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Rakesh Singh Rajput et al. / International Journal of Engineering Science and Technology

Vol. 2(12), 2010, 7797-7811

NON-AXISYMMETRIC DYNAMIC

RESPONSE OF IMPERFECTLY

BONDED BURIED ORTHOTROPIC

THIN EMPTY CYLINDRICAL SHELLDUE TO INCIDENT SHEAR WAVE (SV

WAVE)

RAKESH SINGH RAJPUT*

Reader, Mechanical Engineering, Directorate of Technical Education,

Bhopal, M. P., India

[email protected]

SUNIL KUMARProfessor, Mechanical Engineering, Rajeev Gandhi Technical University

Bhopal, M. P., India

[email protected]

ALOK CHAUBEY

Professor, Mechanical Engineering, Rajeev Gandhi Technical University

Bhopal, M. P., [email protected]

J. P. DWIVEDI

Professor, Mechanical Engineering, IT-BHU

Varanasi, U. P., India

Abstract

This paper is deals with the non-axisymmetric dynamic response of imperfectly bonded buried orthotropic thin

empty pipelines subjected to incident shear wave (SV-wave). In the thin shell theory the effect of sheardeformation and rotary inertia is not considered. The pipeline has been modeled as an infinite thin cylindrical

shell imperfectly bonded to surrounding. A thin layer is assumed between the shell and the surrounding medium

(soil) such that this layer possesses the properties of stiffness and damping both. The degree of imperfection of

the bond is varied by changing the stiffness and the damping parameters of this layer. The non axisymmetric

results are comparing with axisymmetric.

Key words: - Shear wave, Orthotropic, Imperfect bond, Seismic Wave, Non-Axisymmetric, DynamicResponse, Buried Pipelines, and Thin Shell.

1. IntroductionDuring past few years a number of studies like Cole Ritter and Jordon (1979) and Singh et al (1987) on the

axisymmetric dynamic response of buried orthotropic pipe/shells have been reported. Later Chonan (1981);Dwivedi and Upadhyay (1989; 1990; 1991); and Dwivedi et al (1991) have analyzed the axisymmetric problems

of imperfectly bonded shell for the pipesmade of orthotropic materials.

*Corresponding Author

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Upadhyay and Mishra (1988) have presented a good account of work on non-axisymmetric response of buried

thick orthotropic pipelines under seismic excitation. Again Dwivedi et al (1992a; 1992b); Dwivedi et al(1993a; 1993b; 1996); and Dwivedi et al (1998) have analyzed the non-axisymmetric problems of imperfectly

bonded buried thick orthotropic cylindrical shells. Kauretzis et al (2007) have presented analytical calculations

of blast induced strains on buried pipe lines. Hasheninajad and Kazemirad (2008) have reported dynamicresponse of eccentric tunnel in poro-elastic soil under sesmic excitation. Lee et al (2009) have done the risk

analysis of buried pipelines using probabilistic method. But in all these analyses pipelines have been modeled

as thick shell. As far as the non-axisymmetric dynamic response of thin shell is concerned, no work has beenreported so far. There is no work available discussing the effect of bond imperfection on the non axisymmetric

response of buried thin pipes made of orthotropic materials. Therefore, in present paper, the effect of imperfect

bond on the non- axisymmetric dynamic response of buried orthotropic thin pipelines has been analyzed.

2. Basic Equations and FormulationThe cylindrical pipeline has been modeled as an infinitely long cylindrical shell of mean radius R and

thickness h. It is considered to be buried in a linearly elastic, homogeneous and isotropic medium of infinite

extent. Basic approach of the formulation is to obtain the mid plane displacements of the shell by solving theequations of motion of the orthotropic shell. Traction terms in the equations of motion are obtained by solving

the three-dimensional wave equation in the surrounding medium. Appropriate boundary conditions are applied

at the shell surfaces. Equations arising out of boundary conditions along with the equations of motion of the

shell are simplified to yield a response equation in matrix form. Equation governing the non axis-symmetric

motion of an infinitely long orthotropic cylinder has been derived following the approach of Herrmann andMirsky (1957).

The cylindrical shell buried in a linearly elastic, homogeneous and isotropic medium of infinite medium, a thin

layer is assumed between the shell and the surrounding medium (soil). The degree of imperfection of the bond is

varied by changing the stiffness and the damping parameters of this layer. The shell is excited by a shear

vertical wave (SV- wave). A wavelength (=2/) is considered which strikes the shell at an angle withthe axis of the shell. Let a cylindrical polar co-ordinate system (r,, x) which is defined in such a way that xcoincides with the axis of the shell and, in addition, z is measured normal to the shell middle surface, which is

given as

h/2zh/2-, Rrz (1)The equation of motion of shell in the matrix form is given as Rajput et.al (2010)

[{L} {U}] + {P*} = 0 (2)

Where [L] is 3 3 a matrix and {U} = [w v u] T

With w, v and u as the displacement components of the middle surface of the shell in the radial, tangential andaxial directions respectively, the elements of {P*} are given by Herrman and Mirsky (1957) as:

,1P,1

h/2

h/2-

*

2

2/

_h/2

*

1 zh

zz

R

z

R

zP

,1P,1

h/2

h/2-

*

4

2/

2/

*

3 zx

h

hzR

z

R

zzP

2/

2/

*

5 1h

hzxR

zzP

For the evaluation of {P*}, ij at z = (h/2) must be determined in the terms of incident and scattered field in thesurrounding ground. The total displacement field in the ground is written as

d=d(i) + d(s)

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Where i and s represents the incident and scattered parts respectively. By solving the wave equation in the

surrounding infinite medium the components of incident and scattered fields can be written as (Chonan, S.,1981):

)(expcos

1

5

3

'

11

'

)( ctxin

BR

r

Ir

R

n

BR

rIB

R

rI

d

n

nn

i

r

)(expsin

)(

5

'

311

)( ctxin

BR

rI

BR

rI

r

RinB

R

rI

r

Rrn

d

n

nn

i

)(expcos3211)( ctxinxBR

rIB

R

rIid

nn

i

x

(3)

Where B1 ='1B /R, B3 =

'3B /R

2 and B5 ='5B /R. (

) denotes differentiation with respect to the argument of

the Bessel functions. The constants B1, B3 and B5 depend on the parameters of the incident wave and may be

expressed as:

3n52

2n3

1

11n1

A)1(B,

Ai)1(B,

Ai)1(B (4)

)ctx(iexpncosBR

rK

r

RnB

R

rKiB

R

rKd 6n4

'n12

'n

)s(r

)ctx(iexpnsinB

R

rK

BR

rK

r

RinB

R

rK

r

Rn

d

6'n

4n12n

)s(

)ctx(iexpncos

BR

rKBR

rKid 4n

22n1)s(

x

(5)

Where dr, d, dx are components of displacement vector,

321 ;; AAA

are amplitudes of P, SV and SH wave

respectively.

Here ./Band/,/ '662'

44

'

22 RBRBBRBB are constants.Stress fields due to the incident wave can be obtained by plugging above equations into the stress-displacement

relations of the medium, and are given by:

)ctx(iexpncos

BR

rI

r

R

R

rI

r

Rn2

BR

rIi2

BR

rI2

R

rI2

R

5n'n

3

"

n

2

1

1"n

2n

22

21

)i(

rr

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)(expsin2

2

5

2

'"2

3

'

1

1

'

)(ctxin

B

R

rI

r

nR

R

rI

r

R

R

rI

BR

rI

r

R

R

rI

r

Rin

BR

rI

R

rI

r

R

r

Rn

R

nnn

nn

nn

i

r

)(expcos

22

51

'2

2

2

11

'

1

)(ctxin

BR

rI

r

Rin

R

rIB

R

rIi

Rn

nn

i

rx

)(expcos

2

2

22

6

'

4

"2

1

2

"22

2

2

1

)(ctxin

BR

rK

r

R

R

rK

r

Rn

BR

rKi

BR

rK

R

rK

R

nn

n

nn

s

r

)(expsin2

2

6

2

'"2

4

'

1

2

'

)(ctxin

BR

rK

r

nR

R

rK

r

R

R

rK

BR

rK

r

R

R

rK

r

Rin

BR

rK

R

rK

r

R

r

Rn

R

nnn

nn

nn

s

r

)(expcos

22

61

4

'2

2

2

12

'

1

)(ctxin

BR

rK

r

Rin

BR

rKB

R

rKi

Rn

nn

s

rx

(6)

WhereIn ( ) Modified Bessel function of first kind

Jn ( ) Bessel function of first kind

Kn ( ) Modified Bessel function of second kindWith the help of above equations the stresses at the outer surface of the shell (z = h/2 or r = R + h/2) can be

obtained. Thus {P*} in Eq. (2) can be determined.

Now the mid plane displacement and slopes are assumed to be of the form;

0ww cosn exp[i(x-ct)]

0vv sinn exp[i(x-ct)]

0uu cosnexp[i(x-ct)] (7)

Plugging Eq. (7) in Eq. (2) and (6) along with the expression for {P*}, a set of six simultaneous algebraicequations are obtained.Three more equations are obtained by imposing the boundary conditions at the inner and

outer surfaces of the shell: i.e.,

2/

)()()( hRr

s

r

i

r ddw

2/

)()()()2/( hRr

siddhv

2/

)()( )()2/(hRr

s

x

i

xx ddhu (8)Boundary conditions at the outer surface of the shell (r = R + h/2) are obtained by assuming that the shell and

the continuum are joined together by a bond which is thin, elastic and inertia less. This implies that the stress at

the shell-soil interface is continuous. To take the elasticity of the bond into account, the stresses in the bond are

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assumed proportional to relative displacements between the shell and continuum. shear modulus of medium

and density of shell material.

2/2/ ])()([()( hRrxs

xi

xxxhRrrx Rrut

ZS

2/2/ )])([()( hRrsr

irrrhRrrr w

tZS

2/2/ ])()([()( hRrsi

hRrr Rrut

ZS

(9)

,.RSr

R

RS .

and RSxx

.

are the non-dimensionalized stiffness coefficient of the bond in

radial , tangential and axial direction respectively.

1cZrr

1cZ

and1cZx

x

are the non-dimensionalized damping coefficient of the bond in

radial, tangential and axial direction respectively.

Thus a total of six algebraic equations are obtained. These six equations when simplified give the final response

equation, which may be put into the form

3523110 BBFB}}{{ FFUQ (10)

;6423

6

3

4

3

2

3

0

3

0

3

0 T

T

BBBUVWAB

AB

AB

Au

Av

AwU

Where [Q] is a (6x6)) matrix and {F1}, {F2} and {F3} are (61) matrices. But for the response of shear verticalwave the amplitudes due to shear waves

1B and 5B would be zero so the effect of {F1} and {F3} matrices

would be eliminated. Putting values of 1B = 5B = 0 and substituting values of 3B Eq. (6) becomes as

22

20 F)1(}}{{

AiUQ

n

(11)Elements of Matrix F

;222

1 12

122

21

31 nn II

hF

;

21

2

21

2

11132

h

IInhF nn

;22

1 113

3 nIih

F

;

21

, 13513

4

h

InFIF n

n

)](2)(){2[)({ 1"2

1

2

2

2

1

1

1

'3

6

nn

rr

rrn II

iIF

Elements of matrix Q in Eq. (11) and matrix L in Eq. (2) are same as provided in Rajput et.al (2010).

3. Results and DiscussionsResults are presented for a transversely isotropic shell with r- as the plane of isotropy.Consequently zEE , xxz GG , xzx , zz , )1(2/EG zz . Thus wehave 23 , and )1(2/E/G z1xz4 . In addition 3.0xz has beentaken in the numerical calculations. Different values of 1 and 2 used are as follows 1 = 0.5, 0.1, 0.05 and

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2 = 0.1,05,.02, has been varied between 0.1 to 10.0 to take into account different soil conditions around thepipe, = 0.1 corresponds to the soft soil , =1 for medium soil and for hard rocky soil = 10.0 For all the

values of , m = 0.25 has been assumed. Thickness to radius ratio of the shell (h ) has been taken as 0.01

and the density ratio of the surrounding medium to the shell ( ) has been taken as 0.75. Non-dimensional

amplitude of the middle surface of the shell in the radial and axial directions (W andU) have been plottedagainst the non-dimensional wave number of the incident shear vertical wave (SV-wave) ( = 2R/). The shellresponse has been shown non-axisymmetric (n = 1) flexural mode and axisymmetric mode taking stiffness

coefficient (x

r ), damping coefficient ( x r ) as bond parameter and 1 , 2 as the shell

orthotropic parameters.

The bond parameters have been varied between zero and infinity,x =

= r = x = = r = 0

corresponds to perfect bonding between the shell and surround shell.

Figure 1 to figure 3 shows the effect of stiffness coefficient on the axial displacement of the shell at different

angle of incidence of wave, at different condition of the soil with increasing wave number

Figure1. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as

Parameter

10x 100x

1.0x

1.0x

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Parameter


Parameter

100,10,1.0x

1.0x

100x

1.0x

100x

1.0x

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Figure 4to figure 6 shows the effect of damping coefficient on the axial displacement of the shell at different


Figure4. Axial displacement )(U vs. ( ) withx = 0.1, 10,100 as Parameter


Parameter

100,10,1.0x

100x

1.0x

100,10,1.0x

100,10,1.0x

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Parameter

Figure 7 to figure 10 shows the effect of stiffness coefficient on the radial displacement of the shell at different


Figure7. Radial displacement )W( vs. wave number ( ) withr = 0.1, 10,100 as

Parameter

100,10,1.0x

100x

100,10,1.0x

1.0r

100r

100,10,1,0r

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Figure8 Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as

Parameter

Figure9 Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as

Parameter

100r1.0r

100,10,1,0r

100,10,1.0r

100,10,1.0r 1.0r

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Figure10. Radial displacement )(W vs. wave number ( ) with r = 0.1, 10,100 asParameter

Figure11. Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as

Parameter

100,10,1.0r

100,10,1.0r

100,10,1.0r

100,10,1.0r

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Figure12. Radial displacement )(W vs. ( ) with r = 0.1, 10,100 as Parameter.

Conclusions

Based on the results presented following general conclusions could be drawn-.

Unlike the behavior observed in the non-axisymmetric mode a loose contact between the shell andthe surrounding soil does not always give more shell displacement as compared to those for a

perfectly bonded shell. Therefore assuming a perfect bond may not always lead to a safe and

conservative estimate of displacements.

Effects of bond parameters depend upon the soil conditions and the incidence angle andwavelength of the incident wave. In hard and rocky surroundings bond imperfections show more

prominent effects on the shell response.

The flexural mode response assumes considerable importance in soft soil condition and at higher

apparent wave speed. Flexural mode response due to incident shear vertical wave is significant only at large angle of

incidence.

For large angle of incidence radial deflection is higher in flexural modes at larger wavelength but atthis value it is the axial deflection in axisymmetric mode, which is higher. Thus for larger wavelength

axisymmetric mode is more important because the most common cause of pipeline failure is excessive

axial deformation, while at smaller wavelength the flexural mode has much importance for axialdisplacement.

Both the shell orthotropic parameters influence the radial displacement equally well but 2 has astronger influence on the axial displacement than 1 . As the Stiffness and the damping coefficient isincrease the axial and radial response increase.

Axial deflection and radial deflection both are increase when the value of bonding parameter stiffnesscoefficient (

x r), damping coefficient (

x r) increase from zero to infinity (perfect

to imperfect bonding) as parameters.

Nomenclature

A Amplitude of the plane wave

321 ;; AAA

Amplitudes of P-SV-SH wave respectively

B1 .,B6 Arbitrary constants

'6

'1 B.......B Arbitrary constants

100,10,1.0r

100r

1.0r

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c Apparent wave speed along the axis of the shell

dr, d, dx Components of displacement vector

ER, E, EX Young modulus of the shell.

xr eee ;;

Unit vector in co-ordinate direction

{F1}, {F2}, {F3} Column vector

Gx, Gxz, Gz Shear moduli of the shell

H Vector displacement in the medium

H'

Displacement potential corresponding to SV wave

Hr, H, HX Components of vector potential(Hx corresponding to SH wave)

h Thickness of the shell

h (=h/R) Non dimensional thickness of the shellIn ( ) Modified Bessel function of first kind

Jn ( ) Bessel function of first kindKn ( ) Modified Bessel function of second kind

kx, k Shear correction factor

{L} Matrix operator

xxxx MMMM ;;;

Stress resultant moments

xxxx NNNN ;;; Stress resultantsn Mode shape number in the tangential direction{P*} Column matrix

R Mean radius of the shell

r Radial coordinate

t Time

U Non-dimensional amplitude of the shell in axial directionu Displacement of the shell middle surface in the axial directionu0 Displacement amplitude of the shell middle surface in the axial direction

uz, u, ux Displacement component of a point in the shell

V Non-dimensional amplitude of the shell in the tangential directionv Displacement of the shell middle surface in the tangential direction

v0 Displacement amplitude of shell middle surface in tangential direction

W Non-dimensional amplitude of the shell in the radial directionw Displacement of the shell middle surface in the radial directionw0 Displacement amplitude of the shell middle in the radial directionx Coordinate along the shell axis

z Coordinate normal to middle surface of the shell

Angle of incidence of the wave(=2R/) Non-dimensional wave number of incident wave1, 2, 3, 4 Non-dimensional shell orthotropic parameters of the shellz Normal to middle surface of the shell Tangential direction Wave length of the incident wave Lames constant Modulus of rigidity

xzGNon-dimensional modulus of rigidity of medium

m Poisson ratio the mediumx, x, z, zx, xz Poisson ratios of the shell

(= 2 cos / ) Apparent wave number Density of the shell materialm Density of the medium

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m Non-dimensional density of the medium

ij Components of stress tensor Scalar displacement potential in the mediumx Angle of rotation in r-x plane Symmetry constant =1for n=0, =2 for n=1xo Amplitude ofx Angle of rotation in r- plane

Subscripts

m Medium

r Radial directionx Axial direction

z Normal to middle surface of the shell

Tangential direction

Superscripts

i Incident wave

s Scattered wave

REFERENCES

[1] Ariman T and Muleski GE (1988). Review of the response of buried pipelines under seismic Excitation. Int. J. Earthquake Engg. and

Structural Dynamics. 9: 133-151.

[2] Ariman, T., Liu, S. C., and Nickell, R. E. (Eds.),Life Earthquake Engineering Buried Pipelines, Seismic Risk and Instrumentation

New York: ASME, pp. 114-131.

[3] Chonan S (1981). Response of a pre-stressed, orthotropic thick cylindrical shell subjected to pressure pulses. J. Sound and Vibration.78: 257-267.

[4] Cole BW, Ritter CJ and Jordon S (1979). Structural analysis of buried reinforced plastic mortar pipe. Lifeline earthquake engineering-

buried pipelines, seismic risk and instrumentation. T. Ariman, S. C. Liu and R. E. Nickell, eds., ASME.

[5] Dwivedi JP and Upadhyay PC (1989) Effect of imperfect bonding on the axisymmetric dynamic response of buried orthotropic

cylindrical shells. J. Sound and Vibration. 135: 477-486.[6] Dwivedi, J. P. and Upadhyay, P. C. 1990. Effect of fluid presence on the dynamic response of imperfectly bonded buried orthotropic

cylindrical shells. J. Sound and Vibration. 139(II): 99-110.

[7] Dwivedi JP and Upadhyay PC (1991). Effect of imperfect bond on the dynamic response of buried orthotropic cylindrical shells under

shear-wave excitation. J. Sound and Vibration. 145(2): 333-337.[8] Dwivedi JP, Singh VP and Upadhyay PC (1991). Effect of fluid presence on the dynamic response of imperfectly bonded cylindrical

shells due to incident shear-wave excitation. Computer and Structures. 40(4): 995-1001.

[9] Dwivedi JP, Mishra BK. and Upadhyay PC (1992). Non-axisymmetric dynamic response of imperfectly bonded buried orthotropic

pipeline due to an incident shear wave. J. Sound and Vibration. 157(1): 81-92.

[10] Dwivedi JP, Mishra BK and Upadhyay PC (1992b). Non-axisymmetric dynamic response of imperfectly bonded buried orthotropicpipelines due to incident shear wave (SH-wave). J. Sound and Vibration. 157(1): 177-182.

[11] Dwivedi JP, Singh, VP and Upadhyay PC (1993a). Non-axisymmetric dynamic response of imperfectly bonded buried fluid-filled

orthotropic cylindrical shells due to incident shear wave. J. Sound and Vibration. 167(2): 277-287.

[12] Dwivedi JP, Singh VP and Upadhyay PC (1993b). Effect of fluid presence on the non-axisymmetric dynamic response of imperfectly

bonded buried orthotropic pipelines due to incident shear wave. Computer and Structures. 48(2): 219-226.

[13] Dwivedi JP, Singh VP and Upadhyay PC (1996). Non-axisymmetric dynamic response of imperfectly bonded buried fluid-filledorthotropic cylindrical shells. ASME J. of Press. Vessel Techno. 118(1): 64-73.

[14] Dwivedi JP, Mishra BK and Upadhyay PC (1998). Non-axisymmetric dynamic response of imperfectly bonded buried orthotropic

pipelines. Struct. Engg. and Mechanics. 6(3): 291-304.[15] Hasheminajad SM. and Kazemirad S (2008). Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under

seismic excitation. Soil Dynamics and Earthquake Engg. 28: 277-292.

[16] Herrman G and Mirsky J (1957). Non axially Symmetric Motion of Cylindrical Shell. J. Acoustical Society America. 29(10): 1116-

1123.[17] Jones RM (1975) Mechanics of Composite Materials. McGraw Hill Publications, Newyork.

[18] Kouretzis GP, Bouckovalas GD and Gantes CJ (2007). Analytical calculation of blast-induced strains on buried pipe lines. Int. J.Impact Engg.34: 1683-1704.

[19] Lee DH, Kim BH, Lee H and Kong JS (2009) Seismic behavior of buried gas pipelines under earthquake excitation. Engg. Structure.

31: 1011-1023.

[20] Muleski GE, Ariman TS and Ariman CP (1979). A Shell Model of a Buried Pipe in a Seismic environment. ASME J. Press. VesselTechno. 101: 44-50.

[21] Rajput RS, Sunil Kumar, Chaubey A, and Dwivedi JP. (2010). Comparison of Non-Axisymmetric Dynamic Response of Imperfectly

Bonded Buried Orthotropic Thick and Thin Fluid filled Cylindrical Shell due to Incident Shear Wave (SH Wave). IJEST.2: 5845-5871

[22] Rajput RS, Sunil Kumar, Chaubey A. and Dwivedi JP. (2010). Non-Axisymmetric Dynamic Response of Imperfectly Bonded Buried

Orthotropic Thin Fluid Empty Cylindrical Shell due to Incident Compressional Wave. EJSR.10:443-464

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[23] Upadhyay PC and Mishra BK (1988). Non Axisymmetric Dynamic Response of buried Orthotropic shells.J. Sound and Vibration.121: 149 -160.

[24] Singh VP, Upadhyay PC and Kishore B (1987). On the dynamic response of Buried Orthotropic Cylindrical Shells. J. Sound and

Vibration. 113: 101-115.

Biographical notes

Rakesh Singh Rajput received M. Tech. from NIT Allahabad, India and pursuing Ph.D. from Rajeev Gandhi Technological University,

Bhopal, India. He is a Reader in Mechanical Engineering, and working with Department of Technical Education (DTE), Govt. of M. P.,

India. His research interests include machine design and mechanics. He has written book on Engineering Mechanics.Dr. Sunil Kumar is a Professor of Mechanical Engineering, Rajeev Gandhi Technological University, Bhopal, India. He has received M.

Tech from IT-BHU, Varanasi, India and Ph. D. from IIT Delhi, India. He has more than 20 years of experience in teaching and research. His

current area of research includes Combustion mechanics and System Dynamics.

Dr. Alok Chaube is a Professor of Mechanical Engineering, Rajeev Gandhi Technological University, Bhopal, India. He has received M.Tech from IIT Delhi, India and Ph. D. from IIT Roorkee, India. He has more than 20 years of experience in teaching and research. His

current area of research includes Fluid mechanics and System Dynamics.

J P DWIVEDI

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