Risk and Reliability of Offshore Structures

Prof. Srinivasan Chandrasekaran

Department of Ocean Engineering
Indian Institute of Technology, Madras

Module – 02
Reliability theory and Structural Reliability
Lecture – 19
Risk and Reliability

(Refer Slide Time: 00:21)

Welcome friends to the 19 lecture on module-2, where we are focusing on reliability

theory and structural reliability. In this lecture, which is the 19th lecture, we will try to
talk about some extension of reliability in terms of a fatigue studies. We will continue in
series of lectures now. So, here we are trying to compare the risk and reliability again as
we discussed in the earlier module lectures.

In the last couple of lectures, we gave an example study of how to understand the
performance of a specific system structural system and reversed combination of
environmental loads. We took example of a tension leg platform which is designed to be
a compliance system with large displacements permitted in the system, where the relative
displacement between the water particle velocity and that of the structural velocity is
reduced, therefore structure has gained a very good re centering ability under the lateral
forces caused by waves and wind.
When you look at the system under the reverse combination of forces, where in the
second case we talked about distinctly high sea waves and seismic loads where the
seismic loads cause indirect change in tension and that affects the system reliability. In
the earlier case, we spoke about the math stability issue on tension leg platform. We
picked up a real example and tried to compare the stability of the system, which is the
part of the reliability study, which essentially checks the performance of the system and
its intended function under the given environmental loads for a specific period of time.

So, now I will get back to the concepts of comparing risk and reliability and safety, then
we will move onto these studies slightly more in a mathematical manner. We all know
that environmental loads that act on offshore structures are not completely deterministic;
only a few features of the loads are known, maybe that is one of the interesting critical
features of analyzing offshore structures for reliability study.

(Refer Slide Time: 03:59)

We can always quote certain examples to verify this statement some of the examples
such as extreme waves distinctly high sea waves, the earthquake forces, sea bed
movement, wind loads on super structure, ice loads, shock and impact loads. Interesting
examples of extreme combinations which can always ascertain this statement that,
environmental loads acting on offshore structure or not all the time completely
deterministic.
(Refer Slide Time: 05:31)

Now there are issues related to this particular combination of forces which we will say
apart from being random in nature one issue is they are random in nature. The main issue
which is very critical as far as we are concerned is they have a high probability of
exceedance on the safe design values which is one of the very serious botheration as far
as reliability is concerned, because they will lead to what we call overloading situation.
Now, how to encounter this particular issue, because we know that the loads acting on
offshore structural system are not completely deterministic you do not know the
complete prescribed loading of the system because apart from being random in nature,
the very serious problem in these kind of load combinations or loads is that they always
have a very high probability of exceedance from their safe designed values.

So, now to counteract this we generally look at the safety. So, check for safety is a tool to
counteract the above critical issues, but unfortunately safety is actually a statistical
judgment. Therefore, I can confidently say reliability is also based on engineering
judgment. Therefore, one can say in general in reliability studies the experience plays a
very important role in the accuracy of the results. Apart from the loads being
nondeterministic in nature, apart from the loads have high probability of exceedance
from the safe design values, there is one more complication.
(Refer Slide Time: 08:53)

The complication is while applying reliability to offshore structures. We will say while
applying reliability to offshore structures, it is important to include complexities let say
all complexities that are of economic importance what we call as level 4 of reliability.
So, a reliability applied offshore structure is not only an engineering judgment of
ascertaining the performance of a function intended for a specific purpose under the
given specific time for a specific period.

In addition to that is having one more class of importance where economic perspective
needs to be also addressed. We also know that a variety of uncertainties as explained in
earlier lectures; do not guarantee a deterministic approach; presence of variety of
uncertainties we can say it does not guarantee a deterministic approach for reliability
analysis. These uncertainties clearly indicate that the reliability assessment offshore
structures shall be done only in probabilistic point of view.
(Refer Slide Time: 11:34)

As said earlier, one can say that risk is an extension of reliability because it also includes
the consequences of failure, whereas, reliability stops at or after identifying the
probability of failure we know that risk actually is a product of probability of failure of
occurrence of an even into consequence of failure. More interestingly, when reliability or
risk is applied to offshore structures the consequences should also include the economic
perspective. So, therefore, in engineering practice, one routinely encounters situations
that involve some event which has got a particular probability of occurrence; and if it
occurs has a specific set of consequences. With experienced engineering judgment, one
should be of course, able to assign a suitable probability to the occurrence of an event
and of course, some quantified magnitude or the cost effect to the consequences if that
event is occurred.
(Refer Slide Time: 13:46)

So, for risk assessment, we have two parameters; one is the probability of failure of a
specific event which is undesired; the second could be the consequences of the
occurrence of the event; if the event occurred what could be the consequences. So, based
on engineering judgment and experience, one should be able to assign probability of
failure of any specific event in a given system and also one should be able to correlate
the consequence of failure if the event occurs.

So, essentially you know reliability study or risk is probabilistic is engineering judgment
based on experience. And of course, it has got a method of assigning probability of
failure and consequence of failure as said by 2011, 13 and Cornell 1969. Therefore, in
system reliability or in general in reliability theory, the combination of uncertain events
and the adverse consequences is a determinant of risk.
(Refer Slide Time: 16:16)

Determinant of risk depends on is the combination of uncertain event and the adverse
consequences. Alternatively, to express the condition of a structure in a positive manner
reliability is considered. So, alternatively as we understand now to consider the
performance or assessment of a structure in a positive manner reliability studies are
conducted.

(Refer Slide Time: 17:46)

Now, the term risk has a hidden meaning. The main focus of risk is consequences of
failure; the hidden focus is chance of failure. So, therefore, one can say reliability index
of any system is an indicator of safety. So, one can say that risk is an extension of
reliability which addresses the consequences of failure which is the focus of risk
analysis, whereas reliability stops at identifying the chances of failure. So, reliability
terminates at the stage of assessing the probability of failure while risk continues to
address beyond this point. Therefore, it is important for us to know that risk conveys the
financial component of unsafeness that is very important.

(Refer Slide Time: 19:33)

Risk conveys the financial component of unsafeness, whereas reliability does not
conveys that is why generally friends risk analysis is more popular on engineering
structures compared to reliability. Of course, we agree now that risk is an extension of
reliability which focuses on the consequences of failure; and to a greater extent on
economic perspective that is why risk assessments are essentially more popular and more
authenticate compared to reliability analysis. As a common practice in oil and gas
industries and offshore structures in general, risk assessment is very important because
an oil and gas industries work towards what we call risk mitigation or risk reduction,
they do not work towards reliability analysis at all.

So, in reliability, the focus is probability of failure in risk the focus is consequences of
failure. Reliability does not give me the economic perspective, whereas risk focuses on
economic perspective. Then one can ask me a question how about level 4 reliability;
level 4 reliability is related to those type of structures whose economic importance is
higher; however, whatever studies you conduct on level 4 reliability analysis on
structures of higher importance, but still in those analysis also the economic perspective
of the consequences is not addressed. So, more or less risk analysis invades supersedes
reliability assessment.

(Refer Slide Time: 22:32)

There are reasons for this. Risk covers a broad spectrum of adverse effects or you can
say better estimates in comparison to reliability analysis, because they cover a broad
spectrum of adverse effects like adverse effects on the society what we call a societal
risk; individual, what we call personal risk or individual risk. It also talks about adverse
effects on financial status; it also talks about adverse effects on processing plant on its
asset management. So, risk analysis addresses towards the CAPEX investment on a
project, whereas reliability does not touch the CAPEX part. Reliability focuses on only
the operational expenditure or the failure towards operation, whereas risk analysis can
even address to some extent the asset management which related to the CAPEX
investment of the whole project.

Having said this that reliability and risk are comparable, risk has a precedent stage of
reliability study because probability of failure needs to be assessed for a given structural
system. So, reliability is a focus on assessing the performance failure of a given system
against it is intended function over a specific period of time under specific conditions.
So, you always pick up a structural system impose certain combination of forces or loads
or certain degradation effects or aging effects on material then asses the performance of
the structure. The moment I say the performance of the structure, we always talk about
the load carrying capacity, the sustaining on material, displacements or stress induced by
large vibrations and displacements all are essentially engineering properties. Whereas we
all agree these properties cannot be linked directly to the cost effect at all.

Whereas, risk picks up the probability of failure as assessed from the reliability analysis
identifies the consequence of this failure in economic perspective as well and try to give
overall picture about the asset management of the given project therefore, risk analysis
generally supersedes the reliability studies. So, risk analysis more or less in the financial
perspective reliability is more or less in engineering perspective now when we talk about
reliability analysis there are two types or extreme types of structural elements which are
common.

(Refer Slide Time: 26:32)

So, when we talk about let us say structural reliability, there are two types of
fundamental analysis one can carry on structures, because structures in general or
structural elements in general can be divided into two namely brittle members, ductile
members. An element is considered to be a brittle element, which becomes completely
ineffective after it fails or those elements which become completely ineffective after
failure. While ductile elements are those elements which is able to maintain which can
maintain its load carrying capacity or I can say in general intended function even if fails.
Now, in general a special case involves a system in which the correlation coefficient is
same for all pairs of elements.

(Refer Slide Time: 29:02)

Here let us take a special case. The special case in sense it involves a system containing
pairs of elements whose correlation coefficient is same, and lies in the range 0 and 1.
Now, let us consider as system with n number of elements. One can say strength of the
ith element where i is equals 1 to n be denoted as R i. To estimate the probability of
failure we have to make certain assumptions. One, strength of all elements is normally
distributed.
(Refer Slide Time: 31:28)

Two - strength of all elements are equally correlated; three - all applied loads are
deterministic and time invariant; and lastly all elements are designed to have the same
reliability index which we call as beta. So, now, we are moving from element level or
component level analysis to the structure level analysis, we are talking about structural
reliability. So, these assumptions are generally made when you perform structural
reliability on a given system whereas a focus is not on the elements now we are talking
about the overall failure of the entire system under the given combination of forces.

(Refer Slide Time: 33:11)

Now there are two systems possible system in series system in parallel. Let us say
system in series, which has got equally correlated elements. Now, for n elements of a
system in series probability of failure as given by Stuart 1958 is probability of failure is 1
minus integral of minus infinity plus infinity phi of beta of the element plus t root p
divided by 1 minus this is rho root raise to the power n phi of e dt - equation 1. In this
case, beta e is a reliability index of each element phi indicates the cumulative distribution
function and probability density function; rho of course indicates the correlation
coefficient.

(Refer Slide Time: 35:40)

Alternatively, we talk about system in parallel with equally correlated ductile elements.
One can say resistance of a parallel system with n ductile elements is given by R equals
summation of i equals i to n of R i because we know ductile elements can perform the
intended function that is the load carrying capacity even after they fail ,because they
have a sufficient reserve capacity. Now we can say here R i is the resistance of each
element; and R is the resistance of the system with n elements.
(Refer Slide Time: 37:30)

Now, once we assume that resistance of elements follow the same cumulative
distributive function resistance of all elements follow the same cumulative distributive
function then mean and variance of the system resistance can be expressed in terms of
element parameters like mu R will be n mu R i is nothing but n into mu e.

(Refer Slide Time: 38:54)

The variance sigma square R can be i equals 1 to n j equals 1 to n rho i j the correlation
coefficient R j which can be said as i is equal to 1 to n sigma of the element square plus
double summation i equals 1 to n i not equals j 1 to n rho sigma e square. This is 4 (a),
this is 4 (b), which can be said as n variance of the elements plus rho n n minus 1
variance 4 b sorry 4 c, so which now amount to n sigma e square of 1 minus rho plus n
rho I call this as equation number 5.

So, the variance of the system is given by the variance of the elements if I know the
correlation coefficient between the elements which are assumed to be equal as in one of
the; be in assumptions in the whole derivation where n is a number of elements in
parallel. Therefore, to determine the reliability index for the entire system it is important
to assess the relationship of reliability of elements beta e to the mean and standard
deviation of each element.

(Refer Slide Time: 41:18)

Therefore, reliability index of the system demands estimate of reliability index of the
element which of course, depends on mean and standard deviation of each element. So,
friends, we are able to compare the risk and reliability analysis in general with a
conceptual idea; and we also explained how risk analysis can supersede the reliability
analysis to a greater extent at level 4 reliability demands in terms of offshore structures.
We have also said how system reliability now can be interpreted if the element reliability
is known to us for a system in series and system in parallel.

We will continue the discussion in the next lecture to really understand how to get the
extension of reliability of the system, if I know the reliability index of the element
whether the elements are in series or in parallel.
Thank you very much.