Sgma J Eng & Nat Sc 36 (1), 218, 191-26 Sgma Journal of Engneerng and Natural Scences Sgma Mühendslk ve Fen Blmler Dergs Research Artcle PARAMETRIC EIGENVALUE ANALYSIS OF MINDLIN PLATES RESTING ON WINKLER FOUNDATION WITH SECOND ORDER FINITE ELEMENT Yaprak Itır ÖZDEMİR* Department of Cvl Engneerng, Karadenz Techncal Unversty, TRABZON; ORCID:-2-658-1366 Receved: 15.12.217 Accepted: 23.2.218 ABSTRACT The purpose of ths paper s to study free vbraton analyss of thck plates restng on Wnkler foundaton usng Mndln s theory wth shear lockng free fourth order fnte element, to determne the effects of the thckness/span rato, the aspect rato, subgrade reacton modulus and the boundary condtons on the frequency paramerets of thck plates subjected to free vbraton. In the analyss, fnte element method s used for spatal ntegraton. Fnte element formulaton of the equatons of the thck plate theory s derved by usng second order dsplacement shape functons. A computer program usng fnte element method s coded n C++ to analyze the plates free, clamped or smply supported along all four edges. In the analyss, 8-noded fnte element s used. Graphs are presented that should help engneers n the desgn of thck plates subjected to earthquake exctatons. It s concluded that 8-noded fnte element can be effectvely used n the free vbraton analyss of thck plates. It s also concluded that, n general, the changes n the thckness/span rato are more effectve on the maxmum responses consdered n ths study than the changes n the aspect rato Keywords: Free vbraton parametrc analyss, thck plate, mndln s theory, second order fnte element, wnkler foundaton. 1. INTRODUCTION Plates are structural elements whch are commonly used n the buldng ndustry. A plate s consdered to be a thn plate f the rato of the plate thckness to the smaller span length s less than 1/2; t s consdered to be a thck plate f ths rato s larger than 1/2 [1]. The dynamc behavor of thn plates has been nvestgated by many researchers [2-17]. There are also many references on the behavor of the thck plates subjected to dfferent loads. The studes made on the behavor of the thck plates are based on the Ressner-Mndln plate theory [18, 19, 2, 21]. Ths theory requres only C contnuty for the fnte elements n the analyss of thn and thck plates. Therefore, t appears as an alternatve to the thn plate theory whch also requres C 1 contnuty. Ths requrement n the thn plate theory s solved easly f Mndln theory s used n the analyss of thn plates. Despte the smple formulaton of ths theory, dscretzaton of the plate by means of the fnte element comes out to be an mportant parameter. In many cases, numercal soluton can have lack of convergence, whch s known as shear-lockng. Shear lockng can be avoded by ncreasng the mesh sze,.e. usng fner mesh, but f the * Correspondng Author: e-mal: yaprakozdemr@hotmal.com, tel: (462) 377 4 18 191
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 thckness/span rato s to o small, convergence may not be acheved even f the fner mesh s used for the low order dsplacement shape functons. In order to avod shear lockng problem, the dfferent methods and technques, such as reduced and selectve reduced ntegraton, the substtute shear stran method, etc., are used by several researchers [22, 23, 24, 25, 26]. The same problem can also be prevented by usng hgher order dsplacement shape functon [27, 28]. Wanj and Cheung [29] proposed a new quadrlateral thn/thck plate element based on the Mndln-Ressner theory. Soh et al. [3] mproved a new element ARS-Q12 whch s a smple quadrlateral 12 DOF plate bendng element based on Ressner-Mndln theory for analyss of thck and thn plates. Brezz and Marn [31] developped a lockng free nonconformng element for the Ressner-Mndln plate usng dscontnuous Galarkn technques. Belounar and Guenfound [32] mproved a nev rectangular fnte element based on the stran approach and the Ressner-Mndln theory s presented for the analyss of plates n bendng ether thck or thn. Vbraton analyss made by Senjanovc et al. [33], n the paper the modfed Mndln theory s used for the constructon of the dynamc stffness matrx, the flexblty matrx, and the transfer matrx of a thck plate smply supported at two opposte edges. They presented natural frequences and modes of panel Mndln plates. S et al. [16] studed vbraton analyss of rectangular plates wth one or more guded edges va bcubc B-splne method, Cen et al. [34] developped a new hgh performance quadrlateral element for analyss of thck and thn plates. Ths dstngushng character of the new element s that all formulatons are expressed n the quadrlateral area co-ordnate system. Shen et al. [35] studed free and forced vbraton of Ressner-Mndln plates wth free edges restng on elastc foundatons. Woo et al. [36] found accurate natural frequences and mode shapes of skew plates wth and wthout cutouts by p- verson fnte element method usng ntegrals of Legendre polynomal for p=1-14. Qan et al. [37] studed free and forced vbratons of thck rectangular plates usng hgher-order shear and normal deformable plate theory and meshless Petrov-Galarkn method. Özdemr and Ayvaz [38] studed shear lockng free earthquake analyss of thck and thn plates usng Mndln s theory. GuangPeng et al. [39] studed free vbraton analyss of plates on Wnkler elastc foundaton by boundary element method. Fallah et. al. [4] analyzed free vbraton of moderately thck rectangular FG plates on elastc foundaton wth varous combnatons of smply supported and clamed boundary condtons. Governng equatons of moton were obtaned based on the Mndln plate theory. Jahrom et al. [41] analyzed free vbraton analyss of Mndln plates partally restng on Pasternak foundaton. The governng equatons whch consst of a system of partal dfferental equatons are obtaned based on the frst-order shear deformaton theory. Özgan and Daloğlu [42] studed free vbraton analyss of thck plates on elastc foundatons usng modfed Vlasov model wth hgher order fnte elements, also same wrters [43] studed the effects of varous parameters such as the aspect rato, subgrade reacton modulus and thckness/span rato on the frequency parameters of thck plates restng on Wnkler elastc foundatons. The purpose of ths paper s to study free vbraton analyss of thck plates restng on Wnkler foundaton usng Mndln s theory wth second order fnte element, to determne the effects of the thckness/span rato, the aspect rato, subgrade reacton modulus and the boundary condtons on the frequency paramerets of thck plates subjected to free vbraton. A computer program usng fnte element method s coded n C++ to analyze the plates free, clamped or smply supported along all four edges. In the program, the fnte element method s used for spatal ntegraton. Fnte element formulaton of the equatons of the thck plate theory s derved by usng hgher order dsplacement shape functons. In the analyss, 8-noded fnte element s used to construct the stffness and mass matrces of the thck and thn plates [27]. 2. MATHEMATICAL MODEL The governng equaton for a flexural plate (Fg. 1) subjected to free vbraton wthout dampng can be gven as 192
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 Mw K w (1) where [K] and [M] are the stffness matrx and the mass matrx of the plate, respectvely, w and w are the lateral dsplacement and the second dervatve of the lateral dsplacement of the plate wth respect to tme, respectvely., z y x b t w x y k w a Fgure 1. The sample plate used n ths study The total stran energy of plate-sol-structure system (see Fg. 1) can be wrtten as; П= П p + П s + V (2) where П p s the stran energy n the plate, T x y x y x y x y p = 1 E d A 2 x y y x x y y x A T k w w w w + x y E x y d A 2 x y x y - (3) A where П s s the stran energy stored n the sol, H 1 s = jj (4) 2 and V s the potental energy of the external loadng; V=- A q wda (5) In ths equaton E and E are the elastcty matrx and these matrces are gven below at Eq. (17), q shows appled dstrbuted load. 2.1. Evaluaton of the stffness matrx The total stran energy of the plate-sol system accordng to Eq. (2) s; 193
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 T U e = 1 x y x y x y x y E d A 2 x y y x x y y x + A T k w w w w x y E x y d A 2 x y x y + A 1 T w x,y w x,y d A 2 (6) A At ths equaton the frst and second part gves the conventonal element stffness matrx of the plate, [k e p ], dfferentaton of the thrd ntegral wth respect to the nodal parameters yelds a matrx, [k e w ], whch accounts for the axal stran effect n the sol. Thus the total energy of the plate-sol system can be wrtten as; 1 T e e U e = w e k p k w w e d A (7) 2 where w w... w T e 1 y1 x1 n yn xn (8) Assumng that n the plate of Fg. 1 u and v are proportonal to z and that w s the ndependent of z [21], one can wrte the plate dsplacement at an arbtrary x, y, z n terms of the two slopes and a dsplacement as follows; u ={w, v, u}={w (x,y,t), zφ y (x,y,t), -zφ x (x,y,t)} (9) where w s average dsplacement of the plate, and φ x and φ y are the bendng slopes n the x and y drectons, respectvely. The nodal dsplacements for 8-noded fnte element (MT8) (Fg. 2) can be wrtten as follows; w= 8 w 8 y= z h y, u=-zφ x = z h x 1 1 1 1,...,8 for 8 - noded element, (1) The dsplacement functon chosen for ths element s; 2 2 2 2 w c1 c2r c3s c4r c5rs c6s c7r s c8rs (11) From ths assumpton, t s possble to derve the dsplacement shape functon to be [38]; h [h1, h 2, h3,h 4, h5, h6, h7,h8] (12) 194
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 w x y 4 7 z 3 y 4 7 s 3 1 8 6 8 r 6 x w 1 5 2 5 2 [B] Fgure 2. 8- noded fnte element(second order),used n ths study [47]. Then, the stran-dsplacement matrx [B] for ths element can be wrtten as follows [44]: h x h y h y h x -... 1,...,8 for 8 - noded element (13) h h - x h y - h............ 5x21 The stffness matrx for MT8 element can be obtaned by the followng equaton [44]: 1 1 K 1 J drds (14) 1 A T T B DB da. B DB whch must be evaluated numercally [26]. As seen from Eq. (14), n order to obtan the stffness matrx, the stran-dsplacement matrx, [B], and the flexural rgdty matrx, [D], of the element need to be constructed The flexural rgdty matrx, [D], can be obtaned by the followng equaton. Ek [D] E γ. (15) In ths equaton, [ E k ] s of sze 3x3 and [ E ] s of sze 2x2. [ E k ], and [ E ] can be wrtten as follows [45, 47]: 195
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 2 2 E 1 1 3 [ E k ]= t ; [ E ]= 2. 41 k t 12 2 2 1 1 (16) E 2. 4 1 2 1 where E, υ, and t are modulus of the elastcty, Posson s rato, and the thckness of the plate, respectvely, k s a constant to account for the actual non-unformty of the shearng stresses. By assemblng the element stffness matrces obtaned, the system stffness matrx s obtaned. 2.2. Evaluaton of the mass matrx M The formula for the consstent mass matrx of the plate may be wrtten as H T H d. (17) In ths equaton, s the mass densty matrx of the form [46]: m1 m 2, (18) m3 where m 1 = p t, m 2 =m 3 = 1 3 p t, and p s the mass denstes of the plate. and H can be 12 wrtten as follows, H dh / dx dh / dy h 1... 8.. (19) It should be noted that the rotaton nerta terms are not taken nto account. By assemblng the element mass matrces obtaned, the system mass matrx s obtaned. 2.3. Evaluaton of frequency of plate The formulaton of lateral dsplacement, w, can be gven as moton s snusodal; w= W sn ωt (2) Here ω s the crcular frequency. Substtuton of Eq. (2) and ts second dervaton nto Eq. (1) gves expresson as; [K- ω 2 M] {W}= (21) Eq. (21) s obtaned to calculate the crcular frequency, ω, of the plate. Then natural frequency can be calculated wth the formulaton below; f= ω /2π (22) 196
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 3. NUMERICAL EXAMPLES 3.1. Data for numercal examples In the lght of the results gven n references [27, 28, 38], the aspect ratos, b/a, of the plate are taken to be 1, 1.5, and 2.. The thckness/span ratos, t/a, are taken as.1,.5,.1,.2, and.3 for each aspect rato. The shorter span length of the plate s kept constant to be 1 m. The mass densty, Posson s rato, and the modulus of elastcty of the plate are taken to be 2.5 kn s 2 /m 2,.2, and 2.7x1 7 kn/m 2. Shear factor k s taken to be 5/6. The subgrade reacton modulus of the Wnkler-type foundaton s taken to be 5 and 5 kn/m 3. For the sake of accuracy n the results, rather than startng wth a set of a fnte element mesh sze, the mesh sze requred to obtan the desred accuracy were determned before presentng any results. Ths analyss was performed separately for the mesh sze. It was concluded that the results have acceptable error when equally spaced 4x4 mesh sze for 17-noded elements are used for a 1 m x 1 m plate. Length of the elements n the x and y drectons are kept constant for dfferent aspect ratos as n the case of square plate. In order to llustrate that the mesh densty used n ths paper s enough to obtan correct results, the frst sx frequency parameters of the thck plate wth b/a=1 and t/a=.5 s presented n Table 1 by comparng wth the result obtaned SAP2 program and the results Özgan and Daloğlu [215]. In ths study Özgan and Daloğlu used 4-noded and 8-noded quadrlateral fnte element wth 1x1 and 5x5 mesh sze. It should be noted that the results presented for MT8 element are obtaned by usng equally spaced 2x2, 4x4 and 8x8 mesh szes. As seen from Table 1, the results obtaned by usng 8-noded quadrlateral fnte element have excellent agreement wth the results obtaned by Özgan and Daloğlu [215] and SAP2 even f 2x2 mesh sze s used for MT8 element. 3.2. Results The frst sx frequency parameters of thck plate restng on Wnkler foundaton wth free edges are compared wth the same thck plate modeled by Ozgan and Daloğlu (21) and SAP2 program and t s presented n Table1. The subgrade reacton modulus of the Wnklertype foundaton for ths example s taken to be 5 kn/m 3. Ths thck plate s modeled wth MT8 element 8x8 mesh sze for b/a=1., t/a=.5 ratos. As seen from Tables 1, the values of the frequency parameters of these analyses are so close even f ths study mesh sze s so poor. And also SAP2 analyss made by frst order fnte element, wrter s use second order fnte element as before explaned. Then wrters enlarged parameters of aspect rato, b/a, thckness/span rato, t/a, for help the researchers. The frst sx frequency parameters of thck plates restng on Wnkler foundaton consdered for dfferent aspect rato, b/a, thckness/smaller span rato, t/a, are presented n Table 2 for the wth free edges, n Table 3 for thck clamped plates. In order to see the effects of the changes n these parameters better on the frst sx frequency parameters, they are also presented n Fgs 3-4 for the thck free plates, n Fgs 5-6 for the thck clamped plates. 197
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 Table 1. The frst fve natural frequency parameters of plates for b/a=.1 and t/a=.5. λ =ω 2 Ozgan and Daloğlu, 215 PBQ8(FI) MT8 (4 element) Ths Study MT8 (16 element) MT8 (64 element) SAP2 1 399.42 3997.94 3851.64 3962.35 4. 2 399.42 3997.94 3862.49 3962.35 4. 3 4.4 44.4 3862.51 3963.71 4. 4 8676. 8813.62 8489.34 8597.96 8619.6 5 13957.64 16817.97 13672.78 13798.2 13292.31 6 17252.34 22889.28 16822.21 16967.53 1638.24 As seen from Tables 2, and Fgs. 3, except the value of t/a.5,the values of the frst sx frequency parameters for a constant value of t/a decreases as the aspect rato, b/a, ncreases. For t/a.5 the results are the opposte. Table 2. Effects of aspect rato and thckness/span rato on the frst sx frequency parameters of the thck free plates restng on elastc foundaton. a)subgrade reacton modulus k=5 λ = ω 2 k b/a t/a λ 1 λ 2 λ 3 λ 4 λ 5 λ 6.5 373 39 39 542 1254 13428 1..1 218 218 225 17542 3764 49454 5.2 69 69 73 58673 12671 164565.5 369 377 386 187 1532 5752 2..1 218 223 226 2999 4558 2459.2 67 72 72 1634 15462 68549 b)subgrade reacton modulus k=5 λ = ω 2 k b/a t/a λ 1 λ 2 λ 3 λ 4 λ 5 λ 6.5 3962 3962 3964 8598 13798 16968 1..1 1996 1996 26 19316 39376 51187 5.2 937 937 967 59525 127495 165364.5 3962 3962 3963 4664 512 9318 2..1 1993 22 23 477 6335 22226.2 934 959 964 11498 16326 69399 198
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 Table 3. Effects of aspect rato and thckness/span rato on the frst sx frequency parameters of the thck clamped plates restng on elastc foundaton. a)subgrade reacton modulus k=5 λ = ω 2 k b/a t/a λ 1 λ 2 λ 3 λ 4 λ 5 λ 6.5 29759 12226 12226 252899 377489 3821 1..1 1244 376857 376857 745179 15457 169678 5.2 278637 86419 86419 15688 253714 211595.5 1425 23434 4562 9194 92812 113158 2..1 49981 81613 155628 29587 296925 357465.2 27837 861812 863825 1566356 245759 213845 b)subgrade reacton modulus k=5 λ = ω 2 k b/a t/a λ 1 λ 2 λ 3 λ 4 λ 5 λ 6.5 33343 123791 123791 256447 38127 38554 1..1 13782 378568 378568 746871 15214 171364 5.2 279493 86525 86525 1569623 254527 2111418.5 1784 2718 49199 9376 96382 116723 2..1 51728 8335 157352 297524 298634 359176.2 146447 231679 418269 694223 742664 829834 As also seen from Tables 2, and Fg.. 3, the values of the frst three frequency parameters for a constant value of b/a decrease as the thckness/span rato, b/a, ncreases up to the 3 rd frequency parameters, but after the 3 rd frequency parameters, the values of the frequency parameters for a constant value of b/a ncrease as the thckness/span rato, t/a, ncreases. The decreases n the frequency parameters wth ncreasng value of b/a for a constant t/a rato gets less for larger values of b/a up to the 3 rd frequency parameters. After the 3 rd frequency parameters, the decrase n the frequency parameters wth ncreasng value of b/a for a constant t/a rato gets also less for larger values of b/a. The changes n the frequency parameters wth ncreasng value of b/a for a constant t/a rato s larger for the smaller values of the b/a ratos. Also, the changes n the frequency parameters wth ncreasng value of b/a for a constant t/a rato s s less than that n the frequency parameters wth ncreasng ncreasng t/a ratos for a constant valueof b/a. These observatons ndcate that the effects of the change n the t/a rato on the frequency parameter of the plate are generally larger than those of the change n the b/a ratos consdered n ths study. 199
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 Fgure 3. Effects of aspect rato and thckness/span rato on the frst sx frequency parameters of the thck free plates wth subgrade reacton modulus k=5. As also seen from Tables 3, and Fgure. 4, the curves for a constant value of b/a rato are farly gettng closer to each other as the value of t/a ncreases up to the 3rd frequency parameters. 2
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 Ths shows that the curves of the frequency parameters wll almost concde wth each other when the value of the rato of t/a ncreases more. Fgure 4. Effects of aspect rato and thckness/span rato on the frst sx frequency parameters of the thck free plates wth subgrade reacton modulus k=5. 21
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 After the 3 rd frequency parameters, the curves for a constant value of t/a rato are farly gettng closer to each other as the value of b/a ncreases. Ths also shows that the curves of the frequency parameters wll almost concde wth each other when the value of the rato of b/a ncreases more. In other words, up to the 3 rd ferquency parameters, the ncrease n the t/a rato wll not affect the frequency parameters after a determned value of t/a. After the 3 rd ferquency parameters, the ncrease n the b/a rato wll not affect the frequency parameters after a determned value of b/a. As seen from Tables 2, and 3, and Fgures. 3, and 4, the values of the frequency parameters for a constant value of t/a decrease as the aspect rato, b/a, ncreases. Ths behavor s understandable n that a thck plate wth a larger aspect rato becomes more flexble and has smaller frequency parameters. The decreases n the frequency parameters wth ncreasng value of b/a rato gets less for a constant value of t/a. As seen from Tables 3, and Fgures. 4,, the values of the frequency parameters for a constant value of b/a ncrease as the thckness/span rato, t/a, ncreases. Ths behavor s also understandable n that a thck plate wth a larger thckness/span rato becomes more rgd and has larger frequency parameters. The ncreases n the frequency parameters wth ncreasng value of t/a rato gets larger for a constant value of b/a. The frst mode shape ( 1 =3962.35) The second mode shape ( 2 =3962.35) The thrd mode shape ( 3 =3963.71) The fourth mode shape ( 4 =8597.96) The ffth mode shape ( 5 =13798.2) Fgure 5. The frst sx mode shapes of the thck free plates for b/a=1. and t/a=.5 wth subgrade reacton modulus k=5. It should be noted that the ncrease n the frequency parameters wth ncreasng t/a ratos for a constant value of b/a rato gets larger for larger values of the frequency parameters. 22 The sxth mode shape ( 6 =16967.53)
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 These observatons ndcate that the effects of the change n the t/a rato on the frequency parameter of the thck plates clamped along all four edges are always larger than those of the change n the aspect rato. The frst mode shape ( 1 =33343.5) The second mode shape ( 2 =12379.49) The thrd mode shape ( 3 =12379.51) The fourth mode shape ( 4 =256446.91) The ffth mode shape ( 5 =38126.62) The sxth mode shape ( 6 =385539.73) Fgure 6. The frst sx mode shapes of the thck clamped plates for b/a=1. and t/a=.5 wth subgrade reacton modulus k=5. As also seen from Fgure 4, the curves for a constant value of the aspect rato, b/a are farly gettng closer to each other as the value of t/a decreases. Ths shows that the curves of the frequency parameters wll almost concde wth each other when the value of the thckness/span rato, t/a, decreases more. In other words, the decrease n the thckness/span rato wll not affect the frequency parameters after a determned value of t/a. In ths study, the mode shapes of the thck plates are also obtaned for all parameters consdered. Snce presentaton of all of these mode shapes would take up excessve space, only the mode shapes correspondng to the sx lowest frequency parameters of the thck plate free, clamped along all four edges for b/a = 1 and t/a =.5 are presented. These mode shapes are gven n Fgures 5, and 6, respectvely. In order to make the vsblty better, the mode shapes are plotted wth exaggerated ampltudes. As seen from these fgures, the number of half wave ncreases as the mode number ncreases. It should be noted that appearances of the mode shapes not gven here for the thck plates clamped along all four edges are smlar to those of the mode shapes presented here. 23
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 4. CONCLUSIONS The purpose of ths paper was to study parametrc free vbraton analyss of thck plates usng hgher order fnte elements wth Mndln s theory and to determne the effects of the thckness/span rato, the aspect rato and the boundary condtons on the lnear responses of thck plates subjected to vbraton. As a result, free vbraton analyze of the thck plates were done by usng p verson serendpty element, and the coded program on the purpose s effectvely used. In addton, the followng conclusons can also be drawn from the results obtaned n ths study. The frequency parameters ncreases wth ncreasng b/a rato for a constant value of t/a up to the 3 rd ferquency parameters, but after that the frequency parameters decrases wth ncreasng b/a rato for a constant value of t/a. The frequency parameters decreases wth ncreasng t/a rato for a constant value of b/a up to the 3 rd ferquency parameters, but after that the frequency parameters ncreases wth ncreasng t/a rato for a constant value of b/a. The effects of the change n the t/a rato on the frequency parameter of the thck plate are generally larger than those of the change n the b/a ratos consdered n ths study. REFERENCES [1] Ugural A.C. (1981), Stresses n Plates and Shells, McGraw-Hll., New York. [2] Lessa A. W. The free vbraton of rectangular plates. J Sound Vb 1973; 31 (3): 257-294. [3] Lessa A.W. Recent research n plate vbratons, 1973 1976: classcal theory. The Shock and Vbraton, Dgest 1977; 9 (1): 13 24. [4] Lessa A.W. Recent research n plate vbratons, 1973 1976: complcatngeffects. The Shock and Vbraton, Dgest 1977; 9 (11): 21 35. [5] Lessa A.W. Plate vbraton research, 1976 198: classcal theory. The Shock and Vbraton Dgest 1981; 13 (9): 11 22. [6] Lessa A.W. Plate vbraton research, 1976 198: complcatngeffects. The Shock and Vbraton Dgest 1981; 13 (1): 19 36. [7] Lessa A.W. Plate vbraton research,1981 1985 part I: classcal theory. The Shock and Vbraton Dgest 1987; 19 (2): 11 18. [8] Lessa A.W. Plate vbraton research, 1981 1985 part II: complcatngeffects. The Shock and Vbraton Dgest 1987; 19 (3): 1 24. [9] Provdaks C. P., Beskos D. E. Free and forced vbratons of plates by boundary elements. Comput Meth Appl Mech Eng 1989;, 74: 231-25. [1] Provdaks C. P., Beskos D. E. Free and forced vbratons of plates by boundary and nteror elements. Int J Numer Meth Eng 1989; 28: 1977-1994. [11] Warburton G. B. The vbraton of rectangular plates. Proceedng of the İnsttude of Mechancal Engneers 1954; 168: 371-384. [12] Caldersmth G. W. Vbratons of orthotropc rectangular plates. ACUSTICA 1984; 56: 144-152. [13] Grce R. M., Pnnngton R. J. Analyss of the flexural vbraton of a thn-plate box usng a combnaton of fnte element analyss and analytcal mpedances. J Sound Vb 22; 249(3): 499-527. [14] Sakata T., Hosokawa K. Vbratons of clamped orthotropc rectangular plates. J Sound Vb 1988; 125 (3): 429-439. [15] Lok T. S., Cheng Q. H. Free and forced vbraton of smply supported, orthotropc sandwch panel. Comput. Struct 21; 79(3): 31-312. [16] S W.J., Lam K. Y., Gang S. W. Vbraton analyss of rectangular plates wth one or more guded edges va bcubc B-splne method. Shock Vb 25; 12(5). 24
Parametrc Egenvalue Analyss of Mndln Plates / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 [17] Ayvaz Y., Durmuş A. Earhquake analyss of smply supported renforced concrete slabs. J Sound Vb 1995; 187(3): 531-539. [18] Ressner E. The Effect of Transverse Shear Deformaton on the Bendng of Elastc Plates. J Appl Mech (ASME) 1945; 12: A69-A77. [19] Ressner E. On Bendng of Elastc Plates. Quarterly Appl Math 1947; 5(1): 55-68. [2] Ressner,E. On a Varatonal Theorem n Elastcty. J Math Physcs 195; 29: 9-95. [21] Mndln RD. Influence of rotatory nerta and shear on flexural motons of sotropc, elastc plates. J Appl Mech 1951; 18(1): 31-38. [22] Hnton E., Huang HC. A famly of quadrlateral Mndln plate element wth substtute shear stran felds. Comput Struct 1986; 23(3): 49-431. [23] Zenkewch OC. Taylor RL., Too JM., Reduced ntegraton technque n general analyss of plates and shells. Int J Numer Meth Eng 1971; 3: 275-29. [24] Bergan PG., Wang X. Quadrlateral Plate Bendng Elements wth Shear Deformatons. Comput Struct 1984; 19(1-2): 25-34. [25] Özkul T. A., Ture U. The transton from thn plates to moderately thck plates by usng fnte element analyss and the shear lockng problem. Thn-Walled Struct 24; 42: 145-143. [26] Hughes T. J. R., Taylor R. L., Kalcja W. Smple and effcent element for plate bendng. Int J Numer Meth Eng 1977; 11: 1529-1543. [27] Özdemr Y. I., Bekroğlu S., Ayvaz Y. Shear lockng-free analyss of thck plates usng Mndln s theory. Struct Eng Mech 27; 27(3): 311-331. [28] Özdemr Y. I. Development of a hgher order fnte element on a Wnkler foundaton. Fnte Elem Anal Des 212; 48: 14-148. [29] Wanj C., Cheung Y. K. Refned quadrlateral element based on Mndln/Ressner plate theory. Int J Numer Meth Eng 2; 47: 65-627. [3] Soh A. K., Cen S., Long Y., Long Z. A new twelve DOF quadrlateral element for analyss of thck and thn plates. Eur J Mech ; A/Solds 21; 2: 299-326. [31] Brezz F., Marn L. D. A nonconformng element for the Ressner-Mndln plate. Comput Struct 23; 81: 515-522. [32] Belounar L., Guenfound M. A new rectangular fnte element based on the stran approach for plate bendng. Thn-Wall Struct 25; 43: 47-63. [33] Senjanovc I., Tomc M., Hadzc N., Vladmr N. Dynamc fnte element formulatons for moderately thck plate vbratons based on the modfed Mndln theory. Eng. Struct. 217; 136:1-113. [34] Cen S., Long Y.Q., Yao Z.H., Chew S.P. Applcaton of the quadrlateral area coordnate method: Anew element for Mndln-Ressner plate. Int J Numer Meth Eng 26; 66: 1-45. [35] Shen H. S., Yang J., Zhang L. Free and forced vbraton of Ressner-Mndln plates wth free edges restng on elastc foundaton. J Sound Vb 21; 244(2): 299-32. [36] Woo K.S., Hong C.H., Basu P.K. and Seo C.G. Free vbraton of skew Mndln plates by p-verson of F.E.M. J Sound Vb 23; 268: 637-656. [37] Qan, R.C. Batra, L.M. Chen. Free and forced vbraton of thck rectangular plates usng hgher-orde shear and normal deformable plate theory and meshless Petrov-Galerkn (MLPG) method. Comput Modelng Eng & Scences 23; 4(5): 519-534. [38] Özdemr Y. I., Ayvaz Y. Shear lockng-free earthquake analyss of thck and thn plates usng Mndln s theory. Struct Eng Mech 29; 33(3): 373-385. [39] Gunagpeng Z., Tanxa Z., Yaohu S. Free vbraton analyss of plates on Wnkler elastc foundaton by boundary element method. Opt Elect Materals Applcat II 212; 529: 246-251. 25
Y.I. Özdemr / Sgma J Eng & Nat Sc 36 (1), 191-26, 218 [4] Fallah A., Aghdam M. M. and Kargarnovn M.H. Free vbraton analyss of moderately thck functonally graded plates on elastc foundaton usng the extended Kantorovch method. Arch Appl Mech 213; 83(2): 177-191. [41] Jahrom H. N., Aghdam M. M., Fallah A. Free vbraton analyss of Mndln plates partally restng on Pasternak foundaton. Int J Mech Scences 213;75: 1-7. [42] Özgan K., Daloglu A. T. Free vbraton analyss of thck plates on elastc foundatons usng modfed Vlasov model wth hgher order fnte elements. Int J Eng Materals Scences 212; 19: 279-291. [43] Özgan K., Daloglu A. T. Free vbraton analyss of thck plates restng on Wnkler elastc foundaton. Challenge J Struct Mech 215; 1(2): 78-83. [44] Cook R.D., Malkus D. S., Mchael E. P. Concepts and Applcatons of Fnte Element Analyss. John Wley & Sons, Inc., Canada; 1989. [45] Bathe KJ. Fnte ElementProcedures. Prentce Hall, Upper Saddle Rver, New Jersey; 1996. [46] Tedesco J. W., McDougal W. G., Ross C.A. Structural Dynamcs. Addson Wesley Longman Inc., Calforna; 1999. [47] Weaver W., Johnston PR. Fnte Elements for Structural Analyss. Prentce Hall, Inc., Englewood Clffs, New Jersey; 1984. 26