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