The city with a metropolitan population of over 20 million is sinking at a rate of almost 50 centimeters (20 inches) per year — and this isn’t stopping anytime soon.
At first glance, you’d be inclined to attribute this to the strong earthquakes that sometimes strike Mexico City. But while earthquakes can cause their own damage, they’re not the main culprit here. Instead, it’s something much more inconspicuous: subsidence.
You can read it all here. Put into geotechnical terms, the bed of old Lake Texcoco has some very high void ratio soils, and as a large city puts pressure on them the void ratio decreases as the voids between the soil grains shrink. Thus the entire city has severe settlement, total and differential.
A diagram, from the Swedish geotechnical engineer and academic Bengt Broms, showing how we consider the volume and mass/weight relationships in soil. The particulate matter of the soil means that the soil mass has three components: solid (particles,) water (in the voids) and gas/air (also in the voids.) That simplification is shown above, along with the definition of void ratio.
A diagram, again from Bengt Broms, illustrating the problem in Mexico City and whenever what we call consolidation settlement takes place. The soil particles have been combined into one mass (hatched area.) As pressure is applied, the particles come closer to each other and the volume of the voids decreases, thus we have settlement.
A photo from Mexico City showing the effects of subsidence many years ago. The top of the pole was originally the ground surface before structures were built on it and subsidence started. The photo and an explanation can be found in the textbook Soils in Construction. Needless to say, it’s only gotten worse in the intervening years. Photo courtesy of J.R. Bell.
There are some things in geotechnical engineering that don’t get really good (if any) coverage in many textbooks, which means that those who go on into that part of civil engineering are blindsided by their appearance. One of these is the “line of optimums” approach for compaction evaluation. The only formal textbook I know of that covers it is Soils in Construction, for which I must credit my co-author, Lee Schroeder. It also appears in the Soils and Foundations Reference Manual.
The line of optimums approach seeks to answer a key question in compaction: how much compactive energy is necessary to effect a given compaction? We have the Standard Proctor and the Modified Proctor test, but when we’re trying to determine a specific compactive effort for a particular soil and project, we need more flexibility.
Compaction Chart with Multiple Compactive Energies and Line of Optimums, from Rebrik (1966)
The lines on the chart are as follows:
Lines 1, 2, 3 and 4 represent compaction curves for a soil, but with a different number of blows (25, 50, 100 and 150, as shown in the chart.) As is customary, the plot is the water content (x-axis) against the dry density (y-axis,) although in American practice this is usually the dry unit weight.
Line 5 is the “zero air voids curve,” i.e., the curve where the degree of saturation S = 100%.
Line 6 is a “trendline” of the peak compaction dry densities, a “line of optimums.”
At this point, we will show a more “contemporary” approach to the line of optimums method than what’s in the book, which dates back to the semilog paper era. The first thing we do is to switch the x-axis from the water content to the number of blows for each compaction. We then plot this against the y-axis of the maximum dry density for each compaction, the result is tabulated as follows:
As it happens the power correlation turns out to have the highest R2 value. The book’s graph implies an exponential fit, but the variance in R2 between them is not great. Using the spreadsheet’s trendline feature gives the designer more flexibility in reducing the data. An explanation of that trendline feature–and curve fitting in general–can be found at Least Squares and Curve Fitting.
At this point we have a problem: we do not have the compactive effort for the differing blow counts. This is easily remedied because Equation 8.5 in Soils in Construction shows that, if the compaction mould, volume of soil, number of lifts and impact energy are all the same (a reasonable assumption in this case,) then the number of compactive blows is directly proportional to the compactive energy.
With that information in hand, we can take an actual compaction method and lift depth and estimate the maximum dry density we can expect. If it is not adequate for the task, we can either use a different compaction method or revise our design for a lower compactive dry density. We also need to determine a reasonable relative compaction, which will reduce this value to one which we can expect to happen during actual compaction.
The line of optimums method is a good one for compaction evaluation, and we hope that this little presentation helps you to understand it.
Reference
Rebrik, B.M (1966) Vibrotekhnika v burenii (Vibro-technology for Drilling.) Moscow, Russia: Nedra.
In the course of teaching my Soil Mechanics class, I’ve tried numerous different charts and methods for teaching the Unified system of soil classification. Probably the most success I’ve had is with the one from NAVFAC DM 7, and it’s below. (I’ve included the plasticity chart for completeness.)
This chart is reproduced (with better typography) in my book Soils in Construction. Unfortunately ASTM has been messing with this procedure, and for that reason I have had to shift to it in the last years of teaching Soil Mechanics. I still prefer this NAVFAC chart because it reduces soil classification to a straight-up process of elimination rather than the “decision tree” approach ASTM apparently prefers. NAVFAC DM 7.1, the newer edition, has gone with a narrative description of the system (except for the plasticity chart) which is even harder to follow for those just learning the system.
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