Theoretical Models

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

RefnameTM:factOfSafety
LabelFactor 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 (\({\text{N}}\))
Sourcefredlund1977
RefByGD:mobShr
RefnameTM:equilibrium
LabelEquilibrium
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 \(0\).
Sourcefredlund1977
RefByGD:normForcEq, GD:momentEql, and GD:bsShrFEq
RefnameTM:mcShrStrgth
LabelMohr-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.
Sourcefredlund1977
RefByGD:resShr
RefnameTM:effStress
LabelEffective 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
Sourcefredlund1977
RefByGD:effNormF
RefnameTM:NewtonSecLawMot
LabelNewton’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
RefByGD:weight