94
Z. Liu et al.
Let x 1 , x 2 , …, x n be n independent random variables, if they are from the same
parent, which have the same distribution and the same finite mean and variance,
denoted by μ and σ
2 , respectively. Then, for any ε > 0, we have Eq. (8.2) below.
lim
n→∞
P
1
n
n
i=1
x i − μ
≥ ε
= 0
( 8 . 2 )
Bernoulli Trials: If the probability of occurrence of random event A is P(A), in n
independent experiments, the frequency of event A occurs is m, and the frequency is
W (A) = m/n. Then for any ε > 0, we have Eq. (8.3) below.
lim
n→∞
P
m
n
− P(A)
< ε
= 1
(8.3)
The core content of the Monte Carlo method is to obtain the ratio of the number of
structural failures to the total number of samples by a large number of samples, which
is the failure probability of the structure. As the number of simulations increases, the
Monte Carlo method will gradually become more accurate. Therefore, in structural
reliability calculations, Monte Carlo calculations are considered to be quasi-accurate
calculations, while the accuracy of the other approximation method is often verified
by the Monte Carlo method.
8.2.3 Calculation Model
The pressurizer spray nozzle and the adjacent structure mainly include the upper
head, the surfacing layer, the thermowell, the safety end, the spray head connecting
pipe and the forging, etc. Considering that the analyzed object has the axisymmetric
geometric characteristics, the calculation is performed using a two-dimensional
axisymmetric model in ANSYS. Two models are built in total. The first is the thermal
analysis model, which is established by the two-dimensional 8-node thermal element
(PLANE77). The second is the thermal stress analysis model, which is established by
the 8-node harmonic element (PLANE83), which is used to calculate the secondary
stress due to thermal gradient and different material expansion coefficients. Both
models contain non-pressure boundary portions such as nozzle tips and heat pipes,
while also considering the effect of the weld overlay on heat transfer.
In order to eliminate the influence of the boundary effect, the nozzle and the lower
part of the cylinder should be extended. Equation (8.4) defines the extension length.
L = 2.5
R m t
(8.4)
where R m is the average radius of the nozzle or the head, and t is the corresponding
minimum wall thickness. After calculation, the extension length of the nozzle and
Z. Liu et al.
Let x 1 , x 2 , …, x n be n independent random variables, if they are from the same
parent, which have the same distribution and the same finite mean and variance,
denoted by μ and σ
2 , respectively. Then, for any ε > 0, we have Eq. (8.2) below.
lim
n→∞
P
1
n
n
i=1
x i − μ
≥ ε
= 0
( 8 . 2 )
Bernoulli Trials: If the probability of occurrence of random event A is P(A), in n
independent experiments, the frequency of event A occurs is m, and the frequency is
W (A) = m/n. Then for any ε > 0, we have Eq. (8.3) below.
lim
n→∞
P
m
n
− P(A)
< ε
= 1
(8.3)
The core content of the Monte Carlo method is to obtain the ratio of the number of
structural failures to the total number of samples by a large number of samples, which
is the failure probability of the structure. As the number of simulations increases, the
Monte Carlo method will gradually become more accurate. Therefore, in structural
reliability calculations, Monte Carlo calculations are considered to be quasi-accurate
calculations, while the accuracy of the other approximation method is often verified
by the Monte Carlo method.
8.2.3 Calculation Model
The pressurizer spray nozzle and the adjacent structure mainly include the upper
head, the surfacing layer, the thermowell, the safety end, the spray head connecting
pipe and the forging, etc. Considering that the analyzed object has the axisymmetric
geometric characteristics, the calculation is performed using a two-dimensional
axisymmetric model in ANSYS. Two models are built in total. The first is the thermal
analysis model, which is established by the two-dimensional 8-node thermal element
(PLANE77). The second is the thermal stress analysis model, which is established by
the 8-node harmonic element (PLANE83), which is used to calculate the secondary
stress due to thermal gradient and different material expansion coefficients. Both
models contain non-pressure boundary portions such as nozzle tips and heat pipes,
while also considering the effect of the weld overlay on heat transfer.
In order to eliminate the influence of the boundary effect, the nozzle and the lower
part of the cylinder should be extended. Equation (8.4) defines the extension length.
L = 2.5
R m t
(8.4)
where R m is the average radius of the nozzle or the head, and t is the corresponding
minimum wall thickness. After calculation, the extension length of the nozzle and