The following content is
provided under a Creative Commons license. Your support will help MIT
OpenCourseWare continue to offer high-quality educational
resources for free. To make a donation or view
additional materials from hundreds of MIT courses, visit
MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Ladies and gentlemen,
welcome to this course on nonlinear finite
element analysis of solids and structures. A few years ago, we produced at
MIT a course on the linear analysis of solids
and structures. That course was quite
well-received. And we obtained a number of
requests to also produce a course on the nonlinear
analysis of solids and structures. The present course is the answer
to those requests. In this course, we will be using
my book as a textbook. You might be already a bit
familiar with this book. We will be referring to
some of the sections in quite some detail. In addition, of course, you
also have the study guide. Let me now share with you,
briefly, some thoughts that I had while designing
this course. The field of nonlinear, finite
element analysis is a very large field. And, in fact, four large fields
come to mind as feeding into nonlinear, finite element
methods, continuum mechanics, finite element discretizations,
numerical algorithms, software
considerations. Because the field of nonlinear,
finite element analysis is such a large field,
I had to select certain topics as the topics of this
finite series of lectures. I believe that the lectures
provide a good introduction and foundation to a nonlinear,
finite element analysis. Of course, the lectures cannot
answer all the questions that you might have regarding
nonlinear, finite element analysis. I believe, though, that they
will be answering some of the questions and, hopefully,
stimulate discussions that you might have while watching
these lectures with your colleagues. We, at MIT, continue to
work in nonlinear, finite element analysis. And we also offer, from time
to time, short courses. It would be very nice to meet
you at one of these short courses and to discuss any
questions that you might have regarding these lectures, once
you have seen them, and regarding your work on
nonlinear, finite element analysis in general. Let me now return to
what I like to talk about in this course. In this course, we want to
concentrate on methods that are generally applicable,
modern techniques and practical procedures. I believe it is important that
we discuss in this course practical and effective
procedures. In particular, methods that
are or are now becoming an integral part of computer-aided
design, computer-aided engineering
software. In this course, I would like to
discuss with you geometric and material nonlinear analysis,
static and dynamic solutions, basic principles and
their use, and share with you example solutions. I believe that the course will
be of interest in many branches of engineering
throughout the world. And in this spirit,
we have designed a logo for this course. The logo is shown here. It shows the whole world as an assemblage off finite elements. If you're watching me from
Chicago, you're watching me from about here. If you're watching from
Munich, you're watching about there. And if you are in Tokyo,
you're watching me from about there. Welcome to this course. I'd like to now start with the
first lecture, which has the title Introduction to
Nonlinear Analysis. And let me take off my jacket,
because we have a lot of work ahead, and summarize to you what
I would like to present to you in this first lecture. We discuss, first, some
introductory view graphs and show some short movies. We then classify nonlinear
analyses. We then discuss the
basic approach of an incremental solution. And we share some example
solutions. Let me walk over to my view
graphs which I've prepared for this lecture so that we can
start with a discussion of this material. Finite element nonlinear
analysis in engineering mechanics can be an art, but it
can also be a frustration. For those of you who have been
doing some nonlinear analysis already, I think you will value
that it can be an art and it can be a frustration
because it can be a very difficult matter. But it's always provides
a great challenge. And that, of course, is the
exciting part of working in nonlinear, finite element
analysis. Some important engineering
phenomena can only be assessed using nonlinear analysis
techniques. For example, the collapse or
buckling of structures due to sudden overloads-- I'm thinking here, typically
of shell structures, say-- the progressive damage behavior
due to long-lasting severe loads such as, for
example, high-temperature loads in nuclear reactor
components and, for certain structures such as cables and
transmission towers, nonlinear phenomena need be included in
the analysis, even for service load calculations. The need for nonlinear analysis
has certainly increased in recent years due
to the need for the use of optimized structures, the use
of new materials that are being introduced or have been
introduced over the recent years, the addressing of
safety-related issues of structures. There's more rigorous attention
being given to such safety-related issues now and
certain the corresponding benefits can be most
important. Problems that are addressed by
nonlinear, finite element analysis can be found in many
branches of engineering. And I've listed here some such
branches, nuclear engineering, earthquake engineering, the
automobile industries, defense industries, in aeronautical
engineering, mining industries, offshore engineering
and so on. I'd like to now show you some
movies that we have assembled regarding some finite element
analysis in certain fields. And let us start with a movie
related to the defense industries, then go on to a
movie related to some work in the automobile industries, and
then look at a movie related to some work in earthquake
engineering. And finally, I'd like to also
show you an interesting movie regarding a structural
engineering problem. So let me now introduce you to
the first movie, which is a movie in which a tank is modeled
using a geometric model first and then setting
up a finite element mesh. Here we see a tank
in a maneuver. You surely have seen such
structure before. And here you see, in the top
part of the picture, the geometric model of a tank and,
in the bottom part of the picture, a finite element
discretization. The top model was generated
using a geometric modeler. And the bottom model was
obtained using a finite element mesh generator. These models were constructed
by Structural Dynamics Research Corporation. The second movie relates to a
problem that has obtained much attention in the automobile
industry, namely the problem of what happens when a motor
car crashes against another motor car. And tests have been performed of
motor cars crashing against rigid walls. Test results have
been assembled. And finite element methods may
well be used in the analysis of such types of problems. Here we see a test performed by
the Ford Motor Company of a car crashing against
a rigid wall. And here, a close-up of what
happens to the passengers not wearing seat belts, and
another such test. Here now, in more detail,
what happens to the front part of the car. There is since some time much
interest in modeling these phenomena on the computer. And finite element methods can
be applied and are very valuable here. This certain movie relates to
the use of finite element methods in earthquake
engineering. Here we look at the finite
element model of a tank and the dynamic motions
of that model. We are surely all aware that
earthquake motions can cause severe structural vibrations
and possibly collapse of structures. Here we see a transformer and
the tank to be modeled on it. Here a close-up view
of the tank. The finite element model of the
tank was constructed by Asea, a Swedish company. And here we see some vibratory
motions of the model. The fourth movie shows the
dynamic response and collapse of the Tacoma Narrows
bridge in 1940. The Tacoma Narrows bridge
collapsed on November 7, 1940, about four months after its
opening in winds of 40 to 45 miles per hour. Here you see the dynamic motions
of the bridge prior to its collapse, a side view,
and now a view along its center line. Notice the high torsion of
vibrations of the bridge. The bridge went through large
dynamic motions for hours until its collapse. We only show a very small
segment of that time. And here you see how the
bridge collapsed. Of course, these movies only
indicate, to some extent, where finite element methods
might be applied in engineering practice, but I
thought you might like, you might enjoy, seeing
the movies. Let me now continue with the
material that I have documented on the view graphs. For an effective nonlinear
analysis, a good physical and theoretical understanding
is most important. You want to have some good
physical insight in the problem, setup, and mathematical
formulation of finite element model. Solve that model, and
that will enrich your physical insight. It is this interaction and
mutual enrichment between the physical insight and
mathematical formulation that can be most valuable. The best approach for a
nonlinear finite element analysis is to use reliable
and generally applicable finite elements. With such methods, we can
establish models that we can understand, that we have
confidence in. We start with simple
models of nature. And we find these
as need arises. In fact, I like to think of an
engineer as developing a first model on the back of an envelope
using, of course, simple equations. And these equations will give
some insight into the structural response. If necessary, then the finite
element model is set up. The first finite element
model is set up. And this finite element
model is then refined as need arises. To perform a nonlinear
analysis, we want to, altogether, stay then with
simple, relatively small and reliable models. We always want to perform
a linear analysis first. I will show examples in this
course particular related to this item here. It is very important, in my
view, to first do a linear analysis and then only go on
to a nonlinear analysis. As I mentioned already, we want
to refine the model by introducing nonlinearities as
desired and, once again, use reliable and well-understood
models, obtain accurate solutions of the models. This is very important for
possible proper interpretation of the results. Here we show schematically
the finite element modeling process. We have a problem in nature. We model that problem, the
kinematic conditions, the constitutive relations, the
boundary conditions, the loads, and so on, using
finite elements. We solve the model and interpret
the results. Now this model surely can
only approximate the actual problem in nature. And on interpretation of the
results, we may find that we really should refine
our model. And we do so, set up a new
model, solve again. And like this, we may cycle
a number of times through this process. Of course, traditionally, test
results have been sought for very complex problems,
laboratory test results have been sought. And this may still
be necessary now. However, the finite element
modeling process will certainly compliment these
laboratory test results. Let's look at a typical
nonlinear problem. Here we have a bracket
of mild steel subjected to the load shown. And the possible questions that
we might ask are, what is the yield load of
this bracket? In other words, at what load
do we see first plastic deformations? What is the limit load? What is the maximum load that
this bracket can take? What are the plastic zones? What are the residual stresses
when the load is removed? Is there adhering where
the loads are applied right around here? What is the creep response of
the bracket when the bracket is subjected to high temperature
conditions and these loads? And what are the permanent
deflections of the bracket, et cetera. There are many more questions
that could be asked. And certainly, these questions
here can only be answered by a nonlinear analysis. Possible analyses that we might consider are the following. First we, of course, would
always perform, as I mentioned earlier, a linear elastic
analysis. The linear elastic analysis
would determine the total stiffness and the yield
load of the bracket. Notice here we show the
displacements in red. Actually, these displacements
would be very small. And they are magnified
in this view graph. Another analysis now would be to
perform a plastic analysis, but assuming still small
deformations. This analysis would determine
sizes and shapes of the plastic zones. The displacements here
are still small. We have not shown them
at all actually here. They are very small, however. Then we might want to
go on to a large deformation plastic analysis. This analysis would determine
the ultimate load capacity of the bracket. And notice the displacements
are now large, they are actually large. Here, we have had small
displacement assumptions. Here, we have included the large
deformation effects as also shown on the view graph. For analysis, it is very good
to actually classify all the types of analysis that one
might want to perform. And the first category of
nonlinear analysis is the one that we call
Materially-nonlinear-only analysis, MNO analysis. In this analysis, we assume
that the displacements are infinitesimal, the strains
are infinitesimal. In other words, both
of these quantities are very, very small. And the stress-strain
relationship is nonlinear. So all nonlinearities lie,
really, in here, in the stress-strain relationship. Here, I'm showing a
schematic example. You're looking at a four-node
element that is subjected to the loads shown, and delta
is the displacement. Notice that delta/l
is very small. In fact, even this four is
already relatively large. We should really have here,
possibly, a two. So nonlinearity lies in the
material description. The material is an
elasto-plastic material, Young's modulus, E, yield
stress, sigma y and strain-hardening modulus, Et. As long as the stresses do not
exceed sigma y, we really have a linear analysis. So if you use a computer
program with an MNO formulation and you subject your
model to forces such that the stresses are below sigma
yield everywhere in the model, then you should really obtain
the linear analysis results. The next category of problems
is the one of large displacements, large rotations,
but small strains. Here, in other words, we still
keep the assumption of small strain, but we allow large displacements and large rotations. The stress-strain relations can
be linear or nonlinear. Schematically here, once again,
our four-node element. This four-node element now would
move, as shown, going through large displacements
and large rotations. But the strains in the element,
expressed by delta prime over l are still small. Once again, you may actually
want to make this 0.02. As long as the displacements are
very small, we have now, really, an MNO analysis. The third category of problems
is the one of large displacements, large rotations
and large strains. Here we have included all
kinematic nonlinearities. And the stress-strain relation
is probably also nonlinear because we are dealing
with large strains. As an example, we have, again,
our four-node element here moving as shown. Notice material fibers here,
displaced by a large amount, rotate and are also stretched
by a large amount. This, of course, is
the most general formulation of a problem. However, still not considering
any nonlinearities in the boundary conditions,
nonlinearities in the boundary conditions provide
contact problems. And here, we look schematically
at a simple contact problem. The four-node element, once
again, subjected to loads, there's a gap here between this
element and the spring. And as soon as this gap is
closed, the spring, of course, provides stiffness into the
system, the system being now the four-node element. And, well, that difference has
to be taken into account in the analysis. And there are procedures to
solve contact problems. This is, of course, a very
simple contact problem. But there are procedures to
solve contact problems. Contact problems are very
difficult problems to handle, particularly if you're dealing
with frictional effects. We will, in this course, in this
set of lectures, really not address contact problems,
except that, in the last lecture, we actually
look at one particular contact problem. We use a computer program for
the analysis of a contact problem there. Let's turn to an example
analysis. And I'd like to consider
with you, briefly, the analysis of a bracket. Here is a bracket of
the kind that we looked earlier at already. Notice the dimensions
are as shown. The thickness of the bracket
is one inch. And this bracket is going to
be loaded to large loads. We make a material assumption,
namely the one of an elasto-plastic material
with isotopic hardening as shown here. The yield stress
is 26,000 psi. Here you have the
Young's modulus. And here you have the
strain-hardening modulus. We use symmetry conditions to
analyze the bracket the same way as we would do it, of
course, in a linear analysis. And with the symmetry
conditions, we consider on this view graph just the lower
part of the bracket. You see a pin here,
rollers there. And we're using eight-node,
isopolymetric elements to model this part of
the bracket. We apply the load
as shown here. Notice no special considerations
to the hole which the load is applied. Our interest really lies in
predicting the stresses and strains in this region and to
also predict the overall collapse of the bracket. We will use three kinematic
formulations for the analysis. First of all, we use the
Material-nonlinear-only analysis assumption. In other words, small
displacements, small rotations and small strains are
assumed in the analysis of this bracket. Then we will perform an analysis
using the total Lagrangian formulation. We will discuss this formulation
quite extensively in later lectures
of the course. This formulation assumes large
displacements, large rotations and large strains
kinematically. Kinematically, these are
the assumptions. However, we will point out that
the material law that we are using, or the material law
description that we are using, is only applicable with this
formulation to small strains. So the overall analysis, using
this total Lagrangian formulation, is then only
applicable to model large displacements, large rotations, but only small strains. I get back to that just now. We will also use an updated
Lagrangian formulation which kinetically includes large
displacements, large rotations and large strains. And on the material model level,
we also include there large strain effects. So once again here, as
summarized on this view graph, the material used in conjunction
with the total Lagrangian formulation is
actually not applicable to large-strain situations, but
only to large displacements, rotation in small-strain
conditions. So once again, our total
analysis is only applicable to this kind of situation. Whereas, the updated Lagrangian
formulation does model, kinematically and on the
stress-strain level, large displacements, large rotations
and large strains. The analysis results obtained
are shown here. Notice we are plotting here
the force applied and the total deflection between points
of load application. The three analyses give us three
distinct curves at large displacements. But for small displacements,
of course, these curves are indistinguishable. Notice here we have about 10%
strain at point A. I will show you just now where point A is. The important point to notice
is that these are three distinct curves. They are distinct because
we have made different assumptions, kinematically
and on the stress-strain load level. We will talk about such
assumptions in the later lectures of this course. On this view graph, now you
see this point A, which I mentioned earlier. And you see also the original
mesh shown in dashed, black lines, and the deformed mesh,
shown in red, corresponding to a level of load of
12,000 pounds. I'd like to now look with you
at two animations regarding problems that show
nonlinearities typical of the nonlinearities that we are
talking about in this course. The first animation shows a
plate with a hole that is subjected to high
tensile forces. The plate undergoes plastic
deformations. And, in fact, the load becomes
so large that the plate basically ruptures. Let's look at this
animation now. Here we see one quarter
of the plate. The plate is subjected to a
uniformly distributed tensile load along its upper edge. For the quarter model of the
plate, symmetry boundary conditions are imposed along
the left vertical and lower horizontal edges. The time code given above
the plate gives the time of the load step. Also given is the load
applied at that time. However, note that we perform
a static analysis. Therefore, the time code merely
denotes a load level, as we will discuss
later in detail. Each time increment of one milli
corresponds to a load increment of 12.5 megapascal. We will increase the
load monotonically. The plate will become
plastic at the hole. And this plasticity will then
spread until, in essence, the plate ruptures. We used 288 eight-node plane
stress finite elements for the quarter of the plate. And we show to spread of
plasticity in the plate by darkening the areas
that are plastic. Here now you see the time
and load increasing. In the first load steps, the
plate remains elastic and the deformations are small. Now you see the first
plasticity developing near the hole. The plastic zone increases
rapidly as the higher load levels are reached until a large
portion of the plate is plastic and, in essence, the
maximum load-carrying capacity has been reached. We will consider the analysis
of this plate in more detail later in the course. The second animation shows a
frame that is subjected to forces such that the frame
undergoes very large displacements. Here we see the frame model,
using beam elements that we discuss later in the course. The frame is loaded, as shown,
by the force arrows. There are pin connections
at the points of load application. The frame is assumed to be of an
elastic material, hence, no plasticity is assumed to develop
or that the frame will be subjected to very large
displacements. The indicated loads will push
the top of the frame down and the bottom up to such an extent
that the points of load application cross over. You see above the frame
a time and a corresponding load level. This is, again, a
static analysis. And each time increment of one
corresponds merely to a load increment of 250 pounds. Here now, you see the
deformations of the frame develop as the load
is increased. The displacements are
very large at the higher load levels. Of course, this is only a
numerical experiment at the high load levels, but an
interesting one that indicates in a simple manner what
can all be done in finite element analysis. We will consider this problem
solution also in more detail later in the course. Let us now look at the
basic approach of an incremental solution. We consider a body, a structure
or solid subjected to force and displacement
boundary conditions that are changing, and we describe the
externally applied forces and the displacement boundary
conditions as a function of time. Schematically, here we show a
body, of course, supported as shown and subjected to a force
varying as a function of time and a prescribed displacement
varying as a function of time. The time variation of the force
is shown down here. And the time variation of the
prescribed displacement is shown here. Notice that here we have a
particular time point, T. And here we have a time point
T plus delta t. Here we have, similarly, the
time point T and here, also, the time point T plus delta t. Since we anticipate
nonlinearities, we use an incremental approach measured
in load steps or time steps. And this means that the loads,
the prescribed loads, the imposed loads and the prescribed
displacements, are discretized as a function
of time, as shown on this view graph. Notice here we have, at time
T, the impulse load TRI. This upper, superscript T means
at time T then delta t, in advance, time
T plus delta t. We would have, in other words,
T plus delta t ri. In other words this T would
now be replaced by T plus delta t and so on. This is how we are inputting
or prescribing the impulse loads and also the prescribed
displacements. When they apply forces and
displacements very slowly, meaning that the frequencies of
the loads are much smaller than the natural frequencies
of the structure, we have a static analysis. This, of course, means that the
periods of the loads are very long when measured
on the natural periods of the structure. In other words, you have a
spring and you apply a load that varies very slowly. And this means we have
a static analysis. If the load's very fast, and
by that we mean that the frequencies of the loads are
in the range of the natural frequency of the structure,
then we have a dynamic analysis. Let us look a bit closer at the meaning of the time variable. Time is a pseudo-variable, only
denoting the load level in nonlinear static analysis
with time-independent material properties. As an example. here we have a cantilever
subjected to a load, R, a tip load, R. And if we were to
perform a run, 1, in which we prescribe the loads
as shown here-- and notice, at time 1, a
load of 100, at time 2, a load of 200-- delta t is equal to 1. If we were to perform this
analysis and, in addition, this analysis where delta t is
equal to 2 and the load at time 2 is equal to 100, the
load at time 4 is equal to 200, then we would obtain
identically the same results in run 1 and run 2 because it's
a static analysis and the material properties are
time-independent. So in this particular case,
certainly, time is a pseudo-variable. And we can design
a time-stepping. We can design many different
time-stepping schemes using different time steps and
always obtain the same results, provided, at the end
of the first time step, we have the load 100 and at the end
of the second time step we have the load 200. However, time is an actual
variable in dynamic analysis and in nonlinear static analysis
with time-dependent material properties, for
example, when the material contains grid conditions. Now delta t must be chosen very
carefully with respect to the physics of the problem, with
respect to the numerical techniques used and the
costs involved. If delta t is not chosen
appropriately, you may have a very high cost for
the analysis. And on the other hand, you may,
with improper choice of delta t, also obtain
very bad results. So it is very important to
choose delta t judiciously for an accurate and cost-effective
analysis. At the end of each load or time
step, we need to satisfy the three basic requirements
of mechanics, equilibrium, compatibility and the
stress-strain law. These are the three fundamental
requirements to be satisfied in mechanics. This is achieved in finite
element analysis in an approximate manner using
finite elements by the application of the principle
of virtual work. Now there is a lot
of information on this view graph. And we don't have time in this
lecture to go into any depth. There is, of course, quite a bit
of mathematics that has to be introduced for the discussion
of all of what we see here on the view graph. And that's what we do in
the later lectures. I do not have time now to go
into these mathematics, but let us just very briefly
look at the basic procedure that we using. We are saying that, at any time
step, T plus delta t, the externally applied loads, the
vector of externally applied loads and this vector, includes
pressure loads. Concentrated loads and, in the
dynamic analysis also inertial forces, this vector must at any
time T plus delta t equal to the vector F at time T plus
delta t where this vector F corresponds to the internal
element stresses at time T plus delta t. We will talk in the later
lectures about how we calculate this vector
T plus delta tF. Of course, there are some very
important considerations in the proper calculation
of this vector F. Let us now assume that the
solution at time T is known. Hence, the stresses, at
time T, are known. The volume surface area and so
on, all of the information corresponding to the body
at time T is known. And we now want to obtain the
solution corresponding to time T plus delta t, that is for the
loads applied at time T plus delta t. Of course, this is
a typical step of the incremental solution. Once we have the solution for
time T plus delta t, we can use the same scheme to calculate
the solution for time T plus 2 delta
t and so on. For this purpose, we
solve, in static analysis, this set of equations. TK is a tangent stiffness
matrix of the system. Delta u is a vector of
increments in the nodal point displacements. On the right-hand side, we have
the externally applied loads corresponding to time T
plus delta t in this R vector. And here, TF corresponds to or
is the vector of nodal point forces that correspond to the
internal element stresses at time T. Notice this is an out-of-balance
load vector. And this out-of-balance load
vector gives us an increment in displacements. This increment in displacements
is added to the displacements at time T. And
that gives us a displacement vector corresponding to
time T plus delta t. Notice I have an approximation
sign there. This approximation sign is a
result of the fact that we obtained this set of equations
by linearization. We will talk about how these
equations are obtained in the later lectures. We will start from the basic
principle of virtual work and use continuum mechanics
considerations to derive, in a very consistent manner,
this set of equations. Of course, there are many
details involved. We will talk about the total
Lagrangian formulations, the updated Lagrangian
formulation, the Material-nonlinear-only
analysis, the details that go into these formulations
in terms of kinematic approximations, kinematic
approximations related to large displacements, large
rotations, large strains. We also will talk about the
constitutive relations that enter in this tangent stiffness
matrix and in this force vector and so on. Anyway, this set of equations
gives us an approximation to the displacements at time T plus
delta t because of the linearization process used to
arrive at these equations. And, more generally, we want
to solve this set of equations, tangent stiffness
matrix, still on the left-hand side, times a displacement
increment corresponding to the iteration, I. And on the
right-hand side, we have the load vector corresponding to the
loads at time T plus delta t and the nodal point force
vector corresponding to the internal element stresses at
time T plus delta t and at the end of iteration I minus 1. This is here the out-of-balance
load vector corresponding to the start
of iteration I. This out-of-balance load vector,
and with that tangent stiffness matrix, gives us an
increment in the nodal point displacement vector
corresponding to iteration I. We take this one, add it to
the previous displacements that we had already, namely
those corresponding to time T plus delta t at the end of
iteration I minus 1, and we obtain a better approximation to
the displacements at time T plus delta t. Notice, in this iteration, we
have initial conditions that I've given down here in
iteration I equal to 1. On the right-hand side we need,
of this equation here, we need this vector. And that vector is given by the
vector TF, the nodal point force vector corresponding to
the internal element stresses at time T. We also need initial
conditions on the displacement vector, and those
are given right here. Notice that, when I is equal
to 1, this set of equations reduces to the equations
that we had on the previous view graph. Once again, we will derive these
equations from basic continuum mechanics principles
in the later lectures. And we will also talk about
iterative schemes to accelerate the convergence
in the solution of these equations. This then brings me to the end
of what I wanted to discuss this you on the view graphs. I'd like to now turn to an
example solution, an interesting example solution. But, moreover, it displays the
kinds of nonlinearalities that we will be talking about in
this set of lectures. This example solution is
documented on slides. So let me walk over here and
put on the first slide. Here we show the structure
that we want to consider. It's a spherical shell subjected
to pressure loading. The material data for the
shell are given here. Notice we also include the mass
density because we will be talking about dynamic
analysis of this shell as well. We will also introduce an
imperfection of the shell. And that imperfection
is given down here. Notice it is given as a function
of the angle phi, which you see up here, as well
as a thickness, H, of the shell, the Legendre polynomial
given here and parameter data that we will be varying. First, we will consider a static
analysis of the shell and then the dynamic analysis
of the shell. On the next slide, now you see
the model that was used for the analysis of the shell,
twenty 8-node elements, axisymmetric elements subjected
to the pressure loading, and, we are modeling,
the pressure loading in deformation-dependent loading. That means that, as the shell
deforms, the pressure will remain perpendicular to
the shell surface to which it is applied. On the next slide now, we show
the first analysis results. On this axis, we plot P/PCR
where PCR is the buckling pressure of the shell, assuming
elastic conditions, and calculated using analytical
solution. On this axis, we plot the rate
of displacement of the shell at phi equal to zero. The dotted line here shows the
linear elastic solution to the shell problem. Notice that these dots,
of course, could be continued here. The total Lagrangian
formulation, which includes large displacement and large
rotation effects, gives us a buckling pressure of 0.98 PRC. A Material-nonlinear-only
analysis, including the elasto-plastic material
conditions, but not including, in other words, large
displacement, large rotation effects, gives us this
solution here. Notice that, of course, this MNO
analysis does not give us a proper bucking prediction
for the shell. To obtain the elastic-plastic
buckling load of the shell, we have to perform a total Lagrangian formulation solution. And we obtain this load level
here as the bucking pressure. This analysis was performed
with no imperfections on the shell. The next slide now shows
the results of the same kind of analyses. But at an imperfection of delta
0.1, notice here the linear analysis, denoted as E,
the Material-nonlinear-only analysis results, denoted as
EP, the large displacement, large rotation analysis
results. But assuming an elastic
condition for the shell, denoted as E, TL, TL standing
for total Lagrangian formulation and the
elasto-plastic large displacement solution given
here, the results given here and denoted as EP, TL. Notice that, because of the
imperfection, the load, the maximum load-carrying capacity
of the shell, is decreased. On the next slide now we show
the maximum load-carrying capacities for delta equal to
zero, delta equal to 0.1, delta equal to 0.2, delta
equal to 0.4. In each case, we have used the
total Lagrangian formulation including elasto-plastic
conditions. In other words, of course, we
had to include the large displacement effects in order to
pick up the proper buckling load or load-carrying capacity
of the shell, so these are the load-carrying capacities of
the shell for different imperfection levels. The next slide now shows the
results obtained using a dynamic analysis. At this time, we apply the
pressure instantaneously as a step load, constant in time. Notice we are plotting here now
mean displacement data of the shell and time
along this axis. The Etl solution results
are very close to the E solution results. I think you know now what
I mean by E and Etl. The E Ptl solution results
are quite close to the EP solution results. This means that the shell is
stable under this load application. In other words, there is no
buckling, no increase in deformations as time
progresses. The next slide now shows, as we
increase the imperfection level from delta equal to zero
to delta equal to 0.1, the displacements of the shell using
the E Ptl solution or formulation increase
with time. Therefore, the shell is unstable
under this pressure application. Of course, dynamic pressure
application, no. Notice the other solution
results are given here. Therefore, we now can play with
the imperfection level and with the load level in
order to estimate the load-carrying capacity of the
shell for each of these cases. And on this slide, you're
seeing for a constant imperfection level, we
increase the load applied to the shell. And you can see that,
at 0.4 PCR, the shell is still stable. At 0.5 PCR, the shell
is unstable. On the final slide now, we show
all the information that we have obtained in
the analysis. On this axis, we are plotting
the buckling load, on this axis, the amplitude
of imperfection. This curve here corresponds
to the static backing of the shell. And this curve corresponds
to the dynamic buckling of the shell. Of course, we could not discuss
all the details regarding these analyses
in this short time. Please refer to the study guide
in which you'll find a reference to the paper in which
we have discussed the details of these analyses. This then brings me to the
end of this lecture. And I hope you enjoy
the course.
