Environmental Engineering Reference
In-Depth Information
distribution curve
R
for the values
R
= 0:
0
∫
V
=
p dR P
(0),
=
[3.6]
R
R
−∞
Here
V
is the probability of fracture.
For the normal law of distribution of
R
formula [3.6) has the form
according to Ref. 24
m
D
R
R
2
2
m
m
u
1
1
1
−
R
R
∫
V
= +Φ −
= −
Φ
= −
e du
.
[3.7]
2
2
2
D
D
R
R
0
mD
is denoted by γ and termed the safety characteristic
/
The value
R
R
g=
mD A
/
1/
,
=
[3.8]
A
R
here denotes the coefficient of variation of the indestructibility function
R
.
The probability of failure is then expressed by the formula
R
R
R
V
=
1 / 2
( ),
−Φ g
[3.9]
i.e., it is functionally linked to only one value, namely safety characteristic γ.
The safety can be defined as the ratio of the standard of the strength function
to its expected value. In comparison with quantity
V
, it has the advantage
that it can be expressed in ordinary cases by a simple numeric value of the
order of 1-5 and not by a very small fraction like
V
. As an example, some
values of
V
as a function of γ are given in Table 3.1.
In many cases, the indestructibility function can be expressed by the
following simple formula:
R = r - q,
[3.10]
Here
r
is the strength of the installation, measured in some units of the scale
x
, for example, in kg/cm
2
of the strength of the material of the structure;
q
is the load on the installation, measured in units of the same scale
x
, for
example, in kg/cm
2
, the stress in a dangerous cross section caused by external
forces.
If the distribution functions
r
and
q
are known and can be represented
with sufficient accuracy by the distribution normal law, the distribution
curve of
R
is also normal with the centre
Table 3.1
Values of
V
as a function of
g
V
0.1
0.01
0.001
0.0001
3.2·10
-
5
3·10
-
6
2.9·10
-
7
γ
1.28
2.32
3.15
3.77
4.00
4.50
5.00
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