Effect of Winkler foundation, inhomogeneity and orthotropy on the frequency of plates

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1 Journl of Structurl Engineering & Applied ecnics 8 Volume Issue Pges -5 ttps://doi.org/.346/jsem.8.5.goldenligtpulis.com RESEARCH ARTICLE Effect of Winkler foundtion inomogeneity nd ortotropy on te frequency of pltes V.C. Hciyev *.R. irzeyev nd A.I. Siriyev Ntionl Acdemy of Sciences of Azerijn Institute of temtics nd ecnics Bku Azerijn Astrct Tis study investigtes te frequencies of inomogeneous ortotropic pltes on elstic foundtion. Te inomogeneity of ortotropic mterils vries linerly it te coordintes of lengt nd tickness. Te sic eqution is derived pplying te Donnell-ustri teory nd solved using te lerkin metod. Te effects of inomogeneity Winkler elstic foundtion nd ortotropy on te frequencies re investigted in detil. Keyords Free virtion; Inomogeneous ortotropic mterils; Pltes; Winkler elstic foundtion Received: 4 rc 8; Accepted: rc 8 ISSN: (online) 8 olden Ligt Pulising All rigts reserved.. Introduction Inomogeneous structures re idely used in vition erodynmic structure spce veicles ligt-lloy structure of crs nd in oter engineering structures. Te increse in te numer of constructive vriles extends te possiilities of dvnced composite mterils s ell s stility nd virtion eviors my e significntly ltered. Te reson for te ppernce of eterogeneity of te mteril cn e mnufcturing tecnology terml nd mecnicl tretment eterogeneity of compositions nd numer of oter resons []. Significnt contriutions to consider different types of inomogeneity re given in references [-7]. In mny prcticl pplictions composite pltes re in contct it soils or oter solid prticles nd cn ve significnt nd unvoidle effect on teir eviors. A compreensive revie of elstic foundtion models is discussed in te ork y orunov-posdov et l. [8]. Virtion of omogeneous nd nonomogeneous ortotropic pltes resting on elstic foundtion ic s prcticl pplictions in civil mecnicl mrine nd erospce engineers ve een extensively studied y using vrious nlyticl nd numericl metods [9-5]. In mjority of te ove-mentioned-studies te elstic properties of inomogeneous ortotropic mterils ere vried depending on te coordinte of te tickness or te coordintes in te plne seprtely. Te proposed study descries te free virtion of eterogeneous ortotropic pltes on te Winkler elstic foundtion in ic te inomogeneity linerly vries long te lengt nd tickness togeter.. Formultion of te prolem Te configurtion of rectngulr inomogeneous ortotropic plte it te lengt te idt nd te tickness resting on Winkler elstic * Corresponding utor E-mil: vgif.ciyev.imm@gmil.com

2 Effect of Winkler foundtion inomogeneity nd ortotropy on te frequency of pltes foundtion is illustrted in Fig.. Te plte referred to system of rectngulr coordinte system Oxyz. Te mid-plne eing z = nd te origin is t one corners of te ortotropic plte. z =. Te x nd y xes re tken long te principle directions of ortotropy nd z xis is norml to tem. It is ssumed tt Young's moduli nd ser modulus of te eterogeneous ortotropic plte vry linerly it te coordintes x nd z s x z E E ( ) x z E E ( ) x z ( ) ile density nd Poisson's rtios ssumed to e constnt [-7]. Here E te Young s moduli in x nd y directions nd nd E () re re is te ser modulus nd is te density of te omogeneous ortotropic plte. Here X = x / nd Z = z / re te dimensionless vriles in x nd z directions; nd re non-omogeneity prmeters vried from zero to unity. As nd te eterogeneous mteril is converted into omogeneous mteril. 3. Bsic reltions Bsed on te clssicl sell teory te reltionsips eteen te stresses nd strins t n ritrry point of inomogeneous ortotropic pltes re ritten in te folloing form [-7]: Fig.. Te inomogeneous ortotropic rectngulr plte on te elstic foundtion nd te coordinte system x z E ( ) e e z x y x z E ( ) e e z y x x z ( ) e z xy () ere is te smll displcement in z direction nd eij ( i j ) re te strins in te middle plne of inomogeneous ortotropic plte. Te force nd moment resultnts re expressed y te folloing reltions [5 6]: T T T / ( z z z ) dz / (3) Since tere re no externl forces in te plne of ( T T T ) it is terefore ssumed tt plte te resultnt forces re everyere is equl to zero. In tis cse te folloing conditions cn e ritten: A e e A A e e A Ae A ere / (4) A A nd ( i j ) re te curvtures of te deformed plte. Tking into ccount Eq. () in te expression (3) fter integrting e otin te folloing expressions for te moments: x x y D ( ) x y x D ( ) x x y D ( ) ij (5)

3 3 Hciyev et l. ere D D D re flexurl rigidities of te omogeneous ortotropic plte nd re defined s: D E D E D ( ) ( ) (6) Te prtil differentil eqution of te motion of ortotropic plte resting on te Winkler elstic foundtion cn e ritten s [5]: K x xy y t (7) ere K (in N / m 3 ) is te modulus of sugrde rection for te Winkler elstic foundtion [56]. Sustituting Eq. (5) into Eq. (7) fter elementry trnsformtions e otin te eqution of motion of inomogeneous ortotropic plte resting on te Winkler elstic foundtion s: 4 4 x ( ) D D 4 4 x y 4 D D 4D x y D 4 x y x xy D K t 4. Te solution of eqution of motion (8) We ssume tt te oundry conditions for te ending of continuous inomogeneous ortotropic pltes coincide it te usul ones in te omogeneous isotropic plte. We tke te rmonic solution of Eq. (8) in te form [56] mx ny it ( x y t) Asin sin e (9) ic stisfies te simply-supported oundry conditions edges of te inomogeneous ortotropic rectngulr pltes ere i m nd n re ve numers nd A is te unknon mplitude. Sustituting Eq. (9) into Eq. (8) ten pplying te lerkin s metod to te resulting eqution for m = n = te expression for te frequency of free virtion of inomogeneous ortotropic pltes resting on te Winkler elstic foundtion is otined s: K ( ) ( ) D D D D 4D ere 4 4 / nd /. / () Te vlues of te dimensionless frequency prmeter for inomogeneous ortotropic pltes resting on te Winkler elstic foundtion is found s: / E () As nd K te frequency coincides it te frequency of te omogeneous ortotropic plte itout n elstic foundtion nd cn e expressed s: 4 4 D D D D 4D () 5. Results nd discussion Tle presents comprtive study of te dimensionless frequency prmeters for te flexurl modes of tin omogeneous isotropic squre pltes on Winkler foundtion. Dimensionless prmeters re specified s 4 K / D K.3 (3) D Tle. Comprison of frequency prmeters for te flexurl modes of tin squre pltes on omogeneous Winkler elstic foundtion ( / / ) K Reference Leiss [5].4 Zou et l. [].43 Present study.4 5 Leiss [5] 3. Zou et l. [] 3.4 Present study 3.

4 Effect of Winkler foundtion inomogeneity nd ortotropy on te frequency of pltes 4 Tle. Vrition of frequency prmeters of te inomogeneous ortotropic rectngulr pltes on inomogeneous viscoelstic nd elstic foundtions versus ( /.5 m / 5 ) K K 5 (N/m ) Tle 3. Vrition of frequency prmeters of te inomogeneous ortotropic rectngulr pltes on Winkler elstic foundtion ( / 5 m ) / K / K 5 (N/m ) Tese vlues re tken from te study of Leiss [5]. As seen te nturl frequencies re in good greement it te results of Leiss [5] nd Zou et l. [] for tinner pltes. Vrition of te dimensionless frequency prmeters for omogeneous nd inomogeneous rectngulr pltes it nd itout Winkler elstic foundtion ginst E/ Ertio for / =.5 = m nd / =.5 re presented in Tle. Te mteril properties of ortotropic plte re 5 4 nd 5 E 6.9 P.5 3 kg/m.5e. As Tle sos te dimensionless frequency prmeters for te considered cses re ekly decresed en rtio increses. Since te soil effect is not tken into ccount te eterogeneity effect is pproximtely independent of rtio ltoug te eterogeneity effect is considerle. For exmple if te foundtion effect is not tken into ccount te effect of non-omogeneity is significnt (.47%) en te Young s nd ser moduli cnge only in x direction eres tis effect is found to e -4.3% en te Young moduli cnge only in z direction nd te effect of eterogeneity is %7.3 s te Young's moduli cnge in x nd z directions togeter. Here te negtive sign indictes tt te vlue for te inomogeneous plte is smller tn te omogeneous plte. Considering te foundtion effect significnt cnges occur in te effects of non-omogeneity on te frequencies. For exmple s te Winkler 6 foundtion coefficient is tken to e K 5 N/m 3 te effect of non-omogeneity on te frequencies increses y up to 9% s te Young s nd ser moduli cnge only in x direction; tis effect increses from 3.7% to 4.4% en te Young nd ser moduli vry only in z direction te effect of non-omogeneity ek increse from 3.83% to 4.47% s te Young nd ser moduli vry in x nd z directions togeter s E/ E rtio increses from to 55. Vritions of te dimensionless frequency prmeters of omogeneous nd inomogeneous rectngulr pltes it nd itout Winkler elstic foundtion ginst / rtio for / = 5 nd = m re presented in Tle 3. Te mteril properties of ortotropic plte re E = 6.9P E =.69P = 6.9P nd = 95kg/m 3 Tle 3 sos tt te vlues increse it incresing /. Te effect of te Winkler foundtion on te frequency vlues first increses

5 5 Hciyev et l. nd ten decreses s te / increses from.5 to. For exmple te effect of Winkler foundtion on te frequencies of omogeneous ortotropic pltes cnces eteen 8.43% nd 3.55%. For te inomogeneous ortotropic pltes te effect of elstic foundtion cnges in te rnge of 9.48% nd 33.76% s te Young s nd ser moduli cnge only in x direction. Tis effect cnges in te rnge of 4.54% nd 9.76% en te Young s nd ser moduli vry only in te z direction nd te rnge of te influence of te inomogeneity is 4.53% to 9.% s te Young nd ser moduli vry in te x nd z directions togeter since te rtio / 6. Conclusions increses from.5 to. Te purpose of tis study is to investigte te effects of te inomogeneous nd elstic foundtion on te frequency of ortotropic pltes. Te inomogeneity of ortotropic mterils vries linerly it te coordintes of te lengt nd tickness. Te sic eqution is derived pplying te Donnell-ustri teory nd solved using te lerkin s metod. Te effects of inomogeneity Winkler elstic foundtion nd ortotropy on te frequencies re investigted in detil. References [] Kolcin AS Fvrion EA (977) Teory of Elsticity of Inomogeneous Bodies. Cisinu USSR. [] Hciyev VC Sofiyev AH irzoyev RD (996) Free virtion of non-omogeneous elstic rectngulr pltes. Proceedings of Institute of temtics nd ecnics Acdemy of Sciences of Azerijn IV (XIII) 3-8 Bku (in Russin). [3] Sofiyev AH Aksogn O (999) Dynmic stility of non-omogeneous ortotropic elstic cylindricl sell under time dependent externl pressure. Tecnicl Journl Turkis Cmer of Civil Engineers :67-6. [4] Fres E Zenkour A (999) Buckling nd free virtion of non-omogeneous composite crossply lminted pltes it vrious plte teories. Composite Structures 44: [5] Sofiyev AH Omurtg H Scnck E (9) Te virtion nd stility of ortotropic conicl sells it non-omogeneous mteril properties under ydrosttic pressure. Journl Sound nd Virtion 39(3-5): [6] Xi P Long SY Cui HX Li Y (9) Te sttic nd free virtion nlysis of nonomogeneous modertely tick plte using te mesless locl rdil point interpoltion metod. Engineering Anlysis it Boundry Elements 33: [7] Ll R Kumr Y (3) Trnsverse virtions of nonomogeneous rectngulr pltes it vrile tickness. ecnics of Advnced terils nd Structures : [8] orunov-possdov N likov TA Solomin VI (984) Design of Structures on Elstic Foundtion. Strojizdt osco Russi. [9] Omurtg H Kdioğlu F (998) Free virtion nlysis of ortotropic pltes resting on Psternk foundtion y mixed finite element formultion. Computers nd Structures 67: [] Zou D Ceung YK Lo SH Au FTK (4) Tree-dimensionl virtion nlysis of rectngulr tick pltes on Psternk foundtion. Interntionl Journl for Numericl etods Engineering 59: [] orimoto T Tnig Y (7) Elstic stility of inomogeneous tin pltes on n elstic foundtion. Arcives Applied ecnics 77(9): [] Hsu H () Virtion nlysis of ortotropic rectngulr pltes on elstic foundtions. Composite Structures 9: [3] Ll R R Sini (5) Buckling nd virtion nlysis of non-omogeneous rectngulr Kircoff pltes resting on to-prmeter foundtion. eccnic 5(4): [4] en SS Al-Kouz W (6) Virtion nlysis of non-uniform ortotropic Kircoff pltes resting on elstic foundtion sed on nonlocl elsticity teory. Interntionl Journl of ecnicl Sciences 4:-. [5] Leiss AW (973) Te free virtion of pltes. Journl of Sound Virtion 3: [6] Reddy JN (4) ecnics of Lminted Composite Pltes nd Sells: Teory nd Anlysis. nd ed. CRC Press.