Thin Plate Bending Analysis Using an IndirectTrefftz Collocation Method

Carlos M. Tiago†, Vitor M. A. Leitao‡ and Anna VasarhelyiDepartment of Civil EngineeringTechnical University of Budapest

H-1521 Budapest, Hungary†on leave from:

‡Departamento de Engenharia CivilInstituto Superior Tecnico1049-001 Lisboa, Portugal

Abstract

In this work, the application of an indirect Trefftz collocation method tothe analysis of bending of thin plates (Kirchhoff’s theory) is described. Thedeflection field approximation is obtained with the use of a set of functionssatisfying a priori the homogeneous part of the differential equation of theproblem. Each of the approximating functions is derived from a known thickplate solution. The boundary conditions are imposed by means of continuous(integral) and discrete (collocation) least squares methods. Numerical exam-ples are presented and the accuracy of the proposed technique is assessed.

Keywords: Trefftz method, thin plates, point collocation, complex trial functions,least squares.

1 Introduction

In the engineering practice, the finite element method (FEM) is used widely toobtain numerical solutions of problems that are formulated in terms of differentialequations and the corresponding boundary conditions (BC). The search for accurateand better performing elements lead some researchers to use, as trial functions for theinternal fields, functions that are themselves solutions of the differential equationsof the problem a priori. The interelement continuity and the BC may, then, beenforced pointwise or in an integral weighted residual sense.

While the conventional conforming FEM formulations may be viewed as par-ticular applications of the Rayleigh-Ritz method, the alternative method of usingactual solutions of the governing equations as trial functions is closely related to themethod presented by Trefftz in 1926 [1]. Since 1977, when Jirousek and Leon [2]

1

published a thin plate bending element based on the variational Trefftz method, therange of applications has been extended to a broad variety of problems. Reviews onthe subject may be found in Zielinski [3] and Kita and Kamiya [4].

It is well known that the thin plate theory, due to the assumptions that are made,requires C1 continuity. That leads to difficulties when generating conventional FE(finite elements). On the other hand, for the usual practice of civil engineering itis not necessary to take into account the effect on bending of the transverse (shear)deformation. Ref. [7] shows that it is possible to obtain quality results for Kirchhoffplates with the variational Trefftz method.

The formulations of Trefftz-type elements (T-elements) can include an auxiliardisplacements frame defined in terms of the same degrees of freedom (DOF) usedin the conventional FE. The main advantage is that these elements can be used bya standard FE program. On the other hand, the frameless T-elements (in whichthe DOF are only the parameters of the set of functions chosen to approximatethe internal field) are easier to generate. The formulation described in this papercan be seen as a form of frameless T-elements. The choice of how to impose theBC is also an important one. Two options are available: point collocation or someform of weighted residuals (integration) on the boundary. A review of both wasmade by Eason [12], where references to applications in atomic physics, elasticity(composite and homogeneous materials, stress concentrations, thermal loadings),electromagnetic fields, fluid flow, heat transfer and mathematical are given.

A review of the analysis of problems of shells and plates using the boundarycollocation method (BCM) was made by Hutchinson [9]. Discussion of several caseswas made: thin plate bending, buckling and vibration, thick plate vibration, shallowspherical shells and thick shells of revolution. A extensive review of the applicationof the BCM to mechanics of continuous media is given by Kolodziej [8]. The possi-bility of using influence functions and collocation outside the domain of the problemunder analysis was exploited by Wu and Altiero [13, 14] to solve thin plate bendingproblems.

The structure of this work includes a brief description of the Kirchhoff platebending theory, followed by the definition of the complex polynomial set of functionsused to approximate the deflection field. The technique proposed here, together withthe definition of the BC, is then described. Reference to implementation aspects,numerical applications and corresponding comments and conclusions complete thework.

2 Classic formulation of thin plate theory

Consider the plate represented on figure 1, where “−” stands for imposed quantities.The following notation will be used:

w - deflection of the middle surface of the plate;

∂w∂n

- slope of the middle surface of the plate;

Mn - bending moments of a section of the plate perpendicular to the n direction;

2

∂w∂n

w

Mn

V n

Ω

Γ

P (x, y)

p(x, y)

t n

sα

x

y

z

Figure 1: Arbitrary thin plate.

Mnt - twisting moments of a section of the plate perpendicular to the n direction;

Qn - shearing force parallel to z axis of a section of the plate perpendicular to then direction;

Vn = Qn + ∂Mnt

∂s- effective shear force parallel to z axis of a section of the plate

perpendicular to the n direction;

x, y, z - cartesian co-ordinate system;

t - thickness of the plate;

E - Young’s modulus;

ν - Poisson’s ratio;

p - uniform load;

P - concentrated load.

The differential equation of the deflection surface is the well known Lagrangeequation:

∇4w =p

D, (1)

where:

∇4 is the biharmonic operator;

D = Et3

12(1−ν2)is the flexural rigidity of the plate.

3

By neglecting the effect of deformation due to the shear stresses it is possible toidentify the dependency between an additional shear force, Q′

n and ∂Mnt

∂t.

The domain of the problem, Ω, may be split into several regions featuring somespecific behaviour. For the i-th region the BC may be classified as being of thesupport or continuity type. Each point on the boundary of this region should fulfiltwo support BC: deflection or effective normal shear force and normal slope ornormal bending moment:

wi = wi t ∈ Γiw, (2a)

∂wi

∂n=

∂w

∂n

i

t ∈ Γi∂w∂n

, (2b)

M in + GJ

∂3wi

∂t2∂n= M

i

n t ∈ ΓiMn

, (2c)

Qin +

∂M int

∂t+ EI

∂4wi

∂t4= V

i

n t ∈ ΓiVn

, (2d)

The existence of beams on the boundary is allowed by means of the flexuralrigidity, EI, and torsional rigidity, GJ .

The enforcement of continuity and equilibrium between the regions requires fourboundary conditions to be satisfied at the interface between Ωi and Ωi+1, that is,ΓI = Γi ∩ Γi+1.

wi − wi+1 = 0, (3a)

∂wi

∂n+

∂wi+1

∂n= 0, (3b)

M in + GJ

∂3wi

∂t2∂n− M i+1

n = 0, (3c)

V in + EI

∂4wi

∂t4+ V i+1

n = 0. (3d)

Once the displacement field is known it is straightforward to evaluate the bendingand twisting moments

Mx = −D

(∂2w

∂2x+ ν

∂2w

∂2y

), (4a)

My = −D

(∂2w

∂2y+ ν

∂2w

∂2x

), (4b)

Mxy = Myx = −D(1 − ν)

(∂2w

∂x∂y

)(4c)

and the shear forces

Qx = −D∂

∂x∇2w, (5a)

Qy = −D∂

∂y∇2, (5b)

4

where ∇2 is the harmonic operator.Noticing that ∂

∂n= ∂

∂xcos α + ∂

∂ysin α equation (2b) can be written in the form

∂wi

∂xcos α +

∂wi

∂ysin α =

∂wi

∂n. (6)

The integration of the normal component of the stresses parallel to n direction,σn, and the shearing stresses perpendicular to n direction, τnt, gives:

Mn = Mx cos2 α + My sin2 α + 2Mxy sin α cos α, (7a)

Mnt = (My − Mx) sin α cos α + Mxy

(cos2 α − sin2 α

). (7b)

The shear forces acting perpendicular to the middle plane are:

Qn = Qx cos α + Qy sin α. (8)

Substituting (4) in (7a), and then in (2c), the following equation is obtained:

− D

ν∇2wi + (1 − ν)

(cos2 α

∂2wi

∂x2+ sin2 α

∂2wi

∂y2

+ sin 2α∂2wi

∂x∂y

)+ GJ

∂3wi

∂t2∂n= M

i

n. (9)

Proceeding in the same way, substituting (5) in (8) and (4) in (7b), and thenin (2d), leads to the following equation

− D

cos α

∂

∂x∇2wi + sin α

∂

∂y∇2wi+

+ (1 − ν)∂

∂t

[cos 2α

∂2wi

∂x∂y+

1

2sin 2α

(∂2wi

∂y2− ∂2wi

∂x2

)]+ EI

∂4wi

∂t4= V

i

n, (10)

where ∂∂t

= − sin α ∂∂x

+ cos α ∂∂y

. By using equations (2a), (6), (9) and (10) all

the support boundary conditions can be written in terms of wi and its derivativesalone.

The same procedure is used for continuity type BC.

3 Approximating functions of the deflection field

The Trefftz method uses, as trial functions, actual solutions of the differential equa-tion. The complex form representation is a very convenient way of describing theinfinite set of linearly independent trial functions which form the complete basis ofsolutions of the governing differential equations of the plate bending problem. Dueto the linearity of the differential equation it is possible to express the deflection

5

function (the solution) as the sum of the solution of the homogeneous part (wh) and

a particular solution of the nonhomogeneous part (w):

w =w + wh, (11)

A thick plate solution is given in [6]. From this it is possible to obtain anappropriate thin plate solution after neglecting warping and change of thicknessterms (all terms which involve h2, z2 and z3), as follows,

wh =1

2µ [

ζΦ + χ], (12)

where 2µ = E1+ν

, ζ = x + iy, ζ = x − iy, Φ = Φ (ζ) and χ = χ (ζ). One possiblechoice for functions Φ and χ can be the following complex power series:

Φ (ζ) =m∑

j=1

aj (ζ)j , (13a)

χ (ζ) =m∑

j=2

bj (ζ)j , (13b)

where aj = αj + iβj and bj = γj + iδj. The term δ0 is dropped out because itscoefficient in the expansion (12) is zero. The terms γ1 and δ1 are, also, excludedsince they represent rigid body motions that are already taken into consideration inthe terms α0 and β0. Expression (12) assumes the following matrix form:

wh =m∑

j=1

Nj cj = Nw c, (14)

where:

Nw =[

N0 N1 · · · Nm

],

cT =[

c0 c1 · · · cm

],

N0 =1

2µ

[x y 1

],N1 =

1

2µ

[r2

],

Nk =1

2µ

[r2 (ζ)k−1 −r2 (ζ)k−1 (ζ)k − (ζ)k

],

c0 =[

α0 β0 γ0

], c1 =

[α1

],

ck =[

αk βk γk δk

], where r2 = x2 + y2 and k = 2, 3 . . . m.

The number of terms in c is 4 m.The particular solution,

w, for a uniform load, p, and for a concentrated load,

P , located at (xP , yP ), may be taken as

w =

p r4

64D+

P r2P ln r2

P

16πD(15)

6

where r2P = (x − xP )2 + (y − yP )2.

If it is of interest to analyse the local effects of point loads may be advantageousto replace the actual load by a uniform one acting in a small circle of radius b.Taking d as an optional constant and using the definitions, ρ = r

dand β = b

d, the

solution for an infinite plate loaded uniformly over a circle with static resultant Pis

if ρ ≤ β

w =

[β2

64(4 ln β − 3) +

1

16+

ρ2

32

(4 ln β − β2

)+

ρ4

64β2

]Pd2

πD, (16a)

if ρ ≥ β

w =

[1

32

(2 + β2

) (1 − ρ2

)+

1

16

(β2 + 2ρ2

)ln ρ

]Pd2

πD. (16b)

Solutions for linear loads acting on a patch domain or straight line are given byFernandes [10].

For regions with singularities due to the geometry (regions containing holes, skewplates, etc) alternative trial functions for homogeneous and particular solutions maybe used.

The region (or regions) may have an arbitrary shape as long as it is simplyconnected. For regions that are not simply connected, such as the ones with holes, itshould be wise to use different sets of solutions for that specific case or to decomposethe region into a number of simply connected ones.

4 Imposition of the boundary conditions

At each point located on the boundary interface two four boundary conditionsshould be satisfied.

The deflection field, w, is expressed by substituting (14), (15) and (16) in (11).Substituting w in (2a), (6), (9) and (10), the equations relating the values of

the boundary conditions to the undetermined parameters, c, are obtained. In the

process derivatives ofw and N arise. After symbolically evaluating the derivatives,

the equations for the i-th region can be put in the form:

w

i+ Ni

w ci = wi, (17a)

∂w

i

∂n+ Ni

∂w∂n

ci =∂w

∂n

i

, (17b)

M i

n + NiMn

ci = Mi

n, (17c)

V in + Ni

Vnci = V

i

n, (17d)

The corresponding equations for interfaces are derived in the same way.

7

Writing these equations at selected points along the boundary (collocation pointsor CP) the governing system can be represented in the form:

Dc =d −d (18)

where matrix D contains the expansion terms of the series Nw, N ∂w∂n

, NMn ,

NVn and the right hand side d −d contain the imposed values and the particular

solutions, w − w, ∂w

∂n− ∂

w

∂n, Mn −

Mn and V n −

V n.

A determined system of equations may be obtained when the number of condi-tions imposed equals the number of terms of the displacement field expansion. Anon symmetric square system matrix, D, is, thus, obtained.

As the boundary conditions are imposed at points, the residuals (the differenceto actual BC values) may be unacceptable or exhibit unreasonable variation. Todecrease the errors, it is possible to collocate at more points than the strictly nec-essary ones leading, in this manner, to an overdetermined system of equations. Theleast squares criteria is used here to define the optimal solution whereby the unde-termined coefficients, ci, of region i are found from the minimisation of the squareof the boundary residuals.

If a continuous approach is used instead of a discrete one, follows that:Minimize I(ci)

I (c) = W 2w

∫Γi

w

(wi − wi

)2dΓi

w

+ W 2∂w∂n

∫Γi

∂w∂n

(∂wi

∂n− ∂w

∂n

i)2

dΓi∂w∂n

+ W 2Mn

∫Γi

Mn

(M i

n + GJ∂3wi

∂t2∂n− M

i

n

)2

dΓiMn

+ W 2Vn

∫Γi

Vn

(V i

n + EI∂4wi

∂t4− V

i

n

)2

dΓiVn

+ W 2w

∫ΓI

(wi − wj

)2dΓI

+ W 2∂w∂n

∫ΓI

(∂wi

∂n+

∂wj

∂n

)2

dΓI + W 2Mn

∫ΓI

(M i

n + GJ∂3wi

∂t2∂n− M j

n

)2

dΓI

+ W 2Vn

∫ΓI

(V i

n + EI∂4wi

∂t4+ V j

n

)2

dΓI ,

(19)

where dΓI = Ωi⋂

Ωj and Ww, W ∂w∂n

,WMn and WVn are weights. The j superscript

refers to a neighbouring region.The first four terms represent the support BC’s and the other four express the

continuity and equilibrium between the i and j regions.The weights are used to restore the homogeneity of the physical dimensions and

the relative strength of the different boundary conditions. In this work equal weightsare assumed.

8

By setting the first derivative of (19) with respect to c to zero, the followingequation arises

Kc = f

where K is a square symmetric definite positive matrix.The contribution of the i-th region to this system of equations is

[Kii Kij

]ci

cj

= fi ,

.If summation at collocation points is used instead of integrations then the discrete

approach is reached.A method for finding the optimal set of weights is presented and illustrated in

Ref [11].

5 Remarks on implementation

An important issue in the implementation of the formulation is the choice of thenumber and the position of the collocation points. If some part of the boundary ismore important to the problem (in the sense that some local effect occurs such asa load or a change in the geometry or the boundary conditions), then the densityof CP’s located on that part, compared to the density on the remaining boundary,should be increased. Another way of achieving this is to increase the relative weightsof the CP’s. In this case the weights should be moved inside the integration operatorsin (19). Also equations (17) should be written in a local co-ordinate system locatedat the region centroid and in dimensionless (or normalized) form,

ξ =x

d, η =

y

d

where d is a characteristic distance of the problem, e. g., the length of the largestside of all regions. Once the system is solved the effect of that normalizing factorhas to be removed in order to determine all the relevant results.

6 Numerical examples

The cases of a simply supported rectangular plate and a clamped circular platesubjected to various types of loading are considered.

In all cases the following numerical data used is: E = 1.0, p = 1.0, a = 1.0,ν = 0.3 and t = a

10. The values of the weights are as follows: Ww = 1.0, W ∂w

∂n= 1.0,

WMn = 1.0 and WVn = 1.0.

6.1 Simply supported rectangular plate

Consider a simply supported plate with dimensions b × a × t, material propertiesE, ν.

9

a2

b2

E, ν, t

A B

C D

x

y

(a) Geometry and dimensions.

1

2

3

4

1 2

34

1

x

y

(b) Discretization adopted.

Figure 2: Thin rectangular simply supported plate with symmetry simplification.

The Navier solution for this case is

w(x, y) =1

π4D

∞∑m=1,2,3,...

∞∑n=1,2,3,...

am n(m2

b2+ n2

a2

)2 sinmπx

bsin

nπy

a. (20)

where the coefficients am n depend on the applied load.The discretization adopted for the three loads treated is show in figure 2(b).

6.1.1 Uniform load

The numerical example concerns a rectangular simply supported plate under uniformload. The results obtained by collocation and with the direct approach are comparedto the exact solution.

Only a quarter of the plate was analysed (figure 2(a)).The boundary conditions (in the local centroid co-ordinate system, figure 2(b))

are:i) at symmetry axes

∂w

∂n

∣∣∣∣x=0, 0≤y≤a

2

= 0,∂w

∂n

∣∣∣∣0≤x≤ b

2, y=0

= 0,

V n

∣∣x=0, 0≤y≤a

2

= 0, V n

∣∣0≤x≤ b

2, y=0

= 0.

ii) at simply supported edges

w|x= b2, 0≤y≤a

2= 0, w|0≤x≤ b

2, y= a

2= 0,

Mn

∣∣x= b

2, 0≤y≤a

2

= 0, Mn

∣∣0≤x≤ b

2, y= a

2

= 0.

The coefficients for the Navier solution for this case [5] are

am n =16 p

π2 m n, (21)

10

10 wadim A −Qadim Bx −Qadim C

y

m Disc. Cont. Disc. Cont. Disc. Cont.6 0.103029 0.104508 0.3425 0.3164 0.4694 0.47678 0.101157 0.101246 0.3787 0.3793 0.4629 0.460910 0.101237 0.101291 0.3727 0.3740 0.4649 0.466412 0.101280 0.101286 0.3672 0.3670 0.4645 0.4638

Navier 0.101287 0.3696 0.4650

Table 1: Values of wadim A, Qadim Bx and Qadim C

y for a simply supported plate sub-jected to uniform load.

10 Madim Axx Madim A

yy −10 Madim Dxy

m Disc. Cont. Disc. Cont. Disc. Cont.6 0.47312 0.47417 0.10355 0.10508 0.48884 0.528938 0.46149 0.46435 0.10165 0.10180 0.48206 0.4788110 0.46342 0.46385 0.10164 0.10171 0.46945 0.4686912 0.46349 0.46354 0.10168 0.10169 0.46579 0.46574

Navier 0.46350 0.10168 0.46267

Table 2: Values of Madim Axx , Madim A

yy and Madim Dxy for a simply supported plate sub-

jected to uniform load.

where m = 1, 3, 5, . . . and n = 1, 3, 5, . . ..The results obtained considering b = 2a are presented in tables 1 and 2. The

Navier solution was found using 5000 terms in each direction.The CPU time in creating the system matrices for both, discrete and continuous,

approaches are listed in table 3. The discrete approach is, as expected, faster thanthe continuous one as no integrations are required.

6.1.2 Uniform load in a rectangular area

Consider the plate represented in figure 3, where b = 2a.The coefficients for the Navier solution for this case are

am n =16

a b

P

u vsin

mπξ

bsin

nπη

asin

mπu

2bsin

nπv

2a(22)

where m = 1, 2, 3, . . ., n = 1, 2, 3, . . . and constants u, v, ξ and η are represented infigure 4. P denotes the total load, P = u v p.

m Disc. Cont.

6 0.05 0.68 0.06 1.1510 0.11 1.8112 0.17 1.98

Table 3: CPU time (sec) in creating the system matrices.

11

a

b

a2

a2

a4

a4

x

yE, ν, t

E

Figure 3: Simply supported rectangular plate subjected to uniform load applied ontwo rectangular areas.

a

b

u

v

ξ

η

x

yE, ν, t

Figure 4: Simply supported rectangular plate subjected to uniform load applied onrectangular area, see (22).

12

100 wadim A 100 Madim Exy

m Disc. Cont. Disc. Cont.10 0.139705 0.134997 0.73549 0.7202914 0.134859 0.134675 0.75186 0.7512118 0.134665 0.134659 0.75783 0.7584522 0.134657 0.134659 0.75715 0.75726

Navier 0.134659 0.75727

Table 4: Values of wadim A e Madim Exy for a simply supported rectangular plate sub-

jected to uniform load applied on rectangular areas.

10 Madim Exx 10Madim E

yy

m Disc. Cont. Disc. Cont.10 0.49685 0.50000 0.78522 0.7724214 0.49590 0.49531 0.80496 0.8042718 0.49570 0.49567 0.80591 0.8060322 0.49585 0.49586 0.80604 0.80603

Navier 0.49593 0.80601

Table 5: Values of Madim Exx e Madim E

yy for a simply supported rectangular plate sub-jected to uniform load applied on rectangular areas.

The results are listed in tables 4 and 5. Point E is denoted in figure 3.This case was studied by Venkatesh and Jirousek [15] and similar results were

obtained.The principal bending directions are represented in figure 5.

6.1.3 Bilinear load

The model analised is represented in figure 6.If the load, p, is expressed in the global reference system by p(x, y) = p0 + p1x +

p2y, then the series coefficients am n that describe the load in the Navier method aream n = ap0

m n + ap1m n + ap2

m n, where ap0m n is given by (22) and

ap1m n =

8 p1

π3m2nsin

nπη

asin

nπv

2a

[2b cos

mπξ

bsin

mπu

2b

+2ξmπ sinmπξ

bsin

mπu

2b− umπ cos

mπξ

bcos

mπu

2b

],

ap2m n =

8 p2

π3mn2sin

mπξ

bsin

mπu

2b

[2a cos

nπη

asin

nπv

2a

+2ηnπ sinnπη

asin

nπv

2a− v nπ cos

nπη

acos

nπv

2a

].

where u, v, ξ and η are the same as in the previous example and P = u v p0.

13

Figure 5: Principal bending directions for a simply supported rectangular plate sub-jected to uniform load applied on rectangular areas.

a

b

b2

p

x

y

E, ν, t

Figure 6: Simply supported rectangular plate subjected to a bilinear load.

14

The derivation of these expressions, as well as the particular solutions used inthis case and in the previous example can be found in Fernandes [10].

The results obtained in the center of the plate for several a, b ratios are shownin table 6.

The Navier solution was found using 500 terms in each direction. The discreteprocedure was applied assuming m = 10 and m = 30 (m is the order of the approx-imation considered).

15

100w

adim

A10

Madim

Axx

10M

adim

Ayy

b am

=10

m=

30N

avie

rm

=10

m=

30N

avie

rm

=10

m=

30N

avie

r1.

00.

2624

860.

2627

440.

2627

440.

3406

300.

3404

980.

3404

980.

3164

280.

3170

950.

3170

951.

10.

3155

250.

3158

350.

3158

350.

3562

640.

3560

650.

3560

650.

3677

750.

3684

600.

3684

601.

20.

3674

260.

3677

510.

3677

510.

3677

730.

3675

180.

3675

180.

4170

600.

4176

990.

4176

991.

30.

4172

290.

4175

710.

4175

710.

3760

550.

3757

790.

3757

790.

4636

330.

4642

220.

4642

221.

40.

4643

390.

4647

290.

4647

290.

3818

570.

3816

180.

3816

180.

5071

610.

5077

290.

5077

291.

50.

5084

400.

5089

340.

5089

340.

3857

780.

3856

490.

3856

490.

5475

280.

5481

200.

5481

201.

60.

5494

130.

5500

890.

5500

890.

3882

840.

3883

500.

3883

500.

5847

620.

5854

380.

5854

381.

70.

5872

730.

5882

310.

5882

300.

3897

270.

3900

850.

3900

850.

6189

760.

6198

130.

6198

131.

80.

6221

090.

6234

790.

6234

790.

3903

670.

3911

240.

3911

240.

6503

270.

6514

260.

6514

261.

90.

6540

510.

6560

040.

6560

030.

3903

920.

3916

700.

3916

690.

6789

900.

6804

820.

6804

822.

00.

6832

380.

6859

980.

6859

970.

3899

260.

3918

690.

3918

680.

7051

340.

7071

930.

7071

933.

00.

8483

590.

8868

440.

8868

440.

3690

170.

3883

670.

3883

890.

8553

200.

8847

280.

8847

16

Tab

le6:

Val

ues

ofw

adim

A,M

adim

Axx

and

Madim

Ayy

for

asi

mply

suppor

ted

pla

tesu

bje

cted

toa

bilin

ear

load

.

16

The results achieved prove the excellent agreement of the two different solutions.

6.2 Circular clamped plate

Consider a circular clamped plate of radius a, thickness t and material propertiesE, ν.

In this example, the description of the problem geometry, as well as the gener-alized forces, will be made in a cylindrical coordinate system.

The boundary conditions in the centroid local system of coordinates are:

w|r=a = 0, (23a)

∂w

∂n

∣∣∣∣r=a

= 0. (23b)

The exact solution for the deflection of the middle surface of the plate subjectedto a central point load, p, is [5]:

w =P

8πDr2 log

r

a+

P

16πD

(a2 − r2

), (24)

where r2 = x2 + y2.Since this is a case of symmetrical bending of circular plates Mrθ, Qθ and θθ are

zero.The bending moments Mr and Mθ for points located far from the point where

the load is applied are

Mr =P

4π

[(1 + ν) ln

a

r− 1

](25a)

Mθ =P

4π

[(1 + ν) ln

a

r− ν

](25b)

and the shear force is

Qr = − P

2πr(26)

In the center of the plate the maximum bending moment can be evaluated as-suming the load is uniformly distributed in a circle of small radius, c, by [5]:

Mmax =P

4 π

[(1 + ν) ln

a

c− (1 − ν)c2

4a2

]. (27)

The discretization used, a single region with four sides (cubic splines), is repre-sented in figure 7.

The results, using the actual concentrated load, are in tables 7 and 8, wherepoints A and B are located in the center and in the boundary of the plate, respec-tively.

The bending moments in the center of the plate were found using the particularsolution (16), and the results are in table 9.

17

x

y

1

2

3

1 2

3

4

5

1

Figure 7: Discretization used for a circular clamped plate.

10 wadim A −Qadim Br

m nequations DOF Cont. Disc. Cont. Disc.1 8 4 0.19826 0.19826 0.15915 0.159153 24 12 0.19923 0.19942 0.15915 0.159155 40 20 0.19925 0.19921 0.15268 0.15374

Exact 0.19894 0.15915

Table 7: Values of wadim A and Qadim Br for a circular clamped plate subjected to a

central point load.

−10 Madim Br −10Madim B

θ

m nequations DOF Cont. Disc. Cont. Disc.1 8 4 0.79756 0.79756 0.24052 0.240523 24 12 0.79502 0.79452 0.23798 0.237485 40 20 0.78015 0.78269 0.23171 0.23279

Exact 0.79577 0.23873

Table 8: Values of Madim Br and Madim B

θ for a circular clamped plate subjected to acentral point load.

Madim Ar

10=

Madim Aθ

10

m nequations DOF Cont. Disc.1 8 4 0.119084 0.1190843 24 12 0.119110 0.1191155 40 20 0.119110 0.119109

Exact 0.119102

Table 9: Values of Madim Ar and Madim A

θ for a circular clamped plate subjected to acentral load distributed in a circle of radius c = a

1.0·105 .

18

0,000,020,040,060,080,100,120,140,160,180,20

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,010

wadim

Trefftz m = 10, 10 PC/side

ra

expression (24)

(a) Deflection, wadim.

-0,30

-0,25

-0,20

-0,15

-0,10

-0,05

0,00

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Qa

dim

r 10

Trefftz m = 10, 10 PC/side

ra

expression (26)

(b) Shear force, Qadimr .

Figure 8: Results for a circular clamped plate subjected to a central concentratedload.

In figure 8 it is possible to compare the solution presented with the solution givenby Timoshenko [5].

7 Conclusions

In this work an Indirect Trefftz formulation was applied to the analysis of bendingof thin plates by using the Kirchhoff plate bending theory.

An assumed displacement field (the primary variable in the irreducible thin plateapproximation) that satisfies a priori the governing differential equation of the prob-lem (Trefftz method) ensures that all the requirements (equilibrium, compatibilityand stress-strain relationship in solid mechanics problems) in the domain are ful-filled. In case the domain under analysis is discretized into more than one regions,

19

of arbitrary shape, and this may occur for some specific problems, an assumed dis-placement field must be defined for each region and extra continuity/compatibilityconditions must be enforced at the interfaces.

Other type of problems (domain configuration and/or boundary conditions) existfor which it is more convenient to use specifically devised displacement field approx-imation. These displacement fields are obtained in such a way that some prescribedboundary conditions are directly enforced. This is the case for skewed or perforatedplates for example.

The approximating displacement field is expressed by a truncated series. Thenumber of terms of the series which is used is variable. It depends mainly on thegeometry of the domain and on the boundary conditions/loads.

In the present study the governing differential equation is of the fourth order.This implies that in the finite element analysis the primary variables are, at least,the displacement and the rotations of the middle surface. The present formulationonly requires the use of the natural unknown, thus reducing the dimension of thesought solution.

The boundary conditions of each region are expressed in terms of the primaryvariable approximation and imposed by minimizing, in a conveniently weighed way,the errors of the generalized displacements and forces. To achieve this, a discrete(or collocation) and a continuous (or integration) approach for the minimization ofthe residues were studied.

The minimization process based on least squares proved to be an appropriate wayof imposing the boundary conditions. On the other hand, the collocation procedureis more efficient as it requires less numerical calculations for similar accuracy.

The resulting system of equations (either obtained by collocation or by integra-tion) is full (in case only one region is used) or almost full (as, in general, only afew regions are used) and symmetric. To avoid ill conditioning appropriate scalingshould be used.

As for the applications are concerned, the results show that good approximationscan be obtained using an assumed displacement field that a priori satisfy the dif-ferential equation of the problem. The comparison with analytical (when available)or numerical (Navier series) solutions reveal, for the cases tested, an almost perfectagreement.

Acknowledgements

The first and second authors acknowledge the financial support of the TEMPUSprogramme, research contract TEMPUS JEP 11236/96., of PRAXIS, through grantPRAXIS/2/2.1/CEG/33/94, and of ICIST.

References

[1] Trefftz, E., Ein Gegenstuck zum Ritzschen Verfharen, Proc. 2nd Int. Cong.Appl. Mech., Zurich, pp. 131-137, (1926).

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[2] J. Jirousek and N. Leon, A powerful finite element for plate bending, Comp.Meth. Appl. Mech. Eng., 1977, 12, 77-96.

[3] Zielinski, A.P., On trial functions applied in the generalized Trefftz method,Advances in Engineering Software, 1995, 24, pp. 147-155.

[4] Kita, E., Kamiya, N., Trefftz method: an overview, Advances in EngineeringSoftware, 1995, 24, pp. 3-12.

[5] Timoshenko and S. Woinowsky-Krieger, Theory of plates and shells. 2nd ed.McGraw-Hill, 1959.

[6] Piltner, R., Recent developments in the Trefftz method for finite element andboundary element applications, Advances in Engineering Software, 1995, 24,107-115.

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[8] J. A. Kolodziej. Review of application of boundary collocation methods in me-chanics of continuous media. Solid Mechanics Archives, 12:187–231, 1987.

[9] J. R. Hutchinson. Analysis of plates and shells by boundary collocation. Bound-ary Element Analysis of Plates and Shells, editor: D. F. Beskos, Springer-Verlag, pages 341–368, 1991.

[10] Carlos M. T. T. Fernandes.Utilizacao e Desenvolvimento de uma Formula-cao Indirecta de Trefftz na Analise de Lajes Finas. Thesis, Instituto SuperiorTecnico, Universidade Tecnica de Lisboa, 1999, MSc (in portuguese).

[11] Carlos Fernandes and Vitor Leitao. On a least squares Trefftz collocationmethod for plate bending. Editors: Vladimır Kompis, Milan Zmindak eBranislav Hucko, Numerical Methods in Continuum Mechanics, pages 7–12,High Tatras, Slovak Republic , 6-9 th October, 1998. University of Zilina.

[12] E. D. Eason. A review of least-squares methods for solving partial differentialequations. International journal for numerical methods in engineering, 10:1021–1046, 1976.

[13] B. C. Wu and N. J. Altiero. A boundary integral method applied to plates of ar-bitrary plan form and arbitrary boundary conditions. Computers & Structures,10:703–707, 1979.

[14] B. C. Wu and N. J. Altiero. A new numerical method for the analysis ofanisotropic thin-plate bending problems. Computer methods in applied mechan-ics and engineering, 25:343–353, 1981.

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