This section collects and defines all the data needed to build the instance models.
| 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 |
| 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 | |
| Source | fredlund1977 |
| RefBy | GD:resShearWO, IM:nrmShrForNum, GD:normForcEq, GD:momentEql, GD:mobShearWO, GD:bsShrFEq, DD:lengthLb, DD:convertFunc2, and DD:convertFunc1 |
| 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 | |
| Source | fredlund1977 |
| RefBy | GD:resShearWO, IM:nrmShrForNum, GD:normForcEq, GD:momentEql, GD:mobShearWO, GD:bsShrFEq, and DD:lengthLs |
| 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 |
| 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 | |
| Source | fredlund1977 |
| RefBy | GD:resShr, GD:resShearWO, GD:mobShr, and GD:baseWtrF |
| 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 | |
| Source | fredlund1977 |
| RefBy | GD:srfWtrF |
| 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 | |
| Source | fredlund1977 |
| RefBy | IM:nrmShrForNum and GD:momentEql |
| 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, GD:effNormF, and TM:effStress |
| 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 |
| 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 |
| 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 |
| 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 | |
| Source | chen2005 and karchewski2012 |
| RefBy | IM:intsliceFs, IM:fctSfty, and DD:convertFunc2 |
| 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 | |
| Source | chen2005 and karchewski2012 |
| RefBy | IM:intsliceFs and IM:fctSfty |
| 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 | |
| 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 | |
| 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 |
| 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 |