Shear strength models

Every method of slices needs the shear strength available on the base of each slice. OGR Slip2D offers 22 strength models, from Mohr–Coulomb and undrained strength to Hoek–Brown, Barton–Bandis, anisotropic and tabulated envelopes. The chapter first shows how a curved envelope becomes a cohesion and a friction angle on every slice, then describes each model: the expression the program evaluates, its default parameters and the source of the formula.

3.5.1How a strength model enters the analysis#

A strength model is a function $\tau_f(\sigma'_n)$ that returns the shear strength available on a plane under an effective normal stress $\sigma'_n$. It enters the analysis through the mobilised shear force on the base of each slice, Eq. (3.1.2). The methods of slices are written for a base with a cohesion and a friction angle (Eq. (3.2.1)), so a model whose envelope is not a straight line is replaced, on each slice, by its tangent at the stress the base carries:

where

$c_i,\ \phi_i$

instantaneous cohesion (kPa) and friction angle (°) of slice $i$

$\bar\sigma$

effective normal stress at which the envelope is read on that slice, taken as zero when it is negative (kPa)

Figure 3.5.1 shows the construction on the default Hoek–Brown envelope. Where it can, the program uses the derivative the model supplies; otherwise it differentiates the envelope numerically with a centred step of $\max(10^{-3}\,\text{kPa},\ 10^{-4}\bar\sigma)$, one-sided near zero. For Mohr–Coulomb the construction returns $c'$ and $\phi'$ themselves, whatever $\bar\sigma$ is.

Where the envelope is read#

The stress $\bar\sigma$ is not known before the method has been solved. The first solve reads every envelope at the estimate

where

$W$

total vertical load on the slice: its weight, the loads on its top, any water standing on it, the vertical load of any support crossing it and, under seismic loading, its vertical seismic force (kN/m)

$\alpha,\ l,\ u$

inclination (°) and length (m) of the base, and pore pressure on it (kPa)

A tangent taken anywhere other than at the stress the method finally resolves is not the envelope there: it lies above a concave curve, and on an envelope that passes through the origin a tangent read at zero stress carries no strength at all. When at least one slice of the surface has an envelope whose slope changes with the stress, OGR Slip2D therefore solves again with each such envelope read at the effective normal stress of the method's own solution, and repeats until the two agree:

Slices whose envelope is a straight line keep the estimate of Eq. (3.5.3), which makes no difference to them. A surface that has not settled after 50 solves is reported as not converged rather than given a factor of safety. How this iteration sits inside each method's own iteration on $F$ is described in Section 3.3.7.

Models that read more than the normal stress#

Some models depend on where the base is and how it is oriented. For them the solver passes, with $\sigma'_n$, a description of the base: its inclination $\alpha$, the depth of its centre below the ground, the elevation of the top of the material layer it lies in, its distance to the nearest point of the ground surface, the local orientation of the bedding when the material is linked to an anisotropic surface, and the vertical effective stress

where

$W,\ W_w$

vertical load of the slice (its weight and the vertical loads applied to it) and weight of any water standing on it, with no seismic coefficient (kN/m)

$b$

slice width (m)

The tables of this chapter say which of these each model reads. A model that needs the description but is evaluated without one (for instance when an envelope is plotted) states below what it falls back on.

Tensile strength and partial factors#

Two further properties of a model are used elsewhere. Its tensile strength is the tension the Tensile Stress Check tolerates on a slice base (Section 3.3.6); it is zero for every model except the two Hoek–Brown models, for which it is given in Eq. (3.5.19). Its partial-factor category says what a design standard divides when it is active (Section 3.12): $c'$ and $\tan\phi'$, the undrained strength $c_u$, or the shear strength $\tau$ as a whole. Table 3.5.1 lists the category of every model.

3.5.2The 22 models at a glance#

Table 3.5.1 lists the models in the order of this chapter, with the name shown in the program, the identifier a project file, the Python API and the command line use, the parameters with their default values, what the model reads besides $\sigma'_n$, and its partial-factor category.

3.5.3Soil models#

Mohr–Coulomb#

The linear envelope of Coulomb (1776), written in effective stresses:

A tensile normal stress contributes no friction: below $\sigma'_n = 0$ the strength stays at $c'$. Strength gained from matric suction above the water table is added to $c'$ as described in Section 3.5.8.

Undrained strength (φ = 0)#

The total-stress analysis of a saturated clay loaded faster than it can drain, in which the shear strength is the undrained strength and does not depend on the normal stress (Skempton, 1948):

The single parameter is cohesion = $c_u$ (default 50 kPa).

Undrained strength varying with depth#

In a normally consolidated clay the undrained strength is proportional to the vertical effective stress (Ladd & Foott, 1974), so under level ground it grows linearly with depth, and published soft-ground profiles are stated that way (Duncan & Wright, 2005). Three models implement the straight line

where

$c_0$

undrained strength at the reference, $z = 0$ (kPa)

$\Delta c$

rate of change with $z$ (kPa/m)

$c_{cut}$

cutoff, applied only when it is enabled: a maximum for a rising profile and a minimum for a falling one (kPa)

The three models differ only in where $z$ is measured from (Table 3.5.2). The line is not bounded below zero: where it reaches zero inside a material, that part of the material has no strength, and the analysis notes give the elevation at which this happens.

Drained–undrained#

A drained Mohr–Coulomb envelope capped by a constant strength, equal to the drained strength at a threshold normal stress $\sigma_t$:

Its parameters are cohesion = $c'$ (5 kPa), phi = $\phi'$ (28°) and sigma_threshold = $\sigma_t$ (100 kPa). The threshold is a stress, not a strength, and a design standard does not factor it.

SHANSEP#

The undrained strength normalised by the vertical effective stress and the overconsolidation ratio, after Ladd & Foott (1974):

where

$S$

normally consolidated strength ratio $s_u/\sigma'_v$ (default 0.25)

$m,\ \mathrm{OCR}$

exponent (0.8) and overconsolidation ratio (1)

$A,\ s_{u,min}$

strength added to the normalised term, and a floor on the result (both 0 kPa)

$\sigma'_v$

vertical effective stress at the base, Eq. (3.5.5)

The strength does not depend on $\sigma'_n$, so the methods read it as a cohesion with $\phi_i = 0$. Where $\sigma'_v$ is zero the strength is $\max(A,\ s_{u,min})$. The pore pressure in $\sigma'_v$ includes any excess pore pressure from an undrained load ($\bar B$, Section 3.6): an undrained load does not consolidate the soil, so it leaves the stress the strength depends on unchanged. Evaluated without a slice, the model uses $\sigma'_n$ in place of $\sigma'_v$.

Vertical stress ratio#

A strength proportional to the vertical effective stress at the base, with a floor:

with K = 0.25 and min_strength = $\tau_{min}$ = 0 kPa by default. It is the SHANSEP model of Eq. (3.5.10) with $S = K$, $\mathrm{OCR} = 1$ and $A = 0$, and like it the methods read it as a cohesion; without a slice it uses $\sigma'_n$ in place of $\sigma'_v$.

3.5.4Rock models#

Hoek–Brown and Generalized Hoek–Brown#

The Hoek–Brown criterion gives the strength of a jointed rock mass in terms of the principal effective stresses. OGR Slip2D offers it in its original form (Hoek & Brown, 1980), model hoek_brown_classic,

and in its generalised form (Hoek, Carranza-Torres & Corkum, 2002), model hoek_brown,

where

$\sigma'_1,\ \sigma'_3$

major and minor principal effective stresses at failure (kPa)

$\sigma_{ci}$

uniaxial compressive strength of the intact rock (sigci, kPa)

$m,\ m_b,\ s,\ a$

rock-mass constants; the classic model is the generalised one with $a = 1/2$

For the generalised model the three constants can be derived from the Geological Strength Index, the intact-rock constant $m_i$ and the disturbance factor $D$ with the relations of Hoek et al. (2002):

The methods need the strength on the plane of the slice base, that is, the Mohr envelope of the criterion. For any criterion $\sigma'_1(\sigma'_3)$ it follows from the relations of Balmer (1952), as given by Hoek et al. (2002):

where

$k$

slope of the criterion, $d\sigma'_1/d\sigma'_3 = 1 + a\,m_b\left(m_b\sigma'_3/\sigma_{ci} + s\right)^{a-1}$, at the same $\sigma'_3$

OGR Slip2D does not evaluate Eq. (3.5.16). It evaluates, for both models, the expressions

where $\hat\sigma = \max(\sigma'_n,\ 10^{-6}\ \text{kPa})$, with $a = 1/2$ and $m_b = m$ for the classic model; $\tau_f = 0$ when the bracket is not positive.

Both models have a finite tensile strength, the root of the bracket of Eq. (3.5.13), which is the biaxial tensile strength of Hoek et al. (2002) and does not depend on $a$:

(with $m$ for the classic model). It is the tension the Tensile Stress Check allows on a slice base in these materials, and it is zero when $\sigma_{ci}$, $m_b$ or $s$ is not positive. The tensile strength comes from the criterion itself and is not affected by the approximation of Eq. (3.5.18).

Barton–Bandis#

The strength of a rock joint from its joint roughness coefficient and joint wall compressive strength, in the form with the residual friction angle of Barton & Choubey (1977) reviewed by Barton & Bandis (1990). OGR Slip2D evaluates

where

$\hat\sigma$

$\max(\sigma'_n,\ 10^{-6}\ \text{kPa})$

$\phi_r$

residual friction angle (phi_r, default 30°)

$\mathrm{JRC},\ \mathrm{JCS}$

joint roughness coefficient (8) and joint wall compressive strength (50 000 kPa)

$\phi_{max}$

cap on the total friction angle (max_total_friction, 75°)

Above $\sigma'_n = \mathrm{JCS}$ the roughness term is zero and the joint mobilises $\phi_r$ alone. The cap keeps the angle finite as the stress tends to zero; with the default parameters it acts only below about 0.12 kPa.

Power curve#

An envelope made of a cohesion intercept, a power of the normal stress and a waviness angle:

The power term is taken as zero where $\hat\sigma + d \le 0$. The parameters are a (0.7), b (1.0), c (3 kPa), the offset d (0 kPa) and the waviness angle waviness = $W$ (0°), which adds the contribution of joint waviness rather than a friction angle of the material. With the default values the curve is the straight line $\tau_f = 3 + 0.7\,\sigma'_n$. The model supplies its exact derivative $a\,b\,(\hat\sigma + d)^{b-1} + \tan W$ for the linearisation of Eq. (3.5.2).

Hyperbolic#

An envelope that starts from the origin with slope $\tan\phi_0$ and tends to a limiting strength $c_\infty$ at high stress:

with c_inf = $c_\infty$ = 150 kPa and phi_0 = $\phi_0$ = 20° by default; its exact derivative is $c_\infty^2\tan\phi_0/(c_\infty + \hat\sigma\tan\phi_0)^2$.

Figure 3.5.3 plots the four envelopes of this section and the Mohr–Coulomb envelope of Section 3.5.3 with their default parameters, as the program evaluates them, and Table 3.5.3 gives the values at five stresses.

3.5.5Anisotropic models#

In a bedded or foliated material the strength depends on the orientation of the slip surface relative to the layering. The four anisotropic models read the inclination $\alpha$ of each slice base, measured counter-clockwise from the horizontal; the first two compare it with a bedding direction, the last two use it directly.

Anisotropic linear#

A Mohr–Coulomb material whose cohesion and friction change linearly from a bedding strength $(c_1,\ \phi_1)$ to a rock-mass strength $(c_2,\ \phi_2)$ as the base turns away from the bedding (Mercer, 2012). With $\delta$ the acute angle between the base and the bedding,

where

$\alpha_b$

inclination of the bedding (bedding_angle, 0°), or its local value at the slice when the material is linked to an anisotropic surface

$A,\ B$

angle up to which only the bedding strength applies (10°), and angle beyond which only the rock-mass strength applies (30°); $A = B$ is a step

The tangent of the friction angle is interpolated, not the angle itself. Figure 3.5.4 shows the strength of the default material at $\sigma'_n = 100$ kPa as the base rotates. Evaluated without a slice, the model returns the weaker of the two extreme orientations, $\delta = 0$ and $\delta = 90^\circ$.

Snowden modified anisotropic linear#

The anisotropic linear model with two extensions (Mercer, 2012). The transition may differ on either side of the bedding, and the bedding and rock-mass strengths are functions of the normal stress, each either a shear/normal function or a C/Phi function (Section 3.5.6):

where $\alpha'$ is $\alpha - \alpha_b$ folded into $(-90^\circ,\ 90^\circ]$, and $t$ is given by Eq. (3.5.24) with $|\alpha'|$ in place of $\delta$. By default $A_1 = A_2 = 10^\circ$, $B_1 = B_2 = 30^\circ$, and the two functions are single-row C/Phi functions equal to the default $(c_1,\ \phi_1)$ and $(c_2,\ \phi_2)$ of the anisotropic linear model, so the two default materials are the same material. Without a slice the model returns the weakest of the strengths at $\alpha' = 0$, $-90^\circ$ and $+90^\circ$.

Anisotropic strength function#

A table of ranges of the base inclination itself, each with a constant cohesion and friction angle. With rows $(\alpha_k,\ c_k,\ \phi_k)$ in increasing $\alpha_k$ and the last one at $90^\circ$,

An inclination that falls exactly on a limit belongs to the lower range. The default table is $(-30^\circ,\ 10\ \text{kPa},\ 35^\circ)$, $(0^\circ,\ 1\ \text{kPa},\ 20^\circ)$, $(90^\circ,\ 5\ \text{kPa},\ 10^\circ)$. Without a slice the model returns the weakest row at the given stress. A table saved by a version before 0.1.218, when the same numbers were interpolated points, is kept in the file but refused by the analysis until it has been reviewed.

Generalized anisotropic#

Any strength model, or the strength of another material, assigned to a range of base inclination. The ranges are absolute inclinations folded into $(-90^\circ,\ 90^\circ]$; the first rule whose range holds the inclination supplies the strength, a limit belongs to the lower range, and the chosen model receives the same description of the base, so a child model that reads $\sigma'_v$ or a depth gets it. Weight, water and every other property are those of the parent material. A set of rules that leaves an inclination uncovered, or whose model cannot be built, is refused before the analysis starts. Without a slice the model returns the weakest of its rules' models.

3.5.6Tabulated functions#

Three models take the envelope from a table, typically fitted to laboratory results. They involve no formula of their own beyond the interpolation.

For the C/Phi function the program supplies the exact derivative of $\tau_f$, including the change of $\phi$ with stress, to the linearisation of Eq. (3.5.2); the other two are differentiated segment by segment. A design standard divides the $\tau$ of every point of the first two tables, and the interpolated $c$ and $\tan\phi$ of the third.

3.5.7Infinite strength and no strength#

infinite_strength models a material that the slip surface cannot shear, such as sound bedrock or a structure; in the linearisation of Eq. (3.5.2) it becomes a cohesion of $10^{12}$ kPa with $\phi_i = 0$, and a design standard does not factor it. Supports whose capacity depends on the strength of the surrounding ground receive none from it, and the analysis notes say so (Section 3.10). no_strength returns $\tau_f = 0$, for water or open voids.

3.5.8Unsaturated strength#

Above the water table a seepage analysis returns negative pore pressures, and matric suction adds to the strength of the soil. Fredlund, Morgenstern & Widger (1978) extended Mohr–Coulomb with a second friction angle $\phi^b$ for the suction:

OGR Slip2D applies it as a bilinear envelope through two properties of the material, $\phi^b$ and the air-entry value $\psi_{ae}$. With the suction $s = -u$: when $s \le \psi_{ae}$ the negative pore pressure is kept and acts through $\tan\phi'$; when $s > \psi_{ae}$ the pore pressure is taken as $-\psi_{ae}$ and the excess suction adds the cohesion

to $c_i$ of Eq. (3.5.2), for every strength model. Both properties default to zero, which clips negative pore pressures to zero so that suction adds nothing. The suction can also be capped by a negative pore-pressure cutoff. The pore pressures themselves are the subject of Section 3.6.

3.5.9References#

• Balmer, G. (1952). A general analytical solution for Mohr's envelope. Proceedings of the American Society for Testing Materials, 52, 1260–1271.
• Barton, N. & Bandis, S. (1990). Review of predictive capabilities of JRC-JCS model in engineering practice. In N. Barton & O. Stephansson (eds), Rock Joints, 603–610. Balkema, Rotterdam.
• Barton, N. & Choubey, V. (1977). The shear strength of rock joints in theory and practice. Rock Mechanics, 10(1–2), 1–54.
• Coulomb, C.A. (1776). Essai sur une application des règles de maximis et minimis à quelques problèmes de statique, relatifs à l'architecture. Mémoires de Mathématique et de Physique présentés à l'Académie Royale des Sciences par divers Savans, 7, 343–382.
• Duncan, J.M. & Wright, S.G. (2005). Soil Strength and Slope Stability. Wiley, Hoboken.
• Fredlund, D.G., Morgenstern, N.R. & Widger, R.A. (1978). The shear strength of unsaturated soils. Canadian Geotechnical Journal, 15(3), 313–321.
• Hoek, E. & Brown, E.T. (1980). Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, ASCE, 106(GT9), 1013–1035.
• Hoek, E., Carranza-Torres, C. & Corkum, B. (2002). Hoek–Brown failure criterion – 2002 edition. Proceedings of the 5th North American Rock Mechanics Symposium, Toronto, 1, 267–273.
• Ladd, C.C. & Foott, R. (1974). New design procedure for stability of soft clays. Journal of the Geotechnical Engineering Division, ASCE, 100(GT7), 763–786.
• Mercer, K.G. (2012). The history and development of the anisotropic linear model: part 1. Australian Centre for Geomechanics Newsletter, Vol. 38 (July 2012), 13–16. Part 2 followed in the same newsletter: Mercer, K.G. (2013), Vol. 40 (July 2013), 1–5.
• Skempton, A.W. (1948). The φ = 0 analysis of stability and its theoretical basis. Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, 1, 72–78.