dynamic analysis of thick plates on elastic foundations using winkler foundation...

Scientia Iranica A (2014) 21(1), 11{18

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Dynamic analysis of thick plates on elastic foundationsusing Winkler foundation model

K. Ozgan� and A.T. Daloglu

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

Received 9 July 2012; received in revised form 28 January 2013; accepted 8 June 2013

KEYWORDSFinite elementmethod;Thick plate;Elastic foundation;Winkler model;Shear lockingproblem;Dynamic analysis.

Abstract. Dynamic analysis of rectangular thick plates on elastic foundations underpartially distributed and centrally concentrated impulsive load is presented using Winklerfoundation model. An 8-noded (PBQ8) Mindlin plate element are adopted for modelingthe plate to account the transverse shear deformation e�ects neglected in the classical thinplate theory. Selective reduced integration technique is used to avoid shear locking problemwhich occurs as thickness/span ratio decreases. The results obtained from the study arecompared with the solutions obtained by SAP2000 software to show the validity of theelements. The variation of the maximum displacement with various values of subgradereaction modulus, aspect ratios, the ratio of plate thickness to shorter span of the plateand loaded area are investigated. Numerical examples show the applicability of the 8-nodedelement to dynamic analysis of thick plates on elastic foundation subjected to external loadsusing Winkler model.c 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Vibration problems of plates on elastic foundationoccupy signi�cant place in many �elds of structuraland foundation engineering. Kirchho� plate theoryis used in most of the studies because of its simplic-ity. Although Kirchho� plate theory works well forthe thin plates, the result obtained by the Kirchho�theory may not be accurate enough as the plate getsthicker since the shear deformation e�ect is ignoredin the theory. Mindlin plate theory accounts forthe angle changes within a cross section of the plateand assumes that transverse shear deformation occurs.Mindlin plate elements based on Mindlin plate theorycan be used for the analysis of both thin and thickplates when reduced or selective reduced integrationtechnique are used to avoid shear locking problem forcalculation of the sti�ness matrix. On the other hand,

*. Corresponding author. Tel.: +90 462-377-2662;Fax: +90 462-377-2606E-mail address: [email protected] (K. Ozgan)

the real behavior of the foundation-plate system isquite complex. Various foundation models have beendeveloped to simplify the complex phenomenon. Thesimplest and most widely used is Winkler model thatthe subsoil is represented by a set of discrete springelements and the vertical displacement is assumed tobe proportional to the contact pressure at an arbitrarypoint.

For the past twenty years, a lot of studies con-cerning dynamic analysis of plates on elastic foundationare performed. Omurtag et al. [1] developed a mixed�nite element formulation based on Gateaux di�er-ential for free vibration analysis of Kirchho� plateson elastic foundation. Daloglu et al. [2] examineddynamic analysis of plates on elastic foundation usingmodi�ed Vlasov model and thin plate theory. Shen etal. [3] performed the free and forced vibration analysisof Reissner-Mindlin plates resting on Pasternak-typeelastic foundation. Huang and Thambiratnam [4]discussed dynamic response of plates on elastic foun-dation subjected to moving loads and the investigatede�ects of velocity, subgrade reaction, moving path and

12 K. Ozgan and A.T. Daloglu/Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 11{18

distance between multiple moving loads on responses.Hasemi et al. [5] studied free vibration analysis of ver-tical rectangular Mindlin plates resting on Pasternakelastic foundation and fully or partially in contact with uid on their one side for di�erent combinations ofboundary conditions. Hsu [6] presented the vibrationresponses of orthotropic plates on nonlinear elasticfoundations using the di�erential quadrature method.Celep and G�uler [7] studied the static behavior andforced vibration of a rigid circular plate supported bya tensionless Winkler elastic foundation by assumingthat the plate is subjected to a uniformly distributedload and a vertical load having an eccentricity. Er�ozand Yildiz [8] presented a �nite element formulation offorced vibration problem of a prestretched plate restingon rigid foundation. Yu et al. [9] presented the dynamicresponse of Reissner-Mindlin plate resting on an elasticfoundation of the Winkler-type and Pasternak-typeusing an analytical-numerical method. Tovstik [10]studied vibration and stability of a prestressed plate onelastic foundation. Ozgan and Daloglu [11] applied themodi�ed Vlasov model to the free vibration analysisof thick plates resting on elastic foundation usingPBQ4 Mindlin plate element with selective reducedintegration technique. Wen and Aliabadi [12] usedboundary element method for the analysis of Mindlinplates on elastic foundation subjected to dynamicload.

In this study, 8-noded (PBQ8) Mindlin plateelements are adopted for the dynamic analysis of thickplates resting on Winkler-type foundation under par-tially distributed impulsive load and centrally concen-trated impulsive load. A computer program is codedin Fortran for the purpose. Newmark-� method isused for time integration. Selective reduced integrationtechniques are used to obtain sti�ness matrix of plates.The results of the study using the presented elementsare compared with the solutions obtained by SAP2000software to show the validity of the elements. Later themaximum displacements of the plate are examined forvarious values of modulus of subgrade reaction, aspectratios, loaded area and the ratio of plate thickness toshorter span of the plate.

2. Mathematical model

The dynamics of elastic structures include the kineticenergy of the plate in addition to the strain energyand work of external forces. Applying the Hamilton'sprinciple, the equation of motion for dynamic of plateson elastic foundation acted on by external loads isderived from the below equation (Figure 1).

�t2Zt1

Yk

�U!dt = 0; (1)

Figure 1. A plate resting on elastic foundation.

where �k is the kinetic energy and given as:Yk

=12

Z

f _wgT [�]f _wgd; (2)

and U = �p + �s + V in which �p is the strain energystored in the plate, �s is the strain energy stored in thesubsoil and V is the potential energy of external loads.These expressions can be written as:Y

p

=12

Z

[B]T [D][B]d; (3)

Ys

=12

Z

[w(x; y)]T k[w(x; y)]d; (4)

V = �Z

NT qd; (5)

where is the domain of the plate, [�] is the massdensity matrix, fwg the vector of generalized dis-placement components relevant to internal forces, thedot denotes the partial derivative of the vector ofgeneralized displacement with respect to time variable,[D] is the material matrix relating the stresses tostrains, [B] is the strain-displacement matrix of theplate, k is the subgrade modulus of the subsoil, [N ]is de�ned as displacement shape functions and q isexternal loads [13].

As mentioned before, in this study, rectangular�nite elements based on Mindlin plate theory consid-ering the transverse shear e�ects are used for �niteelement formulation. PBQ8 element has 8 nodes-24degrees of freedom (Figure 2). At each node of theelement, one displacement and two rotations are takenas unknowns.

w;'x; 'y: (6)

K. Ozgan and A.T. Daloglu/Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 11{18 13

Figure 2. The �nite elements used in this study.

The displacement function is:

w = [N ]fweg: (7)

The matrix [N ] contains the displacement shape func-tions found in explicit form in reference Weaver andJohnston [14]. The sti�ness matrices, mass matrix andload vector can be derived by substituting Eq. (7) intoEqs. (2) to (5) and by using the standard procedure inthe �nite element methodology.

Finally the equation for the plate-soil system tobe solved is:

[M ]f �wg+ [K]fwg = fFg; (8)

where [K] is the sti�ness matrix of the plate-soilsystem, [M ] is the mass matrix of the plate-soil system,fFg is the applied load vector, w and �w are thedisplacement and acceleration vector of the plate,respectively [13].

In this study the Newmark-� method is used forthe time integration of Eq. (8) by using the averageacceleration method [15]. Value of � is considered as 1

4 .Evaluation of the element sti�ness and mass matricesis performed using the displacement function given inEq. (7) [11].

It is well known that when Mindlin plate elementsare used, very sti� results may be obtained for thesolution of thin plates. This phenomenon is called shearlocking problem. The selective reduced integration ruleis used to obtain the element sti�ness matrix of theplate to avoid shear locking. In the selective reduceintegration, the bending terms of plates is integratedwith the normal rule and the shear terms of plate witha lower-order rule.

3. Numerical examples

For the numerical example, a rectangular plate freelyresting on elastic foundation subjected to partiallydistributed and centrally concentrated impulsive loads

is considered. At �rst the results obtained in this studyare compared with the solutions obtained by SAP2000software to show the validity of the elements. Thenthe maximum displacements of the plate are examinedfor various values of subgrade reaction modulus, aspectratios, loaded area and the ratio of plate thickness toshorter span of the plate since the maximum displace-ments are the most important for the design. Theresults are presented in graphical and tabular form.

The properties of the plate-soil system are asfollows. The modulus of elasticity of the plate isEp = 27000000 kN/m2, Poisson's ratio of the plateis �p = 0:20. The mass densities of the plate istaken to be � = 2500 kg/m3. Longer span length isconsidered as 10, 15 and 20 m for ly=lx = 1:0, 1.5and 2.0 respectively while the shorter span length ofthe plate is kept constant at 10 m. The thickness ofthe plate is considered as 0.5 m, 1.0 m and 2.0 m.The analysis is performed for three di�erent values ofsubgrade reaction modulus of subsoil, k = 500, 5000and 50000 kN/m3. The impulsive pressure q(�t) =q0:F (�t) is applied on the top surface of the plate,in which q0 is the maximum amplitude which is 30kN/m2 for partially distributed load and is 10 kN forconcentrated load at the center of the plate, F (�t) isa unit function in time domain which has the type inFigure 3. Partially distributed load is applied centrallyto the plate and loaded area is a=lx = b=ly = 0:5 asshown in Figure 4. The values for initial displacement,velocity and acceleration are assumed to be equal tozero.

For the sake of accuracy in results, a convergencestudy is performed before analysis. First the timeincrement is �xed and the mesh density is increased un-til the maximum displacements converge satisfactorily.Then the time increment is increased for the constantvalue of mesh density determined in the previous stepuntil the maximum displacements converge satisfac-torily. It is concluded that the result is acceptablewhen equally spaced 6 elements for 10 m length forconcentrated load and 8 element for 10 m length forpartially distributed load if a 0.000025 s time incrementare used.

The maximum displacements for ly=lx = 1:0 and2.0, h = 0:5 m and k = 500 kN/m3 have beencompared �rst with the results obtained from SAP2000software for veri�cation of the present formulation as

Figure 3. Shape of transverse impulsive load.


Table 1. Comparison of maximum displacements of plate on elastic foundation.

Finiteelement

Partially distributed load Centrally concentrated load10 m �10 m

plate10 m �20 m

plate10 m �10 m

plate10 m �20 m

platewmax

(mm)wmax

(mm)wmax

(mm)wmax

(mm)SAP2000 0.05202 0.06519 0.00107 0.00072

PBQ8(SRI) 0.05624 0.06828 0.00120 0.00080

Figure 4. Plate subjected to partially distributed load.

presented in Table 1. It should be noted that Sap2000software uses only 4 noded thick plate element while 8noded Mindlin plate element is used in the computerprogram coded in this study. Nevertheless, all resultsare in good agreement with SAP2000 solutions for allload cases.

The results obtained for various values of ly=lx,h=lx and k are given in Table 2. The time historiesof the maximum displacements for only two casesare presented in Figure 5 for partially distributedload and in Figure 6 for centrally concentrated loadbecause presentation of all of the time histories ofthe displacements at di�erent points on the platewould take up excessive space. The time histories ofthe center displacement of the plate di�er from eachother depending on the dynamic characteristics of thesystem.

In the case of partially distributed load, the vari-ation of the maximum displacement of the plate withvarious values of aspect ratio (ly=lx) for di�erent valuesof the ratio of plate thickness to shorter span length ofthe plate (h=lx) and subgrade reaction modulus (k) isplotted in Figures 7-9 in order to show the e�ects ofthe changes in these parameters better. The maximumdisplacement of the plate decreases as the subgradereaction modulus (k) increases.

Table 2. Maximum displacements of plates on elasticfoundation for PBQ8(SRI) element.

k(kN/m3)

ly=lx h=lx

Distributedload

Concentratedload

wmax

(mm)wmax

(mm)

500

1.00.05 0.05624 0.001200.10 0.03103 0.000580.20 0.02001 0.00035

1.50.05 0.06031 0.000940.10 0.03337 0.000390.20 0.02055 0.00021

2.00.05 0.06828 0.000800.10 0.03618 0.000390.20 0.02112 0.00019

5000

1.00.05 0.02659 0.000890.10 0.01270 0.000360.20 0.00724 0.00018

1.50.05 0.03207 0.000650.10 0.01443 0.000220.20 0.00763 0.00010

2.00.05 0.03574 0.000650.10 0.01676 0.000280.20 0.00823 0.00014

50000

1.00.05 0.01338 0.000720.10 0.00651 0.000300.20 0.00303 0.00014

1.50.05 0.01557 0.000470.10 0.00688 0.000170.20 0.00363 0.00006

2.00.05 0.01592 0.000610.10 0.00817 0.000280.20 0.00420 0.00014

As expected, the maximum displacement of theplate increases as the value of aspect ratio (ly=lx)increases. The increase in the value of maximumdisplacement of the plate with increasing the aspectratio (ly=lx) is less for the larger value of subgrade


Figure 5. The time histories of the center displacementsof the plate subjected to uniformly distributed load.

reaction modulus (k). It should be said that aspectratio (ly=lx) does not a�ect the maximum displacementof the plate after a certain value of subgrade reactionmodulus (k).

The maximum displacement of the plate decreasesas the ratio of plate thickness to shorter span lengthof the plate (h=lx) increases. This behavior is un-derstandable in that a plate with a larger thicknessbecomes more rigid. But the decrease in the valueof maximum displacement of the plate with increasingthe ratio of plate thickness to shorter span length ofthe plate (h=lx) is less for the larger value of subgradereaction modulus (k). It should be said that the ratioof plate thickness to shorter span length of the plate(h=lx) does not a�ect the maximum displacement of theplate after a certain value of subgrade reaction modulus(k).

Figure 10 shows the e�ect of loaded area onthe time histories of the displacement for thick plateon Winkler elastic foundation subjected to partiallydistributed impulsive load. As expected, results show

Figure 6. The time histories of the center displacementof the plate subjected to concentrated load.

that the central displacements increase when the areaof the loaded region increases.

In the case of concentrated load, the variation ofthe maximum displacement of the plate with the aspectratio (ly=lx), the ratio of plate thickness to shorter spanlength of the plate (h=lx) and the subgrade reactionmodulus (k) is plotted in Figures 11-13 in order to showthe e�ect of the changes in these parameters better.The maximum displacement of the plate decreases asthe value of aspect ratio (ly=lx) increases. An increasemay be expected as the span of the plate gets larger butit should be noted that the load on per unit area getssmaller because the concentrated load is kept constant.The decrease in the value of maximum displacement ofthe plate with increasing aspect ratio (ly=lx) is less forthe larger value of subgrade reaction modulus (k). Itshould be said that aspect ratio (ly=lx) does not a�ectthe maximum displacement of the plate after a certainvalue of subgrade reaction modulus (k).

The maximum displacement of the plate decreasesas the ratio of plate thickness to shorter span length of


Figure 7. The variation of the maximum displacement with ly=lx for partially distributed load for h=lx = 0:05,h=lx = 0:10 and h=lx = 0:20.

Figure 8. The variation of the maximum displacement with h=lx for partially distributed load for ly=lx = 1:0, ly=lx = 1:5,and ly=lx = 2:0.

Figure 9. The variation of the maximum displacement with k for partially distributed load for h=lx = 0:05, h=lx = 0:10,and h=lx = 0:20.

the plate (h=lx) increases. This behavior is understand-able in that a plate with a larger thickness becomesmore rigid. But the decrease in the value of maximumdisplacement of the plate with increasing the ratioof plate thickness to shorter span length of the plate(h=lx) is less for the larger value of subgrade reactionmodulus (k). It should be said that the ratio of platethickness to shorter span length of the plate (h=lx) doesnot a�ect the maximum displacement of the plate aftera certain value of subgrade reaction modulus (k).

The maximum displacement of the plate decreasesas the value of subgrade reaction modulus (k) increases.The decrease in the maximum displacement decreaseswith increasing subgrade reaction modulus (k).

The e�ect of the subgrade reaction modulus (k)on the maximum displacement is larger than that ofthe other parameters.

4. Conclusions

The dynamic analysis of thick plates on Winkler-typeelastic foundation subjected to impulsive load is carriedout using an 8-noded Mindlin plate element in whichtransverse shear e�ect is taken into consideration anda parametric study is performed to show the e�ectof the thickness/shorter-span ratio, the aspect ratio,loaded area and the subgrade reaction modulus on themaximum displacement of the plate. The following


Figure 10. E�ect of loaded area on dynamic behaviors ofthick plate for ly=lx = 1:0, h=lx = 0:05 and k = 500kN/m3, for a=lx = b=ly = 0:2, a=lx = b=ly = 0:4, anda=lx = b=ly = 0:6.

conclusions can be drawn from the results obtained inthis study.

� The maximum displacement of the plate decreaseswith increasing the subgrade reaction modulus forboth loading cases.

� The maximum displacement of the plate decreases

with increasing thickness/span ratio for loadingcases.

� The maximum displacement of the plate increaseswith increasing aspect ratio for partially distributedload.

� The maximum displacement of the plate decreaseswith increasing aspect ratio for centrally concen-trated load while an increase is expected. But itshould be noted that the load on per unit areagets smaller because the concentrated load is keptconstant.

� The changes in the maximum displacement de-creases for larger value of the thickness/span ra-tio and the subgrade reaction modulus while thechanges in the maximum displacement increases forlarger value of aspect ratio.

� The e�ects of thickness/span ratio and aspect ratioon the maximum displacement of the plate decreaseafter a certain value of subgrade reaction modulusof the subsoil.

� The subgrade reaction modulus has stronger e�ecton the maximum displacement of the plate com-pared to the other parameters.

� Results show that the central displacements in-creases with increasing the loaded area.

Figure 11. The variation of the maximum displacement with ly=lx for centrally concentrated load for h=lx = 0:05,h=lx = 0:10, and h=lx = 0:20.

Figure 12. The variation of the maximum displacement with h=lx for centrally concentrated load for ly=lx = 1:0,ly=lx = 1:5, and ly=lx = 2:0.


Figure 13. The variation of the maximum displacement with k for centrally concentrated load for h=lx = 0:05,h=lx = 0:10, and h=lx = 0:20.

References

1. Omurtag, M.H., �Oz�utok, A., Ak�oz, A.Y. and�Oz�celik�ors, Y. \Free vibration analysis of Kirchho�plates resting on elastic foundation by mixed �niteelement formulation based on gateaux di�erential",International Journal for Numerical Methods in En-gineering, 40, pp. 295-317 (1997).

2. Daloglu, A., Dogangun, A. and Ayvaz, Y. \Dynamicanalysis of foundation plates using a consistent Vlasovmodel", Journal of Sound and Vibration, 224(5), pp.941-951 (1999).

3. Shen, H.S., Ynag, J. and Zhang, L. \Free and forcedvibration of Reissner-Mindlin plates with free edgesresting on elastic foundations", Journal of Sound andVibration, 244(2), pp. 299-320 (2001).

4. Huang, M.H. and Thambiratnam, D.P. \Dynamicresponse of plates on elastic foundation to movingloads", Journal of Engineering Mechanics, 128(9), pp.1016-1022 (2002).

5. Hasemi, Sh.H., Karimi, M. and Taher, H.R.D. \Vibra-tion analysis of rectangular Mindlin plates on elasticfoundations and vertically in contact with stationary uid by the Ritz method", Ocean Engineering, 37, pp.174-185 (2010).

6. Hsu, M.H. \Vibration of orthotropic rectangular plateson elastic foundations", Composite Structures, 92, pp.844-852 (2010).

7. Celep, Z. and G�uler, K. \Static and dynamic responsesof a rigid circular plate on a tensionless Winklerfoundation", Journal of Sound and Vibration, 276, pp.449-458 (2004).

8. Er�oz, M. and Yildiz, A. \Finite element formulation offorced vibration problem of a prestretched plate restingon rigid foundation", Journal of Applied Mathematics,2, pp. 1-12 (2007).

9. Yu, L., Shen, H.S. and Huo, X.P. \Dynamic responsesof Reissner-Mindlin plates with free edges resting ontensionless elastic foundations", Journal of Sound andVibration, 299, pp. 212-228 (2007).

10. Tovstik, P.Ye. \The vibration and stability of aprestressed plate on elastic foundation", Journal ofApplied Mathematics and Mechanics, 73, pp. 77-87(2009).

11. Ozgan, K. and Daloglu, A.T. \Application of themodi�ed Vlasov model to the free vibration analysisof thick plates resting on elastic foundation", Shockand Vibration, 16, pp. 439-454 (2009).

12. Wen, P.H. and Aliabadi, M.H. \Boundary elementformulations for Mindlin plate on an elastic founda-tion with dynamic load", Engineering Analysis withBoundary Elements, 33, pp. 1161-1170 (2009).

13. Kolar, V. and Nemec, I., Modeling of Soil StructureInteraction, Amsterdam, Elsevier (1989).

14. Weaver, W. and Johnston, P.R., Finite Elements forStructural Analysis, Englewood Cli�s, NJ, Prentice-Hall, Inc (1984).

15. Humar, J.L., Dynamic of Structures, Englewood Cli�s,NJ, Prentice-Hall (1990).

Biographies

Korhan Ozgan is an Assistant Professor in CivilEngineering Department at the Karadeniz TechnicalUniversity, Turkey. He received his MS and PhDdegrees in Structural Engineering in 2000 and 2007respectively, from Karadeniz Technical University. Hisresearch �elds are static and dynamic analysis of beamsand plates and soil-structure interaction.

Ayse T. Daloglu is a professor of Civil Engineeringat Karadeniz Technical University. She received herPhD degree in 1992 from Texas Tech University. Soilstructure interaction, structural optimization usinggenetic algorithm and snow loads on steel structuresare among her research interest.

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