A macroelement for waffle slab analysis

Cnmputn

& Srrww~s

Vol

IS. No. 2. pp. II:-122.

1982

001.c7w91821Mo117M103.0010 0 1982 Pergamon Press Ltd.

PnntedinGnalBritain

A MACROELEMENT

FOR WAFFLE SLAB ANALYSIS

WORSAKKANOK-NUKULCHAltand AHMADKAMAL GILANI~ Division of Structural Engineering and Construction, Asian Institute of Technology. Bangkok, Thailand (Receioed 4 December 1980;received for publication 2 Ju1.r 1981)

Ahstraet-Behaviour of a waffle slab under transverse loading is investigated through a series of three-dimensional finite element analyses, and modal analysis of a deformed waffle slab cell. “Macroelement” concept is introduced, taking an advantage of the repetitive nature of waffle slab cells and the fact that only few deformation modes participate significantly in sustaining transverse loads. Less significant deformation modes are removed with little consequence by appropriate kinematic constraints. The result is a macroelement cell with fewer degrees-offreedom. The macroelement is shown to retain ail the dominant deformation modes of a waffle slab cell. Results from tested example show good performance of the proposed macroelement with considerable amounts of computer .

time and storagesaved.

INTRODUCTION

A present trend in the development of finite element method is the search for specialized elements which represent a particular behaviour of structural component. Many finite element approaches [ l-31 have been proposed for the transverse load analysis of a wafie slab. Although quite accurate. direct beam/plate element assemblage often requires large computer storage, which makes the analysis too expensive. Several general purpose programs thus provide certain options for reducing the size of system degrees-of-freedom. For example, the master-slave option allows the user to eliminate near dependent degrees-of-freedom through assumed “rigidlink” constraints in the system. However, implicit constraints other than the rigid link cannot be handled in general for a possible further reduction. In most cases, validity of the solution depends on a proper selection of the master and slave degrees-of-freedom, which must be comprehended correctly by the user. Alternative approach for waffle slab analysis is to model the slab with orthotropic plate elements. An element of considerable accuracy, based on the equivalent stiffness of a “smeared” orthotropic plate[4], has proved to be economical. It fails, however, to give an accurate stress distribution at the level of individual plate and beams. No method of waffle slab analysis, to the authors’ knowledge, has yet been presented which yields accurate stresses at the component level and is economical in terms of computer time and storage. The present study strives therefore to rectify this situation. In the first part, we examine behaviour of a waffle slab under uniform transverse load. Deformation modes analysis is conducted for a typical wafhe slab cell and the conclusions drawn from this part serve later as a basis for deriving the waffle slab cell“macroelement”. The second part begins with concept of macroelement, by which a macroelement for representing waffle slab cell is developed. This involves reducing the threedimensional problem of waffle slab analysis to a twodimensional one, by imposing certain kinematic constraints to eliminate unimportant deformation modes.

The much smaller number of generalized degrees-offreedom saves considerable amount of solution time and computer storage. BEHAVlOUR OF WAFFLESLAB

Stiffness contribution of individual wage slab components In general, the response of a waffle slab to transverse loads is a result of the mutual interplay of three distinct structural components. (1) beam grillage, (2) plate bending and (3) plate membrane. To understand the integrated action of these structural components, we analyse the waffle slab shown in Fig. 1 with a three-dimensional finite element model under a uniform transverse load of 10lb/in’. To study the effect of individual components, the analyisis is divided into three parts. (a) Analysis of beam grillage alone. (b) Plane stress element is added at the level of plate. This analysis gives an insight into the beam-membrane interaction and the mechanism by which additional stiffness is imparted to the structure. (c) Finally, plate bending stiffness is superimposed on the plane stress element for a fully compatible threedimensional model of wafhe slab. The analysis of this

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/l’:Ol5 tAssistant Professor. SGraduate Student and Research Associate.

r3” thick place

,-x1

8

8

36

36

6

6

6

8

Fig. 1. A waffle slab problem; n, and a2 are numbers of cells in x, and x2 directions. 117

118

W. KANOK-NUKLJLCHAI and A. K. GILANI

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MACROELEMENT FOR WAFFLE SLAB

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Boom grill*

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Fig. 2. Deflection response of waffle slab under uniform transverse load due to (a) beam grillage (b) griilage t plane membrane and (c) complete waffle slab.

model should yield a complete solution for the problem. Deflection curves along the slab centre by the above three analyses are shown in Fig. 2. Considerable increase in the overall structural stiffness is obtained with the addition of plate membrane to the beam grillage. This is attributed to the great in-plane stiffness of the plate that restrains the top fibres of each beam from moving freely in the horizontal plane and thus limits the amount of bending and twisting of the beam grillage. On this account, the horizontal movements of the beam grillage at its top nodes are infinitesimal and can be disregarded in the interest of economy. In case (c), the inclusion of plate bending stiffness only provides local coupling between the plate and beam degree-of-freedom. Increase in the overall structural stiffness due to this effect is rather nominal (Fig. 2). Deformation modes of wage slab cell A waffle slab can be divided into identical cells by imaginary lines passing through the middle vertical planes of beams, whereby each cell will consist of a plate surrounded by beams on the four sides. Deformation of a typical waffle slab cell can generally be represented by the superposition of independent modes of deformation. However, under transverse loading (which is the normal form of loading for watiie slabs), the participation of many deformation modes is unimportant. In an attempt to isolate the dominant modes of the deformation, modal analysis is conducted for a typical waffle slab cell modelled with one bilinear degenerated shell (BDS) elementf51 for plate and four modified BDS elements for the beams. Using a transformation[6] X=QY

where K= g”K4 and R= 4’R. Solution of eqn (2) gives the participation factors y. each of which defines the relative significance of an independent deformation mode of the system. Neglecting the three rigid body modes, the mode shaped corresponding to the five highest values of y are shown in Fig. 3. These correspond to the M,, MY,M,, bending modes and the in-plane extension and the twisting modes’. However, the in-plane extension and the twisting modes will be restrained by the higb in-plane rigidity of the waffle slab. Hence, we can conclude that the behaviour of a wafRe slab under transverse load is predominantly governed by the combination of the three bending modes,

Representation of waffle slab as consisting of identical cells prompts the idea of applying an extension of the finite element method, in which each cell is considered as a “macroelement”. If the behaviour of each wattle slab cell under transverse loading can be accurately represented by a macroelement, the need to work with each segment of beams and plate is diminished. The structure is thus reduced to a number of macroelements, each of which represents the combined behaviour of all the components within the macroelement. In a conventional finite element analysis, each cell of waffle slab is divided into a number of plate and beam elements. Let the degree-of-freedom of a typical plate element “j” in a cell be vP’j)and those of a beam element “k” be ~b(~‘.The corresponding load vectors are VP(i) and Vr,lk).respectively. Now consider the whole cell as an element and let r be its chosen master degrees-offreedom and R the corresponding load vector. If a system of virtual displacement 6r is imposed on the macroelement, the virtual work associated with the master degrees of freedom is WE = 6rrR;

y = 2.690x165

Y = 1.909

x

16.

(3)

Y - 3.635

x 16’

Y = 1.203r,$

(I)

in the stiffness equation Kx = R, where x is the vector of waffle slab cell degrees-of-freedom, 4 is an orthogonal transformation matrix containing eigenvectors of the system (K- AI)4 = 0, and R is the load vector corresponding to a uniform transvers load, leads to Ity=ii.

(2)

Fig. 3. Five most significant deformation ment.

modes of a macroele-

A macroclement

whereas the virtual work associated with the degrees-offreedom of individualplate and beam elements is N

w, =

St

j-1

(4)

k-1

119

for waflle slab analysis

where N, and Nb are respectively the number of plate and beam elements in the macroelement. If Kp”’ and Kb(*)are the stiffness matrices of plate element “j” and beam element “k” respectively, eqn (4) can be expressed as

degrees-of-freedomcan thus be reduced to a minimum, i.e. those just sufficientfor reproducingthe three necessary modes. Such kinematicassumptionsare the following: (1) Plate component of wattle slab is rigid in plane; thus, beams are fully restrained horizontally at their top edges. (2) Beams are subject to the usual kinematicassumptions of deep beam theory, i.e. beam cross sections remain plane under bendingand twisting. (3) Corresponding top and bottom edges of beams undergo the same transverse displacement. Configuration of macroeiement

Certain kinematic constraints are introduced so that the degrees-of-freedom of individual plate and beam elements are dependent on the master degrees-offreedom. i.e. yPu’= a,“+

(6a)

and y*(k)= ab (k)r,

(6b)

where aPu’and abgk’are transformation matrices. Equation (5) can then be written in terms of r as W, = Srr (K1,t Kb)r.

0)

where

and Pb(k)Kb

(kl

ab

(k)

.

(7c)

Equating the virtual works by master and slave degrees-of-freedom,and eliminatingSr7 from both sides lead to (K: + Ka)r = R.

Figure 4(c) shows a basic macroelementdeveloped for wafRe slab ceil. It consists of one plate and four surroundingbeam elements. Given the large numberof ceils normally constitutinga waffleslab, such a crude mesh in the macroelement is considered su6cient for achieving desired accuracy. Four master nodes are assignedat the plate comers of the macroelement. Due to the above kinematicconstraints, only three degrees-of-freedomare found to be necessary. They are the transverse displacement u3 and two rotations, 8, and & about two orthogonal axes in the plane of the plate. Horizontal displacement degrees-of-freedom are not included on account of the Kinematic Assumption 1. Beam kinematics are restrained to bending and twisting about horizontal axes lying in the plane of the plate, and are hence governed by 8, and 6,. The Hughes’ bilinear quadrilateral eiement(7J is chosen to model the plate. This element, based on the Mindiin plate theory, contains four nodes, each having three degrees-of-freedom(Fig. 4a). To model each beam, the same Hughes’ element superposed by a modified plane-stress stiffness is used. Thus, the final beam element contains five degrees-of-freedom per node: three displacements and two rotations (Fii. 4b). The plane stress stiffness is added to govern the two in-plane displacementdegrees-of-freedom.The shear part of this stiffness is computed with reduced integration to avoid shear-lockingphenomenon when the element is used for thin beams. In this case, a 2-by-1Gaussianquadrature (2 along the beam depth and 1 along its axis) is

(8)

+ FULLY RfSTRAlNEC f CONDENSED

The effectiveness of a macroelementthus depends on: (1) a proper choice of r which ,couid uniquely and suficientiy represent the overall macroelement strain energy associated with the important modes of deformation, and (2) validity of the kinematic constraints imposed on the macroelement by the transformation equations, viz. eqns (6a) and (6b). To a certain extent, the accuracy of macroelementalso depends on the number of plate and beam elements used.

indicates that only three bending modes play significant role in its response to transverse loads. With this consideration, kinematic field of a waffle slab cell may be simplifiedto allow exclusively the three dominantmodes in addition to the rigid body ones. In other words, specific kinematic constraints can be introduced to eiiminate all minor modes. The number of generalized

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Kinematic assumptions for deriving a macroelement The mode superposition study of a waffle slab ceil

for

A Waffle

(b ) Bcom

Slob

Element “k”

Call

slab macroelement with its constituent plate and beam elements.

120

W.KANOK-NUKULCHAI and A. K. GLANI

recommended[5] for the evaluation of in-plane shear energy.

eqn (9) into eqn (74 gives the total contributionof beams to the macroelementstiffness.

Sti&ess formulation of macroelement The plate contribution to the macroelement stiffness

Deformation

NUMERJCAL EXAMPLES

modes of the macroelement

This problem serves to examinedeformation modes of can be obtained in a straightforwardmanner as only one plate element is employed. The transformationmatrix ap the proposed wahle slab macroelement. The modal in eqn (6a) is simply an identity matrix. Thus, the con- analysis previously used for the three-dimensionalfinite tribution from the plate to the macroelementstiffness is element model of waffleslab cell is followed. Deformation modes are displayed in Fig. 5. As expecdirectly the stiffness of the plate. For beams, special considerationshould be given first ted, the macroelement, despite its fewer degrees-offreedom, produces all the three complete bendingmodes to the use of appropriate width in the stiffness computation. As macroelements are assembled, two beam of the wafle slab cell. elements merge along the joining edge of the two neighbouring macroelements,and jointly represent a segment Simply supported wafle slab under unifromly distrubuted of the actual rib of waffle slab. Stiffness contribution load The simply supported waffleslab of Fig. 1 is analysed from each should therefore be half of the expected final usingthe proposed macroelement.Each wattleslab cell is stiffness. The plane stress stiffness is a linear function of the represented by a single macroelement.Since all the cells width: thus half of the actual width is used in its com- are identical, the stiffness matrix of a typical macroeleputation.The out-of-planestiffness,on the other hand, is ment needs to be formed once and can be used a function of t’ where t is the actual width of the rib. repeatedly for all identical macroelements. Boundary Effective width of the beam panel is therefore taken as conditions along the edges in x and )’ directions are respectively u3= 8, = 0 and u3= 0, = 0. t/‘d2 in the out-of-planestiffness computation. Deflection curve at the plate centre-line, obtained by Procedure to obtain the beam contribution to macrothe macroelementanalysis, is compared with the threeelement stiffness is explainedas follows (a) For each beam element, stiffness is formed with dimensional analysis and the analytical solution by Hovichitr et al.[8] in Fig. 6. Stress resultant curves of respect to the five degrees-of-freedomat each node. (b) The element stiffness matrix is modifiedso that all plate and beam grid are presented in Figs. 7 and 8 horizontal displacementdegrees-of-freedomof the beam at the top nodes are zero in accordance with Kinematic Assumption1. (c) The transformation matrix ab which relates the beam degrees-of-freedom and the master degrees-offreedom is derived subject to Kinematic Assumption2. The beam degrees-of-freedom which have no direct couplingwith the master degrees-of-freedom,i.e. in Fig. 4(b), ~9,and 0, at all nodes, are released by static condensation before the transformation.Displacementut at bottom nodes are also condensed to comply with Kinematic Assumption3. This implies that discontinuity of these released beam degrees-of-freedomover the intermacroelement boundary is allowed. The situation is found to have little consequence however. The coupling between beam degrees-of-freedom and the master degrees-of-freedomis described by eqn (6b), n or explicitly,

, _____-&fy j I

.is =4900rIOb

&4 for the beam element “k” shown in Fig. 4(b), where d is the depth ofbeam measuredfrom plate mid-surface. -is = I 412XIOS .1,,= I 440r10s (d) Similar transformation matrices are obtained for the other _three beams after condensing the stiffness Fig. 5. Deformation moses of three dimensional finite element -. matrtces of each individualbeam element. Substituting model of a watTlIeslab cell. Duplicate modes are omitted.

A macroelement

for waffieslab analysis

121

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0.25

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Hovichitr at al Simple 3-D arolyrir

1

Refined

A Simple mocroelement s Refined 3-D molyris

Fii.

6.

I ’

20

nwcroelrmnt

_

Comparison of slab deflections between macroelement and finite element models.

*

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‘-. “C

k

respectively. The result indicated that the overall performance of this macroelement, despite its coarseness and small number of degrees-of-freedom, is very satis-

factory. A more accurate macroelement, based on the similar approach, can be developed by dividinga wafee slab cell into nine plate and twelve beam elements. The master nodes are therefore increased from four to twelve. Transformationmatricesfor the plate and beam elements are similar to those employed for the coarse mesh. However, internal degrees-of-freedomof plate elements are released by static condensation before assembling into the global stiffness matrix. This macroelement can yield a distributionof stress resultants within each waffle slab cell, which is not possible with the simple macroelement. Deflection and stress resultant curves of the same problem obtained by this refined macroelement are presented in Figs. 68. Three-dimensionalsolution obtained with a finer mesh of nine plate and twelve beam elements for each cell, using BDS element, is also included. Comparisonwith the simple macroelement solutions shows that the refined macroelementyields higher accuracy and a more realistic picture of stress distribution in each cell. However, the great computational effort and core storage required for the refined macroelement solution may not justify its use. The simple macroelementis therefore recommendedfor the general analysis of wame slabs.

,&O/

0.25

x,/b

C

- -----

= 0.0625

--

0.25

Fig.7.

0.5

Stress resultants in plate component

of the waf&slab.

0.25

05

x2/b

x2/b

-.. -----.-

CONCLUSIONS

macroelementconcept proposed in this paper can be summarizedby the following: (1) For structures consisting of a great number of identical substructures, the conventional finite element analysis can be very uneconomical. This enhances the need to represent individual substructure by a macroelement. (2) Normally, a structure is designed to sustain a particular pattern of load. Deformation of a typical substructure under this load is a linear combination of all possible, orthogonal modes of deformation. Usually, only a few of these deformation modes participate significantly,whereas participationof many other modes is negligible. (3) A set of appropriate kinematic assumptions is then introduced in an attempt to remove all the minor

S~mfh 3- D analysts Slnpk mocrohmant Rofimd 3-D onolyur Rhmd mocroolwnw

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The

025

0.5 x2/b

Fig.8. Stress resultants in ribs of the waffleslab.

modes. The result is a simple macroelement which contains much fewer generalizeddegrees-of-freedom. Macroelement of a waffte slab is shown to perform exceptionally well in the analysis of wafee slab, with a phenomenalsavingin computertimeand storage.Resultof

122

W. KANOK-NUKUL-CHAI and A. K. GUANI

stresses can be determined accurately at the component level, despite the much reduced degrees-of-freedom. This macroelement should therefore serve as an efficient basis for the transverse load analysis of waffle slabs.

1. R. P. McBeam, Finite element analysis of stiffened plates. Ph.D. Dissertation.Stanford University, Stanford, California (1968). 2. M. M&rain. Finite elementanalysis of skew compositegirder bridges, SE.94 Rep. No. 67-28. University of California, Berkeley,California(1%7). 3. P. G. Beraan. R. W. Clouah and S. Moitahedi.Analvsis of

stiffened plates using the finite element method, Rep. No. KSESM 70-I. University of California, Berkeley. California (1970). 4. V. Ravindran. Finite element formulation for waffle slabs treated as orthotropic plates. Master of EngineeringThesis. Asian Institute of Technology,Bangkok(1978). 5. W. Kanok-Nukulchai,A simpleand efficientfinite elementfor generalshell analysis. Jnt. /. Numer.Met/t.Engng 14, 17%200 (1979). 6. K-J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall,EngfewoodCl&, New Jersey (1976). 7. T. J. R. Hughes, R. L. Taylor and W. Kanok-Nukulchai,A simple and klcient finite element for plate bending. fnt. 1. Numer.Enenn 11. 1529-1543(1977). 8. I. Hovichi~,-kn analysis of waf& slabs. Ph.D. Dissertation No 17 A&n ln.h~tr of Trchnnlncw RanolmLr11977\