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Theoretical Models

This section focuses on the general equations and laws that SSP is based on.

Factor of safety

Refname	TM:factOfSafety
Label	Factor of safety
Equation	${F_{\text{S}}}=\frac{P}{S}$
Description	${F_{\text{S}}}$ is the factor of safety (Unitless) $P$ is the resistive shear force ( ${\text{N}}$ ) $S$ is the mobilized shear force ( ${\text{N}}$ )
Source	fredlund1977
RefBy	GD:mobShr

Equilibrium

Refname	TM:equilibrium
Label	Equilibrium
Equation	$\displaystyle\sum{{F_{\text{x}}}}=0$ $\displaystyle\sum{{F_{\text{y}}}}=0$ $\displaystyle\sum{M}=0$
Description	${F_{\text{x}}}$ is the $x$ -coordinate of the force ( ${\text{N}}$ ) ${F_{\text{y}}}$ is the $y$ -coordinate of the force ( ${\text{N}}$ ) $M$ is the moment ( $\text{N}\text{m}$ )
Notes	For a body in static equilibrium, the net forces and moments acting on the body will cancel out. Assuming a 2D problem (A:Effective-Norm-Stress-Large), the $x$ -coordinate of the force ${F_{\text{x}}}$ and $y$ -coordinate of the force ${F_{\text{y}}}$ will be equal to $0$ . All forces and their distance from the chosen point of rotation will create a net moment equal to $0$ .
Source	fredlund1977
RefBy	GD:normForcEq, GD:momentEql, and GD:bsShrFEq

Mohr-Coulumb shear strength

Refname	TM:mcShrStrgth
Label	Mohr-Coulumb shear strength
Equation	${τ^{\text{f}}}={σ_{N}}‘\,\tan\left(φ’\right)+c’$
Description	${τ^{\text{f}}}$ is the shear strength ( ${\text{Pa}}$ ) ${σ_{N}}‘$ is the effective normal stress ( ${\text{Pa}}$ ) $φ’$ is the effective angle of friction ( ${{}^{\circ}}$ ) $c’$ is the effective cohesion ( ${\text{Pa}}$ )
Notes	In this model the shear strength ${τ^{\text{f}}}$ is proportional to the product of the effective normal stress ${σ_{N}}‘$ on the plane with its static friction in the angular form $\tan\left(φ’\right)$ . The ${τ^{\text{f}}}$ versus ${σ_{N}}‘$ relationship is not truly linear, but assuming the effective normal forces is strong enough, it can be approximated with a linear fit (A:Surface-Base-Slice-between-Interslice-Straight-Lines) where the effective cohesion $c’$ represents the ${τ^{\text{f}}}$ intercept of the fitted line.
Source	fredlund1977
RefBy	GD:resShr

Effective stress

Refname	TM:effStress
Label	Effective stress
Equation	$σ’=σ-u$
Description	$σ’$ is the effective stress ( ${\text{Pa}}$ ) $σ$ is the total normal stress ( ${\text{Pa}}$ ) $u$ is the pore pressure ( ${\text{Pa}}$ )
Notes	$σ$ is defined in DD:normStress.
Source	fredlund1977
RefBy	GD:effNormF

Newton’s second law of motion

Refname	TM:NewtonSecLawMot
Label	Newton’s second law of motion
Equation	$\boldsymbol{F}=m\,\boldsymbol{a}\text{(}t\text{)}$
Description	$\boldsymbol{F}$ is the force ( ${\text{N}}$ ) $m$ is the mass ( ${\text{kg}}$ ) $\boldsymbol{a}\text{(}t\text{)}$ is the acceleration ( $\frac{\text{m}}{\text{s}^{2}}$ )
Notes	The net force $\boldsymbol{F}$ on a body is proportional to the acceleration $\boldsymbol{a}\text{(}t\text{)}$ of the body, where $m$ denotes the mass of the body as the constant of proportionality.
Source	–
RefBy	GD:weight