Academic Issues – Page 9 – vulcanhammer.net

A Quick Preliminary Way to Determine Slope Stability

Posted on 16 June 2022 by Don Warrington

Slope stability for circular failure surfaces is one of those topics where, for a complete solution, solving the problem has involved a computer solution before there were computers. The problem is that it is necessary to a) discretise the problem before solving it, in this case using slices, and b) try a large number of circle centre locations before finding the right one. The problem is illustrated below.

Prediction of slope stability by circular-cylindrical slip surfaces (from Tsytovich (1976))

One common method has been to use charts. A commonly used group is the Janbu series of charts, one of which is shown below.

Janbu Chart for cohesive soil (from NAVFAC DM 7.01)

A simpler solution is presented by Tsytovich (1976). The equation proposed there is this:

${\it FS}=\tan(\phi)A+{\frac {cB}{{\it \gamma}\,h}}$ (1)

where

$FS =$ factor of safety
$\phi =$ internal frictional angle of the soil
$c =$ cohesion of the soil
$\gamma =$ unit weight of the soil
$A,B =$ coefficients given below, functions of $e$ and $h$
$e$ = distance from slope toe to hard underground layer (see Figure 78(b))
$h$ = height of slope from toe to top (see Figure 78(b))
$m$ = second number of slope ratio rise:run (see Figure 78(b))

It is also possible to solve for a maximum height, thus:

$h={\frac {cB}{{\it \gamma}\,\left ({\it FS}-\tan(\phi)A\right )}}$ (2)

Coefficients of A and B for Approximate Prediction of Stability of Slopes (from Tsytovich (1976))

As an example, consider a slope with an angle of 35 degrees and a height of 20′. The soil has an internal friction angle of 15 degrees, a cohesion of 600 psf and a unit weight of 120 pcf. Estimate the factor of safety against slope failure using the method described. For this slope e = 0, so assume the slip surface passes through the lower edge (toe) of the slope.

The slope ratio 1:m is the reciprocal of the tangent of the slope given. Taking the tangent and inverting it gives a slope ratio of 1:1.43. It can be seen by inspection that A = 2.64 and through interpolation B = 6.38. Direct substitution yields a factor of safety of 2.3.

As a comparison we use the Slope program from our Soil Mechanics Course, the results are shown below using Fellenius’ Method.

The factor of safety is very close. If we used Bishop’s Method in the program, we could expect the factor of safety to increase. One thing absent from this problem is the presence of water, which always complicates slope stability analysis.

If we consider the example problem from the Janbu chart above, Equation (1) reduces to the equation for FS on the chart provided that $B = N_o$ . Doing the linear interpolation yields $B = 6.05$ , which is higher than the $N_o = 5.8$ on the chart, or $FS = 1.26$ .

Obviously the method is not suitable for final design but it is interesting for producing preliminary results for a problem which has traditionally been computationally difficult.

Foundation Design and Analysis: Shallow Foundations, Bearing Capacity

Posted on 14 June 202229 August 2026 by Don Warrington

Other resources:

Note: this is an update from an earlier lecture, which actually combines two lectures as well. Some new equipment was used; however, the “live screen” method didn’t quite work out, which meant that I ended up putting the slides in the video during video editing. Since I point to the slides from time to time, this may end up with some awkward moments. I apologise for any inconvenience.

Academic Issues, Soil Mechanics

Manual For The Systematic Study Of The Regime Of Underground Waters – Altovskij, Konopljantsev — Mir Books

Posted on 13 June 2022 by Don Warrington

In this post, we will see the book Manual For The Systematic Study Of The Regime Of Underground Waters edited by M. Altovskij; A. Konopljantsev. About the book The manual, is divided into four chapters. The first chapter-general theoretical problems-treats underground water and its behaviour as a natural historical process reflecting certain peculiarities attendant upon […]
Manual For The Systematic Study Of The Regime Of Underground Waters – Altovskij, Konopljantsev — Mir Books

Academic Issues

Another Featured Student: Ogheneruona Uwusiaba

Posted on 2 June 2022 by Don Warrington

My institution has featured yet another one of my students who graduated last month, Ogheneruona Uwusiaba, from Nigeria.

“Ruona” (as we called her) was a very diligent student, and took all of the classes I taught (Soil Mechanics, Foundations and Fluid Mechanics Laboratory.) To be a student athlete, mother, engineering student and graduate Magna Cum Laude takes a special type of person, and Ruona is that.

She mentioned her senior design project for African housing. Another one of her teammates was Arielle Scalioni, who was featured here.

This website–and by extension my teaching–goes around the world to the extent that more visits come from outside the U.S. than inside. Sometimes though the world comes to my classroom, and Ruona was a part of that.

Academic Issues, Geotechnical Engineering

The “Quick and Dirty” Way to Derive Boussinesq’s Point Load Stress Equations

Posted on 1 June 20221 June 2022 by Don Warrington

In another post we show the formal, nice way to derive the equations for stresses under point loads. Here we’re going to show a “quick and dirty” way to derive them (or at least some of them.) It’s based on Tsytovich (1976) but I have made some changes to tighten up the theory behind them and make them a little more comprehensible. They don’t derive all of the equations, but the method gives a better physical understanding of what’s going on when we apply such a load to the ground surface.

Let’s start with Tsytovich’s diagram, shown below.

Radial stresses under the action of a concentrated force (from Tsytovich (1976))

It can be shown that this state exists due to static equilibrium. The point load P forms a sphere around itself; the principal stresses $\sigma_R$ radiate from the load, forming a sphere around the point of radius R. The magnitude of the stress is given by the equation

$\sigma_R = A \frac{\cos(\beta)}{R^2}$

By same static equilibrium, the vertical force of the stresses and P are thus

$P-\int_{0}^{2\,\pi}\!\sigma_{{R}}\cos(\beta){dF}=0$

The infinitesimal surface area of the stress can be defined as

$dF = 2\pi(R \sin\beta)(R d\beta)$

Substituting this into the integral yields

$P-\int_{0}^{1/2\,\pi}\!2\,\sigma_{{R}}\cos(\beta)\pi\,{R}^{2}\sin(\beta){d\beta}=0$

Making appropriate substitutions, the integral evaluates to

$P-2/3\,A\pi =0$

and thus

$A=3/2\,{\frac {P}{\pi }}$

Substituting,

$\sigma_R = 3/2\,{\frac {P\cos(\beta)}{\pi \,{R}^{2}}}$

We want the vertical and shear stresses at this point. What we need is a conversion from the polar to cylindrical coordinates, which are given by the equations (Timoshenko and Goodier (1951))

$\sigma_{{z}}=\sigma_{{R}}\left(\cos(\beta)\right)^{2}+\sigma_{{T}}\left(\sin(\beta)\right)^{2}$
$\tau_{{\it rz}}=\left(\sigma_{{R}}+\sigma_{{T}}\right)\sin(\beta)\cos(\beta)$

These are coordinate transformations for plane stress equations and are discussed in detail in Boresi et.al. (1993) in terms of the direction cosines of the stress vectors. It may seem odd to see sine terms for these but $\cos\beta = \sin(\beta-\frac{3\pi}{2})$ . If we also assume that $\sigma_T = 0$ , again substituting we have

$\sigma_{{z}}=3/2\,{\frac {P\left (\cos(\beta)\right )^{3}}{\pi \,{R}^{2}}}$
$\tau_{{{\it rz}}}=3/2\,{\frac {P\left (\cos(\beta)\right )^{2}\sin(\beta)}{\pi \,{R}^{2}}}$

Since

$\beta = \arccos({\frac {z}{\sqrt {{z}^{2}+{r}^{2}}}})$

and

$R = \sqrt {{z}^{2}+{r}^{2}}$

substituting both of these yields our desired result

$\sigma_{{z}}=3/2\,{\frac {P{z}^{3}}{\pi \,\left ({z}^{2}+{r}^{2}\right )^{5/2}}}$
$\tau_{{{\it rz}}}=3/2\,{\frac {P{z}^{2}r}{\pi \,\left ({z}^{2}+{r}^{2}\right )^{5/2}}}$

Many textbooks (such as Verruijt) state these equations in terms of R, but in general problems such as this are defined most simply in terms of the depth of the point of interest z and the horizontal distance from that point r.

For the vertical stresses, if we define an influence coefficient

$K=3/2\,{\frac {1}{\pi \,\left (1+(\frac{r}{z})^{2}\right )^{5/2}}}$

then

$\sigma_z = K \frac{P}{z^2}$

and we can use the following table to determine the influence coefficients K.

Influence Coefficient K to calculate vertical stresses from a concentrated force for a given r/z ratio (from Tsytovich (1976))

The reason we’ve skipped the lateral stresses is because they’re dependent upon the elastic properties of the soil, and also because the vertical stresses are of greater interest.

The point load problem is an important one because many of the area load problems are based on its solution. It can also be used in other ways in spite of the fact that the solution is singular at the point where the load is applied.

Other References

Boresi, A.P., Schmidt, R.J., and Sidebottom, O.M. (1993) Advanced Mechanics of Materials. Fifth Edition. New York: John Wiley and Sons.
Timoshenko, S., and Goodier, J.N. (1951) Theory of Elasticity. New York: McGraww-Hill Book Company, Inc.