Civil Engineering Reference
In-Depth Information
10.4 DISTRIBUTION PATTERN
Most investigators agree that strength is at least approximately a normally
distributed variable. This means that it can be completely described by a
mean strength and a standard deviation, that is, the percentage of results
lying above or below any particular value can be calculated from the mean
strength, the standard deviation, and a table of values from a statistical
textbook as seen in Figure 10.4. The authors have found this assumption to
be well justified in practice except that only about half the results theoreti-
cally expected to be below the mean minus 1.64 σ usually occur in practice.
The formula used is
X
=
F
+ kσ, where
X
is the required average strength,
F
the specified strength, σ the standard deviation, and
k
is a constant
depending on the proportion of results permitted to be below
F
. In fact
the overall distribution is likely to consist of a number of subdistributions,
each with a slightly different mean strength but probably with a similar SD
slightly lower than the overall SD as shown in Figure 10.5. The assumption
is that there is a basic variability in the process with hopefully infrequent
occurrences of an unusual factor.
The SD can be calculated in at least two different ways. The “tradi-
tional” way (by which it is defined, and which the typical calculator uses) is
by calculating the mean, totaling the squared differences of each individual
result from it, and then finding the square root of that total.
A second way (referred to herein as the “basic” method) is to average the
difference between successive results and divide by 1.13. If (as assumed)
the mean has continued to be the same during the entire string of results,
Mean strength
Speci
�
ed
characteristic
strength
5% defectives
1.64σ
25
30
35 40
Compressive Strength (N/mm
2
)
45
50
55
Figure 10.4
The normal distribution.
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