Compurers & Srrucrures Vol. 43. No. 1. pp. lfA179. 1992 W45-7949/92 55.00 + 0.00 Printed in Great Britain. 0 1992 Pergamon Press plc

HYBRID-TREFFTZ p-METHOD ELEMENTS FOR ANALYSIS OF FLAT SLABS WITH DROPS

J. JIROUSEK~ and M. N’DIAYE$

tLSC, Swiss Federal Institute of Technology, Lausanne, Switzerland ~Structural Division of Thies Institute of Technology, Senegal

(Received 15 January 1991)

Abstract-A simple method is presented by which various types of internal displacement fields involved in the Hybrid-Trefftz (HT) element formulation for flat slabs with drops can be generated in a unified way. Apart from a standard set of displacement functions for an isotropic plate element, such fields involve, in particular, a special purpose set of functions representing the local solution in the vicinity of a column. The elastic properties of the column and the effect of the sharp radial variation of the plate thickness in the drop area are properly accounted for. The presented functions extend the library of optional internal displacement fields with which an advanced HT element subroutine is typically provided. Its use along with the p-extension of the HT elements enables the local effects in flat slabs with or without drops to be efficiently handled through crude unrefined finite element meshes by using just a single element for each column head as well as a large portion of the surrounding slab. The accuracy and the practical efficiency of the approach are assessed on a series of numerical examples involving a comparison with a three-dimensional analysis.

1. INTRODUCTION

A reluctance to abandon the obvious advantages of the well-established conventional assumed displace- ment elements has often led to a sacrifice of both the accuracy and the economy. In the case of the flat slabs with drops, an accurate solution to the stress concentration generated in the vicinity of the column head is difficult to obtain and even if a plate theory rather than a three-dimensional (3-D) solution is adopted, a tedious local mesh refinement is required to obtain reliable moment concentrations. A further costly refinement may be necessary if a reliable prediction of shear forces is an important issue of the analysis. Such refined discretisation is hardly applicable for a full structure analysis of the slab.

For some time now, it has become obvious that various problems involving stress singularities or stress concentrations can efficiently be handled if use is made of an alternative finite element (FE) formu- lation now known as the Hybrid-Trefftz (HT) model [ 11. Here the intraelement displacement field accurately fulfils the governing differential equations and the interelement continuity is enforced by a stationary principle involving a conforming frame function field. Different aspects of this comparatively recent FE model involving p-method capabilities [2], implementation of a curvilinear geometry [3], error estimation [4], adaptivity [S] and other relevant topics have been studied during the last five years. Though initially [6-81 the HT elements have not been explic- itly designed to handle areas of high stress gradients, it has soon become obvious [9] that such problems can be solved with a surprising efficiency if the

element subroutine is associated with a library of optional special purpose functions representing various local solutions applicable in the vicinity of, for example, a singular corner, a crack, a circular or elliptic hole [lO-131, etc. and involving both the isotropic and the anisotropic materials [14, 151. This attractive possibility has motivated the present tendency to further extend the libraries of such functions and to progressively building up a collec- tion of local solutions covering the current needs of the engineering practice. The present contribution falls into this category and Fig. 1 shows the types of solutions that will be provided.

Throughout this paper it will be assumed that the reader is familiar with the HT element formulation of which a short description has been presented in one of the previous issues of this journal [15] and further relevant details have been given elsewhere [l-3]. This will make it possible to concentrate on the generation of the element (index ‘e’) optional internal displace- ment field. For a HT element based on the classical thin plate theory governed by the Lagrange plate equation

where D = Et3/12(1 - v’) is the plate rigidity, this internal field will conventionally be assumed in the form

m

w, = iJe + 1 QjC, = Ge + Qec, (2) j=l

163

164 J. JIROUSEK and M. N’DIAYE

displacement Z and that with M = 3, for example, the normal slope will be cubic and the transverse displacement quartic along the element side.

For practical implementation of the HT elements it is useful to remember [l] that the necessary (but not sufficient) condition (also known as ‘stability’ condition) for the resulting stiffness matrix to have correct rank is

Fig. 1. Discretization of a plate into HT p-method elements: A, elements with standard set of internal functions. B, C, elements with optional special purpose

functions.

where tie stands for a particular solution to (1)

V4Ge=% on R,,

@, is a suitably truncated part of a complete set of homogeneous solutions

V4@,= 0 on f13 (2b)

and c, are underdetermined coefficients. As in [15], a p-extension of the HT FE model will

be used for all numerical applications. As shown in Fig. 2, such elements can have an optional number of curved sides and the final unknowns-the element DOF-include the conventional DOF (w, w,, wY) at corner nodes 0, along with an optional number M of hierarchic DOF (a,, u2, , a,+,) belonging to the element sides but associated for convenience with the mid-side nodes, A. These DOF are used to define along the element boundary an independent con- forming frame function field including transverse displacement 6, and Cartesian components of rotation 3,,, @Y;e (appropriate frame function defini- tions are given for example in [2, 151). In the process of the element formulation (variational principle [ 11) the undetermined coefficients cj of the internal field w, are expressed in terms of the frame function DOF and since the resulting stiffness matrix is symmetric and positive definite, the element can be implemented without difficulty into the conventional FEM codes. For the specification of the boundary conditions note that, at A, the odd and the even DOF alternatively concern the normal slope e,, and the transverse

3 DOF (w,wx,w,)

M DOF (a,,a, ,..., aM 1 (M optional)

Fig. 2. A HT p-version element.

m > NDOF - NRIG. (3)

Here m is the number of independent homogeneous solutions in (2), NDOF is the number of elements degrees of freedom (frame function in Fig. 2) and NRIG is the number of possible rigid body motion modes of the element. Although (3) is the only necessary condition, correct rank may always be achieved by suitably augmenting m. Generally an optimal value of m (not only warranting correct rank but also optimally compensating the excess of rigid- ity, associated with a fixed number of frame function DOF, by suitably relaxing the conformity) will be used based on numerical experiments.

The following two sections present a unified approach to the generation of the standard and special purpose internal displacement functions (2) for elements of either a constant (Sec. 2) or a radially variable (Sec. 3) thickness. The special purpose func- tions include the plate-column and perforated plate types (elements B and C in Fig. 1). These functions have been implemented in the HT plate bending element subroutine of the program SAFE [16] and all the numerical examples and the assessment studies of Sec. 4, including the comparison with a 3-D analysis of a flat slab with drops, have been obtained with this software. Section 5 brings concluding remarks.

2. DISPLACEMENT FUNCTIONS FOR HT PLATE ELEMENT OF CONSTANT THICKNESS

For the class of problems considered in the present paper a unified approach to generation of internal displacement functions can be based on the general solution of the Lagrange plate equation (1) written in a local polar coordinate system (Fig. 3). Here the non-dimensional radius

/)=;

6 0 F X

r-ap

Y

Fig. 3. Local polar coordinate system.

Hybrid-Trefftz element formulation 165

has been normed by the average distance a between the origin 0 and the corner nodes of the element. For a standard element, 0 is the centre of gravity of the element while for a plate-column element (Fig. 4) or a perforated element, 0 is situated at the column axis or at the hole centre. The main reason for using a normed radius (presenting a value close to 1 at the element boundary) is to avoid an over/underflow. Indeed such problem may otherwise arise due to the presence of large powers of the radius involved in the boundary integrals used to evaluate the element stiffness matrix.

Based on the Lagrange plate equation in polar coordinates [17], the general solution in the p, 9 may conveniently be written as

k = 0,1.2....

+ k =T2.,,, fk(P)Bk sin kg, (4)

where wP stands for the particular solution (V’w, =p/D), which may also be written as

wp = c w;&) COS k9 k = 0.1.2s..

+ 1 w$(p) sin k9, (4a) k = I,Z,...

and where the homogeneous solution involves for each k two vectors of four integration constants

Bk = {B,k 3 Bx 9 & t &k )

a)

(4b)

and a matrix of four fundamental solutions

f,(p)=[l p2 lnp P21np] in k=O

f,(p)=[p p’ l/p ~lnp] ifk=l

f,(p)=[p’ pZfk p-k p2-k] if k > 1.

Based on (4) the expression of the form

w,= fi,,+ f ojcj= 3, + @$, J=I

W

(5)

of the internal displacement field (2) of the element will be generated through the use of suitable sub- sidiary conditions depending on the type of the element.

2.1. Standard HT element

In the case of a standard element where in (4) the constants A,,, Ati and B,,, B4k have to be set to zero to prevent the displacement w, the rotation i3w /ar and the shear force Q, to go to infinity as r = ap -+ 0, the remaining constants may simply be assimilated to the undetermined coefficients cj

(6)

Consequently, the functions G)e and Gj are readily identified as

G’, = wp

[@,+I @j+2I=&QfCOSks,

[@j+3@j+4] = f,Qfsin k9,

(7)

(8)

where

,:=,=[A y i :]’ fork> 1 @a)

b)

Fig. 4. HT plate-column element (a) without and (b) with drops.

166 J. JIROUSEK and M. N’DIAYE

and (since the rigid body motion modes have to be removed [I] from we) the cases k = 0 and k = 1 yield the three following independent functions only

@r=$ @,=gcos9, @,=gsin9. (8b)

Note that the particular term 3, = ids, e.g. for a uniform load p = const is

and for a concentrated load P the singular solution is

P 0 w =-s’lns’. ’ 16nD

where s stands for the distance between the point of application of P and the point considered. For a load uniformly distributed over a given circular area (not necessarily centred at the origin 0) the corresponding 3, may be found in [l]. More elaborate expressions for an arbitrary patch load and line load are also available and have been presented in [18].

For practical implementation of the above func- tions, it is useful to note that the minimum (but not necessarily sufficient) number of c$-functions in w, is defined by the stability condition (3) with NRIG = 3.

2.2. Plate-column element

A typical HT platecolumn element is shown in Fig. 4a. The plate is assumed free of membrane forces (the small horizontal actions exerted by the columns are neglected) and the plate displacements are as- sumed to vanish at middle surface level. The elastic properties of the column are characterized by its

where I,, I, and A are the principal moments of inertia and area of the cross-section. Although the cross-section may be arbitrary, it is assumed that the short plate column intersection has a circular cross- section of 2r, in diameter, where the equivalent radius r0 is equal to

r. = J(+n), (11)

where A, = A if the column section is full and A, is the area of a full section presenting the same external contour if the actual section is hollow.

No attempt will be made to analyse the complex 3-D behaviour at the platecolumn intersection and the cylindrical extension of the column through the plate thickness will be considered as rigid. If at the same time the plate behaviour in the neighbourhood of the intersection is assumed to obey the classical Kirchhoff plate theory, it becomes easy to formulate the kinematic and static conditions involved in representing the plate-column interaction.

To derive from the general solution (4) a set of special purpose functions for a plate-column element, suitable kinematic and static conditions (which represent the platecolumn interaction) have to be accounted for at r = r, = up,. With the notation of Fig. 5, the following conditions will be specified:

k = 0 (simple compression of the column)

dw

( > ar ,=,,,=O

2nro(Q,),=,, = K(wL=~~, (114

k = 1 (column flexure in the X-Z plane)

dW (-> ar r=r0.9=0

=h4r=,,).:9=o r.

~ k = 1 (column flexure in the y-z plane)

dW ( > ar =‘(wL,.s=r.2 r=ro.g=n:2 r.

s 277

r0 0

(-M,sin9-M,,~0~9+Q,r~sin3),=,“d~=K~~(w),=,.,~~-~:~,

k>l flexural and extensional rigidities which for example in the case of a column fixed at its base (as in Fig. 4) are equal to

(w),=,=O

(lib)

(1 lc)

aw ( > ar I=,0

= 0. (114

Hybrid-Trefftz element formulation 167

M,dr

Fig. 5. Polar coordinate components of internal forces in plate under bending.

If the moments M,, M,, and the shear force Q, in these conditions are expressed in terms of w [17]

M,,_D E!+y idw+ia’w [ ( a2 ap2 pap p2aP2 )I ’

A4,8=-.$~~-~,$$~-~w), Q, = -; -$ V’w, (12)

and the general solution (4) is further substituted for w, it becomes possible to eliminate again the con- stants A,k, Alk and Bjk, BJk, or alternatively, express for each k the vectors A,, B, in terms of the linearly independent constants (6)

(1% b)

The expressions of the 4 x 2 matrices Qf, Q,” and of the load dependent vectors QkA, Qka are listed in Appendix I. Finally, the substitution of (13a, b) into (4) makes it easy to present the element internal displacement functions ti’, and @, in the following simple form

*‘, = WP + Ir=J2..,fktiCoSk9 . , .

+ 1 fkqf sin k9 (14a) k= 1.2,.

[~j++1j+2I=fkQ:COSk9,

[C~,+~@~+J=f~Qfsink9. (14b)

Note that the rigid body motion terms in (4), i.e. the terms 1, p cos 9 and p sin 9, should be conserved in w,. Indeed such terms result in non-vanishing column strains and as such are now essential in representing the plate-column interaction. Also the number NRIG in the stability condition (3) has now to be set to zero since the resulting plate-column element has no rigid body motion modes.

CAS 43/l-L

2.3. Perforated element

It is interesting to point out that a similar type of formulation can also be used to generate special purpose functions for a HT element with a circular hole. If the hole boundary r = r,, = up, is free, then the two conditions to be satisfied are, for example

The corresponding matrices Qf , Q,” and qf , qf are listed in Appendix II.

Special purpose functions for other possible combi- nations of the boundary conditions around the circu- lar hole can be obtained in the same manner.

3. EXTENSION TO HT ELEMENTS WITH RADIALLY VARIABLE THICKNESS

The variation of the thickness considerably compli- cates the generation of the internal displacement field since the Lagrange equation (1) has to be replaced by a more general plate equation with variable coefficients [ 171. For the class of application addressed in this paper, an approximate but efficient solution of this problem can be based (Fig. 6) on replacing the continuous variation of the plate thick- ness by a stepwise one. This makes it possible to assume, for any particular interval (p,, p, + , ) present- ing a constant thickness t,, the general solution of the form (4) with integration constant

‘b = (‘A,, 9 ‘A,, 1 ‘A,, > ‘A,, >,

% = {‘BM , $2, iB,3, ‘B,, }, (16)

.- n /

/,- I

’ ’ _- I ,I,/ ,’ ,I j , , / /- ,’ I//,‘/ 6

1 ’ I I ((

;:i-

X 1’ \ \ \I \\\\ r=ap \“\ ,. \‘\ ’

\ \ ‘.

‘.

h Fig. 6. HT plate element with radially variable thickness.

168 J. JIROUSEK and M. N’DIAYE

and to link these constants to the constants ‘-‘A

k, j- lBk of the previous interval (p,_ , , pi) by the following four transition conditions written for

P =P,

lw=‘-Iw

a. a --lw=---~w ap ap

iR, = i - ‘R, + R. (17)

A similar approach has in the past been success- fully applied by Benda [19] and Jirousek [20] for the analysis of circular plates of variable thickness. They have found that: l a relatively low number of constant thickness

intervals yields already a good approximation of the displacements and the internal forces of a plate with a continuous variation of the thickness, l the application of the conditions (17) makes it

possible to derive a simple chain rule by which the integration constants of all intervals are expressed in turn in terms of the independent constants of the interval (pO, pi).

A similar technique can also be applied here and it leads with

cj+I= ‘Ak,, c,+z=‘&

cj+3= Oh,, 5+4= O&z, (18)

for any i > 0, to

lAk=iQ; ‘/+I +I$,

{ 1 c/+2

i~~=tQf 9+3 +1qk8,

i 1

(1% W cj+41

which is a form analogous to the relations (13a, b) of Sec. 2. The chain rule used to evaluate from the known expressions (Sec. 2 and Appendix I) of OQf=Qf, OQf=Qf, ‘d=qkA and Oqf=QkB the expressions of the same quantities for the following intervals i = 1,2, . . . , is of the form

i+is~-*QkA+is,4, iQks=iski-l~+tsf,

(20)

where iSk and ‘si, isf are the transfer matrices listed in Appendix III.

With (19a, b) the internal displacement (2) can finally be defined again in a form similar to relations (14a, b) of Sec. 2

G’, = ‘WP + c fkiq; cos k9 k = 0,1,2,..

+ 1 f,‘$sin k9 (21a) k = I.Z....

[@,+ ,aj+,] = fkiQ; cos k9,

(21b)

The only difference is that the expressions of G’, and @, are now different for each interval i.

Though in the structural applications (Fig. 4b) the plate thickness is seldom varied symmetrically with respect to the assumed middle surface plane x-y, this fact, under thin plate theory, is considered of little importance. As a consequence, the thickness vari- ation is assumed symmetric with respect to x-y plane. In the following section the admissibility of this assumption is assessed by means of a 3-D analysis.

4. NUMERICAL STUDIES AND ASSESSMENT

4.1. Standard HT element

The HT p-method elements have originally been presented [2] based on the biharmonic polynomials. The aim of this subsection is to ascertain that the use of the alternative set (8) of T-functions leads to comparable results and to complete the study of the convergence properties of these elements. Both the h- and the p-refinement methods will be considered. The latter will be characterized by the number M of hierarchic DOF associated with the mid-side nodes. Only the uniform p-refinement with the same M for all mid-side nodes will be used. As shown in [2] only the odd values M = 1,3,5,7, . should be considered since they preserve the desirable uniform- ity in the degree of the polynomial interpolation of the normal and the tangential slope of the frame functions 9, and G,.

An important question which arises during the implementation of the HT element into a standard FE code is the number m of Trefftz functions 3 (j = 1,2, . , m) to be used with a given p-refinement level M. Numerical studies have shown that for quadrilateral HT plate elements with curved sides the minimum m as given by the necessary but not suffi- cient ‘stability’ condition (3) always yields a stiffness matrix with correct rank. This minimal numbers (m = 13,21,29, 37 for M = 1,3, 5, 7) incidentally also preserve the desirable geometric isotropy (invari- ance of element properties with respect to coordinate axes orientation since then for any k9 both the terms involving cos k9 as well as sin k9 are always preserved in the set of @-functions of the element).

Hybrid-Trefftz element formulation 169

al ln Na

22 3 L 5 6 1 8

- & p- convergence la'

Ine

- 10”

Na

hl

-b-

1 5 50

t/t,

Fig. 7. HT h- and p-method (a) convergence rates and (b) study of discretization minimizing computer time.

Figure 7(a) presents a numerical h- and p-conver- gence study of a centrally loaded square plate sup- ported at corners. Only a symmetric plate quadrant has been discretized and the error in energy norm defined as

(22)

(U = exact value of strain energy and U,, = HT element approximation) has been plotted on a log-log scale against the total number N, of active DOF of the FE mesh. The superiority of the p- method over the h-method is immediately obvious if the accuracy is considered in terms of N,. This measure, however, disregards the actual cost of the computation which is adversely affected by the in- creasing cost of evaluation of element matrices and a more densely populated stiffness matrix of the assem- bled elements. Therefore in Fig. 7(b) the error ‘e’ has also been plotted against the computer time? to indicate that as a rule, even in simple cases a 3 x 3 FE mesh will computationally be more efficient that just a single element representing the whole domain.

4.2. HT elements with radially variable thickness

The number of constant thickness subintervals necessary to adequately represent a smooth thickness variation has been studied by considering a symmet- rically loaded circular plate represented by a single HT element with just two curved sides (Fig. 9) used to represent the circular boundary. The lowest p- refinement level M = 0 (no hierarchic DOF) is adopted since both the frame function definition as presented in [ 151 and the internal displacement field

The same results as shown in Fig. 7(a) have also been obtained with the alternative @-function set of [2] (biharmonic polynomials). The small differences observed in the last decimal digits of the results have been due to the rounding-off errors and a close inspection has shown that each of the two alternative

M DOF

t Program SAFE [16] with standard skyline equation Fig. 8. Square plate (different boundary conditions) solver. represented by a single HT p-method element.

sets of functions is actually a linear combination of the other.

It should be pointed out that the results in Fig. 7(a) have been obtained by specifying in the customary FE way the concentrated load P as a nodal load. Though highly accurate, still better results may be achieved if the concentrated loads are represented by the particular load term (9b) rather than as nodal loads. The additional advantage is that the load location is not confined to the mesh nodes. As an example, Tables l-3 show some results of engineering interest (displacements, moments, shear forces) obtained in the extreme case of a single HT element (Fig. 8) covering the whole square plate area (no symmetry condition used) with different boundary conditions.

It is worth noting that the accuracy of HT element may further be improved if an optimal rather than minimal number m of @-functions is associated with each p-refinement level M. The use of unnecessarily large m deteriorates the accuracy: m + 00 restores perfectly the conformity, but the beneficial effect of compensating the excessive rigidity of the frame by a slight lack of conformity between the frame and the internal field is lost. On the other hand, the minimum m (warranting that the stiffness matrix has full rank) may allow too much of non-conformity. Extensive numerical studies have, however, show that the improvement of results due to an optimally tuned m is not sufficiently significant to overweigh the increased computational cost associated with a larger H matrix. Furthermore, the optimal number m has been found mesh dependent and for finer meshes (3 x 3 and more) likely to be used in practice, the difference between the optimum and minimum m tends to disappear.

170 J. JIROUSEK and M. N’DIAYE

Table 1. Study of p-convergence on a simply supported square plate (Fig. 8). M = number of hierarchic mid-side DOF (constrained DOF included). A single HT p-element used over the whole plate

Finite element/theoretical value

Load Point Quantity

C Dw: pa4 M,: pa2

Uniform p B Q,: ia A M, : pa2

M2: pa2 V: pa2

Concent. central P

C Dw: Pa2 B Q,: Pia A M,: P

M*: P v: P

M=l 3 5 7 Theoretical value

1.053 1.002 1.000 1.000 0.00406235 1.025 1.000 1.000 1.000 0.0478864 0.740 1.038 1.019 0.997 0.337657 0.331 0.856 0.937 0.960 0.032483 1 2.313 0.900 0.937 1.004 -0.0324831 1.322 1.078 1.023 0.982 0.06496618

1.025 0.998 1.000 1.000 0.0116008 0.763 1.126 0.980 1.022 0.417307 0.812 1.234 1.002 0.972 0.0609527 1.686 0.551 0.953 1.087 -0.0609527 1.249 0.892 0.977 1.030 0.121905

Table 2. Study ofp-convergence on a clampled square plate (Fig. 8). M = number of hierarchic mid-side DOF (constrained DOF included). A single HT p-element used over the whole plate

Finite element/theoretical value

Load Point Quantity M=l 3 5 7 Theoretical value

C Dw: pa4 1.470 1.036 1.001 1.000 0.00126532 Uniform M,: pa* 1.182 1.017 1.001 1.000 0.022905 1

p B M,y: pa2 0.568 0.888 1.005 1.014 -0.0513338 Q,: pa 0.566 0.962 1.002 1.021 0.441301

C Dw: Pa2 1.194 1.021 1.003 1.000 0.00561202 Concent. central P B M,: P 0.592 0.823 0.929 0.972 -0.1257705

0,: Pla 0.401 0.790 0.839 0.970 0.793827

Table 3. Study of p-convergence on a square plate supported at corners (Fig. 8). M = number of hierarchic mid-side DOF (constrained DOF included). A single HT p-element used over the whole plate

Finite element/theoretical value

Load Point Quantity M=l 3 5 7 Theoretical value

C Dw : pa4 0.988 1.000 1.000 1.000 0.0255065 Uniform M,: pa2 0.988 1.039 1.000 1.000 0.111711

p B Dw : pa’ 0.988 1.004 1.000 0.999 0.0177474 M,: pa2 0.976 1.024 1.002 0.993 0.150439

C Dw: Pa2 0.981 1.000 1.000 1.000 0.0391419 Concent. central P B Dw: Pa* 0.972 1.003 1.000 1.000 0.0229130

My: P 0.925 1.012 0.995 0.994 0.203004

M=O M=O

Fig. 9. Symmetrically loaded circular plate represented by a single HT element. The lowest p-refinement level (M = 0) is sufficient to accurately represent any case of symmetry of

revolution.

are capable of accurately representing any axisym- metric behaviour with the minimum number of DOF. Two cases of radially variable thickness and of boundary conditions have been considered as shown in Fig. 10. The plate with exponentially varying rigidity (Fig. 10a) has been taken from [19] and the converged results for n = 500 subintervals have been used as reference solution. For the plate with linearly variable thickness (Fig. lob) the theoretical solution has been taken from Timoshenko and Woinowsky- Krieger [17]. Tables 4 and 5 show how the accuracy in displacement w and moments M, and MS improves

Hybrid-Trefftz element formulation

1, r=af, _.j

a a a aa n n n 2n

Fig. 10. Two cases of circular plate of variable thickness: (a) exponential variation and (b) linear variation.

with increasing number n of constant thickness subintervals over the element. Though the linear variation of t appears more demanding, even in this case, for n = 50 the error in the largest IU, is inferior to 0.5%. Since the chain rule used to generate the @-fur&ions along the set of subintervals is very fast, a default value of n = 50 has been adopted for all applications with radially variable thickness.

4.3. Applicability of plate theory to the analysis of

slabs on columns

To facilitate the generation of reliable reference results, two cases of centrally supported and

b)

Fig. 11. Uniformly loaded @ = 10 kN/m2) centrally sup- ported circular slab (a) with drop and (b) without drop.

uniformly loaded circular slabs have been con- sidered: slab with a drop (Fig. lla) and without drop (Fig. 1 lb). In both cases, the HT element solution consisted in using a single element with two semicircular sides each with vanishing number of hierarchic DOF (M = 0) at the mid-side nodes A.

Table 4. Circular plate (v = 0) with experimental variation of thickness (Fig. 10a)

D,w: pa4

n p=o.o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10 0.1254 0.1241 0.1204 0.1141 0.1053 0.0939 0.0798 0.0632 0.0440 0.0227 0.0000 20 0.1253 0.1240 0.1203 0.1140 0.1052 0.0938 0.0798 0.063 1 0.0440 0.0227 0.0000 40 0.1252 0.1240 0.1203 0.1140 0.1052 0.0938 0.0797 0.063 1 0.0440 0.0227 0.0000

500 0.1252 0.1240 0.1203 0.1140 0.1052 0.0938 0.0797 0.063 1 0.0439 0.0227 0.0000

M,: paa2

n 0 = 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10 0.2484 0.2451 0.2368 0.223 1 0.2042 0.1804 0.1521 0.1196 0.0832 0.0433 0.0000 20 0.2484 0.2455 0.2371 0.2234 0.2045 0.1807 0.1523 0.1197 0.0833 0.0433 0.0000 40 0.2484 0.2456 0.2372 0.2235 0.2045 0.1807 0.1523 0.1198 0.0833 0.0433 0.0000

500 0.2484 0.2456 0.2373 0.2235 0.2046 0.1807 0.1524 0.1198 0.0833 0.0433 0.0000

MS: paa2

n p=o.o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10 0.2484 0.2454 0.2355 0.2197 0.1992 0.1753 0.1495 0.1231 0.0973 0.0732 0.0513 20 0.2484 0.245 1 0.2352 0.2194 0.1989 0.1750 0.1492 0.1228 0.0971 0.073 1 0.0512 40 0.2484 0.2450 0.235 1 0.2193 0.1988 0.1750 0.1492 0.1228 0.0971 0.0730 0.0512

500 0.2484 0.2450 0.2351 0.2193 0.1988 0.1749 0.1491 0.1228 0.0971 0.0730 0.0512

172

n

10 20 40 50

Theory

J. JIROUSEK and M. N’DIAYE

Table 5. Circular plate (v = l/3) with linearly variable of thickness (Fig. lob)

D,w: pa4

p = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1ozY

0.3785 0.3481 0.298 1 0.2453 0.1930 0.1423 0.0932 0.0458 0.0000 0.3750 0.3459 0.2967 0.2443 0.1923 0.1418 0.0929 0.0456 0.0000 0.3738 0.3452 0.2962 0.2439 0.1921 0.1416 0.0928 0.0456 0.0000 0.3737 0.3451 0.296 1 0.2439 0.1920 0.1416 0.0927 0.0456 0.0000

0.3734 0.3449 0.2960 0.2438 0.1920 0.1415 0.0927 0.0456 0.0000

IV,: pa2

n p = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10

10 0.0735 0.0516 0.0403 0.0344 0.0290 0.0229 0.0157 0.0089 0.0000 20 0.078 I 0.0546 0.0434 0.0357 0.0299 0.0236 0.0172 0.0090 0.0000 40 0.0795 0.0557 0.0440 0.0364 0.0302 0.024 1 0.0173 0.0094 0.0000 50 0.0797 0.0558 0.0440 0.0365 0.0303 0.024 1 0.0173 0.0094 0.0000

Theorv 0.0800 0.0560 0.0442 0.0366 0.0303 0.0242 0.0173 0.0094 0.0000

n

iVg: paa2

n = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10

10 0.0245 0.0466 0.0906 0.1430 0.1746 0.2070 0.2817 0.3612 0.4002 20 0.0261 0.0545 0.0882 0.1291 0.1742 0.2250 0.2791 0.3377 0.3989 40 0.0265 0.0539 0.0882 0.1285 0.1741 0.2243 0.2788 0.3369 0.3984 50 0.0266 0.0525 0.0882 0.1313 0.1740 0.2206 0.2788 0.3413 0.3983

Theory 0.0267 0.0539 0.0882 0.1284 0.1740 0.2243 0.2787 0.3368 0.3982

(a) Slab with drop (Fig. 11~). Since the thick- ness of the slab in the region of the slab is important and the mid-surface is not plane (the slab is eccen- tric) some of the basic assumptions of the classical thin plate theory have been violated. To assess the practical consequence of this situation, the HT single element solution has been compared with results of a 3-D analysis performed by using a fine mesh of 328 axisymmetric isoparametric quadratic elements leading to a total of 2430DOF. To analyse the effect of eccentricity, two 3-D analyses have been performed wherein the geometry

cm1

t? M,( kNm/ml

of the drop is either taken as shown in Fig. 11(a) or with the same thickness variation applied sym- metrically with respect to the middle plane of the slab. The results of the two 3-D analyses are displayed in Fig. 12 along with the HT element solution (a single element with 50 constant thickness subintervals in the drop region). The inspection of the results shows that the engineering accuracy of the simple HT element solution is quite satisfactory.

(b) Slab without drop (Fig. llb). A single HT element now yields the exact Kirchhoff theory

ml

- 3-O with actual exrentrlc drop geometry

-Q- 34~1th drop symmetnc wth

respect to the rnlddk plane of the slab

--v-- HTP-extension With rpeclal purpose

T- functom

0 ml

Fig. 12. Comparison of results for circular slab with drop of Fig. Ila.

__+L__ Slab wth drop-HT soluhon _-Q__ Slab ulthout drop-HTsolutmn

- Slab without drop-Plate theory

udh tranvene shear defnrmatwn

Fig. 13. Influence of drop on distribution of (a) displace- ments and (b) moments. Comparison of cases of Fig. 1 I(a)

and (b).

solution of annular plate with the following boundary conditions

aw QA intemalboundaryr=r,,...,--0, w=2- & Era

external boundary r = a, . . . , w = M, = 0, (23)

where r. and 1 are respectively the radius of cross- section and the length of the column. As a matter of

Fig. 15. Example of HT element discretization of rectangular flat slab with drops.

interest some results of the cases in Fig. 1 l(a) and (b) have been compared in Fig. 13 to highlight the influence of the drops on the slab behaviour.

To ascertain the applicability of the classical Kirchhoff plate theory which disregards the trans- verse shear deformations, the case of Fig. 1 l(b) with boundary conditions (22) has also been solved follow- ing the Mindlin plate theory. The comparison of the results in Fig. 14 shows that the Kirchhoff theory if sufficiently accurate in such applications for engineering practice.

The Mindlin plate theory raises the interesting question of modelling slabs on columns in FE analy- sis based on the popular isoparametric Co plate elements. Indeed, in sharp contrast with the classical Kirchhoff plate theory, a concentrated load now produces infinite displacements and as a consequence,

Hybrid-Trefftz element formulation 173

)

twtmm)

-6

-L

-2

0

- plate thecq wth transverse shear deform&on

_.w_ Mesh of 156 Conventional lsoparametw quadratIc M~ddlm plate elements.

Column zore represented as slab on

elarhr found&on

I _-_ S,ngle HT 2 rIded element ~11th

spew1 purpose T- functiom

Fig. 14. Comparison of different solutions for centrally loaded circular slab of Fig. 11(b).

174 J. JIROUSEK and M. N’DIAYE

1 - -

----

-----. ‘\ --y, ‘\,,

Hybrid-Trefftz element formulation

M,/M, (kN m/m)

175

- - - Poshve I I I 0 IO 20

--- Negative kN m/m

Fig. 17. Principal moments in the slab with drops of Fig. IS. Uniform load p = 10 kN/m2.

the use of point supports is, in principle, illicit. Whereas in a crude mesh of Co Reissner-Mindlin elements a point support leads to results close to those of the Kirchhoff theory (where a point support is licit), the solution becomes unpredictable if the mesh is strongly refined in the vicinity of the support. To avoid this situation, a possible alternative consists in simulating the columns by using over the area corresponding to the column cross-section one or more Co plate elements on elastic (Winkler) founda- tion. (This amounts to augmenting the customary plate element stiffness matrix by the following area integral: C, jA N%‘dx dy, where N is the matrix of customary Co shape functions and C,,. is the Winkler coefficient.) As a matter of interest, Fig. 14 shows the results of such a calculation (156 isoparametric quadratic Co plate elements over a symmetric plate quadrant) which however exibits now a marked difference with respect to the earlier exact Mindlin theory calculation following the boundary conditions (22). This difference is due to the fact that the Winkler foundation leads to somewhat unrealistic deforma- tion of the area simulating the cross-section of the column. The remedy is to suitably increase the Young modulus of the plate elements covering this subregion. However, in contrast with the HT approach outlined in this paper, this technique cannot correctly rep- resent the effect of the flexural rigidity of the column.

4.4. Practical application

A symmetric quadrant of a rectangular slab with drops has been solved by using a crude 5 x 7 mesh of

HT plate elements (Fig. 15). The four elements including a drop and a column use the optional special purpose Trefftz functions following the Sets 2.2 and 2.3 with 50 constant thickness subintervals in the zone of the drop. The remaining HT elements use the standard set of Trefftz functions (Sec. 2.1). Some results for a uniformly distributed load are displayed in Figs 16 and 17.

5. CONCLUDING REMARKS

The concept of optional special purpose Trefftz functions has been used along with p-extension of the Hybrid-Trefftz (HT) plate elements to provide a highly efficient approach to the analysis of flat slabs with drops. The slab is discretized by a crude mesh of HT p-method element where each column with the adjacent part of the slab including the drop are represented by a single element. Suitable set of Trefftz functions accounts for the plate-column interaction and accurately represents the local effect in the vicin- ity of the column head. As a consequence accurate moment concentrations are obtained without mesh refinement, as a part of the full structure analysis of the slab.

Examples assessing the practical efficiency of the advocated approach have been provided. Apart from considerable economy in computer and data preparation costs, the excellent accuracy and high p-convergence rate (generally associated with HT model) are valuable assets. The possibility of a simple adaptive reliability assurance based on a uniform

176 J. JIROUSEK and M. N’DIAYE

p-refinement and specially devised for use with 9. J. Jirousek, A cqntribution to finite element and associ-

the standard FE codes, has been presented ated techniques for the analysis of problems with stress

elsewhere [4, 51. singularities. Proc. 2nd Int. Conf. Num. Methods Engng

The implementation of the presented HT elements (GAMNI), Vol. 2, pp. 719-729. Dunod, Paris (1980).

10. J. Jirousek and P. Teodorescu. Laree finite element is simDle since a single HT element subroutine Dro- method for the solution of problems-in the theory of

1 1

vided with a library of optional Trefftz functions elasticity. Compuf. Sfruct. lj, 575-587 (1982). _

covers the whole problem. As shown in the paper, a 11. R. Piltner, Spezielle finite Elemente mit Liichern,

variety of cases including a standard plate element Ecken und Rissen under Verwendung von analytischen Teilliisungen. Fortschritt-Berichte der VDI-

plate-column elements (with or without drops) Zeitschriften, Reihe 1, Nr. 96. VDI, Diisseldorf (1982). and an element with a circular hole. have been 12. J. Jirousek, Implementation of local effects into conven-

obtained in a unified manner from a complete set of biharmonic functions in polar coordinates.

tional and non-conventional finite element formu- lations. In Local Eficts in the Analysis of Structures (Edited by P. LadevBze), pp. 279-298. Elsevier (1985).

Acknowledgement-M. N’Diaye was supported by the Swiss 13. i. Piltner, Special finite elements with holes and Confederation. internal cracks. Inr. J. Numer. Meth. Enpnp 21.

1.

2.

3.

4.

5.

6.

7.

8.

“Y .

1471-1485 (1985). REFERENCES 14. T. G. Gerhardt, A hybrid/finite element approach for

stress analysis of notched anisotropic materials. Trans. J. Jirousek and Lan Guex. The Hvbrid-Trefftz finite ASME 51. 804-810 (19841. element model and its application to plate bending. Inf.

J. Jirousek, Hybrid-Trefftz plate bending elements with p-method capabilities. Int. J. Numer. Meth. Engng 24,

J. Numer. Meth. Engng 23, 651-693 (1986).

1367-1393 (1987). J. Jirousek and A. Venkatesh, Implementation of curvi- linear geometry into p-version HT plate elements. Inf. J. Numer. Meth. Engng u), 431-443 (1989). J. Jirousek and A. Venkatesh, A simple stress error estimator for Hybrid-Trefftz p-version elements. Int. J. Numer. Meth. Engng 28, 21 l-236 (1989). J. Jirousek and A. Venkatesh, Adaptivity in Hybrid- Trefftz finite element formulation. In!. J. Numer. Meth. Engng 29, 391-405 (1990). J. Jirousek and N. Leon, A powerful finite element for plate bending. Comp. Meth. Appt. Mech. Engng 12, 77-96 (1977).

klement model. -Cornput. Strucf. 34, 51-62 (1990). 16. J. Jirousek, Manuel d’utilisateur du programme

15. J. Jirousek and M. N’Diaye, Solution of orthotropic

SAFE-RCdaction provisoire,

plates based on p-extension of the Hybrid-Trefftz finite

LSC-DGC, Swiss Federal Institute of Technology, Lausanne (1990).

17. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd Edn. McGraw-Hill, New York (1969).

18. A. Venkatesh and J. Jirousek, Accurate FE analysis of thin plates under concentrated loads. IREM Internal Report 89/6, Swiss Federal Institute of Technology, Lausanne (1989).

19. J. Benda, Circular plates of variable thickness (in Czech). Bulletin of the Brno Institute of Constructional Engineering and Architecture, Czechoslovakia, Nos 39 and 47 (1954).

J. Jirousek, Basis for development of large finite el- 20. J. Jirousek, Solution of arbitrarily loaded circular plates ements locally satisfying all field equations. Comp. of variable thickness (in Czech). Bulletin of the Brno Meth. Appl. Mech. Engng 14, 65-92 (1978). Institute of Constructional Engineering and Architecture, A. P. Zielinski and 0. C. Zienkiewicz, Generalized finite Czechoslovakia, No. 106 (1956). element analysis with T-complete boundary solution 21. A. K. Noor and I. Babuska, Quality control and functions. Inr. J. Numer. Meth. Engng 21, 509-528 assessment of finite element solutions. Finite Element (1985). Analysis and Design 3, l-26 (1987).

k=O

APPENDIX I

AUXILIARY MATRICES FOR GENERATION OF INTERNAL DISPLACEMENT FIELD OF HT

PLATE-COLUMN ELEMENT

c 0

0 c

16~~0 P: -$(I +2lnp,) ~--

a2DKz D2

I

D’P, $(l-2lnp,)

0

0 1 q,” =!

c -g(l+2lnp,)-+&$(l-2lnp,)-g(l +4lnp,) I : ,

PO x+%(1-4lnp,)+*

z - :

177

where

2p, In *p,, 8n

c=Dz-- a2DK, PO

J? = uniform load, P,, = xa2p~p + P, P = concentrated load (Fig. 4a).

Hybrid-Trefi element formulation

k=l

1 Q: or Qf=-

c

where

C 0

0 C

K -Kpi(l -2lnp,)-8nDp2

2 ->K -4K

PO

e=i[K(l +2lnp,)-8nD]

K =K, or K, for Tf or Tf

q!=qf=[O 0 0 oy

k>l

_kp’#- 1) -(k + 1)~~ ;

where

APPENDIX H

AUXILIARY MATRICES FOR GENERATION OF INTERNAL DISPLACEMENT FIELD OF HT ELEMENT

WlTH CIRCULAR HOLE (RADIUS r,=crp,)

k=O

Q,“=

I !

1 0

0 1

0 2+?$

0 0

0

0

qt = t a’pg

1 f&p+v +4(* +v)lnp,]

-1 1

178 J. JIROUSEK and M. N’DIAYE

k=l

q: =qf= [O 0 0 O]T.

k>l

Q;=Qf=&

k -2v 0

0 k - 2v

(k - l)(k --2h$ ;(2+k)(k’-2-2v)p;~k+”

I

-k(k*-2+2v)p;“-” y&k + 1)(2+k)‘d’ J

qf=l$=[O 0 0 O]?

APPENDIX III

TRANSFER MATRICES ‘S, AND ‘s,

k=O

D -pf(D - 1)(2lnp,- l)(l +v)/2 (D - l)(21npi- 1)(1 -v)/4 -pf(D - I)[1 - v +4(1 + v) ln2p,]/4

‘S, = 0 [D(l -v) + 1 + v]/2 (D - 1)(1 - v)l(4d) (D - I)(1 - v)(l + 2 In p,)/4

0 p;(D - l)(l + v) [D(l + v) + 1 - v)]/2 pf(D - 1)(1 + v)(l + 2 In p,)/2

0 0 0 1

k=l

‘S, =

D pf(D - 1)(3 + v)/2 (D - 1)(1 - vY(2d) (D - I)(1 +v +4lnp,)/4

0 [D(l -v) + 3 + v]/4 -(D - 1)(1 - v)l(4d) (D - 1)(1 -v)/(W)

0 -pf(D - I)(3 + v)/4 [D(3+v)+l-v)]/4 -p;(D - 1)(3 + v)/8

0 0 0 0

k>l

[D(3+v)+l-v]/4 pf(D-l)(l+v) p;%(D - I)(1 -v) pf” -“(D - 1)[2(1 + v)

x (k + 1)/W) x (k + I)/4 + k2(l - v)]/(4k)

0 [D(l - v) + 3 + v]/4 _p,-Y’+k)(D - 1) -p;%(D - l)(l - v)(k - I)/4

is,= x(1 -v)k/4

-p”(D - 1)(1 -v) --P;“+~‘(D - 1)[2(1 + v) [D(3 + v) + I - v]/4 x (k - 1)/4 +(l - v)k2/(4k)

p:(D - l)(l + v)(k - 1)/(2k)

p;u’ +(D _ 1) p”(D - 1)(1 - v)(k + 1)/4 0 x (1 - v)k/4

[D(l -v) + 3 + v]/4

Hybrid-Trefftz element formulation

with

'D D=i-‘D

-‘x (1 + 2 1x1 p,)Ap -z 1-W + ap,(l + In p,)RI

‘st= [~;;~;~l-21npi)K+api(lnpi- l)Rl 1 ,

179

where