As has been discussed, reliability block diagram (RBD) allows you to graphically represent how the components within a system are reliability-wise connected. In most cases, independence is assumed across the components within the system. For example, the failure of component A does not affect the failure of component B. However, if your system consists of components that are sharing a load, then the assumption of independence no longer holds true.
If one component fails, then the component(s) that are still operating will have to assume the failed unit's portion of the load. Therefore, the reliabilities of the surviving unit(s) will change. Calculating the system reliability is no longer an easy proposition. In the case of load sharing components, it is required to know the change of the failure distributions of the surviving components in order to determine the system's reliability.
To illustrate this, consider the a system of two units connected reliability-wise in parallel (Figure 5.16).
Figure 5.16: Two units reliability-wise in parallel.
Assume that the units must supply an output of 8 volts and if both units are operational, then each unit is to supply 50% of the total output. If one of the units fails, then the surviving unit supplies 100%. Furthermore, assume that having to supply the entire load has a negative impact on the reliability characteristics of the surviving unit. Since the reliability characteristics of the unit change based on whether both or one is operating, a life distribution along with a life-stress relationship (as discussed in the Statistical Background section of this on-line reference) will be needed to model each component.
To illustrate the steps needed, we will create the model starting from raw data. Assume that a total of 20 units were tested to failure at 7, 10 and 15 volts. The test data set is presented in the next table.
For this example, Units 1 and 2 are the same component. Therefore, only one set of data was collected. However, it is possible that the load sharing components may not be the same. If that were the case, data would need to be collected for each component.
The data set in Table 1 was analyzed using ReliaSoft's ALTA software (as shown in Figure 5.17) with the Inverse Power Law as the underlying life-stress relationship and Weibull as the life distribution.
Figure 5.17: Using ALTA to calculate component parameters.
The estimated model parameters, b, K and n, are shown next.
Or:
(20)
(21)
And for this case:
Figure 5.18 shows a plot of Eqn. (20).
Figure 5.18: Reliability curves for different voltage output conditions.
Now that the failure properties have been determined using the test data, the reliability of the system at some time, t, can be calculated using the following equation:
(22)
Where:
And:
S is the total load (or required output).
P1 and P2 are the portion of the total load that each unit supports when both units are operational. In this case, P1 = P2 = 0.5 = 50%.
S1 and S2 represent the portions of the load that Unit 1 and Unit 2 must support when both units are operational.
is
the equivalent operating time for Unit 1 if it had been operating at S instead of S1.
A graphical representation of the equivalent time is shown in Figure 5.19,
where the red line represents the low stress (load) and the green line
represents the high stress (load).
Figure 5.19: Illustrating te
can be calculated by:
can be calculated the same way, or:
In this example, the reliability equations for Unit 1 and Unit 2 are
the same since they are the same type of component and demonstrate the
same failure properties. In addition, the total output is split equally
between the two units (when both units are operating), therefore and will also be the same.
The next step is to determine the reliability of the system after 8,760 hours, R(t = 8,760). Using Eqn. (22) the system reliability is found to be:
BlockSim uses this formulation when computing reliabilities of units in a load sharing configuration. When using the algebraic solution viewer, BlockSim returns a single token for the reliability of units in a load sharing configuration (as well as in the case of standby redundancy, discussed in the next section of this on-line reference). As an example, consider the following RBD with unit 1 in series with a container that includes two load sharing units.
BlockSim will return the system equation as:
Where RLS implies a form similar to Eqn. (22). BlockSim allows for k-out-of-n units in a load sharing configuration.
A component has five possible failure modes, A, BA, BB, BC and C, and the B modes are dependent. The system will fail if mode A occurs, mode C occurs or two out of the three B modes occur.
Modes A and C
have a Weibull distribution with a 
= 2 and 
= 10,000 and 15,000 respectively. Events BA,
BB
and BC
have an exponential distribution with a mean of 10,000 hours.
If any B event occurs (i.e. BA, BB or BC), the remaining B events are more likely to occur. Specifically, the mean times of the remaining B events are halved. Determine the reliability at 1000 hours for this component.
The first step is to create the RBD. Modes A and C and a load sharing container with the Bi modes must be drawn in series, as illustrated next.
The next step is to define the properties for each block including those of the container. Setting the failure distributions for modes A and C is simple. The more difficult part is setting the properties of the container and the contained blocks. Based on the problem statement, the B modes are in 2-out-of-3 load sharing redundancy. When all three are working (i.e. when no B mode has occurred), each block has an exponential distribution with μ = 10,000. If one B mode occurs, then the two surviving units have an exponential distribution with μ = 5,000.
Let's assume a power life-stress relationship for the components. Then:
(23)
(24)
Substituting μ1 = 10,000 and V1 = 1 in Eqn. (23) and casting it in terms of K yields:
(25)
Substituting μ2 = 5,000, V2 = 1.5 (because if one fails, then each survivor takes on an additional 0.5 units of load) and Eqn. (25) for K in Eqn. (24) yields:
This also could have been computed in ALTA, as shown in Figure 5.20, or with the Load & Life Parameter Experimenter in BlockSim, as shown in Figure 5.21.
Figure 5.20: Calculation performed in ALTA.
Figure 5.21: BlockSim's Load and Life Parameter Experimenter.
At this point, the parameters for the load sharing units have been computed and can be set, as shown in Figure 5.22.
Figure 5.22: Load sharing parameters.
The next step is to set the weight proportionality factor. This factor defines the portion of the load that the particular item carries while operating, as well as the load that shifts to the remaining units upon failure of the item. To illustrate, assume three units (1, 2 and 3) are in a load sharing container with load proportionality factors of 1, 2 and 3 respectively (and a 1-out-of-3 requirement).
Unit 1 carries 
or 16.6% of the total load.
Unit 2 carries 
or 33.3% of the total load.
Unit 3 carries 
or 50% of the total
load.
The actual load on each unit then becomes the product of the entire load defined for the container times the portion carried by that unit. For example, if the load is 100 lbs, then the portion assigned to unit 1 will be 100 • 0.166 = 16.6.
In the current example, all units share the same load, thus they have equal weight proportionality factors. Because these factors are relative, if the same number is used for all three items then the results will be the same. For simplicity, we will set the factor equal to 1 for each item.
The last items that need to be defined are the total load and the 2-out-of-3 redundancy. The total load is dependent on how the parameters were computed. In this case, we assumed that the total load was 3 when we computed the parameters (i.e. the load per item was 1 when all worked and 1.5 when two worked). This is defined at the container level, as shown in Figure 5.23.
Figure 5.23: Defining total load for load sharing units.
Once the problem has been set up in BlockSim, the reliability at 1,000 hours can be determined. From the Analytical QCP, this is found to be 93.87%.
See Also:
Time-Dependent System Reliability (Analytical)
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