Sunday, 19 November 2017

WHAT IS FLUID MECHANICS AND ITS APPLICATION?

We were discussing the various basic concepts of strength of materials and also the basic concepts of thermal engineering in our previous posts. After looking strength of materials and thermal engineering topics, we will go ahead to start a new subject i.e. Fluid mechanics with the help of this post.

Today we will start here with very basic term of fluid mechanics i.e. meaning and significance of fluid mechanics with the help of this post.

Fluid Mechanics

Fluid mechanics is basically defined as the branch of science which deals with the behavior, role in engineering life and flow of fluids i.e. Liquids and gases.

 
We have used a term here i.e. fluid. Let me first provide you a brief introduction for fluid.

Fluids are divided in to liquid and gases. Liquid is hard to compress and it will secure the shape according to the shape of its container with an upper free surface.

While gases are easy to compress and gases will expand to fill its container. Therefore there will not be any upper free surface in case of gases.

So, the basic difference between liquids and gases are nature of compressibility.

Let us come to the main subject i.e. Basic definition and significance of fluid mechanics

So as we have discussed above, very simple definition of fluid, Fluid mechanics is basically defined as the branch of science which deals with the behavior and flow of fluids i.e. Liquids and gases.

Science of flow has been classified in two types i.e. hydraulics and hydrodynamics. Hydraulics is born from experimental studies and hydrodynamics is developed from theoretical studies.

Hydraulics and hydrodynamics both are merged now and termed as Fluid mechanics. Therefore, we will further study hydraulics and hydrodynamics in the category of fluid mechanics.

In simple, fluid mechanics will have following parts that we will have to study in our upcoming posts.

 

Fluid Statics

A part of fluid mechanics where we study fluid at rest will be termed as fluid statics. We will analyze the fluid under static condition i.e. fluid will not be in motion in this case.

Fluid kinematics

A part of fluid mechanics where we study fluid in motion without considering the effect of pressure forces will be termed as fluid kinematics. We will analyze here the fluid under motion, but we will neglect the effect of pressure forces.

Fluid Dynamics

A part of fluid mechanics where we study fluid in motion with considering the effect of pressure forces will be termed as fluid dynamics. We will analyze here the fluid under motion and we will also consider the effect of pressure forces too.

 
So we have discussed here the basic definition of fluid mechanics and we have also seen here three components of fluid mechanics.

Now we will go ahead to discuss the characteristics of a fluid in our next post.
Do you have suggestions? Please write in comment box.

Reference:

Fluid mechanics by Y. Nakayama and R F Boucher
Image Courtesy: Google

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Thursday, 16 November 2017

STRESS IN THIN SPHERICAL SHELL

We were discussing the basic concept of thin cylindrical and spherical shells, stresses in thin cylindrical shells and derivation of expression for circumferential stress or Hoop stress and longitudinal stress developed in the wall of cylindrical shell in our previous posts.

Today we will derive here the expression for stress developed in the wall of thin spherical shell, with the help of this post.

Before going ahead, we will first remind here the fundamental of a thin spherical shell

Thin spherical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Air receiver tank is one of the best examples of thin spherical shells.

 
A spherical shell will be considered as thin spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Circumferential stress or Hoop stress

Stress acting along the circumference of thin spherical shell will be termed as circumferential stress or hoop stress.
Let us consider here following terms to derive the expression for circumferential stress or hoop stress developed in the wall of thin spherical shell.

P = Internal fluid pressure
d = Internal diameter of thin spherical shell
t = Thickness of the wall of thin spherical shell
σ = Circumferential stress or hoop stress developed in the wall of thin spherical shell

Thin spherical shell bursting will take place if force due to internal fluid pressure, acting on the wall of thin spherical shell, will be more than the resisting force due to circumferential stress or hoop stress developed in the wall of thin spherical shell.

 
In order to secure the expression for circumferential stress or hoop stress developed in the wall of thin spherical shell, we will have to consider the limiting case i.e. force due to internal fluid pressure should be equal to the resisting force due to hoop stress developed in the wall of thin spherical shell.

Force due to internal fluid pressure = Internal fluid pressure x Area on which fluid pressure will be acting

Force due to internal fluid pressure = P x (π/4) d2
Resisting force due to hoop stress = σ x π d t

As we have seen above, we can write following equation as mentioned here.

Force due to internal fluid pressure = Resisting force due to longitudinal stress
P x (π/4) d2 = σ x π d t

σ = P x d / (4 t)

Do you have suggestions? Please write in comment box.

 
We will now discuss the basic concept of thick cylindrical and spherical shell, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

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Sunday, 12 November 2017

LONGITUDINAL STRESS IN THIN CYLINDER

We were discussing the basic concept of thin cylindrical and spherical shells, stresses in thin cylindrical shells and derivation of expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell in our previous posts.

Today we will derive here the expression for longitudinal stress developed in the wall of thin cylindrical shell, with the help of this post.

Before going ahead, we will first remind here the fundamental of a thin cylindrical shell

 
Thin cylindrical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Boilers, LPG cylinders, Air receiver tanks are the best examples of thin cylindrical shells.

A cylindrical or spherical shell will be considered as thin cylindrical or spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin cylindrical and spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Longitudinal stress

Stress acting along the length of thin cylinder will be termed as longitudinal stress. 

If fluid is stored under pressure inside the cylindrical shell, pressure force will be acting along the length of the cylindrical shell at its two ends. Cylindrical shell will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as longitudinal stress. 
Let us consider here following terms to derive the expression for longitudinal stress developed in the wall of cylindrical shell. 

P = Internal fluid pressure
d = Internal diameter of thin cylindrical shell
t = Thickness of the wall of the cylinder
σL = Longitudinal stress developed in the wall of the cylindrical shell

 
Cylindrical shell bursting will take place if force due to internal fluid pressure, acting on the ends of the cylinder, will be more than the resisting force due to longitudinal stress developed in the wall of the cylindrical shell.

In order to secure the expression for longitudinal stress developed in the wall of the cylindrical shell, we will have to consider the limiting case i.e. force due to internal fluid pressure, acting on the ends of the cylinder, should be equal to the resisting force due to longitudinal stress developed in the wall of the cylindrical shell.

Force due to internal fluid pressure = Internal fluid pressure x Area on which fluid pressure will be acting 

Force due to internal fluid pressure = P x (π/4) d2
Resisting force due to longitudinal stress = σL x π d t 

As we have seen above, we can write following equation as mentioned here.

Force due to internal fluid pressure = Resisting force due to longitudinal stress
P x (π/4) d2 = σL x π d t
σL = P x d / (4 t)

Do you have suggestions? Please write in comment box.

 
We will now derive the expression for stress developed in the wall of thin spherical shells, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

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Saturday, 11 November 2017

DERIVATION OF HOOP STRESS IN THIN CYLINDER

We were discussing the basic concept of thin cylindrical and spherical shells and stresses in thin cylindrical shells in our previous posts.

Today we will derive the expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell, with the help of this post.

Before going ahead, we will first remind here the fundamental of a thin cylindrical shell

 
Thin cylindrical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Boilers, LPG cylinders, Air receiver tanks are the best examples of thin cylindrical shells.

A cylindrical or spherical shell will be considered as thin cylindrical or spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin cylindrical and spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Circumferential stress or Hoop stress

Stress acting along the circumference of thin cylinder will be termed as circumferential stress or hoop stress.

If fluid is stored under pressure inside the cylindrical shell, pressure will be acting vertically upward and downward over the cylindrical wall. Pressure vessel will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as circumferential stress or Hoop stress.
Let us consider here following terms to derive the expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell.

P = Internal fluid pressure
d = Internal diameter of thin cylindrical shell
t = Thickness of the wall of the cylinder
L = Length of the cylindrical shell
σH = Circumferential stress or hoop stress developed in the wall of the cylindrical shell

 
Cylindrical shell bursting will take place if force due to internal fluid pressure will be more than the resisting force due to circumferential stress or hoop stress developed in the wall of the cylindrical shell.

In order to secure the expression for circumferential stress or hoop stress developed in the wall of the cylindrical shell, we will have to consider the limiting case i.e. force due to internal fluid pressure should be equal to the resisting force due to circumferential stress or hoop stress.

Force due to internal fluid pressure = Internal fluid pressure x Area on which fluid pressure will be acting
Force due to internal fluid pressure = P x (d x L)
Force due to internal fluid pressure = P x d x L

Resisting force due to circumferential stress = σH x 2 L t

As we have seen above, we can write following equation as mentioned here.
Force due to internal fluid pressure = Resisting force due to circumferential stress
P x d x L = σH x 2 L t
σH = P x d / (2 t)
Do you have suggestions? Please write in comment box.

 
We will now derive the expression for longitudinal stress developed in the wall of cylindrical shell, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

STRESSES IN A THIN CYLINDER SUBJECTED TO INTERNAL PRESSURE


Today we will see here the stresses in thin cylindrical shells, subjected with internal pressure, with the help of this post.

Before going ahead, we will first remind here the fundamental of a thin cylindrical shell

 
Thin cylindrical shell is also termed as a pressure vessel and such vessels are usually used in various engineering applications such as for storing the fluid under pressure. Boilers, LPG cylinders, Air receiver tanks are the best examples of thin cylindrical shells.

A cylindrical or spherical shell will be considered as thin cylindrical or spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin cylindrical and spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Stresses in a thin cylindrical shell

Let us consider one cylindrical shell, as displayed here, subjected with internal fluid pressure P.
There will be two types of stresses, which will be developed in the wall of thin cylindrical shell and these stresses are as mentioned here.

Circumferential stress or Hoop stress
Longitudinal stress

Circumferential stress or Hoop stress

Stress acting along the circumference of thin cylinder will be termed as circumferential stress or hoop stress.
 
If fluid is stored under pressure inside the cylindrical shell, pressure will be acting vertically upward and downward over the cylindrical wall. Pressure vessel will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as circumferential stress or Hoop stress.
Circumferential stress or hoop stress developed in the wall of cylindrical shell could be easily determined with the help of following formula.

Longitudinal stress

Stress acting along the length of thin cylinder will be termed as longitudinal stress.

If fluid is stored under pressure inside the cylindrical shell, pressure force will be acting along the length of the cylindrical shell at its two ends. Cylindrical shell will tend to burst as displayed here in following figure and stresses developed in such failure of cylindrical shell will be termed as longitudinal stress. 
Longitudinal stress developed in the wall of cylindrical shell could be easily determined with the help of following formula.
Longitudinal stress = (1/2) x Circumferential stress
σL = 1/2 x [p x d]/2t
σL =  p.d/4t

 
Do you have suggestions? Please write in comment box.

We will now derive the expression for circumferential stress or Hoop stress developed in the wall of cylindrical shell, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Wednesday, 8 November 2017

THIN CYLINDRICAL AND SPHERICAL SHELLS


Today, we will start here another important topic of strength of material i.e. thin cylindrical and spherical shells with the help of this post.

Thin cylindrical and spherical shells

 
Thin cylindrical and spherical shells are also termed as pressure vessels and such vessels are usually used in various engineering applications such as for storing the fluid under pressure.

Boilers, LPG cylinders, Air receiver tanks are the best examples of thin cylindrical shells.

A cylindrical or spherical shell will be considered as thin cylindrical or spherical shell, if the wall thickness of shell is very small as compared to the internal diameter of the shell.

Wall thickness of a thin cylindrical and spherical shell will be equal or less than the 1/20 of the internal diameter of shell.

Let us consider one cylindrical shell as displayed here in following figure. Fluid is stored here under pressure within the cylindrical shell. We will first find out here the condition to consider the shell as thin cylindrical or spherical shell.
d = Internal diameter of the shell
t = Wall thickness of the shell
l = Length of the cylinder
P = Internal pressure of the fluid stored inside the cylinder

Condition for thin cylindrical or spherical shell

Wall thickness of thin cylinder < [(1/20) x Internal diameter]
t < d/20
t/d < 1/20
d/t > 20
 
This is the condition that we must have to note it to consider a cylindrical or spherical shell as thin cylindrical or spherical shell.

As we have recently discussed that pressure vessels are used to store the fluid under pressure. Therefore, there will be two cases of failure of cylindrical vessel as mentioned here.

Case: 1

As fluid is stored under pressure inside the cylindrical shell, pressure will be acting vertically upward and downward over the cylindrical wall. Pressure vessel will tend to burst as displayed here in following figure.

Case: 2

As fluid is stored under pressure inside the cylindrical shell, pressure will be acting at the both ends of the cylindrical wall. Pressure vessel will tend to burst as displayed here in following figure.
Do you have suggestions? Please write in comment box.

 
We will now discuss another topic, stresses in a thin cylinder subjected to internal pressure in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

Continue Reading

Tuesday, 31 October 2017

DERIVE THE EXPRESSION FOR DEFLECTION AND STIFFNESS IN CLOSED COIL HELICAL SPRING

We were discussing the basic concept of spring in strength of materialvarious definitions and terminology used in springs,  importance of spring index  and expression for maximum bending stress, deflection developed in the plate of leaf spring and basic difference between open coiled and closed coiled helical spring in our previous posts.

Today, we will find out here the expression for deflection of spring under the applied load with the help of this post.

Let us brief here first helical spring

 
Helical springs are usually used in number of applications due to their shock absorption and load bearing properties. There are two types of helical spring i.e. Open coiled helical spring and closed coiled helical spring. We will be concentrated here on closed coil helical spring.

Closed coiled helical spring

Closed coiled helical springs are also termed as tension springs as such springs are designed to resist the tensile load and twisting load. In simple, we can say that closed coiled helical springs are those springs which are used in such applications, where tensile or twisting loads are present.

In case of closed coiled helical spring, spring wires are wound tightly. Hence such springs will have very small pitch. Closed coiled helical springs wires are very close to each other and hence, spring turns or coils will lie in same plane.

In case of closed coiled helical spring, turns or coils of such spring will be located at right angle to the helical axis.

Closed coiled helical spring, as displayed here, carrying an axial load W. In case of closed coiled helical spring, helix angle will be small and it will be less than 100. Therefore, we will neglect the bending effect on spring and we will only consider the effect of torsional stresses on the coils of closed coiled helical spring.
Let us consider the following terms from above figure of closed coil helical spring.
d = Diameter of spring wire or coil
p = Pitch of the helical spring
D = Mean diameter of spring
R = Mean radius of spring
n = Number of spring coils
W = Load applied on spring axially
C = Modulus of rigidity
τ = Maximum shear stress developed in the spring wire
θ = Angle of twist in wire of spring
L = Length of the spring
δ = Deflection of spring under axial load

As spring is loaded by an axial load W, therefore work will be done over the spring and this work done will be stored in the form of energy in spring.

So we will determine here the work done by axial load W over the spring and we will also determine the strain energy stored in the spring. 

Expression for deflection developed in spring under axial load could be derived by equating the energy stored in spring with work done on spring.

Each section of spring will be subjected with torsion and hence strain energy stored in the spring will be determined as mentioned here

Strain energy stored in the spring = (τ 2/4C) x Volume of the spring

Volume of spring = Area of cross section (V) x Length of the spring (L)
V = (П/4) x d2
L = 2ПRn
 
Strain energy stored in the spring = (τ 2/4C) x Volume of the spring
Strain energy stored in the spring = (τ 2/4C) x 2ПRn

Let us recall the expression for shear stress developed in spring under axial loading and we will have following result for shear stress τ.
Therefore, Strain energy stored in the spring will be given as

Work done on spring could be determined as mentioned here

Work done on spring = (1/2) W x δ 

As we know that expression for deflection developed in spring under axial load could be derived by equating the energy stored in spring with work done on spring and therefore we will have following equation as mentioned here.

Stiffness of spring

 
Stiffness of spring could be easily determined by dividing the load with deflection
Stiffness of spring = Load (W) / Deflection (δ)
Do you have suggestions? Please write in comment box.

We will now discuss another topic i.e. thin cylindrical and spherical shells, in the category of strength of material, in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

Also read

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