0045.75749/89 13.00 + 0.00 0 1989 Papmon Pms plc

A NEW DISCRETE KIRCHHOFF PLATE/SHELL ELEMENT WITH UPDATED PROCEDURES

M. FAFARD~, G. DHA*$ and J. L. BATOZ$

Wcpartment of Civil Engineering, Lava1 University, Quebec, Canada GlK 7P4 SDivision M N M, UniversitC de Technologie de Compiegne. 60206 Compibgne, France

(Receiued 25 Febnurry 1988)

Abstract-We present a new six-node triangular plate/shell element (called DLTP) for geometrical and material nonlinear analysis. The formulation is based on small clasto-plastic strains and updated Lagrangian formulation (ULF). Two updated procedures arc compared in the elastic and plastic range. The DLTP element is obtained by superposition of a discrete Kirchhoff model (DKTP) for bending and a linear strain triangular element for membrane (LST). Several numerical examples are presented employing special solution strategies for tracing the pm- and post-buckling curves.

INTRODUClION

A large number of structural systems in civil, mech- anical and aerospace engineering are composed of thin plate/shell components. A reliable and econom- ical design of such structures requires a detailed evaluation of pre- and post-buckling configurations which may exist in the range of normal and extreme load conditions. The finite element method is a powerful analytical tool to study complex behaviour of thin structures under large displacements and rotations. It is possible also to predict correctly the ultimate load capacity by including plastic yield criteria in the constitutive relations.

The success of a finite element procedure for nonlinear analysis of thin structures is related to a proper choice of:

(1) a plate/shell element to represent coupled membrane-bending effects;

(2) a reference configuration to represent the deformation and stress states (updated configur- ation);

(3) a technique or criteria to represent the progres- sive plastification through the thickness;

(4) a strategy for solving highly nonlinear relations with multiple solutions.

We employ a flat triangular plate;shell element with a discrete Kirchhoff formulation to represent the bending behaviour. The simplest element in this family is a three-node triangle (DCT) composed of the DKT (discrete Kirchhoff triangle) element for bending and the CST (constant strain triangle) for membrane (I-61. This element gives good results for linear and nonlinear analysis if the contribution of in-plane displacement to membrane stresses is rela- tively small as compared with that of transverse displacement. As soon as one component of in-plane

displacement becomes important (i.e. a sort of ori- ented structure behaviour in the plane), the DCT element gives very poor results, as shown in (51, for certain nonlinear situations. This has been observed also in linear buckling analysis using a four-node quadrilateral element for wide I-beams [7j.

In this study, we employ the DLTP element (a six-node triangle) composed of DKTP (DKT plus [8]) for bending and the classical LST (linear strain triangle-six nodes) for the membrane. This element has been successfully employed for studying various linear and geometrically nonlinear elastic structures [S-lo].

The updated Lagrangian formulation is best adapted to study oriented structures undergoing large displacements and rotations. Due to the hypothesis of oriented structures (plane stress), the choice of different approximations for in-plane and transverse displacements and the particular nature of strain- displacement relations, the total Lagrangian finite element formulation fails to capture correct solutions. In this study, we employ two variants of updated Lagrangian formulation (ULF). The procedure ULFl corresponds to a total Lagrangian formulation within each step size with the configuration at the beginning of the step taken as reference. The proce- dure ULF2 is a fully updated formulation, the esti- mated configuration at each iteration is taken as reference. We find the application of TLF, ULFl and ULF2 using DCT element in [5,6].

The material nonlinearity is defined by von Mises type yield criteria with a hardening rule [ 11). A special technique to consider the progressive plastification of thickness under combined membrane-bending stresses is developed. The Newton-Raphson method coupled with the arc length [12] type step incre- ment is employed to obtain pre- and post-buckling configurations.

591

592 M. FAFARD cr al.

VARIATIONAL MODEL

For an oriented structure, the variational represen- tation of equilibrium relations in the current con- figuration C(r) using a local orthogonal coordinate system is written as (Fig. 1)

w= trW:l b,l) d v - We,, = 0 (1)

for all admissible test functions (virtual displace- ments) (u*), where [a,] represents components of the Cauchy stress tensor (a, = 0 along the thickness n), [OF] is the virtual deformation gradient tensor, (u *) = (u* L’* H’*) is the virtual displacement field, and W,,, represents the virtual work due to applied forces.

The choice of oriented coordinate system permits us to choose finite element approximations of in- plane and bending variables in a simple manner. Moreover, the hypothesis related to thin structures such as plane stress and negligible transverse shear strains are introduced explicitly.

For flat element formulation, the local coordinates become the Cartesian coordinates (x,~, z) over each element. leading to

P:l= PC* Y.-- C)

(2)

From an algorithmic point of view, we indicate the known equilibrium configuration at the beginning of a step by C’ and the estimation ‘of desired configuration C(r) by C2. The variational problem in an incremental form becomes

Wr = WI,: + A WI,: = 0. (3)

where WI,: defines the contribution of residual forces in the estimated configuration C’ and AU’/,: is the improvement to obtain an equilibrium for the step C(r) - c:.

r Fig I, Definition of the Cauch) stress approximation for a

thin plate.

Updated Lagrangian formulation ULFI

The expression W for the C’ configuration is

w:= I

tr([DXe?l) dV2 - W,,,,, (4) P

where W: corresponds to W for the configuration C2 (superscript) using C’ geometrical space descrip- tion (subscript). In ULFl, Wi is written in the C’ description:

W= W;+AW,=O

where [Sfi is the PK-2 stress tensor, [E:] is the virtual Green-Lagrange deformation tensor, and

El = detVlVl-‘[e7VTr. (6)

with the deformation gradient [fl between C2 and C’ defined by

= VI + [A 1 (7)

u = u:, 1’ = r:, H‘ = )v;

W:l = [FlT[D*l[Fl = f W+l?FJ + VlqF*l), 03)

with [F*] = [A*] and u = u* in the matrix [A] (a common coordinate system is used to define these quantities). For an oriented structure, we employ local coordinates in C’ and C2 configurations to evaluate various tensor components. In the following sections, all quantities are in local coordinates unless specified otherwise.

One may show from (6) that, for small strains,

,J??c, = ,&?c?* (9)

since

det [F] z 1

and by polar decomposition [13]

PI = [RI[U = VI

PiI = [RIT[~‘l[Rl = ,J$lc~ (10)

where [R] is the rotation matrix between local coor- dinates of C’ and C:. The constitutive relations are

Discrete Kirchhoff plate/shell element with updated procedures 593

written in the incremental form using Updoted formulation VLF2

[sn = [c’]+ [S:-‘I.

For elastic behaviour

(11) In this formulation, qn. (3) is written in the C’

description. However, for update of stresses, [ofi is obtained from [Sa using qns (9). (11) and (15) [5,6, 141. We thus have

with

{S:-‘) = [HI{+‘} (12) W= W;+(AW),=O, (18)

where W: is given by (4) and

and {S:-I}, {E:-‘} are vector organization of com- ponents with [H] representing the elasticity matrix.

For plastic material, the increment [Sf-‘1 is cor- rected according to flow rules to respect the yield criteria.

AW,= 5

ttiW:lbfl + VVIP~I) do2 - w,,,. ol

(19)

Particular aspects for thin shells

The expression AW in qn (3) defines the tangent stiffness matrix for the Newton type solution method. In a general form, it is obtained by performing the first variation of W for displacement components:

In local coordinates of flat elements, thin plate/shell theory assumes linear variation of dis- placements and strains along the thickness ( - r/2 Q z G r/2, where f is the thickness). Assuming moderate rotations and representing u: by u for simplicity,

(AU’), = I

tr([AL:l K8l) dr ’ 1.1

+ 5

tr([E:][AS$)dc’ -AW,,,, (13) 1.1

with

with

AE: = i (F’I’WI + Wl~Wv (14)

where AF = AA [u in eqn (7) is replaced by Au]. In the case of elasto-plastic behaviour, we use the

following relation to evaluate the incremental stresses:

IS:-‘} = [H,,l{Ei-‘J, (15)

where [H,] represents a first-order approximation of plastic yield-criteria and [ET-‘] is a function of the displacement u: at the iteration i. The incremental stress [ASa is obtained by eqn (11):

[ASa = [AS;-‘]. (16)

since [Au’] = 0. From eqns (15) and (16) we obtain

where

{AS:} = W,l(AE:-‘1, (17)

[AE:-I] = $([AF]T[F] + [F]T[AF]).

One may remark that ULFl is the total Lagran- gian formulation if the configuration at the beginning of each step is taken as reference or initial configuration.

where (u. v) characterize the in-plane behaviour and H‘, j’?x, & (transverse displacement and rotations) rep- resent the bending.

(20)

594 hi. FATARD et al.

The membrane stresses and moments are defined as usual. Appendix A gives a summary of the case of plate/shell relations.

FINITE ELEMENT MODEL

The variational relations are discretized by finite element approximation to give

IV= x w’. (21) ckmna

where IP are integration points and matrices [E] and [E,] depend on the finite element approximation and the matrix organization. [Q represents the appropriate organized form of constitutive proper- ties. [u] correspond to stresses defined in a desired configuration.

In this study, we employ the DLTP element with six nodes having (u, L’, H.) a1 mid-side nodes and (u, L’, W, jX, /I,) at comer nodes [g-lo] (Fig. 2). The approximations are given by

The general form of eqn (3) in discretized form becomes u = (NXuJ

or

Wrl{AuI = (RI.

The standard form of residual vector and the tangent matrix using numerical integration is

{TJ =;[WI

M = c [W[Fi Ml. (231 IP

o = (N){c,} (24)

B* = W*XH’J

SY = W,>{H’J, (25)

with (H’,) = (w,, 8,,, 0,., i w2 i w3, Or,, 6,,, . . .>. The functions (N, . NJ, representing classical

quadratic approximations [ 11, 15,161, and (If,), (H,) are given in Appendix B.

Z,W t Global coordirmte system

8, for osaembly

M Local coordinate system

(membrone,LST)

haI (plate bending,DKTP),

I

+ I

WI 8x1

2: 8 Y3

8 23

lk8,3 [kl

II 1, (rotational) (flat ahell,DLTP) 8,

J

FIN. 2. Construction of a DLTP flat shell element

Discrete Kirchhoff plate/shell element with updated procedures 595

For linear problems, it is not necessary to define w explicitly over the element. However, for nonlinear problems, one needs the approximation for w, and H*,~ over the element. Numerical experience has shown that the interpolation choice for HI is very important for obtaining an efficient nonlinear element. This is primarily due to membrane locking introduced by choice of (u, u, w).

For the DCT element, the linear interpolation instead of a cubic one for H’ seems to give better results. For the DLTP element, the Green-Lagrange tensor is defined with quadratic variation for u and u, and linear for w, for calculating nonlinear compo- nents. The geometrical stiffness matrix in the tangent matrix is obtained with a linear interpolation of 7~. L’, H’) since quadratic interpolation gave stiff re- sults. The finite element expressions for calculating {r) and [k,] are given in Appendix C.

In order to avoid singularity in the assembled matrix using flat elements in a global coordinate system, the local element matrices are augmented by a fictitious local variable 8, at each comer node with the corresponding rigidity matrix [I 11: I -+ -f

Ktl=rEAf

[ 1 1 -f . (26)

1

where E is the elasticity modulus, A the element area and t the thickness. z is taken as 10e4 in this study.

PLASTICITY MODEL

The von Mises yield criteria with normality flow rule are employed to represent the material plasticity:

f([U’]. 6, K) = v’i(i tr[O’]‘) - 6(K) = 0, (27)

where [a’] is the deviatoric stress, a’ (x) is the uniaxial yield stress, and K is the hardening parameter.

For subsequent yielding, the isotropic hardening rule is employed. Thus plasticity matrix [H,] using first-order plastic consistence is used for calculating

W,l:

(28)

with

a=z+ g {V}, ( > where {H,} is the elastic matrix and 3; is the slope of the uni-axial stress-strain curve.

A layered approach is employed to consider the plastification of thickness. The element thickness at

Fig. 3. Layers integration technique.

an integration point is defined by a number of equally spaced nodes; a layer being defined by two nodes with a linear stress variation. The stresses at each node are calculated according to the plasticity criteria [eqn (2711 for plane stress state. The resultant in-plane forces and moments are (Fig. 3)

{Iv) _f (@}I +p.+1+ i (uJi) i-2

[Hz] = f ( [Hepl’ +Fln + I + i [&Ii) i-2

+ {UL - {ul,+, 6 >

+ W,l,+, ; + V&l, $

where (n + 1) is the number of nodes.

NONLINEAR SOLUTION STRATEGY

The algebraic relations representing the behavi.our of oriented structures are highly nonlinear. The load-displacement curves had multi-solution configurations with pre-buckling and post-buckling zones representing stable and unstable situations. The existence of limit and bifurcation points further com- plicates the solution methods. In order to analyse the nonlinear structures, one needs to employ various strategies in a solution procedure to include:

5% M. FAFARD CI al.

(1) a choice for update of tangent matrix; (2) a choice of solution technique for each itera-

tion such as triangularization or conjugate gradient with pretonditioning;

(3) a choice of control techniques such as load increment, displacement increment or arc length method;

(4) criteria for detection of critical points; (5) a proper estimation of solution directions via

eigenvectors.

A description of these aspects may be found in (171, along with the expert system program organization.

In this study, we are basically using the Newton-Raphson method with matrix triangular- ization at each iteration. We also employ the load, dominant displacement or arc length control strategy to capture pre- and post-buckling zones. It is possible to trace curves with limit points using these control strategies; however, the bifurcation points require special attention for predicting solution paths. In this study, we make use of eigen- vector and eigenvalue to analyse the bifurcation zones. The calculation of lowest eigenvector and eigenvalue is obtained using an inverse iteration scheme [ 151 which is available in almost all computer codes.

We employ three possibilities for bifurcation zones which require the calculation of lowest eigenvalue and eigenvector. In the first method, the bifurcation path is sought by employing an extra concentrated load P, corresponding to the predomin- ant value of the eigenvector. The value of P, is adjusted relative to total load value (0.1-l%). In another method, the geometry of structure is per- turbed by the normalized eigenvector, requiring a new calculation of residual vector and the tangent matrix. Finally. one may choose the normalized eigenvector as an improvement of known solution to detect bifurcation paths. In ULF, the last two methods are identical since the geometry update is implicit.

NCMERKAL EXAMPLES

Various examples have been studied to assess the performance of DLTP element for geometrical and material nonlinearities. It is of interest to evaluate the reliability of ULFI and ULFZ. Furthermore, tests are performed to assess the most suitable w-approxima- tion which avoids the buckling behaviour in geo- metrical nonlinear formulations.

For comparison purposes, two elements, DCT (DKT bending + constant membrane strains) and DLTP (DKTP bending + linear membrane strains) are employed in this study. Due to different variants of these elements, we introduce the following element classification:

Element Formu- lation

DLTPI-L DLTPI-Q DLTPl-L-Q DLTPZ-L DLTPZ-Q DCTI-L Dcrl-c

ULFI ULFl ULFI ULF2 ULF2 ULFl ULFl

w-Approximation (same for Y, u) for

GrUrl- Geometrical Lwaw matrix

Linear Linear Quadratic Quadratic Linear Quadratic Linear Linear Quadratic Quadratic Linear Linear Cubic cubic (14, u hear)

The choice of w-approximation for nonlinear terms does not influence the linear formulation of DCT and DLTP elements.

Elastic cantilever plate

A cantilever plate under a tip concentrated load (Fig. 4) forms the benchmark test to assess the influence of w-approximation for DCT and DLTP elements. This example exhibits the interesting phen- omenon of membrane locking due to a high trans- verse displacement contribution to in-plane stresses. Moreover, an analytical solution based on beam theory is available for comparison [ 181. Results are presented in Tables 1 and 2 showing the relative convergence properties of various elements. Typical load displacement results are shown in Fig. 5.

The numerical experimentation permits us to make the following remarks:

(1) DCTI-L gives better results than DCTI-C, which had severe locking;

(2) DLTPl-Q and DLTPl-L-Q give almost identi- cal results of good accuracy;

Lgz+ t so.1 E = 1.2. IO‘ II = 0.3

‘t

l--j7 ”

16 elements DCT or DLTP

Fig. 4. Cantilever plate.

Discrete Kirchhoff plate/shell &ment with updated proc&m

Table I. Rciativc error (%) of finite element results for c~ntitevtr place for the load (PL’/W) = IO

Element 5 steps 10 steps 40 steps 100 steps

w IL U/L w/L 4L V/L U/L W/L UlL

DLTPI-L -0.4 0.7 -0.1 - 1.3 -0.1 -0.9 0.0 DLTPI-Q -7.8 - 18.0 -3.1 -7.6 -0.2 -1.3

::

DLTPI-L-Q -0.2 1.2 -0.1 -0.8 0.2 -0.8 0:s -0.2

0.1 DLTP2-Q I) * -0.1 -1.1 0.7 -0.4 I.1 0.7 DCTI-L

-4G 0.1 0.7 -0.9 0.3 -1.2 -1.2

DCTI-c -79.8 - 30.0 -60.2 -2.3 -7.6 -8:: -2.6

* No convergence after 200 iterations at the first load step. Theoretical mults[18]: at (PL2/N)= 10, w/L =0.811, u/L =O.W.

597

WA and UiL displvrn?WS

Fig. 5. Transverse and axial displacements at free end of a fixed-end plate.

(3) DLTPZ-Q also gives good results; (4) convergence is faster if the geometrical matrix

is evaluated with linear (u, v’, H’) approximations for DLTP;

(5) linear approximations are recommended for DCT and DLTP.

Etasric thin cylindrical shett

A quarter of a thin cylindrica1 shell under a central load with curved sides simply supported is studied (Fig. 6) using DCT and DLTP elements. The solution is obtained using the displacement increment method. The central displa~ment w,/t is increased from 0 to H’JI = 140 in 70 equal steps. We can observe various limit points along with large displacements and rotations for the zone of study.

Table 2. Average number of iterations per step for cantilever plate

Element 5 steps IO steps 40 steps 100 steps

DLTPl-L 8.6 7.3 4.1 DLTPI-Q 11 6.7 4.2 3:; DLTPI-L-Q 10.2 8.1 4.4 3.2 DLTPZ-Q t 13.6 5.6 3.5 DCTi-L 5.7 4.4 4.1 DCTI-c 14.5 7.5 5.3

l No convergence after 200 iterations at the first load step.

In Fig. 7, we give typical results for DCT and DLTP elements using different mesh sizes and step increments. Table 3 gives some statistical results. The following remarks may be made regarding the be- haviour of these elements.

(1) Results for DCTl-L and DCTI-C differ starting with H.,/I 2 80. However, if the displacement increments are smaller for DCTI-C (large number of steps: MO), the results are similar since locking due to cubic approximation is reduced.

(2) DLTPl-L and DLTPZ-L are quite similar in behaviour. There is a convergence problem for DLTP2-Q, but DLTPI-Q seems to give good preci- sion and convergence.

(3) From an efficiency point of view, DCT-L per- forms better than DLTP-L since nonlinear terms in the memb~ne stresses are mainly due to transverse displacements for this problem.

I-Beam buckling

The buckling of a tide I-beam under axial load with one fixed end is studied using DCT and DLTP elements (Fig. 8).

This example exhibits in-plane shear locking and di~cujties in initializing the buckfing mode. In [S], DCT2-L and DQT2 (DKT element with quadratic

598 M. FAFAIU et al.

240 l lemontl OK1 OI OLTP

Fig. 6. Thin cylindri~l shell.

Table 3. Statistical results for the cvlindrkal shell

Element Av. no. Relative CPU No. of

iterations by step time DOF

DLTPI-L 2.9 5.5 1883 DLTPI-Q 3.3 6.6 1883 DLTPZ-L 3.1 5.6 1883 DLTPZ-Q 3:2 1883 DCTI-L I.0 746 DCTI-C 3.3 1.1 746 DCTI-Lt 3.1 8.9 2932 DCTI-C: 2.9 1.8 746

* No convergence after 20 iterations at step 46. t 480 elements. :I40 equal incremental displacement.

Fig. 8. X-&am under axial load.

membrane strain) elements have beem employed to study the pre- and post-buckling zones.

The post-buckling path may be initialized by load-perturbation [5] or by initial displacement tech- nique (using lowest eigenvector). Pre- and post-buck- ling paths are traced using an arc length control technique.

Results are shown in Fig. 9, which indicate the stiff character of the DC? element with non-convergence in the post-buckling range (severe shear locking).

For the load perturbation, we apply Pp = P/IO00 for the first 10 steps. The solution starts with an initial load of P = 0.3PE (P,= R2EIj4L2) followed by arc length control in the following steps. For the displacement perturbation, we impose an initial displacement profile corresponding to the Euler eigenvector with an amplitude of L./WOO t 0.3. The solution starts with an initial toad of P = Pt., folIo~ng the arc length control to trace the complete path.

3.0

+ DlJ’Pl Q 70 stmps 240 rlomrnts 0 DLTPP-L 70 mops 240 elemsntr A DCTld. 70tiopr 240r*mrmr 0 0CTI-C ?Ost*pr 24Or*ment~ s 0CTl-C 140 step8 240 rkmrnls

Fig. 7. Transverse displacement at the center of the cylindrtcal shell

Dherete Kidhoff plate/shell ekment with updated pm&ure~

1 DLTP1.LpMuhtbn lo66 A DLTPl-Lpmtihlbn diaplaamrnl _

3 - 0 Tebot(S)DDlY a Tabol(5) DC?2

I .

$2 Bifwatiin point . I l

OO 0.2 0.4 0.6 0.6 VA

Fig. 9. Lateral displacement at free ad of I-beam axially loaded.

Plate with plastifcation

0

A square plate under uniform load with simply supported edges is studied using linear plate theory with von Mises type plastic yield criteria (Fig. 10). We study one-quarter of the plate with different layers along the thickness to represent the plasticity. The results are compared with those obtained by Dinis and Owen [19] using global plasticity criteria with isoparametric element and by Horrigmoe and Eidshein (201 with a hybrid triangle using global

f l 6.69 1. to’ MPo criteria. Kikuchi and Ando (211 used 20 layers along v 8 0.3 thickness for plasticity. Results are presented in

Mp- 400 N-m/m

t 9 11.2Smm Fig. 11 using DLTP with 28 layers for comparison.

ur - 12.6 MPa Our study indicated that seven layers are sufficient

to obtain precise limit load and plasticity representa-

Fig. 10. Simply supported square plate under uniform load. tion along the thickness (Fig. 12).

Fig. I I. Loadcentral deflection response of a simply supported square plate.

600

Fig. 12. Load/central deflection response of a simply supported square plate, layers integration technique.

Biaxial bending of beam-column

We study a beam-column undergoing biaxial bending with geometrical and material non-lineari- ties. This example (Fig. 13) is of interest since exper- imental results [22] are available for comparison. Moreover, Akoussah [14] has studied the same prob- km with a torsional-bending beam element in three- dimensional space.

+Pinod vt

We employ DLTPZ-L and DLTPl-L elements with eight layers over the flange plate. Results are shown in Figs. 14 and 15 for load-rotation of mid-point about the x-axis of the mid-height point and for load-displacements (0, w) at the same point.

The ultimate load obtained is identical to the cxper- imental load. The results compare quite well with those obtained by using a beam element [14]. It is shown that von Mises yield criteria derived from the uniaxial tests are valid for predicting ultimate load.

- Pined

Rosidualr stnss 44 [kico]

View A-A ;~101oooMPa

E,a5l75MPO

12 l lqmmnWgroup

16 groups on the length of the beam-column

Fig. 13. Geometrical and mechanical characteristics of the beam-column.

Lwretc Kirchltofr plate/sheh element with updated procedures

4w I I I I I

o Akoussah (14) . DLTF&L r*IMnt 0 DLlPt-L &mmt

601

Fig. 14. Rotation about x-axis at mid-height of the beam-column in biaxial bending.

CONCLUSION

The recently developed DLTP element has per- formed well for studying problems with geometrical and material nonlinea~ties.

It is obvious that the DCT element is very efficient if the in-plane strain nonlinearity is mainly due to transverse displacements. However, if the contribu- tion of a membrane displacement component be- comes important (structure reiatively oriented in the plane due to particular stress state and loading patterns) DLTP has superior convergence and preci- sion properties.

It has been shown that the updated Lagrangian formulation is least adapted to oriented structures due to the particular nature of in-plane and trans- verse displacement approximations. For the prob- lems studied, ULFI and ULF2 seem to be quite similar. The advantage of ULF2 may be more evident

if there are large dispia~ents and important though moderate rotations within each iteration.

Another important aspect of this study has been to assess the role of w-approximation for evaluating geometrical stiffness matrix and Green-Lagrange ten- sor components (for w,, wy). The membrane locking effect due to nonlinear terms is very severe for DCT-C. It seems that the best choice is DCT-L and DLTP-L. However, DLTP-Q may be accepted as well.

We have developed a layer representation of plas- ticity along the thickness. The special technique of explicit integration for each layer is very well adapted for calculating correctly the bending terms, which are cubic functions of thickness. For a composite ma- terial, one may easily vary thickness to represent correctly different materials.

The technique of load perturbation and initial displacement represented by eigenvector profile

Fig. 15. Transverse displacements at mid-height of the beam-column in biaxial bending.

602 M. FAFARD l I al,

performed well in the limit point and bifurcation point zones. The displacement and arc length control technique, coupled with Newton-type solution method, is capable of tracing most complex load4isplacement paths for oriented structures.

Finally, we may recommend that the flat DLTP-L using ULF and layer representation of plasticity provides an excellent tool for analysing complex plate/shell structures. Of course, DCT-L is to be preferred for problems with in-plane nonlinearity dominated by transverse displacements only (w,)‘,

(Q’.

Acknowledgemenr-We thank the National Sciences and Engineering Research Council of Canada and Le Fonds FCAR of Quebec Ministry of Education for financial support. Esenam Akoussah, a graduate student at Lava1 University Civil Engineering Department, has contributed indirectly through his advice and experience lo this project.

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APPENDIX A

The updated Lagrangian formulation adapted to plate elements is defined in terms of in-plane forces {NJ and moments (M}:

I2 IW = {a} dz; z{o} dz - I>‘2

(N) = Wx.. N,, NJ; CM > = CM,, M,, MS,.)

(AlI

(a) may represent Cauchy or PK.2 stresses. From eqns (AI), (15) and (20), we can write

iN1 = Wli4 + WGblb)

{M) = Wi”#4 + [HS,lb j3 642)

where

r’2 V-f;1 = I

’ 2 [H,] dz; [Hz] =

s 2 V&l dz

-12 -1’2

” [HR] =

s z’[H,] d;.

-12

(a) Variational model expression for ULF 1

w= s

(<e:,>IN:I + (z*){M:j)dA - W,,,,. (A3) AI

(N:> and (Mi) are defined by PK-2 stresses {Si).

Discrete Kirchhoff plate/shCll &nEtlt with updated proc&rcs 603

(ef) - ( au* au+ au* au’ axdy,z+ax

I I >

r au* au au+ do au-* aw --+--+-- ax, ax, ax, ax, ax, ax, 1 au+ au + ar+ au + a0 aw -- -- -_ ah ah ah b, ah ah

bell = au* au

i I at+ do + ad aw -- -- -- ax, ay, + ax, a~, ax, ah

+au* au +av* au +aw*aw -- -- -- ah ax, ay, ax, ay, ax,

(A4) wiih {h’;}, {M:}, d&cd by {a2}. The tangent stiffness matrix is obtained from

AW- 5

(<e?XANj + <P)<AW AI

+ <c:X~{AcJ) dA’. (AM

(A% All quantities are defined in C2 description with oriented coordinates (x2, y2). One simply replaces

hi,) by (x~,Y~) in eqm WXAW. In finite element implementation, residual vector

{r } and tangent stiffness matrix [k J are calculated by replacing directly A’ and A ’ and {N:}, {M:} by

(A6) (IV:}, {M:} (ULFl case). This is acceptable as long as the strains remain small. However, the components of {N} are supposed to be defined in oriented coordi-

u =u:;v = ~1:; w = wf in the local coordinate in C’. For the tangent matrix

Aw= (<e:,)IANj + WXAW +

<t ;>[~{A+ dA ‘3 (~47)

where

be,,,, is similar to (A4) and (AS); u*, c*, W* being replaced by Au, Ar, AH.. Ax is obtained from (A6) by replacing B:. 8: by A&. AD,.

[q corresponding to geometrical stiffness is

with N,, = (N&X, etc.

H, = 64a,,(Nk - Nk, J

H, = 64e,(N, - Nk+ J.

The indices i, j, k I, m and p for 12 components of (AIO) <I-J,) and (H,) are presented in Table Bl.

The (N) functions given in [16] are

(b) Variational expression for ULF2 r

N,=f[a(-1+3a)(-2+3a)-9P]

w= J (<eWNil+ <z*){W)dA2 - W2,,,, N, = f [2ab( - 1 + 3a) + 3P]

& (Al 1) N,==$[2ab(-1+36)+3P],

nates of C’ configuration. Thus the global transfor- mation matrices for ULFl and ULF2 are not identical since the oriented coordinates in C’ and C’ differ.

APPENDIX B

Refer to Fig. 2 for element definition. The interpo- lation functions on the reference element are:

at each comer (node 1, 2 or 5):

Hx, = -a,+68N, + 4Nk+ ,) + a,,(4N, + 68N,+ J

I-IX2 = e,(26N, + ION,,,) + C,,(ION, + 26NP+,)

H x3 = NI- I - b,$‘k - 4?L, - 4,Np - b,,N,+,

HJI = -e,,(68N, + 4Nk,,) + e,,(4N, + 68N,+ ,)

42=-N,-,+~,N,+g,N~+,+g,,N,+f,,N,+,

tA9) H,, = -c,,{26N, + ION,,,) - c,,(lON,,+ 26N,+,)

at each mid-node (2, 4 or 6):

604 M. FAFARD cr al.

Table Bl

Node label W,); W,.) bl-;

0 > (Fig. 2) components i,j;k l,WP

Hr, H.tz Hr, Comer: 1 1,3;2 5.1; 8

H, I H,l H,., H..

Mid: 2 .-

1,3;2 - H!4

H,, H, Hr, cy = ’ xiPv

Comer: 3 3,5; 5 1,3; 2 27 1;

H,, H,6 H,., H,,

Mid: 4 3,5;5 - H)+ d,=;

H, H,,, 4, II (

-;y:+;x:

Comer: 5 5, 1;8 3,5; 5 H,, H, I o H, I I

H 112 Mid: 6 5,1;8 -

H 112

The [Bb] matrix can be written

where

P=i.,q; ;.=I-t-q. hW,.$ +_hWx.J

PI = hW,.d +hW,.J A proper choice of coefficient a and b permits us to define nine functions which are given in Table B2. I

j~lWx.;) +_kWx.,) +j,,(H,.:) +MH,.rl) 1 The coefficient a,,. b,,, etc., are (Fig. Bl):

APPENDIX C

Table B2

The Green-Lagrange strain tensor can be written, for the ULF2 between C’ and C’ configuration,

Sij ’ sin 8ij s Xij /Pij

Cij ’ COS8,j “y,j/Pij

FIN. Bl. Real and reference elements

Discrete Kirchlloff plate/shell eknetn with updrttd pK4a!dwa 605

where

{UJ = Ml.1

Pbl =

0 0 0 N, 0

0 0 0 0 N2.s

0 0 0 N2# N2=

w,N,, . . - . .

w,NIJ . . . . .

+

0 . . .

0 . . .

0 . . . I

0 0 Hx,.x Hx2.x H.r>, 0 0 0 Hx4-x . * *

0 0 H,,, H.,.2J H,,,. 0 0 0 H,d,. * . *

0 0 HA,. HA,, HA, 0 0 0 Hx4.t . * *

+ + + +

H &1.X H,2s H,.L. H,, . . .

(u,) = (u, t, w, ex, e,, e,, u2 t2 w2 * . .).

We obtain the incremental in-plane forces and incremental moments from eqn. (At) for the two formulations.

The [E$j matrix will be different if we use a linear. quadratic or cubic interpolation to evaluate the non- linear terms.

The residual vector for ULFl corresponding to eqn (A3) can be written

The residual (C2), where

In the same

+ s

([Bb]T{Mi}) dA’. (C2) A’

vector for ULF2 is also given by eqn

[B;,] E 0. (C3)

manner, the tangent stiffness matrix can be written for ULFl, using eqns (A7), (A8), and

(Cl),

[Krl = (PVI + kC’,lYIH~lW’l + N’,l) dA ’

(C4)

For the ULFZ, the [B:,] matrix is not included in eqn (C4).

A transformation or rotation matrix is used to express the local variables in the global system before assemblage.

In both ULFl and ULF2 we c6nsider the [RI] matrix (direction cosines of local axes in Ci) to compute the local displacements (u,):, but, for the assemblage, we use the [RI] matrix for ULFl and the [R’J matrix (direction cosines of local axes in C’, an estimated configuration of C’) for ULFZ:

{RjJow = [R’]{rjb, for ULFl.

IRj&i,.l= WIIrLl for ULF2

606 M. FAFARD CI al.

We resume the ULFI and ULF2 algorithmic schema as follows:

C’ known: global coordinates x’

r

local stress [a’] at each integration point

New step

{Vi}, = 0 initialize the global displace- ments to zero

Iterative loop (II)

{R)” = 0; [KJ” = 0 initialize global residual vector and global tangent matrix

Element loop (fe)

Calculate the rotation matrix [R] from configuration C’

Calculate the local coordinate and dis- placement

{XI), = [clip,%; id), = [RI

{r)” = 0; [k,]” = 0

Integration loop (IP)

Calculate the incremental and total local stress

is;-‘) eqns (A2) and (Cl)

{Si) = {a’} + {Si-‘) [eqn (II)]

Evaluate local residual vector and tan- gent stiffness matrix

using for ULFl: using for ULFZ:

W: [eqn (WI W: {eqns (C2) with

NJ= 01

AI+’ hn (C3)l AW {qns (C3) with

Wil = 01 Calculate a new rota- tion matrix [R] with configuration [C’]

{r}b = {r}‘< + (r}Ip

[kJ” = [k,]” + [kJ”

Calculate the global residual vector and global tangent stiffness matrix

{r}: = [RIT{r}”

kl: = Wkl”~Rl

Assemblage of the residual vector and tangent matrix

{RJ = iRJ + Ir)f

WA = WTI + [klf -

Evaluate the new incremental displace- ment

[Kr]{AU}“= {R}

{ U;}‘d = { C’;}$ + {AL’}:

Convergence test

Update the stress and coordinates

\ , ‘CT’\ = i’“‘) + {ST-‘}

{X’} = {X’} + {UT}.