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).
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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 sub- jected 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
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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
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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
An J 1ðanr Þ sinh an z þh
2
z þ
h
2
ea2
nkt 1
r an
!(
K X1
n¼1
An J 0ðanr Þ sinh an z þh
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
An J 1ðanr Þ sinh an z þh
2
z þ
h
2
ea2
nkt 1
r an
!(
K z þh
2
X1
n¼1
An J 0ðanr Þ ea2nkt 1
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
An J 0ðanr Þ sinh an z þh
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
An J 0ðanr Þ ea2nkt 1
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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