Mohr – Coulomb failure criterion continued

Things to remember when using the Mohr – Coulomb failure criterion:

• The linear failure envelope is just an approximation to simplify calculations
• The failure envelope is stress dependent and will produce some kind of curvature if shear strength tests are executed in much different confining stresses (fig 1, from Duncan and Write, 2005).

• According to Lade, 2010 the failure envelope is curved and at low effective stresses which can be found in superficial failures on slopes, the use of linear Mohr – Coulomb may be in the unsafe side. Soils without cementation do not provide any effective cohesion in very low effective stresses (fig 2, from Lade, 2010).

• When the linear Mohr – Coulomb criterion is used it must be evaluated for the expected stress range in the field.
• Small cohesion values will not produce significant errors when high effective stresses are anticipated in the calculation.
• In low effective stress even minimum values of effective cohesion (in cohesionless soils) can produce significant errors in factor of Safety (FS) calculations.

References:

Duncan J. M.,  Wright S. G., (2005). “Soil Strength and Slope Stability”. Wiley, New York.

Lade P. V. (2010). “The mechanics of surficial failure in soil slopes”. Engineering Geology 114, pp 57-64.

Slope stability and scale effects

In previous entries the issue of stiff fissured clays and the time to failure was briefly touched. The design of such slopes is not a trivial matter and requires significant knowledge of soil mechanics, geology, hydrogeology etc. One additional issue mentioned (one that sometimes is neglected) is the scale effect. This was presented in the previous entry for a very deep mine in rock. This issue of scale effect in relation to stress field will be briefly presented for the case of stiff fissured clays and hard soils.

In the following picture a large highway cut of about 30m is shown. For a civil engineering project this is a significantly high cut. The effective stress filed in this cut can range from of 50 – 500kPa which is the normal range for laboratory testing.

In the second picture a large excavation for a lignite mine is presented. The depth of excavation of this multi bench cut is around 135m. The excavation of this type needs to consider bench stability of slopes with heights of around 18m and also overall slope stability for highs above 135m. In the second case a large part of a possible failure surface could be in a stress field of around 1500-2000kPa or even more.

In the following figure the two types of cuts are compared and one can easily understand the significance of scale effects in the design of the different cuts.

The scale of the mine excavation is such that even in one cross section, one has to consider besides the stress field, differing geology (pic 4), presence of faults, ground water locations and pore pressures etc. We will focus on the stress dependency at this point.

According to Stark et al, (2005) both fully softened and residual failure envelopes are stress dependent. In this work Stark et al provides an empirical graph regarding the stress dependency until 700kPa of normal stress for residual friction angle and 400kPa for fully softened friction angle.

Shear strength information for higher effective stresses >1MPa are not readily available. Furthermore execution of such tests in very high effective loads is not easy for most commercial laboratories. It may even be very difficult to execute ring shear tests in very high loads due to sample thickness and squeezing out from the sides.

In such high slopes the failure surface can pass from a number of soil layers with different shear strength properties. It is not easy to evaluate the “average” shear strength of layers involved in a possible failure surface. Unfortunately a rule of thumb for selecting shear strength parameters for such slopes cannot be provided. Engineering judgment is required in selecting such parameters and the stress conditions must not be ignored. Shear strength tests should be evaluated in relation to the expected stress field.

Slope failures, landslides and mines

On 11 of April 2013, around 9:30 p.m. a large slide (maybe the largest) in the northeast section of the Kennecott mine occured (fig 1). The slide was preceded by slope movements that reached ~50mm per day. Two major questions could be raised, why this slide occurred and could it have been predicted before hand and remediated?

These are very difficult questions and require significant knowledge of the geology, geotechnical conditions of the area, operational practices, climatic conditions etc. In the following paragraphs some initial ideas regarding the stability of high mine slopes and some interesting references will be provided for interested individuals. The incident in Kennecott is an important lesson of how important continuous monitoring of slopes is in such mine operations.

I would like to start with a very interesting graph published by Hoek et al (2000), “Large-scale slope Design – A Review of the State of the Art”.

This chart presents slope height versus overall angle with solid markers representing unstable slopes and open markers represent stable slopes. This chart is for copper porphyry open pits. In this graph the Kennocott mine (Bingham Canyon) is also shown but not the April 2013 one.

It is very interesting to note that most of the unstable markers are located in a range between 35 and 45 degrees of slope angle. Although much information is required for detail evaluation of each point and why instability occurred, a trend can be seen. Can we assume that slopes designed bellow 32-35o would not provide stability problems?

In the next figure I would like to focus on scale effects when dealing with mine slopes in rock or even hard rock materials. In the down left side of the figure 2 a slope with 30 meters height is depicted. In the upper left one of 90m with the same spacing of joints and finaly on the right a slope of 500m again with the same spacing and trance length of discontinuities (figures adopted from Sjoberg, 1996).