a normal component $E$ and a shear component $X$ .
Limit equilibrium	The state in which the shear strength divided by the factor of safety just balances the forces that drive the sliding mass.
Line of thrust	The line through the points of application of the interslice resultants.
Material boundary	An open polyline, from edge to edge of the external boundary, that divides the mass into regions; a closed one inside the mass is a lens.
Matric suction	Pore-air pressure minus pore-water pressure; in the program, the negative pore pressure of an unsaturated zone, taken as positive.
Method of slices	The division of a sliding mass into vertical slices whose equilibrium is written one by one.
Mobilised shear strength	The shear strength divided by the factor of safety: the part of it needed for equilibrium.
Newmark displacement	The permanent displacement of a rigid block sliding on the slip surface, the double integral of the part of an acceleration record above the critical acceleration.
Over-design factor	What the analysis reports when partial factors are applied to its inputs: it must exceed 1, and it is not a factor of safety.
Partial factor	A factor that divides a strength, or multiplies an action, in a design to a standard.
Passive support	A support whose force is added to the resisting force, and so is divided by the factor of safety.
Phreatic surface	The surface at which the pore pressure is zero (atmospheric); in a section, the water table.
Piezometric line	A line whose height above a point gives the pore pressure there, $u = \gamma_w H_u h$ , for the materials assigned to it.
Pore pressure	The pressure of the water in the pores of the ground.
Probability of failure	The fraction of the Monte Carlo or Latin hypercube samples whose factor of safety is below 1.
Pseudo-static analysis	A representation of earthquake loading by constant forces proportional to the static weight, $k_h W$ and $k_v W$ .
Rapid drawdown	A lowering of the water level against a slope faster than the soil can drain, so that its pore pressures do not follow the new level.
Region	An area of the model bounded by the external and material boundaries and filled by one material; regions are computed, not drawn.
Reliability index	The number of standard deviations that separate the mean factor of safety from 1.
Seepage face	The part of the boundary where water leaves the ground at atmospheric pressure; its extent is part of the solution.
Sensitivity analysis	The factor of safety of the critical surface recomputed as one input runs through its range, the others held at their mean values.
Slice	A vertical strip of the sliding mass, narrow enough for its base to be treated as straight.
Slip surface	The surface along which the mass is assumed to slide: a circle, or a polyline for a non-circular surface.
Slope limits	The stretch of ground surface within which a trial surface must start and end.
Stale result	A result computed on a model that has changed since.
Tension crack	A vertical crack at the crest, with no tensile strength, that may hold water.
Transient analysis	A groundwater analysis in stages over time, with a factor of safety for each stage that asks for one.
Water table	The line of the phreatic surface; below it, $u = \gamma_w H_u h$ for the materials assigned to it.
Weak layer	A polyline with a strength of its own, along which a non-circular slip surface can travel instead of crossing it.

The sources of the methods and models named on this page are listed in Section 5.7; the theory behind each symbol is in chapter 3, starting with Section 3.1.