Hello to all,
Italy has just issued a construction law based on limit state design
(similar to LRFD- load &resistace factor design- as known in USA and
Canada).
This entails a probabilistic analysis of capacity and demand to define
probability of failure as P(f) = P[(C-D) < 0] where C-D is the performance
distribution of the structural system subject to loads and resistances. Now,
this law allows also a semiprobabilistic assessment of stability, taking
point estimates of the distributions called "characteristic values"
(adopting the definition of Eurocode 7). These point estimates are the 95th
percentile for loads, the 5th percentile for resistances. Problems arise in
the definition of the characteristic value for soils. A current
interpretation of EC7 would be that characteristic values for soils would be
the 5th percentile of the distribution of the mean, when failure surfaces
are larger than the typical scale of fluctuation of the investigated soil
property. Assuming we are in this condition, if I have, say, 5 samples with
mean 38, STD 4.85 (assuming a normal distribution of the population):
5th percentile of the mean = 38-2.132(4.85/2) = 32.8
Now some field tests, called penetrometers, measure soil strenght at
discrete intervals, by the hammer blows needed to ensure a certain
penetration into the soil (let's say 20 cm to 1 foot). This is similar to
some sort of a continuos, discrete sampling of a signal or random process
(soil properties along a profile may be similar to stock-market
fluctuations).
In this case I have, as a dataset, a collection of spatial means. But this
will also smooth my signal (smaller variance), so if I follow the above way
to determine the 5th percentile of the mean, I'll come up with an higher
value, which in fact is the characteristic value not of the soil, but of a
smoothed property of the soil.
At last my question: do you know if there is a way around this problem
(which I have to tackle also in other areas, such as vibration
measurements - instrumentation circuits sample the signal with user -defined
temporal means). Changing the field tests is often not an option.
I hope I've been clear enough.
Best to all
Luca Nori
<At last my question: do you know if there is a way around this problem
(which I have to tackle also in other areas, such as vibration
measurements - instrumentation circuits sample the signal with user -defined
temporal means). Changing >
Have you tried a kriging program with a nugget effect (e.g.,
Best,
-- Tony
At 10:45 AM 3/6/2006 +0100, Nori Luca wrote:
>A current interpretation of EC7 would be that characteristic values for
>soils would be the 5th percentile of the distribution of the mean, when
>failure surfaces are larger than the typical scale of fluctuation of the
>investigated soil property. Assuming we are in this condition, if I have,
>say, 5 samples with mean 38, STD 4.85 (assuming a normal distribution of
>the population): 5th percentile of the mean = 38-2.132(4.85/2) = 32.8
This formula looks like that for a lower 95% confidence limit--the -2.132
evidently is the 5th percentile of Student's t with four df--except that
sqrt(5) has been replaced by 2. What is the theory behind this
formula? And what exactly do you mean by "the distribution of the
mean?" Are you trying to compute a confidence limit for the mean
characteristic value of all soils within the study region, or are you
trying to estimate a percentile of the sampling distribution of the mean of
five independent random samples of soils in that region?
>Now some field tests, called penetrometers, measure soil strenght at
>discrete intervals, by the hammer blows needed to ensure a certain
>penetration into the soil (let's say 20 cm to 1 foot). This is similar to
>some sort of a continuos, discrete sampling of a signal or random process
>(soil properties along a profile may be similar to stock-market fluctuations).
>
>In this case I have, as a dataset, a collection of spatial means. But this
>will also smooth my signal (smaller variance), so if I follow the above
>way to determine the 5th percentile of the mean, I'll come up with an
>higher value, which in fact is the characteristic value not of the soil,
>but of a smoothed property of the soil.
>
>At last my question: do you know if there is a way around this problem
If I have understood this correctly, what you are saying is that you view
every penetrometer value as measuring a vertically averaged quantity, but
you would like to estimate statistical properties of the "soil"
itself. This is known as the "change of support" problem in
geostatistics. It is well discussed in Journel and Huijbregts' classic
text "Mining Geostatistics" (c. 1978). The solution requires estimating
the vertical component of the variogram (which is indeed very much akin to
using the correlogram of a time series). The extreme cases are (i) a "pure
nugget" effect, having no vertical spatial correlation, and (ii) perfect
correlation. In case (i) the solution is a deconvolution and in case (ii)
the average quantity coincides with the point values (so that no numerical
adjustment is necessary). The variogram describes all intermediate
cases. The change of support methodology accounts for varying amounts of
vertical correlation in order to perform a kind of deconvolution of the
averages.
In many cases, however, one does want to compute average or bulk properties
of the soil. For instance, to assess its permeability to vertical flow
through a given horizon, we would want to estimate a suitable mean of the
vertical permeabilities throughout that horizon. Therefore it is not
immediately clear whether it is appropriate or not to use a "smoothed"
property such as a blow count. The answer depends on how the blow counts
will be used for subsequent modeling or decision making.
--Bill Huber
Tony, Bill,
thanks for your tips, you have independently suggested to me the same
solution, and I'm going right into the study of spatial statistics and
variograms. Hope deconvolution won't be a major problem. Tony, I just saw
the freeware on the SADA website, I'm going to download it right away.
Besides:
Bill Huber wrote:
> This formula looks like that for a lower 95% confidence limit--the -2.132
> evidently is the 5th percentile of Student's t with four df--except that
> sqrt(5) has been replaced by 2. What is the theory behind this
> formula? And what exactly do you mean by "the distribution of the
> mean?" Are you trying to compute a confidence limit for the mean
> characteristic value of all soils within the study region, or are you
> trying to estimate a percentile of the sampling distribution of the mean
of
> five independent random samples of soils in that region?
>
I'm sorry I wasn't more specific, I'm trying to estimate, from the sampled
values (along a vertical profile beneath or nearby the foundation
structure), the 5th percentile of the distribution of the population mean.
This would be the Eurocode7 "characteristic value". The formula is just what
you say, only using sqrt(n-1) rather then sqrt
as the st.deviation
scaling factor, my source for that being David Vose's book on QRA . If you
have a reason to believe sqrt
may be more appropriate for this problem,
I'll trust you and just change it.
> If I have understood this correctly, what you are saying is that you view
> every penetrometer value as measuring a vertically averaged quantity, but
> you would like to estimate statistical properties of the "soil"
> itself. This is known as the "change of support" problem in
> geostatistics. It is well discussed in Journel and Huijbregts' classic
> text "Mining Geostatistics" (c. 1978). The solution requires estimating
> the vertical component of the variogram (which is indeed very much akin to
> using the correlogram of a time series). The extreme cases are (i) a
"pure
> nugget" effect, having no vertical spatial correlation, and (ii) perfect
> correlation. In case (i) the solution is a deconvolution and in case (ii)
> the average quantity coincides with the point values (so that no numerical
> adjustment is necessary). The variogram describes all intermediate
> cases. The change of support methodology accounts for varying amounts of
> vertical correlation in order to perform a kind of deconvolution of the
> averages.
Your understanding is right, I'm interested in estimating the soil
properties knowing only the vertically averaged quantities. Are you speaking
about the real variogram of the soil properties (coming from not-averaged
measurements)? We would not know it, although the vertical spatial
correlation can be estimated from the literature for various soil groups.
> In many cases, however, one does want to compute average or bulk
properties
> of the soil. For instance, to assess its permeability to vertical flow
> through a given horizon, we would want to estimate a suitable mean of the
> vertical permeabilities throughout that horizon. Therefore it is not
> immediately clear whether it is appropriate or not to use a "smoothed"
> property such as a blow count. The answer depends on how the blow counts
> will be used for subsequent modeling or decision making.
As a matter of fact, Eurocode7 suggests that, for large volumes of soil
involved (in the failure process) the bulk property to use is "a cautious
estimate of the mean value". This rather obscure sentence is mainly
construed as referring to the 5th percentile of the population mean. So,
knowing only the vertically averaged quantities (blow counts), I should
estimate the soil properties (not averaged) and from there figure out the
5th percentile of the population mean, to comply with EC7 specs.
Interestingly, if the model does not involve large failure surfaces (i.e.
isolated foundation piles base failure), some codes and authors reccomend to
use as a characteristic value the 5th percentile of the sample itself
(usually parameterized as a normal pdf). The rationale would be that in
large volumes-profiles random fluctuations around the trend (noise) would
cancel out and the arithmetic mean of the properties would govern the
failure mechanism. Conversely, the fluctuation itself would govern, hence we
would be in a more unfavourable condition (the 5th percentile of the sample
mean is lower than the 5th percentile of the population mean ).
(David Gillette writes
I would like to caution you about using statistical analysis of soil
properties to infer the mechanical properties of the soil mass on a gross
scale. This is a topic that has occupied much of my attention for the last
25 years.
For a compressibility problem such as settlement of a building, the
vertically averaged stiffness can be a pretty good representation.
However, for strength problems, it is the weakest link in the chain (the
weakest layer) that governs. What's needed is an average over a potential
failure plane for a slope or foundation. Vertical averaging can be
dangerously unconservative.
Penetration testing cannot measure strength. It must be interpreted using
correlations and/or theoretical analysis of the soil shearing around the
tip of the penetrometer. Impact-type penetration testing ("standard"
penetration test, Becker-hammer penetration test, or the Japanese large
penetration test), even more than steady-push testing (cone penetrometer
test), is heavily dependent on empirical adjustments and correlations based
on very heterogeneous data sets. For a given soil, the SPT measurement is
actually a function of the lifting mechanism for the hammer, the operator's
skill and fatigue level, the condition of the pulleys, the style of hammer
(even among "standard" hammers), the type of drill rods, whether the
sampler is used with a liner, drilling fluid, and plenty of other things.
The CPT has fewer things that can go wrong with it (and is preferred
whenever soil conditions permit), but it too relies on semi-empirical
correlation factors that are sensitive to stress history of the soil,
small-scale layering (such as varves), plasticity index (essentially a
measure of how much water a clay can absorb and still act as a solid), and
anisotropy. With any soil strength testing, statistical variation in the
raw data (SPT blow counts or laboratory strength measurements) generally
does not come anywhere close to capturing the uncertainty in the soil
strength. (Sixteen years ago, I fiddled with a Dempster-Shafer
weight-of-evidence approach, incorporating the engineer's judgment/voodoo
of what the test data actually mean, and conflicting/corroborating results
from different types of test.)
I don't have serious heartburn with the 5th percentile/95th percentile
approach sometimes being a reasonable and inexpensive way to reach a usable
result, but the 5th percentile MUST come from strengths determined for the
materials actually involved in the potential failure mechanism, NOT from
vertical averaging of the raw data.
Best regards,
DRG (Dirt, Rocks - Geotech)
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