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Data Definitions

This section collects and defines all the data needed to build the instance models.

Interslice normal water forces

Refname	DD:intersliceWtrF
Label	Interslice normal water forces
Symbol	$\boldsymbol{H}$
Units	$\frac{\text{N}}{\text{m}}$
Equation	$\boldsymbol{H}=\begin{cases}\frac{\left({\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}+\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slope},i}}\right)^{2}\,{γ_{w}}, & {\boldsymbol{y}_{\text{wt},i}}\geq{}{\boldsymbol{y}_{\text{slope},i}}\\\frac{\left({\boldsymbol{y}_{\text{wt},i}}-{\boldsymbol{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}, & {\boldsymbol{y}_{\text{slope},i}}\gt{}{\boldsymbol{y}_{\text{wt},i}}\land{}{\boldsymbol{y}_{\text{wt},i}}\gt{}{\boldsymbol{y}_{\text{slip},i}}\\0, & {\boldsymbol{y}_{\text{wt},i}}\leq{}{\boldsymbol{y}_{\text{slip},i}}\end{cases}$
Description	$\boldsymbol{H}$ is the interslice normal water forces ( $\frac{\text{N}}{\text{m}}$ ) ${\boldsymbol{y}_{\text{slope}}}$ is the $y$ -coordinates of the slope ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{y}_{\text{slip}}}$ is the $y$ -coordinates of the slip surface ( ${\text{m}}$ ) ${γ_{w}}$ is the unit weight of water ( $\frac{\text{N}}{\text{m}^{3}}$ ) ${\boldsymbol{y}_{\text{wt}}}$ is the $y$ -coordinates of the water table ( ${\text{m}}$ )
Source	fredlund1977
RefBy	GD:resShearWO, IM:nrmShrForNum, and GD:mobShearWO

Base angles

Refname	DD:angleA
Label	Base angles
Symbol	$\boldsymbol{α}$
Units	${{}^{\circ}}$
Equation	$\boldsymbol{α}=\arctan\left(\frac{{\boldsymbol{y}_{\text{slip},i}}-{\boldsymbol{y}_{\text{slip},i-1}}}{{\boldsymbol{x}_{\text{slip},i}}-{\boldsymbol{x}_{\text{slip},i-1}}}\right)$
Description	$\boldsymbol{α}$ is the base angles ( ${{}^{\circ}}$ ) ${\boldsymbol{y}_{\text{slip}}}$ is the $y$ -coordinates of the slip surface ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{x}_{\text{slip}}}$ is the $x$ -coordinates of the slip surface ( ${\text{m}}$ )
Notes	This equation is based on the assumption that the base of a slice is a straight line (A:Surface-Base-Slice-between-Interslice-Straight-Lines).
Source	fredlund1977
RefBy	GD:resShearWO, IM:nrmShrForNum, GD:normForcEq, GD:momentEql, GD:mobShearWO, GD:bsShrFEq, DD:lengthLb, DD:convertFunc2, and DD:convertFunc1

Surface angles

Refname	DD:angleB
Label	Surface angles
Symbol	$\boldsymbol{β}$
Units	${{}^{\circ}}$
Equation	$\boldsymbol{β}=\arctan\left(\frac{{\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{slope},i-1}}}{{\boldsymbol{x}_{\text{slope},i}}-{\boldsymbol{x}_{\text{slope},i-1}}}\right)$
Description	$\boldsymbol{β}$ is the surface angles ( ${{}^{\circ}}$ ) ${\boldsymbol{y}_{\text{slope}}}$ is the $y$ -coordinates of the slope ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{x}_{\text{slope}}}$ is the $x$ -coordinates of the slope ( ${\text{m}}$ )
Notes	This equation is based on the assumption that the surface of a slice is a straight line (A:Surface-Base-Slice-between-Interslice-Straight-Lines).
Source	fredlund1977
RefBy	GD:resShearWO, IM:nrmShrForNum, GD:normForcEq, GD:momentEql, GD:mobShearWO, GD:bsShrFEq, and DD:lengthLs

Base width of slices

Refname	DD:lengthB
Label	Base width of slices
Symbol	$\boldsymbol{b}$
Units	${\text{m}}$
Equation	$\boldsymbol{b}={\boldsymbol{x}_{\text{slip},i}}-{\boldsymbol{x}_{\text{slip},i-1}}$
Description	$\boldsymbol{b}$ is the base width of slices ( ${\text{m}}$ ) ${\boldsymbol{x}_{\text{slip}}}$ is the $x$ -coordinates of the slip surface ( ${\text{m}}$ ) $i$ is the index (Unitless)
Source	fredlund1977
RefBy	GD:sliceWght, IM:nrmShrForNum, IM:nrmShrForDen, GD:momentEql, DD:lengthLs, and DD:lengthLb

Total base lengths of slices

Refname	DD:lengthLb
Label	Total base lengths of slices
Symbol	${\boldsymbol{L}_{b}}$
Units	${\text{m}}$
Equation	${\boldsymbol{L}_{b}}={\boldsymbol{b}}_{i}\,\sec\left({\boldsymbol{α}}_{i}\right)$
Description	${\boldsymbol{L}_{b}}$ is the total base lengths of slices ( ${\text{m}}$ ) $\boldsymbol{b}$ is the base width of slices ( ${\text{m}}$ ) $i$ is the index (Unitless) $\boldsymbol{α}$ is the base angles ( ${{}^{\circ}}$ )
Notes	$\boldsymbol{b}$ is defined in DD:lengthB and $\boldsymbol{α}$ is defined in DD:angleA.
Source	fredlund1977
RefBy	GD:resShr, GD:resShearWO, GD:mobShr, and GD:baseWtrF

Surface lengths of slices

Refname	DD:lengthLs
Label	Surface lengths of slices
Symbol	${\boldsymbol{L}_{s}}$
Units	${\text{m}}$
Equation	${\boldsymbol{L}_{s}}={\boldsymbol{b}}_{i}\,\sec\left({\boldsymbol{β}}_{i}\right)$
Description	${\boldsymbol{L}_{s}}$ is the surface lengths of slices ( ${\text{m}}$ ) $\boldsymbol{b}$ is the base width of slices ( ${\text{m}}$ ) $i$ is the index (Unitless) $\boldsymbol{β}$ is the surface angles ( ${{}^{\circ}}$ )
Notes	$\boldsymbol{b}$ is defined in DD:lengthB and $\boldsymbol{β}$ is defined in DD:angleB.
Source	fredlund1977
RefBy	GD:srfWtrF

$y$ -direction heights of slices

Refname	DD:slcHeight
Label	$y$ -direction heights of slices
Symbol	$\boldsymbol{h}$
Units	${\text{m}}$
Equation	$\boldsymbol{h}=\frac{1}{2}\,\left({{\boldsymbol{h}^{\text{R}}}}_{i}+{{\boldsymbol{h}^{\text{L}}}}_{i}\right)$
Description	$\boldsymbol{h}$ is the $y$ -direction heights of slices ( ${\text{m}}$ ) ${\boldsymbol{h}^{\text{R}}}$ is the heights of the right side of slices ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{h}^{\text{L}}}$ is the heights of the left side of slices ( ${\text{m}}$ )
Notes	This equation is based on the assumption that the surface and base of a slice are straight lines (A:Surface-Base-Slice-between-Interslice-Straight-Lines). ${\boldsymbol{h}^{\text{R}}}$ and ${\boldsymbol{h}^{\text{L}}}$ are defined in DD:sliceHghtRightDD and DD:sliceHghtLeftDD, respectively.
Source	fredlund1977
RefBy	IM:nrmShrForNum and GD:momentEql

Total normal stress

Refname	DD:normStress
Label	Total normal stress
Symbol	$σ$
Units	${\text{Pa}}$
Equation	$σ=\frac{{F_{\text{n}}}}{A}$
Description	$σ$ is the total normal stress ( ${\text{Pa}}$ ) ${F_{\text{n}}}$ is the total normal force ( ${\text{N}}$ ) $A$ is the area ( ${\text{m}^{2}}$ )
Source	huston2008
RefBy	GD:resShr, TM:effStress, and GD:effNormF

Tangential stress

Refname	DD:tangStress
Label	Tangential stress
Symbol	$τ$
Units	${\text{Pa}}$
Equation	$τ=\frac{{F_{\text{t}}}}{A}$
Description	$τ$ is the tangential stress ( ${\text{Pa}}$ ) ${F_{\text{t}}}$ is the tangential force ( ${\text{N}}$ ) $A$ is the area ( ${\text{m}^{2}}$ )
Source	huston2008
RefBy	GD:resShr

Torque

Refname	DD:torque
Label	Torque
Symbol	$\boldsymbol{τ}$
Units	$\text{N}\text{m}$
Equation	$\boldsymbol{τ}=\boldsymbol{r}\times\boldsymbol{F}$
Description	$\boldsymbol{τ}$ is the torque ( $\text{N}\text{m}$ ) $\boldsymbol{r}$ is the position vector ( ${\text{m}}$ ) $\boldsymbol{F}$ is the force ( ${\text{N}}$ )
Notes	The torque on a body measures the tendency of a force to rotate the body around an axis or pivot.
Source	–
RefBy	GD:momentEql

Interslice normal to shear force ratio variation function

Refname	DD:ratioVariation
Label	Interslice normal to shear force ratio variation function
Symbol	$\boldsymbol{f}$
Units	Unitless
Equation	$\boldsymbol{f}=\begin{cases}1, & \mathit{const_f}\\\sin\left(π\,\frac{{\boldsymbol{x}_{\text{slip},i}}-{\boldsymbol{x}_{\text{slip},0}}}{{\boldsymbol{x}_{\text{slip},n}}-{\boldsymbol{x}_{\text{slip},0}}}\right), & \neg{}\mathit{const_f}\end{cases}$
Description	$\boldsymbol{f}$ is the interslice normal to shear force ratio variation function (Unitless) $π$ is the ratio of circumference to diameter for any circle (Unitless) ${\boldsymbol{x}_{\text{slip}}}$ is the $x$ -coordinates of the slip surface ( ${\text{m}}$ ) $i$ is the index (Unitless) $n$ is the number of slices (Unitless) $\mathit{const_f}$ is the decision on f (Unitless)
Source	fredlund1977
RefBy	IM:nrmShrForDen, GD:normShrR, DD:convertFunc2, and DD:convertFunc1

First function for incorporating interslice forces into shear force

Refname	DD:convertFunc1
Label	First function for incorporating interslice forces into shear force
Symbol	$\boldsymbol{Φ}$
Units	Unitless
Equation	$\boldsymbol{Φ}=\left(λ\,{\boldsymbol{f}}_{i}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}$
Description	$\boldsymbol{Φ}$ is the first function for incorporating interslice forces into shear force (Unitless) $λ$ is the proportionality constant (Unitless) $\boldsymbol{f}$ is the interslice normal to shear force ratio variation function (Unitless) $i$ is the index (Unitless) $\boldsymbol{α}$ is the base angles ( ${{}^{\circ}}$ ) $φ’$ is the effective angle of friction ( ${{}^{\circ}}$ ) ${F_{\text{S}}}$ is the factor of safety (Unitless)
Notes	$\boldsymbol{f}$ is defined in DD:ratioVariation and $\boldsymbol{α}$ is defined in DD:angleA.
Source	chen2005 and karchewski2012
RefBy	IM:intsliceFs, IM:fctSfty, and DD:convertFunc2

Second function for incorporating interslice forces into shear force

Refname	DD:convertFunc2
Label	Second function for incorporating interslice forces into shear force
Symbol	$\boldsymbol{Ψ}$
Units	Unitless
Equation	$\boldsymbol{Ψ}=\frac{\left(λ\,{\boldsymbol{f}}_{i}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}}{{\boldsymbol{Φ}}_{i-1}}$
Description	$\boldsymbol{Ψ}$ is the second function for incorporating interslice forces into shear force (Unitless) $λ$ is the proportionality constant (Unitless) $\boldsymbol{f}$ is the interslice normal to shear force ratio variation function (Unitless) $i$ is the index (Unitless) $\boldsymbol{α}$ is the base angles ( ${{}^{\circ}}$ ) $φ’$ is the effective angle of friction ( ${{}^{\circ}}$ ) ${F_{\text{S}}}$ is the factor of safety (Unitless) $\boldsymbol{Φ}$ is the first function for incorporating interslice forces into shear force (Unitless)
Notes	$\boldsymbol{f}$ is defined in DD:ratioVariation, $\boldsymbol{α}$ is defined in DD:angleA, and $\boldsymbol{Φ}$ is defined in DD:convertFunc1.
Source	chen2005 and karchewski2012
RefBy	IM:intsliceFs and IM:fctSfty

Sums of the interslice normal forces

Refname	DD:nrmForceSumDD
Label	Sums of the interslice normal forces
Symbol	${{\boldsymbol{F}_{\text{x}}}^{\text{G}}}$
Units	${\text{N}}$
Equation	${{\boldsymbol{F}_{\text{x}}}^{\text{G}}}={\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}$
Description	${{\boldsymbol{F}_{\text{x}}}^{\text{G}}}$ is the sums of the interslice normal forces ( ${\text{N}}$ ) $\boldsymbol{G}$ is the interslice normal forces ( $\frac{\text{N}}{\text{m}}$ ) $i$ is the index (Unitless)
Source	fredlund1977
RefBy

Sums of the interslice normal water forces

Refname	DD:watForceSumDD
Label	Sums of the interslice normal water forces
Symbol	${{\boldsymbol{F}_{\text{x}}}^{\text{H}}}$
Units	${\text{N}}$
Equation	${{\boldsymbol{F}_{\text{x}}}^{\text{H}}}={\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}$
Description	${{\boldsymbol{F}_{\text{x}}}^{\text{H}}}$ is the sums of the interslice normal water forces ( ${\text{N}}$ ) $\boldsymbol{H}$ is the interslice normal water forces ( $\frac{\text{N}}{\text{m}}$ ) $i$ is the index (Unitless)
Source	fredlund1977
RefBy

Heights of the right side of slices

Refname	DD:sliceHghtRightDD
Label	Heights of the right side of slices
Symbol	${\boldsymbol{h}^{\text{R}}}$
Units	${\text{m}}$
Equation	${\boldsymbol{h}^{\text{R}}}={\boldsymbol{y}_{\text{slope},i}}-{\boldsymbol{y}_{\text{slip},i}}$
Description	${\boldsymbol{h}^{\text{R}}}$ is the heights of the right side of slices ( ${\text{m}}$ ) ${\boldsymbol{y}_{\text{slope}}}$ is the $y$ -coordinates of the slope ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{y}_{\text{slip}}}$ is the $y$ -coordinates of the slip surface ( ${\text{m}}$ )
Source	fredlund1977
RefBy	DD:slcHeight

Heights of the left side of slices

Refname	DD:sliceHghtLeftDD
Label	Heights of the left side of slices
Symbol	${\boldsymbol{h}^{\text{L}}}$
Units	${\text{m}}$
Equation	${\boldsymbol{h}^{\text{L}}}={\boldsymbol{y}_{\text{slope},i-1}}-{\boldsymbol{y}_{\text{slip},i-1}}$
Description	${\boldsymbol{h}^{\text{L}}}$ is the heights of the left side of slices ( ${\text{m}}$ ) ${\boldsymbol{y}_{\text{slope}}}$ is the $y$ -coordinates of the slope ( ${\text{m}}$ ) $i$ is the index (Unitless) ${\boldsymbol{y}_{\text{slip}}}$ is the $y$ -coordinates of the slip surface ( ${\text{m}}$ )
Source	fredlund1977
RefBy	DD:slcHeight