Table of Symbols

The symbols used in this document are summarized in the Table of Symbols along with their units. Throughout the document, a subscript $i$ indicates that the value will be taken at, and analyzed at, a slice or slice interface composing the total slip mass. For vector quantities, the units shown are for each component of the vector.

Symbol	Description	Units
$A$	Area: A part of an object or surface.	${\text{m}^{2}}$
$\boldsymbol{a}\text{(}t\text{)}$	Acceleration: The rate of change of a body’s velocity.	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{b}$	Base Width of Slices: The width of each slice in the $x$ -direction.	${\text{m}}$
${\boldsymbol{C}_{\text{den}}}$	Proportionality Constant Denominator: Values for each slice that sum together to form the denominator of the interslice normal to shear force proportionality constant.	${\text{N}}$
${\boldsymbol{C}_{\text{num}}}$	Proportionality Constant Numerator: Values for each slice that sum together to form the numerator of the interslice normal to shear force proportionality constant.	${\text{N}}$
$c’$	Effective Cohesion: The internal pressure that sticks particles of soil together.	${\text{Pa}}$
$\mathit{const_f}$	Decision on F: A Boolean decision on which form of f the user desires: constant if true, or half-sine if false.	–
${F_{\text{n}}}$	Total Normal Force: Component of a force in the normal direction.	${\text{N}}$
${F_{\text{rot}}}$	Force Causing Rotation: A force in the direction of rotation.	${\text{N}}$
${F_{\text{S}}}$	Factor of Safety: The global stability metric of a slip surface of a slope, defined as the ratio of resistive shear force to mobilized shear force.	–
${F_{\text{t}}}$	Tangential Force: Component of a force in the tangential direction.	${\text{N}}$
${F_{\text{x}}}$	$x$ -coordinate of the Force: The force acting in the $x$ -direction.	${\text{N}}$
${F_{\text{y}}}$	$y$ -coordinate of the Force: The force acting in the $y$ -direction.	${\text{N}}$
$\boldsymbol{F}$	Force: An interaction that tends to produce change in the motion of an object.	${\text{N}}$
${{\boldsymbol{F}_{\text{x}}}^{\text{G}}}$	Sums of the Interslice Normal Forces: The sums of the normal forces acting on each pair of adjacent interslice boundaries.	${\text{N}}$
${{\boldsymbol{F}_{\text{x}}}^{\text{H}}}$	Sums of the Interslice Normal Water Forces: The sums of the normal water forces acting on each pair of adjacent interslice boundaries.	${\text{N}}$
$\boldsymbol{f}$	Interslice Normal to Shear Force Ratio Variation Function: A function of distance in the $x$ -direction that describes the variation of the interslice normal to shear ratio.	–
$\boldsymbol{G}$	Interslice Normal Forces: The forces per meter in the $z$ -direction exerted between each pair of adjacent slices.	$\frac{\text{N}}{\text{m}}$
$\boldsymbol{g}$	Gravitational Acceleration: The approximate acceleration due to gravity on Earth at sea level.	$\frac{\text{m}}{\text{s}^{2}}$
$\boldsymbol{H}$	Interslice Normal Water Forces: The normal water forces per meter in the $z$ -direction exerted in the $x$ -direction between each pair of adjacent slices.	$\frac{\text{N}}{\text{m}}$
$h$	Height: The distance above a reference point for a point of interest.	${\text{m}}$
$\boldsymbol{h}$	$y$ -direction Heights of Slices: The heights in the $y$ -direction from the base of each slice to the slope surface, at the $x$ -direction midpoint of the slice.	${\text{m}}$
${\boldsymbol{h}^{\text{L}}}$	Heights of the Left Side of Slices: The heights of the left side of each slice, assuming slice surfaces have negative slope.	${\text{m}}$
${\boldsymbol{h}^{\text{R}}}$	Heights of the Right Side of Slices: The heights of the right side of each slice, assuming slice surfaces have negative slope.	${\text{m}}$
${\boldsymbol{h}_{\text{z}}}$	Heights of Interslice Normal Forces: The heights in the $y$ -direction of the interslice normal forces on each slice.	${\text{m}}$
${\boldsymbol{h}_{\text{z,w}}}$	Heights of the Water Table: The heights in the $y$ -direction from the base of each slice to the water table.	${\text{m}}$
$i$	Index: A number representing a single slice.	–
$\boldsymbol{\hat{j}}$	Unit Vector: A vector that has a magnitude of one.	–
${K_{\text{c}}}$	Seismic Coefficient: The proportionality factor of force that weight pushes outwards; caused by seismic earth movements.	–
${\boldsymbol{L}_{b}}$	Total Base Lengths of Slices: The lengths of each slice in the direction parallel to the slope of the base.	${\text{m}}$
${\boldsymbol{L}_{s}}$	Surface Lengths of Slices: The lengths of each slice in the direction parallel to the slope of the surface.	${\text{m}}$
$M$	Moment: A measure of the tendency of a body to rotate about a specific point or axis.	$\text{N}\text{m}$
$m$	Mass: The quantity of matter in a body.	${\text{kg}}$
$\boldsymbol{N}$	Normal Forces: The total reactive forces per meter in the $z$ -direction for each slice of a soil surface subject to a body resting on it.	$\frac{\text{N}}{\text{m}}$
$\boldsymbol{N’}$	Effective Normal Forces: The forces per meter in the $z$ -direction for each slice of a soil surface, subtracting pore water reactive force from total reactive force.	$\frac{\text{N}}{\text{m}}$
$n$	Number of Slices: The number of slices into which the slip surface is divided.	–
$P$	Resistive Shear Force: The Mohr Coulomb frictional force that describes the limit of mobilized shear force that can be withstood before failure.	${\text{N}}$
$\boldsymbol{P}$	Resistive Shear Forces: The Mohr Coulomb frictional forces per meter in the $z$ -direction for each slice that describe the limit of mobilized shear force the slice can withstand before failure.	$\frac{\text{N}}{\text{m}}$
$p$	Pressure: A force exerted over an area.	${\text{Pa}}$
$\boldsymbol{Q}$	External Forces: The forces per meter in the $z$ -direction acting into the surface from the midpoint of each slice.	$\frac{\text{N}}{\text{m}}$
$\boldsymbol{R}$	Resistive Shear Forces Without the Influence of Interslice Forces: The resistive shear forces per meter without the influence of interslice forces in the $z$ -direction for each slice.	$\frac{\text{N}}{\text{m}}$
$r$	Length of the Moment Arm: The distance between a force causing rotation and the axis of rotation.	${\text{m}}$
$\boldsymbol{r}$	Position Vector: A vector from the origin of the Cartesian coordinate system defined to the point where the force is applied.	${\text{m}}$
$S$	Mobilized Shear Force: The shear force in the direction of potential motion.	${\text{N}}$
$\boldsymbol{S}$	Mobilized Shear Force: The mobilized shear force per meter in the $z$ -direction for each slice.	$\frac{\text{N}}{\text{m}}$
$\boldsymbol{T}$	Mobilized Shear Forces Without the Influence of Interslice Forces: The mobilized shear forces per meter without the influence of interslice forces in the $z$ -direction for each slice.	$\frac{\text{N}}{\text{m}}$
${\boldsymbol{U}_{\text{b}}}$	Base Hydrostatic Forces: The forces per meter in the $z$ -direction from water pressure within each slice.	$\frac{\text{N}}{\text{m}}$
${\boldsymbol{U}_{\text{g}}}$	Surface Hydrostatic Forces: The forces per meter in the $z$ -direction from water pressure acting into each slice from standing water on the slope surface.	$\frac{\text{N}}{\text{m}}$
$u$	Pore Pressure: The pressure that comes from water within the soil.	${\text{Pa}}$
$\boldsymbol{u}$	Displacement: The change in an object’s location relative to a reference point.	${\text{m}}$
$V$	Volume: The amount of space that a substance or object occupies.	${\text{m}^{3}}$
${\boldsymbol{V}_{\text{dry}}}$	Volumes of Dry Soil: The amount of space occupied by dry soil for each slice.	${\text{m}^{3}}$
${\boldsymbol{V}_{\text{sat}}}$	Volumes of Saturated Soil: The amount of space occupied by saturated soil for each slice.	${\text{m}^{3}}$
$v$	Local Index: Used as a bound variable index in calculations.	–
$W$	Weight: The gravitational force acting on an object.	${\text{N}}$
$\boldsymbol{W}$	Weights: The downward force per meter in the $z$ -direction on each slice caused by gravity.	$\frac{\text{N}}{\text{m}}$
$\boldsymbol{X}$	Interslice Shear Forces: The shear forces per meter in the $z$ -direction exerted between adjacent slices.	$\frac{\text{N}}{\text{m}}$
$x$	$x$ -coordinate: The $x$ -coordinate in the Cartesian coordinate system.	${\text{m}}$
${{x_{\text{slip}}}^{\text{maxEtr}}}$	Maximum Entry $x$ -coordinate: The maximum potential $x$ -coordinate for the entry point of a slip surface.	${\text{m}}$
${{x_{\text{slip}}}^{\text{maxExt}}}$	Maximum Exit $x$ -coordinate: The maximum potential $x$ -coordinate for the exit point of a slip surface.	${\text{m}}$
${{x_{\text{slip}}}^{\text{minEtr}}}$	Minimum Entry $x$ -coordinate: The minimum potential $x$ -coordinate for the entry point of a slip surface.	${\text{m}}$
${{x_{\text{slip}}}^{\text{minExt}}}$	Minimum Exit $x$ -coordinate: The minimum potential $x$ -coordinate for the exit point of a slip surface.	${\text{m}}$
${\boldsymbol{x}_{\text{cs}}}\text{,}{\boldsymbol{y}_{\text{cs}}}$	Critical Slip Surface Coordinates: The set of $x$ -coordinates and $y$ -coordinates that describe the vertices of the critical slip surface.	${\text{m}}$
${\boldsymbol{x}_{\text{slip}}}$	$x$ -coordinates of the Slip Surface: $x$ -coordinates of points on the slip surface.	${\text{m}}$
${\boldsymbol{x}_{\text{slope}}}$	$x$ -coordinates of the Slope: $x$ -coordinates of points on the soil slope.	${\text{m}}$
${\boldsymbol{x}_{\text{wt}}}$	$x$ -coordinates of the Water Table: X-positions of the water table.	${\text{m}}$
$y$	$y$ -coordinate: The $y$ -coordinate in the Cartesian coordinate system.	${\text{m}}$
${{y_{\text{slip}}}^{\text{max}}}$	Maximum $y$ -coordinate: The maximum potential $y$ -coordinate of a point on a slip surface.	${\text{m}}$
${{y_{\text{slip}}}^{\text{min}}}$	Minimum $y$ -coordinate: The minimum potential $y$ -coordinate of a point on a slip surface.	${\text{m}}$
${\boldsymbol{y}_{\text{slip}}}$	$y$ -coordinates of the Slip Surface: Heights of the slip surface.	${\text{m}}$
${\boldsymbol{y}_{\text{slope}}}$	$y$ -coordinates of the Slope: $y$ -coordinates of points on the soil slope.	${\text{m}}$
${\boldsymbol{y}_{\text{wt}}}$	$y$ -coordinates of the Water Table: Heights of the water table.	${\text{m}}$
$z$	$z$ -coordinate: The $z$ -coordinate in the Cartesian coordinate system.	${\text{m}}$
$\boldsymbol{α}$	Base Angles: The angles between the base of each slice and the horizontal.	${{}^{\circ}}$
$\boldsymbol{β}$	Surface Angles: The angles between the surface of each slice and the horizontal.	${{}^{\circ}}$
$γ$	Specific Weight: The weight per unit volume.	$\frac{\text{N}}{\text{m}^{3}}$
${γ_{\text{dry}}}$	Soil Dry Unit Weight: The weight of a dry soil/ground layer divided by the volume of the layer.	$\frac{\text{N}}{\text{m}^{3}}$
${γ_{\text{sat}}}$	Soil Saturated Unit Weight: The weight of saturated soil/ground layer divided by the volume of the layer.	$\frac{\text{N}}{\text{m}^{3}}$
${γ_{w}}$	Unit Weight of Water: The weight of one cubic meter of water.	$\frac{\text{N}}{\text{m}^{3}}$
$λ$	Proportionality Constant: The ratio of the interslice normal to the interslice shear force.	–
$π$	Ratio of Circumference to Diameter for Any Circle: The ratio of a circle’s circumference to its diameter.	–
$ρ$	Density: The mass per unit volume.	$\frac{\text{kg}}{\text{m}^{3}}$
$σ$	Total Normal Stress: The total force per area acting on the soil mass.	${\text{Pa}}$
$σ’$	Effective Stress: The stress in a soil mass that is effective in causing volume changes and mobilizes the shear strength arising from friction; represents the average stress carried by the soil skeleton.	${\text{Pa}}$
${σ_{N}}’$	Effective Normal Stress: The normal stress in a soil mass that is effective in causing volume changes; represents the average normal stress carried by the soil skeleton.	${\text{Pa}}$
$τ$	Tangential Stress: The shear force per unit area.	${\text{Pa}}$
${τ^{\text{f}}}$	Shear Strength: The strength of a material against shear failure.	${\text{Pa}}$
$\boldsymbol{τ}$	Torque: A twisting force that tends to cause rotation.	$\text{N}\text{m}$
$\boldsymbol{Φ}$	First Function for Incorporating Interslice Forces Into Shear Force: The function for converting resistive shear without the influence of interslice forces, to a calculation considering the interslice forces.	–
$φ’$	Effective Angle of Friction: The angle of inclination with respect to the horizontal axis of the Mohr-Coulomb shear resistance line.	${{}^{\circ}}$
$\boldsymbol{Ψ}$	Second Function for Incorporating Interslice Forces Into Shear Force: The function for converting mobile shear without the influence of interslice forces, to a calculation considering the interslice forces.	–
$\boldsymbol{ω}$	Imposed Load Angles: The angles between the external force acting into the surface of each slice and the vertical.	${{}^{\circ}}$

Table of Symbols

Software Requirements Specification for Slope Stability analysis Program

Table of Symbols