A99924663 AIAA-99-1275

AN ELEMENT STRESS RECOVERY TECHNIQUE WITH EQUILIBRIUM CONSTRAINT IN LAMINATED COMPOSITE

PLATES

Maenghyo Cho * and Soon Chang Yeon t

Department of Aerospace Engineering, Itiha University, 253 Yong-Hyun Dong Nam-Ku, Inchon 402-751, Republic of Korea

Abstract A new stress recovery technique in laminated com-

posite plates including transverse stresses is proposed to recover boundary stress accurately. In the present method, in order to obtain improved bending moment and shear resultant fields, equilibrium constraints are imposed using two residuals of equilibrium in varia- tional form. Independent stress resultant field is as- sumed for all stress resultant. components within an element. This element stress recovery technique with equilibrium constraint resulted in accurate boundary resultant stress extraction regardless of the values of penalty parameters. In the laminated composite plate model, this stress recovery technique has been ap- plied and the transverse stresses are recovered by using three dimensional stress equilibrium equations. For the accurate prediction of in-plane and out-of-plane stresses through the thickness in thick multilayered plate, a postprocess method has been applied. Deriva- tives of stresses can be accurately estimated in the element level. Numerical examples demonstrated that the accurate boundary stresses can be recovered by this method. The present method should work as an efficient tool for the localized-boundary stress extrac- tion.

I. Introduction Although displacement-based finite element method

is a conventional tool for structural analysis, stresses are only accurate at some special points such as Gauss integration points and so-called superconver- gent points. Furthermore, the discontinuity of stresses across the interfaces between~elements makes an ef- ficient stress recovery not easy. Recently, many re- searchers try to develop more accurate stress recov- ery techniques since the patch recovery technique(ZZ method) by Zienkiwicz and Zhu’ has been proposed.

*Assistant Professor, Member ZA t Graduate Research Assistant

Copyright 01999 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved

Starting from ZZ method, researchers attempted to improve the accuracy of the recovered stress within a patch by increasing order of the assumed stress field. However, this may result in ill-conditioned ma- trix in the final stage of patch rccovcry routine To overcome this drawback, they introduced equilibrium constraints into the ZZ method in various ways and increased order of the assumed stress polynomials.2-6

An element stress recovery technique has been pro- posed by Park and Shin7 in which two equilibrium constraints are derived from virtual work principle and these two equilibrium residuals were inchudcd in the error functional for minimization. This tcchniquc provides satisfactory enhancement in accuracy of rc- covered boundary stresses.

In the present study, we aimed at the recovery of m-plane boundary stresses and interlaminar stresses of laminated composite plates. For the improved in-plane stress recovery, independent stress field is assumed in each elements. The recovery technique developed by Park and Shin7 in two and three dimensional elasticity elements are extended to the nine-noded plate bend- ing elements. Independently assumed in-plant stress field can be constructed in nine-noded quadrilateral cl- ement. The constructed in-plane stress field enhances the stress prediction in the x-y plane. However, this improved in-plane response of first order shear dcfor- mation plate model is not sufficient for the prediction of through-the-thickness behaviors of thick multilay- ered composite plates.

Thus to obtain improved through-the-thickness be- haviors of deformations and stresses, the postprocess method developed by Cho and Kim’ is applied. The method uses the shear strain energy equivalence bc- tween first order plate theory and higher order plate theory. The detailed procedures can be found in the Ref.a In the reconstruction of displacement of higher

-order theory, the first order derivative of w is-required. In the in-plane stress reconstruction, the second order derivatives of w are required. The interlaminar shear stress reconstruction requires third order derivatives of w. In the conventional isoparametric nine-noded

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element, the third order derivatives of w cannot be calculated within an element. However, if we utilize the stress recovery technique proposed by Ref,7 in the nine-noded element, element-wise stress recovery can be performed and thus resultant forces and the higher order derivatives can be obtained at the boundary of the plates.

II. Element Resultant Stress Recovery Formulation

Assumed Stress Resultant Fields

The stress resultant field is assumed in each element as follows

where {M}, is the assumed stress resultant vector in an element, [P] is the polynomial matrix, and {a} is the coefficient vector to be determined. For plate- bending problems, with five components of the stress resultant field, the [P] matrix can be expressed as equation (2).

PI 0

[P] i-- : :

0 0 0

= A

0 0

0 0 0 0 0 Fd 0 0 0 0 0 r; I

The stress resultant vector is expressed as

= [Q {fG}

where

Dll

i D31

D12 D13 0 0 D21 022 023 0 0

@,I = 032 033 0 0 , 0 0 0 G,,t 0 0 0 0 0 G,,t

1 The nine-noded assumed strain plate bending ele-

ment is employed’ instead of using the conventional isoparametric element to avoid shear locking and spu- rious modes of an element. Each pi is assumed as equation (4). Order of each pi can be selected as fol- lows.

$ t2 5% i = 1,2,3 i=4,5 (4 ,

In other words, order of each assumed stress re- sultant component is equal to that of the assumed displacement in the finite element used for analysis.

Residual Error At The Superconvergent Point

General expression for the difference between the the stress resultant calculated from the assumed recov- ery stress resultant field {M}, and the stress resultant obtained by finite element analysis {M}f at the i-th superconvergent point inside of an element can be ex- pressed as

{el = {Ml, (t8, GJ - tWr (Es7 d (5) = Pig {al - Wf CL, d

where (&, as) represents the coordinates of the super- convergent point in the local coordinates.

The energy error norm is defined as

11412 = MT PI {el (6)

w-here [,!?I is compliance matrix which satisfies {K} = ISI WI Residual Error Computation Using Boundary Integral

The total potential energy of plate bending problem is

1 =- 2 J A {K}~ @,I {K} dA - / @I: WI, ds (7) B

A plate in an equilibrium state satisfies the following equation, which stands for the well-known principle of virtual work that ignores body force:

&r = J 6 {K}T{M} dA -

=OA J 6 {u};{M}, ds (8)

*

where a(n) is virtual curvature vector, {M} is the stress resultant vector and s is the surface integral un- der the external force. The traction vector {M}, in equation (8) along the boundary is expressed in terms of the element resultant {M},. Coordinate transform relation along the boundary of an element is given as,

71 0 [

case sin0 2 - s - 11 I -sine co&J y (9)

where n, s are normal and tangential coordinates along the boundary. The external force vector {M}, is ex- pressed in terms of the assumed stress resultant vector PO,.

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where

= [LIM WI, (10)

0 0

I anx and boundary displacements {u}~ is expressed in terms of nodal displacements {u}

Since the actual stress resultants are unknown, the actual stress resultants are replaced by the assumed stress resultant field at each element, and surface inte- gration is replaced by equations (10) and (11). Then equation (8) is expressed as:

where A, is the area of an element and boundary shape transform matrix [L] = [L]T[L]M. Now, sum of each element residual of equilibrium (6rrres)i does not ap- proach to zero until the assumed stress resultant field is very close to the actual stress resultant.

The virtual displacement vector 6 {u} can be in- terpolated by using nodal degree of freedom vector 6 {Ue}, and the virtual curvature vector b {K} can be obtained by differentiating the virtual element nodal degree of freedom vector 6 {Ue} in the following way:

b(u) = [N]6(Ue} 6 {K} = [B] b(Ue} (13)

where [N] is the shape function matrix and [B] is the operation matrix that relates curvature and displace- ment. Introducing the equations (1) and (13) into the first term in the right hand side of equation (12), the following expression is obtained:

J 6 {K}~{M), dA = 6 {u,I~J [@PI (al dA (14) A. A. The second term in right hand side of equation (12) can be written as:

J6 tu)TU WI, ds = 6 lu,lT$JI%lPl {al ds (15) s 8

Finally, using equations (14) and (15), the residual of equilibrium (&r,,,)i can be rewritten as:

(~T~,,)I = 6 {Ue>‘(Ce - De) {a) 1 0 (16)

where

c, = J PIT WA A. D, = J [NITIL][P]ds 9

Residual Error Computation Using Nodal Forces

The equilibrium residual of an element can bc de- rived in a different way by representing the left hand side of equation (14) using stiffness matrix of the ele- ment as follows:

J 6 {~}‘rB,l {K} dA = J 6 {K}~{M}, dA (17) A. A.

Introducing equations (13) and (17) into the equation (12), another form of equilibrium residual (6~r,~)2 can be derived as

(~~,e,)z = J 6 {~VI TV) dA - J 6 {K}T(M), dA c = s;rr,me {UC} - c, r$ = 0 08)

where K, = JA,[B]T[fi][B]dA is the element stiffness matrix and the nodal force vectors {Fe} are defined as

{Fe) = Ke {ue) (19)

where {F,} is the nodal force vector,

Recovery of Resultants

The coefhcient vector {a} is determined from con- dition that strain energy error norm J(ei]], the first residual vector (&)I, and the second residual vector (Reg)2 are minimized in the least squares sense. The error functional F( {a}) is constructed as

F(ta)) = ak lle~l12 + PWi$)l(fLqh +Y(R~~)~UL)Z i=l

= *{e}T [i?] {e} + P[(G - De> Ia)ITKG - De) {all + r[G ial - WeHT[G {al - {Fell (201

where n is the total number of sampling points within an element, and Q, p, y are penalty parameters. In the

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above equation, the first term of F({a}) corresponds to the least squares functional used in the ZZ method,’ and the third term is the same form with the residual functional introduced in the LP method.6 By employ- ing the second term in addition to the third term, the independent resultants can be assumed in an element.

The coefficient vector {a} in the equation (1) is de- termined by minimizing the functional in the equation (20) with respect to the coefficient vector. Thus, the coefficient vector {a} is chosen such that the equation (21) is satisfied.

mta)> = 0

ata (21)

This condition can be rewritten as follows after rear- rangement:

VW {al = {W (22)

where [RM] = a e[P]T[S][P]

i=l

+,O(Ce - De>T(Ce -De) + rC:Ce n

i=l

The equation (22) is a system of well-conditioned lin- ear algebraic equations and should be solved in each elements where the stress recovery is desired.

III. Stress Recovery Formulation With Postprocess Method

After obtaining the improved stress resultants in the x-y plane, the curvatures and mid-plane strains are obtained in the bi-quadratic polynomial form. Thus pointwise stresses through the thickness are expressed in terms of 5 and 7. In the present section, deforma- tions and stresses in the thickness direction are refined from the nine-noded finite element solutions by a post- process method. The primary variables of both an efficient higher order plate theory(EHOPT)l’ and first order plate theory(FOPT) are matched by assuming the transverse shear strain energies of both theories to be equivalent. Then the primary variables of EHOPT can be expressed in terms of those of FOPT. If the solutions of FOPT are available, the EHOPT defor- mation field can be easily constructed from the known FOPT variables. However, in the displacement field of EHOPT, the first derivatives of w appear and in the in-plane stress field the second derivatives of w. Even in the transverse shear stress recovery, an approach by integrating equilibrium equations requires calcula- tions of the third derivatives of transverse deflection w. This situation makes the application of postpro- cess method difficult in the lower order finite element

framework(such as three-noded triangular element or four-noded quadrilateral element). Implementation of postprocess method on higher order element like nine- noded element can be found in Ref.‘l However, in the Ref.ll the finite difference method between adja- cent elements was employed. In the present method, the resultants are computed within an element and from the shear force resultants the second derivatives of shear angle ya = +a + w,~ can be computed within the element.

The postprocess method is outlined here.

The displacement fields of an FOPT and an EHOPT as a postprocessor are given as follows:

ua = ‘11, “( Xl, x2) + ~dh(~l,~Z)

u3 = =7(x1, x2) (23)

U - ‘11:(X1,x2) + qL&l,ZZ) a-

423 N/Z-l

-- 3h2 w,, -I- ti’a + c 4i,hb + W,Y)

k=l 1 N/Z-l

(q-y + w,y)[(z - Zk)H(Z - zk) k=l

-(-z - Zk)ff(-Z - zk)]

U3 = +a, 22) (24)

where N is the number of layer, H(z - %k) is the Heav- iside unit step function, and a& can be determined from layer transverse material properties, and layer thickness. A detailed derivation of equation (24) can be found in Ref.‘O

Matching the Rotational Variables The variables of the EHOPT can be obtained in

terms of the variables of FOPT by assuming the trans- verse shear energy of EHOPT and FOPT to be equiv- alent :

plmayPT - UFOPT - shear (25)

where Ushear denotes the transverse shear strain en- ergy of the laminated composites plate.

This relationship can be expressed as

(& + w,a)(EHoPT) = C,(& + w,,)(~~~~) (26) (No sum on o)

where C, is the matching factor. It is assumed that the out-of-plane deflection w of

EHOPT is the same as that of FOPT. The derivation of equation (25) and (26) can be found in Ref.*

Finally, higher order in-plane displacement field can be reconstructed by the FOPT primary variables in symmetric lamination layups.

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uEHOPT cr

= 2 {(C, - l)($J~OPT + uJ;?r, + $EOPT)

-g [C, (w;o” + $,“““) N/2-1

I;=1

N/2--1

k=l

X(Z - zk)ft(z - zk) - (-2 - 2k)H(-2 - zk)l

(No sum on CE) (27)

From the recovered assumed stress resultant field, the curvature and shear angle at the midplane can be ob- tained.

(28)

11,,, = fh = p11 + Pl2E + Pm + P14671

+P15t2 + PlS$

+y,y = ny = p21 + 0225 + p2371 + P24h’

+P25t2 + P2im2 (30) A,Y + $Y,X = b = /331 + P326 + p33q + p34th

+P35J2 + P3sq2

ti,, + w,z = yz = p41 + P42t + p4371 + p44571

$y + w,y = ?‘y = /351 + @52t + p537] + !%4’6q

where ,&j are known paramet-em. The first derivatives of curvature n,, /cy, fizy and

shear angle y5 ,yV are directly computed from equa- tion (30). From the reconstructed displacement field in equation (27), the in-plane stresses through the thick- ness can be computed. The interlaminar stresses are computed by using three-dimensional stress equilib- rium equations. This procedure requires 2nd deriva- tives of yz and yy. The first derivatives of yz and yy are constants within elements. Thus to obtain the second derivatives of yz and yr,, the shear angle yL+ is interpolated within an element considering adjacent element informations of ^(a. The second derivatives of 7a can be directed computed from the interpolated y,(see Fig. 1).

The flowchart of the present stress recovery routine is given in Fig. 2.

Numerical Examples To evaluate the performance of the element stress

recovery technique with equilibrium constraints, an isotropic plate bending problem under sinusiodal load- ing is considered(see Fig. 3).

Isotropic material properties and geometric proprr- ties of the plate are given as follows

E = 25 x 106psi, Y = 0.25, a/h = b/h = 100

The nine-noded assumed strain plate bending clc- ment is employed for the robust finite clement analysis. The convergence of the transverse deflection w of the assumed strain element is shown in Fig. 4. The analyt- ical solution of classical plate theory is used as the ref- erence for the performance comparisions. For the given mesh configuration(see Fig. 5), the moment resultants and shear force resultants calculated from the present method are compared to the conventional boundary stress resultants and analytical solutions which is given in the Figs. 6 and 7. The stresses of the conventional recovery method are obtained by extrapolating from the Gauss point stresses of the element. As shown in the Figs. 6 and 7, the proposed element stress rccov- ery method works well compared to the conventional stress recovery method. While the conventional stress extrapolation met.hod provides inaccurate shear force resultants in the coarse mesh, it should be emphasized that the present recovery technique provides very accu- rate boundary shear force resultants even in the coarse mesh(see Fig. 7).

To evaluate the performance of the present recovery technique in the thick laminated plates, we consider the symmetric cross-ply laminated plate problems. The three-dimensional elasticity solutions of Paganor for simply supported rectangular plates under sinu- soidal loading are reproduced here for comparison. Two examples are considered: case 1, a three-layered square laminate[0/90/0] with uniform ply thickness; and case 2, a nine-layered square laminate[O/gO/... /90/01~

Both cases are under sinusoidal loading conditions. The material properties for the 0-deg layer and geo- metric properties are given as

El = 25 x 106psi, Ez = lO”psi, Gi2 = Grs = 0.5 x 106psi, G23 = 0.2 x lo’psi, ~12 = ~13 = ~23 = 0.25, a/h = b/h = 4

To facilitate the comparison with other theories, the following nondimensional parameters arc defined:

(522, eyy, c,y) = h” g(%z,~yyJJ2y 1

(‘Tyx 7 Em) = ~(Uyz,%z) (31)

($4 = f$aw,7 5 - 10;.$t4haw (1

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Taking advantage of the symmetry of loading, boundary conditions, and lamination configurations, only quadrants of square plate are used in the finite element analysis. The 4 x 4 mesh configuration is used for the whole computation. Figs. 8-14 represent the recovered stresses through the thickness of case 1. The in-plane displacements and in-plane stresses are accurately recovered compared to the exact elasticity solutions. In Figs. 13 and 14, the transverse shear stresses are recovered. Both of them show reasonable accuracy.

Figs. 15-21, depict the recovered stresses through the thickness of case 2. All the stresses and the the in-plane displacements are recovered accurately com- pared to the 3-D elasticity solutions.

Conclusions An efficient element stress recovery method with

equilibrium constraints is proposed. It consists of two steps. The first step is to recover the moment and shear force resultants from the independently assumed resultant fields. The superconvergent stress error and two equilibrium residual are defined and the moment and shear forces are obtained by the least square mini- mization of the proposed penalty functional. This step makes the enhancement of in-plane stress possible. The second step is to employ a postprocess method which makes possible the improvement of through-the thickness stress distributions of thick laminates. The numerical examples demonstrate the efficiency and ac- curacy of the proposed method. Especially, it should be emphasized that the sensitive boundary shear re- sultants are accurately recovered by the present stress recovery technique with equilibrium constraint, which cannot be expected in the conventional stress extrap- olation method.

The present stress recovery method may work as an element postprocessor to improve the prediction of the boundary stresses of thick laminated plate. The present method can be extended to the thick laminated shell problem and this work is in progress.

References ‘O.C. Zienkiewicz and J.N. Zhu, “The Superconvergent

Patch Recovery and a Posteriori Error Estimate. Part 1, 2: The Recovery Technique, ” International Journal for Numeri- cal Methods in Engineering,33, 1987, pp. 1331-1364.

2N.E. Wiberg, and F. Abdulwahab, “Patch Recovery Based on Superconvergent Derivatives and Equilibrium,” International Journal for Numerical Methods in Engineering, 36, 1993, pp. 2703-2724.

3N.E. Wiberg, F. Abdulwahab and S. Ziukas, “Enhanced Superconvergent Patch Recovery Incorporating Equilibrium and Boundary Conditions,” International Journal for Numerical Methods in Engineering, 37,1994, pp.3417-3440.

4T. Blacker and T.Belytschko, “Superconvergent Patch Re- covery with Equilibrium and Conjoint Interpolant Enhance-

merit,” International Journal for Numerical Methods in Engi- neering, 37, 1994, pp.517-536.

5Tessler, A.Riggs, H.R.,and Mac, S.C., “A Variational Method for Finite Element Stress Recovery Technique and Er- ror Estimation,” Comp. Meth. Appl. Mech. Eng., 111, 1994, pp.369-382.

6T. Lee, H.C. Park and S.W. Lee, “A Superconvergent Stress Recovery Technique with Equilibrium Constraint,” ac- cepted for International Journal for Numerical Methods in En- gineering, 1997.

7H.C. Park and S.H. Shin, “An Element Stress Recov- ery Technique with Equilibrium Constraint for Superconvergent Boundary Stress Extraction,” S.D.M. Conference, part 1, 1998, pp.116-123.

‘M. Cho and J.H. Kim, “Postprocess Method Using a Dis- placement Field of Higher-Order Laminated Composite Plate Theory,” AZAA Journal , 34, 1996, pp.362-368.

QC.H.Yeom and S.W.Lee, “An Assumed Strain Finite El- ement Model for Large Deflection Composite Shells,” Intema- tional Journal for Numerical Methods in Engineering, 28, 1989, pp.1749-1768.

I’M. Cho, and R.R. Parmerter, “An Efficient Higher Order Plate Theory for Laminated Composites,” Composite Struc- tures, 20, 1992, pp.113-123.

“M. Cho, and J.S. Kim, “Improved Mindlin Plate Stress Analysis for Laminated Composites in Finite Element,” AIAA Journal, 35, 1997, pp.587-590.

12N. J.Pagano, “Exact Solutions for Rectangular Bidirec- tional Composites and Sandwich Plates,” Journal of Composite Materials, 4, 1970, pp.20-34.

13G. Rengarajan, M.A. Aminpour and N.K. Knight, Jr, “Im- proved Assumed-Stress Hybrid Shell Element with Drilling De- gree of Freedom for Linear Stress, Buckling and Free Vibration Analysis,” ,Intemational Journal for Numerical Methods in En- gineering, 38, 1995, pp.1917-1943.

/

Target element Adjacent element

Recovered element

Fig. 1 Shear force interpolation for the computa- tion of the second derivatives of ys, yV.

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4 Solving the FOPT

bv finite clement

stress resultant rec0”ay along the in-plane direction

Calculstian of two residuals of equilibrium

Element stress resultant field recovery by nCm&ing least square error functional

to obtain ‘m-plane displacement of bigha

Matching the rotational variables using the trimsverse shear seain energy equiva.leace

Fig. 4 Deflection convergence for isotropic plate

I [Ij :Stress recovery zone I Qy UYZ

Fig. 5 Mesh configuration and stress recovery zone and recovery boundary points

Fig. 2 Flow chart of the present element stress recovery routine

Fig. 3 Plate configuration and loading condition

A : Fern Extrapolation - . : Present Method , -. : Analytical Solution 1

No. of element along oneside

Fig. 6 Moment M,, convergence for isotropic plate

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I ‘o, I

No: of eleient alo& oneside I

Fig. 7 Shear Force Qz convergence for isotropic plate

Fig. 8 In-plane displacement C(0, 4) through the thickness for [O/90],

. . 0.4 ‘.._ ._ 03

I \.. ‘_ . . ‘_

Fig. 9 In-plane displacement fi( g, 0) through the thickness for [0/9013

Fig. 10 In-plane stress LTzz(%, g) through the thick- ness for [O/90],

Fig. 11 In-plane stress (Tyy(4, g) through the thick- ness for [O/901s

Fig. 12 In-plane stress ii,,(O,O) through the thick- ness for [O/90],

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Fig. 13 Transverse shear stress Czz(O, i) through the thickness for [O/90],

Fig. 14 Transverse shear stress Cyz( ;, 0) through the thickness for [O/90],

Fig. 15 In-plane displacement G(O, $) through the thickness for [O/90/0/90/0],

Fig. 16 In-plane displacement a( %, 0) through the thickness for [O/90/0/90/0],

Fig. 17 In-plane stress a,,(;, $j through the thick- ness for [O/90/O/90/O]s

Fig. 16 In-plane stress &yV( 4, k) through the thick- ness for [O/90/0/90/0],

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Fig. 19 In-plane stress (Tzy(O,O) through the thick- ness for [O/90/0/90/0],

Fig. 20 Transverse shear stress azz(O, c) through the thickness for [O/90/O/90/O]S

Fig. 21 Transverse shear stress a,, (f , 0) through the thickness for [O/90/0/90/0],

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