Environmental Engineering Reference
In-Depth Information
written as:
Permitted service
Stress
−
strain state
≤
characterisics of the
of structural element
stress
−
strain state
The simplest example of such a condition could be the formula of the strength
of a tensile-loaded bar
N
R
=s− >
0,
[3.1]
T
F
where
N
is tension;
F
is the cross-sectional area of the bar; σ
T
is the tensile
strength of the bar material.
In general, the condition [3.2] can be written as:
(
)
Rxx x
>
,
,...
0,
[3.2]
12
n
where
x
l
,
x
2
,...,
x
n
are some calculated quantities.
Each of the calculated quantities
x
l
,
x
2
,...,
x
n
can deviate from its mean
(expected) values, and these deviations can be characterised as a function
of their distribution
p
z
(
x
l
,
x
2
,...,
x
n
) obtained statistically or on the basis of
theoretical considerations. From this function, one can go to the distribution
curve
R
using the formula
52
∞
∂
R
(
)
[3.3]
∫∫ ∫
p
=
...
p x x
,
,...,
x dx dx
,...,
dx
.
R
z
12
n
2 3
n
∂
x
1
−∞
−
( )
n
1 times
If the law of distribution of
p
g
is normal and the function
R
(
x
l
,
x
2
,...,
x
n
)
is linear or if the function
R
(
x
l
,
x
2
,...,
x
n
) is non-linear but the variance of
the distribution of
p
R
is so small that within the square root of variances,
multiplied by a small number (two or three) the function
R
with sufficient
accuracy can be replaced by a linear function, the curve
p
R
will be expressed
by the normal distribution
2
(
Rm
−
)
1
Rm
−
1
R
−
p
=
Φ
'
=
R
e
.
2
D
[3.4]
R
D
D
2
π
D
R
R
R
where
m
R
and
D
R
are determined by replacing
u
by
R
52
.
It remains to determine the probability of default of inequality [3.2] or,
equivalently, the probability of fulfilling the fracture condition
Rx x x
<
(
,
...,
)
0,
[3.5]
Knowing the curve
p
R
, this can be done very easily. It is enough to integrate
it from minus infinity to zero, i.e., determine the ordinate of the integrated
12
n
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