Theory of Metal Cutting

Tool Geometry for Single Point Tool

Tool Designation (ASA)

\[\alpha_b, - \alpha_s, - \gamma_e, - \gamma_s, - C_e, - C_s, - R\]

$\alpha_b$ : back rake angle

$\alpha_s$ : Side rake angle

$\gamma_e$ : End relief angle

$\gamma_s$ : Side relief angle

$C_e$ : End cutting edge angle

$C_s$ : Side cutting edge angle

R: Nose radius

Tool Designation (Orhtogonal Rake system)

\[i - \alpha - \gamma - \gamma_1 - Ce - \lambda - R\]

i : Inclination angle

$\alpha$ : Side rake

$\gamma$ : Side relief

$\gamma_1$ : End relief

Ce : End cutting edge angle

$\lambda$ : Approach angle

R : Nose radius

Orthogonal Cutting

$t_o$ : Uncut chip thickness

$t_c$ : Produced chip thickness

α : Rake angle

$\phi$ : Shear angle

V : Cutting velocity

Cutting Ratio:

It is defined as uncut chip thickness to produced chip thickness.

\[r = \frac{t_o}{t_c} = \frac{sin \phi}{cos(\phi - \alpha)}\]

\[tan(\phi) = \frac{r cos \alpha}{1 - r sin \alpha}\]

Shear strain in chip:

\[\gamma = cot(\phi) + tan(\phi - \alpha)\]

Velocity relation:

• The velocity of the tool relative to workpiece is called cutting velocity (V).
• The velocity of the chip relative to the work in shear plane is called shear velocity(Vs).
• The velocity of chip on the rake face of tool is called chip velocity(Vc). 

By Triangle Rule,

\[\boxed{\frac{V}{cos(\phi - \alpha)} = \frac{V_s}{cos \alpha} = \frac{V_c}{sin \phi}}\]

Forces in Orthogonal Cutting

Assumptions:

• 2-D cutting process and plane strain process
• $t_o << w$
• Infinite sharp cutting edge
• Continuous chip with no BUE (Built Up Edge)
• Uniform shear and normal stresses along shear plane and tool chip interface.

$F_t$ : Thrust force

$F_c$ : Cutting  force

R : Resultant force

$F_s$ : Shear force (in shear plane)

$N_s$ : Normal force on shear plane

F : Friction force at tool chip interface

N : Normal force at tool chip interface normal to rake face of tool

Average coefficient of friction (μ) is given by:

\[μ = tan( \beta) = \frac{F}{N}\]

Force Circle

You don't need to remember following equations just remember above 'Force Circle' and derive:

\[F = F_c sin \alpha +  F_t cos \alpha \]

\[N = F_c cos \alpha -  F_t sin \alpha \]

\[F_s = F_c cos \phi -  F_t sin \phi  = R cos(\phi + \beta - \alpha)\]

\[N_s= F_c sin \phi +  F_t cos \phi  = R sin (\phi + \beta - \alpha)\]

If τ is the ultimate shear stress of work material and As is shear plane area then shear force can be written as:

\[F_s = τ * A_s = \frac{τ w t_o}{sin \phi}\]

Cutting Force:

\[F_c = R cos(\beta - \alpha) = \frac{F_s cos(\beta - \alpha) }{cos(\phi + \beta - \alpha)} = \frac{τ w t_o cos(\beta - \alpha) }{sin \phi cos(\phi + \beta - \alpha)}\]

Merchant’s theory

Shear angle $\phi$ takes value such that least amount of energy is consumed or minimize work done.

\[ \frac{d F_c}{d \phi} = 0\]

\[\boxed{\phi = \frac{\pi}{4} - \frac{(\beta - \alpha)}{2}}\]

Above relationship is OK for plastics but does not hold well for metals.

Modified Merchant’s Relationship : \( 2 \phi + \beta - \alpha = C\)

C is property of work material but it tends to increase with cold work.

NOTE: Remember that the value of $\phi$ obtained from Merchant's theory is always higher that actual value than the exact value in case of metals, therefore the forces calculated are lower.

Energy Dissipation

 Total power consumption $P_c = F_c * V$

The major plastic deformation take places in shear plane, this is primary heat source (Ps). Frictional energy loss due to sliding motion between chip and tool rake face, called secondary heat source( $P_f = F * V_c$).
\[P_c  \approx  P_s + P_f\]
\[P_s = P_c - P_f = F_c*V - F*V_c\]

Mean Temperature rise of material passing through shear zone:
\[\theta_s = \frac{(1-\Gamma)P_s}{\rho c V w t_o}\]

$\Gamma$ : Fraction of shear zone heat which goes to workpiece

$\rho$: density of material

c : specific heat

Mean Temperature rise of the chip due to frictional zone:
\[\theta_f = \frac{P_f}{\rho c V w t_o}\]

The final temperature is is given $\theta = \theta_o + \theta_s + \theta_f$ where $\theta_o$ is initial temperature of workpiece.

Mechanism of Tool Wear

Loss of material from surface is called wear.

1. Adhesive Wear: Due to high pressure and temperature wear particles transfer from softer surface to harder body.
2. Abrasive Wear: If one of the surface contains very hard particles then these particles during the processes of sliding may dislodge material from the other surface by abrasine action.
3. Fatigue Wear: loss of material from a surface due to fracture under cyclic loading conditions.
4. Diffusion: Atoms in a metallic crystal move from a region of high concentration to that of low concentration.
5. Corrosive Wear: It is due to chemical reactions between the surface and environment (like water, oxygen etc.). 

Types of Tool Wear

• Crater Wear : Wear on the rake face of tool. Its main reason id diffusion along with abrasion.
• Flank Wear : Wear on flank face of tool due to work hardening. The main reason is adhesion and abrasion.

Types of tool wear

Taylor's Tool Life Equation

Tool life is mainly affected by the cutting, higher the speed smaller the life. Relationship between cutting speed and tool life is given by:

\[\boxed{V T^n = C}\]

where C is constant based on tool and work and cutting condition.

Extended Taylor's equation which includes feed (f) and depth of cut (d) is given by:

\[VT^n f^x d^y = C'\]

List of NPTEL courses for ESE/GATE ME

 I am listing all relevant NPTEL courses and standard book's name for reference:

Subject	NPTEL Course Link	Book  Author Name
Strength of Materials	Strength of Materials(WEB)	E.J Hern , Gere & Timoshenko
Thermodynamics	BASIC THERMODYNAMICS(WEB) Video lectures	Moran and Shapiro,   P K Nag
Production Engineering	Manufacturing Process I (WEB) - Forming, casting, Welding Manufacturing Process I (Video) - Forming, casting, Welding Manufacturing Process II (WEB) - Machining	Amitabh Ghosh & Malik,  Kalpkjian Schmid
Material Science	Material Science(WEB)	WD Callister
Industrial Engineering	Industrial Engineering	O P Khanna, Buffa & Sarin
Operations Research	Introduction to Operations Research (Video)	TAHA Hira and Gupta
Fluid Mechanics	Fluid Mechanics (WEB) Turbomachinery (web)	R. K. Bansal, Modi & Seth
Heat Transfer	Heat and Mass Transfer (Video) Video lecture by SK Som	Cengel and Boles, Incropera
Refrigeration & Air Conditioning	Refrigeration & Air Conditioning(WEB) Refrigeration & Air Conditioning (Video)	CP Arora
I C Engines		Sharma Mathur,  V. Ganeshan
Theory of Machines		RS Khurmi , S.S Rattan
Machine Design	Machine Design (WEB)	V.B Bhandari, Shigley

> ALL eBooks of Mechanical Engineering : Download

                              

Springs

Springs are energy absorbing units and used to store energy and to release it slowly or rapidly depending on application.

Closed Coiled Helical Spring under axial pull:

Consider one half turn of helical spring as shown   

Every cross section would be under torsion F*R

thus, maximum stress by torsion theory:

\[\tau_{max} = \frac{16 T}{\pi d^3}\]

\[\boxed{\tau_{max} = \frac{8FD}{\pi d^3}}\]

where D = 2*R = diameter of coil

d = diameter of spring wire 

If one cross section twists angle θ relative to other then:

\[θ = \frac{TL}{GJ} = \frac{FR(\pi R)}{GJ}\]

\[\delta' = Rθ = \frac{\pi F R^3}{GJ}\]

Total deflection \( \delta = 2N* \delta' = \frac{2N F \pi R^3}{GJ} = \frac{8NF D^3}{G d^4}\)

Spring Stiffness(K) :

\[\boxed{K = \frac{F}{\delta} = \frac{G d^4}{8 N D^3}}\]

where, N : number of turns

                  G : Modulus of rigidity 

NOTE: Stiffness is inversely proportional to the number of coils, therefore when a spring cut into half its stiffness becomes double for each part.

Wahl's correction factor( $k_w$ )

The simple torsion theory gives inappropriate results, thus for accurate result multiply by Wahl's factor.

\[\tau_{max} = \frac{k_w 8FD}{\pi d^3}\]

Wahl's correction factor is defined as:

\[k_w = \frac{4C-1}{4C-4} + \frac{0.615}{C}\]

where C is spring index = D/d

  

Springs in series:

When different stiffness springs joined such that shares common load, said to be in series. 

Total deflection \( \delta = \frac{F}{K_{eq}} = \delta_1 + \delta_2 + ..... + \delta_n = \frac{F}{K_1} + \frac{F}{K_2} + .....+ \frac{F}{K_n}\)

\[\boxed{\frac{1}{K_{eq}} =  \frac{1}{K_1} + \frac{1}{K_2} + .....+ \frac{1}{K_n}}\]

Springs in parallel: 

When different stiffness springs joined such that shares common deflection, said to be in parallel. 

\[ \delta = \frac{F}{K_{eq}} = \delta_1 = \delta_2 = ..... = \delta_n = \frac{F_1}{K_1} = \frac{F_2}{K_2} = .....= \frac{F_n}{K_n}\]

Total load \(F = F_1 + F_2 + ..... + F_n\)

\[\boxed{K_{eq} = K_1 + K_2 + ...... + K_3}\]

Spiral Spring

Consider a rectangular cross sectional (breadth B and thickness t) spiral spring is given by following equation:  

\[ r = \frac{b}{2} + \frac{a-b}{4 \pi N}θ \]

Springs will be subjected to uniform bending due to action of central moment which tends to reduce the radius of curvature at all points.

When winding couple M is applied to the spindle, resisting force F will be set up at the pin such that 

M = F * R

Now, Consider two small elements of length dl at distance x each side of center. 

For small deflection change in slope is = $\frac{M dl}{EI}$.

for left side element change in slope,

\[dθ_1 = \frac{F(R+x) dl}{EI}\]

for right side element change in the slope,

\[dθ_2 = \frac{F(R-x) dl}{EI}\]

sum of these two slopes = $dθ_1 + dθ_2 = \frac{2FR dl}{EI}$

Total angle of twist = $\frac{1}{2} \int_{0}^{L} \frac{2FR}{EI}dl = \frac{FRL}{EI} = \frac{ML}{EI}$

Length of spiral $ L = \frac{\pi N}{2}(a+b) $

We get wind up angle of spiral spring is :

\[θ = \frac{M}{EI} \Bigg[ \frac{\pi N}{2} (a+b) \Bigg] \]

Maximum Bending moment = F*a

Maximum bending stress = $\frac{M y}{I}$

\[ \sigma_{max} = \frac{F a(t/2)}{Bt^3/12}\]

\[ \sigma_{max} = \frac{6 F a}{Bt^2}\]

Fluid Kinematics

Fluid kinematics involves velocity, acceleration of the fluid, description and visualisation of fluid motion without considering forces that produce motion. There are two general approaches in analysis fluid mechanics:

1. Eulerian Approach: Fluid motion is given by prescribing the necessary properties (p, V, T etc) in terms of space and time. 
2. Lagrangian Approach: Information of the fluid in terms of what happens at fixed points in space. 

• Steady Flow: A flow is said to be steady if fluid properties do not vary with respect to time.
• Unsteady Flow: Fluid properties vary with time.
• Uniform Flow: if fluid properties do not change point to point at any instant of time.
• Non-Uniform flow: When fluid properties changes from point to point at any instant of time, flow is defined as non-uniform flow.
• Laminar flow: Fluid particles moves in layers or lamina. No mixing in normal direction.
• Turbulent Flow: Fluid particles have random movement, intermixing in layers.
• Incompressible Flow: Density variation is negligible.

Stream Line

An imaginary line or curve such that it is tangent to the velocity field at a given instant, given 

Equation of streamline is given by:

\[\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}\]

for 2D flow : vdx - udy = 0

Path Line

It is the path traced by a single fluid particle at different instant of time. 

Streak Line 

It is defined as locus of various fluid particles that have passed through a fixed point.

NOTE: In steady flow, streakline, pathline and streamline are identical.

Continuity Equation (Conservation of mass)

\[ρAV = Constant\]

where, ρ = Density

A = Area

V = Velocity

For incompressible fluid it reduces to $A_1V_1 = A_2V_2$.

Generalised Continuity Equation

\[\frac{\partial ρ}{\partial t} + \frac{\partial (ρ u)}{\partial x} + \frac{\partial (ρ v)}{\partial y} +\frac{\partial (ρw)}{\partial z} = 0\]

where u,v,w are component of velocity in x, y, z direction.

• If flow is steady $\frac{\partial ρ}{\partial t} = 0$.
• For steady, imcompressible flow continuity equation $\boxed{\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} = 0}$.

Total acceleration of fluid

\[\frac{D\vec{V}}{Dt} = a_x i + a_y j + a_z k = \frac{Du}{Dt}i + \frac{Dv}{Dt}j + \frac{Dw}{Dt}k\]

$\frac{D}{Dt}$ is called Material Derivative or Substantial Derivative or Total Derivative with respect to time.

• In steady flow, local acceleration will be zero.
• In uniform flow, convective acceleration will be zero.

Rotational Compenent ($\omega$)

\[ \omega =  \frac{1}{2} \nabla \times \vec{V} = \frac{1}{2}
\begin{vmatrix}
 i & j & k \\
 \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}
\\ u & v & w
\end{vmatrix} \]

\[\omega_z = \frac{1}{2}(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})\]

• Vorticity ($\Omega$) is as the vector that is twice the rotational vector.

\[\Omega = 2 \omega = \nabla \times \vec{V}\]

• Flow said to be irrotational if vorticity or rotational vector is zero at all points in region.
• Circulation ($\Gamma$) is line integral of tangential component of velocity around a closed curve.

$\Gamma$ = Vorticity X Area 

Velocity Potential Function ($\phi$)

Velocity potential function exist only for irrotational flow i.e. the existence of velocity potential function implies the flow is irrorational.

\(u = \frac{\partial \phi}{\partial x}\),    \(v = \frac{\partial \phi}{\partial y}\),     \(w = \frac{\partial \phi}{\partial z}\)

Stream Line Function ($\psi$)

     \(u = \frac{\partial \psi}{\partial y}\),       \(v = -\frac{\partial \psi}{\partial x}\)

• Volume flow rate between $\psi_1$ and $\psi_2$ is equal to $\psi_2 - \psi_1$.
• Velocity potential function can be defined for 3D flow but stream line function is defined only for 2D.

Along the stream line $d\psi = 0$,

\[d\psi = \frac{\partial \psi}{\partial x}dx + \frac{\partial \psi}{\partial y}dy = 0\]

\[-vdx + udy = 0 \]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\psi = const.} = \frac{v}{u}\]

Along Equipotential line $\phi(x,y)$ is constant,

\[d\phi = \frac{\partial \phi}{\partial x}dx + \frac{\partial \phi}{\partial y}dy = 0\]

\[udx + vdy = 0 \]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\phi = const.} = - \frac{u}{v}\]

\[\Bigg( \frac{dy}{dx}\Bigg)_{\psi = const.} *\Bigg( \frac{dy}{dx}\Bigg)_{\phi = const.} = -1\]

Hence, the streamlines and equipotential lines are orthogonal to each other except stagnation point.

PERT And CPM

PERT and CPM techniques are used for planning and scheduling large projects.

PERT :  Program Evaluation and Review Technique.
CPM: Critical Path Method.

Terminology:

Network: It consist of series of activities arranged in a logical sequence and shows inter relationship between them.

Activity: It is time consuming effort which is required to perform part of work. It is represented by an arrow ($\to$).

Event: It is the start or completion point of an activity. It is represented by a circle.

Predecessor and Successor Activities: Activities that must be completed before another activity can be started called its predecessor activities. Successor activity occurs following predecessor activities.

Dummy Activity: An activity that consumes no time but shows precedence or relationship among activities. It is represented by dotted line in network.

Each event has two important times associated with it:

Earliest Time: Time when an event can occur when all predecessor activities finished at the earliest possible time.

Latest Time: It is the latest time when each activity can start without delaying the total project.

Forward Pass Computation/ Earliest Start Time

Earliest Start Time (EST): It is the earliest start time of an activity when all predecessor activities are started at their earliest time. 

Earliest Finish Time (EFT): if activity time is 't' then EFT = EST + t

Backward Pass Computation / Latest Time

Latest Finish Time (LFT): The latest finish time for an activity without delaying the project.

Latest Start Time (LST)  = LFT - t

Example:

Example 

Forward Pass:

Starts from initial event to move to end event.

$EST_1$ = 0

$EST_2$ = $EST_1$ + t = 0 + 12 = 12

$EST_3$ = $EST_2$ + 8 = 20

$EST_4$ = $EST_2$ + 10 = 22

$EST_5$ = max{ ($EST_3$+14) , ($EST_4$ + 8) } = max{34, 30} = 34

$EST_6$ = $EST_5$ + 4  = 38

Backward Pass:

Starts from end event to come to first event.

$LST_6$ = 38

$LST_5$ = $LST_6$ - 4 = 34

$LST_4$ = $LST_5$ - 8 = 26

$LST_3$ = $LST_5$ - 14 = 20

$LST_2$ = min{ ($LST_3$ - 8), ($LST_4$ - 10)} = min{12, 16} = 12

$LST_1$ = $LST_2$ - 12 = 0

Slack: Difference between the latest time and earliest time of an event is called slack. An activity with zero slack is known as critical activity.

Critical Path : Critical path is the on the network of project activities which takes longest time from start to finish.
1-2-3-5-6

Total Float: It is the time which completion of an activity can be delayed beyond earliest expected completion time without affecting total project time. 

Total float = Latest start time - Earliest start time

Free Float: It is the time which the completion an activity can be delayed without delaying the earliest start of any succeeding activity.

Free Float = Total Float - Head Event Slack

Independent Float: It is the time which the start of an activity can be delayed without affecting the earliest start time of any immediately following activities assuming that the preceding activity has finished its LFT.

Independent Float = Free Float - Tail Event Slack

PERT :  Program Evaluation and Review Technique

It is used when activity times are not known with certainty. 

The fundamental assumption in PERT is that the three estimates of time ($t_m, t_o, t_p) follow β - distribution curve.

Optimistic Time ($t_o$): It is an estimate of minimum possible time to complete the activity under ideal condition. 

Most Likely time ($t_m$): This lies between optimistic and pessimistic time estimates.

Pessimistic Time ($t_p$): It is the longest time taking into consideration odds.

 Expected time (mean) :

\[t_e = \frac{t_o + 4t_m + t_p}{6}\]

Standard deviation        \( σ = \frac{t_p - t_o}{6}\)

Variance                        \( σ^2 = (\frac{t_p - t_o}{6})^2\)

• Expected time of project ($T_{e}$) is the sum of the expected time of all activities lies on critical path.
• Expected time of activity is assumed to be β - distribution and  expected time for project is normally distributed.
• Variance of the expected time of project ($σ_{cp}^2$) is the sum of the variance of the expected time of all activities along the critical path.
• Probability that the project will be completed  in a given time (T)

Normal distribution $ Z = \frac{T - T_e}{ σ_{cp}}$

Probability P = φ(Z).

For a normal deviate of Z = +1, the corresponding probability is 84.1% , for Z = –1 corresponding P = 15.9 % and for Z = 0 corresponding P = 50%.

Difference Between CPM and PERT:

CPM	PERT
It is activity oriented	It is event oriented
Activity time is deterministic	Activity time is probabilistic
One time estimate	Three time estimates

Pressure Vessels

When the thickness of the wall of the shell in less than $\frac{1}{10}$ to $\frac{1}{15}$ of its diameter, then shell is called thin shell.
\[t < \frac{D}{10} \quad to \quad \frac{D}{15}\]
When the thickness of the wall of the shell in greater than $\frac{1}{10}$ to $\frac{1}{15}$ of its diameter, then shell is called thick shell.
\[t > \frac{D}{10} \quad to \quad \frac{D}{15}\]

Thin Cylinders

Say L length, diameter d and t thickness of cylinder is subjected to internal pressure P. Due to this pressure, three type of stresses are developed at any point on the wall of cylinder -

1. Hoop Stress / Circumferential stress ($\sigma_h$) will be tensile in nature.
2. Longitudinal stress / Axial stress ($\sigma_L$) will be tensile in nature.
3. Radial stress ($\sigma_R$) will be compressive in nature. 

Analysis of thin cylinder

• It is assumed that stresses are uniformly distributed through the thickness of the wall.
• Radial stress varies from P at inner surface to atmospheric pressure at the outside of surface.
• If the internal pressure is very low, radial stress is negligible compared to axial and hoop stress. 

1. Hoop or circumferential stress ($\sigma_h$):

This stress is directed along to the tangent to the circumference of the cylinder. It resist the bursting effect due to internal pressure. 

At the equilibrium,

$P*(dL) = σ_h*(2tL)$

Hoop stress,                                       $\boxed{σ_h = \frac{Pd}{2t}}$

2. Longitudinal Stress ($\sigma_L$):

This stress is directed along the length of the cylinder and it tends to increase the length. 

At the equilibrium,
\(P*(\frac{\pi d^2}{4}) = σ_L*(\pi dt)\)

Longitudinal stress                  \(\boxed{σ_L = \frac{Pd}{4t}}\)

3. Longitudinal strain

\[ε_L = \frac{σ_L}{E} - μ\frac{σ_h}{E}\]

\[\boxed{ε_L = \frac{Pd}{4tE}(1 - 2μ)}\]

4. Hoop Strain

\[ε_h = \frac{σ_h}{E} - μ\frac{σ_L}{E}\]

\[\boxed{ε_h = \frac{Pd}{4tE}(2 - μ)}\]

5. Volumetric Strain

Volumetric strain = Longitudinal strain + 2*hoop strain
\[\boxed{ε_v = \frac{Pd}{4tE}(5 - 4μ)}\]

Thin Sphere

Hoop Stress/ Longitudinal Stress  

\[σ_h = σ_L = \frac{Pd}{4t}\]

Hoop Strain/ Longitudinal Strain

\[ε_h = ε_L = \frac{Pd}{4tE}(1 - μ)\]

Volumetric Strain

                                            \(ε_v = 3*ε_L = \frac{3Pd}{4tE}(1 - μ)\)

Thick Cylinders

• Radial stress in thin cylinder is neglected but it is of significant magnitude in case of thick cylinders.
• Tangential stresses assumed uniformly distributed over the wall in thin cylinder while it changes gradually from inner surface to outer surface in case of thick cylinders.  

• Axial Stress \( \sigma_z = \frac{P_i r_i^2}{r_o^2 - r_i^2}\)

• Radial Stress \(\sigma_r = A - \frac{B}{r^2}\)

• Circumferential or hoop stress   \(\sigma_h = A + \frac{B}{r^2}\)

A and B are constant which can be determine by boundary conditions:

$\sigma_r = - P_i$  at $r = r_i$

$\sigma_r = - P_o$  at $r = r_o$

Columns: Buckling Failure

STRUT: A structural member which carries an axial compressive load.

COLUMN: A vertical strut is known as column.

A long column becomes unstable when its axial compressive load reaches a limit called critical buckling load. Its lateral deflection called buckling.

Load carrying capacity of columns in depend upon -

• Material
• End connections
• dimension or slenderness ratio

Euler's Theory

Assumptions:

• Column is perfectly straight and uniform cross section.
• Applied compressive load is perfectly axially.
• Stresses are within elastic limit.
• The material is homogenous and isotropic.

Maximum allowable buckling load or Euler's critical load is given by:

\[\boxed{P_e = \frac{{\pi}^2EI_{min}}{L_e^2}}\]

where, $I_{min}$ = Moment of inertia about centroids axis

Le = Effective length                            

NOTE: This formula doesn't take account the axial stress. It is applicable for long columns where effect of crushing is neglected. 

Slenderness ratio (S)

Slenderness ratio is defined as the ratio of its effective length to least radius of gyration. 

\[S = \frac{L_e}{k}\]

\[ K = \sqrt{\frac{I_{min}}{A}}\]

\[Buckling \quad Stress \quad \sigma_b = \frac{P_e}{A} = \frac{{\pi}^2EI_{min}}{A L_e^2} = \frac{{\pi}^2E}{S^2}\]

Rankine's Theory

\[\frac{1}{P_R} = \frac{1}{P_C} + \frac{1}{P_e}\]

$P_R$ = Rankine Load or Crippling load

$P_C$ = crushing load = $\sigma_C$

$P_e$ = Buckling Load

\[P_R = \frac{\sigma_c A}{1 + K'(\frac{L_e}{k})^2}\]

K' = Rankkine's constant = $\frac{\sigma_c}{\pi^2E}$

• Effect of crushing and buckling considered in this formula.
• This formula is applicable to any column.

Theories of Failure

1. Maximum Principal Stress theory (RANKINE’S THEORY)
2. Maximum Shear Stress theory (GUEST AND TRESCA’S THEORY)
3. Maximum Principal Strain theory (St. VENANT’S THEORY)
4. Total Strain Energy theory (HAIGH’S THEORY)
5. Maximum Distortion Energy theory (VONMISES AND HENCKY’S THEORY)

Maximum Principal Stress Theory (MPST)

According to MPST, failure occurs when the value of maximum principal stress is equal to that of yield point stress. 

Condition for failure is,

Maximum principal stress ($\sigma$) > failure stresses (Syt)

Condition for safe design,

Maximum principal stress  ≤  Permissible stress 

where, permissible stress = failure stress / Factor of Safety =\( \frac{Syt}{N}\)

\[\boxed{\sigma ≤  \frac{Syt}{N}}\]

NOTE:

• This theory is suitable for brittle materials under all loading conditions (bi axial, tri axial etc.) because brittle materials are weak in tension.
• This theory is not suitable for ductile materials because ductile materials are weak in shear.
• This theory can be suitable for ductile materials when state of stress condition such that maximum shear stress is less than or equal to maximum principal stress i.e. 

1. Uniaxial state of stress( $τ_{max} = \frac{\sigma}{2}$)
2. Biaxial loading when principal stresses are like in nature. ( $τ_{max} = \frac{\sigma}{2}$)
3. Under hydrostatic stress condition (shear stress in all the planes is zero).

Maximum Shear Stress Theory (MSST)

According to this theory, failure occurs when maximum shear stress at any point reaches the yield strength. 

Condition for safe design,

\(\boxed{\tau_{max} ≤  \frac{Sys}{N} = \frac{Syt}{2N}}\)

For tri-axial state of stress,

Max{\(|\frac{σ_1 - σ_2}{2}|, |\frac{σ_2 - σ_3}{2}|, |\frac{σ_3 - σ_1}{2}|\)} ≤  $\frac{Syt}{2N}$

For bi-axial state of stress,

Max{\(|\frac{σ_1 - σ_2}{2}|, |\frac{σ_2}{2}|, |\frac{σ_1}{2}|\)} ≤  $\frac{Syt}{2N}$

NOTE:

• This theory is well suitable for ductile materials.
• MSST and MPST will give same results for ductile materials under uniaxial state of stress and biaxial state of stress when principal stresses are like in nature.
• MSST is not suitable for hydrostatic loading.

Maximum Principal Strain theory (M P St T)

According to this theory, failure occurs when maximum principal strain reaches strain at which yielding occurs in simple tension.

Condition for safe design,

 \(\boxed{ε_1 ≤  \frac{Syt}{EN}}\)

\(\frac{1}{E}[σ_1 - μ(σ_2 + σ_3)] ≤  \frac{Syt}{EN} \)

for biaxial state of stress, $σ_3$ = 0

\(σ_1 - μ(σ_2) ≤  \frac{Syt}{N} \)

Total Strain Energy theory (T St E T)

According to this theory, failure occurs when total strain energy per volume is equal to strain energy per volume at yield point in simple tension.

Condition for safe design,

 Total Strain Energy per unit volume  ≤  Strain energy per unit volume at yield point under tension test.

 Total Strain Energy per unit volume = $\frac{1}{2}σ_1ε_1$ + $\frac{1}{2}σ_2ε_2$ + $\frac{1}{2}σ_3ε_3$ 

$ε_1 = \frac{1}{E}[σ_1 - μ(σ_2 + σ_3)]$

$ε_2 = \frac{1}{E}[σ_2 - μ(σ_1 + σ_3)]$

$ε_3 = \frac{1}{E}[σ_3 - μ(σ_2 + σ_1)]$

we get,

\[\frac{TSE}{Vol} = \frac{1}{2E}[σ_1^2 + σ_2^2 + σ_3^2 - 2μ(σ_1σ_2 + σ_2σ_3 + σ_3σ_1)]\]

\[\frac{TSE}{Vol}\Bigg]_{Y.P.} = \frac{1}{2E}(\frac{Syt}{N})^2 \]

\[[σ_1^2 + σ_2^2 + σ_3^2 - 2μ(σ_1σ_2 + σ_2σ_3 + σ_3σ_1)] ≤ (\frac{Syt}{N})^2 \]

for bi axial case $σ_3 = 0$,

\(σ_1^2 + σ_2^2 - 2μσ_1σ_2  ≤ (\frac{Syt}{N})^2 \)

Above Equation is an equation of ellipse whose semi major axis is $\frac{Syt}{\sqrt{1-μ}}$ and minor axis is $\frac{Syt}{\sqrt{1+μ}}$

NOTE: This theory is suitable for hydrostatic stress condition.

Maximum Distortion Energy Theory (M D E T)

According to this theory, failure occurs when strain energy of distortion per volume is equal to strain energy of distortion per unit volume at yield point in simple tension.

Total strain energy/Vol = Volumetric strain energy/vol + distortion energy / volume

Volumetric Strain Eenrgy /vol = $\frac{1}{2}$ (average stress)(Volumetric strain)

\[Vol SE/Vol = \frac{1}{2}\frac{σ_1 + σ_2+ σ_3}{3}[\frac{1-2μ}{E}(σ_1 + σ_2+ σ_3)] = \frac{1-2μ}{6E}(σ_1 + σ_2+ σ_3)^2 \]

DE/vol = TSE/vol - Vol SE/vol

\[\boxed{DE/vol =  \frac{1 + μ}{6E}[(σ_1 - σ_2)^2 + (σ_2 - σ_3)^2 + (σ_3 - σ_1)^2]}\]

\[DE/vol]_{YP} =  \frac{1 + μ}{6E}[2(\frac{Syt}{N})^2]\]

Condition for safe design,

$DE/vol  ≤  DE/vol]_{YP}$

\[[(σ_1 - σ_2)^2 + (σ_2 - σ_3)^2 + (σ_3 - σ_1)^2] ≤  2(\frac{Syt}{N})^2\]

for bi axial case $σ_3 = 0$,

\(σ_1^2 + σ_2^2 - σ_1σ_2  ≤ (\frac{Syt}{N})^2 \)

This Equation is an equation of ellipse whose semi major axis is $\sqrt{2}Syt$ and minor axis is $\sqrt{2/3}Syt$

NOTE:

• This theory is best for ductile materials.
• It can not be applied materials under hydrostatic stress condition. 

Comparison among the different failure theories

Comparison of different failure theories

Deflection Of Beams

The deflection is measured from original neutral surface of the beam to neutral surface of deformed beam. The configuration of neutral surface of deformed beam is known as elastic curve of beam.

Differential Equation of Elastic Curve for the loaded beam

Assumptions:

• Stress is proportional to strain i.e. hooks law applies.
• Small deflection
• pure bending or Any deflection resulting from the shear deformation of the material or shear stresses is neglected.

from analytic geometry, curvature of line is given by,

\[k = \frac{1}{R} = \frac{\frac{d^2y}{dx^2}}{\Big[1 + \frac{d^2y}{dx^2}\Big]^{3/2}} \approx \frac{d^2y}{dx^2}\]

from simple bending theory equation,

\[\frac{σ}{y} = \frac{M}{I} = \frac{E}{R} \quad or \quad \frac{1}{R} = \frac{M}{EI}\]

So basic differential equation governing deflection of beam is

\[\boxed{M = EI \frac{d^2y}{dx^2}}\]

since shear force is \(\frac{dM}{dx}\) thus,

\[\boxed{Slope = \frac{dy}{dx}}\]

\[\boxed{Bending \quad Moment = EI\frac{d^2y}{dx^2}}\]

\[\boxed{Shear \quad Force = EI\frac{d^3y}{dx^3}}\]

\[\boxed{Load \quad Distribution = EI\frac{d^4y}{dx^4}}\]

Deflection for Common Loadings

1. Moment load at the free end of cantilever beam

• Maximum Bending Moment       M = -M   
• Slop at end     \(θ = \frac{ML}{EI}\)
• Maximum deflection (at end)     \(δ = \frac{ML^2}{2EI}\)
• Deflection Equation    \(EI y = -\frac{Mx^2}{2}\)

2. Concentrated load at the free end of cantilever beam

• Maximum Bending Moment       M = -PL   
• Slop at end     \(θ = \frac{PL^2}{2EI}\)
• Maximum deflection (at end)     \(δ = \frac{PL^3}{3EI}\)
• Deflection Equation    \(EI y = -\frac{Px^2}{6}(3L-x)\)

3. Cantilever Beam subjected to a Uniformly distributed load

• Maximum Bending Moment       \(M = -\frac{qL^2}{2}\)   
• Slop at end     \(θ = \frac{qL^3}{6EI}\)
• Maximum deflection (at end)     \(δ = \frac{qL^4}{8EI}\)
• Deflection Equation    \(EI y = -\frac{qx^2}{120L}(6L^2 -4Lx + x^2)\)

4. Cantilever Beam subjected to a Uniformly Triangular distributed load

• Maximum Bending Moment       \(M = -\frac{q_oL^2}{6}\)   
• Slop at end     \(θ = \frac{q_oL^3}{24EI}\)
• Maximum deflection (at end)     \(δ = \frac{q_oL^4}{30EI}\)
• Deflection Equation    \(EI y = -\frac{q_ox^2}{120L}(10L^3 -10L^2x + 5Lx^2 - x^3)\)

5. Concentrated load at the mid-span of simply supported beam

• Maximum Bending Moment       \(M = -\frac{PL}{4}\)   
• Slop at end     \(θ_A = θ_B = θ = \frac{PL^32}{16EI}\)
• Maximum deflection (at midspan)     \(δ = \frac{PL^3}{48EI}\)
• Deflection Equation (for 0 < x < L/2) \(EI y = -\frac{Px}{48}(3L^2 - 4x^2)\)

6. Uniformly distributed load of simply supported beam

• Maximum Bending Moment       \(M = -\frac{qL^2}{8}\)   
• Slop at end     \(θ_A = θ_B = θ = \frac{qL^3}{24EI}\)
• Maximum deflection (at mid span)     \(δ = \frac{5qL^4}{384EI}\)
• Deflection Equation    \(EI y = -\frac{qx}{24}(L^3 -2Lx^2 + x^3)\)

Strain Energy Method (Castigliano’s Theorem)

Castigliano’s Theorem states that the partial derivative of the strain energy with respect to an applied force is equal to the displacement of the force along its line of action.

\[\boxed{δ = \frac{\partial U}{\partial P}}\]

The strain energy is given by

\[U = \int\limits_{0}^{L} \frac{M^2}{2EI} dx\]

\[\boxed{δ = \int\limits_{0}^{L} \frac{\partial M}{\partial P}\frac{M}{EI} dx}\]

Example : Take a cantilever beam subjected to concentrated load P at free end

M = P(L - x)

\(δ = \int\limits_{0}^{L} \frac{\partial M}{\partial P}\frac{M}{EI} dx\)

\(δ = \int\limits_{0}^{L} (L - x)\frac{P(L-x)}{EI} dx\)

\(δ = \frac{PL^3}{3EI}\)

Heat Exchangers

A heat exchanger is an adiabatic steady flow device in which two or more flowing stream of fluids exchange heat between themselves due to temperature difference without loosing or gaining any heat from the surroundings.

Classification of heat Exchangers

Classification of Heat Exchangers

1. Direct transfer type heat exchanger : Both fluids could not come into contact with each other but exchnage heat through the pipe wall of separation. 

Examples:

• Economiser
• Air preheater
• Concentric type heat exchanger
• Pipe in pipe heat exchnager
• Super heaters

1. Parallel flow heat exchanger : Both fluids flow in same direction. 
   
2. Counter flow heat exchanger : Both fluids flow in opposite direction. 
   
3. Cross flow heat exchanger : both fluids flow in perpendicular direction with respect to each other. example : Automobile radiator  
   

2. Direct Contact type heat exchanger : Both hot and cold fluids mixed up with each other in order to exchange heat between themselves. These type of heat exchnagers are used when the mixing of fluids is harmless or desirable.

Examples:

• Cooling tower
• Jet Condonser

3. Regenerative type heat exchanger : Both hot and cold fluids alternatively pass through the heat exchanger i.e. high thermal capacity cellulose matraix, one heating it and other picking up from it.

Example : Ljungstorm air preheater use in gas turbine power plants.

Heat Exchanger Analysis

Let m =  Mass flow rate, kg/s

      Cp = Specific heat of fluid at constant pressure,  J/Kg-K

      Ti = inlet temperature 

      Te = exit temperature

      h in subscript represents hot fluid

      c in subscript represents cold fluid

Assuming that there is no heat loss to the surroundings and changes in potential and kinetic energy are negligible, from energy balance

 Rate of enthalpy decrease of hot fluid = Rate of enthalpy increase of cold fluid

   - (ΔH )hot fluid =(ΔH)cold fluid

\[\boxed{m_h C_{ph} (T_{hi} - T_{he}) = m_c C_{pc} (T_{ce} - T_{ci})} \]

Overall heat transfer coefficient (U)

\[\frac{1}{U} = \frac{1}{h_1} + \frac{1}{h_2} + F_1 + F_2 \]

Fouling factor (F) : It takes into account the thermal resistance offered by any scale or deposite formed on the either side of wall due to the chemical reaction between the flowing fluid and pipe material. Its unit is $m^2 K / watt$.

Temperature profile of fluids in heat exchanger

Variation of temperature when one of the fluids condense or boils:

Log Mean Temperature Difference (LMTD) Method

LMTD is defined as

\[ΔT_m = \frac{ΔT_i - ΔT_e}{ln(\frac{ΔT_i}{ΔT_e})}\]

• For Parallel flow HE

              \(ΔT_i = T_{hi} - T_{ci}\)

              \(ΔT_e = T_{he} - T_{ce}\)

• For Counter flow HE

              \(ΔT_i = T_{hi} - T_{ce}\)

              \(ΔT_e = T_{he} - T_{ci}\)

Heat transfer rate between hot and cold fluid $= Q = m_h C_{ph} (T_{hi} - T_{he}) = m_c C_{pc} (T_{ce} - T{ci})$

Area of the heat exchanger $ A = \frac{Q}{U ΔT_m}$

Note:

• For the same hot and cold fluids and for the same mass flow rate of both the fluids and for specified inlet and outlet temperatures, LMTD value for a counter-flow heat exchanger is always greater than that for a parallel-flow heat exchanger i.e. for the same heat transfer rate required the area of counter flow heat exchanger shall be lesser than parallel flow heat exchanger.
• When both the fluids have equal capacity rates (i.e. $m_h C_{ph} = m_c C_{pc}$) in counter flow heat exchanger then the temperature difference between the hot and the cold fluids will remain constant along the heat exchanger and that is equal to LMTD.

Effectiveness of heat exchanger ε

It is defined as the ratio of actual heat transfer rate between hot and cold fluid to the maximum possible heat transfer rate. 

\[ε = \frac{Q_{actual}}{Q_{max}}\]

where,

$Q_{actual} = m_h C_{ph} (T_{hi} - T_{he}) = m_c C_{pc} (T_{ce} - T{ci})$

$Q_{max} = (mC_p)_{small} * (T_{hi} - T_{ci})$

$(mC_p)_{small}$ is the smaller capacity rate between $m_h C_{ph}$ and $m_c C_{cp}$.

NTU Method

Number of Transfer Units (NTU) is defined as 

\[NTU = \frac{UA}{(mC_p)_{small}}\]

Since NTU is proportional to area of HE, it signifies the overall size of heat exchanger.

Capacity Rate Ratio (C)

It is defined as the ratio of smaller capacity rate to bigger capacity rate. 

\[C = \frac{(mC_p)_{small}}{(mC_p)_{Big}}\]

Note: C = 0 if one of fluid changes its phase.

ε = f(NTU, C)

• For parallel flow heat exchanger,

\[ε_{parallel} = \frac{1 - e^{- (1+C) NTU}}{1 + C}\]

• For Counter flow heat exchnager

\[ε_{counter} = \frac{1 - e^{-(1-C) NTU}}{1 - Ce^{-(1-C) NTU}}\]

If C = 0 then $ε_{parallel} = ε_{counter} = 1 - e^{-NTU}$

Note:

• When all the inlet and outlet temperatures are specified, the size of the heat exchanger can easily be determined using the LMTD method.
• NTU method is used when exit temperatures are not known.