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)

12 Limit state design concepts
This lower strength limit, below which not more than 5% of the test
results can be expected to fall, is referred to as the characteristic
strength in Section 1.4, and is equal to the mean strength less 1.64
times the standard deviation.
Example 1.3-5
A random variable x is known to be normally distributed. Explain how the
probability of x assuming any value between the limits x 1 and x 2 can be
determined.
SOLUTION
Let P(x 1 :5 x :5 x2 ) denote the probability of x assuming any value
between the limits x1 and x2•
Using the normal probability distribution as defined by eqn (1.3-7),
P(x1 :5 x :5 x2) = J: 2 a~I2n) exp{ -!(x - .X) 2 /if} dx
The integral on the right-hand side of the above equation cannot be
evaluated by elementary means. However, if we set z = (x- .X)! a, then x
= az + .X so that dx = adz; also z 1 = (x 1 - x)/a and z 2 = (x 2 - x)/a.
Then the above equation becomes
P(x1 :5 x :5 x2) = ( 2 a~I2n) exp( -!z 2 )adz
fz 2 1 I 2
= z, ~(2n) exp( -"2z )dz
In other words, the required answer is obtained by integrating the area
under the standardized normal probability distribution curve, as defined by
eqn (1.3-9). The integral is conveniently evaluated using Table 1.3-3. Of
course, it is first necessary to express x 1 and x2 in terms of z, as illustrated
by Example 1.3-3.
1.4 Characteristic strengths and loads
As stated earlier, limit state design is based on statistical concepts and
on the application of statistical methods to the variations that occur in
practice. These variations may affect not only the strength of the materials
used in the structure, but also the loads acting on the structure. Indeed,
the strength of concrete itself provides a good example of the variations
that can occur. Past experience has shown, for example, that the
compressive strength of concrete test specimens, which have been made as
identically as possible under laboratory conditions, may have a coefficient
of variation (see eqn 1.3-6) of as much as ±10%. In reinforced or
prestressed concrete construction, therefore, it is not practicable to specify
that the concrete should have a certain precise cube strength, or that the
reinforcement should have a particular yield stress or proof stress.
Limit state design uses the concept of characteristic strength, [k, which
means that value of the compressive strength of concrete, the yield or