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