NASA Technical Memorandum 109149

iii .d/

J a'/ / _i

Accurate Interlaminar Stress RecoveryFrom Finite Element Analysis

Alexander Tessler

Langley Research Center, Hampton, Virginia

H. Ronald Riggs

University of Hawaii at Monoa, Honolulu, Hawaii

(NASA-TM-I09149) ACCURATE N95-I1815

INTERLAMINAR STRESS RECOVERY FROM

FINITE ELEMENT ANALYSIS (NASA.

Langley Research Center) 13 p unclas

G3/39 0022767

September 1994

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19950005402 2018-07-30T02:43:27+00:00Z

ACCURATE INTERLAMINAR STRESS RECOVERY

FROM FINITE ELEMENT ANALYSIS

A. Tessler

NASA Langley Research Center, Mail Stop 240, Hampton, VA 23681-0001

H. R. Riggs

University of Hawaii at Manoa, Honolulu, HI 96822

Introduction

The successful design of advanced com-

posite aerospace structures requires a detail

knowledge of strains and stresses that are in-

duced by the applied loads. The inherent

strength weakness between adjacent plies con-stitutes a weak link in a laminate which under

the action of interlaminar stresses may result

in the dominant interlaminar failure mode

know as delamination. The delamination may

then yield the ultimate laminate failure. Con-

sequently, predicting accurate interlaminarstrains and stresses is an essential facet of

composites analysis. An comprehensive ac-

count of the currently available methods for

the interlaminar stress recovery is presented

by Byun and Kapania. 1

The first-order shear deformation theory

with the use of constitutive relations produces

highly inaccurate interlaminar stresses that

exhibit nonphysical discontinuity at the ply

interfaces of composite laminates. This diffi-

culty is overcome by integrating three-dimen-

sional equilibrium equations of elasticity the-

ory to recover the interlaminar stresses. 2 This

alternative procedure, which is not entirely

consistent with the assumptions of the first-or-

der shear deformation theory, provides rea-

sonably accurate predictions for the interlami-

nar stresses for closed form analytic solutions.

When the procedure is applied to the results of

finite element analysis, the accuracy of the

interlaminar stress predictions tends to be

rather poor. The main reason for this is that

the integration of three-dimensional equilib-

rium equations involves expressions of

strain gradients whose recovery from the finite

element shape functions is known to be infe-

rior.

This report discusses an accurate recovery

procedure of interlaminar shear stresses fromfinite element solutions to laminated sandwich

and composite plates. The procedure is gen-

eral enough to be useful as a full-field post-

processor of all finite element response quan-tities.

Approach

The preferred procedure for computing

through-the-thickness distributions of inter-

laminar (transverse) stresses in laminated

plates and shells is by means of integrating

three-dimensional equilibrium equations of

elasticity theory. The integrals contain partial

derivatives of strain measures of plate/shell

theory. These quantities are independent of the

thickness coordinate and are commonly at-

tributed to a reference plate/shell surface. For

example, upon integration of the elasticity

equilibrium equations, the interlaminar shear

stress, rx_ , corresponding to the first-order

shear-deformation plate theory can be written

as

Zxz =QI 'e,x + Q6"e,y (1)

where a comma stands for partial differentia-

tion; e is the vector of strain measures,

e={E_o,e>,o,Y_,o,pc:o,lCyo,r,_,o} (la)

Qj are the vectors whose components are

functions of the thickness coordinate, z,

Qj = {qu,q2j,q6j,pu,p2j,p6j} (lb)

where j = 1,6 and

qo=-SzC_k)dz, pij=-SzCbk'zdz (lc)

with C_k) denoting the elastic stiffness coeffi-

cients of the k-th ply.

The difficulties in obtaining reliable pre-

dictions of strain gradients from a finite ele-

ment analysis are well established. For exam-

ple, in first-order shear deformation elements

based on Reissner-Mindlin type theories, the

displacement (kinematic) fields are approxi-

mated with Co -continuous shape functions en-

suring continuity of displacements along ele-

ment boundaries. The strains are represented

by the first gradients of displacements; hence,

they are only C-j continuous, exhibiting non-

physical discontinuities along element bound-

aries. Only at a limited number of discrete lo-

cations within the interior of a finite element

reasonably accurate strain results can be calcu-

lated, these are referred to as optimal points.

The first gradients of strain are obviously evenless accurate, and hence reliable strain deriva-

tives cannot be obtained directly from the el-

ement interpolation functions.

Tessler and co-workers 3-6 recently devel-

oped a smoothing variational formulation

which combines discrete least-squares and

penalty-constraint functionals in a single

variational form. The strain (or stress) compo-

nent to be smoothed and its Cartesian gradi-

ents enter in the variational principle as inde-

pendent variables. For a sufficiently large

value of the penalty number, the approach en-

ables the resulting smooth strain field to be

practically C _-continuous throughout the do-

main of smoothing, exhibiting superconver-

gent properties of the smoothed quantity. The

continuous strain gradients are also obtained

directly from the solution. The use of the re-

covered stress (strain) gradients has been

demonstrated for assessing errors in finite el-

ement analysis by computing the residual in

the strong form of the equilibrium equation.

Also, the method has the versatility of being

applicable to the analysis of rather general and

complex structures built of distinct compo-nents and materials, such as found in aircraft

design. For these types of structures, the

smoothing is achieved with "patches", each

patch covering the domain in which the

smoothed quantity is physically continuous.

The mathematical basis of the smoothing

methodology is briefly summarized below.

Let kp=k(xp) (xp=(x_,xp2), p=l,2 ..... N)

represent a set of discrete optimal values for a

specific strain component defined on the two-

dimensional domain O=[x=(x_,x2)c_2],

with these strains resulting from a finite ele-

ment analysis. We seek a smooth, continuous

strain e(x) and its gradients el---7_e (i=1,2)which can best represent the discrete data.

The problem can be cast as a minimization

of a least-squares/penalty-constraint func-tional, such that

_q_=O

where q_ is given by

N

q,=! y [_p _ e(xp)l 2N p=l

(2)+,_, j" [(e, 1 _ 01 )2 + (e,2 _ 02 )2 ld.O

where N is the total number of data points,

xp denotes the position vector of the data

point, ,1, is a penalty number, and 01 and 02

are independent continuous functions also de-

fined on t2. The minimization of • is per-

formed with respect to the coefficients in e

and 0i which serve as the unknowns. The first

term in Eq. (2) represents a discrete least-

squares functional with the term

2

Ae = kp - e(Xp) (3)

signifying the error in kp as compared with

the smooth solution e(xp). The second term in

Eq. (2) is a penalty constraint functional

which, for sufficiently large values of ,_,

yields the following constraint relations

e i _ 0i (i= 1,2) on t2 (4)

The proper interpretation of Eq. (4) is that for

all practical purposes the 0, (i = 1,2) represent

the gradients of e. The variational principle

in Eq. (2) is readily discretized with C°-con -

tinuous element approximations, forming a

mesh of "smoothing" elements. The smooth-

ing finite element analysis then provides con-tinuous and reliable solutions for the strain

and its gradients. The latter quantities are then

employed in Eq. (1) for the recovery of inter-

laminar shear stress distributions through the

thickness.

In this paper, the accuracy and robustness

of the two-dimensional smoothing discretiza-

tion is examined for the problem of recovering

accurate strain gradients, and subsequent inte-

gration of equilibrium equations to computeinterlaminar shear stress distributions in lami-

nated composites. The model problem is a

simply-supported rectangular plate under a

doubly sinusoidal transverse pressure load.

The problem has an exact analytic solution

which serves as a measure of goodness of therecovered interlaminar shear stresses.

The following charts describe the essential

theoretical features of the approach and sum-

marize the numerical results.

2. Reddy, J. N., "A simple higher-order the-

ory for laminated composite plates," ASME

Journal of Applied Mechanics, 51, (1984)

745-752.

3. Tessler, A., Freese, C., Anastasi, R., Ser-

abian, S., Oplinger, D. and Katz, A., "Least-squares penalty-constraint finite elementmethod for generating strain fields from moirefringe patterns," Proceedings of SPIE: Pho-tomechanics and Speckle Metrology, 814

(1987) 314-323.

4. Tessler, A., Riggs, H. R., and Macy, S. C.,

"Application of a variational method for com-puting smooth stresses, stress gradients, anderror estimation in finite element analysis,"

MAFELAP Highlights 1993, John Wiley &Sons, U.K. (1994).

5. Tessler, A., Riggs, H. R., and Macy, S. C.,"A variational method for finite element stress

recovery and error estimation," ComputerMethods in Applied Mechanics and Engineer-

ing, 111 (1994) 369-382.

6. Riggs, H. R., and Tessler, A., "Continu-

ous versus wireframe variational smoothing

methods for finite element stress recovery,"

Proceedings of the Second International

Conference on Computational Structures

Technology, Athens, Greece (1994)

References

1. Byun, C., and Kapania, R. K., "Prediction

of interlaminar stresses in laminated plates

using global orthogonal interpolation polyno-

mials," AIAA Journal, 30(11), (1992) 2740-

2749.

3

Content

• FSDT review

• Recovery Procedure- Integration Scheme

- Smoothing Analysis

• Numerical results

• Summary

Plate Theory (FSDT) Notation

5-Variable Approximation

ux(x,y,z) = u + h_ Oy I

lUy X,y,z,=v+h¢ Iz lU_(x,y,z)=w I

W lt]z

4

Eight Strain Measures:First Partial Derivatives of Kinematic Variables

t- °tK"° =

_70

U,x

V,y

U,y + V x

y,x

Ox,y

Ox. x -F Oy,y

W,x -t" Oy

W,y + Ox

Inplane Strains

Bending Curvatures

Transverse Shear Strains

(uniform through thickness)

FEM: C°-continuousinterpolations u, v, w, e,, 0,.

Integration Scheme

(1) 1:xz=Ql.e,x+Q6.e v

(la) e ---- {Exo,Eyo, 7x3,, o, Kxo, ICy o, _xy.o}

(lb) Qj = {qlj,q2j,q6j,Ptj,Pej,P6j}

(lc) qij =-Iz _ "k)dz' P(i=-Iz _*)zdz

Shear Stress

Strain Vector

Material-ThicknessCoordinate Vector

Qj components

5

Smoothing Analysis

(1) _p = _(Xp)

1 N(2) 4_=-_ _ [_p- e(xp)] =

_= /7=1

+_"I [(ea - 01)2+(e, 2- 02)2]dg2[2

(3) e,i-+O i (i=1,2)

SampleoptimalFEMdiscretestrainvaluesat Gausspointsof finiteelements

ApplyLeast-squares/penalty-constraintfiniteelementmethodto smoothFEM strainsfor entiredomain

RecoverC°-continuous,smoothedstraingradientsthroughoutFEM domain

i iii

,I

Numerical Examples

SS Square Plate Under Doubly SymmetricSinusoidal Normal Pressure

Y

L _=_" x

SS

• Loading: q sin(Fix/L) sin(Hy/k)

• Span/rhickness=30, 10, and 5

Lamination

• Gr/Ep:

• GrlEp:• Sandwich:

[0/90/0]s 6-ply

[03/903/0a]s 18-ply

[(01g0)s/hJs 21-ply

6

1

o.s-

zlh

o-

-0.5-

-1

FSDT vs. 3D ElasticityInterlaminar Shear Stresses

Thin, 6 Ply Gr/Ep Laminate

[0 / 90 / 0]s,. L / 2h = 30

.... , .............. , .... L .... , ....

I-- 30EUAST,C_I /

0.0S 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05

1 .... i .... J , . - ......

• O.S_zlh °

-0.5_-1 . , . . .

0.1 0.15

Ty.

0,2

FSDT: First-order Shear Deformation Theory Solution3D ELASTICITY: 3D Elasticity Theory Solution

FSDT vs. 3D ElasticityInterlaminar Shear Stresses

1

O.S-

zlh

o.

-0.5,

-1

Moderately Thick, 18 Ply Gr/Ep Laminate

[03/903/03] s, L/2h=lO

• FSDT (integrated) I

I- 3DE_sT_rrY I

..11.5]

.,--, .... i -li

0.05 O, 1 O. 15 0.2 0.25 0.3

, , , , . . . , .... , ....

0.05 O.1 0.15 0.2

T_

7

FSDT vs. 3D ElasticityInterlaminar Shear Stresses

1

0.S-

zlh

0-

-0.5-

-1

Thick, 21 Ply Sandwich Laminate

[(0/90)5 / hc]s, L/2h = 5

, , , , L ¢ -- . = -- , _ ....

..... FSDT (Integrated) I

3D ELASTICITY J

.... 0._35 .... 0'.1 .... 0.'15 .... O12

!

0.5-

zlh

o-

-0.5.

-1

I l

O.OS 0.1 0.1S

(k)----f[G (k) + ¢(_)v ]dZ Ixz -- J ¢r,x ,

0.2

FEM and Smoothing Meshes

FINITE ELEMENT MESH9-NODE ANS SHELL ELEMENTS

÷ 4.

O4. 4.

o u,v,W, 0x, 0y DOF

+ Optimal Curvatures @2x2 Gauss Points

SMOOTHING ELEMENTMESH3-NODE SMOOTHING ELEMENTS

;KXXXXXXX

XXXXXXXX

• _:,Ol, O2 DOF

8

Finite Element Results

Thin, 6 Ply Gr/Ep Laminate

[0/90/0] s, L/2h=30

Response % Error inQuantity Max. Value

Deflection 0.02Rotation 0.03Curvature 4.69

Interlaminar Shear 1;xz 8.80

Interlaminar Shear l:yz 8.20

1

0,5-

zlh

0-

-0.5-

-1

FEM vs. RecoveredInterlaminar Shear Stresses

Thin, 6 Ply Gr/Ep Laminate

• [0/90/0] s, LI2h=30

.... i .... , .... h.... , .... , .... , ..... 11 .... i , , • • , .... , ....

0.05 0.1 0.15 0.2 0.25 0.3 0.3S O.OS 0.1 0.1S 0.2

9

FEM vs. RecoveredInterlaminar Shear Stresses

Thick, 21 Ply Sandwich Laminate[(0/90) 5/hc] S, LI2h=5

o,,............i...........!....................

z/h I_FSDT.... - .... FEM ......

o I---Recovery

-o.s ......................................l....

O.OS 0.1 0.15

%0.2

i i =i i ,

o. -......... ..--........... !............. ;--

.... - .... FEM :,.-_-_-,

I - - "Recovery

?-o.s-I.......................i............. !'"

I _

-1 I_

O.OS 0.1 0,15

t

I

I

r

I

I

I

I

I

I

I

I

I'---

I

|i

O.Z

Results Significance

• Direct evaluation of interlaminar shearstresses from FEM underestimates their value

considerably (unconservative for design)

• FEM-based smoothing method recovers

improved interlaminar shear stresses that areslightly overestimated (conservative fordesign)

• Recovered interlaminar shear stresses are

improved by at least a factor of 3 [Error of2.8%(Recovery) vs. 8.2%(FEM)]

10

Summary

• Strain smoothing produces high-accuracystrains and their gradients that are C °-continuous over full domain

• 'Smoothed' strain gradients yield superiorrecovery of interlaminar shear stresses overdirect FEM recovery

• Effective for laminated composite and sandwich

plates

• Viable full-field post-processor of all strains &stresses in general purpose FE software

11

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I. AGENCY USE ONLY (Leave blank) 2. P,,E,PORT DATE 3. REPORT TYPE AND, i_ATES COVEI_ED_eptemoer 1994 i ecnmcal Memoranaum

4. TITLE AND SUBTITLE

Accurate Interlaminar Stress Recovery from FiniteElement Analysis

6. AUTHOR(S)

A. Tessler and H. R. Riggs

7. PERFORMING ORGANIZATION NAME(S) AND AODRESS(ES)

NASA Langley Research Center, Hampton, VA 23681-0001

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space AdministrationWashington, DC 20546-0001

S. FUNDING NUMBERS

5( 5-63-53-01

8. PERFORMING ORGANIZATIONREPORT NUMBER

10. SPONSORING / MONITORINGAGENCY REPORT NUMBER

NASA TM-109149

11. SUPPLEMENTARY NOTES

A. Tessler: Langley Research Center, Hampton, VAH. Ron Riggs: University of Hawaii at Monoa, Honolulu, Hawaii

1Za. DISTRIBUTION / AVAILABILITY STATEMENT

Unclassified- Unlimited

Subject Category 39

13. ABSTRACT (Maximum 200 words)

12b. DISTRIBUTION CODE

The accuracy and robustness of a two-dimensionalsmoothing methodologyis examinedfor theproblem of recoveringaccurate interlaminarshear stress distributions in laminated composite and sandwich plates. The smoothingmethodology is based on a variational formulation which combines discrete least-squaxes and penalty-constraintfunctionals in a single variational form. The smoothing analysis utilizes optimal swains computed at discretelocationsin a finite element analysis. These discrete strain data are smoothed with a smoothing element discretization,producing superior acctracy swains and their firstgradients. The approach enables the resulting smooth swain field tobe practically C: -continuous throughout the domainof smoothing, exhibiting superconvergent properties of thesmoothed quantity. The continuous swaingradients are also obtaineddirectly from the solution. The recovered swaingradients are subsequently employed in the integration of equilibrium equations to obtain accurate interlaminar shear

istr_.. The model problem is a simp!y-s .uppo..rted .r_tangular plate under a doubly sinusoidal wansverse pressure•The problem has an exact analyuc soitmon which serves as a measme of goodness of the recovered intedaminarshear su'ess_. The method has the vematility of being applicable to the analysis of rathergeneral and complexsOxc.tmv.sbuilt of distinct components and materials, such as found in aircraft design. For these types of structures,the smoothing is achieved with "patches", each patch covering the domain in which the smoothed quantity isphysically continuous.

14. SUBJECTTERMS

Finite Element Analysis

Sandwich Plates

17. SECURITY CLASSIFICATIONOF REPORT

Unclassified

NSN 7540-01-280-5500

InterlaminarStresses

Variational Principle

18. SECURITY CLASSIFICATIONOF THIS PAGE

Unclassified

Laminated Compos

• Stress Recovery

19. SECURITY CLASSIFICATIONOF ABSTRACT

Unclassified

15. NUMBER OF PAGES

12_. PRICE CODE

A03

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev 2-89)Prescribed by ANSi Std Z39-18298-102