# Combination of families of fragility curves (cumulative probability functions)

#### sandreev

##### New Member
Hi guys, I am a geotechnical engineer and currently involved in a project for risk calculation of an embankment dam.
It's my first post here, I am still novice to probability theory and statistics. I hope I will explain well what my problem is and I'll be glad if you can give me advices. Sorry for the long explanation below but I assume some of you are not involved in engineering statistics and probability/risk calculation so I want to make things as clear as possible if you have experience in this field, please skip the text in italics.

In engineering practice we present the conditional probability of failure of a structure for a given failure mode with respect to a loading parameter (earthquake magnitude or flood water level etc) with the so-called fragility curves - log-normal cumulative probability function of the load parameter (A) with median capacity (let's say Am) and and error term eR (accounting for aleatory uncertainty of the process) and:
A=Am.eR
eR is lognormal random variable with median value 1 and standard deviation βR - the above equation gives a median fragility curve.
When taking into consideration the epistemic uncertainty the above equation becomes
A=Am.eR.eU
eU is also an error term (lognormal random variable with median value 1 and standard deviation βU).
With these three Am, eR, eU a family of fragility curves is defined for whatever confidence levels (subjective probabilities) we need.

In this particular case we have two failure modes with two different fragility curves. Our guideline says we can combine them using the Uni-modal limits theorem:
max[p1,p2]<pf<1-(1-p1)*(1-p2), where p1 and p2 are the conditional probabilities for failure modes 1 and 2.
The above is true for the median (50%) fragility curves. My question is when I want to calculate a family of combined fragility curves what should I do?
If I combine the 95% fragility for failure mode 1 with the 95% fragility for failure mode 2 is this the 95% combined fragility or it's something else?