NONDETERMINISTIC SYSTEMS
2.4 REDUNDANCY FACTOR OF SYSTEM WITH MANY COMPONENTS
2.4.2 Using the MCS-based Program
In some practical cases, a structure may consist of more than 200 components, such as a truss bridge or a high-rise building. Therefore, it is necessary to study the redundancy factors of systems that have a high number of components (N ≥ 200).
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Since the computational time needed by RELSYS to obtain the probability of failure for systems with more than 200 components is excessive, the Monte Carlo Simulation-based program is used herein to find the probability of failure, Pf, of the N-component systems (N = 100, 300, and 500). In this subsection, the mp×ns and ms×np SP systems where m equals to 5, 10 and 20 are investigated.
The algorithm of the MCS-based program for the calculation of Pf using MATLAB is described as follows:
1. Give the mean value of the load effect E(P), coefficients of variation of resistance and load effect V(R) and V(P), correlation between the resistances of components ρ(Ri, Rj), probability distribution types of resistance and load, number of components N, number of simulation samples w, and the initial guess for the mean value of component resistance Ecs(R);
2. Generate the random samples of resistance Ri and load effect P based on the above parameters, and the dimensions of the Ri and P vectors are w × 1;
3. Obtain the performance function for each component gi (i=1, 2, …, N); the Ri P dimensions of g is also w × 1; i
4. For series system, define a w × 1 zero vector L, and the ratio of the number of
>
L| g1 0 |...| gN 0@
to the total sample size w represents the failure probability of series system (“|” is logical OR in MATLAB; it refers to union herein); for the parallel system, define a w × 1 unit vector Q, and the ratio of the number of>
Q& g1 0 &...& gN 0@
to the sample size w is the Pf of parallel system (“&”is logical AND in MATLAB; it refers to intersection herein); for the mp × ns SP
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system, define a w × 1 zero vector L and a w × 1 unit vector Q, and the ratio of the number of
^
L|>
Q&g1 0
&...&
gm 0
@
|...|>
Q&gm(n1)1 0
&...&
gmn 0
@ `
to the sample size w is the Pf of the SP system; and for the ms × np SP system, define a w × 1 zero vector L and a w × 1 unit vector Q, and the ratio of the number of
^
Q&>
L|g1 0
|...| gm 0
@
&...&>
L|gm(n1)1 0
|...|
gmn 0
@ `
to the sample size w is the Pf of the SP system; it should be noted that in the series-parallel systems, n × m is equal to the number of components N.5. Repeat steps 1 to 4 for t times (e.g., t = 50) to obtain the average probability of failure of the system; then, convert it to the reliability index.
When using the MCS-based program to find the reliability index of systems, it is noticed that as N increases, the computational time required increases dramatically.
Therefore, the aforementioned search algorithm that requires a group of initial values is not efficient when combined with the MCS-based program. In order to reduce the computing time, a simple algorithm based on the effects of the number of components on the redundancy factor is used herein in combination with the MCS-based program to find Ecs(R) and ηR. The procedure of this algorithm is as follows:
1. Determine an initial guess value of Ecs(R) based on the effects of number of components N on the redundancy factors. For example, it was found previously that Ecs(R) associated with series (or series-parallel) system increases as N increases;
however, this increase is less significant as N becomes larger. Therefore, the initial guess of Ecs(R) for the 300-component series system can be obtained by increasing the Ecs(R) of 200-component series system by ∆ percent (0.5 ≤ ∆ ≤ 1). On the contrary, increasing N leads to lower Ecs(R) in parallel systems. Hence, the initial
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guess of Ecs(R) for the 500-component parallel system can be determined by reducing the Ecs(R) of 400-component parallel system by ∆ percent (0.5 ≤ ∆ ≤ 1).
2. Substitute the initial value to the MCS-based method described above to obtain the system reliability index βsys;
3. Checkpoint: if |Esys3.5 |dTol (Tol is set to be 10-4 herein), then return this initial value; otherwise go to the next step;
4. Checkpoint: if the βsys < 3.5, increase the initial value by δ percent (0.1 ≤ δ ≤ 0.3); if βsys > 3.5, reduce the initial value by δ percent (0.1 ≤ δ ≤ 0.3);
5. Repeat steps 2-4 until Ecs(R) is found.
Ecs(R) can usually be found within four loops. A flowchart for this algorithm combined with the MCS-based program is presented in Figure 2.12. It is seen that this algorithm is similar to the search algorithm that is combined with RELSYS; however, since the initial values in this algorithm are selected based on the conclusions from the effects of N on the redundancy factors, they are much closer to the final value of Ecs(R) than those in the search algorithm; therefore, the number of trials is drastically reduced and, therefore, the computational time is decreased.
As stated previously, the coefficients of variation of resistance and load are 0.05 and 0.3, respectively. The mean value of load acting on each component E(P) is assumed to be 10. Three correlation cases (ρ(Ri,Rj) = 0, 0.5 and 1.0) among the resistances of components and two probability distribution types (normal and lognormal) of the loads and resistances are investigated herein. Based on these parameters, the mean values of resistance associated with a single component for the normal and lognormal distribution are found to be Ec(R) = 21.132 and Ec(R) = 27.194,
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respectively. By combining the MCS-based program with the simple algorithm, the redundancy factors of different types of N-component systems (N = 100, 300 and 500) are evaluated. The mean resistances of components and redundancy factors associated with the no correlation (ρ(Ri,Rj) = 0) and partial correlation (ρ(Ri,Rj) = 0.5) cases are presented in Table 2.4 and Table 2.5.
It is observed that in the no correlation and partial correlation cases (a) ηR of the series and mp×ns SP systems that have the same number of parallel components (i.e., m is the same in these SP systems) becomes larger as N increases; however, the contrary is observed in the parallel and ms×np SP systems which have the same number of series components (i.e., m is the same); and (b) the redundancy factors associated with the normal and lognormal distributions are very close; this indicates that the effect of distribution type on the redundancy factor is not significant.
In the perfect correlation case (ρ(Ri,Rj) = 1.0), ηR = 1.0 for different types of systems with different number of components associated with both normal and lognormal distributions. This was expected since for systems whose components are identical and perfectly correlated, the system can be reduced to a single component;
therefore, the redundancy factors in the perfect correlation case do not change as the system type and number of components vary.
For the investigated systems associated with different correlation cases, the component reliability indices βcs can be found after Ecs(R) is obtained. Figure 2.13 illustrates the variations of the component reliability index and redundancy factor in the series and parallel systems due to the increase in the number of components. It is noticed that (a) as the number of components increases, the component reliability
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increases in series systems, while it decreases in parallel systems; (b) for the series systems, the component reliability associated with the normal distribution is higher than that associated with the lognormal distribution in the no correlation and partial correlation cases; however, contrary conclusion is found in the parallel systems; (c) the effect of the probability distribution type of R and P on ηR is not significant, especially in the series systems; and (d) in the perfect correlation case, the component reliability index is equal to 3.5 and the redundancy factor equals 1.0.
2.5 REDUNDANCY FACTORS OF SYSTEMS CONSIDERING