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, India b 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 to arbitrary initial temperature on the upper face with lower face at zero temperature and the fixed circular edge thermally insulated. 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, nonisothe rmal problems of the theory of elasticity became increasingly important. This is due to their wide application in diverse fields. The high velocities of modern aircraft give rise to aerodynamic heating, which produces intense thermal stresses that reduce the strength of 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. Roy Choudhary [2,3] and Wankhede [4] determined Quasi-static thermal stresses in thin circular plate. Gogulwar and 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] studied the behaviour of thermoelas tic thick plate under lateral loads and obtained the results for radial and axial dis- placeme nts and tempera ture change have been computed numerically and illustr ated graphically for different theories of generalized thermoelasticity. Also Nasser [9,10] solved two-dimensional problem of thick plate with 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 * Correspond ing author. E-mail addresses: vinayakskulkarn i1@rediffmail.com(V.S. Kulkarni), kcdeshmuk h2000@rediffmail.com(K.C. Deshmukh). Applied Mathematical Modelling xxx (2006) xxx–xxx www.elsevier.com/locate/apm ARTI CLE IN PRESS
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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.
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.
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
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
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.
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.