Advanced Marine structures

Prof. Dr. Srinivasan Chandrasekaran

Department of Ocean Engineering
Indian Institute of Technology, Madras

Lecture - 19
Plastic design-IV - Example problems - I

Now, we continue discussing on the lectures on module 1 of advanced marine structures

course. So, let us quickly see what we have so far discussed in a brief summary in this
course. We started to understand, what are the different kinds of marine structures for its
application for specific purpose in oil exploration? How do they react for the
environmental loads on which they are acted upon? How these structures are designed or
evolved based on form special modifications? The form modification, the geometric
modifications make the structure suitable for different levels of oil exploration from
shallow water to ultra deep waters. So, it is form based design and it is not a function
based design.

Therefore, we were critically understanding or trying to understand, what are the

different varieties of environmental loads, which act on these kinds of structures. So, we
classified them in five permanent P, P category, live load L category, deformation loads
D category, environmental loads E category and accidental loads A category. We
understood that these loads, they are various series available for computing let us say
wave load, wind load, current load, earthquake load, different methodologies etcetera.
But in nut shell, all of them do have lot of variabilitys and uncertainties in their
estimates.

Therefore, ultimately if you work out a load encounting a marine structure, it should be
deterministic value, but does not remain deterministic, it becomes probabilistic. Because
the probability of exceedance of any specific value, let us say you say you want to design
a marine structure for wave height of 15 meters and a wave period of 15 seconds for a
given sea state. What is the guarantee that the wave height and wave period at that sea
state, where your structure is located within the service life of the structure will not be
exceeded by these values?
So, we are looking for a question of probability of exceedance. The moment the
probability term comes in, we associate this with the value called characteristic load.
Then, we said even though the load and the strength of the load and the resistance can be
normally distributed, but the normal distribution will not help us. Because I would like to
see what would be those cut off values of the distribution of the load and the resistance in
terms of S and P value, which is connected by a factor called gamma, which we call as a
safety factor.

So, we said that the characteristic value of the load means 95 percent, this value will not
be exceeded. The probability of exceedance will leave only 5 percent. If I say the
characteristic value of the strength of the material, why characteristic value of the
strength of the material, because strength of the material has got lot of imperfections,
residual stresses set in the material or in the member, because of welding process,
because of flame cutting, because of geometric un equality, because of manufacturing
defects, because of construction processes errors etcetera.

So, we have to also account for these variations, these uncertainties in the strength of the
material also. So, we say characteristic strength of the material, which we say as gamma
M and we say 5 percent is the variation. So, we account for these and ultimately we
connect these two and say there is a factor of safety or a margin of safety, which should
account for these uncertainties and variabilitys present in the given system. Once, we
said this we discussed about ultimate limit states.

Different varieties of limit states were limit state of serviceability and limit state of
ultimate that is ULS and SLS are very important. They have got to be met as a mandate
in a given design. If you follow ULS even then, limit state of serviceability for control or
deflection should be checked for its safety of serviceability. So, we said that ultimate
limit state is one process where we allow the structure to the load level beyond which the
structure tend to collapse or we say the structure will get into a mechanism. Now, the
question comes what is a mechanism?
(Refer Slide Time: 04:42)

In general, if a body is subjected to any load, the body will offer resistance which is
called as internal reaction to the applied loads. This process of offering resistance to the
applied load is what we call as load carrying capacity. This is general behavior. Now, if
the body, if the body does not offer resistance to the load anymore, then this is called a
mechanism. So, mechanism is a system of structural members which does not offer
resistance to any external load applied on to it.

Now, obviously you cannot have a structural system, which does not offer resistance at
all. So, to be very implicit we must say that beyond a specific load level. It will not offer
resistance beyond a specific applied load level. That load level is what we call as
collapse load. Remember, very importantly mechanism is not the collapsed state of a
structure. The structure has not collapsed. Structure has stopped offering resistance to the
load at that instant. Once, the load exceeds even that then only it will collapse.

There is always a difference between a mechanism and a collapsed state of the structure.
Mechanism is that state of the structural system or the body which does not offer any
resistance or resistance any more to the applied load. And this applied load will have a
level beyond which this will be implemented. That load level is what we call as
collapsed load or ultimate load. Mechanism is not the collapsed state of the structure. If
the load level exceeds even beyond that then it will collapse.
So, the structure becomes a mechanism in ultimate limit state design methodology. There
are two theorems available with help of which we can easily find out what is the plastic
moment of resistance of a given section or what is the collapse load the structure can
sustain? First let us understand the difference between analysis and design.

(Refer Slide Time: 09:01)

Let us understand this first because I am talking about plastic analysis here whereas;
ultimate limit state is a design mechanism. So, what is the difference between these two?
Interestingly, we have undergone both these terminology commonly people mix this.
Analysis deals with estimate of forces and reactions of the structural system. The
reactions may not necessarily be only the forces; the reactions can be rotations,
displacements also.

You can react in any manner. Not necessarily only the force will be pumped back it can
be any reactions. Design is to determine the cross sectional dimensions of the member,
the structural geometry which talks about arrangement of members etcetera to safely
disburse the forces. So, in any process the first step is to find the analysis, do the
analysis. Then, the second step is to do the design. But interestingly, for doing the
analysis, since we are interested to know what is the reactions offered by the member
you need to know the member dimensions for example, you need to know I E, you need
to know A, area of cross section, moment of inertia which are all geometric characters of
a given structure or a member, is it not?
So, if you do not know the cross sectional dimension you cannot actually find out the
reaction to the forces on the member. You can find the forces, but the reaction cannot be
found out. Now, see these two are in a close loop, they are interconnected, they cannot
be separated, they cannot be separated. So, what people generally do is assume certain
dimension, find the reactions, then find the forces. Check those dimensions, whether it is
safe. If they are not safe repeat the process again.

That is what the general theory about analysis and design is. In this context, let us see
what are you talking about plastic analysis and plastic design? In plastic analysis I must
find out the forces, what are the forces? The forces which will cause or tend to cause
collapse to my system. In plastic design, we can check whether the moment coming in
the cross section anywhere, anywhere in the given system because of these collapse
forces does it exceed the capacity or not?

We are checking, but M p is a function of shape factor and shape factor is a function of
b, d etcetera cross section dimension. Therefore, what you generally do is equate the
maximum moment to M p and for that M p find the optimum dimensions of the section.
So, that is what we say as plastic design. So, interestingly if you do not have the collapse
forces to which the structure will subjected to, I cannot do the design, is it clear? So, far
we were talking about the design principles where given that the collapse load is known
to me, I can find the optimum dimensional section to safely distribute that load coming
on to the section.

Now, I am going to talk about how to get that collapse load itself for a given structural
system, are we clear? This is what we are trying to do. One may wonder sir, why we
discussed about the design first then the analysis later. Why we did not talk about the
analysis first and then the design? Because that is the sequential step we do. As long as
you do not appreciate what is M p as long as you do not know how M p depends on the
sectional characteristics. There is no point and there is no sense in making and finding W
c. It is very simple. As long as they do not have a section with you, you cannot do an
analysis. So, first we found out the section, we understood the section first. Now, we will
talk about analysis. To do the analysis we have got two theorems available which is
called static theorem and kinematic theorem.
(Refer Slide Time: 15:18)

Before going to these theorems, we will quickly compare the elastic analysis and the
plastic analysis let us quickly compare this. In elastic analysis, you must fulfill three
conditions. One, equilibrium condition; this condition says the structure under any load
should remain in its equilibrium. To be very clear, it should remain in static equilibrium.
There is another you call dynamic equilibrium. We are talking about static equilibrium
here.

The second condition, which is to be met for a successful elastic analysis is that
continuity or compatibility condition. This says the deformation of various fibers of the
cross section, the deformation of various layers of the cross section should be compatible
with the adjacent ones. If there are two layers, which are touching each other, then the
deformation on the upper layer A should be compatible with the lower layer B. We have
got to check that; that is called continuity or compatibility.

The third condition is very important which is called limit stress condition. This
condition says that the stress in any fiber or layer should not exceed the yield value.
These are essentially three conditions, which are to be satisfied for a successful
deployment of elastic analysis. Taking it granted that you all know how to do an elastic
analysis because elastic analysis is also not that easy. It is complicated when the structure
becomes statically indeterminate of a very high order you have got to use different tools.
There are various methods.
We will not focus those methods in this course, because it is purely dedicated to
something else on the area of topic, but you can refer to standard text books which you
have studied already in your undergraduate program. We will touch upon some
important aspects in the example problems later just to re brush whether you have
understood them or not? But my focus is not explaining the elastic analysis in this course
at all. We all think that we know this.

Since, we know this we can quickly compare the equivalency of this with the new
method, which you are studying now. The first condition, which must satisfy in plastic
analysis, what we call mechanism condition. It says that ultimate load is reached or
collapse load is reached when mechanism is formed. The number of plastic hinges I am
introducing a new term, the number of plastic hinges should be just sufficient to form a
mechanism.

We already know, what is a mechanism? We already defined what is mechanism.

Mechanism is a structural system or assembly of members, which does not offer
resistance to external loads at all. Obviously, the load level cannot be 0 because
structural will have an internal resistance offered to any load. Therefore, inherently we
say that the load level is beyond a specific value, that value is what we call as a collapse
load. I have introduced a new term plastic hinges. We will talk about this slightly later.
So, the number of plastic hinges introduce should be just sufficient to convert the given
structure to a mechanism.
(Refer Slide Time: 21:22)

It is assumed that redistribution of moments will occur on the basis of mechanism

condition. What does it mean? It is a very interesting statement. I will explain this
statement. Once, a mechanism is formed it means plastic hinges will be formed at critical
sections in a given structure. What is a plastic hinge? We will come to that. Let us
understand; we know what is a plastic hinge? Plastic hinge form at a given section or
sections which converts the structure into a mechanism, it does not mean that the
structure will collapse.

Once, it will become as mechanism redistribution will start taking place. So, the highly
stress sections will no more take any load, whatever load comes on to section will be
redistributed to the successive highly stress section which enables sequence of formation
of plastic hinges, one by one. When all possible hinges are formed completely, in totality
then the structure will collapse, is that clear? That is the mechanism condition. The
second condition I think I will, can I rub this and write here? I will rub this.
(Refer Slide Time: 23:01)

The second condition is equilibrium condition, which is similar to elastic analysis. Same,
the structure should remain in static equilibrium under the given set of loads. Third
condition which is called plastic moment condition. At any cross section, moment should
not exceed the plastic moment carrying capacity of the section where M p is plastic
moment of resistance of the section which is easily given by, is it not? We already know
this.

So, for optimum design the worst scenario is M can be equal to M p. Not necessarily, M
should be less then M p. M is moment at any section, let us say M x, not necessarily M
should be less then M p. M should be not greater then M p, but M can be equal to M p if
you want to really optimize the design. These are the three conditions which one must
satisfy for successful understanding of or deployment of plastic analysis. We can see
here obviously condition one and two they match exactly.

Condition three and three they match similarly, because here it should not exceed sigma
y. It is talking about the stress limitation, is talking about stress limitation. It talks about
moment condition. The condition of compatibly continuity is been replaced by a new
condition called mechanism condition where people say that, if the structure gets enough
number of plastic hinges just sufficient to convert into the mechanism. Then
redistribution will be start taking place and it is believed that the ultimate load has
reached.
(Refer Slide Time: 26:07)

Now, the question is what is the plastic hinge and where are they formed? What is a
plastic hinge? Plastic hinge is a section, it is a section actually in a member where the
moment is equal to the plastic moment of resistance of the member. It is not less then or
not greater than. It is exactly equal to M p. How does it look like? It looks like a filled
dot. How can it be compared with normal hinge, which we call structural hinge? We can
compare this.

(Refer Slide Time: 27:22)

Let us say, I have a simply supported beam where one end is hinged, other end is roller.
This end is on roller, this end is what we call as a hinged joint. Now, here also we say
there is a hinge, here also we say there is a hinge. This is called structural hinge, this is
called plastic hinge. Now, let us see, what is the difference between these two? Structural
hinge and plastic hinge has got explicit differences. At structural hinge the moment is
actually 0, it does not transfer any moment at all, whereas in plastic hinge the moment is
not 0 it is equal to some capacity of M p. Structural hinge is indicated by a circle without
filling. Plastic hinge is indicated with a circle with filling.

(Refer Slide Time: 28:35)

So, I can say plastic hinge is a rusted, which rotates at an applied moment M p. It is a
rusted hinge. It requires some energy, some moment to rotate this, whereas structural
hinge does not require any moment, it is 0. Now, can you tell me in articulated tower you
have a hinge between the deck and the tower legs. What kind of hinge is this? Structural
hinge. In triceratops, you have a hinge between the deck and the buoyant leg structure.
What kind of hinge is this? Structural hinge. These are all structural hinges which does
not transfer any moment at all. Moment is 0 at these joints, whereas plastic hinge is a
section whose moment capacity is not 0, but M p. Now, the question comes where will
they form?
(Refer Slide Time: 30:27)

Plastic hinges, questions. One, where will they form? How many numbers? Both
questions; where will they form? They will form at the following sections. One, at
sections where moment is maximum; under concentrated loads, if you have got any
concentrated load applied on the platform deck, at those points plastic hinges will form.
Three, points of 0 shear force. Four, at fixed supports, five at sections where moment of
inertia changes. How many of them will form?

(Refer Slide Time: 32:10)

If the structure is statically indeterminate to a degree of n, then n plus 1 hinges are
required to form mechanism. I believe, you all know how to find the degree of
indeterminacy for a given structure. There are two kinds of indeterminacy, static
indeterminacy, kinematic indeterminacy, total degree of indeterminacy. For a given
structural system, you must know; how to compute the degree of indeterminacy? If you
know the number n, then n plus 1 hinges are required to convert the structure into a
mechanism.

What is a mechanism? Mechanism is a structure, which does not offer resistance to load
any more. Earlier it was offering, earlier it was offering when the load was applied.
When the load reaches specific state or specific value, now the structure stopped its
reaction of offering resistance. Structure is then called by a new name called mechanism.
So, ideally a structure is called as a mechanism when sufficient number of plastic hinges
are formed in a structure I am redefining the mechanism now.

A structure is called a mechanism when sufficient number of plastic hinges are formed in
a structure. How many n plus 1? n is degree of indeterminacy. Obviously, now you will
thoroughly understand that for plastic analyst and design the structure need to essentially
be a static indeterminacy part. But, if you have determinate system like a simply
supported beam whose static degree of indeterminacy is 0 still then one hinge will be
formed to make it as a mechanism, remember that.

That is very, very interesting. Even determinate structure can also be analyzed using
plastic design because one hinge will form at a specific section of critical location and it
becomes a mechanism. So, plastic design can also be applied to statically determinates
systems. The disadvantage is redistribution of moments will not occur because there is
only one hinge, only one section. Once M p reaches structure is going to collapse that is
the advantage.

Now, there are two theorems which will help us to do plastic analysis. What does it
mean, to determine the collapse load in a given structural system. Now, we all agree that
what is the difference between the load encountering the structure and the collapse load.
What is the difference between these two? What is the difference between the collapse
load and the load, which we have calculated to come on the structure? There are
differing by what we call as a load factor Q, W c by W w; that is what we have seen. Is it
not?

(Refer Slide Time: 35:54)

What you have so far computing will be all working loads. It is not elastic load, it is
working load. The load coming on the structure, you enhance this load, you multiply this
load by a number whose values always more than 1. We obtained collapse load. Now,
you may wonder say then why do we do plastic analysis because we can directly get the
collapse load from this mechanism. The load will call as a collapse load only when the
structure becomes the mechanism.

All loads multiplied by load factor are not collapse loads. The loads which are
responsible to cause collapse are collapse loads. So, how to get this? You must have a
procedure, you must have a methodology to obtain the collapse loads. Is it clear? Then,
the question comes, there are two confusions here. One, the collapse load what again the
directly from this equation. The other is the collapse load, which will do by doing a
plastic analysis from two theorems.

How are they different? They are exactly same, but the catch is here. I do a plastic
analysis using the theorem, get collapse load for a given mechanism condition. From the
collapse load I get the working load. That will be my design load for the structure. Is it
clear? Therefore, I am designing the structure for that design load which has got a factor
of safety or a margin of safety and this margin of safety will never allow the structure to
collapse.

It means even though the structure is designed by a traditional term called collapse
mechanism, the design principle will not lead to actually a collapse. Is that clear? It will
only make the structure to become a mechanism. That is all, is that clear? So, we are not
designing or we are not using a method by which we are purposefully collapsing a
structure. It is not like that. This margin of safety will take care of that, is that clear?

(Refer Slide Time: 38:26)

So, there are two theorems available. One is called static theorem, other is called
kinematic theorem. This otherwise is called lower bond theorem. This otherwise called
upper bond theorem. The statement of this theorem are the following. For a given
structural system and load arrangements for a given, this statement is very important.
You please try to understand these two theorems thoroughly. Both will lead a same
answer, but which is easy is depending upon how you want to practice or how you want
to do the analysis.

The statements once I complete they be will very, very confusing. I am writing slowly to
explain you. These are available almost all text books on structural engineering.
Everybody will be teach this in plastic design courses, but understanding them will be a
great problem because once I complete the statement it will, as if you have got to
memorize this. You will be not able to understand this. So, I am writing in part and
parcel, I am writing it parallelly, so that you can compare in your mind what is
happening?

The difference between these two for a given structural system, so to apply static
theorem you must have a structural system with you. It means plastic analysis does not
tell you what a system you have to have in place. The system is always chosen by the
designer. For example, you want design a TLP, you want to employ a nautical tower, you
want to employee a triceratops, these are all form based systems which is not arrived
from any design or any analysis procedures.

So, you must fix, choose the system readily available to you. So, for a given structural
system and the load arrangements where the load will act for example, on the deck, what
are all the tops side details? Which are all the loads coming on that? That arrangement
you must already have with you. For a given structural system and load arrangements if
there exist any bending moment distribution which is statically admissible. So, for a
given structural system be it indeterminate or be it determinate.

Obviously, one will not look for a determinate system, because of obvious reasons. Can
you give me one reason why I am not interested in looking for a determinate system?
Why I cannot go for a determinate system? Why I should look for indeterminate system?
In a indeterminate system you require n number, large number of plastic hinges to form,
to convert the system to a mechanism, whereas in determinate system you require only
one hinge to form to convert into a mechanism beyond which it will collapse.

So, there is no reserve strength available in the geometry. There are two reserve
strengths. One is from the material; one is from the geometry or arrangement of the
members. Material strength is what we are looking at from the elastic to post elastic or
plastic state. That is called ductility. There is the reserve strength, the material. The
reserve strength in the geometry is arrangement of members, which we call as
indeterminate structure, determinate structure etcetera.

So, even though the given structural system is statically indeterminate for a given load
arrangement, you must know how to get the bending moment distribution. Now, how
many of are very good in finding out the bending moment distribution or drawing a
bending moment diagram for a statically indeterminate structure of a very high order. So,
it is a pre request to apply this procedure of analysis. The pre request is this. You must
have the values of bending moment distribution in advance with you.

So, if that bending moment distribution which is admissible exists, then I have to add one
more adjective here. Statically admissible and safe for the given set of loads Q, then the
value of Q or the value of load let us say the value of load W will be less than or equal to
collapse load. Since, the value of load W is less than or equal to collapse load this
theorem is called lower bound theorem. Therefore, I can say the collapse load the
statement ends here, the collapse load obtained from this theorem will be, is lesser than
or equal to true collapse load.

That is why it is called lower bound theorem? The upper bound theorem has a different
statement, for a given frame and loading. So, in this case also you should have the
structural system in position, you must know the load arrangements.

(Refer Slide Time: 46:23)

Value of load corresponding to any assumed mechanism; any assumed mechanism will
be either equal to or greater than W c. So, what is the catch in both these methods? So, I
should say the collapse load derived from the assumed mechanism will be either equal to
or greater than true collapse load. Now, what is the catch between these two theorems,
which may stick distinctly different? There is a very important statement or very
important catchword here, which makes both this theorem distinctly different.
What is that? Both the theorem bonds structural system and load arrangements, therefore
this cannot be a difference. One distinct difference can be this theorem gives me the
lower bound values; this theorem gives the upper bound values. Of course, that is in the
name itself. So, that cannot be a distinct difference. The distinct difference is this
methods wants you to assume a mechanism. These methods want you to have bending
moment distribution.

Now, what would be or which method would be difficult? For a statically indeterminate
structure from a very high order it is generally difficult to find the bending moment
distribution. Comparatively, it is easy to assume a mechanism, what is a mechanism?
Mechanism is that structure which has got n number of plastic hinges which can convert
the given structure to a mechanism. That number n will be nothing but one value more
than the degree of indeterminacy.

So, plastic hinges; if the degree of indeterminacy is 5 you require 6 hinges, where they
can form? We have got a list. So, keep on selecting these junctions, put the plastic
hinges, assume a mechanism, find the collapse load. So, this method appears to be
simple because this method does not require you to solve the problem to find the bending
moment distribution. If that is so simple then why this method is not popular compare to
this.

The reason is if we miss out any specific mechanism, if we do not know how to assume a
correct mechanism you can skip certain basic critical mechanisms, your collapse load
estimate will be wrong, you land up in a wrong value. That fear is there, in this case the
fear is not there because you are only assuming a bending moment distribution which is
safe and admissible, but here you are not assuming the bending moment distribution.
You are assuming a mechanism directly.

This method though appears as if it is very simple it can lead to wrong estimates of
collapse loads. Then one can ask me a question, sir, will this method always lead to
correct value of class mechanism? The difficulty here is, if you do not know correctly
how to do a bending moment distribution then also you can have an error here. Both
ways there are difficulties. So, let us quickly look at the steps of first theorem, second
theorem.
Then combining these two, there is a theorem available in the literature called
uniqueness theorem. That is also we will see. Then we will do couple of examples by
both the methods and see which is easy and why, is that clear? Possibly, I think this we
should do in the next lecture that will take some time for me. To solve, I cannot rush
through in 5 minutes, I have to complete this. We will do couple of problems to
understand. So, in this lecture we understood very interestingly that why a plastic
analysis and design procedure cannot be employed for a static determinate system.

You can do, there is no, nothing stops you from doing that, but the advantage is it will
not give you enough reserve strength for redistribution of moments which is one of the
important advantage of a plastic design. Second point, what we learnt is plastic analysis
and design is not a theorem or not technique, which leads to collapse of a structure.
Though, you call the load as a collapse load, though you call the load as an ultimate load,
the ultimate term adjective may give a feeling to the people that that is the ultimate load
beyond which the structure will collapse and the loads can exceed etcetera.

All these probabilities are there in back mind, but still this methodology is not leading
towards collapse of a structure because Q is what we call margin of safety which is
generally and comparatively good with respect to working (( )) methodology. Now, a
question comes very simply when this method is economical, when this method is
utilized in the maximum reserve strength of the material, when this method is utilizing
the reverse strength of the geometry. In terms of static indeterminacy, redistribution of
moments, why this method is not popular for a longer time in engineering practice?

Even now it is not very popular. Why this method is not popular? This method has
higher risk if you do not do the design properly. So, as a beginner one is not encouraged
to do a plastic design because he does not do that kind of error checking in the initial
stages of design. It does not mean that plastic design done by a engineer is always
unsafe, but it needs rigorous checking. You may skip certain mechanisms, you may skip
or you may do a wrong distribution of moments, you may land up in a wrong collapse
world value which you think that is an ultimate value; in reality that may not be what is
called as true collapse load, is that clear?

So, there is always a sense of risk available in this. That is where we talking about the
difference between elastic analysis and plastic analysis. We have seen the three
conditions to be satisfied for both the analysis separately. We have also tried to
understand what is the difference between analysis and design in terms of plastic design.
Why do we talk about this? In the next class, we will talk about the example problems on
both the theorems, solve them and try to understand which is comfortable to us and why?

Thanks.