Quasi Static KC

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Quasi-static thermal stresses in a thick circular plate

V.S. Kulkarni a, K.C. Deshmukh b,*

a Department of Mathematics, Govt. College of Engineering, Chandrapur 442 401, Maharashtra, Indiab Post-Graduate Department of Mathematics, Nagpur University, Nagpur 440 010, Maharashtra, India

Received 1 July 2005; received in revised form 1 March 2006; accepted 12 April 2006

Abstract

The present paper deals with the determination of a quasi-static thermal stresses in a thick circular plate subjected toarbitrary initial temperature on the upper face with lower face at zero temperature and the fixed circular edge thermallyinsulated. The results are obtained in series form in terms of Bessel’s functions and they are illustrated numerically. 2006 Elsevier Inc. All rights reserved.

Keywords: Quasi-static; Transient; Thermoelastic problem; Thermal stresses

1. Introduction

During the second half of the twentieth century, nonisothermal problems of the theory of elasticity becameincreasingly important. This is due to their wide application in diverse fields. The high velocities of modernaircraft give rise to aerodynamic heating, which produces intense thermal stresses that reduce the strengthof the aircraft structure.

Nowacki [1] has determined steady-state thermal stresses in circular plate subjected to an axisymmetric tem-perature distribution on the upper face with zero temperature on the lower face and the circular edge. RoyChoudhary [2,3] and Wankhede [4] determined Quasi-static thermal stresses in thin circular plate. Gogulwarand Deshmukh [5] determined thermal stresses in thin circular plate with heat sources. Also Tikhe and Desh-mukh [6] studied transient thermoelastic deformation in a thin circular plate, where as Qian and Batra [7] stud-

ied transient thermoelastic deformation of thick functionally graded plate. Moreover, Sharma et al. [8] studiedthe behaviour of thermoelastic thick plate under lateral loads and obtained the results for radial and axial dis-placements and temperature change have been computed numerically and illustrated graphically for differenttheories of generalized thermoelasticity. Also Nasser [9,10] solved two-dimensional problem of thick platewith heat sources in generalized thermoelasticity. Recently Ruhi et al. [11] did thermoelastic analysis of thick

0307-904X/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2006.04.009

* Corresponding author.E-mail addresses: [email protected] (V.S. Kulkarni), [email protected] (K.C. Deshmukh).

Applied Mathematical Modelling xxx (2006) xxx–xxx

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walled finite length cylinders of functionally graded materials and obtained the results for stress, strain anddisplacement components through the thickness and along the length are presented due to uniform internalpressure and thermal loading.

This paper deals with the realistic problem of the quasi-static thermal stresses in a thick circular plate subjected to arbitrary initial temperature on the upper face with lower face at zero temperature and fixed circular

edge thermally insulated. The results presented here will be more useful in engineering problem particularly inthe determination of the state of strain in thick circular plate constituting foundations of containers for hotgases or liquids, in the foundations for furnaces, etc.

2. Formulation of the problem

Consider a thick circular plate of radius a and thickness h defined by 0 6 r 6 a, h/2 6 z 6 h/2. Let theplate be subjected to the arbitrary initial temperature over the upper surface ( z = h/2) with the lower surface(z = h/2) at zero temperature and the fixed circular edge thermally insulated. Under these more realistic pre-scribed conditions, the quasi-static thermal stresses are required to be determined.

The differential equation governing the displacement potential function /(r, z, t) is given in [12] as

o2/

or 2 þ 1

r

o/

or þ o2/

o z 2 ¼ K s ð1Þ

with / ¼ 0 at t ¼ 0; ð2Þ

where K is the restraint coefficient and temperature change s = T T i. T i is initial temperature. Displacementfunction / is known as Goodier’s thermoelastic potential. The temperature of the plate at time t satisfies theheat conduction equation,

o2

T

or 2 þ

1

r

oT

or þ

o2

T

o z 2 ¼

1

k

oT

ot ð3Þ

with the conditionsT ¼ f ðr Þ for z ¼ h=2; 0 6 r 6 a; for all time t ; ð4Þ

T ¼ 0 on z ¼ h=2; 0 6 r 6 a; ð5Þ

and

oT

or ¼ 0 at r ¼ a; h=2 6 z 6 h=2; ð6Þ

where k is the thermal diffusivity of the material of the plate.The displacement function in the cylindrical coordinate system are represented by the Michell’s function

defined in [12] as

ur ¼ o/

or

o2 M

or o z ; ð7Þ

u z ¼ o/

o z þ 2ð1 mÞr2 M

o2 M

o z 2 : ð8Þ

The Michell’s function M must satisfy

r2r2 M ¼ 0; ð9Þ

where

r

2

¼

o2

or 2 þ

1

r

o

or þ

o2

o z 2 : ð10

Þ

2 V.S. Kulkarni, K.C. Deshmukh / Applied Mathematical Modelling xxx (2006) xxx–xxx

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The component of the stresses are represented by the thermoelastic displacement potential / and Michell’sfunction M as

rrr ¼ 2G o

2/

or 2 K s þ

o

o z mr2 M

o2 M

or 2

; ð11Þ

rhh ¼ 2G 1r

o/or

K s þ oo z

mr2 M 1r

o M or

; ð12Þ

r zz ¼ 2G o

2/

o z 2 K s þ

o

o z ð2 mÞr2 M

o2 M

o z 2

ð13Þ

and

rrz ¼ 2G o

2/

or o z þ

o

or ð1 mÞr2 M

o2 M

o z 2

; ð14Þ

where G and m are the shear modulus and Poisson’s ratio respectively, and for traction free surface stressfunctions

rrr ¼ rrz ¼ 0 at r ¼ a ð15Þ

for the thick plate.Eqs. (1)–(15) constitute mathematical formulation of the problem.

3. Solution

To obtain the expression for temperature T (r, z, t).Assume

T ðr ; z ; t Þ ¼ z þ h

2

X

1

n¼1

f nðt Þ J 0ðanr Þ; ð16Þ

where a1,a2, . . . are roots of the equation J 1(aa) = 0. J n(x) is Bessel function of the first kind of order n and the

function f n(t) is yet to be determined.

Eqs. (3) and (16) gives

f 0nðt Þ ¼ a2nkf nðt Þ: ð17Þ

On integrating f 0nðt Þ one obtains

f nðt Þ ¼ Anea2nkt ; ð18Þ

where An

is a constant. The constant An

can be found from the nature of temperature on upper face.Using Eqs. (4), (16) and (18), one obtains

f ðr Þ ¼ hX1

n¼1

An J 0ðanr Þ: ð19Þ

Hence by theory of Bessel’s function (19) gives

An ¼ 2

a2hJ 20ðanaÞ

Z a

0

rJ 0ðanr Þ f ðr Þdr : ð20Þ

By Eqs. (16) and (18) the required expression for temperature function is obtained as

T ðr ; z ; t Þ ¼ z þ h

2 X

1

n¼1

An J 0ðanr Þea2nkt : ð21Þ

V.S. Kulkarni, K.C. Deshmukh / Applied Mathematical Modelling xxx (2006) xxx–xxx 3

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At t = 0, initial temperature T i is given by

T i ¼ z þ h

2

X1

n¼1

An J 0ðanr Þ: ð22Þ

Hence temperature change s is obtained as

s ¼ z þ h

2

X1

n¼1

An J 0ðanr Þ ea2nkt 1

: ð23Þ

Now suitable form of M satisfying (9) is given by

M ¼X1

n¼1

½ Bn J 0ðanr Þ þ C nanrJ 1ðanr Þ cosh an z þ h

2

; ð24Þ

where B n

and C n

are arbitrary functions.Assuming displacement function /(r, z, t) which satisfies (1) and (2) as

/ðr ; z ; t Þ ¼ X1

n¼1

Dn½ J 0ðanr Þ sinh an z þ h

2 z þ

h

2 ea2

nkt 1 : ð25Þ

Using / in (1), one have

we find Dn ¼ KAn

a2n

:

Thus Eq. (25) become

/ðr ; z ; t Þ ¼ K X1

n¼1

An½ J 0ðanr Þ sinh an z þ h

2

z þ

h

2

ean2kt 1

a2n

!: ð26Þ

Now using Eqs. (23), (24) and (26) in (7) and (8), and (11)–(14), one obtains.The expressions for displacements and stresses respectively as

ur ¼ K X1

n¼1

An J 1ðanr Þ sinh an z þh

2

z þ

h

2

ean2kt 1

an

!( )

þX1

n¼1

a2n Bn J 1ðanr Þsinh an z þ

h

2

X1

n¼1

a3nC nrJ 0ðanr Þ sinh an z þ

h

2

; ð27Þ

u z ¼ K X1

n¼1

An½ J 0ðanr Þ an cosh an z þh

2

1

ean2kt 1

a2n

!( )

X1

n¼1

a2n Bn J 0ðanr Þcosh an z þ

h

2 X1

n¼1

½4ð1 mÞ J 0ðanr Þ anrJ 1ðanr Þa2nC n cosh an z þ

h

2 ; ð28Þ

rrr ¼ 2G K X1

n¼1


2

z þ

h

2

ea2

nkt 1

r an

!(

K X1

n¼1


2

ea2

nkt 1

an

!

þX1

n¼1

an J 0ðanr Þ J 1ðanr Þ

r

a2

n Bn sinh an z þh

2

þX1

n¼1

½ð2m 1Þ J 0ðanr Þ r an J 1ðanr Þa3nC n sinh an z þ

h

2 ); ð29Þ


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rhh ¼ 2G K X1

n¼1


2

z þ

h

2

ea2

nkt 1

r an

!(

K z þh

2

X1

n¼1


þX1

n¼1

a2n Bnð J 1ðanr ÞÞ

r sinh an z þ

h

2

þX1

n¼1

ð2m 1Þa3nC n J 0ðanr Þ sinh an z þ

h

2

); ð30Þ

r zz ¼ 2G K X1

n¼1


2

z þ

h

2

ea2

nkt 1 (

X1

n¼1

a3n Bn J 0ðanr Þ sinh an z þ

h

2

þX1

n¼1

½2ð2 mÞ J 0ðanr Þ anrJ 1ðanr Þa3nC n sinh an z þ

h

2

) ð31Þ

and

rrz ¼ 2G K X1

n¼1

An J 1ðanr Þ an cosh an z þ h

2

1

ea2

nkt 1

an

!þX1

n¼1

a3n Bn J 1ðanr Þ cosh an z þ

h

2

(

X1

n¼1

½2ð1 mÞ J 1ðanr Þ þ anrJ 0ðanr Þa3nC n cosh an z þ

h

2

): ð32Þ

Now using (15) in (29) and (32) one obtains

Bn ¼ K

X1

n¼1

An

a3n

ea2nkt 1

; ð33Þ

and

C n ¼ 0: ð34Þ

Using Eqs. (33) and (34) Eqs. (27)–(32) one obtains the expressions for displacement and stresses respectivelyas

ur ¼ K z þ h

2

X1

n¼1

An J 1ðanr Þ ea2

nkt 1

an

!; ð35Þ

u z ¼ K X1

n¼1

An½ J 0ðanr Þ ea2

nkt 1

a2n

!" #; ð36Þ

rrr ¼ 2GK z þ h

2

X1

n¼1

An J 1ðanr Þr

ea2nkt 1an

!; ð37Þ

rhh ¼ 2GK z þ h

2

X1

n¼1

An

J 1ðanr Þ

r an

J 0ðanr Þ

ea2

nkt 1

; ð38Þ

r zz ¼ 2GK z þ h

2

X1

n¼1


ð39Þ

and

rrz ¼ 2GK X1

n¼1

An J 1ðanr Þ ea2

nkt 1

an !: ð40Þ


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4. Numerical calculations

Setting f ðr Þ ¼ T 0dðr bÞ ða > bÞ in Eq: ð20Þ; ð41Þ

where T 0 is constant and d(r) is well known direct delta function of argument r. One has

An ¼ 2bT 0 J 0ðanbÞa2hJ 20ðanaÞ

: ð42Þ

The numerical calculation have been carried out for steel (SN 50C) plate with parameters

a ¼ 1 m; b ¼ 0:5 m; h ¼ 0:25 m;

thermal diffusivity k = 15.9 · 106 (m2 s1) with

a1 ¼ 3:8317; a2 ¼ 7:0156; a3 ¼ 10:1735; a4 ¼ 13:3237; a5 ¼ 16:470; a6 ¼ 19:6159;

a7 ¼ 22:7601; a8 ¼ 25:9037; a9 ¼ 29:0468; a10 ¼ 32:18

are the roots of transdental equation J 1(aa) = 0. For convenience setting A = T 0, B = KT 0 and C = 2GKT 0 in

the expressions (23) and (35)–(40).The numerical expressions for temperature change, displacement and stress components are obtained as

s

A ¼

1

h z þ

h

2

X1

n¼1

J 0ðan=2Þ

J 20ðanÞ J 0ðanr Þ ea2

nkt 1

; ð43Þ

ur

B ¼

1

h z þ

h

2

X1

n¼1

J 0ðan=2Þ J 1ðanr Þ

J 20ðanÞ

ea2nkt 1

an

!; ð44Þ

u z

B ¼

1

h X1

n¼1

J 0an

2

½ J 0ðanr Þ

J 20ðanÞ

ea2nkt 1

a2n

!" #; ð45Þ

rrr

C ¼

1

h z þ

h

2

X1

n¼1

J 0an

2

J 1ðanr Þ

rJ 20ðanÞ

ea2nkt 1

an

!; ð46Þ

rhh

C ¼

1

h z þ

h

2

X1

n¼1

J 0an

2

J 20ðanÞ

J 1ðanr Þ

r an

J 0ðanr Þ

ea2

nkt 1

; ð47Þ

r zz

C ¼

1

h z þ

h

2

X1

n¼1

J 0an

2

J 20 anð Þ

J 0ðanr Þ ea2nkt 1

; ð48Þ

and

rrz

C ¼

1h

X1

n¼1

J 0 an2

J 20ðanÞ

J 1ðanr Þ ea2nkt 1

an

!" #: ð49Þ

The numerical variations are shown in the following figures with the help of computer programme.

5. Concluding remarks

In this paper, a thick circular plate is considered and determined the expressions for temperature change,displacements and stress functions due to arbitrary heat supply on the upper surface. As a special case math-ematical model is constructed for f (r) = T 0d(r 0.5) and performed numerical calculations. The thermoelasticbehaviour is examined such as temperature change, displacements and stresses with the help of arbitrary initial

heat supply on the upper surface.


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From Figs. 1 and 2, temperature change increases with time.From Figs. 3 and 4, radial displacement function increases with the time within the circular region

0 6 r 6 0.5 and decreases within annular region 0.5 6 r 6 1 in radial direction, where as in axial directionit is increases with the time.

From Fig. 5, axial displacement function increases with the time within the circular region 0 6 r 6 0.2 and

it remains constant within annular region 0.26 r 6

1, where as in axial direction it remains constant.From Figs. 6 and 7, radial stress function rrr

decreases with the time within the circular region 0 6 r 6 0.5and increases within annular region 0.5 6 r 6 1, where as in axial direction it is decreases with the time.

From Figs. 8–11, the stress function rhh and axial stress function rzz decreases with the time, where as in

axial direction it is increases with the time.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1r

t=4

Fig. 1. The temperature change s A

on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.125 -0.075 -0.025 0.025 0.075 0.125z

t=4

Fig. 2. The temperature change sð AÞ

on r = 0.5 in axial direction at t = 1, 2, 3 and 4.

-0.0084

-0.0063

-0.0042

-0.0021

0

0.0021

0.0042

0.0063

0.0084

0 0.2 0.4 0.6 0.8 1

r

t=4

Fig. 3. The radial displacement function ur/B on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.


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0

0.0005

0.001

0.00150.002

0.0025

0.003

0.0035

0.004

0 0.2 0.4 0.6 0.8 1

r

t=4

Fig. 5. The axial displacement function uz/B on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.2 0.4 0.6 0.8 1

r

t=4

Fig. 6. The radial stress function rrr

/C on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.

0

0.00005

0.0001

0.00015

-0.125 -0.075 -0.025 0.025 0.075 0.125

z

t=4

Fig. 7. The radial stress function rrr/(C ) on r = 0.5 in axial direction at t = 1, 2, 3 and 4.

0

0.00002

0.00004

0.00006

0.00008

-0.125 -0.075 -0.025 0.025 0.075 0.125

z

t=4

Fig. 4. The radial displacement function ur/B on r = 0.5 in axial direction at t = 1, 2, 3 and 4.


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0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1r

t=4

Fig. 8. The stress function rhh/(C ) on upper surface of plate z = 0.125 in radial direction radial at t = 1, 2, 3 and 4.

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.125 -0.075 -0.025 0.025 0.075 0.125

z

t=4

Fig. 9. The stress function rhh/C on r = 0.5 in axial direction radial at t = 1, 2, 3 and 4.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

r

t=4

Fig. 10. The axial stress function rzz

/(C ) on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.125 -0.075 -0.025 0.025 0.075 0.125z

t=4

Fig. 11. The axial stress function rzz/C on r = 0.5 in axial direction at t = 1, 2, 3 and 4.


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From Fig. 12, stress function rrz

increases with the time within the circular region 0 6 r 6 0.5 and decreaseswithin annular region 0.5 6 r 6 1, where as in axial direction it remains constant.

It means we may find out that displacement and stress components occurs near heat source (at r = 0.5).Also radial stress component r

rr develops compressive stress within the circular region 0 6 r 6 0.5 and tensile

stress within annular region 0.5 6 r 6 1, where as axial stress component rzz and stress component rhh devel-ops tensile stress near heat source and at center.

From figures of radial and axial displacements it can observe that the radial displacement occur away fromthe center (r = 0) where as axial displacement is maximum at centre. so it may conclude that due arbitrary heatsupply the plate bends concavely at the center.

The results obtained here are more useful in engineering problems particularly in the determination of stateof strain in thick circular plate. Also any particular case of special interest can be derived by assigning suitablevalues to the parameters and function in the expression (35)–(40).

Acknowledgement

The authors express their sincere thanks to Prof. P.C. Wankhede for his valuable guidance while preparingthis manuscript. Also the authors are thankful to University Grants Commission, New Delhi to provide thepartial financial assistance under major research project scheme.

References

[1] W. Nowacki, The state of stresses in a thick circular plate due to temperature field, Bull. Acad. Polon. Sci., Ser. Scl. Tech. 5 (1957)227.

[2] S.K. Roy Choudhary, A note of quasi static stress in a thin circular plate due to transient temperature applied along the circumferenceof a circle over the upper face, Bull. Acad. Polon Sci. Ser. Scl. Tech. 20–21 (1972).

[3] S.K. Roy Choudhary, A note on quasi-static thermal deflection of a thin clamped circular plate due to ramp-type heating of aconcentric circular region of the upper face, J. Franklin Inst. 206 (3) (1973).

[4] P.C. Wankhede, On the quasi static thermal stresses in a circular plate, Indian J. Pure Appl. Math. 13 (11) (1982) 1273–1277.

[5] V.S. Gogulwar, K.C. Deshmukh, Thermal stresses in a thin circular plate with heat sources, J. Indian Acad. Math. 27 (1) (2005).[6] A.K. Tikhe, K.C. Deshmukh, Transient thermoelastic deformation in a thin circular plate, J. Adv. Math. Sci. Appl. 15 (1) (2005).[7] L.F. Qian, R.C. Batra, Transient thermoelastic deformation of a thick functionally graded plate, J. Therm. Stresses 27 (2004) 705–

740.[8] J.N. Sharma, P.K. Sharma, R.L. Sharma, Behavior of thermoelastic thick plate under lateral loads, J. Therm. Stresses 27 (2004) 171–

191.[9] M.EI-Maghraby Nasser, Two dimensional problem with heat sources in generalized thermoelasticity with heat sources, J. Therm.

Stresses 27 (2004) 227–239.[10] M.EI-Maghraby Nasser, Two dimensional problem for a thick plate with heat sources in generalized thermoelasticity, J. Therm.

Stresses 28 (2005) 1227–1241.[11] M. Ruhi, A. Angoshatari, R. Naghdabadi, Thermoelastic analysis of thick walled finite length cylinders of functionally graded

material, J. Therm. Stresses 28 (2005) 391–408.[12] Naotake Noda, Richard B. Hetnarski, Yoshinobu Tanigawa, Thermal Stresses, second ed., Taylor and Francis, New York, 2003, pp.

259–261.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1

r

t=4

Fig. 12. The stress function rrz

/C on upper surface of plate z = 0.125 in radial direction at t = 1, 2, 3 and 4.


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