The Design of Modern Steel Bridges - TEDI

82 The Design of Modern Steel Bridges
will thus be proportional to
e ð1=2Þo2 R e ð1=2Þo2 S ¼ e ð1=2Þðo2 R þo2 S Þ
In the space of the reduced variables, the locus of the point of equal frequency
will thus be a circle around the origin, larger radius representing lower frequency.
These circles are shown in Fig. 4.3.
In Fig. 4.3 a vector OA is drawn from the origin normal to the failure
boundary. OA is thus the shortest distance from the origin to the failure line.
Point A represents the most likely set of values of oR and oS for the occurrence
of failure and is thus called the ‘design point’. It can be shown that the length
of the vector OA is given by
mR mS
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2 R þ s2 p
S
and is thus numerically equal to the reliability index b.
The coordinates of the point A in the space of oR and oS can be shown to be
ðmR mSÞsR
ðs2 R þ s2 SÞ , ðmR mSÞsS
ðs2 R þ s2 SÞ which can also be expressed as ðaRb, aSbÞ when
aR ¼
@g
@oR
@g
@oR
2
þ @g
@oS
2 1=2 , aS ¼
@g
@oR
@g
@oS
2
þ @g
@oS
2 1=2
The design of new structures can be performed by considering any point on
the horizontal axis of R or S in Fig. 4.1 and its associated probability density
values of R and S. Alternatively, the distance of the point from the mean values
of R and S in multiples of the respective standard deviation may be used. For
the sake of convenience, the ‘design point’ can be chosen for this purpose, as
this point represents the most likely situation at failure.
Converting the reduced variables oR and oS to basic variables R and S, the
values of the latter at the ‘design point’ are
Rd ¼ mR þ oRsR ¼ mR þ aRbsR
Sd ¼ mS þ oSsS ¼ mS þ aSbsS
Usually the resistance and action-effect variables R and S of a structure are
functions of a number of variables x 1, x 2, ..., xn, and the probability density
functions f(R) and f(S) depend upon the probability density functions of the
individual variables x 1, etc. and how they are related in the functions R and S.
Some of the variables may be common to both, causing some correlation; this
has to be taken into account by modifying the statistical parameters by the
correlation coefficient. All the basic variables may be replaced by a new set of