Methods of slices — OGR Slip2D documentation

OGR Slip2D implements nine methods of slices. All of them cut the sliding mass into the same slices, apply the same loads and pore pressures and read the strength of each slice base in the same way; they differ in the equilibrium conditions they satisfy and in what they assume about the forces between slices. The equations of each method are given as the program solves them, with the original source of each formula.

3.2.1Common formulation#

The forces on a slice are those of Figure 3.1.2, and the mobilised shear force on its base is Eq. (3.1.2). In OGR Slip2D the base of every slice is the straight chord between its two base corners, on a circular surface as on any other; $\alpha$ is the inclination of that chord, $b$ the width of the slice and $l = b\sec\alpha$ the length of its base. $W$ is the whole vertical load the slice carries: the weight of its soil, the vertical component of any load on its top and the weight of water ponded on it. The pore pressure $u$ is taken at the middle of the base.

Each method finds the sense in which the mass slides from the sign of its own driving sum, $\sum W\sin\alpha$ ( $\sum W\tan\alpha$ for Janbu's methods), and writes its equations in that sense. In this chapter the angles $\alpha$ , $\beta$ (the inclination of the ground surface over a slice) and $\theta$ (the inclination of an interslice force) are therefore positive where the line dips in the direction of sliding (over the upper, driving part of the surface) and negative where it rises towards the exit. Interslice normal forces $E$ are positive in compression, and with $\theta > 0$ the resultant on the up-slope face of a slice pushes it towards the toe and downwards. The equations are written for the static case; seismic, water and support forces enter every method as described in Section 3.9, Section 3.6 and Section 3.10, and the In OGR Slip2D boxes below say where.

Strength of a slice base#

All nine methods read the shear strength of a base as a straight line: the tangent to the envelope of its material at an estimate of the effective normal stress on the base,

\tau_f(\sigma'_n) \approx c_i + \sigma'_n\tan\phi_i, \qquad \tan\phi_i = \left.\dfrac{\mathrm{d}\tau_f}{\mathrm{d}\sigma'_n}\right|_{\sigma'_{n0}}, \qquad c_i = \tau_f(\sigma'_{n0}) - \sigma'_{n0}\tan\phi_i

which every method except the Ordinary one first evaluates at

\sigma'_{n0} = \dfrac{\max\left(0,\; W\cos\alpha - u\,l\right)}{l}

where

$c_i,\ \tan\phi_i$: intercept and slope of the tangent, neither allowed below zero (kPa, —)
$\sigma'_{n0}$: first estimate of the effective normal stress on the base (kPa)

For a Mohr–Coulomb material the tangent is the envelope itself, $c_i = c'$ and $\phi_i = \phi'$ , and in the rest of this chapter $c'$ and $\phi'$ stand for $c_i$ and $\phi_i$ . The Ordinary method reads the envelope at its own normal stress, Eq. (3.2.3). For a curved envelope the tangent depends on where it is taken, and the program repeats the solution at the stress each method resolves until the two agree (Section 3.3.7); the envelopes themselves are the subject of Section 3.5.

In OGR Slip2D

The methods are chosen in Analysis → Project Settings → Methods, where several can be ticked for one analysis. The tangent of Eq. (3.2.1) is computed in one place for all nine methods: analytically when the strength model provides its derivative, by a finite difference otherwise. The matric-suction cohesion of an unsaturated base is added to $c_i$ . A surface on which no base has any shear strength gets $F = 0$ and is reported as a failed calculation, with the same message whatever the method.

Source: ogr_slip2d/methods/base.py · ogr_slip2d/methods/bishop.py

Table 3.2.1 lists what the program solves for in each method; the equilibrium conditions each one satisfies are in Table 3.1.2.

Table 3.2.1 The nine methods: identifier used in project files, scripts and the MCP server, what the program solves for, and how a surface that is not a circle is treated.

Method	Identifier	Solved for	Non-circular surface
Ordinary (Fellenius)	`ordinary_fellenius`	nothing: closed form	moment balance about an axis
Bishop simplified	`bishop_simplified`	$F$ , fixed-point iteration	moment balance about an axis
Janbu simplified	`janbu_simplified`	$F$ , fixed-point iteration	same equations
Janbu corrected	`janbu_corrected`	$F_0$ as above, then $F = f_0 F_0$	same equations
Spencer	`spencer`	$F_f$ and $F_m$ at each $\lambda$ , and $\lambda$	$F_m$ about an axis
GLE / Morgenstern–Price	`gle_morgenstern_price`	$F_f$ and $F_m$ at each $\lambda$ , and $\lambda$	$F_m$ about an axis
Corps of Engineers #1	`corps_engineers_1`	$F$ , root of the interslice force recursion	same equations
Corps of Engineers #2	`corps_engineers_2`	$F$ , root of the interslice force recursion	same equations
Lowe–Karafiath	`lowe_karafiath`	$F$ , root of the interslice force recursion	same equations

3.2.2Ordinary method (Fellenius)#

The Ordinary method, also called the Fellenius or Swedish method, ignores the interslice forces altogether (Fellenius, 1927). The normal force on each base is then the component of the slice's own load perpendicular to it, and moment equilibrium of the whole mass about the centre of a circular surface gives the factor of safety directly, without iteration:

F = \dfrac{\sum\left[c'\,l + \left(W\cos\alpha - u\,l\cos^2\alpha\right)\tan\phi'\right]}{\sum W\sin\alpha}

where

$l$: length of the base (m)
$W$: vertical load on the slice (kN/m)
$\alpha$: inclination of the base (°)
$u$: pore pressure at the middle of the base (kPa)
$c',\ \phi'$: strength of the base, Eq. (3.2.1) (kPa, °)

The pore-water term is $u\,l\cos^2\alpha$ and not $u\,l$ . Resolving the effective weight of the slice, $W - u\,b$ , perpendicular to the base gives $(W - u\,l\cos\alpha)\cos\alpha$ : this is the correction of Turnbull & Hvorslev (1967). The two forms coincide when $u = 0$ or the base is horizontal, and part company as the pore pressure and the inclination of the base grow, the uncorrected one giving the smaller effective normal force.

In OGR Slip2D

One pass, no iteration. An effective normal force that comes out negative is set to zero before the envelope is read, so that base contributes its cohesion only. $\sum W\sin\alpha$ is evaluated with the true lever arm of each load about the centre, $(x_m - x_O)/R$ with $x_m$ the abscissa of the middle of the slice: the base is the chord of the arc, so its own inclination is not the tangent at $x_m$ . The pseudo-static force $k_h W$ acts at mid-height of the slice; it is resolved onto the base and adds its moment about the centre to the driving side, and horizontal water forces enter by their moments. On a surface that is not a circle the same balance is taken about a moment axis (Section 3.2.3), still without iteration. Because its normal force does not depend on $c'$ or $\phi'$ , this is the one method the curved-envelope iteration of Section 3.3.7 leaves alone, and the m-alpha check does not apply to it: it forms no such denominator.

Source: ogr_slip2d/methods/ordinary.py

3.2.3Bishop's simplified method#

Bishop (1955) assumes that the interslice forces are horizontal, $X = 0$ . Vertical equilibrium of each slice, with the mobilised shear force of Eq. (3.1.2), then gives the base normal force

N = \dfrac{W - \left(c'\,l - u\,l\tan\phi'\right)\sin\alpha\,/\,F}{m_\alpha}

m_\alpha = \cos\alpha + \dfrac{\sin\alpha\,\tan\phi'}{F}

and moment equilibrium of the whole mass about the centre of the circle, in which $N$ has no moment because it acts along a radius, gives

F = \dfrac{\sum\left[c'\,b + (W - u\,b)\tan\phi'\right]/m_\alpha}{\sum W\sin\alpha}

where

$N$: total normal force on the base (kN/m)
$m_\alpha$: the denominator of the base normal force (—)
$b$: width of the slice (m)

Because $F$ appears on both sides of Eq. (3.2.6), the program solves it by fixed-point iteration: the right-hand side is evaluated with the current value of $F$ and the result becomes the next value, from the initial factor of safety set in the project (1.0 by default) until two successive values differ by less than the convergence tolerance. The iteration, its limits and its acceleration are described in Section 3.3.1 and Section 3.3.2.

Surfaces that are not circles#

Eq. (3.2.6) is a ratio of moments only because every base of a circle has the same lever arm, the radius, which cancels. On any other surface each base has its own arm and the base normal force no longer passes through the point moments are taken about, so it has a moment of its own (Fredlund & Krahn, 1977). OGR Slip2D then takes the moment of every force about an axis $O$ as a cross product, and solves

F = -\,\dfrac{\sum_i M_O\!\left(Q_i\,\mathbf{t}_i\right)}{\sum_i\left[M_O\!\left(\mathbf{W}_i\right) + M_O\!\left(N_i\,\mathbf{n}_i\right) + M_O\!\left(\mathbf{H}_i\right)\right]}, \qquad Q_i = \dfrac{c'\,b + (W - u\,b)\tan\phi'}{m_\alpha}

where

$M_O(\mathbf{P})$: anticlockwise moment about $O$ of a force $\mathbf{P}$ applied at $(x, y)$ : $(x - x_O)P_y - (y - y_O)P_x$ (kN·m/m)
$Q_i$: shear resistance of base $i$ with $F$ divided out, $Q = F\,S_m$ (kN/m)
$\mathbf{t}_i$: unit vector along the base chord, opposing the movement of the base
$N_i\,\mathbf{n}_i$: base normal force of Eq. (3.2.4), acting at the middle of the base along the normal that points into the mass
$\mathbf{W}_i$: vertical load of the slice, on the vertical through the middle of its base
$\mathbf{H}_i$: horizontal forces (seismic at mid-height of the slice, water at their own elevation) and support forces

The sense of rotation is that of the moment of the weights. On a circle, with its centre as the axis, the base normal forces have no moment and Eq. (3.2.7) differs from Eq. (3.2.6) only in taking the shear along each chord rather than along the arc. OGR Slip2D keeps Eq. (3.2.6) for circles and uses Eq. (3.2.7) for every other surface, iterating on $F$ with the damped update $F_{k+1} = \tfrac12\left[F_k + g(F_k)\right]$ , where $g(F_k)$ is the value Eq. (3.2.7) returns.

A non-circular surface has no centre, so the axis has to be chosen. Unless the user places one, OGR Slip2D builds it for each surface from the chord $\mathrm{AB}$ joining its two ends:

\mathbf{O} = \tfrac{1}{2}\left(\mathbf{A} + \mathbf{B}\right) + \mathbf{R}_{90}\left(\mathbf{B} - \mathbf{A}\right) = \left(\tfrac{x_A + x_B}{2} - (y_B - y_A),\ \ \tfrac{y_A + y_B}{2} + (x_B - x_A)\right)

with $\mathrm{A}$ the left end, $\mathrm{B}$ the right end and $\mathbf{R}_{90}$ an anticlockwise rotation by 90°: the axis lies above the middle of the chord, at a distance from it equal to the chord's length. A composite surface takes moments about the centre of the circle it was built from, and a surface clipped by a weak layer keeps the axis of the surface it was clipped from (the centre of a circle, or the constructed axis of a polyline) unless the user has placed one.

Note

On a surface that is not a circle, the factor of safety of a method that satisfies moment equilibrium only (Ordinary and Bishop) depends on the axis; that of Spencer and GLE, which satisfy force equilibrium as well, does not. The program's own record of the effect: a deep circle analysed as a polyline of 192 chords inscribed in it gives, with the automatic axis, 1.2427 by the Ordinary method and 1.2500 by Bishop's against 1.3043 on the arc (−4.7 % and −4.2 %), while Spencer gives 1.3032; with the axis placed on the centre of the circle all three return 1.3043.

In OGR Slip2D

Circles. Eq. (3.2.6) with the true lever arm of each load, $(x_m - x_O)/R$ , in place of $\sin\alpha$ , as in the Ordinary method. The seismic force adds $k_h W\,(y_O - y_g)/R$ to the driving side, with $y_g$ at mid-height of the slice; horizontal water forces enter by their moments about the centre; active and passive supports as in Section 3.10. The iteration is undamped, can be accelerated (Section 3.3.2) and never stops on its first step. If $|m_\alpha| < 10^{-6}$ on any slice the calculation fails with the message that $m_\alpha$ collapsed.

Other surfaces. Eq. (3.2.7), damped as above, starting from the larger of 0.05 and the initial factor of safety, never accelerated. The user's axis is set with Surfaces → Slope Limits → Define Moment Axis… and applies to every surface except circles and composite surfaces, which always turn about their own centre; Reset Moment Axis returns to one automatic axis per surface.

Source: ogr_slip2d/methods/bishop.py · ogr_slip2d/moment_balance.py · ogr_slip2d/surface.py

3.2.4Janbu's simplified and corrected methods#

Janbu's simplified method also neglects the interslice shear, so its base normal force is that of Eq. (3.2.4), but it satisfies horizontal force equilibrium of the whole mass instead of moment equilibrium (Janbu, 1973):

F_0 = \dfrac{\sum\left[c'\,b + (W - u\,b)\tan\phi'\right]/n_\alpha}{\sum W\tan\alpha}

n_\alpha = \cos^2\alpha\left(1 + \dfrac{\tan\alpha\,\tan\phi'}{F_0}\right) = m_\alpha\cos\alpha

It is solved by the same fixed-point iteration as Bishop's equation. It needs no centre of rotation and applies unchanged to any shape of surface. Janbu compared it with his generalised procedure, which includes the interslice shear, and gave an empirical factor that corrects for its omission:

F = f_0\,F_0

f_0 = 1 + b_1\left[\dfrac{d}{L} - 1.4\left(\dfrac{d}{L}\right)^{2}\right]

where

$F_0$: factor of safety of Janbu's simplified method (—)
$L$: length of the chord joining the two ends of the slip surface (m)
$d$: largest perpendicular distance from that chord to the slip surface (m)
$b_1$: coefficient for the type of soil, Table 3.2.2 (—)

The correction curves are Janbu's (1973); Eq. (3.2.12) is the expression fitted to them given by Abramson et al. (2002). Janbu computed his curves for homogeneous slopes. Figure 3.2.1 shows $L$ and $d$ on a circle.

Table 3.2.2 Coefficient

b_1

of Eq. (3.2.12) for the three soil types of Janbu's curves.

Soil type on the slip surface	Coefficient
Cohesion only ( $\phi' = 0$ )	0.69
Friction only ( $c' = 0$ )	0.31
Cohesion and friction	0.50

Figure 3.2.1 The chord

L

between the two ends of a slip surface and the largest perpendicular distance

d

from it to the surface, on the circle of Table 3.2.4 cut into 25 slices. The program measures

d

at the corners of the slice bases. With

d/L = 0.128

and a soil with cohesion and friction, Eq. (3.2.12) gives

f_0 = 1.052

. Computed with OGR Slip2D 0.1.235.

In OGR Slip2D

The sense of sliding is the sign of $\sum W\tan\alpha$ , ponded water included. The driving side of Eq. (3.2.9) adds the horizontal seismic forces $\sum k_h W$ and the net horizontal water force in the sense of sliding; supports enter with a factor $\sec\alpha$ (Section 3.10). If $|n_\alpha| < 10^{-6}$ on any slice the calculation fails. The acceleration of Section 3.3.2 applies to both methods, on $F_0$ .

For the correction, $L$ joins the first and last points of the base polyline and $d$ is the largest distance from it to the corners of the slice bases. The soil type is read on every base: a Mohr–Coulomb material by its $c'$ and $\phi'$ ; the undrained models (constant, varying with depth, SHANSEP) and Vertical Stress Ratio as cohesion only; any other envelope by its shape: flat is cohesion only, through the origin is friction only, anything else both. Bases with no strength, or with infinite strength, take no part. If every base that takes part is of one type, $b_1$ is that type's; if they are of two or more types, $b_1 = 0.50$ , the cohesion-and-friction value, a rule adopted because Janbu's curves are for homogeneous slopes. The result records $b_1$ and $f_0$ . Everything computed per slice for Janbu corrected (base forces, the admissibility checks of Section 3.3) is that of the equilibrium solved at $F_0$ , not at $f_0F_0$ , because the correction adjusts the number and re-solves nothing.

Source: ogr_slip2d/methods/janbu.py

3.2.5Spencer's method#

Spencer (1967) assumes that all the interslice resultants are parallel, at an inclination $\theta$ that is not known in advance,

X_i = \lambda\,E_i, \qquad \lambda = \tan\theta

and satisfies both force and moment equilibrium, with $F$ and $\lambda$ as the two unknowns. OGR Slip2D solves it as the general limit-equilibrium (GLE) formulation of Fredlund & Krahn (1977) with a constant interslice function, so Spencer and GLE/Morgenstern–Price share every equation of this section and differ only in Eq. (3.2.18).

Equations of a slice#

Number the slice boundaries from the down-slope free end, 0, to the up-slope one, $n$ , so that slice $i$ lies between boundaries $i-1$ and $i$ . Vertical equilibrium of the slice, now with the interslice shear, gives the base normal force, and horizontal equilibrium carries the interslice normal force from one boundary to the next:

N_i = \dfrac{W_i + X_i - X_{i-1} - \left(c'\,l - u\,l\tan\phi'\right)_i\sin\alpha_i\,/\,F}{m_{\alpha,i}}

E_i = E_{i-1} - N_i\sin\alpha_i + \dfrac{S_i}{F}\cos\alpha_i - H_i, \qquad S_i = c'\,l_i + \left(N_i - u_i\,l_i\right)\tan\phi'

where

$E_i,\ X_i$: interslice normal and shear forces on boundary $i$ ; $E_0 = X_0 = X_n = 0$ at the free ends (kN/m)
$S_i$: shear resistance of base $i$ , $F$ times the mobilised shear (kN/m)
$H_i$: net external horizontal force on slice $i$ , positive in the sense of sliding (kN/m)

The factor of safety then comes out twice, once from each equilibrium condition:

F_f = \dfrac{\sum S_i\sec\alpha_i}{\sum\left[\left(W_i + X_i - X_{i-1}\right)\tan\alpha_i + H_i\right]}

F_m = \dfrac{\sum S_i}{\sum W_i\sin\alpha_i}

Eq. (3.2.16) is horizontal equilibrium of the whole mass, $E_n = 0$ , after $N_i\sin\alpha_i$ has been replaced by its value from vertical equilibrium; at $\lambda = 0$ it is Janbu's simplified equation, Eq. (3.2.9), term by term. Eq. (3.2.17) is moment equilibrium about the centre of a circle, where the interslice forces cancel in pairs and act only through $N_i$ ; at $\lambda = 0$ it is Bishop's equation. On a surface that is not a circle $F_m$ is the moment balance of Eq. (3.2.7) about the same axis as Bishop's, with $N_i$ from Eq. (3.2.14).

For each trial $\lambda$ the program iterates $F_f$ and $F_m$ to two separate fixed points, marching Eq. (3.2.14) and Eq. (3.2.15) across the slices on every pass. The solution is the $\lambda$ at which the two agree, $F = F_f = F_m$ . How the program searches for it, and when it rejects a value of $\lambda$ , is the subject of Section 3.3.3, where Figure 3.3.1 shows both factors against $\lambda$ .

Figure 3.2.2

F_f

and

F_m

against

\lambda

on the critical circle of the two-layer slope of the quick start, computed with the program's own inner solve. At

\lambda = 0

the force curve gives Janbu's simplified factor, 1.183, and the moment curve Bishop's, 1.300; they cross at

\lambda = 0.343

F = 1.290

, which is the Spencer solution of the quick start.

In OGR Slip2D

Spencer is the GLE engine with $f = 1$ at every boundary. The march starts at the down-slope free end. On a circle the denominator of Eq. (3.2.17) uses the true lever arms and carries the seismic, water and support moments, as Bishop's does; supports also enter both slice equations (Section 3.10). The result carries $\lambda$ (for Spencer, $\tan\theta$ ), the interslice forces $E$ and $X$ at every boundary and the verdict on their admissibility (Section 3.3.4).

Source: ogr_slip2d/methods/spencer.py · ogr_slip2d/interslice.py

3.2.6GLE / Morgenstern–Price#

Morgenstern & Price (1965) let the ratio of the interslice forces vary along the surface as a chosen function $f(x)$ scaled by $\lambda$ ; Fredlund & Krahn (1977) wrote the procedure in the general form of Eq. (3.2.16) and Eq. (3.2.17):

X_i = \lambda\,f(x_i)\,E_i

With $f \equiv 1$ it is Spencer's method; in OGR Slip2D that identity holds by construction, since the two run the same code. The function is evaluated at the slice boundaries, in terms of the abscissa normalised over the surface,

t = \dfrac{x - x_0}{x_n - x_0}

where $x_0$ and $x_n$ are the abscissae of the left and right ends of the surface: $t = 0$ is the left end whatever the direction of sliding. The program offers the four functions of Table 3.2.3, drawn in Figure 3.2.3. All four are symmetric about $t = \tfrac12$ , so for them that convention does not depend on which way the slope faces. A function that vanishes at both ends, such as the half sine, allows no interslice shear near them; the clipped sine keeps a ratio of $0.2\lambda$ there.

Table 3.2.3 Interslice force functions offered for GLE / Morgenstern–Price, with

t

from Eq. (3.2.19).

Function	Identifier	Expression
Half Sine (default)	`half_sine`	$\sin(\pi t)$
Constant	`constant`	$1$
Trapezoidal	`trapezoidal`	$4t$ for $t < 0.25$ ; $1$ for $0.25 \le t \le 0.75$ ; $4(1 - t)$ for $t > 0.75$
Clipped Sine	`clipped_sine`	$0.2 + 0.8\sin(\pi t)$

Figure 3.2.3 The four interslice force functions of Table 3.2.3, plotted from the functions of the program itself. The half sine, the default, is drawn in vermilion.

In OGR Slip2D

The function is chosen in Analysis → Project Settings → Methods → Interslice force function, which is active when GLE / Morgenstern–Price is ticked; the default is Half Sine. The range searched for $\lambda$ , shared with Spencer, is set on the Advanced page as Minimum lambda and Maximum lambda (−0.1 and 6.0 by default). Only these four functions can be chosen in the program; a Python script that builds the method object itself may pass any function of $(x, x_0, x_n)$ (Section 5.3). The result carries $\lambda$ and the ratios $\lambda f(x_i)$ at every boundary.

Source: ogr_slip2d/methods/gle.py

3.2.7Methods with a prescribed interslice inclination#

Corps of Engineers #1, Corps of Engineers #2 and Lowe–Karafiath satisfy force equilibrium only. Instead of solving for the inclination of the interslice forces they prescribe it from the geometry, which leaves the factor of safety as the only unknown. The three share one engine, the numerical solution of the Modified Swedish method published by the U.S. Army Corps of Engineers (USACE, 2003), and differ only in where $\theta$ comes from.

The inclination of the interslice forces#

Corps of Engineers #1 uses one inclination for the whole surface, that of the line joining its two ends. USACE (2003, §C-4a) takes the side forces parallel to the average slope of the embankment, usually the line from crest to toe; OGR Slip2D draws that line between the points where the slip surface meets the ground and, where a tension crack closes the mass, from the top of the crack.
Corps of Engineers #2 takes $\theta_i = \beta_i$ , the inclination of the ground surface over slice $i$ . It is zero wherever the ground is horizontal, so the interslice shear vanishes there.
Lowe–Karafiath takes the average of the inclinations of the ground surface and of the slip surface (Lowe & Karafiath, 1960):

\theta_i = \tfrac{1}{2}\left(\alpha_i + \beta_i\right)

Each rule defines $\theta$ per slice, from the chords of its top ( $\beta_i$ ) and of its base ( $\alpha_i$ ). The force on an interior boundary acts at the average inclination of the two slices that share it,

\theta_{i,i+1} = \tfrac{1}{2}\left(\theta_i + \theta_{i+1}\right)

which changes nothing for Corps of Engineers #1, whose $\theta$ is constant. Figure 3.2.4 shows the three choices on the same surface.

Figure 3.2.4 Inclination of the interslice forces at the interior boundaries of the same circle, cut into ten slices, as the program gives it for (a) Corps of Engineers #1, (b) Corps of Engineers #2 and (c) Lowe–Karafiath. The value printed is the inclination at the third boundary. The circle is that of Table 3.2.4; computed with OGR Slip2D 0.1.235.

Force equilibrium#

Let $Z$ be the interslice resultant, inclined at $\theta$ . Writing the horizontal and vertical equilibrium of a slice and eliminating its base normal force gives a linear relation between the forces on its two faces:

Z_{\mathrm{d}}\,n_\alpha(\theta_{\mathrm{d}}) = Z_{\mathrm{u}}\,n_\alpha(\theta_{\mathrm{u}}) + C

n_\alpha(\theta) = \cos(\alpha - \theta) + \dfrac{\tan\phi'}{F}\sin(\alpha - \theta)

C = W\left(\sin\alpha - \dfrac{\tan\phi'}{F}\cos\alpha\right) + H\left(\cos\alpha + \dfrac{\tan\phi'}{F}\sin\alpha\right) - \dfrac{c'\,l - u\,l\tan\phi'}{F}

where

$Z_{\mathrm{u}},\ \theta_{\mathrm{u}}$: interslice resultant on the up-slope face of the slice and its inclination (kN/m, °)
$Z_{\mathrm{d}},\ \theta_{\mathrm{d}}$: the same on the down-slope face
$H$: net external horizontal force on the slice, positive in the sense of sliding (kN/m)

With $\theta$ constant the relation becomes $Z_{\mathrm{d}} = Z_{\mathrm{u}} + C/n_\alpha$ , Eq. C-19 of USACE (2003). The program marches Eq. (3.2.22) across the slices from one free end, where $Z = 0$ ; the factor of safety is the value of $F$ that leaves no force at the other end, $Z_n(F) = 0$ . The base normal force then follows from vertical equilibrium of each slice (USACE 2003, Eq. G-16):

N = \dfrac{W + Z_{\mathrm{u}}\sin\theta_{\mathrm{u}} - Z_{\mathrm{d}}\sin\theta_{\mathrm{d}} - \left(c'\,l - u\,l\tan\phi'\right)\sin\alpha\,/\,F}{m_\alpha}

The root is found by bracketing. $Z_n$ is evaluated at $F$ = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, 2.0, 2.5, 3.0, 4.0 and 5.0, and between the first two values of opposite sign the root is refined by the secant method, falling back to bisection whenever a step leaves the bracket, until $|Z_n| < 10^{-6}$ kN/m or the bracket is narrower than the convergence tolerance. A trial $F$ at which $n_\alpha \le 10^{-6}$ on any face is refused, because a vanishing denominator creates a spurious root at low values of $F$ .

Note

The bracket is searched between $F = 0.2$ and $F = 5.0$ . A surface whose factor of safety lies outside that range cannot be bracketed by these three methods: the program then returns the sampled value with the smallest residual, marks the calculation as not converged ("no F-bracket") and does not use it.

Total or effective interslice forces#

Where water pressure acts on the vertical faces between slices, the force transmitted across a face is the sum of an effective part, carried by the soil skeleton, and the integral of the water pressure, which is horizontal. A method that solves for the inclination of the interslice forces, or assumes them horizontal, is insensitive to that split; a method that prescribes the inclination is not. USACE (2003, §C-4a) accepts both readings and notes that the factor of safety differs between them.

In OGR Slip2D

Analysis → Project Settings → Methods → Interslice forces, active when one of the three methods is ticked: Total (water thrust included), the default, or Effective (water thrust separated). With Effective, the water pressure on each vertical face is integrated from the project's pore-pressure model and applied to the two slices that share the face as a horizontal load, and $\theta$ is prescribed for the effective force. Total is the convention in which the published values of these methods are given; Effective is the reading to use for a slope with water standing over it, where with total forces the factor of safety depends on the depth of the water.

The march is tried with the geometry as drawn and, if no root is bracketed, mirrored; the first orientation that brackets a root is used. The root refinement takes at most the larger of Maximum iterations and 60 steps. The acceleration of Section 3.3.2 does not apply, and neither does the m-alpha check, since these methods divide by $n_\alpha(\theta)$ of Eq. (3.2.23) and not by $m_\alpha$ . The line of thrust is available for all three (Section 3.3.8).

Source: ogr_slip2d/methods/modified_swedish.py · ogr_slip2d/methods/lowe_karafiath.py

3.2.8Choosing a method#

No method of slices is exact: each closes the statically indeterminate problem of Section 3.1.3 with an assumption. What the methods share and where they differ can be summarised as follows, after Duncan, Wright & Brandon (2014).

Complete equilibrium. Spencer and GLE/Morgenstern–Price satisfy every equilibrium condition, and procedures that do so give factors of safety that differ little from one another (Fredlund & Krahn, 1977). Of the methods that use moment equilibrium, they are the only ones whose answer on a non-circular surface does not depend on the moment axis.
Bishop's simplified method gives values close to those of the complete-equilibrium procedures on circular surfaces. On other surfaces its result depends on the moment axis (Section 3.2.3).
The Ordinary method needs no iteration and always gives an answer, but in effective-stress analyses with high pore pressures its factor of safety can be well below that of the complete-equilibrium procedures.
Force-equilibrium methods (Janbu's two and the prescribed-inclination family) depend on the inclination they assume for the interslice forces: horizontal for Janbu's, fixed by the geometry for the others.

Running several methods on the same model, as OGR Slip2D allows, shows how much the answer depends on those assumptions. Table 3.2.4 gives the nine factors of safety on one circle of a simple slope.

Table 3.2.4 The nine methods on one circle: the circle of lowest Bishop factor found by a grid search (centre 38.0, 36.0; radius 26.10 m) on a dry homogeneous slope 10 m high at 2H:1V, with

c' = 10

kPa,

\phi' = 30°

and

\gamma = 20

kN/m³, cut into 25 slices, at the project's default settings. Computed with OGR Slip2D 0.1.235; an illustration, not a benchmark.

Method	$F$	vs Spencer	Note
Ordinary (Fellenius)	1.816	−4.1 %
Bishop simplified	1.896	+0.2 %
Janbu simplified	1.799	−5.0 %
Janbu corrected	1.894	0.0 %	$f_0 = 1.052$ , $b_1 = 0.50$
Spencer	1.893	—	$\lambda = 0.388$ ( $\theta = 21.2°$ )
GLE / Morgenstern–Price	1.893	0.0 %	half sine, $\lambda = 0.475$
Corps of Engineers #1	1.905	+0.6 %
Corps of Engineers #2	1.920	+1.4 %
Lowe–Karafiath	1.904	+0.5 %

Engineering judgement

The spread between methods depends on the problem, and a dry homogeneous slope on a circle is a favourable case. A large spread is a reason to look at the forces on the slices (Section 3.3) before quoting any of the values.

3.2.9References#

Abramson, L.W., Lee, T.S., Sharma, S. & Boyce, G.M. (2002). Slope Stability and Stabilization Methods, 2nd ed. Wiley, New York.
Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique, 5(1), 7–17.
Duncan, J.M., Wright, S.G. & Brandon, T.L. (2014). Soil Strength and Slope Stability, 2nd ed. Wiley.
Fellenius, W. (1927). Erdstatische Berechnungen mit Reibung und Kohäsion (Adhäsion) und unter Annahme kreiszylindrischer Gleitflächen. Ernst & Sohn, Berlin.
Fredlund, D.G. & Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal, 14(3), 429–439.
Janbu, N. (1973). Slope stability computations. In R.C. Hirschfeld & S.J. Poulos (eds.), Embankment-Dam Engineering: Casagrande Volume, 47–86. Wiley, New York.
Lowe, J. & Karafiath, L. (1960). Stability of earth dams upon drawdown. Proceedings of the 1st Pan-American Conference on Soil Mechanics and Foundation Engineering, Mexico City, 2, 537–552.
Morgenstern, N.R. & Price, V.E. (1965). The analysis of the stability of general slip surfaces. Géotechnique, 15(1), 79–93.
Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique, 17(1), 11–26.
Turnbull, W.J. & Hvorslev, M.J. (1967). Special problems in slope stability. Journal of the Soil Mechanics and Foundations Division, ASCE, 93(SM4), 499–528.
U.S. Army Corps of Engineers (2003). Slope Stability. Engineer Manual EM 1110-2-1902. Washington, DC.