Environmental Engineering Reference
In-Depth Information
m mm
=
−
[3.11]
R
r
q
and dispersion
DDDD
=−
2
+
.
[3.12]
R
r
rq
q
Usually, the strength
r
and the load
q
can be considered independent,
random variables. Then
D
rq
= 0, and formula [3.12] is simplified:
DDD
=
+
.
[3.13]
R
r
q
The safety characteristic in this case is:
mm
DDD
−
r
q
g=
.
[3.14]
−
2
+
r
rq
q
Dispersion
D
rq
is not equal to zero only in the presence of a correlation
between strength and load. This relationship is generally absent or has a
negligible value. In this case, the safety characteristic can be determined
by a simple formula
mm
DD
−
r
q
g=
.
[3.15]
+
r
q
In the general case of the arbitrary distribution functions
r
and
q
, the
probability of failure
V
can be determined by the formula
∞
∫
p
=
pR qqdq
(
,) ;
+
[3.16]
R
−∞
∞
∞∞
∞
rq
=
(
)
( )
∫
∫∫
∫∫
V P
= =
(0)
=
p dR
+
p R q q dqdR
=
,
p r q drdq
,
.
R
R
−∞
−∞ −∞
−∞ −∞
Thus, the distribution function
p
(
r, q
) should be integrated in the plane
r, q
over the area lying below the line
r = q
(Fig. 3.1).
In practice, it is easy to do this numerically, as values of
p
(
r, q
) in all
directions decrease rather quickly.
An example of calculation of steel structures
In steel structures, fracture stresses are represented by yield strength σ
T
.
The permissible stress in structures [σ] is defined by the safety factor which
in various regulatory documents may have different meanings, such as for
steel St3 it is 1400 kgf/cm
2
and then increases to 1600 kgf/cm
2
. It was
assumed that in the former case, the safety factor with respect to the minimum
yield strength was 2200 kgf/cm
2
: 2200/1400 = 1.57 and 2200/1600 = 1.37,
respectively.
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