Free vibration analysis of thick plates on elastic ...nopr.niscair.res.in/bitstream/123456789/14656/1/IJEMS 19(4) 279-291... · Free vibration analysis of thick plates on elastic

Indian Journal of Engineering & Materials Sciences Vol. 19, August 2012, pp. 279-291

Free vibration analysis of thick plates on elastic foundations using modified Vlasov model with higher order finite elements

Korhan Ozgan* & Ayse T Daloğlu

Department of Civil Engineering, Karadeniz Technical University, Trabzon 61080, Turkey

Received 12 January 2011; accepted 11 July 2012

This study is concerned with the free vibration analysis of thick plates on elastic foundation using modified Vlasov model. The effects of the subsoil depth, aspect ratios, the ratio of subsoil depth to span of the plate and the value of the vertical deformation parameter within the subsoil on the frequency parameters of plates on elastic foundations are investigated. An eight-noded, 24 degrees of freedom quadrilateral finite element (PBQ8) based on Mindlin plate theory is used for the analysis. The first six natural frequency parameters are presented to show the effects of the parameters considered in the study. It is concluded that the subsoil depth is more effective on the frequency parameters of plates resting on elastic foundation than the other parameters considered in this study.

Keywords: Finite element, Elastic foundation, Thick plate, Free vibration, Vlasov model

The behavior of plates resting on elastic foundations has wider interest because of being an important model for many engineering applications. Natural frequencies and corresponding mode shapes are guide engineers to investigate the behavior of the soil structure system with time. Therefore, free vibration analysis constitutes a considerable part of research done on this topic.

Many researches use the classical thin plate theory in which shear deformation effect is ignored. Consequently, the results evaluated by using classical thin plate theory may not be reliable especially as the plate gets thicker. Another important point in these kinds of problems is the selection of a realistic subsoil model. As is known, Winkler model in which it is assumed that the foundation consists of linear elastic springs is often preferred because of its simplicity. The main deficiency of this model is that it assumes no interaction between the springs. Some researchers developed two parameter models to take the interaction in the springs into account. Although the interaction is provided by using shear deformation parameter in these models, there is a drawback about how to establish the subsoil parameters. In order to perform better models, researchers tried to enhance two parameter models introducing a third parameter that characterizes the vertical deformation profile within the subsoil1.

Omurtag and Kadıoğlu2 investigated the free vibration analysis of orthotropic Kirchhoff plates

resting on elastic foundation. Daloğlu et al.3 applied the consistent Vlasov model to dynamic analysis of plates resting on elastic foundations subjected to external loads using classical plate theory. Shen et al.4 examined the free and forced vibration analysis for a Reissner-Mindlin plate with four free edges resting on Pasternak type elastic foundation. They investigated the effects of foundation stiffness, transverse shear deformation and plate aspect ratio on dynamic response of Reissner-Mindlin plates. Malekzadeh and Karami5 used differential quadrature method for vibration of thick plates on elastic foundation and considered continuously varying thickness of the plate. Leung and Zhu6 presented an analytical trapezoidal hierarchical element for the transverse vibration of Mindlin plates resting on two parameter foundation using Legendre orthogonal polynomials to avoid shear locking problem. Omurtag et al.7 studied the free vibration analysis of Kirchhoff plates resting on elastic foundation using Gateaux differential. Zhou et al.8 analyzed the three dimensional vibration problems of thick circular plates resting on elastic foundation and investigated the effects of various thickness-radius ratios, foundation stiffness parameters and boundary conditions on the dynamic behavior of the thick circular plates on elastic foundation. Yu et al.9 presented the dynamic response of Reissner-Mindlin plate resting on an elastic foundation of the Winkler-type and Pasternak-type using an analytical-numerical method. Jedrysiak10

____________ *Corresponding author (E-mail: [email protected])

INDIAN J ENG. MATER. SCI., AUGUST 2012

280

calculated frequencies of thin plates interacting with an elastic periodic foundation. Zhong and Yin11 investigated the free vibration behavior of plate on elastic foundation by finite integral transform method using Winkler foundation method.

The purpose of this study is to examine the free vibration behavior of thick plates on elastic foundations using both of Mindlin thick plate theory that includes the effects of shear deformation and modified Vlasov model that includes shears strain in the subsoil. For the purpose, stiffness and mass matrices were calculated by a computer program coded in FORTRAN using finite element method, and then the eigenvalue solution was carried out using MATLAB for Windows 5.3. Carrying out a parametric study, the effects of the subsoil depth, aspect ratio, the ratio of subsoil depth to span of the plate and the vertical deformation parameter within the subsoil on the frequency parameters of plates are investigated. Mathematical Model

The equation of motion for a plate-soil system subjected to free vibration with no damping is

[ ] [ ] 0=+ wKwM … (1)

where [K] is the stiffness matrix of the plate-soil system, [M] is the mass matrix of the plate-soil system, w and w are the displacement and acceleration of the plate, respectively.

Governing equations of plate-soil system

Total potential energy of the plate-soil system, Fig. 1, can be written as

( ) 24 21 12

2 2D wdxdy kw t w

dxdy qwdxdy

+∞ +∞

Ω −∞ −∞

Ω

∏ = ∇ + + ∇

−

∫ ∫ ∫

∫ … (2)

where w and [D] are displacement of the plate in the vertical direction and flexural rigidity of the plate, respectively. Soil parameters, k and 2t, in above expression can be defined as

( )( )( )

( )

2

0

2

0

1

1 1 2

22 1

H

s s

s s

Hs

s

Ek dz

z

Et dz

ν ϕ

ν ν

ϕν

− ∂ =

+ − ∂

=+

∫

∫ … (3)

where sE , H, and sν are modulus of elasticity of the

soil, subsoil depth and Poisson ratio of the soil, respectively. )(zφ in the above equation is the mode shape that defines the variation of the deflection in the vertical direction and can be evaluated using the following equations depending on γ parameter that denotes the vertical deformation parameter within the subsoil12.

( )sinh 1

sinh

z

Hz

γ

ϕγ

−

=

… (4)

( )

( )

∫ ∫

∫ ∫∞+

∞−

∞+

∞−

+∞

∞−

+∞

∞−

∇

−

−=

dxdyw

dxdyw

H s

s

2

2

2

12

21

ν

νγ

Stiffness matrix

As mentioned before, an 8-noded, 24 degrees of freedom rectangular finite element (PBQ8) based on Mindlin plate theory that includes the effects of shear deformation through the plate thickness are used to developed the element stiffness matrices (Fig. 2). Nodal displacements at each node are

wi, ϕxi and ϕyi i=1, 2, …, 8 … (5) and the displacement function is

ewNw ][=

… (6)

The matrix [N] contains the displacement shape functions given elsewhere13. By using the standard procedure in the finite element methodology for the

Fig. 1—A sample plate on elastic foundation

OZGAN & DALOGLU: FREE VIBRATION ANALYSIS OF THICK PLATES

281

assemblage of elements, global stiffness matrix is constructed as follows

[ ] [ ] [ ] [ ]( )∑=

++=en

i

twp kkkK1

2 … (7)

where ne is the number of plate finite elements, and [kp], [kk] and [k2t] are element stiffness matrix of the plate, vertical deflection element stiffness matrix of the foundation and shear deformation element stiffness matrix of the foundation, respectively. The element stiffness matrix can be seen in explicit forms for plate14, vertical deflection of foundation15 and the shear deformation of foundation16.

The selective reduce integration technique on the shear terms is used to obtain the element stiffness matrix of the plate to avoid shear locking problem that occurs under the thin plate limit.

The effect of the soil surrounding the plate is considered and applied as boundary conditions to the nodal points on the boundary of the plate as equivalent stiffness parameters before solving the plate-soil system. The equivalent forces are computed as a function of the displacement on the boundary and are added to corresponding terms in the stiffness matrix17. Mass matrix

The dynamics of elastic structures is based on Hamilton’s variational principle with the kinetic energy of

[ ] Ω= ∫Ω

dwwT

k 2

1 µπ … (8)

where w represents the partial derivative of the vector of generalized displacement with respect to time variable and [ µ ] is the mass density matrix of the form

[ ] 3

3

10 0

31

0 012

10 0

12

p s

p

p

h H

h

h

ρ ρ

µ ρ

ρ

+

=

… (9)

where pρ is the mass density of the plate, h is

thickness of the plate and sρ is the mass density of the

soil. The consistent mass matrix is obtained by

substituting ewNw 1= into Eq. (8).

[ ] [ ] ∫Ω

Ω= dNNMT

11 µ … (10)

The following equation can be written for each finite element

[ ]T

dx

dN

dy

dNNN

=1 … (11)

The consistent mass matrix of the plate-soil system can be computed by substituting Eq. (11) into Eq. (10). By assembling the element mass matrix obtained, the system mass matrix is evaluated. The element mass matrix is given in Appendix.

After substituting tWw ωsin= into the governing equation for a plate subjected to free vibration with no damping given in Eq. (1), one can obtain

0])[]([ 2 =− WMK ω … (12)

where W is a vector of mode shape of vibration and λ (=ω2, ω is the circular frequency) is the frequency parameter. The eigenvalue solution of this equation yields frequency parameters and the corresponding mode shapes. Numerical Examples

A parametric study for the free vibration analysis of the plates on elastic foundations is carried out using the various values of H, ly/lx, H/ly and γ. The value of the vertical deformation parameter, γ, are taken as 1,

Fig. 2—A 8 noded, 24 degrees of freedom PBQ8 Mindlin plate element


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2,3,4,5,6,7 and 8. The depths, H, of the subsoil are taken to be 5, 10 and 15 m for each γ parameter considered, and the ratios ly/lx used are 1,0; 1,5 and 2,0 while lx remains constant for each subsoil depth. In the calculation of the mass matrix, the mass densities of plate and subsoil are taken to be 2500 and 1700 kg/m3, respectively. The properties of the plate-soil system are as follows: The length of the plate in the x direction, 10 m; the thickness of the plate, 0.5 m; the modulus of elasticity of the plate, 27000 MPa and the Poisson's ratio of the plate, 0.20. The subsoil is composed of loose sand. The modulus of elasticity of the subsoil is 20 MPa and the Poisson's ratio of the subsoil equals to 0.25.

The mesh size was determined for the desired accuracy in the results before analysis. It is concluded that the result has accurate enough when equally spaced 5 elements for 10 m length is used in each direction. The element length is kept constant for various dimension of the plate.

The same example was also studied by Ayvaz and Oğuzhan18. They used MZC rectangle finite element based on classical Kirchhoff plate theory in their study. Kirchhoff plate theory ignores the shear strain effects throughout the plate thickness. The results obtained in this study for γ=1 and H=5 m are compared with those reported by Ayvaz and Oğuzhan in Table 1. As seen from the table, agreement is good for low frequency parameters but results begin to diverge slightly with increasing frequency parameter. The difference in the values of the frequency parameters obtained from each element is expected to increase as the frequency parameter increases.

The six frequency parameters of plates obtained for various values of subsoil depth (H), aspect ratios (ly/lx) and the value of the vertical deformation parameter within the soil (γ) are presented in Table 2. The variation of the first six frequency parameters with

various values of ly/lx for three different values of subsoil depth are given in Figs 3-5 for γ=1, 4 and 8, respectively, in order to show the effects of the changes in these parameters better.

Authors solved the same example in their previous study using 4-noded Mindlin plate element and this study is an extension of that work19. The frequency parameters calculated in the previous study are also shown in the plots in Fig. 3 in order to compare the frequency parameters obtained with higher order finite element in this study. As seen from plots, the frequency parameters obtained for both of PBQ4 and PBQ8 element are very close to each other. It should be noted that selective reduced integration techniques is used for both elements. The frequency parameters obtained by PBQ8 element are smaller than that obtained by PBQ4 element for larger frequency parameters while the frequency parameters obtained by both the elements nearly overlap for small frequency parameters.

As seen from tables and figures, the frequency parameters decrease with increasing aspect ratio (ly/lx) for a constant value of subsoil depth (H) and the vertical deformation parameter (γ). The frequency parameters decrease with increasing subsoil depth (H) for a constant value of aspect ratio (ly/lx) and the vertical deformation parameter (γ).

The decrease in the frequency parameters decreases with increasing subsoil depth (H). When the subsoil depth (H) increases, the curves tend to get close to each other. This means that the frequency parameters will not change considerably after a certain value of subsoil depth (H).

It is note that the decrease in the frequency parameters with increasing aspect ratio (ly/lx) for a constant value of subsoil depth (H) and the vertical deformation parameter (γ) increases for larger frequency parameters.

Table 1—Comparison of the first six frequency parameters of plates for γ=1 and H=5 m

Frequency Parameters ly(m) H/ly Reference

λ1 λ 2 λ3 λ 4 λ5 λ6

P. Study 1571,50 1987,10 2752,40 2911,40 3552,60 5047,80 20 0,250

Ref18 1571,98 1989,22 2758,50 2918,06 3569,62 5071,84 P. Study 1639,60 2334,50 2856,80 4060,20 4206,20 7788,90

15 0,333 Ref18 1640,12 2338,66 2863,57 4073,73 4233,23 7920,53

P. Study 1802,70 3080,90 3080,90 5851,20 7727,80 9285,10 10 0,500

Ref18 1803,90 3089,09 3089,09 5918,41 7765,27 9327,86

P. Study 2282,80 3766,70 5710,10 9385,80 12902,60 34484,40 5 1,000

Ref18 2285,14 3776,76 5742,41 9434,44 13362,06 34948,61


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Table 2—The first six frequency parameters of plates on elastic foundation for various values of H, H/ly, ly/lx and γ

Frequency Parameters γ H(m) ly(m) H/ly

λ1 λ 2 λ3 λ 4 λ5 λ6

20 0,250 1571,5 1987,1 2752,4 2911,4 3552,6 5047,8

15 0,333 1639,6 2334,5 2856,8 4060,2 4206,2 7788,9

10 0,500 1802,7 3080,9 3080,9 5851,2 7727,8 9285,1 5

5 1,000 2282,8 3766,7 5710,1 9385,8 12902,6 34484,4

20 0,500 604,9 913,7 1503,7 1640,5 2145,1 3265,3

15 0,667 651,1 1173,5 1594,0 2558,4 2672,4 5358,2

10 1,000 755,6 1790,0 1790,0 4004,0 5375,0 6388,6 10

5 2,000 1070,8 2395,6 4277,9 6939,4 9723,3 23747,7

20 0,750 365,0 637,5 1152,6 1292,2 1746,9 2713,2

15 1,000 402,6 864,3 1240,1 2119,6 2234,0 4574,9

10 1,500 484,5 1430,1 1430,1 3467,2 4655,9 5307,3

1

15

5 3,000 733,8 2020,3 3881,0 6224,1 8786,5 19799,6

20 0,250 1742,9 2122,1 2792,8 2949,7 3521,2 4925,6

15 0,333 1806,2 2432,2 2883,3 3980,4 4112,0 7472,7

10 0,500 1958,8 3076,1 3076,1 5602,3 7383,4 8842,2 5

5 1,000 2399,2 3662,2 5256,3 8800,8 12033,1 33233,7

20 0,500 639,5 908,0 1410,5 1531,6 1962,3 2967,8

15 0,667 681,0 1132,3 1484,9 2318,6 2413,7 4804,8

10 1,000 776,3 1645,2 1645,2 3552,9 4786,7 5770,2 10

5 2,000 1059,5 2137,0 3627,1 6047,8 8455,9 21867,8

20 0,750 371,2 599,1 1029,6 1144,3 1520,8 2365,1

15 1,000 404,0 789,4 1099,3 1835,4 1923,2 3953,0

10 1,500 476,8 1250,0 1250,0 2940,8 3966,4 4668,3

2

15

5 3,000 696,2 1715,2 3170,4 5200,8 7326,7 17596,0

20 0,250 2153,6 2510,9 3121,6 3272,6 3798,1 5129,1

15 0,333 2214,5 2798,1 3202,5 4220,1 4344,2 7541,4

10 0,500 2363,0 3373,8 3373,8 5724,4 7434,8 8806,2 5

5 1,000 2785,1 3892,6 5233,8 8676,1 11706,6 32605,4

20 0,500 750,4 991,6 1431,7 1538,2 1915,2 2826,9

15 0,667 789,1 1191,1 1494,2 2225,5 2308,9 4473,9

10 1,000 879,8 1627,9 1627,9 3302,3 4436,5 5367,2 10

5 2,000 1145,2 2036,0 3224,3 5461,2 7588,2 20529,9

20 0,750 416,5 612,7 980,1 1073,3 1390,8 2132,0

15 1,000 446,2 776,3 1036,0 1656,3 1726,6 3489,2

10 1,500 513,8 1156,4 1156,4 2574,9 3479,7 4183,1

3

15

5 3,000 714,9 1526,0 2659,0 4441,9 6231,1 15912,2

20 0,250 2685,8 3031,7 3608,3 3754,8 4253,8 5537,9

15 0,333 2745,6 3306,3 3683,5 4652,5 4773,5 7872,9

10 0,500 2892,9 3842,3 3842,3 6088,8 7755,5 9066,0 5

5 1,000 3307,5 4322,4 5509,3 8891,8 11804,4 32517,0

20 0,500 901,3 1127,3 1529,3 1627,0 1971,7 2823,8

15 0,667 938,6 1312,1 1584,6 2253,0 2330,6 4360,1

10 1,000 1027,4 1702,4 1702,4 3236,6 4311,8 5192,4 10

5 2,000 1284,0 2060,5 3060,5 5193,5 7154,7 19818,5

20 0,750 483,5 661,5 989,4 1070,0 1351,6 2026,3

15 1,000 511,6 809,1 1037,0 1585,1 1647,0 3247,8

10 1,500 576,6 1139,1 1139,1 2392,6 3226,3 3903,9

4

15

5 3,000 767,9 1451,1 2379,1 4020,6 5609,2 14941,2

(Contd.)


284

Table 2(Contd.)—The first six frequency parameters of plates on elastic foundation for various values of H, H/ly, ly/lx and γ

Frequency Parameters γ H(m) ly(m) H/ly

λ1 λ2 λ3 λ4 λ5 λ6

20 0,250 3257,3 3596,5 4151,9 4295,8 4778,4 6032,7

15 0,333 3316,5 3863,5 4223,7 5162,5 5282,0 8321,6

10 0,500 3463,4 4374,9 4374,9 6557,7 8196,8 9466,3 5

5 1,000 3874,0 4831,4 5921,4 9269,1 12110,5 32708,7

20 0,500 1066,1 1283,1 1661,0 1753,7 2078,4 2893,3

15 0,667 1102,5 1458,9 1711,9 2341,5 2416,1 4364,1

10 1,000 1190,4 1819,9 1819,9 3269,3 4308,8 5151,0 10

5 2,000 1442,1 2147,6 3029,3 5104,0 6966,4 19466,2

20 0,750 558,6 725,8 1028,5 1102,1 1361,6 1994,6

15 1,000 585,6 863,6 1071,1 1574,9 1632,4 3136,2

10 1,500 649,3 1162,0 1162,0 2316,1 3107,5 3757,0

5

15

5 3,000 835,0 1439,2 2237,7 3800,2 5269,8 14397,7

20 0,250 3839,3 4174,1 4715,3 4857,5 5329,2 6563,4

15 0,333 3898,1 4436,0 4784,8 5703,4 5822,2 8822,4

10 0,500 4044,8 4930,9 4930,9 7071,8 8692,2 9933,2 5

5 1,000 4453,0 5371,8 6396,7 9722,6 12517,4 33030,1

20 0,500 1235,2 1446,4 1808,0 1897,7 2209,1 2999,3

15 0,667 1271,1 1616,1 1855,9 2460,1 2533,0 4428,6

10 1,000 1358,5 1957,6 1957,6 3352,0 4368,3 5181,7 10

5 2,000 1607,0 2265,4 3068,1 5107,5 6906,6 19302,3

20 0,750 636,6 797,0 1082,5 1152,0 1396,8 2002,3

15 1,000 663,0 928,2 1121,8 1596,8 1651,8 3093,5

10 1,500 725,9 1205,5 1205,5 2295,6 3060,0 3685,5

6

15

5 3,000 1607,0 2265,4 3068,1 5107,5 6906,6 19302,3

20 0,250 4424,1 4755,9 5286,7 5427,9 5891,8 7111,6

15 0,333 4482,7 5014,1 5354,6 6258,9 6377,2 9349,8

10 0,500 4629,3 5497,2 5497,2 7608,5 9215,4 10435,6 5

5 1,000 5035,7 5927,1 6905,2 10216,4 12978,7 33423,5

20 0,500 1405,9 1613,1 1962,9 2050,7 2352,7 3125,5

15 0,667 1441,4 1778,5 2008,8 2595,2 2667,0 4526,3

10 1,000 1528,5 2106,0 2106,0 3462,2 4462,3 5253,9 10

5 2,000 1774,8 2399,9 3146,6 5162,8 6918,3 19243,1

20 0,750 715,8 871,5 1144,8 1211,5 1446,1 2032,2

15 1,000 741,8 998,1 1181,8 1636,6 1690,0 3089,4

10 1,500 804,3 1260,4 1260,4 2306,4 3052,2 3658,2

7

15

5 3,000 983,8 1499,2 2150,1 3634,3 4976,8 13890,8

20 0,250 5009,9 5339,4 5862,4 6002,8 6460,9 7669,8

15 0,333 5068,2 5594,9 5929,1 6822,7 6940,7 9892,8

10 0,500 5214,9 6069,0 6069,0 8158,3 9755,0 10959,3 5

5 1,000 5620,0 6490,8 7433,7 10734,3 13472,6 33868,0

20 0,500 1577,3 1781,6 2122,5 2209,1 2504,0 3263,9

15 0,667 1612,6 1943,8 2167,0 2740,2 2811,3 4644,1

10 1,000 1699,5 2260,8 2260,8 3589,0 4577,2 5351,8 10

5 2,000 1944,1 2544,5 3249,5 5249,4 6973,3 19246,8

20 0,750 795,8 948,1 1212,0 1277,0 1503,9 2075,8

15 1,000 821,4 1071,3 1247,4 1687,4 1739,8 3108,4

10 1,500 883,7 1322,3 1322,3 2336,2 3068,5 3658,7

8

15

5 3,000 1061,3 1549,5 2155,0 3617,9 4923,3 13775,9


285

As seen from Figs 3-5, the change occurring in the frequency parameters with increasing subsoil depth (H) for a constant value of aspect ratio (ly/lx) or ratio of subsoil depth to span of the plate (H/ly) is larger than the change occurring in the frequency parameters with increasing aspect ratio (ly/lx) or ratio of subsoil depth to span of the plate (H/ly) for a constant value

of the subsoil depth (H). This is seen more clearly when the value of the vertical deformation parameter gets higher.

The frequency parameters increase with increasing ratio of the subsoil depth to span of the plate (H/ly) for a constant value of subsoil depth (H) and the vertical deformation parameter (γ).

Fig. 3—The variation of the first six frequency parameters of the plate on elastic foundation with various values of H and ly/lx for γ=1. −∆−, H=5 m; −◊−, H=10 m; −ο−, H=15 m; ···×···, H=5 m [18] ; ···+···, H=10 m [18] ···∗···, H=15 m [18]


286

The observations indicate that the effects of the subsoil depth on the frequency parameters of the plate are always larger than those of the other parameters.

The mesh size used in the analysis for PBQ8 element is approximately half of that for PBQ4 element.

In this study, the mode shapes of the plate on elastic foundation are also obtained for all parameters considered. Since presentation of all of these mode shapes would take up excessive space, only the mode shapes corresponding to the six lowest frequency parameters of the plate for γ=1, H=5 m, ly/lx=1 and γ=1, H=5 m, ly/lx=2 and γ=1, H=15 m, ly/lx=1 and γ=8,

Fig. 4—The variation of the first six frequency parameters of the plate on elastic foundation with various values of H and ly/lx for γ=4. −∆−, H=5 m; −◊−, H=10 m; −ο−, H=15 m


287

H=5 m, ly/lx=1 are presented in Figs 6-9, respectively. In order to get better visibility, the mode shapes are plotted with exaggerated amplitudes. As seen from these figures, the number of half wave increases as the mode number increases.

Conclusions The purpose of this study is to examine the effects

of the aspect ratio of the plate, the subsoil depth, the

ratio of the subsoil depth to the span of the plate and the vertical deformation parameter within the subsoil on the frequency parameters of the thick plate resting on elastic foundation. This analysis is carried out by using Matlab for Windows 5.3 for the solution of the generalized eigenvalue problem including stiffness and mass matrices that are evaluated by a computer program coded for this purpose using finite element method. In this study, the soil parameters (k and 2t)

Fig. 5—The variation of the first six frequency parameters of the plate on elastic foundation with various values of H and ly/lx for γ=8. −∆−, H=5 m; −◊−, H=10 m; −ο−, H=15 m


288

Fig. 6—The first six mode shapes of the plate on elastic foundation for γ=1, H=5 m, ly/lx=1



289




290

are not given as input data, but they are calculated depending on a third soil parameter (γ) that represents the vertical deformation profile within the subsoil. Following conclusions can be drawn from the results obtained in the study.

The mesh size required to obtain the desired accuracy for 8-noded, 24 degrees of freedom Mindlin plate element (PBQ8) is approximately half of that for 4-noded, 12 degrees of freedom Mindlin plate element (PBQ4).

The frequency parameters always increase as the ratio of subsoil depth to span of the plate (H/ly) increases for any values of the subsoil depth (H) and the vertical deformation parameter (γ ) within the subsoil.

The frequency parameters always decrease as the subsoil depth (H) increases for any value of aspect ratio (ly/lx) and the ratio of subsoil depth to span of the subsoil (H/ly).

The frequency parameters always decrease as the aspect ratio (ly/lx) increases for any value of subsoil depth (H) and the vertical deformation parameter (γ ) within the subsoil.

In general, the subsoil depth (H) is more effective on the frequency parameters of plates resting on elastic foundation than the other parameters considered in this study.

References 1 Selvaduari A P S, Elastic Analysis of Soil-Foundation

Interaction, (Elsevier Scientific Publishing Company, Amsterdam), 1979.

2 Omurtag M H & Kadıoğlu F, Comput Struct, 67 (1988) 253-265. 3 Daloğlu A, Doğangün A & Ayvaz Y, J Sound Vib, 224(5)

(1999) 941-951. 4 Shen H S, Yang J & Zhang L, J Sound Vib, 244(2) (2001)

299-320. 5 Malekzadeh P & Karami G, Eng Struct, 26 (2004)

1473-1482. 6 Leung A Y T & Zhu B, J Eng Mech, 131(11) (2005)

1140-1145. 7 Omurtag M H, Özütök A, Aköz A Y & Özçelikörs Y, Int J

Numer Methods Eng, 40 (1997) 295-317. 8 Zhou D, Lo S H, Au F T K & Cheung Y K, J Sound Vib, 292

(2006) 726-741. 9 Yu L, Shen H S & Huo X P, J Sound Vib, 299 (2007)

212-228. 10 Jedrysiak J, Int J Mech Sci, 45 (2003) 1411-1428. 11 Zhong Y & Yin J H, Mech Res Commun, 35 (2008) 268-275. 12 Vallabhan C V G & Daloglu A T, J Struct Eng, 125(1)

(1999) 108-113. 13 Weaver W & Jonhston P R, Finite elements for structural

analysis, (Prentice-Hall, New Jersey), 1984. 14 Ozgan K, Finite Element Analysis of Thick Plates on Elastic

Foundation Using Modified Vlasov Model and

Determination of the Effective Depth of the Soil Stratum, Ph. D. Dissertation, Karadeniz Technical University, Turkey, 2007(in Turkish).

15 Ozgan K & Daloglu A T, Struct Eng Mech, 26 (1) (2007) 69-86.

16 Ozgan K & Daloglu A T, Thin Walled Struct, 46 (2008) 1236-1250.

17 Turhan A, A Consistent Vlasov Model for Analysis of Plates

on Elastic Foundations Using the Finite Element Method, Ph. D. Thesis, The Graduate School of Texas Tech. University, Lubbock, Texas, 1992.

18 Ayvaz Y & Oğuzhan C B, Struct Eng Mech, 28(6) (2008), 635-658.

19 Ozgan K & Daloglu A T, Sound Vib, 16 (2009), 439-454.


291

Appendix

The element mass matrix is given into sub matrices of 6×6 as follows

[ ]

=

5634

6543

3412

4321

MMMM

MMMM

MMMM

MMMM

abM

These sub matrices are

=

3

2

1

33

22

11

1

00000

00000

000003

0000

03

000

003

00

15

2

m

m

m

mm

mm

mm

M

=

33

22

11

33

22

11

2

003

200

0003

20

00003

23

20000

03

2000

003

200

15

1

mm

mm

mm

mm

mm

mm

M

−=

33

22

11

33

22

11

3

0000

0000

00003

40000

03

4000

003

400

15

2

mm

mm

mm

mm

mm

mm

M

−=

3

400

3

400

03

400

3

40

003

400

3

4

003

400

0003

40

00003

4

15

2

33

22

11

33

22

11

4

mm

mm

mm

mm

mm

mm

M

=

5

800000

05

80000

005

8000

005

800

0005

80

00005

8

9

4

3

2

1

33

22

11

5

m

m

m

mm

mm

mm

M

=

5

40000

05

4000

005

400

005

400

0005

40

00005

4

9

4

33

22

11

33

22

11

6

mm

mm

mm

mm

mm

mm

M

where Hhm sp ρρ3

11 += and 3

3212

1hmm pρ==

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