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Comparison of Rectangular Glass Plates withAreaequivalent Trapezoidal Plates

selected with aspect ratios equal to 0.5, 1.0, 1.5,and 2 respectively. The dimensions of these platesare given in Fig. 1. The corresponding four area-equivalent trapezoidal plates are also shown inFig. 1. Here both the rectangular and trapezoidalplates are analyzed by the same finite elementprogram. This way any possible errors due todiscretization and or the size of load increment etc.used in the nonlinear finite element model can beminimized. The maximum displacements andprincipal tensile stresses in both plates arecomputed and compared.

Prof. C. V. Girija Vallabhan and Mostafa Morsi El-ShamiTexas Technical University

Keywords1 = rectangular 2 = trapzoidal 3 = glass plates 4 = nonlinear5 = finite elements 6 = stress analysis 7 = wind loads

AbstractA comparison is made between nonlinear

displacements and stresses in rectangular andarea-equivalent trapezoidal glass plates of samethickness. A finite element code based on anextended nonlinear Mindlin theory of plates is usedfor the analysis. Nine-noded quadrilateral plateelements are used in the finite element code. Anincremental iterative procedure is used to obtainthe convergent results for each load increment.Four different sizes of rectangular plates areselected with aspect ratios such as 0.5, 1.0, 1.5and 2.0. The corresponding four area-equivalenttrapezoidal plates are also analyzed. The resultsare compared and presented.

IntroductionDesigners of glass panels are sometimes

confronted with simply supported plates that aretrapezoidal or having other complex shapes. In thecase of a trapezoidal glass plate, for simplicity orwant of a simple nonlinear analysis, they assumethat the trapezoidal plates act like a rectangularplate with equal area and thickness. Thisprocedure is adopted because the characteristicsof rectangular plates subjected to uniformpressures are widely available in the literature.Here a special type of trapezoidal plate isconsidered where the two interior angles are rightangles with one side inclined at angle of 60degrees. In this analysis, the side opposite to theinclined side in the trapezoidal plate is kept thesame A special quadrilateral finite element methodwith 9 nodes and having 45 degrees of freedomis used in the analyses. Since the trapezoidalplates are also simply supported, the rotationnormal to the inclined boundary is constrained asequal to zero.

Four sample sizes of rectangular plates are

Fig. 1 (a)

Fig. 1 (b)

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3Brief Summary of PreviousResearch on Nonlinear Analysisof Plates

In the past many researchers have studied thenonlinear analysis of plates and majority of themhave used von Karman theory of plates. Someused finite difference methods (Vallabhan 1983)and others have used finite element methods(Zienkiewicz, 1977). But prior to all these analyses,Levy (1942) had solved nonlinear plate problemsusing Fourier series. Variational principles such asGalerkin methods have also been used by a few(Beason 1980, Vallabhan and Ku,1984) for solvingthe same problem. Whatever methodology isadopted, since the problem is nonlinear one has

to use an iterative procedure for obtaining the finalresults. One of the interesting features of all thesemethodologies is that engineers could compareresults obtained from different techniques.

Now, for solving plate problems with complexgeometry using finite element methods, manyresearchers have preferred to use quadrilateralelements with Mindlin theory of plates as opposedto the classical Kirchhoff theory. In Mindlin theory,the deformations due to vertical shear stresses areincluded while the Kirchhoff theory ignores suchshear deformations. Both theories are linear, butare extended to consider the geometricallynonlinear behavior of the plate. El-Shami andVallabhan (1997) has shown that for thinrectangular simply supported plates the resultsobtained from using a finite element model withnonlinear Mindlin theory compared well with thosefrom using the von Karman theory.

A Brief Review of the FiniteElement Model

A geometrically nonlinear quadrilateral finiteelement model is developed using the Mindlinplate theory. Nine nodes are used in thequadrilateral element with 5 degrees of freedomper node. The displacements u, v, and w at a point(x,y,z) in the plate can be calculated as a functionof the middle surface displacements u v0 0, andw0

such that:

( ) ( )( ) ( )

),(),,(,,),,(,,),,(

yxwzyxw

yxzyxvzyxv

yxzyxuzyxu

xo

yo

=

=

+=

(1)

where, are the rotations of the normal of theundeformed plate in the yz and zx planesrespectively. In the finite element model, thedisplacements are represented using theLagrangian type shape functions such as

( ) ,,9

1i

ii uNu

=

=

and similar equations for other displacements. Thedetails of similar finite element formulations aregiven in many finite element text books such asCook et.al. (1989), Zienkiewicz (1977), Weaver andJohnson (1984) etc. and therefore not repeatedhere. Also El-Shami and Vallabhan (1997) hadpresented their formulation of the element stiffnessmatrix in detail in their paper. Here in this paper,the glass plates are subjected to a uniformpressure approximately 10.5 kPa. maximum.Problems of this type are very sensitive to valueof load increments and in fact, these methodscould develop unwanted cumulative errors. Toavoid these problems, the uniform pressure is

Fig. 1 (c)

Fig. 1 (d)

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applied on to the plate in about 200 equal loadincrements. For each load increment, the newstiffness matrix is computed at

u uik

ik

++1

21

where k defines the increment used. An iterativeprocedure is used where the new and olddisplacements match fairly well with a relativetolerance equal to 0.0001.

Results obtained from the new model matchedextremely well with results obtained using finitedifference models (Vallabhan, 1983) in the caseof rectangular plates.

Boundary ConditionsSimply supported boundary conditions are

used in the analyses. The reason for such aboundary condition is that majority of glass platesare supported in some rubber sockets that arekept in some relatively rigid frames and such aboundary is more close to a simply supportedcondition. In other words, the displacement w onthe edges and the rotation normal to theboundaries are zero. This boundary condition hasto be implemented for the inclined side of thetrapezoidal glass plates. This poses a smallproblem because the rotation normal to theinclined side in the trapezoidal plate is linearlyconnected together, i.e.,

n , the rotation normal

to the inclined side is

0sincos =+= yxn(3)

There are many techniques to implement thiscondition in the finite element model, however, apenalty function method with a large rotationalspring constant in the direction of the rotation isused here. The derivation of the correspondingfield equations are determined using the principleof minimum potential energy theorem and thestiffness coefficients in the global matrixcorresponding to the node i has to be modified asindicated by adding the four stiffness coefficientswith the new coefficient C representing therotational restraint in the direction of

n

++

++

216,1626,26

16,262

26,26

CKCK

CKCK

iiii

iiii

(4)

where sin,cos == , C = a largeconstant in the penalty function method. In Eq.(4)the subscript i represents the node where therotation normal to the boundary is set equal tozero. The corresponding component of the forcevector should be zero also for each point along theboundary.

Results of Selected Rectangularand Trapezoidal Plates

Four different sizes of rectangular plates areselected with aspect ratios 0.5, 1.0,1.5, and 2. Thedimensions of these plates are given in Fig. 1(a), (b),(c) and (d). Each plate has a thickness equal to5.715 mm. (.225 inch). These plates are subjectedto a uniform lateral pressure equal to 10.5 kPa (1.5psi.) in 200 equal pressure increments with iterationsat every increment to obtain the closest convergentvalue of displacements. Even though for rectangularplates, solutions can be obtained from publisheddata, new solutions are obtained for the maximumdeflections using the new computer code. Theresults of the maximum displacements for therectangular and trapezoidal plates subjected to auniform lateral pressure up to 10.5 kPa. are givenin Fig. 2 for comparison. In particular, two contourplots for displacements and maximum principalstresses for a trapezoidal plate are presented in Fig.3 (a) and (b) respectively.

Fig. 2 Comparision of dispalcements

Fig. 3(a) Dispalcement contour

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It is interesting to note that the values of themaximum displacements for the selecteddimensions of the plates match fairly well for bothrectangular and trapezoidal plates for aspectratios 0.5, 1.0 and 1.5 (see Fig. 3). However forexample 4 with an aspect ratio equal to 2.0, themaximum displacement in the trapezoidal plate issmaller than its counterpart rectangular plate. Butthe difference is relatively small for practicalconsiderations.

Most important from a design consideration, isthe magnitude of maximum principal stress in a

plate. It is observed from the results that themaximum principal tensile stresses for trapezoidalplates were occurring at the corner area where theinterior angle is 1200. This phenomenon isobserved even for very small applied pressures,in other words, the geometry of the plate makesthe maximum stress to be in that area. Thecomparisons of the maximum principal stressesbetween the rectangular and the area-equivalenttrapezoidal plates are shown in Figs. 4 (a), (b), (c)and (d). For the first example with aspect ratioequal to 0.5, the maximum principal tensile

Fig. 4. Comparison of Stresses in Rectangular and TrapezoidalPlates

Fig. 3 (b) Stress contour

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stresses in rectangular and trapezoidal plates arethe same for pressures from zero to 10.5 kPa. Butfor example 2, i.e., with aspect ratio 1.0, thetrapezoidal plate had higher maximum principalstresses than the corresponding stresses inrectangular plate. The same is true for example 3with an aspect ratio equal to 1.5. But, for example4, with higher aspect ratio equal to 2, thetrapezoidal plate had still higher maximumprincipal stress, but the difference is not very highas can be seen in the Fig. 4 (d). It should be notedthat nondimensional parameters are not used inthese figures because the authors feel that theyhave not solved enough problems to justify a finalpresentation of results using nondimensionalparameters.

Summary and ConclusionsA sophisticated nonlinear finite element

program using quadrilateral plate elements with 45degrees of freedom is made to analyze thegeometric nonlinearity in plates of arbitrary shapesand boundary conditions. For each increment ofloading, an iterative procedure is adopted toachieve converged solutions. The results obtained

from using the new code are almost identical tothose obtained by previous investigators in solvingrectangular plates. Since previous nonlinearsolutions for trapezoidal plates were not available,a comparison could not be made.

Since the strength of a glass plate is associatedwith the maximum tensile stress when it issubjected to a uniform lateral pressure such windloads, strength of plates that are trapezoidal orwith other complex geometry requires specialanalysis. Results presented here are only for somespecial sizes and angles for trapezoidal plates,and a general set of curves for the computationof stresses in them are rather too complicated, inother words, special individual attention isnecessary in such instances. Nonlinearquadrilateral finite element codes are very usefulfor such analyses.

AcknowledgmentThe following research was made possible by

a generous financial support for the Post DoctoralFellowship for Dr. Mostafa El-Shami from theMenoufia University, Menoufia, Egypt.

References

1. Beason, W. L., (1980), A Failure Prediction Model forWindow Glass, Ph.D. Dissertation, Texas TechUniversity, Lubbock, Texas.

2. Cook, R. D., Malkus, D. S., and Plesha, M. E., (1989),Concepts and Application of Finite Element Analysis,John Wiley & Sons, Inc.

3. El-Shami, M. M., Vallabhan, C. V. G., and Kandil, S.A.,(1997 ),Comparison of Nonlinear Von Karman andMindlin Plate Solutions, Proc. of the Texas ASCE Conf.,American Society of Civil Engineers, Spring Meeting,April 4-9, Houston, Texas.

4. Levy, S.,(1942), Square Plate with Clamped EdgesUnder Normal Pressure Producing Large Deflections,NACA, Report No. 740.

5. Vallabhan, C.V.Girija, (1983), Iterative Analysis ofNonlinear Glass Plates, J. Struc. Engr., ASCE, Vol. 109,pp 2416-2426.

6. Vallabhan, C.V.Girija, and Ku, Fu., (1984), NonlinearAnalysis of Rectangular Glass Plates by GalerkinMethod, 4th Inter. Conf. on Applied NumericalModelling, Taiwan, R.O.C, pp 135-142.

7. von Karman, Th.,(1910), Festigkeitspobleme inMaschinenbau, Vol. IV of Encyk. Der Math. Wiss., pp.349.

8. Weaver, Jr., W. and Johnston, P.R., (1984), FiniteElements for Structural Analysis, Prentice Hall,Englewood Cliffs, New Jersey.

9. Zienkiewicz, O.C, (1977), The Finite Element Method,McGrawHill Book Co., New York.