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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.

RefnameIM:fctSfty
LabelFactor of safety
Inputxslope, yslope, ywt, c, φ, γdry, γsat, γw, xslip, yslip, constf
OutputFS
Input Constraints
Output Constraints
EquationFS=n1i=1Rin1v=iΨv+Rnn1i=1Tin1v=iΨv+Tn
Description
  • FS is the factor of safety (Unitless)
  • R is the resistive shear forces without the influence of interslice forces (Nm)
  • i is the index (Unitless)
  • Ψ is the second function for incorporating interslice forces into shear force (Unitless)
  • v is the local index (Unitless)
  • n is the number of slices (Unitless)
  • T is the mobilized shear forces without the influence of interslice forces (Nm)
Notes
Sourcechen2005 and karchewski2012
RefByIM:nrmShrFor, IM:intsliceFs, IM:fctSfty, FR:Display-Interslice-Shear-Forces, FR:Display-Interslice-Normal-Forces, FR:Display-Factor-of-Safety, and FR:Determine-Critical-Slip-Surface

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:

(WiXi1+Xi+Ug,icos(βi)+Qicos(ωi))sin(αi)(KcWiGi+Gi1Hi+Hi1+Ug,isin(βi)+Qisin(ωi))cos(αi)=Nitan(φ)+cLb,iFS

An expression for the effective normal forces, 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):

Ni=(WiXi1+Xi+Ug,icos(βi)+Qicos(ωi))cos(αi)+(KcWiGi+Gi1Hi+Hi1+Ug,isin(βi)+Qisin(ωi))sin(αi)Ub,i

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

(WiXi1+Xi+Ug,icos(βi)+Qicos(ωi))sin(αi)(KcWiGi+Gi1Hi+Hi1+Ug,isin(βi)+Qisin(ωi))cos(αi)=((WiXi1+Xi+Ug,icos(βi)+Qicos(ωi))cos(αi)+(KcWiGi+Gi1Hi+Hi1+Ug,isin(βi)+Qisin(ωi))sin(αi)Ub,i)tan(φ)+cLb,iFS

Since the interslice shear forces X and interslice normal forces G are unknown, they are separated from the other terms as follows:

(Wi+Ug,icos(βi)+Qicos(ωi))sin(αi)(KcWiHi+Hi1+Ug,isin(βi)+Qisin(ωi))cos(αi)(Gi+Gi1)cos(αi)+(Xi1+Xi)sin(αi)=((Wi+Ug,icos(βi)+Qicos(ωi))cos(αi)+(KcWiHi+Hi1+Ug,isin(βi)+Qisin(ωi))sin(αi)+(Gi+Gi1)sin(αi)+(Xi1+Xi)cos(αi)Ub,i)tan(φ)+cLb,iFS

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:

(Wi+Ug,icos(βi))sin(αi)(Hi+Hi1+Ug,isin(βi))cos(αi)(Gi+Gi1)cos(αi)+(Xi1+Xi)sin(αi)=((Wi+Ug,icos(βi))cos(αi)+(Hi+Hi1+Ug,isin(βi))sin(αi)+(Gi+Gi1)sin(αi)+(Xi1+Xi)cos(αi)Ub,i)tan(φ)+cLb,iFS

The definitions of GD:resShearWO and GD:mobShearWO are present in this equation, and thus can be replaced by Ri and Ti, respectively:

Ti+(Xi1+Xi)sin(αi)(Gi+Gi1)cos(αi)=Ri+((Xi1+Xi)cos(αi)+(Gi+Gi1)sin(αi))tan(φ)FS

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

Ti+(λfi1Gi1+λfiGi)sin(αi)(Gi+Gi1)cos(αi)=Ri+((λfi1Gi1+λfiGi)cos(αi)+(Gi+Gi1)sin(αi))tan(φ)FS

Rearranging yields the following:

Gi((λficos(αi)sin(αi))tan(φ)(λfisin(αi)+cos(αi))FS)=Gi1((λfi1cos(αi)sin(αi))tan(φ)(λfi1sin(αi)+cos(αi))FS)+FSTiRi

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

GiΦi=Ψi1Gi1Φi1+FSTiRi

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

G1Φ1=Ψ0G0Φ0+FST1R1

G2Φ2=Ψ1G1Φ1+FST2R2

G3Φ3=Ψ2G2Φ2+FST3R3

Gn2Φn2=Ψn3Gn3Φn3+FSTn2Rn2

Gn1Φn1=Ψn2Gn2Φn2+FSTn1Rn1

GnΦn=Ψn1Gn1Φn1+FSTnRn

Applying A:Edge-Slices, which says that G0 and Gn are zero, results in the following special cases: Equation (8) for the first slice:

G1Φ1=FST1R1

and Equation (9) for the nth slice:

(FSTnRnΨn1)=Gn1Φn1

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

G2Φ2=Ψ1(FST1R1)+FST2R2

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

G3Φ3=Ψ2(Ψ1(FST1R1)+FST2R2)+FST3R3

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

Gn1Φn1=Ψn2(Ψn3(Ψ1(FST1R1)+FST2R2)+FSTn2Rn2)+FSTn1Rn1

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

(FSTnRnΨn1)=Ψn2(Ψn3(Ψ1(FST1R1)+FST2R2)+FSTn2Rn2)+FSTn1Rn1

This can be rearranged by multiplying both sides by Ψn1 and then distributing the multiplication of each Ψ over addition to obtain:

(FSTnRn)=Ψn1Ψn2Ψ1(FST1R1)+Ψn1Ψn2Ψ2(FST2R2)+Ψn1(FSTn1Rn1)

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

FS(Ψn1Ψn2Ψ1T1+Ψn1Ψn2Ψ2T2+Ψn1Tn1+Tn)=Ψn1Ψn2Ψ1R1+Ψn1Ψn2Ψ2R2+Ψn1Rn1+Rn

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:

FS=n1i=1Rin1v=iΨv+Rnn1i=1Tin1v=iΨv+Tn

FS depends on the unknowns λ (IM:nrmShrFor) and G (IM:intsliceFs).

RefnameIM:nrmShrFor
LabelNormal and shear force proportionality constant
Inputxslope, yslope, ywt, γw, xslip, yslip, constf
Outputλ
Input Constraints
Output Constraints
Equationλ=ni=1Cnum,ini=1Cden,i
Description
  • λ is the proportionality constant (Unitless)
  • Cnum is the proportionality constant numerator (N)
  • i is the index (Unitless)
  • Cden is the proportionality constant denominator (N)
Notes
Sourcechen2005 and karchewski2012
RefByIM:nrmShrForNum, IM:nrmShrFor, IM:nrmShrForDen, IM:intsliceFs, IM:fctSfty, FR:Display-Interslice-Shear-Forces, FR:Display-Interslice-Normal-Forces, FR:Display-Factor-of-Safety, and FR:Determine-Critical-Slip-Surface

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=Gi(hz,i+bi2tan(αi))+Gi1(hz,i1bi2tan(αi))Hi(13hz,w,i+bi2tan(αi))+Hi1(13hz,w,i1bi2tan(αi))+λbi2(Gifi+Gi1fi1)+KcWihi2+Ug,isin(βi)hi+Qisin(ωi)hi

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

λ=Gi(hz,i+bi2tan(αi))+Gi1(hz,i1bi2tan(αi))Hi(13hz,w,i+bi2tan(αi))+Hi1(13hz,w,i1bi2tan(αi))+KcWihi2+Ug,isin(βi)hi+Qisin(ωi)hibi2(Gifi+Gi1fi1)

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=Gi(hz,i+bi2tan(αi))+Gi1(hz,i1bi2tan(αi))Hi(13hz,w,i+bi2tan(αi))+Hi1(13hz,w,i1bi2tan(αi))+Ug,isin(βi)hibi2(Gifi+Gi1fi1)

Taking the summation of all slices, and applying A:Edge-Slices to set G0, Gn, H0, and Hn 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=ni=1bi(FxG+FxH)tan(αi)+hi2Ug,isin(βi)ni=1bi(Gifi+Gi1fi1)

Equation (16) for λ is a function of the unknown interslice normal forces G (IM:intsliceFs) which itself depends on the unknown factor of safety FS (IM:fctSfty).

RefnameIM:nrmShrForNum
LabelNormal and shear force proportionality constant numerator
Inputxslope, yslope, ywt, γw, xslip, yslip
OutputCnum
Input Constraints
Output Constraints
EquationCnum,i={b1(G1+H1)tan(α1),i=1bi(FxG+FxH)tan(αi)+h2Ug,isin(βi),2in1bn(Gn1+Hn1)tan(αn1),i=n
Description
  • Cnum is the proportionality constant numerator (N)
  • i is the index (Unitless)
  • b is the base width of slices (m)
  • G is the interslice normal forces (Nm)
  • H is the interslice normal water forces (Nm)
  • α is the base angles ()
  • FxG is the sums of the interslice normal forces (N)
  • FxH is the sums of the interslice normal water forces (N)
  • h is the y-direction heights of slices (m)
  • Ug is the surface hydrostatic forces (Nm)
  • β is the surface angles ()
  • n is the number of slices (Unitless)
Notes
Sourcechen2005 and karchewski2012
RefByIM:nrmShrFor

See IM:nrmShrFor for the derivation of Cnum.

RefnameIM:nrmShrForDen
LabelNormal and shear force proportionality constant denominator
Inputxslip, constf
OutputCden
Input Constraints
Output Constraints
EquationCden,i={b1f1G1,i=1bi(fiGi+fi1Gi1),2in1bnGn1fn1,i=n
Description
  • Cden is the proportionality constant denominator (N)
  • i is the index (Unitless)
  • b is the base width of slices (m)
  • f is the interslice normal to shear force ratio variation function (Unitless)
  • G is the interslice normal forces (Nm)
  • n is the number of slices (Unitless)
Notes
Sourcechen2005 and karchewski2012
RefByIM:nrmShrFor

See IM:nrmShrFor for the derivation of Cden.

RefnameIM:intsliceFs
LabelInterslice normal forces
Inputxslope, yslope, ywt, c, φ, γdry, γsat, γw, xslip, yslip, constf
OutputG
Input Constraints
Output Constraints
EquationGi={FST1R1Φ1,i=1Ψi1Gi1+FSTiRiΦi,2in10,i=0i=n
Description
  • G is the interslice normal forces (Nm)
  • i is the index (Unitless)
  • FS is the factor of safety (Unitless)
  • T is the mobilized shear forces without the influence of interslice forces (Nm)
  • R is the resistive shear forces without the influence of interslice forces (Nm)
  • Φ is the first function for incorporating interslice forces into shear force (Unitless)
  • Ψ is the second function for incorporating interslice forces into shear force (Unitless)
  • n is the number of slices (Unitless)
Notes
Sourcechen2005
RefByIM:nrmShrFor, IM:intsliceFs, IM:fctSfty, FR:Display-Interslice-Shear-Forces, FR:Display-Interslice-Normal-Forces, FR:Display-Factor-of-Safety, and FR:Determine-Critical-Slip-Surface

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

GiΦi=Ψi1Gi1Φi1+FSTiRi

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

Gi=Ψi1Gi1+FSTiRiΦi

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 G0 and Gn are zero, to Equation (17). G depends on the unknowns FS (IM:fctSfty) and λ (IM:nrmShrFor).

RefnameIM:crtSlpId
LabelCritical slip surface identification
Inputxslope, yslope, xwt, ywt, c, φ, γdry, γsat, γw, constf
OutputFSmin
Input Constraints
Output Constraints
Equationxslope=yslope
Description
  • xslope is the x-coordinates of the slope (m)
  • yslope is the y-coordinates of the slope (m)
Notes
  • The minimization function must enforce the constraints on the critical slip surface expressed in A:Slip-Surface-Concave and Sec:Properties of a Correct Solution. The sizes of xwt and ywt must be equal and not 1. The sizes of xslope and yslope must be equal and at least 2. The first and last xwt values must be equal to the first and last xslope values. xwt and xslope values must be monotonically increasing. xslipmaxExt, xslipmaxEtr, xslipminExt, and xslipminEtr must be between or equal to the minimum and maximum xslope values. yslipmax cannot be below the minimum yslope value. yslipmin cannot be above the maximum yslope value. All x values of xcs,ycs must be between xslipminEtr and xslipmaxExt. All y values of xcs,ycs must not be below yslipmin. For any given vertex in xcs,ycs the y value must not exceed the yslope value corresponding to the same x value. The first and last vertices in xcs,ycs must each be equal to one of the vertices formed by xslope and yslope. The slope between consecutive vertices must be always increasing as x increases. The internal angle between consecutive vertices in xcs,ycs must not be below 110 degrees.
Sourceli2010
RefByFR:Display-Graph and FR:Determine-Critical-Slip-Surface