Finite element approximation of free vibration

of folded plates

Erwin Hernandez a,1 Luis Hervella-Nieto b

aDepartamento de Matematica, Universidad Tecnica Federico Santa Maria,

Casilla 110-V, Valparaıso, Chile

bDepartamento de Matematica, Facultade de Informatica, Universidade da

Coruna, Campus da Elvira 15071, A Coruna, Spain

Abstract

In this paper a finite element approximation of the free vibration of folded plates isstudied. Naghdi model, including bending, shear and membrane terms for the plate,is considered. Quadrilateral low order MITC (Mixed Interpolation Tensorial Com-ponent) elements are used for the bending and shear effect, coupled with standardquadratic elements enriched with a drilling degree of freedom for the membraneterm. Convergence properties and optimal order error estimates are proved. Nu-merical examples, showing the good behavior of the method, are presented for oneand two folded plates with different thickness and crank angles.

Key words: folded plates, drilling degree of freedom, MITCPACS: 01.30.−y

1 Introduction

This work deals with the finite element approximation of the free vibrationof folded plates. These kind of structures are presented in many practicalapplications, such that roofs, sandwich plate cores, cooling towers, etc.. Theirgreat interest, from an engineering point of view, is reflected in a large numberof works where the structural behavior of folded plates has been studied by

∗ Corresponding author.Email addresses: erwin.hernandez@usm.cl (Erwin Hernandez),

luisher@udc.es (Luis Hervella-Nieto).1 Supported by FONDECYT 1040341 and USM 12.05.26 (Chile)2 Partially supported by FONDECYT 7040212 (Chile).

Preprint submitted to Elsevier 11 January 2008

using a variety of approaches (see, for example, [12,15,16,17]). Nevertheless,to the best of the authors knowledge, the analysis of the convergence of thenumerical methods can not be found in the literature.

In order to model this problem, we can start with Naghdi shell equations(see [5,8]) over each plate. These equations, when applied to plates, lead toa case where the transversal and the in-plane deformation appear uncoupled,following the first one the Reissner-Mindlin equations (see [8]), meanwhile thesecond one is modeled by the elasticity equation, and does not depend on thethickness of the plate (see [18]).

Each of these uncoupled problem is well studied in the literature. By one side,the elasticity problem is classic. By the other side, regarding the Reissner-Mindlin equations, it is now well understood that standard finite elementmethods applied to this problem produce very unsatisfactory results due tothe so called locking phenomenon. Therefore, some special method based onreduced integration or mixed interpolation has to be used. Among these meth-ods, the so called MITC ones, introduced by Bathe and Dvorkin in [4], orvariants of them are very likely the most used in practice. For these methods,convergence and optimal error estimates independent of the thickness of theplate have been proved (see, for example, [3] for the load problem, and [10],and references therein, for the vibration problem).

When folded plates are considered, the membrane and bending terms arecoupled, as can be clearly seen in Figure 3. Even more, since the rotationsof the normal fibers appear as unknowns for the Reissner-Mindlin model, itis necessary to introduce a new unknown for the in-plane rotation (which isredundant for a single plate) called drilling degree of freedom (see [14]).

In this paper, to approximate the free vibrations of folded plates, we considera method consisting of standard quadrilateral elements of order 1 to approx-imate the in-plane deformation enriched with drilling degree of freedom (see[14]), and low order quadrilateral MITC element (called MITC4) to approxi-mate the bending term (see [10])

The outline of the paper is as follows: in Section 2 we present and studyour method to the case of one single plate. We prove optimal order errorestimates for eigenfunctions and eigenvalues in the framework of the spectralapproximation theory stated in [1]. In Section 3 we applied our method to asystem made by two folded plates, showing optimal order error estimates forthis case. In Section 4 we assess the performance of the method by computingthe free vibrations of some Benchmark cases. Finally, some conclusions aregiven in Section 5.

2

2 Vibrations of a single plate

2.1 Naghdi equations

Let Ω×] − t2, t

2[ be the region occupied by an undeformed elastic plate of

thickness t, where Ω is a convex bi-dimensional domain.

Throughout this paper we use the standard notation in functional spaces, withL2 (Ω) denoting the space of functions whose integral of its square is bounded,and H1(Ω) denoting the space of function in L2(Ω) with generalized derivativesalso in L2(Ω). Let ‖·‖1,Ω and ‖·‖0,Ω be the standard norm defined on H1(Ω) andL2(Ω), respectively. Finally, BC symbolically denotes the essential boundarycondition to be imposed on each problem (probably, not always the same).

In order to describe the deformation of the plate, we consider the generalclassical Naghdi’s model, which is written in terms of the vector fields u =(u1, u2, u3), corresponding to the displacement of the mid-surface of the plate,

and ~β = (β1, β2), corresponding to the rotation of the fiber initially normalto the plate’s mid-surface. It is interesting to distinguish between membraneand transverse displacements, then we denote ~u the membrane displacements,~u = (u1, u2).

We emphasize that, as a general rule of our notation, throughout this paperwe use boldface variables to represent three-dimensional vectorial fields, as

u = (u1, u2, u3) , β = (β1, β2, β3) ,

and vectorial variables to represent the two first components of them,

~u = (u1, u2) , ~β = (β1, β2) .

The vibration modes of the Naghdi model for plates are the solution of thefollowing problems (see [5,13]):

Find ω > 0 and (u, ~β) ∈ [H1 (Ω)]5∩ BC such that

A((

u, ~β)

, (v, ~η))

= ω2B((

u, ~β)

, (v, ~η))

∀ (v, ~η) ∈[

H1 (Ω)]5

∩ BC, (1)

where the bilinear forms A and B are given by

A((

u, ~β)

, (v, ~η))

:=t3

12a(

~β, ~η)

+ ta (~u,~v) + kst∫

Ω

(

~∇u3 − ~β)

·(

~∇v3 − ~η)

,

B((

u, ~β)

, (v, ~η))

:= t∫

Ωu · v +

t3

12

∫

Ω

~β · ~η,

3

with ks a correction factor for the shear term and a(

~β, ~η)

(respectively, a (~u,~v))representing the linear-elasticity bilinear form,

a(

~β, ~η)

:=∫

Ωε(

~β)

: C : ε (~η) .

Here ε(

~β)

denotes the classical linear strain tensor, ε(

~β)

= 12

(

(

~∇~β)T

+(

~∇~β)

)

,

C the material tensor defined by the Hooke’s law, and ks is a correction factorfor the shear strain.

2.2 Uncoupled problems

It is clear that the well known Reissner-Mindlin model for the bending plate(see [4,11]) can be seen as a special case of the above general formulation.In fact, if we consider only the plate transversal displacements (which, inthis case, can be dealt separately from in-plane term), we get the Reissner-Mindlin plate theory, which ensures the incorporation of shear deformationeffects, namely:

Find ωT > 0 and (u3, ~β) ∈ V T such that

t3

12a(

~β, ~η)

+ kst∫

Ω

(

~∇u3 − ~β)

·(

~∇v3 − ~η)

= (ωT)2

(

t∫

Ωu3v3 +

t3

12

∫

Ω

~β · ~η

)

∀ (v3, ~η) ∈ V T, (2)

where V T := [H1 (Ω)]3∩ BC.

Moreover, the in-plane deformation, or membrane terms, deals with the fol-lowing standard linear elasticity problem:

Find ωM > 0 and ~u ∈ [H1 (Ω)]2∩ BC such that

t a (~u,~v) = (ωM)2 t∫

Ω~u · ~v ∀~v ∈

[

H1 (Ω)]2

∩ BC. (3)

The problems above have been deeply studied in the bibliography (see [10],and reference therein, for the plate problem, and [2,6] for the linear elasticityproblem).

4

Fig. 1. Drilling degree of freedom.

2.3 The drilling degree of freedom

To describe the membrane deformation (problem (3)) it is not necessary todeal with the in-plane rotations, since it can be described by means of thein-plane displacements, u1 and u2. Anyway, our goal is to describe systemsconsisting on two or more folded plates, and, in this case (as can be clearlyseen in Figure 3) the in-plane rotations of a plate are transformed into normalrotations of another one.

Then, we modify the problem (3) by introducing a redundant in-plane rotationintroduced by Hughes and Brezzi in [14] for linear elasticity problems, whichis called drilling degree of freedom (see Figure 1).

Hence, if we denote rot ~u = ∂u1/∂x2−∂u2/∂x1, the membrane equation readsin our case

Find ωM > 0 and (~u, β3) ∈ V M such that

ta (~u,~v) + kd

∫

Ω( rot~u − β3) ( rot~v − η3) = (ωM)2 t

∫

Ω~u · ~v ∀ (~v, η3) ∈ V M,

(4)

where V M :=(

[H1 (Ω)]2× L2 (Ω)

)

∩BC and kd is a real parameter to be fixed.

It is important to emphasize that the drilling degree of freedom representsthe rotational of the in-plane displacement. Then, since we assume that ~u ∈[H1(Ω)]

2, we have that β3 ∈ L2(Ω).

Joining the uncoupled problems (2) and (4), we write the Naghdi redundantproblem for a plate:

Find ω > 0 and (u,β) ∈ V such that

A ((u,β) , (v,η)) = ω2B ((u,β) , (v,η)) ∀ (v,η) ∈ V , (5)

where V :=(

[H1 (Ω)]5× L2(Ω)

)

∩ BC.

Here the bilinear forms A and B are given by

5

A ((u,β) , (v,η)) :=t3

12a(

~β, ~η)

+ ta (~u,~v) + kst∫

Ω

(

~∇u3 − ~β)

·(

~∇v3 − ~η)

+kd

∫

Ω( rot~u − β3) ( rot~v − η3) , (6)

B ((u,β) , (v,η)) := t∫

Ωu · v +

t3

12

∫

Ω

~β · ~η, (7)

It is important to emphasize that, in the previous formulation, the drillingterm, β3, does not appear in the mass B, as can be clearly seen in matrixformulations (11) and (19).

2.4 Finite element approximation

In this section we present a finite element method to approximate, with opti-mal order, the spectral problems that we have presented previously.

Concerning the Reissner-Mindlin problem (2), we consider an element of thewell known MITC family, the most used methods for bending plate problems.This family, introduced in [4], has a major advantage: they produce a lock-ing free method, it is, the error remains bounded when the thickness of theplate decrease. In particular, we consider the MITC4 element, which is thelowest order element for quadrilateral meshes among the MITC family. It isbased on discretizing the bending terms, u3, β1, and β2, with usual isopara-metric quadratic finite elements, and relaxing the shear term by introducinga reduction operator ~R on that term.

Then, we assume that Th is a family of decompositions of Ω into convex quadri-laterals (see, for example, [6]). If K is an element in Th, we denote by Qi,j(K)the space of polynomials of degree less than or equal to i in the first variableand to j in the second one.

We introduce the reduction operator ~ϕ 7−→ ~R~ϕ, with ~R~ϕ|K ∈ Q0,1(K) ×Q1,0(K), ∀K ∈ Th (see [3,10] for more details). Note that this operator corre-sponds to an interpolation on the well known rotated Raviart-Thomas space(known as edge space).

Then, the finite element approximation of problem (2) with MITC4 elementsreads:

Find ωTh > 0 and(

u3h, ~βh

)

∈ V Th such that

6

t3

12a(

~βh, ~ηh

)

+ kst∫

Ω

(

~∇u3h − ~R~βh

)

·(

~∇v3h − ~R~ηh

)

= (ωTh)2

(

t∫

Ωu3hv3h +

t3

12

∫

Ω

~βh · ~ηh

)

∀ (v3h, ~ηh) ∈ V Th, (8)

where

V Th :=

(

u3h, ~βh

)

∈ [L2(Ω)]3/

u3h|K ∈ Q1,1(K),

~βh|K ∈ [Q1,1(K)]2 , ∀K ∈ Th

∩ BC.

This problem can be written in matrix form as

kstG −kstS

−kstS′ t3

12A

u3h

~βh

= (ωTh)

2

tMu3h0

0 t3

12M~βh

u3h

~βh

, (9)

where

• G represents the matrix coming from∫

Ω

~∇u3h · ~∇v3h,

• A represents the stiffness matrix from the bilinear form a(

~βh, ~ηh

)

,

• S represents the matrix from∫

Ω

~∇u3h · ~R~ηh, S′ its transpose matrix,

• and M~βh, Mu3h

represent, respectively, the mass matrices coming from∫

Ω

~βh · ~ηh and∫

Ωu3hv3h.

MITC plate elements have a solid mathematical basis. They are reliable, effi-cient and locking free. In particular, for the MITC4 element, a mathematicalanalysis of convergence is provided in [3], where uniform meshes of square el-ements are used. This assumption has been weakened in [10], where, by usingmacro-element techniques, optimal H1 and L2 error estimates are proved. How-ever, the L2-estimates are obtained by assuming that the meshes are formedby higher order perturbations of parallelograms (i.e., asymptotically parallel-ogram meshes). All these estimates are independent of the mesh size h andof the plate thickness t. Moreover, in the same reference and under the sameassumptions on the meshes, it is proved the following optimal estimation forthe eigenmodes and eigenfrequencies of the Reissner-Mindlin spectral problem(see Theorem 5.1 in [10]):

Theorem 1 The solution of Problem (2) consists in a sequence of positiveeigenvalues, ωT. Furthermore, let ωT be an eigenvalue of problem (2) with cor-

responding normalized eigenfunction (u3, ~β). Then, for h small enough, there

7

exists an eigenvalue of the approximation problem (8), ωTh, with corresponding

normalized eigenfunction (u3h, ~βh), such that,

|ωT − ωTh| ≤Ch2,

‖u3 − u3h‖i,Ω +∥

∥

∥

~β − ~βh

∥

∥

∥

i,Ω≤Ch2−i, i = 0, 1.

Concerning the membrane terms, the approximation of problem (3) with La-grangian elements is classical and well known. Then, if we discretize the drillingdegree of freedom by the rotational of Lagrangian elements (it is, by piece-wise constant functions) the analysis remains classical, due to the redundancyof such problem. But, since we are interested in the folded plate problem,we should discretize all the rotations in the same finite element spaces, be-cause the drilling in a plate turns normal rotation in another one. Since in (8)we discretize the rotations with isoparametric quadratic finite elements, weshould use the same elements to discretize the drilling (as suggested in [14]for triangular elements).

Then we want to solve the finite element problem

Find ωMh > 0 and (~uh, β3h) ∈ V Mh such that

ta (~uh, ~vh) + kd

∫

Ω( rot~uh − β3h) ( rot~vh − η3h)

= (ωMh)2 t∫

Ω~uh · ~vh ∀ (~vh, η3h) ∈ V Mh, (10)

where

V Mh :=

(~uh, β3h) ∈ [L2(Ω)]3/

~uh|K ∈ [Q1,1(K)]2 ,

β3h|K ∈ Q1,1(K), ∀K ∈ Th

∩ BC.

The matrix formulation of this problem is

tS + kd R −kd C

−kd C′ kd Mβ3h

~uh

β3h

= (ωMh)2

tM~uh0

0 0

~uh

β3h

, (11)

where S and Mβ3hhave been defined previously and

• M~uhstands for the mass matrix of the membrane displacement,

∫

Ω~uh · ~vh,

8

• R represents the matrix coming from∫

Ωrot~uh rot~vh,

• and C represents the matrix from∫

Ωrot~uh η3h, and C′ its transpose.

Note that this approach leads to approximate the skew-symmetrical part ofthe strain tensor. Hence, by adapting the argument for triangles in [14] (seeRemark in pp. 118 of that reference), the finite element scheme obtained isconvergent and stable with respect to the parameter kd, for any kd > 0.

To prove a double order of convergence of the eigenvalues in this spectralproblem, as we state in Theorem 2, it is necessary to extend the result in [14]to quadrilateral meshes and to prove a double order of convergence for thedrilling degree of freedom. These requirements are considered in the followinglemma:

Lemma 1 Let(

~f, θ3

)

∈ [L2 (Ω)]3 be a given data. Let (~u, β3) and (~uh, β3h) bethe continuous and discrete solutions coming from the source problems asso-ciated to (4) and (10), respectively. The following inequality holds:

‖~u − ~uh‖1,Ω + ‖β3 − β3h‖0,Ω ≤Ch, (12)

‖~u − ~uh‖0,Ω + ‖β3 − β3h‖0,Ω ≤Ch2. (13)

Proof:

To obtain (12) is quite straightforward by repeating the proof of Theorem 3.1in [14], using standard arguments to finite element schemes on quadrilateralmeshes (see [6]).

To prove (13), we denote by V ′

M the dual space of V M, and by Akd

M the boundedand elliptic bilinear form in (4) (see [14]),

Akd

M ((~u, β3) , (~v, η3)) := ta (~u,~v) + kd

∫

Ω( rot~u − β3) ( rot~v − η3) .

Since V Mh ⊂ V M, we can obtain the error equation

Akd

M ((~u − ~uh, β3 − β3h) , (~vh, η3h)) = 0 ∀ (~vh, η3h) ∈ V Mh. (14)

For any given data (~g, θ3) ∈ L2 (Ω)3 ∈ V M, let(

~ud, βd3

)

be the unique solution

of the dual source problem associated to (4), it is,

Akd

M

(

(~v, η3) ,(

~ud, βd3

))

=⟨

(~v, η3) , (~g, θ3)⟩

V ′

M×V M

∀ (~v, η3) ∈ V M. (15)

9

Then, choosing as test function in (15) (~v, η3) = (~u − ~uh, β3 − β3h), and takinginto account (14), we obtain

⟨

(~u − ~uh, β3 − β3h) , (~g, θ3)⟩

V ′

M×V M

= Akd

M

(

(~u − ~uh, β3 − β3h) ,(

~ud − ~vh, βd3 − η3h

))

.

(16)

On the other hand, if we assume regularity for the solution(

~ud, βd3

)

, since

‖(~u − ~uh, β3 − β3h)‖V ′

M

= sup(~g,θ3)∈V M

⟨

(~u − ~uh, β3 − β3h) , (~g, θ3)⟩

V ′

M×V M

‖(~g, θ3)‖V M

,

and ‖~u − ~uh‖0,Ω + ‖β3 − β3h‖0,Ω ≤ ‖(~u − ~uh, β3 − β3h)‖V ′

M

, (13) follows from

(16), the continuity of Akd

M , (12), and standard approximation properties.

Then, putting in the context of the Theorem 7.1 in [1], we have the followingoptimal result for the convergence of eigenfunctions and eigenvalues for themembrane problem with the drilling degree of freedom:

Theorem 2 The solution of Problem (4) consists in a sequence of positiveeigenvalues, ωM. Furthermore, let ωM be an eigenvalue of problem (4) with cor-responding normalized eigenfunction (~u, β3). Then, for h small enough, thereexists an eigenvalue of the approximation problem (10), ωMh, with correspond-ing normalized eigenfunction (~uh, β3h), such that,

|ωM − ωMh| ≤Ch2,

‖~u − ~uh‖i,Ω ≤Ch2−i, i = 0, 1,

‖β3 − β3h‖0,Ω ≤Ch2.

Finally, we write down the finite element spectral problem of the Naghdiformulation, with drilling degree of freedom, for one single plate:

Find ωh > 0 and (uh,βh) ∈ V h such that

Ah

(

(uh,βh), (vh,ηh))

= (ωh)2 B

(

(uh,βh), (vh,ηh))

∀ (vh,ηh) ∈ V h,

(17)where

V h :=

(uh,βh) ∈ [L2(Ω)]6/

uh|K ∈ [Q1,1(K)]3 ,

βh|K ∈ [Q1,1(K)]3 , ∀K ∈ Th

∩ BC,

10

B is the mass term defined in (7) and Ah is the stiffness term modified withthe reduction operator,

Ah

(

(uh,βh), (vh,ηh))

:=t3

12

∫

Ωε(~βh) : C : ε(~ηh) + t

∫

Ωε(~uh) : C : ε(~vh)

+kst∫

Ω

(

~∇u3h − ~R~βh

)

·(

~∇v3h − ~R~ηh

)

+kd

∫

Ω( rot~uh − β3h)( rot~vh − η3h). (18)

With the matrices defined previously, we can write this problem as follows:

tS + kd R 0 0 −kd C

0 kstG −kstS 0

0 −kstS′ t3

12A 0

−kd C′ 0 0 kd Mβ3h

~uh

u3h

~βh

β3h

= (ωh)2

tM~uh0 0 0

0 tMu3h0 0

0 0 t3

12M~βh

0

0 0 0 0

~uh

u3h

~βh

β3h

(19)

From Theorems 1 and 2, it is direct to prove the following optimal estimatefor eigenvalues and eigenvectors of the redundant Naghdi problem:

Theorem 3 The solution of Problem (5) consists in a sequence of positiveeigenvalues, ω. Furthermore, let ω be an eigenvalue of problem (5) with cor-responding normalized eigenfunction (u,β). Then, for h small enough, thereexists an eigenvalue of the approximation problem (17), ωh, with correspondingnormalized eigenfunction (uh,βh), such that,

|ω − ωh| ≤Ch2,

‖u − uh‖i,Ω +∥

∥

∥

~β − ~βh

∥

∥

∥

i,Ω≤Ch2−i, i = 0, 1,

‖β3 − β3h‖0,Ω ≤Ch2.

11

3 Folded plate approximation problem

3.1 Statement of the problem

The problem of two (or more) folded plates is more difficult to establish thanthe previous one, although, in practice, takes approximately the same diffi-culty.

For simplicity, we restrict ourselves to the simplest case of two folded plates,but the following analysis can be generalized directly to the case of a finitenumber of folded plates.

We associate the mid-surface of each plate with a plane domain through achange of variables, as it is usual in Naghdi shells. The deformations of theplates will be described by means of local variables. With this technique, allthe computing are made in bi-dimensional domains and the main difficulty isto relate the local variables in the common boundary of each pair of plates.

Let us assume, then, that we have two plain domains, Ω1 and Ω2, and twolocal charts φ1 and φ2 from the plane domains to the plates (see a completeexample in Section 4.2).

In the case we are concerned (it is, with the charts corresponding to plates)the geometrical coefficients are already simplified in the bilinear form a ofthe Naghdi equation (5), which remain unchanged. Then, the charts are onlyuseful to characterize the local variables, ui and βi, on each plate (as canbe seen in the example of Section 4.2) and to relate them on the commonboundary, that we denote by ΓI (see Figure 3).

The variational formulation of the spectral problem involving two folded platesis, then, the addition of the uncoupled spectral problem for each plate, in avariational space where the displacements and angles are rely on the commoninterface, it is,

Find ωF ∈ R and a non-zero field(

u1,β1,u2,β2)

∈ V F such that

Ai((

ui,βi)

,(

vi,ηi))

= ω2FBi

((

ui,βi)

,(

vi,ηi))

i = 1, 2;

∀(

v1,η1,v2,η2)

∈ V F, (20)

where Ai and Bi are the bilinear forms defined by, respectively, (6) and (7),applying over each domain Ωi, and

12

V F :=(

u1,β1,u2,β2)

∈[

H1(

Ω1) ]

5 × L2

(

Ω1)

×[

H1(

Ω2) ]

5 × L2

(

Ω2)

:

β1|ΓI= Dβ β2|ΓI

, u1|ΓI= Du u2|ΓI

∩ CF ,

with Dβ and Du representing, respectively, the coupling conditions betweenangles and displacements of both plates.

Note that matrices Dβ and Du depend on the parametrization of the platesand on the dihedral angle. In Section 4.2 we show some example of them.

We remark that the coupling conditions involving the drilling degree of free-dom (for example, β1

3 |ΓI= β2

2 |ΓI) must be understood in a distributional sense

(it is, in H−1 (ΓI)), since we are assuming that the drilling belongs to L2. Thishas no practical effects, since in the finite element problem we discretize thedrilling by using piecewise linear functions, and, then, this equality takes aclassical sense.

3.2 Finite element approximation

In this subsection we discretize the problem (20) by using the finite elementspaces introduced in the Section 2.4 for each plate.

Then, the discrete problem for two folded plates can be written as

Find ωFh > 0 and a non-zero field(

u1h,β

1h,u

2h,β

2h

)

∈ V Fh such that

Aih

(

(uh,βh), (vh,ηh))

= (ωFh)2 Bi

(

(uh,βh), (vh,ηh))

∀(

v1h,η

1h,v

2h,η

2h

)

∈ V Fh , (21)

where Aih and Bi stand for the bilinear forms in Section 2.4, applied in the

domain Ωi, for i = 1, 2, and

V Fh =

(

u1h,β

1h,u

2h,β

2h

)

∈[

L2(Ω1)]6

×[

L2(Ω2)]6/

uih|K ∈ [Q1,1(K)]3 ,

βih|K ∈ [Q1,1(K)]3 , ∀K ∈ Th, K ⊂ Ωi, i = 1, 2,

and β1h|ΓI

= Dβ β2h|ΓI

, u1h|ΓI

= Du u2h|ΓI

∩ BC.

The practical implementation of this problem is straightforward from matrixformulation (19). The coupling condition between both plates can be taken

13

into account by performing a static condensation, identifying the correspond-ing degrees of freedom on the common boundary, ΓI, according to the couplingequations β1|ΓI

= Dβ β2|ΓIand u1|ΓI

= Du u2|ΓI.

If we assume enough regularity for our problem, by Theorem 3, it is direct toprove the following

Theorem 4 The solution of Problem (20) consist in a sequence of positiveeigenvalues, ωF. Furthermore, let ωF be an eigenvalue of problem (20) withcorresponding normalized eigenfunction (u1,β1,u2,β2). Then, for h smallenough, there exists an eigenvalue of the approximation problem (21), ωFh,with corresponding normalized eigenfunction (u1

3h,β1h,u

23h,β

2h), such that,

|ωF − ωFh| ≤Ch2,∥

∥

∥u1 − u1h

∥

∥

∥

i,Ω1+∥

∥

∥

~β1 − ~β1h

∥

∥

∥

i,Ω1

+∥

∥

∥u2 − u2h

∥

∥

∥

i,Ω2+∥

∥

∥

~β2 − ~β2h

∥

∥

∥

i,Ω2≤Ch2−i, i = 0, 1,

∥

∥

∥β13 − β1

3h

∥

∥

∥

0,Ω1+∥

∥

∥β23 − β2

3h

∥

∥

∥

0,Ω2≤Ch2.

4 Numerical results

In this section we present some numerical examples showing the good behaviorand the numerical performance of the method that we have presented.

4.1 Numerical results for a single plate

In the first experiment we are going to present, we will check that the numericalresults for one single plate do not deteriorate with the inclusion of the drillingdegree of freedom.

We consider a clamped square plate, with 1m length side and 0.01m of thick-ness, and we take as its physical constants

• Young, E = 200 109,• Poisson, ν = 0.3,• density, ρ = 8000.

We have used successive refinements of a uniform mesh as that in Figure 2,the refinement parameter N being the number of element edges on each sideof the square.

14

Fig. 2. Finite element mesh for the square plate: N = 8.

In Table 1 we compare the vibration frequencies, in rad/sec, for the membraneproblem without drilling degree of freedom (numerical solutions of problem(3)), against the solutions of the membrane problem with drilling degree offreedom (problem (10)). We also include the value of the vibration frequen-cies obtained by extrapolating the computed values as well as the estimatedorder of convergence. Such extrapolation has been obtained by means of aleast square fitting. We can see that the performance of the results remainsunchanged with or without the drilling, and does not depend on the choice ofthe positive value kd, as announced in [14].

Table 1Membrane eigenmodes for the clamped plate.

Modo Drilling/no kd N=12 N=16 N=20 Order Extrap

1 No — 18719.899 18682.668 18665.324 1.98 18634.138

1 Drilling E4(1+ν) 18731.598 18689.525 18669.827 1.96 18633.954

1 Drilling E2(1+ν) 18742.945 18696.208 18674.230 1.94 18633.667

1 Drilling E1+ν

18764.883 18709.195 18682.819 1.91 18633.169

2 No — 22489.324 22361.329 22302.251 2.01 22197.853

2 Drilling E4(1+ν) 22591.245 22418.073 22338.415 2.02 22198.414

2 Drilling E2(1+ν) 22692.522 22474.573 22374.461 2.03 22199.775

2 Drilling E1+ν

22893.363 22586.943 22446.253 2.03 22200.667

As a second numerical experiment, we solve the complete Naghdi problem fora plate without drilling (numerical solution of problem (1)) and with drilling(problem (17)). We have taken, as usual for clamped plates, a shear correctionfactor ks = 5

6. The results are shown in Table 2 where it can be seen that the

numbers are almost the same.

4.2 Folded plates

In this section we report numerical results corresponding to the solution ofproblems including two folded plates.

15

Table 2Bending eigenmodes for the clamped plate.

Modo Drilling kd N=12 N=16

1 No — 553.801 549.394

1 Drilling E2(1+ν) 553.801 549.394

2 No — 1162.732 1138.163

2 Drilling E2(1+ν) 1162.732 1138.163

We take in all the cases the values for the physical constants used in Section4.1, with ks = 5

6and kd = E

2(1+ν).

As a first numerical experiment, we test our method by reproducing the resultsin [12]. In this paper, a high precision composite plate-bending element is used,and the results are compared against other methods. Different crank anglesan thickness ratios are considered.

Then, we consider a system made by two folded plates making a π2

+ α angle(see Figure 3), with α = 0, π

6, π

3; i.e. crank angles of 90, 120, and 150,

respectively. In this case, the chart for the first plate is:

φ1 : Ω1 = [0, 1] × [0, 1] −→ R3

(x, y) → (x, y cos α, 1 + y sin α).

Then, the local basis is:

• a11 :=

∂φ1

∂x(x, y) = (1, 0, 0),

• a12 :=

∂φ1

∂y(x, y) = (0, cos α, sin α),

• a13 := a1

1 × a12 = (0,− sin α, cos α).

The angles are defined as rotations between the normal fibers, in such a waythat θ1

1 is the rotation from a13 to a1

1, θ12 the rotation from a1

3 to a12, and θ1

3 therotation from a1

1 to a12.

The chart for the second plate is:

φ2 : Ω2 = [0, 1] × [0, 1] −→ R3

(x, y) → (x, 0, y).(1)

In this case, the local basis is:

• a21 :=

∂φ2

∂x(x, y) = (1, 0, 0),

16

• a22 :=

∂φ2

∂y(x, y) = (0, 0, 1),

• a23 := a2

1 × a22 = (0,−1, 0).

a21

a13

a11

θ21

a23

a22

θ23

θ22

φ2

ΓI

φ1

θ12

θ11

θ13

a12

Ω1 = Ω2

Plate 2

Plate 1

Fig. 3. Folded plates.

The coupling conditions between these folded plates on ΓI are

u11

u12

u13

=

1 0 0

0 sin α − cos α

0 cos α sin α

u21

u22

u23

,

β11

β12

β13

=

sin α 0 − cos α

0 1 0

cos α 0 sin α

β21

β22

β23

.

Since Ω1 = Ω2 = [0, 1] × [0, 1], the meshes that we consider for each plate arethose in Section 4.1, labeled in the same way.

In Table 3 we show a comparison between the results with our method againstthe results in [12] for a cantilever thin folded plate. To show the performanceof the methods we consider a coarser meshes correspond to N = 8. We assumethat the system is perfectly clamped on the edge (0, 0, y) and (0, y cos α, 1 +y sin α), for y ∈ [0, 1] (see Figure 3). In this table, we present the computedfrequencies in the following non-dimensional form:

ωFh

√

ρ1 − ν2

E. (2)

In Figure 4 we show the deformed first and second mode for the cantileverfolded plate with a crank angle of 150.

As a second numerical experiment, we consider a folded plate, with a crankangle of 150, having two inclined edges clamped and the other two straightedges free; i.e. clamped on the edge (0, 0, y),(1, 0, y), (0, y cos α, 1 + y sin α),and (1, y cos α, 1 + y sin α), for y ∈ [0, 1]. In Table 5 and 4, respectively, we

17

Table 3Comparison for a cantilever folded plate.

Crank angle(deg) method Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

90 actual 0.0492 0.0978 0.1835 0.2147 0.3516

LS9RI[12] 0.0488 0.0956 0.1784 0.2071 0.3407

120 actual 0.0492 0.0950 0.1837 0.2132 0.3038

LS9RI[12] 0.0487 0.0930 0.1785 0.2056 0.2847

150 actual 0.0493 0.0826 0.1838 0.2008 0.2256

LS9RI[12] 0.0487 0.0798 0.1785 0.1868 0.2172

0

0.5

1

1.5

2

2.5Deformed plates

0

0.5

1

1.5

2

2.5Deformed plates

Fig. 4. Deformation corresponding to the first and second modes for a cantileverfolded plate

show the non-dimensional frequencies (according to (2)) for plates with thick-ness 0.1m and 0.01m. As proved in Theorem 4, the order of convergence is,approximately, 2. In Figure 5 we show the deformed folded plates for some ofthese vibration modes.

Table 4Incline edges clamped. Thickness=0.1, angle=150.

Mode N=20 N=28 N=36 Order Extrap

1 0.6711 0.6682 0.6668 1.81 0.6644

2 0.6970 0.6944 0.6931 1.62 0.6904

3 1.1735 1.1686 1.1663 1.72 1.1619

4 1.6153 1.6013 1.5953 2.08 1.5862

5 1.6719 1.6584 1.6525 2.02 1.6432

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Table 5Incline edges clamped. Thickness=0.01, angle=150.

Mode N=20 N=28 N=36 Order Extrap

1 0.0766 0.0764 0.0763 1.98 0.0761

2 0.0897 0.0894 0.0893 1.88 0.089157

3 0.1972 0.1953 0.1946 2.16 0.19346

4 0.2052 0.2034 0.2026 2.14 0.20151

5 0.2328 0.2313 0.2307 2.05 0.22974

01

2−0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2First mode

01

2−0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Second mode

Fig. 5. Incline edges clamped. Deformation for the first and second modes.

Finally, in order to assess the quality of the method for very thin plates, weconsider a folded plate with a crank angle of 120, clamped on the whole of itsboundary, with different thickness-to-span ratio. Tables 6, 7 and 8 show thethree lowest computed vibration frequencies. As in the previous case, extrap-olated more accurate values are included. It can be seen that the numericalresults do not deteriorate when the thickness decrease, what shows that themethod is free of locking. Figure 6 shows the first and third deformed mode.

N = 20 N = 28 N = 36 ORD extrap

0.1 2670.702 2648.981 2639.692 1.95 2623.789

0.01 282.228 279.869 278.909 2.18 277.502

0.001 28.239 28.004 27.908 2.18 27.767Table 6Totally clamped folded plate. First mode when decreasing the thickness.

19

N = 20 N = 28 N = 36 ORD extrap

0.1 3171.929 3145.245 3134.305 2.14 3117.847

0.01 381.338 376.292 374.259 2.21 371.326

0.001 38.207 37.699 37.495 2.21 37.200Table 7Totally clamped folded plate. Second mode when decreasing the thickness.

N = 20 N = 28 N = 36 ORD extrap

0.1 3865.709 3145.245 3134.305 1.82 3790.599

0.01 416.3157 412.703 411.229 2.17 409.055

0.001 41.666 41.305 41.158 2.18 40.942Table 8Totally clamped folded plate. Third mode when decreasing the thickness.

0

1

2 −0.50

0.51

0

0.5

1

1.5

First mode of a two−folded plate

0

1

2 −0.50

0.51

0

0.5

1

1.5

Third mode of a two−folded plate

Fig. 6. First and Third eigenmode for a totally clamped folded plate

5 Conclusions

We have considered a finite element scheme for folded plates, coupling MITC4finite elements for the shear and bending part with standard quadratic finiteelements, enriched with drilling degrees of freedom, for the membrane.

In the case of one single plate, we have proved optimal order error estimates foreigenvalues and eigenfrequencies. We have presented numerical results showingthat, in practice, the optimal order is achieved and, also, showing that theinclusion of the drilling degree of freedom does not deteriorate the results.

We have extended this result to a system made by two folded plates, provingagain optimal error estimates for eigenvalues and eigenfrequencies, not de-pending on the thickness. We have presented numerical examples for differentcases of two folded plates, with different thickness, boundary conditions andcrack angles. The order of convergence, in practice, approaches the theoretical

20

one, and the results are completely free of locking in all cases.

We would like to put into account, as a remark, that this coupled method canbe used to solve shell problems when numerical locking (shear and membrane)is presented.

References

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Analysis. Vol II, P.G. Ciarlet and J.L. Lions, eds., North Holland, Amsterdam,1991, 641-787.

[2] K.J. Bathe, Finite Element Procedures. Prentice Hall,Englewood Cliffs, NJ,1996.

[3] K.J. Bathe, F. Brezzi, On the convergence of a four-node plate bendingelement based on Mindlin/Reissner plate theory and a mixed interpolationin Mathematics of Finite Elements an Applications V, J.R. Whiteman, ed.,Academic Press, London, 1985, 491-503.

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[7] C. Chinosi, M.I. Comodi, G. Sacchi, A new finite element with ’drilling’ D.O.F.,Comput. Methods Appl. Mech. Engrg. 143 (1997) 1-11.

[8] D. Chapelle, K.J. Bathe, Fundamental considerations for the finite elementanalysis of shell structures, Compt. Struc. 66 (1998) 19-36.

[9] D. Chapelle, K.J. Bathe, The Finite Element Analysis of Shells: Fundamentals,Springer Verlang, 2003.

[10] R. Duran, E. Hernandez, L. Hervella-Nieto, E. Liberman, R. Rodrıguez, Errorestimates for low-order isoparametric quadrilateral finite element for plates,SIAM J. Numer. Anal. 41 (2004) 1751-1772.

[11] R. Duran, E. Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992) 561-573.

[12] S. Haldar, A.H. Sheikh, Free vibration analysis of isotropic and composite foldedplates using a shear flexible element, Finite Elements in Analysis and Design42 (2005) 208-226.

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[13] E. Hernandez, L. Hervella-Nieto, R. Rodrıguez, Computation of the vibrationmodes of plates and shells by low order MITC quadrilateral finite elements,Compt. Struc. 81 (2003) 615-628.

[14] T.J.R.Hughes, F.Brezzi. On drilling degrees of freedom, Comput. MethodsAppl. Mech. Engrg. 72 (1989) 105-121.

[15] S.-Y. Lee, S.-C. Wooh, Finite element vibration analysis of composite boxstructures using the high order plate theory, Journal of Sound and Vibration277 (2004) 801-814.

[16] L.X. Peng, S. Kitipornchai, K.M. Liew, Free vibration analysis of folded platestructures by the FSDT mesh-free method, Comput. Mech. 39 (2007) 799-814.

[17] L.X. Peng, S. Kitipornchai, K.M. Liew, Bending analysis of folded plates by theFSDT meshless method, Thin-Walled Struc. 44 (2006) 1138-1160.

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