# Time-variant reliability series system

Good Day all,

First, I want to thank the UQLab team for such a great job putting together this software. I had a look at the example on reliability of parallel system as well as time variant reliability. However, I have a few concerns and will be grateful if anyone can be of help as I do not even know where to start. Appreciate any help on below example for the purpose of clarity.

Problem: Considering a simply supported beam of span L= 5m with uniformly distributed load w(t) modeled as a Standard Gaussian Process. The beam is subject to Flexural and Shear failure which are in SERIES. Degradation of the beam with time is gives as

Flexural : M(t) = M exp(-0.0005t)
Shear: V(t) = V exp(-0.0005t)

variable distribution are M (Normal : mean 500kN, CoV 10%) , V (Normal : mean 700kN, CoV 15%)
w(t) which is the load follows a Standard Gaussian Process ( Mean 90kN/m , CoV 25%)

The beam is treated as a SERIES system of the shear and bending failure.
LSF:
g1 = M(t) - (1/8 * w(t)* (L^2))
g2 = V(t) - (1/2*w(t)*L)

Determine the Failure probability for a 10 years design time (interval of 1yr)

Questions:

1. First this system is in SERIES how can this be approached?
2. Also from the uqlab example the deterioration was linear but in this case it is exponential how can this be dealt with?
3. Two resistance variable are being considered M and V and one load variable w(t); how can we approach this given that we have been given 2 separate limit state function for M and V ?
4. How can the out-crossing rate be obtained for the series system
5. How can the failure probability be obtained for year 1,2,3… 10 from the out-crossing rate
6. If we have a complex system ( series and parallel together) can the approach described in the time -variant reliability uqlab example still be applied?

Your help on this is appreciated. I really want to understand how to approach and set-up this problem in Uqlab for series system as well as a combination of parallel and series.

Example Paper Source: An out-crossing rate model and its efficient calculation for time dependent system reliability analysis by C Jiang, X.P.Wei , Z.L.Huang . Yr 2017.

1 Like

Hi @aokoro,

Thanks for your questions. I will try to reply briefly and mix some of the questions:

1. The examples in UQLab is not applicable to series system as it is. PHI2 is one of many approaches to solve time-dependent reliability problems. Basically PHI2 uses a FORM to solve a parallel problem (the two limit-state being one at t and the other at t+\Delta t. It may not be straightforward but if you still want to use PHI2, you would need to reformulate your two-limit states into new ones (For instance using max and min operators to transform two limit-state into one according to whether they are series or parallel) Alternatively, you can use a direct Monte Carlo simulation approach (You can have a look at what we did here).
1. & 4. You indeed need to reformulate the problem for series system. If you use finite difference for the out-crossing rate, you would need to estimate the probability that all limit-states are safe at the instant t and that at least one of the limit-state fail at instant t + \Delta t, i.e.:
\nu^{+} = \frac{Prob( ( g_1(t) \geq 0 \, \cap \, g_2(t) \geq 0 ) \, \cap \, ( g_1(t+\Delta t) < 0 \, \cup \, g_2(t + \Delta t) < 0 ) }{\Delta t}.
If you want to solve it with FORM, I would say the method in the paper you sent is a good start. Otherwise you can always solve it with Monte Carlo simulation if your computational model allows it (if not, a surrogate may also help).
1. I don’t think it is an issue. The only difference with the example in UQLab is that you wouldn’t get the analytical solution as defined with the linear degradation.
1. I don’t think there is a problem with that.

I hope this helps.
Cheers

Thank you so much for the detailed response. I will have a look at the referenced paper for the direct Monte Carlo approach as I need an approach which can solve a combination of series and parallel components. Also, will the use of max , min operator be required using the Monte Carlo Method for the time varying system?

I had an attempt on reference problem using MCS as advised but with below error message. Appreciate any help I can get on this.
Error Message:
Error: Invalid expression. When calling a function or indexing a variable, use parentheses. Otherwise, check for
mismatched delimiters.

Error in uq_initialize_uq_default_model (line 50)
current_model.Internal.fHandle = str2func(current_model.mString) ;

Error in MCSTV (line 27)
myLimitStateFull1 = uq_createModel(ModelFullOpts1);

I have also attached the code from uqlab. Any correction and direction will be appreciated.mcstimevariant.m (1.6 KB)

Hi @aokoro,

I think you have another error in the definition of the limit-states. A paranthesis is missing in the .mString. Here is the fix for the first one (I let you check the other two):

ModelFullOpts1.mString = 'min([ X(:,1).*exp( -0.005*P(1) )- (25/8).*X(:,3) , X(:,2).*exp(-0.005*P(1))- (5/2).*X(:,3)],[],2)';

That being said, I am not sure I understood what you are trying to do in the code you’ve shared. I think it will be easier to write a short algorithm to compute the time-dependent failure probability than just modifying the FORM-based example. I would still suggest you have a look a that paper I mentioned. You will find a short pseudo-algorithm that you can implement.

Cheers,
Moustapha