Welcome to aircraft structures one course myself
professor Anup Ghosh from aerospace engineering
department of iit Kharagpur this is the first
lecture in that series to introduce with you
the aircraft structures better we look into some
video video options like this though it is it may
not look good to bring the detail of aircraft
structure at the first slide but this is the
other way I feel better because it gives us the
insight of the structure in general sense if we
talk about structures this is assembly of mat
elements which the as load externally applied
loads or and self load of weight also or the
weight which it it is having on its own this
particular example if we look at this example
is actually a double decker aircraft fuselage
cross section this may not match exactly with any
commercial aircraft but these aircraft fuselage
section has been created in cad software and
it is created to give a better feeling before
we start the aircraft structures course ya if if
you look at it the most predominant thing is that
The structure is no way solid in nature they
are this is build up thin sections and there
are two decks below portion if you look at
this portion where generally cargo goods
these those things are placed this portion
and as well as this portion if you look at
this and this we usually really passengers are
carried that is why windows are provided here
even doors are also provided here and this is
the normal way it is done so another video I
would like to show you in this regard that is
again a representative wing structure in this
we will see different sections of wing like the
ribs the spars in this what we see is that these
these shapes which are we say airfoil shape
these maintains the shape of the wing these
sections are known as spars and that way these
are all built from thin work sections and this
these are the aim of our study how each and
every part is designed or analyzed but before
we start into those analysis and design we would
like to go to the basics of the solids mechanics
So in that concern to go through the basics of
the solid mechanics it is better to go through
the evolution of solid mechanics on the other
way we may say that it is the history of solid
mechanics and since we cannot disassociate
aircraft structure from solid mechanics so
it is also the history of aircraft structures
as well this is a very interesting one I really
feel encouraged looking at this one this may be
the first point or noted point in the history
while someone did some experiment to find out
properties of material so if you look at the
slide this is introduction to flight vehicle
structures definitely it is a common title for
the this this portion but basically now what
we will do we will consider a brief history of
structural mechanics and or solid mechanics as we
said the history of structural mechanics or solid
mechanics is as old as civilization so this this
this statement I have put at first because you
know anything we do even if we sleep on a cot that
is a structure so if we whatever we do if even
If we think of invention of wheel that is also
a structure anything we can imagine anything if
we think of an axe that axe is also a structure
that handle of the axe has somebody thought of
its strength come somebody thought of its size
then he designed it so if you look think of a
spear that is also a structure so looking at
that point of view it is really as old as the
civilization is but the similar way if we look
at the noted down history or the fast documented
investigation that is done by Leonardo da Vinci
though Leonardo da Vinci is much more famous
with arts and sculptures and paintings all those
things we are familiar with monalisa but see he
is the person who did experiments with these
things also if we look at the set up this is
very nice interesting setup this is a bag c c is
a bag full of sand b is a bucket and the string ab
which is connected or from a bar horizontal bar
is going to be tested so what is happening from
the bucket sands are coming to this from the
bucket c sands are coming to the bucket b and
At some point of time slowly as the load will
increase as the amount of sand will increase
it will exert more force on the string and
we can find out we can estimate a measure
of how much is the strength of the string ab
so this may be considered that the string or
wire whatever we say the wire ab how much is the
strength the so this is the first noted point we
can say that this is the first tensile testing
of material so this is really notable thing we
should honor him with giving him the credit but
after this we will go go to the other portions
other portions like the history how we have come
across all these analysis design and other aspects
of solid mechanics and analysis of aircraft
structure so we have segmented this lecture I
have planned the lecture in three segments first
considering the elasticity stress and strain so
if we look at it if you go to the previous
slide please note the time it is something
in the mid of or late of fifteenth century then
what happens we denoted scientist or physicist
When we say that time everyone was physicists
so Galileo Galilei may introduce the concept of
stress concept of stress through the experimental
observation of tensile testing of bar breaking is
independent of length and dependent on cross
section so it is something considered that
Galileo Galilei is the person who first
established the correlation between the
length and cross section with respect to their
breaking load so he concluded that breaking is
independent of length and it is dependent on the
cross section various experiments on stone beams
also he did he did lot of experiments on stone
beams next if we look at it is say maybe in the
seventeenth century mathematical and physical
studies are first carried out by isaac newton
with introduction of laws of motion this is very
famous we have heard anyways three famous laws
we would not spent much time on that these are
quite available in books we are very aware of it
next if we look at Robert Hooke now we always
say it is hooks law hooks law so hooks what
What did Hooke says Robert Hooke said
Robert Hooke and e morriotte sorry Mariotte
in sixteen eighty observed that displacement
is proportional to the applied load for many
materials so this is a very very common
observation for us nowadays but he they
first noted down this observation about James
Bernoulli he is one of the famous Bernoulli family
physicist James Bernoulli noted down in his last
publication in the year seventeen hundred five
that proper way of describing deformation was to
give force per unit area or stress as a function
of the elongation per unit length or strain of
a material fiber under tension it is something
he is the person first said that there is there
are two terms there may be two terms or two way
we can define one is with respect to the force
per unit area and the other is with respect to
the material fiber under tension so elongation
per unit length then it comes about Leonhard
Euler proposed the linear relationship between
stress and strain in seventeen twenty seven
Sigma is equal to E epsilon and this may be noted
at this point of time that it may be noted that
constraint constant e is named after Thomas young
as youngs modulus in eighteen hundred and seven so
if we see what we generally say stress strain
relation sigma is equal to e epsilon for that
only there are contribution of Leonhard Euler
there are contribution of Bernoulli there are
contributions from Robert Hooke and Mariotte
so later if we look at the internal tension
acting across surfaces in a deformed solid
was expressed by Gottfried Wilhelm Leibniz
in sixteen eight four and James Bernoulli
in sixteen ninety one now with probably you
have come across the mechanics course in your
first year maybe in your first or second second
semester this is quite familiar to us that there
are something internal stresses internal tension
acting across surface there are stresses which
acts internally within a body but see that was
first noted down by Leibniz ian Bernoulli in
sixteen ninety one Bernoulli and Euler introduced
The idea that at a given section along the length
of a beam there were internal tensions amounting
to a net force and a net torque so they tried to
make some correlation between the net force and
torque and internal tensions Euler introduced the
idea of compressive normal stress as the pressure
in a fluid he again did further calculations
for that proposition and he said that there
are something compressive normal stresses also
it is in come it is compressive in nature and
that may be considered may be said as pressure
in case of fluid Charles Augustinecoulomb mom
more popularly known as coulomb was apparently
the first to relate the theory of beam as a bent
elastic line to stress and strain in an actual
beam so he is the person who first assumed that
he proposed that the bent elastic line has
a relation with the stress strain of actual
beam he developed the famous expression sigma
is equal to m cross y by I this expression is
really very very good and we use it we you
have done in your makers mechanics course
All problems with this but it got introduced
sometime in the late eighteenth century for the
stress due to better we repeat he developed
the famous expression sigma equals to m by
I multiplied by y for the stress due to pure
bending of a homogeneous linear elastic beam
so this formula holds for pure bending and he
established this shear stress the concept of
shear stress probably you have now but it was a
say concept by parent later implemented in soil
mechanics by coulomb in seventeen seventy three
then we we say a lot of contribution is done from
Augustin Cauchy Augustin Louis cost cauchy in
eighteen twenty two formalized he formalized the
stress concept in the context of a general three
dimensional theory showed it is proportional as
consisting of three by three symmetric array of
numbers sorry he showed that it is its properties
as consisting of three by three symmetric
array of numbers that transform as a tensor
derived the equation of motion for continuum
in terms of components of stresses and gave
The specific development of the theory of linear
elastic response for isotropic solids this is
really a big sentence for us to study now there
are terms which are probably we are not familiar
with like the continuum equation motion of motion
for a continuum three by three symmetric area of
numbers which is actually a tensor so but see
it is better to get introduced with these things
because we will be using in our later courses or
maybe in a very brief way in this present course
as part of this work cauchy also introduced the
equation which expresses the six components of
strain three compo sorry three extensional
and three shear in terms of derivatives
of displacements for the case when all these
derivatives are much smaller than unity so this is
so del u del x is equals to epsilon of x that is
what he said he in the year eighteen twenty two or
in the close year he proposed that and it
was continuing so this is something we will
come back again about further development of this
continuum equations who did what but before that
We better get introduced with these things
which is related to the beams columns plates
and shells development development in the sense
analysis development how those analysis was
being slowly developed by the famous physicist
james Bernoulli if you look at proposed in his
final paper in the year seventeen o five
that curvature of a beam is proportional
to the bending moment this we have we know that
m by ei is equals to one by rho and he also we
can also see that Euler in seventeen hundred and
forty four and daniel Bernoulli he is also from
that Bernoulli family in nine seventeen fifty
one used the theory to address the transverse
vibration of beam transverse vibration of
beam is why if we have in interest to to
look at the response of a vibrating structure
the first thing we generally study is a single
degree freedom system but invariably it
goes to the beam vibration that means if
it is under dynamic load or it is vibrating
because of its inertia how the response is
All those things were first introduced in
seventeen fifty one Euler gave in seventeen fifty
seven his famous analysis of buckling or initially
buckling of initially straight beam subjected to
a compressive loading this Euler buckling formula
already we are introduced with buckling is a kind
of instability and we we need to understand it
we need to find out the critical load so that
structure does not goes to any instable region
and it serves its purpose of carrying load daniel
Bernoulli and Euler in seventeen forty two and in
forty four introduced the strain energy per unit
length for a beam proportional to the square of
its curvature and regarded the total strain energy
as the equation sorry as the quantity analogous
to the potential energy of a discrete mechanical
system so if we look at this statement this is
a very very important development this may be
the fundamental of todays numerical analysis he
said that the strain energy per unit length is
is analogous to the potential energy so this
this led this is the may be considered as
The first statement for the analysis with
respect to the energy we will see many more
other things are also developed and that slowly
has invented the process of numerical methods
like finite element methods following from the
principle of virtual work as introduced by john
Bernoulli Euler rendered the energy stationary
and in this way developed the calculus of
variations as an approach to to the equations
of equilibrium and motion of elastic structures
so this energy stationary and the calculus of
variation these are this the two fundamental
mathematical tools which further laid down the
process of invention of finite element analysis
or other numerical methods and approximate methods
what we generally learn so we were talking about
the variational approach variational approach
played really a major role in the development
of theory of small transverse displacements and
variations of elastic plates and this point of
time it is better to get introduced with the
plates in structural analysis or in structures
We generally call a structure plate while the any
one of the dimension is very very thin compared
to its other two dimension so if we see if we
consider a rectangle like this and say if the
thickness of this rectangle if the thickness
of this rectangle is very small with respect
to its other two dimensions say this is a this
is b we call this as a plate and curvature we
say it is of infinite curve radius of curvature
while it is having some curvature that case we
call that structure as shell so predominantly
we have much use of plates and shells in our
aircraft structures so so this is important
structure in our case and we will see how
we will analyze those things slowly so this
theory was developed in preliminary form by
Sophie Germain and partly improved upon by Simeon
Denis Poisson in the year eighteen hundred and ten
they considered a flat plate as an elastic plate
which resists curvature that means probably they
try to mean the bending navier gave a definitive
development of correct energy expression and
Governing differential equation a few years
later so if you see already we have come across
with respect to the energy with respect to the
variational calculus many names like Navier like
Poisson like that name Sophie sar sorry Sophie
Germain and before that and before that we
have already heard about Bernoulli Euler and many
more so let us go further problems related to the
definition of twisting moment and shear force was
finally resolved in eighteen hundred and fifty by
by German physicist Gustav Robert Kirchhoff
kirchhoff also has done many works in relation
to plate in some advanced stage we generally
say kirchhoffs plate but those things are much
later stage but he first did a distinguish work in
relation to the twisting moment and shear force so
application of virtual work he did and variational
calculus procedures also he established for
that in the framework of simplifying kinematic
assumptions fiber fibers initially perpendicular
to the plate middle surface remains so
after deformation of that surface this this
This statement has become very very famous with
respect to Kirchhoff and this has led down the
analysis of many many more bigger structures
simplified way of analysis this assumption in
your later stage of study you will understand
and you will learn the first step in the theory
of thin shell was in the year seventeen hundred
and seventies by Euler as we have just now got
introduced what is shell and what is plate shell
is a structure where the third dimension is very
small compared to the other two dimensions and
it is having a curvature we also analysis wise
shells are also in two different ways it is
generally categorized deep shell and shallow
cell but we would not go into those details at
present so for thin shell it was first introduced
by Euler he addressed the deformation of an
initially curved beam as an elastic line and
provided a simplified analysis of vibration of
an elastic bell this is related to a famous for
his analysis of a big charge bell it was
further modified after after a long time
In the year eighteen hundred and seventy three
by h aaron acceptable thin shell theories for
general situations appropriate for cases of small
deformations were developed by a e h love he also
did lot of work for thin shell formulation
and later it got further extended by lamb
in eighteen ninety later many improvements are
suggested for thin shell modeling by Koiter w t
Koiter and Novozhilov did the most significant
contribution in the year nineteen fifties with
this let us try to conclude this part of
of the introduction more elastic general
theories in relation to the detail description
in tensor were formed by Cauchy and then later
by Poisson by Wilhelm Leibniz descriptions green
described in more detailed way anisotropy those
things may be considered later and it may
be noted that it is not much earlier this
history ends it is something around nineteen
fifty if we look at this this slide the linear
elastic elasticity as a general theory
sorry linear elasticity as a general
Three dimensional theory was first proposed by
Cauchy Navier and Poisson this terms in this
sentence linear elasticity is very important our
discussion will be related to linear elasticity
only there is a huge domain of nonlinear
elasticity also we would not go into that
general three dimensional theory this general
three dimensional theory will get introduced
in this course at some advanced stage maybe in the
last few lectures and these things were introduced
in the year eighteen twenty to eighteen thirty in
the isotropic case it predicts that there is only
one elastic constant and that the poissons ratio
has the universal value of one fourth see this is
we know now it is not true but at that point of
time it got proved and people started believing
that and later it was changed and got introduced
in a different way now this is related to some
introduction with the non isotropic material
which has a predominant use in structural
aircraft structural analysis in case of laminated
composite we have many application related to that
So maximum possible number of independent elastic
moduli in the most general anisotropic solid were
established by george green in eighteen thirty
seven existence of elastic strain energy required
that the thirty six elastic constants relating the
six stress components to the stick six 6 strains
at most twenty one could be independent sorry
so this is a famous conclusion drawn by green
thirty six to twenty one and then we will see
later for orthotropic it reduces much more and
use considering orthotropic material we
can analyze many of the applications in
aerospace industry for example the composite
structures with this history let us conclude
todays lecture the first session of the lecture
we will start further in our forthcoming session
but before we finish it it is better to pay
our gratitude to these famous scientists I
have tabulated with their birth and year
and the duration they were on the earth
to facilitate human civilization and they are
without their contribution probably there was
We cannot imagine anything under the earth
so it starts with Leonardo da Vinci and I
have noted as a here as the last person has
w t Koiter but it is not the it is not that
there it ends history is a continuing process
and goes further so you may also look at that
contribution is not from any one region
of the earth it is from various places
like Italy France England Switzerland France and
Scotland Germany and soviet union and Netherland
also so to show them the gratitude let me to
end with let me name them for once Leonardo da
Vinci from Italy Galileo Galilei from Italy e
Mariotte from France Robert Hooke from England
Isaac Newton from England g Wilhelm Leibniz from
Germany James Bernoulli from Switzerland john
Bernoulli from Switzerland Daniel Bernoulli
from Switzerland they are of same family
this is very surprising Leonardo Euler from
Switzerland Charles Augustin coulomb from
France George green from England Thomas young
from England Sophie Germain from France s d
Poisson from France claus louis navier sorry
Claude Louis Navier from France Augustin Louie
Cauchy from France g Pialo from sorry g pialo
piola from Italy Barre de saint Venant from
France lord Kelvin from Scotland Gustav
Kirchhoff from Germany l Pochhammer from
Germany j v Boussinesq from France h lamb from
England v Cerruti from Italy hertz h r hertz on
whose name the frequency hertz unit is there
from Germany he also gave theories related to
impact love he is a e h love his books are
also there nowadays on publication you can
find out he is we call he is the father of
shell theory he is from England Novozhilov
from soviet union and Koiter from Netherland so
with that let us end todays session thank you