Instance Models

This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.

The goals GS:Identify-Crit-and-FS, GS:Determine-Normal-Forces, and GS:Determine-Shear-Forces are met by the simultaneous solution of IM:fctSfty, IM:nrmShrFor, and IM:intsliceFs. The goal GS:Identify-Crit-and-FS is also contributed to by IM:crtSlpId.

The Morgenstern-Price method is a vertical slice, limit equilibrium slope stability analysis method. Analysis is performed by breaking the assumed slip surface into a series of vertical slices of mass. Static equilibrium analysis is performed, using two force equations and one moment equation as in TM:equilibrium. The problem is statically indeterminate with only these 3 equations and one constitutive equation (the Mohr Coulomb shear strength of TM:mcShrStrgth) so the assumption GD:normShrR and corresponding equation GD:normShrR are used. The force equilibrium equations can be modified to be expressed only in terms of known physical values, as done in GD:resShearWO and GD:mobShearWO.

The mobilized shear force defined in GD:bsShrFEq can be substituted into the definition of mobilized shear force based on the factor of safety, from GD:mobShr yielding Equation (1) below:

\[\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{N’}}_{i}\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}\]

An expression for the effective normal forces, \(\boldsymbol{N’}\), can be derived by substituting the normal forces equilibrium from GD:normForcEq into the definition for effective normal forces from GD:resShearWO. This results in Equation (2):

\[{\boldsymbol{N’}}_{i}=\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\]

Substituting Equation (2) into Equation (1) gives:

\[\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}\]

Since the interslice shear forces \(\boldsymbol{X}\) and interslice normal forces \(\boldsymbol{G}\) are unknown, they are separated from the other terms as follows:

\[\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\cos\left({\boldsymbol{ω}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}\]

Applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below:

\[\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)=\frac{\left(\left({\boldsymbol{W}}_{i}+{\boldsymbol{U}_{\text{g},i}}\,\cos\left({\boldsymbol{β}}_{i}\right)\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{H}}_{i}+{\boldsymbol{H}}_{i-1}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)-{\boldsymbol{U}_{\text{b},i}}\right)\,\tan\left(φ’\right)+c’\,{\boldsymbol{L}_{b,i}}}{{F_{\text{S}}}}\]

The definitions of GD:resShearWO and GD:mobShearWO are present in this equation, and thus can be replaced by \({\boldsymbol{R}}_{i}\) and \({\boldsymbol{T}}_{i}\), respectively:

\[{\boldsymbol{T}}_{i}+\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{R}}_{i}+\left(\left(-{\boldsymbol{X}}_{i-1}+{\boldsymbol{X}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)}{{F_{\text{S}}}}\]

The interslice shear forces \(\boldsymbol{X}\) can be expressed in terms of the interslice normal forces \(\boldsymbol{G}\) using A:Interslice-Norm-Shear-Forces-Linear and GD:normShrR, resulting in:

\[{\boldsymbol{T}}_{i}+\left(-λ\,{\boldsymbol{f}}_{i-1}\,{\boldsymbol{G}}_{i-1}+λ\,{\boldsymbol{f}}_{i}\,{\boldsymbol{G}}_{i}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)-\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)=\frac{{\boldsymbol{R}}_{i}+\left(\left(-λ\,{\boldsymbol{f}}_{i-1}\,{\boldsymbol{G}}_{i-1}+λ\,{\boldsymbol{f}}_{i}\,{\boldsymbol{G}}_{i}\right)\,\cos\left({\boldsymbol{α}}_{i}\right)+\left(-{\boldsymbol{G}}_{i}+{\boldsymbol{G}}_{i-1}\right)\,\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)}{{F_{\text{S}}}}\]

Rearranging yields the following:

\[{\boldsymbol{G}}_{i}\,\left(\left(λ\,{\boldsymbol{f}}_{i}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)={\boldsymbol{G}}_{i-1}\,\left(\left(λ\,{\boldsymbol{f}}_{i-1}\,\cos\left({\boldsymbol{α}}_{i}\right)-\sin\left({\boldsymbol{α}}_{i}\right)\right)\,\tan\left(φ’\right)-\left(λ\,{\boldsymbol{f}}_{i-1}\,\sin\left({\boldsymbol{α}}_{i}\right)+\cos\left({\boldsymbol{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}\]

The definitions for \(\boldsymbol{Φ}\) and \(\boldsymbol{Ψ}\) from DD:convertFunc1 and DD:convertFunc2 simplify the above to Equation (3):

\[{\boldsymbol{G}}_{i}\,{\boldsymbol{Φ}}_{i}={\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}\,{\boldsymbol{Φ}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}\]

Versions of Equation (3) instantiated for slices 1 to \(n\) are shown below:

\[{\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}={\boldsymbol{Ψ}}_{0}\,{\boldsymbol{G}}_{0}\,{\boldsymbol{Φ}}_{0}+{F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\]

\[{\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}={\boldsymbol{Ψ}}_{1}\,{\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\]

\[{\boldsymbol{G}}_{3}\,{\boldsymbol{Φ}}_{3}={\boldsymbol{Ψ}}_{2}\,{\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}+{F_{\text{S}}}\,{\boldsymbol{T}}_{3}-{\boldsymbol{R}}_{3}\]

…

\[{\boldsymbol{G}}_{n-2}\,{\boldsymbol{Φ}}_{n-2}={\boldsymbol{Ψ}}_{n-3}\,{\boldsymbol{G}}_{n-3}\,{\boldsymbol{Φ}}_{n-3}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}\]

\[{\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}={\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{G}}_{n-2}\,{\boldsymbol{Φ}}_{n-2}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}\]

\[{\boldsymbol{G}}_{n}\,{\boldsymbol{Φ}}_{n}={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}\]

Applying A:Edge-Slices, which says that \({\boldsymbol{G}}_{0}\) and \({\boldsymbol{G}}_{n}\) are zero, results in the following special cases: Equation (8) for the first slice:

\[{\boldsymbol{G}}_{1}\,{\boldsymbol{Φ}}_{1}={F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\]

and Equation (9) for the \(n\)th slice:

\[-\left(\frac{{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}}{{\boldsymbol{Ψ}}_{n-1}}\right)={\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}\]

Substituting Equation (8) into Equation (4) yields Equation (10):

\[{\boldsymbol{G}}_{2}\,{\boldsymbol{Φ}}_{2}={\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\]

which can be substituted into Equation (5) to get Equation (11):

\[{\boldsymbol{G}}_{3}\,{\boldsymbol{Φ}}_{3}={\boldsymbol{Ψ}}_{2}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{3}-{\boldsymbol{R}}_{3}\]

and so on until Equation (12) is obtained from Equation (7):

\[{\boldsymbol{G}}_{n-1}\,{\boldsymbol{Φ}}_{n-1}={\boldsymbol{Ψ}}_{n-2}\,\left({\boldsymbol{Ψ}}_{n-3}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}\]

Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in:

\[-\left(\frac{{F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}}{{\boldsymbol{Ψ}}_{n-1}}\right)={\boldsymbol{Ψ}}_{n-2}\,\left({\boldsymbol{Ψ}}_{n-3}\,\left({\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-2}-{\boldsymbol{R}}_{n-2}\right)+{F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}\]

This can be rearranged by multiplying both sides by \({\boldsymbol{Ψ}}_{n-1}\) and then distributing the multiplication of each \(\boldsymbol{Ψ}\) over addition to obtain:

\[-\left({F_{\text{S}}}\,{\boldsymbol{T}}_{n}-{\boldsymbol{R}}_{n}\right)={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{1}-{\boldsymbol{R}}_{1}\right)+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{2}-{\boldsymbol{R}}_{2}\right)+{\boldsymbol{Ψ}}_{n-1}\,\left({F_{\text{S}}}\,{\boldsymbol{T}}_{n-1}-{\boldsymbol{R}}_{n-1}\right)\]

The multiplication of the \(\boldsymbol{Ψ}\) terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an \(\boldsymbol{R}\) or a \(\boldsymbol{T}\). The equation can then be rearranged so terms containing an \(\boldsymbol{R}\) are on one side of the equality, and terms containing a \(\boldsymbol{T}\) are on the other. The multiplication by the factor of safety is common to all of the \(\boldsymbol{T}\) terms, and thus can be factored out, resulting in:

\[{F_{\text{S}}}\,\left({\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,{\boldsymbol{T}}_{1}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,{\boldsymbol{T}}_{2}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{T}}_{n-1}+{\boldsymbol{T}}_{n}\right)={\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{1}\,{\boldsymbol{R}}_{1}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{Ψ}}_{n-2}\,{\boldsymbol{Ψ}}_{2}\,{\boldsymbol{R}}_{2}+{\boldsymbol{Ψ}}_{n-1}\,{\boldsymbol{R}}_{n-1}+{\boldsymbol{R}}_{n}\]

Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in IM:fctSfty:

\[{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\boldsymbol{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\boldsymbol{Ψ}}_{v}}}+{\boldsymbol{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\boldsymbol{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\boldsymbol{Ψ}}_{v}}}+{\boldsymbol{T}}_{n}}\]

\({F_{\text{S}}}\) depends on the unknowns \(λ\) (IM:nrmShrFor) and \(\boldsymbol{G}\) (IM:intsliceFs).

From the moment equilibrium of GD:momentEql with the primary assumption for the Morgenstern-Price method of A:Interslice-Norm-Shear-Forces-Linear and associated definition GD:normShrR, Equation (14) can be derived:

\[0=-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+λ\,\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,{\boldsymbol{h}}_{i}}{2}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}\]

Rearranging the equation in terms of \(λ\) leads to Equation (15):

\[λ=\frac{-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}}\,{\boldsymbol{W}}_{i}\,{\boldsymbol{h}}_{i}}{2}+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}+{\boldsymbol{Q}}_{i}\,\sin\left({\boldsymbol{ω}}_{i}\right)\,{\boldsymbol{h}}_{i}}{-\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}\]

This equation can be simplified by applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic and external forces, respectively, are zero:

\[λ=\frac{-{\boldsymbol{G}}_{i}\,\left({\boldsymbol{h}_{\text{z},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{G}}_{i-1}\,\left({\boldsymbol{h}_{\text{z},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)-{\boldsymbol{H}}_{i}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i}}+\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{H}}_{i-1}\,\left(\frac{1}{3}\,{\boldsymbol{h}_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}_{i}}{2}\,\tan\left({\boldsymbol{α}}_{i}\right)\right)+{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)\,{\boldsymbol{h}}_{i}}{-\frac{{\boldsymbol{b}}_{i}}{2}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}\]

Taking the summation of all slices, and applying A:Edge-Slices to set \({\boldsymbol{G}}_{0}\), \({\boldsymbol{G}}_{n}\), \({\boldsymbol{H}}_{0}\), and \({\boldsymbol{H}}_{n}\) equal to zero, a general equation for the proportionality constant \(λ\) is developed in Equation (16), which combines IM:nrmShrFor, IM:nrmShrForNum, and IM:nrmShrForDen:

\[λ=\frac{\displaystyle\sum_{i=1}^{n}{{\boldsymbol{b}}_{i}\,\left({{\boldsymbol{F}_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\boldsymbol{α}}_{i}\right)+{\boldsymbol{h}}_{i}\,-2\,{\boldsymbol{U}_{\text{g},i}}\,\sin\left({\boldsymbol{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\boldsymbol{b}}_{i}\,\left({\boldsymbol{G}}_{i}\,{\boldsymbol{f}}_{i}+{\boldsymbol{G}}_{i-1}\,{\boldsymbol{f}}_{i-1}\right)}}\]

Equation (16) for \(λ\) is a function of the unknown interslice normal forces \(\boldsymbol{G}\) (IM:intsliceFs) which itself depends on the unknown factor of safety \({F_{\text{S}}}\) (IM:fctSfty).

See IM:nrmShrFor for the derivation of \({\boldsymbol{C}_{\text{num}}}\).

See IM:nrmShrFor for the derivation of \({\boldsymbol{C}_{\text{den}}}\).

This derivation is identical to the derivation for IM:fctSfty up until Equation (3) shown again below:

\[{\boldsymbol{G}}_{i}\,{\boldsymbol{Φ}}_{i}={\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}\,{\boldsymbol{Φ}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}\]

A simple rearrangement of Equation (3) leads to Equation (17), also seen in IM:intsliceFs:

\[{\boldsymbol{G}}_{i}=\frac{{\boldsymbol{Ψ}}_{i-1}\,{\boldsymbol{G}}_{i-1}+{F_{\text{S}}}\,{\boldsymbol{T}}_{i}-{\boldsymbol{R}}_{i}}{{\boldsymbol{Φ}}_{i}}\]

The cases shown in IM:intsliceFs for when \(i=0\), \(i=1\), or \(i=n\) are derived by applying A:Edge-Slices, which says that \({\boldsymbol{G}}_{0}\) and \({\boldsymbol{G}}_{n}\) are zero, to Equation (17). \(\boldsymbol{G}\) depends on the unknowns \({F_{\text{S}}}\) (IM:fctSfty) and \(λ\) (IM:nrmShrFor).