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