F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

10 Limit state design concepts
for
x = 50 N/mm 2 , z = 50 S 45 = 1
for
x = 60 N/mm 2 z = 60 - 45 = 3
From Table 1.3-3,
' 5
area between z = 0 and z = 3 is 0.4987
area between z = 0 and z = 1 is 0.3413
Therefore
area between z = 1 and z = 3 is 0.4987 - 0.3413 = 0.1574
(b) We want the limits of z between which the area under the normal
probability curve is 0.999. Because of symmetry, the area between
z = 0 and z = z is half of 0.999, i.e. 0.4995. From Table 1.3-3,
z = 3.3. Therefore
x =.X± 3.3a = 45 N/mm 2 ± 3.3 x 5 N/mm2
= 28.5 N/mm 2 or 61.5 N/mm 2
Ans. (a) There is a probability of 15.74% that a random cube would
have a strength between 50 and 60 N/mm 2 •
(b) There is a 99.9% probability that the whole range of cube
strengths is from 28.5 to 61.5 N/mm 2 •
Level of significance and confidence level
If a set of values are normally distributed, then the probability of any single
value falling between the limits .X ± za is the area of the normal probability
curve (Fig. 1.3-3 and Table 1.3-3) between these limits. This
probability, expressed as a percentage, is called the confidence level, or
confidence coefficient. The limits (.X - za) and (.X + za) are called the
confidence limits and the interval (.X - za) to (.X + za) is called the
confidence interval. The probability of any single value falling outside the
limits .X ± za is given by the areas under the two tails of the normal
probability curve outside these limits; this probability, expressed as a
percentage, is called the level of significance.
Example 1.3-4
It is known that a set of test results (which are normally distributed) has a
mean strength of 82.4 N/mm 2 and a standard deviation of 4.2 N/mm 2 •
Determine:
(a) the 95% confidence limits;
(b) the strength limits at the 5% level of significance; and
(c) the strength below which 5% of the test results may be expected to
fall.
SOLUTION
(a) With reference to Fig. 1.3-3 and Table 1.3-3, we wish to determine
the limits of z such that the area below the normal curve between