A deck carrying a loading crane in a port is supported by 18 columns as shown in Figure 1. The columns are divided in the three groups shown in the figure (A, B and C). Failure of the entire system can occur in any of the following three ways during operation: (i) Group A fails, (ii) Either one of the two sub-groups (B1, B2) of Group B fails, (iii) Both sub-groups (C1 and C2) of Group C fail. The resulting combined series and parallel configuration of the entire system is shown in Figure 2. Group A and sub- groups B1, B2, C1, C2 fail when all their columns fail (as one of the columns fails, the remaining columns must carry additional loads through redistribution). Assume that the total load S is distributed among the three groups as follows: Group A carries 10% of S, Group B carries 40% of S, and Group C carries 50% of S. The coefficient of variation of these three loads is 15% (including the coefficient of variation in S and variations in load distribution). Assume further that the individual column capacity in the different groups is as follows: Group A B с Mean Capacity of Each Column 0.150 S 0.125.S 0.125. S Coefficient of Variation of Capacity of Each Column 20% 20% 20% Assume that the column capacities and the load S are all normally distributed. Compute the probability of failure of the entire deck system assuming that the failures of groups A, B and C are independent of each other and that the failures of sub-groups B1 and B2 are also independent of each other. [Hint for failure probability of group C: P(Fc)=2!P(Fc₂)P(Fc₁|Fc₂)]
A deck carrying a loading crane in a port is supported by 18 columns as shown in Figure 1. The columns are divided in the three groups shown in the figure (A, B and C). Failure of the entire system can occur in any of the following three ways during operation: (i) Group A fails, (ii) Either one of the two sub-groups (B1, B2) of Group B fails, (iii) Both sub-groups (C1 and C2) of Group C fail. The resulting combined series and parallel configuration of the entire system is shown in Figure 2. Group A and sub- groups B1, B2, C1, C2 fail when all their columns fail (as one of the columns fails, the remaining columns must carry additional loads through redistribution). Assume that the total load S is distributed among the three groups as follows: Group A carries 10% of S, Group B carries 40% of S, and Group C carries 50% of S. The coefficient of variation of these three loads is 15% (including the coefficient of variation in S and variations in load distribution). Assume further that the individual column capacity in the different groups is as follows: Group A B с Mean Capacity of Each Column 0.150 S 0.125.S 0.125. S Coefficient of Variation of Capacity of Each Column 20% 20% 20% Assume that the column capacities and the load S are all normally distributed. Compute the probability of failure of the entire deck system assuming that the failures of groups A, B and C are independent of each other and that the failures of sub-groups B1 and B2 are also independent of each other. [Hint for failure probability of group C: P(Fc)=2!P(Fc₂)P(Fc₁|Fc₂)]
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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I was wondering if you could help me understand how to find the probability of failure of the entire deck system assuming that the failures of groups A, B and C are independent of each other and that the failures of sub-groups B1 and B2 are also independent of each other
![A deck carrying a loading crane in a port is supported by 18 columns as shown in
Figure 1. The columns are divided in the three groups shown in the figure (A, B and C).
Failure of the entire system can occur in any of the following three ways during
operation: (i) Group A fails, (ii) Either one of the two sub-groups (B1, B2) of Group B
fails, (iii) Both sub-groups (C1 and C2) of Group C fail. The resulting combined series
and parallel configuration of the entire system is shown in Figure 2. Group A and sub-
groups B1, B2, C1, C2 fail when all their columns fail (as one of the columns fails, the
remaining columns must carry additional loads through redistribution). Assume that the
total load S is distributed among the three groups as follows: Group A carries 10% of
S, Group B carries 40% of S, and Group C carries 50% of S. The coefficient of
variation of these three loads is 15% (including the coefficient of variation in S and
variations in load distribution). Assume further that the individual column capacity in the
different groups is as follows:
Group
A
B
с
B1
Assume that the column
capacities and the load S are all normally distributed.
Compute the probability of failure of the entire deck system assuming that the failures of
groups A, B and C are independent of each other and that the failures of sub-groups B1
and B2 are also independent of each other.
[Hint for failure probability of group C: P(Fc) = 2!P(F₂)P(Fc1|Fc₂)]
C1
Mean Capacity
of Each Column
0.150 S
0.125.S
0.125. S
A
Figure 1
B2
C2
Coefficient of Variation of
Capacity of Each Column
20%
20%
20%
A
B1
B
B2
Figure 2
C
C1
C2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faddade67-6dca-41b9-b5d9-3d81c38171e9%2Fdbd5dfd8-72f6-4b22-a79d-ef8cf3d1a922%2Fwoid12_processed.png&w=3840&q=75)
Transcribed Image Text:A deck carrying a loading crane in a port is supported by 18 columns as shown in
Figure 1. The columns are divided in the three groups shown in the figure (A, B and C).
Failure of the entire system can occur in any of the following three ways during
operation: (i) Group A fails, (ii) Either one of the two sub-groups (B1, B2) of Group B
fails, (iii) Both sub-groups (C1 and C2) of Group C fail. The resulting combined series
and parallel configuration of the entire system is shown in Figure 2. Group A and sub-
groups B1, B2, C1, C2 fail when all their columns fail (as one of the columns fails, the
remaining columns must carry additional loads through redistribution). Assume that the
total load S is distributed among the three groups as follows: Group A carries 10% of
S, Group B carries 40% of S, and Group C carries 50% of S. The coefficient of
variation of these three loads is 15% (including the coefficient of variation in S and
variations in load distribution). Assume further that the individual column capacity in the
different groups is as follows:
Group
A
B
с
B1
Assume that the column
capacities and the load S are all normally distributed.
Compute the probability of failure of the entire deck system assuming that the failures of
groups A, B and C are independent of each other and that the failures of sub-groups B1
and B2 are also independent of each other.
[Hint for failure probability of group C: P(Fc) = 2!P(F₂)P(Fc1|Fc₂)]
C1
Mean Capacity
of Each Column
0.150 S
0.125.S
0.125. S
A
Figure 1
B2
C2
Coefficient of Variation of
Capacity of Each Column
20%
20%
20%
A
B1
B
B2
Figure 2
C
C1
C2
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