Study of Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Varying Thickness and Density

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1 Amerian Journal of Computational and Applied Mathematis, (6: DOI:.59/j.ajam.6.5 Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density Arun Kumar Gupta,*, Pragati Sharma Department of Mathematis, M.S. College, Saharanpur, U.P., India Department of Mathematis, H.C.T.M., Kaithal, Haryana, India Astrat The goal of present investigation is to study the effet of thermal gradient on the virations of non-homogeneous trapezoidal plate hose thikness varies paraolially and density varies linearly. Effet of other plate parameters suh as non-homogeneity onstant, taper onstant and aspet ratios have also een studied. C-S-C-S oundary ondition ith to term defletion is taken into onsideration. Rayleigh-Ritz method is used to find the solution of the prolem. Results are alulated ith great auray and presented in taular form. Comparison of the results ith pulished data has also shon. Keyords Viration, Trapezoidal Plate, Thermal Gradient, Non-Homogeneity, Paraolially Varying Thikness. Introdution Thermal effet on viration of non-homogenous plates are of great interest in the field of engineering suh as for etter designing of gas turines, jet engine, spae raft and nulear poer projets. Suh strutures are exposed to high intensity heat fluxes and thus material properties undergo signifiant hanges, in partiular the thermal effet on the modules of elastiity of the material an not e taken as negligile. Plates of variale thikness are frequently used in order to eonomize the plate material or to lighten the plates, espeially hen it is used in the ings of high-speed and high performane airrafts. The study of viration of plates has aquired great importane in the field of researh, engineering and spae tehnology. In the engineering e annot move ithout onsidering the effet of viration eause almost mahines and engineering struturese xperie nes virations. Strutures of plates have ide appliations in ships, ridges et. In the aeronautial field, analysis of thermally indued virations in non-homogeneous plates of variale thikness has a great interest due to their utility in airraft ings. Many analyses sho that plate virations are ased on non-homogeneity of materials. Non-homogeneity an e natural or art ifiial. Non-homogeneous materials suh as * Corresponding author: gupta_arunnitin@yahoo.o.in (Arun Kumar Gupta Pulished online at Copyright Sientifi & Aademi Pulishing. All Rights Reserved plyood, delta ood, fire-reinfored plasti et, are used in engineering design and tehnology to strengthen the onstrution. Study of the effet of viration annot e restrited only in the field of siene ut, our day to day life is also affeted y it. Viration of plates of various shapes, homogeneous or non-homogeneous, orthotropi or isotropi, ith or ithout variation in thikness, have een studied y various authors, ith or ithout onsidering the effet of temperature. A large numer of researhers have reported aout the viration analysis of different types of plates. Some of them are mentioned here. Sharma et al.[] studied the free transverse virations of non-homogeneous irular plates of linearly varying thikness. Gupta et al.[] orked on the virations of non-homogeneous retangular plate of variale thikness in oth diretions ith thermal gradient effet. Gutierrez and Laura [] alulated the fundamental frequeny of virating retangular, non-homogeneous plates. Gupta and Kumar[] studied the effet of thermal gradient on free viration of non-homogeneous viso-elasti retangular plate of paraolially varying thikness. Gupta et al.[5] oserved the transverse viration of non-homogeneous orthotropi viso-elasti irular plate of varying paraoli thikness. Lal et al [6] studied the transverse virations of non-homogeneous retangular plates of uniform thikness using oundary harateristi orthogonal polynomials. Gupta et al.[7] oserved the effet of non-homogeneity on the free virations of orthotropi viso-elasti retangular plate of paraoli varying thikness. Gupta and Kumar[8] studied the thermal effet on viration

2 66 Arun Kumar Gupta et al.: Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density of non-homogenous viso-elasti retangular plate of linear varying thikness. Gupta et al.[9] orked on the viration of viso-elasti orthotropi parallelogram plate ith paraolially thikness variation. El-Sayad and Ghazy[] studied the Rayleigh-Rit z method for free viration of midline trapezoidal plates. Gupta and Kumar[] did the free viration analysis of non-homogeneous viso-elasti irular plate ith varying thikness sujeted to thermal gradient. Lal and Dhanpati[] orked on the transverse virations of non-homogeneous orthotropi retangular plates of variale thikness. Gupta et al.[] did the viration analysis of non-homogeneous irular plate of nonlinear thikness variation y differential quadrature method. Chakraverty et al.[] notied the effet of non-homogeneity on natural frequenies of viration of ellipti plates. Chen et al[5] studied on free viration of non-homogeneous transversely isotropi magneto-eletro elasti plates. Gupta et al Reinfored onrete slas in uildings Gupta and Sharma[7] oserved the effet of thermal gradient on transverse viration of non-homogeneous orthotropi trapezoidal plate of paraolially varying thikness. Chen et al[8] studied the free viration of antilevered symmetrially laminated thik trapezoidal plates. Bamill et al.[9] studied the transverse virations of retangular, trapezoidal and triangular orthotropi, antilever plates. Gupta and Sharma[] oserved the thermally indued viration of orthotropi trapezoidal plate of linearly varying thikness. Gurses et al.[] analysed the shear deformale laminated omposite trapezoidal plates. Kitipornhai et al.[] disussed a gloal approah for viration of thik trapezoidal plates. Gupta and Sharma [] studied the thermal gradient effet on frequenies of a trapezoidal plate of linearly varying thikness. Gupta and Sharma[] studied the thermal effet on viration of non-homogeneous trapezoidal plate of linearly varying thikness. The authors have so far not ome aross any paper dealing ith paraolially varying thikness and linearly varying density. In the present ork the effet of non-homogeneity, taper onstant, thermal gradient and aspet ratios has een studied. The frequenies for the first to s of viration are otained for C-S-C-S non-homogeneous trapezoidal plate y Rayleigh-Ritz method. The authentiity and auray of numerial results of the present ork has een verified ith the authors pulished paper[5]. Results are presented in taular form. Comparison of results has also een presented.. Theoretial Formulation Steel ridge dek Automoile industry Fi gure. Use of plates [6] oservedthe effet of non-homogeneity on viration of orthotropi viso-elasti retangular plate of linearly varying thikness... Geometry of the Plate The plate under onsideration is shon in figure h ih is thin, symmetri and non-homogeneous trapezoidal plate. Here h is the maximum plate thikness ourring at the left edge and h is the minimum plate thikness ourring at the right edge... Thikness and Density The thikness of the plate is paraoli in x d iretion and is of the form ( = h [ ( ( h ( here h is thikness along the edge = x/a= -/. Non-homogeneity of plate arises due to variation in density hih is linear in x diretion and is of the form = ρ [ ( ( ] ρ ( It is assumed that the plate onsidered here is sujeted to a steady one-dimensional temperature distriution along the length i.e. in the x diretion, ]

3 Amerian Journal of Computational and Applied Mathematis, (6: y, / / / O / x, h ( = h [ ( (.5 ] h x, a/ a/ z Fi gure. Geometry of trapezoidal plate ith variale thikness Nomenlature of used Symols a length of the plate non-dimensional o-ordinate = idth of the plate at left edge non-dimensional o-ordinate = ω angular frequeny T kineti energy idth of the plate at right edge V strain energy x longitudinal o-ordinate λ frequeny parameter y vertial o-ordinate z transverse o-ordinate non-homogeneity onstant defletion funtion h plate thikness D ( flexural rigidity E Young s modulus β thermal gradient υ Poisson s ratio taper onstant τ temperature distriution β thermal gradient taper onstant ρ mass density per unit volume of the plate x a y

4 68 Arun Kumar Gupta et al.: Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density τ τ ( = ( here τ denotes the temperature exess aove the referene temperature at a distane x = a and τ denotes the temperature exess aove the referene temperature at the end =. For most of the elasti materials, modulus of elastiity is desried as E = E ( ( γτ here E is the value of Young s modulus along the referene temperature i.e. at τ = and γ is the slope of the variation of E ithτ. On sustituting value of τ from equation ( into ( E = E ( β ( (5 β = ( β. here γτ.. Equati on of Moti on The governing differential equation for kineti energy T and strain energy V is given y and T = a h( ρ da A ω (6 in hih ( Also a V = D ( da A a ( υ a a (7 D is the flexural rigidity of the plate, hih is given y ( ( = D D (8 E h D = (9 ( υ is the flexural rigidity of the plate at the side =, A is the area of the plate and ρ is the mass density per unit area of the plate. Using equation (9 and (5 in (8 flexural rigidity is given y E = h D ( ( β υ ( ( Using equation ( in (7 & (, ( in (6 a E a h V = ( β da A ( ( υ ( υ a a a T = h ω [ ( ( ][ ( ( ] da.. Defletion Funtion and Boundary Conditi on A to term defletion funtion is taken as ρ ( A

5 Amerian Journal of Computational and Applied Mathematis, (6: = A A ( here A and A are onstants to e evaluated. Eq. ( satisfy oundary onditions and provide a good estimation to the frequeny. Clamped-Simply supported-cla mped-simply supported plate is taken into onsideration as shon in figure. Fi gure. C-S-C-S oundary ondition of a trapezoidal plate Also for the plate onsidered here oundaries are defined y four straight lines = = = = (. Method of Solution Rayleigh-Rit z tehnique is used to find the solution of the prolem. Aording to it maximum kineti energy must e equal to ma ximu m strain energy, so it is neessary for the prolem under onsideration that ( = T V δ (5 Using equation ( in ( and (.V= ( υ β υ d d a a a h E a ( ( (6 T = ρ ω d d h a ] ( ( ][ ( ( ([ (7 No (5 eomes ( = T V λ δ (8 here y x

6 7 Arun Kumar Gupta et al.: Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density V a = and λ a υ a a ( ( β dd T = ω ρa ( ν Eh a ([ ( ( ][ ( ( ] dd = is a frequeny parameter. (9 ( Equation (8 involves the unknon A and A arising due to the sustitution of fro m eq (. These to onstants are to e determined from eq. (8, as follos A A ( V λ T = ( V T = On simplifying (, One gets A m A = m, m ( m =, λ ( m, m =, ( here involve parametri onstant and the frequeny parameter. For a non-zero solution, it is desired that o-effiient of eq. ( must e zero. So one gets the frequeny equation as = ( λ Fro m eq. (, one an otain a quadrati equation in from hih the to values an found hih onstitutes first and seond of viration.. Results and Disussion Frequenies for the first to s of virations are omputed for non-homogeneous trapezoidal plate hose thikness varies paraolially and density varies linearly. Different values of non-homogeneity onstant (, taper onstant (, thermal gradient ( β and aspet ratios (a/, / has een onsidered. All results are presented in taular form. Tale and : These tales shos the effet of taper onstant (. to. on the frequeny parameter λ for β =.,., =.,., a/ =. and / =.5,.. Tale and : These tales shos the ehaviour of the frequeny parameter λ ith thermal gradient β (. to., for =.,., =.,., a/ =. and / =.5,.. Tale 5, 6, 7 and 8: These tales sho the effets of the frequeny parameter λ ith aspet ratio / for different ominations of and β as follo s. (i =., β =. (ii =., β =. (iii =., β =. (iv =., β =.. Non-homogeneity onstant varies fro m. to., aspet ratio a/ varies from.75 to. and aspet ratio / varies fro m.5 to.. Tale 9 and : In these tales effet of non-homogeneity onstant (. to. on frequeny parameter has een shon. Four ominations of and β (as in tale 5 to 8, to values of aspet ratio / (.5,. and one value of aspet ratio a/ (. has een taken. In tale and it is oserved that the frequeny parameter λ inreases ith inreasing values of taper onstant for oth the s of viration. The rate of inrease as ell as the value of frequeny parameter λ is higher in seond in omparison to the first. Further on omparing the results of tale and it is found that on inreasing the value of aspet ratio / fro m.5 to., frequeny parameter dereases for oth the s of viration. In tale and it is seen that frequeny parameter λ dereases ith inreasing values of thermal gradient β hatever e the values of other plate parameters. The rate of derease as ell as the value of frequeny parameter λ is higher in seond in omparison to the first. A lso on inreasing aspet ratio / from.5 to., frequeny parameter dereases for oth the s of viration. It is notied fro m tale 5, 6, 7 and 8 that the frequeny parameter λ dereases ith inreasing values of aspet ratio /. Fu rther if e o mpare tale 5 and 6 (a/ =.75 it is found that the frequeny parameter dereases ith inrease in non-homogeneity onstant i.e. fro m. to.. Similar pattern is oserved on omparing tale 7 and 8 (a/ =.. No if e ompare tale 5 and 7 in hih aspet ratio a/ inreased from.75 to. and non-homogeneity onstant

7 Amerian Journal of Computational and Applied Mathematis, (6: is.. It is oserved that frequeny parameter inreases for oth the of viration. Similar pattern is oserved on omparing tale 6 and 8 in h ih non-homogeneity onstant is.. Tale 9 and learly shos that frequeny parameter λ dereases ith inreasing value of non-homogeneity onstant. Also hen aspet ratio / inreased from.5 to., frequeny parameter dereases for oth the of viration Ta le. Frequeny paramet er ( λ for a trapezoidal plate for different values of taper onstant (, thermal gradient ( β =.,., non-homogeneity onstant ( =.,. and aspet ratios (a/ =., (/ =.5 =. =. β =. β =. β =. β =. Seon d m ode m ode * *8.769 *.555 * *5.7 * *.67 * *5.866 *9.657 * * *6.975 *97.75 *. * *7.9 *.68 *5.5 * *8.9 *.8 *5.759 *9.669 *Comparison ith author s paper[5] Tale. Frequeny parameter ( λ for a trapezoidal plate for different values of taper onstant (, thermal gradient ( β =.,., non-homogeneity onstant ( =.,. and aspet ratios (a/ =., (/ =. =. =. β =. β =. β =. β =. Seon d m ode Seon d m ode *Comparison ith author s paper[5].9956 * * * * * * * * * * * * * * * * * * * * * * * *6.9

8 7 Arun Kumar Gupta et al.: Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density Ta le. Frequeny parameter ( λ for a trapezoidal plate for different values of thermal gradient β, t aper onst ant ( =.,. non-homogeneity onstant ( =.,. and aspet ratios (a/ =., (/ =.5, =. =. β =. =. =. =. Seon d * * * * * * * * * * * * * * * * * * * * *5.65 *Comparison ith author s paper[5] Ta le. Frequeny parameter ( λ for a trapezoidal plate for different values of thermal gradient β, taper onstant ( =.,. non-homogeneity onstant ( =.,. and aspet ratios (a/ =., (/ =., =. =. β =. =. =. = Seon d m ode Seon d m ode * * * * * * * * * * * * * * * * * * * * * * * * *Comparison ith author s paper[5]

9 Amerian Journal of Computational and Applied Mathematis, (6: Ta le 5. Frequeny paramet er ( λ for a trapezoidal plate for different ominations of thermal gradient ( β, taper onstant ( and fixed value of non-homogeneity onstant ( =. & aspet ratio (a/ =.75 / =., β =. =., β =. =., β =. =., β = Ta le 6. Frequeny paramet er ( λ for a trapezoidal plate for different ominations of thermal gradient ( β, taper onstant ( and fixed value of non-homogeneity onstant ( =. and aspet ratio (a/ =.75 =., β =. =., β =. =., β =. =., β =. / Ta le 7. Frequeny paramet er ( λ for a trapezoidal plate for different ominations of thermal gradient ( β, taper onstant ( and fixed value of non-homogeneity onstant ( =. and aspet ratio (a/ =. / =., β =. =., β =. =., β =. =., β = Ta le 8. Frequeny paramet er ( λ for a trapezoidal plate for different ominations of thermal gradient ( β, taper onstant ( and fixed value of non-homogeneity onstant ( =. and aspet ratio (a/ =. / =., β =. =., β =. =., β =. =., β =. Mode

10 7 Arun Kumar Gupta et al.: Study of Thermally Indued Viration of Non-Homogeneous Trapezoidal Plate ith Varying Thikness and Density Ta le 9. Frequeny paramet er ( λ for a trapezoidal plate for different value of non-homogeneity onstant ith different ominations of thermal gradient ( β and t aper onst ant ( and aspet ratios (a/ =., (/ =.5 =., β =. =., β =. =., β =. =., β = Ta le. Frequeny paramet er ( λ for a trapezoidal plate for different value of non-homogeneity onstant ith different ominat ions of t hermal gradient ( β and taper onstant ( and aspet ratios (a/ =., (/ =. =., β =. =., β =. =., β =. =., β = Confirmation of Results The auray of the present omputations is ompared ith the pulished results[5] for C-S-C-S non-homogeneo us trapezoidal plate ith non-homogeneity onstant =., aspet ratio a/ =., / =.5,. and thermal gradient β =. to. and taper onstant =. to.. Tale,, and shos a omparison of the values of frequeny parameter otained in the present prolem and pulished paper of the authors[5]. A very lose agreement is seen eteen the present results and of the pulished paper in hih homogeneous plate ith paraolially varying thikness (density is onstant has een onsidered. 6. Conlusions The main purpose of the present ork is to study the effet of thermal gradient on the frequenies of C-S-C-S non-homogeneous trapezoidal plate ith other plate parameters as taper onstant, non-homogeneity onstant and aspet ratios. Thikness of the plate varies paraolially and density varies linearly. Ray leigh-ritz tehnique is used to find frequenies for first to s of viration. Study shos that frequeny parameters inreases ith inreasing value of taper onstant and aspet ratio a/ hereas it dereases ith inreasing value of non-homogeneity onstant, thermal gradient and aspet ratio /. REFERENCES [] Sharma S, Lal R, Neelam. Free transverse virations of non-homogeneous irular plates of linearly varying thikness. Journal of International Aademy of Physial Sienes ; 5: 87-. [] Gupta AK, Panar V, Vats RP. Virations of non-homogeneous retangular plate of variale thikness in oth diretions ith thermal gradient effet. Int. J. of Appl. Math and Meh. ; 6:9-7. [] Gutierrez RH, Laura PAA. Fundamental frequeny of

11 Amerian Journal of Computational and Applied Mathematis, (6: virating retangular, non-homogeneous plates. Applied Aoustis 985; 8: 7-8. [] Gupta AK, Kumar L. Effet of thermal gradient on free viration of non-homogeneous viso-elasti retangular plate of paraolially varying thikness. Ata Tehnia CSAV 9; 5: [5] Gupta AK, Agaral N, Kaur H. Transverse viration of non-homogeneous orthotropi viso-elasti irular plate of varying paraoli thikness. Mathematial Methods in the Applied Sienes ; : [6] Lal R, Kumar Y, Gupta US. Transverse virations of non-homogeneous retangular plates of uniform thikness using oundary harateristi orthogonal polynomials. Int. J. Applied Mathematis and Mehanis ; 6: 9-9. [7] Gupta AK, Agaral N, Gupta DV, Kumar S, Sharma P. Effet of non-homogeneity on the free virations of orthotropi viso-elasti retangular plate of paraoli varying thik-ness. Adv. Studies Theor. Phys. ; : [8] Gupta AK, Kumar L. Thermal effet on viration of non-homogenous viso-elasti retang-ular plate of linear varying thikness. Meania 8; : 7-5. [9] Gupta AK, Kumar A, Gupta DV. Viration of viso-elasti orthotropi parallelogram plate ith paraolially thikness variation. Annals of Faulty Engineering, Hunedoara, International Journal of Engineering ; : 6-7. [] El-Sayad MA, Ghazy SSA. Rayleigh-Ritz method for free viration of midline trapezoidal plates. Canadian Journal on Siene and Engineering Mathematis ; : [] Gupta AK, Kumar L. Free viration analysis of non-homogeneous viso-elasti irular plate ith varying thikness sujeted to thermal gradient. Ata Tehnia CSAV ; 56: 7-. [] Lal R, Dhanpati. Transverse virations of non-homogeneous orthotropi retangular plates of variale thikness. J. Sound and Viration 7; 6: -. [] Gupta US, Lal R, Sharma S. Viration analysis of non-homogeneous irular plate of nonlinear thikness variation y differential quadrature method. J. Sound and Viration 6; 98: [] Chakraverty S, Jindal R, Agaral VK. Effet of non-homogeneity on natural frequenies of viration of ellipti plates. Meania 7; : [5] Chen WQ, Lee KY, Ding HJ. On free viration of non-homogeneous transversely isotropimagneto-eletroelas ti plates. J. Sound and Viration 5; 79: 7-5. [6] Gupta AK, Agaral N, Gupta DV, Kumar M. Study the non-homogeneity effet on viration of orthotropi viso-elasti retangular plate of linearly varying thikness. Int. Revie of Pure and Applied Mathematis ; 6: 5-8. [7] Gupta AK, Sharma S. Study the effet of thermal gradient on transverse viration of non-homogeneous orthotropi trapezoidal plate of paraolially varying thikness. Applied Mathematis ; : -. [8] Chen CC, Kitipornhai S, Lim CW, Lie KM. Free viration of antilevered symmetrially laminated thik trapezoidal plates. Int. J. Mehanial Sienes 999; : [9] Bamill DV, Laura PAA, Rossi RE. Transverse virations of retangular, trapezoidal and triangular orthotropi, antilever plates. J. Sound and Viration 998; : [] Gupta AK, Sharma S. Thermally indued viration of orthotropi trapezoidal plate of linearly varying thikness. J. Viration and Control ; 7: [] Gurses M, Civalek O, Ersoy H. Analysis of shear deformale laminated omposite trape-zoidal plates. Materials and Design 9; : -5. [] Kitipornhai S, Xiang Y, Lie KM, Lim MK. A gloal approah for viration of thik trapezoidal plates. Computers and Strutures 99; 5: 8-9. [] Gupta AK, Sharma P. Study the thermal gradient effet on frequenies of a trapezoidal plate of linearly varying thikness. Applied Mathematis ; : [] Gupta AK, Sharma P. Thermal effet on viration of non-homogeneous trapezoidal plate of linearly varying thikness. Int. J. of Applied Mathematis and Mehanis ; 7: -7. [5] Gupta AK, Sharma P. Effet of linear thermal gradient on virations of trapezoidal plates hose thikness varies paraolially. J. Viration and Control ; 8: 95-.

Determination of the rotational restraint coefficient and local buckling of orthotropic box columns

Determination of the rotational restraint coefficient and local buckling of orthotropic box columns etermination of the rotational restraint oeffiient and loal ukling of orthotropi o olumns A. A. Maghsoudi, H. Saffari & H. Rezaeian epartment of Civil Engineering, Shahid Bahonar University, Iran Astrat