A Brief Note on Quasi Static Thermal Stresses In A Thin Rectangular Plate With Internal Heat Generation

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1 Americn Journl of Engineering Reserch (AJER) 13 Americn Journl of Engineering Reserch (AJER) e-issn : p-issn : Volume-, Issue-1, pp Reserch Pper Open Access A Brief Note on Qusi Sttic Therml Stresses In A Thin Rectngulr Plte With Internl Het Genertion C. M. Bhongde, nd M. H. Durge 1 Deprtment of Mthemtics, Shri. Shivji College, Rjur, Mhrshtr, Indi Deprtment of Mthemtics, Annd Niketn College, Wror, Mhrshtr, Indi Astrct: - The present pper dels with the determintion of qusi sttic therml stresses in thin rectngulr plte with internl het genertion. A thin rectngulr plte is considered hving zero initil temperture nd sujected to ritrry het supply t x = nd x = where s thin rectngulr plte is insulted t y = nd y =. Here we modify Kulkrni (7). The governing het conduction eqution hs een solved y the method of integrl trnsform technique. The results re otined in series form in terms of Bessel s functions. The results for temperture, displcement nd stresses hve een computed numericlly nd illustrted grphiclly. Keywords: - Qusi sttic therml stresses, thermoelstic prolem, internl het genertion, thin rectngulr plte. I. INTRODUCTION Gogulwr nd Deshmukh (4) determined the therml stresses in rectngulr plte due to prtilly distriuted het supply. Nsser M. et l. (4) solved two dimensionl prolem of thick plte with het sources in generlized thermoelsticity. Khndit nd Deshmukh (1) studied thermoelstic prolem in rectngulr plte with het genertion. Recently Ptil et l. (13) determined the therml stresses in rectngulr sl with internl het source, now here thin rectngulr plte is considered hving zero initil temperture nd nd sujected to ritrry het supply t x = nd x = where s the plte is insulted t y = nd y =. Here we modify Kulkrni (7). To otin the temperture distriution, cosine integrl trnsform nd Lplce trnsform re pplied. The results re otined in series form in terms of Bessel s functions nd the temperture chnge, displcement nd stresses hve een computed numericlly nd illustrted grphiclly. A mthemticl model hs een constructed of thin rectngulr plte with the help of numericl illustrtion y considering steel (.5% cron) rectngulr plte. No one previously studied such type of prolem. This is new contriution to the field. The direct prolem is very importnt in view of its relevnce to vrious industril mechnics sujected to heting such s the min shft of lthe, turines nd the role of rolling mill, se of furnce of oiler of therml power plnt nd gs power plnt. II. FORMULATION OF THE PROBLEM A thin rectngulr plte occupying the spce D: x, y, is considered. A thin rectngulr plte is considered hving zero initil temperture nd sujected to ritrry het supply t x = nd x = where s the plte is insulted t y = nd y =. Here the plte is ssumed sufficiently thin nd considered free from trction. Since the plte is in plne stress stte without ending. Airy stress function method is pplicle to the nlyticl development of the thermoelstic field. The eqution is given y the reltion + U = x y t E + x y T (1) where t, E nd U re liner coefficient of the therml expnsion, Young s modulus elsticity of the mteril of the plte nd Airys stress functions respectively. w w w. j e r. o r g Pge 388

2 Americn Journl of Engineering Reserch (AJER) 13 The displcement components u x nd u y in the X nd Y direction re represented in the integrl form nd the stress components in terms of U re given y u x = 1 U U E y x tt dx () u y = 1 U U E x y tt dy (3) ς xx = U y (4) ς yy = U x nd ς xy = U x y (6) ς xy = t y =. (7) where v is the Poisson s rtio of the mteril of the rectngulr plte. The temperture of the thin rectngulr plte t time t stisfying het conduction eqution s follows, T T T x y k α t (8) T x, y, t = t t = x, y (9) T(x, y, t) = f 1 y, t t x =, y (1) T x, y, t = f y, t t x =, y (11) T y T y = t y =, x (1) = t y =, x (13) q x, y, t = δ y y sin β m x + 1 e t, < y < (14) where α is the therml diffusivity of the mteril of the plte, k is the therml conductivity of the mteril of the plte, q is the internl het genertion nd δ r is well known dirc delt function of rgument r. Eq. (1) to Eq. (14) constitute mthemticl formultion of the prolem. III. SOLUTION To otin the expression for temperture T(x, y, t), we introduce the cosine integrl trnsform nd its inverse trnsform re T x, β m, t = K β m, y T x, y, t dy (15) T(x, y, t) = m =1 K β m, y T x, β m, t (16) where the kernel K β m, y = cos β m y (17) where β m is the m th root of the trnscendentl eqution sin β m =, β m = mπ, m = 1,,. On pplying the cosine integrl trnsform defined in the Eq. (15), its inverse trnsform defined in Eq. (16), pplying Lplce trnsform nd its inverse y residue method successively to the Eq. (1), one otins the expression for temperture s T x, y, t = where n=1 m=1 cosβ m y + α 1 n sin x Q t + α k Q 1 t = t e α β m + π n (t u) 1 + e u + e αβ m u αβ m 1 αβ m αβ m (αβm 1) Q t = t e α β m + π n nd (t u) α sin ( 1) n Q 1 t cosβ m y sin β m x + Q 3 t (18) cosβ m y α sin β m k F β m, u 1 + e u + e αβ m u αβ m 1 αβ m αβ m (αβm 1) Q 3 t = 1 + e t e + αβ m t αβ m 1 αβ m αβ m (αβm 1) F 1 β m, u du cosβ m y α sin β m k du w w w. j e r. o r g Pge 389

3 Americn Journl of Engineering Reserch (AJER) 13 Airy stress function U Using Eq.(18) in Eq.(1), one otins the expression for Airy s stress function U s U = t E m =1 n=1 cos β m y β m + π n α sin ( 1) n Q 1 t + α 1 n sin x Q t + α β m k cos(β m y ) sin β m x + Q 3 t β m + π n (19) Displcement nd Stresses Now using Eqs. (18) nd (19) in Eqs. () to (6) one otins the expressions for displcement nd stresses s u x = t m =1 n=1 cosβ m y cos α k 1+v β m u y = sin β m y t m=1 n=1 β m sin απ n 3 ( 1) n 1+v β m + π n Q 1 t + cos x Q t cos(β m y ) cos β m x + Q 3 t () α ( 1) n β m 1+v π β n Q 1 t + sin x Q t + α k 1 + v cos(β m y ) sin β m x + Q 3 t (1) ς xx = t E n=1 sin ( β m )cos β m y m=1 π β n α ( 1) n Q 1 t + sin x Q t + α kβ m β m + π n cos(β m y ) sin β m x + Q 3 t () ( π n )cos β m y m =1 π β n ς yy = t E n=1 sin + ς xy = t E cos + α β kπ n m + π n β m sin β m y m =1 n=1 π β n α ( 1) n Q 1 t + sin x Q t cos(β m y ) sin β m x + Q 3 t (3) n π α 3 ( 1) n Q 1 t + cos x Q t α β kβ π n cos(β m m y ) cos β m x + Q 3 t (4) IV. SPECIAL CASE AND NUMERICAL CALCULATIONS Setting f 1 y, t = f y, t = δ y y 1 δ t t, y 1, < t < F 1 β m, t = F β m, t = cos(β m y 1 ) δ t t = 1m, = m, t =,, 4, 6, 8 sec nd y = y 1 = 1m. 4.1 Mteril Properties The numericl clcultion hs een crried out for steel (.5% cron) rectngulr plte with the mteril properties defined s Therml diffusivity α = m s 1, Specific het c ρ = 465 J/kg, Therml conductivity k = 53.6 W/m K, Poisson rtio θ =.35, w w w. j e r. o r g Pge 39

4 Americn Journl of Engineering Reserch (AJER) 13 Young s modulus E = 13 G p, Lme constnt μ = 6.67, Coefficient of liner therml expnsion t = K 4. Roots of Trnscendentl Eqution The β 1 = , β = 6.88, β 3 = 9.44, β 4 = , β 5 = 15.77, β 6 = re the roots of trnscendentl eqution sin β m =. The numericl clcultion nd the grph hs een crried out with the help of mthemticl softwre Mt l. V. DISCUSSION In this pper thin rectngulr plte is considered which is free from frction nd determined the expressions for temperture, displcement nd stresses due to ritrry het supply on the edges x = nd x = of plte wheres the plte is insulted t y = nd y =. A mthemticl model is constructed y considering steel (.5% cron) rectngulr plte with the mteril properties specified ove. Fig. 1 Temperture T in Y- direction. Fig. The displcement u x in Y- direction. w w w. j e r. o r g Pge 391

5 Americn Journl of Engineering Reserch (AJER) 13 Fig. 3 The displcement u y in Y- direction. Fig. 4 Therml stresses ς xx in Y-direction Fig. 5 Therml stresses ς yy in Y-direction. w w w. j e r. o r g Pge 39

6 Americn Journl of Engineering Reserch (AJER) 13 Fig. 6 Therml stresses ς xy in Y-direction. From figure 1, it is oserved tht temperture T decreses s the time increses. The overll ehvior of temperture is decresing nd it is symmetric out y= 1 in thin rectngulr plte with internl het genertion long Y-direction. From figure, it is oserved tht displcement u x decreses s the time increses. The overll ehvior of temperture is decresing nd it is symmetric out y= 1 in thin rectngulr plte with internl het genertion long Y-direction. From figure 3, it is oserved tht the displcement u y is incresing for y.5, 1.5 y nd decresing for.5 y 1.5. The overll ehvior of displcement u y is decresing long Y-direction nd it is ntisymmetric out y= 1 in thin rectngulr plte with internl het genertion long Y-direction. From figure 4 nd 5, it is oserved tht therml stresses ς xx, ς yy increses s the time increses. Mximum vlue of stresses ς xx, ς yy occure ner het source nd it is tensile in nture in thin rectngulr plte with internl het genertion long Y-direction. From figure 6, it is oserved tht the therml stresses ς xy is decresing for x.5, 1.5 x nd incresing for.5 x 1.5. The overll ehvior of stresses ς xy is incresing nd it is ntisymmetric out y= 1 in rectngulr plte with internl het genertion long Y-direction. VI. CONCLUSION We cn conclude tht temperture T, displcement u x nd u y re decresing with time in thin rectngulr plte with internl het genertion long Y-direction. The therml stresses ς xx, ς yy re tensile in nture in thin rectngulr plte with internl het genertion long Y-direction. The therml stresses ς xy is incresing nd it is ntisymmetric out y = 1 in thin rectngulr plte with internl het genertion long Y- direction. The results otined here re useful in engineering prolems prticulrly in the determintion of stte of stress in thin rectngulr plte, se of furnce of oiler of therml power plnt nd gs power plnt. REFERENCES [1] V. S. Kulkrni nd K. C. Deshmukh, A rief note on qusi- sttic therml stresses in rectngulr plte, Fr Est J. Appl. Mth.6(3),7, [] V. S. Gogulwr nd K. C. Deshmukh, Therml stresses in rectngulr plte due to prtilly distriuted het supply, Fr Est J. Appl. Mth.16(),4, [3] M. Nsser, EI-Mghry, Two dimensionl prolem in generlized thermoelsticity with het sources, Journl of Therml Stresses, 7, 4, [4] M. V. Khndit nd K. C. Deshmukh, Thermoelstic prolem in rectngulr plte with het genertion, Presented in Indin Mthemticl Society conference, Pune, Indi, 7. [5] V. B. Ptil, B. R. Ahirro nd N. W. Khorgde, Therml stresses of semi infinite rectngulr sl with internl het source, IOSR Journl of Mthemtics, 8(6), 13, [6] M. N. Ozisik, Boundry Vlue Prolems of Het Conduction, Interntionl Text Book Compny, Scrnton, Pennsylvni, w w w. j e r. o r g Pge 393