STIFFNESS, THERMAL EXPANSION, AND THERMAL BENDING ...

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AIAA/ASME/ASCE/AHS/ACS 34 th Structures, Dynamics, & Materials Conference 1993 STIFFNESS, THERMAL EXPANSION, AND THERMAL BENDING FORMULATION OF STIFFENED, FIBER- REINFORCED COMPOSITE PANELS Craig S. Collier * Principal Structures Engineer Lockheed Engineering and Sciences Co. NASA Langley Research Center Hampton, VA ABSTRACT A method is presented for formulating stiffness terms and thermal coefficients of stiffened, fiber-reinforced composite panels. The method is robust enough to handle panels with general cross sectional shapes, including those which are unsymmetric and/or unbalanced. Non-linear, temperature and load dependent constitutive material data of each laminate are used to "build-up" the stiffened panel membrane, bending, and membrane-bending coupling stiffness terms and thermal coefficients. New thermal coefficients are introduced to quantify panel response from through-the-thickness temperature gradients. A technique of implementing this capability with a single plane of shell finite elements using the MSC/NASTRAN TM analysis program (FEA) is revealed that provides accurate solutions of entire airframes or engines with coarsely meshed models. An example of a composite, hat-stiffened panel is included to demonstrate errors that occur when an unsym- metric panel is symmetrically formulated as traditionally done. The erroneous results and the correct ones produced from this method are compared to analysis from discretely meshed three-dimensional FEA. NOMENCLATURE lamina (i = 1, 2, 3) E i Elasticities ν ij Poisson's ratios α i Thermal expansion coefficients ij Q Reduced stiffness terms ij Q Transformed reduced stiffness term i Φ Reduced thermal stiffness terms i Φ Transformed reduced thermal stiffness terms σ i , є i Stresses and strains _____________________ * Senior member AIAA This paper is a work of the U.S. Government and is not subject to copyright protection in the United States. laminate h Thickness (i = 1, 2, 3) * ij Q Effective reduced stiffness terms * i Φ Effective reduced thermal stiffness terms panel H Depth S x Unit width of panel corrugation (i = 1, 2, 3) M i Materials (particular layup) of the upper facesheet, coresheet, and bottom facesheet laminate or panel T In-plane temperature gradient G Through-the-thickness temperature gradient (i = 1, 2, 3) h i Distance from the reference plane A ij , B ij , Membrane, membrane-bending coupling, D ij and bending stiffness terms A i α , B i α , Membrane, membrane-bending coupling, D i α and bending thermal force & moment coefs. NA i α , MSC/NASTRAN FEA membrane, NB i α , membrane-bending coupling, and bending ND i α thermal expansion & bending coefficients (i = x, y, xy) α i , α ci Thermal expansion, expansion coupling, δ i , δ ci bending, and bending coupling coefficients E i Effective engineering elasticities ν ij Poisson's ratios N i , M i Forces and moments є i , κ i Reference plane strains and curvatures superscripts p Panel T Thermal M Mechanical

Transcript of STIFFNESS, THERMAL EXPANSION, AND THERMAL BENDING ...

Page 1: STIFFNESS, THERMAL EXPANSION, AND THERMAL BENDING ...

AIAA/ASME/ASCE/AHS/ACS 34th Structures, Dynamics, & Materials Conference 1993

STIFFNESS, THERMAL EXPANSION, AND THERMAL BENDING FORMULATION OF STIFFENED, FIBER-REINFORCED COMPOSITE PANELS

Craig S. Collier *Principal Structures Engineer

Lockheed Engineering and Sciences Co.NASA Langley Research Center

Hampton, VA

ABSTRACTA method is presented for formulating stiffness terms andthermal coefficients of stiffened, fiber-reinforcedcomposite panels. The method is robust enough to handlepanels with general cross sectional shapes, including thosewhich are unsymmetric and/or unbalanced. Non-linear,temperature and load dependent constitutive material dataof each laminate are used to "build-up" the stiffened panelmembrane, bending, and membrane-bending couplingstiffness terms and thermal coefficients. New thermalcoefficients are introduced to quantify panel responsefrom through-the-thickness temperature gradients. Atechnique of implementing this capability with a singleplane of shell finite elements using theMSC/NASTRANTM analysis program (FEA) is revealedthat provides accurate solutions of entire airframes orengines with coarsely meshed models.

An example of a composite, hat-stiffened panel isincluded to demonstrate errors that occur when an unsym-metric panel is symmetrically formulated as traditionallydone. The erroneous results and the correct onesproduced from this method are compared to analysis fromdiscretely meshed three-dimensional FEA.

NOMENCLATURE

lamina (i = 1, 2, 3) Ei Elasticities

νij Poisson's ratiosα i Thermal expansion coefficients

ijQ Reduced stiffness terms

ijQ Transformed reduced stiffness term

iΦ Reduced thermal stiffness terms

iΦ Transformed reduced thermal stiffness terms σi, єi Stresses and strains

_____________________* Senior member AIAA

This paper is a work of the U.S. Government and is not subjectto copyright protection in the United States.

laminateh Thickness

(i = 1, 2, 3) *

ijQ Effective reduced stiffness terms

*iΦ Effective reduced thermal stiffness terms

panelH DepthSx Unit width of panel corrugation

(i = 1, 2, 3)Mi Materials (particular layup) of the upper

facesheet, coresheet, and bottom facesheet

laminate or panel∆T In-plane temperature gradient∆G Through-the-thickness temperature gradient

(i = 1, 2, 3)hi Distance from the reference plane

Aij, Bij, Membrane, membrane-bending coupling, Dij and bending stiffness terms

Ai α, Bi

α, Membrane, membrane-bending coupling, Di

α and bending thermal force & moment coefs.

NAi α, MSC/NASTRAN FEA membrane,

NBi α, membrane-bending coupling, and bending

NDi α thermal expansion & bending coefficients

(i = x, y, xy)αi, αci Thermal expansion, expansion coupling, δi, δci bending, and bending coupling coefficients

Ei Effective engineering elasticities νij Poisson's ratiosNi, Mi Forces and momentsєi, κi Reference plane strains and curvatures

superscriptsp PanelT ThermalM Mechanical

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INTRODUCTION

External surfaces of many of today's aircraft are designedwith stiffened panels. Some various shaped stiffeningmembers commonly used for panel structural concepts areshown in Fig. 1. The stiffening member provides thebenefit of added load-carrying capability with a relativelysmall additional weight penalty. Though some conceptshave nearly equal stiffness in both directions, moststiffened panel designs provide high bending stiffness inonly one direction. Such unidirectionally designed panelsare easier to manufacture and most applications do notrequire high bending stiffness in both directions. Thermalforces and moments induced from temperature gradientsare smaller for stiffened panels than they are for sandwichtype panels. These reduced thermal loads make themefficient as hot structure on high speed aircraft. By natureof their shapes, stiffened panels are both orthotropic andunsymmetric, even when fabricated with conventionalmetallic materials. (Sandwich panels become unsym-metric whenever the upper and lower facesheets havedifferent stiffness properties, for example due to differentmaterials.) These additional panel behaviors complicatethe formulation of stiffness, thermal expansion, andthermal bending. Quantifying these behaviors isimportant because they significantly alter computed force,moment, curvature, strain, and stress.

Fig. 1 The Formulation can be applied to any stiffened,composite panel concept.

Fig. 2 The Formulation can handle the additionalcomposite material design variables.

Treatment of Composite MaterialsWith the advancement of fiber-reinforced compositematerials, current high speed aircraft designs use thesesame stiffened panel concepts incorporating the newermaterials. These newer materials provide more designvariables to optimize, Fig. 2**, thus improving thechances of minimizing structural weight. By takingadvantage of the beneficial tailoring capability of thematerial, the panel facesheets and coresheet becomeorthotropic by themselves, further complicating stiffness,thermal expansion, and thermal bending formulations.

An expedient solution to formulating composite stiffenedpanel stiffness terms and thermal expansion and bendingcoefficients must be founded on an effective balance be-tween the amounts of lamina and laminate data to include.An attempt to include lamina data into each stiffenedpanel formulation would be too complicated and hencedetrimental to a design process. The challenge is toformulate panel stiffness and thermal expansion andbending behavior without knowing the particular layups orlamina material properties.______________________________________________**Throughout, figures illustrate a trusscore. In this paper the trusscoreis considered to be a stiffened panel because the addition of its bottomfacesheet makes it a more general form of the corrugated shaped panel.

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Purpose of PaperThis paper describes a formulation for stiffness terms andthermal coefficients that accurately handles thesecomplexities. The formulations are introduced inreference 1. Their significance for a typical stiffenedpanel design and an entire aircraft analysis isdemonstrated in references 2 and 3 respectively. Thispaper fully develops the formulations and validates themusing discretely meshed, three-dimensional FEA.

The formulations provide a technique for effectivelyincluding lamina and laminate data in stiffened panelstructural properties, while new thermal coefficientsprovide a means to apply through-the-thickness tempera-ture gradients to a panel, Fig 3. Together they constitute acomplete set of stiffness terms and thermal expansion andbending coefficients for stiffened, fiber-reinforcedcomposite panels. These data are then input to the MSC/-NASTRANTM finite element analysis (FEA) programusing a two-dimensional model that has a single plane offinite elements. See Fig. 4.

2-D FEA results for a typical panel design and loading areshown that demonstrate the differences between resultspredicted for an unsymmetric hat stiffened panel designusing traditional formulation and then correctly predictedusing this method's unsymmetric formulation. Finelymeshed 3-D FEA is used to verify this formulation asimplemented with 2-D coarsely meshed FEA.

Fig. 3 Aerodynamic heating of high speed aircraftproduces two kinds of temperature gradients.

ApplicationsThese formulations are particularly useful for coarselymeshed models of a total structural entity such as anengine or airframe. Models of such large surface areascan only be accomplished with a single plane of shellfinite elements. Too many elements would be necessaryto construct a discrete three-dimensional model thatdefined the panel shapes. 3-D models are desirable

because of their accuracy in capturing unsymmetricstiffness and through-the-thickness temperature gradients

Fig. 4 Accurate results are possible with2-D planar FEM's.

as proven with tests [4]. A planar two-dimensional modelcan capture these same effects by including the completeset of panel thermal and stiffness properties in a shellfinite element. These planar finite element models arewell suited for achieving a multidisciplined designcapability for high speed aircraft.

The formulations of this paper apply to any stiffenedpanel concept and are intended to be coded into computerapplication software. They are being added to the ST-SIZE structural-thermal sizing program [3] which islinked with the MSC/NASTRAN finite element programto provide an analysis and sizing capability that can beiterated automatically until a structure's weight converges.The method could be conveniently applied to other typecodes. The link to FEA is not necessary. However, due tothe level of detail and mathematics involved, thetechniques are not suited for hand analyses. A facility forstoring and retrieving temperature and load dependentlaminate data is necessary for optimization or sizingapplications. The ST-SIZE program uses a materialdatabase.

Limitations and assumptions of the method fall withinthose usually applied in classical lamination theory [5]. Aprimary assumption is that strain variation through thepanel cross section follows the Kirchhoff hypothesis forlaminated plates. This hypothesis maintains that a normalto the midplane remains straight and normal upon paneldeformation and that stresses in the XY plane govern thelaminate behavior. Implications of this hypothesis are: 1)membrane strains vary linearly through the panel crosssection 2) stresses vary in a discontinuous manner throughthe cross section 3) the facesheet laminates are perfectlybonded to the coresheets, and 4) the bonds areinfinitesimally thin and non-shear deformable. Thisimplies that єp

z, γpxz, & γp

yz = 0, in addition to the usualplane stress assumption of σp

z, τpxz, & τp

yz = 0. Forconvenience, matrix terms 16, 26, & 66 are referred to as13, 23, & 33.

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STIFFNESS TERMS AND THERMAL EXPANSION& BENDING COEFFICIENTS

Panel stiffnesses and thermal coefficients are calculatedby extending classical lamination theory to the stiffenedcross section. Stacking sequence and lamina materialproperties are used to calculate orthotropic laminateproperties. These laminate properties are treated as if theywere individual laminae and used in extended classicallamination equations for calculating stiffened panel, ortho-tropic, or more general anisotropic properties. Fig. 5illustrates the technique. Special consideration is given tothe actual shape of the stiffening member and its non-platebehavior. This approach offers the capability to handleunsymmetric and unbalanced stiffened panels. The fullcomplement of membrane, bending, and membrane -bending coupling behaviors are included in the derivationof panel stiffness terms and thermal coefficients.

Fig. 5 Laminate formulation is extended to stiffenedpanels.

A stiffened panel may be formulated with each of itslaminates described with membrane stiffness andmembrane expansion. The benefit of this approach is thatstructural and thermal stiffened panel formulation can beperformed efficiently by not having to work with individu-al ply properties of candidate laminates. Stiffness andthermal properties of candidate laminates are computed inadvance and saved in a materials database. They are thendivided by their thicknesses to arrive at effective stiffnessand thermal terms. Thus, this panel formulation treats thefacesheet and coresheet laminates as if they were laminae.A full explanation of the formulation is presented below.

Stiffness Formulation Laminate: Laminate formulation can be summarizedwith the well known equations of membrane Aij, mem-brane-bending coupling Bij, and bending stiffness Dij,

( ) ( )k1k1k

kijij hhQA −= −=�

η(1)

( ) ( )2k

21k

1kkijij hhQ

21B −−= −

=�

η

( ) ( )3k

31k

1kkijij hhQ

31D −= −

=�

η

These equations differ from others [5,6,7] because of adifferent sign convention. The sign convention of Fig. 5causes the usual terms (hi

k-hik-1) to become (hi

k-1-hik) and

the Bij to be negative. The rest of the laminate stiffnessformulation is the same. Therefore, ijQ are the trans-

formed reduced laminae elasticities. ijQ = Qij[T]4 where[T]4 is a fourth order tensor and Qij are the reducedlaminae elasticities. As an example

( )2112

x11 1

EQνν−

= (2)

Qij are interpolated from a material database using thelaminate's temperature and compression or tension stresscondition.

Panel: Laminate Aij from equation (1) are divided bytheir thicknesses to create new data entities *

ijQ for use intemperature dependent and load dependent panelstiffnesses

(3)

( ) ( )��

���

�−�

�+−��

�= 8h7h3

*ijQ1h0h

1

*ijQAp

ij

��

���

���

�� −�

�+��

�� −�

�−= 28h2

7h3

*ijQ2

1h20h

1

*ijQ

21Bp

ij

��

���

���

�� −�

�+��

�� −�

�= 38h3

7h3

*ijQ3

1h30h

1

*ijQ

31Dp

ij

The hi variables are illustrated in Fig. 5. They are definedas: h0=H/2, h1=h0-t1, h2=h1-Ntt, h3=h1-Ntt/2, h4=0, h8=-H/2, h7=h8+t3, h6=h7+Ntb, and h5=h7+Ntb/2. Shown in Fig.2 are the variables t1, t2, and t3 which are the top facesheet,coresheet, and bottom facesheet thicknesses. Ntt and Ntbare the thicknesses of the coresheet top and bottom joiningnodes. The subscripts 1, 2, and 3 on the *

ijQ termsrepresent the different isotropic materials or compositelayups. Properties of these materials or layups are basedon their non-linear temperature and load dependent data.Equations for longitudinal stiffness terms Ap

11, Bp11, and

Dp11 are expanded to account for additional geometric

variables such as coresheet angle θ, corrugation spacing

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Sx, and widths of the coresheet top and bottom joiningnodes Nwt and Nwb. As an example the equation for Bp

11is shown where the variables are shown in Fig. 5.

(Omitted in this PDF version) (4)

Even though the laminate's X axis is parallel to the panel'scorrugation direction, Fig. 2, the 13 and 23 stiffness termsare non-zero when unbalanced layups are used for thefacesheets. If the bottom facesheet was left off of thepanel, the coresheet's contribution to the 33 terms asdefined by reference 8 is likely to be relatively significant.

The terms ( *ijQ )2t and ( *

ijQ )2b of equation (4) distinguishthe coresheet top and bottom node laminate reducedstiffnesses from the middle coresheet reduced stiffnesses.Even though they are the same material or layup, athrough-the-thickness temperature gradient causes theirproperties to be dissimilar.

Unlike the facesheets, the corrugated coresheet does notbehave as a plate. Its nodes and mid portion strain in thelongitudinal direction like a thin strip of plate or a beam.Because of this, the coresheet does not contribute to thepanel stiffness terms 12, 22, 13, and 23 as omitted fromequation (3). More subtle is the fact that the coresheet'scontribution to panel longitudinal stiffness as included inequation (4) is not effected by plate coupling as identifiedby equation (2). The coresheet's accordion shape allows itto strain unconfined in the transverse direction eliminatingthe need for the familiar plate term (1-ν2). Coresheetlaminate membrane stiffness (A11)2 which includesorthotropic coupling cannot be divided by its thickness toobtain ( *

11Q )2. Instead the coresheet's uncoupled ( *11Q )2

terms equal the effective laminate engineering elasticity,Ex

tyxνxyν111A

xE���

��� −

= (5)

or more generally for unsymmetric or unbalancedlaminates

t111A

1xE −=

(6)

Thermal Coefficient Formulation Laminate: Classical lamination thermal analysis can besummarized by equations from reference 5. First a ply'sthermal stresses are computed in its fiber direction

TQ iijTi ∆= ασ (7)

where i = the 1 and 2 ply expansion coefficients, T = thechange in temperature of the ply, and subscript i refers tothe ply's 1, 2, and 3 directions. Second, the ply's thermalstresses are rotated to the laminate axes using a 2nd ordertensor transformation

[ ]2Ti

Ti Tσσ = (8)

where, on the left i = x, y, & xy; and on the right i = 1, 2,& 3. Third, the ply thermal stresses are integrated withother ply thermal stresses to obtain laminate thermalforces and moments

( ) dz)Z,1()(M,N k

2h

2h

Ti

Ti

Ti −= �

σ (9)

where k = the ply layer and i = x, y, & xy. Equation (9)rewritten to include terms from equations (7) and (8) is

( ) dz)Z,1(]T[)T()Q(M,N 2kkik

2h

2hij

Ti

Ti −∆= �

α (10)

This formulation requires obtaining each ply's propertiesand orientation angle (θ) and performing the integrationincluding the difference between each ply's initialtemperature and its new elevated temperature. Whileintegrating laminae expansion coefficients andtemperature differences through the laminate depthprovides thermal forces and moments, other meansbecome necessary to compute thermal forces andmoments for applications where integration cannot beperformed. It is particularly convenient to be able todefine thermal response of a laminate or panelindependently of its current temperature condition. De-scribed below is a new formulation unlike others[5,6,7,9,10,11] that accomplishes this in two parts. Thefirst is defining equivaSlent plate thermal expansion,bending, and expansion-bending coupling coefficients.The second is identifying two unique gradients such thattheir superposition captures each ply's temperaturedifference.

The first and more common gradient is referred to infor-mally as in-plane (∆T) which designates the laminateschange in temperature at a reference plane. (Strictlyspeaking, an in-plane gradient quantifies the temperaturechange on a surface. This data is captured by the FEMmesh.) The second gradient called through-the-thickness(∆G), defines a linear variation of temperature through a

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SDM Conference, 1993 6

laminate's depth. Therefore a ply's ∆T = f(∆T, Z G) ofthe laminate’s depth. Therefore a ply’s ∆T = f(∆T, Z∆G)of the laminate. By defining the terms Φi = Qijαi and

iΦ = ΦI[T]2, equation (10) can be written as

( ) dz)Z,1)(GZT()(M,N k

2h

2h

iTi

Ti −∆+∆Φ= �

(11)

which, by performing the integration and writing in matrixnotation becomes

(12)

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ��

����

���

���

=��

��

∆−∆

−−�

=Φ−−

=Φ−

−−�

=Φ−−−

GT

3kh

31kh

1k ki3

12kh

21kh

1k ki2

1

2kh

21kh

1k ki2

1

kh1kh1k ki

TiM

TiN

ηη

ηη

Equation (12) is abbreviated to

��

���

∆−∆

��

���

�=

��

��

��

GT

DBBA

TiM

TiN

ii

iiαα

αα

(13)

by identifying thermal force Aiα , moment Di

α , and force-moment coupling Bi

α coefficients(14)

( ) ( )k1k1k

kii hhA −Φ= −=�

ηα

( ) ( )2k

21k

1kkii hh

21B −Φ−= −

=�η

α

( ) ( )3k

31k

1kkii hh

31D −Φ= −

=�η

α

Note the similarity of these to the formulation of the Aij,Dij, and Bij stiffness terms of equation (1). Other than the

substitution of ijQ with iΦ , the only other differencebetween the equations is the treatment of X & Yorthotropic coupling. Stiffness formulation definesseparate terms to account for directional coupling.Thermal formulation defines orthotropic coupling at thelamina level as shown in equation (7), reference 5, andwith the Φi term of this method. Doing this reduces whatwould have been 3x3 matrix coefficients into 3x1 vectorcoefficients.

Thermal forces and moments are added to mechanicalforces and moments to produce the system equations

(15)

���

���

++

��

�=

���

���

+���

���

=���

���

Ti

Mi

Ti

Mi

ijij

ijijTi

Ti

Mi

Mi

i

i

DBBA

MN

MN

MN

κκεε

These combined thermomechanical forces and momentsfollow the same rules and are used the same as mechanical

only loading. Consequently, thermal loads multiplied bythe combined 6x6 inverted stiffness matrix providesthermal strains and curvatures

���

���

��

�=

���

���

Ti

Ti

1

ijij

ijijTi

Ti

MN

DBBA

κε

(16)

By adding equation (13) to equation (16), thermal growthcan be rewritten as

���

���

∆−∆∆−∆

��

�=

���

���

GDTBGBTA

DBBA

ii

ii

1

ijij

ijijTi

Ti

αα

αα

κε

(17)

By setting ∆G = 0, two in-plane gradient, thermal expan-sion coefficients are identified

���

���

��

�=

���

���

α

α

αα

i

i

1

ijij

ijij

ci

i

BA

DBBA

(18)

By setting ∆T = 0, two through-the-thickness gradient,thermal bending coefficients are introduced

���

���

��

�=

���

���

α

α

δδ

i

i

1

ijij

ijij

i

ci

DB

DBBA

(19)

The subscript c indicates expansion-bending coupling.These coupling coefficients delineate the unsymmetricresponse due to ∆T and ∆G temperature gradients. αi andδci quantify thermal strain for a ∆T and ∆G respectively,and αci and δi quantify thermal curvature for a ∆G and ∆Trespectively.

GTi

ci

ci

iTi

Ti ∆

���

���

−∆���

���

=���

���

δδ

αα

κε

(20)

All 12 of these unique thermal coefficients and both of thein-plane and through-the-thickness temperature gradientscontribute to thermal forces and to thermal moments.Together they explicitly quantify unsymmetric and unbal-anced response caused by unsymmetric and unbalancedstiffness and by unsymmetric and unbalanced thermalexpansion and bending.

���

���

∆−∆∆−∆

��

�=

���

���

GTGT

DBBA

MN

ici

cii

ijij

ijijTi

Ti

δαδα

(21)

In contrast to the familiar equation (10), equations (13),(20), and (21) are able to quantify thermal responseindependently of current laminate temperature profile.This is accomplished by taking the ply's T and α1 and α2

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7 SDM Conference, 1993

out of the integration and supplying instead laminateexpansion, bending, and expansion-bending couplingcoefficients and in-plane and through-the-thicknesstemperature gradients. In essence, these thermalcoefficients are smeared equivalent plate values which arebased on a through the depth integration. These smearedequivalent plate stiffnesses and thermal coefficients canthen be accurately used for FEA.

Panel: Panel implementation of thermal laminationtheory follows the approach of panel stiffness formulation.As a consequence, non-linear, temperature and loaddependent laminate force coefficients, Ai

α , are divided bylaminate thickness to give iΦ * for use in developingpanel force and moment thermal coefficients. Note howtheir formulation follows panel stiffness formulation.

( ) ( ) ( ) ( )[ ]873*i101

*i

pi hhhhA −Φ+−Φ=α (22)

( ) ( ) ( ) ( )[ ]28

273

*i

21

201

*i

pi hhhhB

2

1 −Φ+−Φ−=α

( ) ( )���

��

� −��

��Φ+−�

��Φ= 3

8373

*i

31

301

*i

pi hhhh

31D α

Equations for the longitudinal coefficients A1pα , B1

pα , andD1

pα are expanded to account for the corrugated shape asexemplified with equation (4). The complete set of panelthermal force and moment coefficients are then used todefine temperature dependent panel expansion, αp & αc

p,and bending, δp & δc

p, coefficients following equations(18) and (19).

The coresheet's corrugated shape allows it to expandfreely in the transverse direction precluding it fromcontributing to the panel transverse or shear thermal forceand moment coefficients: A2

pα , A3pα , B2

pα , B3pα , D2

pα ,and D3

pα. More subtle however, is the way the corrugatedcoresheet contributes to the longitudinal force andmoment coefficients, A1

pα , B1pα , and D1

pα. Since its nodesand mid section expand almost entirely unrestrained in thetransverse direction as separate strips of plates or beams,its thermal formulation must not contain the embedded Xand Y directional coupling that is innate to classicallamination thermal formulation. This need for a non-platethermal formulation is consistent with the need for a non-plate stiffness formulation as described for equations (5)and (6). The solution for uncoupling the longitudinalstiffness from the transverse stiffness was to omit thefamiliar Poisson's plate term (1-ν2) from the coresheetformulation. Uncoupling of a coresheet's laminatelongitudinal thermal behavior from its transverse behaviorcan be accomplished two ways. First the behavior ofstacked lamina can be computed omitting laminaproperties that contribute to the laminate Y axis(transverse direction). Unfortunately this requiresadditional layup computations and saving more data in a

material database. Another way to arrive at a laminate's

uncoupled iΦ * is by

xx

*1 Ε=Φ α (23)

where αx is from equation (18) and Ex is from equation (6).

APPLICATION WITH FEA

In-plane and through-the-thickness temperature gradientscan be correctly applied and solved foranisotropic/orthotropic, unsymmetric, and unbalancedlaminates or stiffened panels with a single plane of shellelements with the MSC/NASTRAN FEA program. Thisis accomplished by including the full complement ofsmeared equivalent plate stiffness matrices and thermalexpansion and bending coefficient vectors in the FEMdata deck. Stiffness matrices for membrane, bending, andmembrane-bending coupling are entered directly intoMSC/NASTRAN with only minor adjustments [12].Thermal expansion and bending coefficient vectors formembrane, bending, and membrane-bending couplingcannot be entered into MSC/NASTRAN without majoradjustments to their formulation [1].

MSC/NASTRAN Thermal CoefficientsA technique is introduced to identify nine unique MSC/-NASTRAN thermal coefficients, NAi

α , NBiα, and NDi

α inorder to compute thermal loads. MSC/NASTRAN com-putes thermal forces and moments by

TNBBTNAAN iijiijTi ∆−∆= αα (24)

GNDDTNBBM iijiijTi ∆−∆= αα

Where the stiffness matrices and thermal coefficientvectors shown can represent either a laminate or astiffened panel. MSC/NASTRAN uses the same signconvention as the formulation of this paper. Theircoefficients can be found by equating equation (24) toequation (13) and solving for ∆T and ∆G separately

ααi

1iji AANA −= (25)

ααi

1iji BBNB −=

ααi

1iji DDND −=

Because equation (24) does not join thermal force calcula-tion with moment calculation, the ensuing stiffnessmatrices of equation (25) are inverted separately. Theseinverted 3x3 matrices are unlike the 6x6 inverted stiffnessmatrix of the other equations.

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SDM Conference, 1993 8

Temperature and load dependent stiffnesses and thermalcoefficients are placed on the MSC/NASTRAN MAT2data record. A change in the panel's bulk temperature isentered in the FEA by supplying the reference temperatureon the MAT2 record and the loadcase dependenttemperature on the TEMPP1 record. The effect of in-plane temperature gradients is then captured with themodel's discretization. Loadcase dependent through-the-thickness gradients are entered on an element basis withthe TEMPP1 record.

Model Reference PlaneFEM grid points are customarily located at a panel'smidplane as depicted in the 2-D FEM of Fig. 4. Typicallya structural analyst will choose the aerodynamicallydefined outer mold line (OML) as his FEM's surface.This causes the midplane of the structural surface to be inerror; however, this is ordinarily done due to the difficultyof offsetting CAD generated lofted surfaces. This errorcauses an unconservative calculation of a structure'sbending stiffness. While perhaps not significant for anairframe fuselage, this inaccuracy is substantial for wingsand other shallow structural components. Another short-coming of this approach, as displayed in Fig. 6, is thateven though an analyst might go through the effort ofoffsetting his model properly, his offset is usually basedon an assumed panel depth that is likely to change asstrength and stability analyses are performed.

Fig. 6 The best choice of a FEM reference plane is thesurface outer mold line.

A solution to these shortcomings is to always use theOML for the location of FEM grid points. This can beaccurately achieved by redefining the panel's referenceplane from the midplane as established in Fig. 5 to theOML as illustrated in Fig. 7. By using the OML as thepanel's reference plane, the hi variables are calculated as:h0=0, h1=-t1, h2=h1-Ntt, h3=h1-Ntt/2, h4=-H/2, h8=-H,h7=h8+t3, h6=h7+Ntb, and h5=h7+Ntb/2. All of thepreviously defined equations of stiffness and thermalcoefficients can still be used exactly as formulated.Higher panel bending stiffnesses will be calculated thisway but they will be balanced out with higher membrane-bending coupling stiffnesses. Note that a symmetric panelwill now have non-zero membrane-bending coupling data.

Fig. 7 The surface outer mold line can be convenientlyused as the FEM reference plane.

Regardless of choice of reference plane, the resulting FEAthermomechanical forces and moments are used tocompute panel strains and curvatures as defined withequation (15). Strains at any location can easily beestablished since strains are formulated to vary linearlythrough panel depth. A laminate strain is the summationof the panel's reference plane strain and the additive orsubtractive contribution of the curvature

{ } mpi

pimi Hκεε −= (26)

where Hm is the distance between the reference plane andthe panel depth location m.

LOAD DEPENDENT RESIDUAL STRAINSResidual panel strains and stresses caused by thermalgrowth are resolved on a laminate basis for each loadcase.They develop in stiffened panels when the panel laminateswant to elongate non-uniformly when heated. Because thepanel laminates cannot act independently, they developresidual strains and stresses when forced to strain togetheras a unit of the panel. Panel curvature dictates that alllaminate strains follow its through-the-depth strain profile.This profile is linear due to Kirchoff's hypothesis that anormal to the midplane remains straight and normal uponpanel deformation. The residual strain is the differencebetween the strain that occurs in the laminate when madea segment of the stiffened panel's linear strain profile, andthe strain that occurs in the laminate when allowed tothermally grow unattached to the panel.

2-D FEA is able to use smeared equivalent plateproperties for a stiffened panel because of this "planesections remain plane" hypothesis. In principle, duringFEA, panel laminates strain together as a unit providing alinear strain profile through-the-depth, and thus do notinclude residuals. In order to quantify a laminate's"design-to" strains, its residual strains must also bequantified and added to FEA computed strains. Fig. 8summarizes this process.

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9 SDM Conference, 1993

Fig. 8 Three types of computed strain contribute tolaminate "design-to" strains.

Laminate strains from laminate "free-body" thermalgrowth, (left top box of Fig. 8) are computed with eitherequation (16), (17) or (20). In this case ∆T is the lami-nate's change in temperature which includes both panel∆T and ∆G. Laminate strains from panel "free-body"thermal growth are computed first by using, again,equation (16), (17), or (20) (this time with panel datainstead of laminate data) and second, by using these panelstrains and curvatures with equation (26) to obtainlaminate strains. Finally, FEA computed forces and mo-ments that solve the internal load redistribution of theindeterminate structure are used in conjunction withequations (15) and (26) to arrive at thermomechanicallaminate strains. All three of these computed thermalstrains are required to determine "design-to" laminatestrains.

TYPICAL STIFFENED PANEL ANALYSIS

The fuselage and wing skins of high speed vehicles arecommonly designed with stiffened panels. A hatstiffened, fiber-reinforced, metal matrix composite is usedfor this example, Fig. 9. Metal matrix composites arechosen for their high temperature capability, some havinga service use up to 1300oF. When allowing a stiffenedpanel to reach these high temperatures, its large mem-brane, bending, and membrane-bending coupling thermalresponse must be analytically quantified.

Panel Design and TemperaturesThe panel cross section shape, dimensions, and laminatelayups are those which are commonly produced bystructural sizing optimization codes. The hat shape isfabricated by brazing the facesheet to the corrugatedcoresheet. Shown to the left of the section is thetemperature profile which is typical of those analyzed forhigh speed flight. The panel's midplane temperature is

625 F. while its hottest point is on top of the facesheet(850 F.) and its coldest point is on the bottom of thecoresheet (400 F.). These laminate temperatures are wellwithin the material's limit. The shaded rectanglerepresents a uniform in-plane gradient of 555 F (625 F-70F). The double shaded triangles represent a through-the-thickness gradient of 300 F/in. By superimposing the twogradients, the variation of temperature through the panel'sdepth is known, as illustrated with the bold line. Thefacesheet's average temperature of 842.5 F. and thecoresheet's average temperatures of 832.75 F., 617.5 F.,and 402.25 F. are used for interpolating the materialdatabase. Laminate material properties are also retrievedfrom the database according to compression or tensionstress conditions. Compression elasticities of metal matrixcomposites are approximately 35% higher than theirtension elasticities at elevated temperatures. These load

and temperature dependent laminate data *ijQ and

*iΦ are

used for formulating panel stiffness terms and thermalcoefficients as defined with equations (3), (4), and (22).

I/FCSHT2/Collier

FACESHEET [0/45/-45/90/0]s t = .05"

850° F

CORESHEET[0/90/0] t = .015"

400° F

82°

0.8 "

Room temperature = 70° FIn-plane temperature gradient (∆T= 555° F)Through-the-thickness temperature gradient (∆G = 300° F/in)

Metal matrix composite material

2.0 "

1.5 "

TOP NODE

BOTTOM NODE

���������842.5° F

832.75° F

625.0° F

617.5° F

402.25° F

Panel Midplane

Coresheet Midplane

∆T70°F

�������

A Typical Hat Stiffened Panel and Temperature Profile

Fig. 9 Typical hat stiffened panel andtemperature profile.

Panel DataThe typical corrugated stiffened panel has the followingproperties when this general anisotropic formulation isused. The equivalent orthotropic plate data shows the"13" and "23" terms of the stiffness matrices and the "3"terms of the thermal coefficient vectors to be zero becausethe laminates are balanced. If the laminates were not bal-anced,

���

���

=2010100.00.0

0.06971342597400.02597401400599

Apij

���

���

−−−−−

=1457320.00.0

0.05054221883110.0188311549085

Bpij

���

���

=1056980.00.0

0.03665771365800.0136580595340

Dpij

The hat panel stiffness terms.

“Design-To” Laminate Strains

�r = �PFB - �LFB

Laminate strains fromLaminate “free-body”

thermal growth

Laminate strains fromPanel “free-body”

thermal growth

FEA computed thermomechanicallaminate strains

“Design-To” Laminate Strains

� = �FEA + �r

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SDM Conference, 1993 10

6pi 10

0.05833.47562.3

��

��

��

��

=α 6pci 10

0.01006.02700.0

��

��

��

��

�−=α

6pi 10

0.05649.48055.3

��

��

��

��

=δ 6pci 10

0.00353.00946.0

��

��

��

��

�−=δ

The hat panel thermal expansion and bending coefficients.

6pi 10

0.01708.45807.6

A −

��

��

��

��

=α 6pi 10

0.00238.30726.3

B −

��

��

��

��

−−

6pi 10

0.01931.29344.2

D −

��

��

��

��

The hat panel thermal force and moment coefficients.

or if an off axis reference direction was used for the panel,these terms would not be zero and fully populated, equiva-lent plate, anisotropic data would have been produced.The midplane was used as the reference plane in lieu ofthe OML for ease of comparison to traditional methodswhich use neutral axes as reference planes. Note that theabsolute values of the membrane-bending coupling Bijterms are as large as the bending Dij terms and that all ofthe thermal coefficients are different values whichindicates the significance of the unsymmetric nature of thepanel. The highest thermal coefficient is 4.6-6 (in-F /in),less than values of typical metallic materials. Conse-quently, the behavior of this panel with isotropic, metallicmaterials would be similar, yet amplified.

Panel ResultsThermal forces, moments, strains, and curvatures werecomputed to compare results of different analysismethods. It was desired to ground the various analyticalresults to test data. However the author was unable toobtain applicable test data for a stiffened, metal matrixcomposite panel that had a through-the-thicknesstemperature gradient applied to it or some other in-servicetype temperature distribution. In lieu of test data, the bestpossible structural-thermal analysis was performed toarrive at baseline results. Baseline strains, curvatures,forces, and moments were acquired by rigorous analysisof each of the panel's laminates which included executingclassical lamination codes. Each laminate's thermalresponse was used to assemble the panel's responsemaintaining the equilibrium of forces and moments andthe simultaneous compatibility of the six strain andcurvature degrees of freedom of the panel.

3-D FEA of the panel was performed to provide a checkof the baseline results. A finely meshed model consistingof 2400 MSC/NASTRAN CQUAD4 shell elements,similar to the 3-D FEM portrayed in figure 4, was built

and executed. Since elements were included to model thecoresheet corrugation pattern, the unsymmetric nature ofthe panel was captured. Each element used temperaturedependent laminate stiffnesses [A] & [D] and thermalcoefficients {Aα} & {Dα} generated by classicallamination codes. This data was input withoutmodification directly into the MAT2 material data cards.

Two different types of 2-D FEA were performed for thepanel. The first type used this formulation and is laterreferred to as the unsymmetric 2-D FEA. The otheranalysis uses traditional stiffened panel formulation and isreferred to as the symmetric 2-D FEA because it ignoresunsymmetric behavior. The only difference between thesemodels is the stiffness and thermal coefficient input data.All other model data is identical. These models aredifferent from the 3-D FEM in that they have smearedequivalent plate properties and do not have elements thatmodel the coresheet corrugation. However the meshdensity (800 elements) and connectivity of the facesheetsurface is the same.

Two different panel boundary conditions are analyzed.The first condition prevents the panel from straining orcurving upon applied temperature loads. This conditionmay be visualized as restraining the panel's growth withinrigid walls. Consequently the full magnitude of inducedthermal forces and moments develop, Table 1. Thesecond boundary condition allows full thermal growth byconstraining the panel in the FEM only at the center gridto prevent rigid body motion. Thermal forces andmoments are zero for this case, Fig. 10, 11. These twoboundary condition cases are the extremes of in-servicepossibilities and therefore band the level of accuracyexpected for actual conditions. Analysis of theorthotropic panel with these boundary conditions causeγp

xy, κpxy, Nxy, and Mxy to be zero, and to not be part of

the solution. If the panel happened to be anisotropic, or beconstrained in a way to develop shear loads, theseresponses would come into play.

Table 1 Comparison of computed panel forces andmoments to the temperature gradients.

The exact match between 2-D unsymmetric FEAcomputed forces and moments and the baseline results isnot surprising. This happens because the constrainedboundary conditions (rigid walls) do not allow the elementshape functions to come into play. The agreement isactually a substantiation of the defined MSC/NASTRANforce and moment equations (24) and thermal coefficients

Baselineresults

2-D FEAUnsymm.

3-D FEA 2-D FEASymm.

Nx -4574 -4574 -4595 -3652Ny -3222 -3222 -3223 -2315Mx 2586 2586 2581 1891My 2336 2336 2337 1678

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11 SDM Conference, 1993

(25). The second boundary condition where growth ispermitted is more of a measure of the 2-D FEM's abilityto capture the panel's unsymmetric behavior. In somerespects, the good agreement to the 3-D FEM's doublecurvature deformed shape, Fig. 10, gives credence to theKirchoff hypothesis of plane sections remain plane andlinear strain distribution through plate thickness (in thiscase panel depth) as implemented with the smearedequivalent plate approach. The agreement also confirmsthe application of this formulation to unsymmetricstiffened composite panels and the facility to capture boththe in-plane and through-the-thickness temperaturegradients with the newly defined expansion, bending, andexpansion-bending coupling thermal coefficients.Because symmetric 2-D FEA omits this additional data, itsdeformed shape caused by the uniform temperatureincrease exhibits zero curvature.

Symmetric 2-D FEA of traditional methods significantlyunder predicts both mechanical and thermal panelresponse. This is due to dissimilarities in the formulationsof panel stiffness terms and thermal coefficients.Traditional formulations currently being practiced [13,14]require treating each layup type with differentapproximations, calculating neutral axes, effective thick-nesses, effective areas (EA), effective moment-of-inertias(EI), effective elasticities, etc. in both the X and Y direc-tions separately using modular ratios and the parallel axistheorem; thus treating the panel as two detachedperpendicular beams. Doing this omits panel straincompatibility because the X-face unit width section is notcoupled with the Y-face unit width section. Attempts tocouple the directions would be in error since the X-faceneutral axis does not lay on the Y-face neutral axis.Finally, membrane-bending coupling of unsymmetric stiff-ness, [Bp], and the membrane-bending coupling of un-symmetric thermal expansion and bending, {αc

p}, {δp},{δc

p}, and {Bpα } are missing.

In order to get the most accurate solution with the 2-Dsymmetric FEA, the FEM input data included X & Y platecoupling as produced by the 3x3 membrane and bendingstiffness matrices and in the six expansion and bendingthermal coefficients. The 2-D symmetric FEM alsocontained the through-the-thickness gradient. This FEA,although, was not able to include load dependent residualstrains.

Panel laminate strains graphed in Fig. 11 were analyzedfor strength and stability. The incorrect 2-D symmetricmargin-of-safety is +0.23 indicating the panel is slightlyoversized. Correct unsymmetric analysis produces a -0.06margin, signifying panel failure.

The reported moments of Table 1 are those values foundat the panel midplane. Unlike forces, the magnitude ofmoments vary according to the location of their referenceplanes. Traditional symmetric analysis uses the neutral

axes for reporting moment. Accordingly, at the X-faceneutral axis Mx = 792 and at the Y-face neutral axis My= 0.0. However, moments have limited usefulness at theneutral axes because they cannot be coupled.

The unsymmetric 2-D FEA actually compares better to thebaseline results than the 3-D FEA. This could be causedby an innate shortcoming of the 3-D FEM which is due tothe top joining node of the coresheet and the facesheetbeing modeled as if they lay on the same plane. Theactual

Fig. 10 FEA deformed plots of the panel response to auniform temperature increase of 555 F.

Fig. 11 Comparison of computed panel "design-to"strains (with residuals) to the temperature gradients.

separation between them is equal to t1/2 + Ntt/2, see Fig.2. This distance is not contained in the 3-D FEM becausethe same grid points are needed for connecting thecoresheet elements to the facesheet elements. Aftermaking minor adjustments to account for the top of thecoresheet being modeled on the same plane as thefacesheet, the 3-D FEM produces nearly the same resultsas the baseline. However, the 36" square hat panelrequires at least 560 elements in order to capture thecorrugated shape of the coresheet, (one element is neededto span the panel depth). With 560 elements, thermal Mxhas a -2.8% error. By using an equivalent plate 2-D FEM,only one element is needed for the 36" panel while stillmaintaining 0% error.

CONCLUSIONS

Formulations presented in this paper provide the capabili-ty to represent a stiffened composite panel of any

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SDM Conference, 1993 12

geometrical cross sectional shape as an equivalentanisotropic plate. The formulations are able to accept anyapplied thermomechanical load combination and quantifyload dependent residual thermal strains. This capability isfounded on the Kirchoff hypothesis of linear straindistribution through a plate's thickness. The benefit of thehypothesis is provided by:

* extending classical lamination theory to theformulation of stiffened panels

* defining explicit thermal coefficients of membrane,bending, and membrane-bending coupling for bothin-plane and through-the-thickness temperaturegradients

* including temperature dependent, load dependent,non-linear material data in the constitutive

formulation of each laminate's stiffnessterms and thermal coeffi cients that build-upthe stiffened panel properties

A major benefit of formulating stiffened panels withsmeared equivalent plate properties is that a coarselymeshed 2-D FEM with a single plane of shell finite ele-ments can be used to accurately analyze complex thermo-mechanically loaded structures. Traditional methods offormulating equivalent plate panel stiffness and thermalcoefficients, though intuitive, are difficult to use for awide possibility of applications. More importantly theygive incorrect results as demonstrated. 2-D FEA that usesthis formulation correlates very well with 3-D FEA.

ACKNOWLEDGEMENTS

This work was performed for the Systems Analysis Of-fice/National Aero-Space Plane Office under NASAcontract NAS1-19000 at Langley Research Center inHampton, Virginia.

REFERENCES

1 Collier, C.S., COMPOSITE STIFFENED PANELS:Part 1 - Stiffness, Thermal Expansion, and ThermalBending Formulation for Finite Element Analysis, NASPCR 1134, April 1992

2 Collier, C.S., "A General Method for Structural Analy-sis and Sizing of Composite Stiffened Panels Using2-D Finite Element Models", NASP Technology Review,Monterey, CA, April 21-24, 1992, Paper No. 212

3 Collier, C.S., "Structural Analysis and Sizing of Stiff-ened, Metal Matrix Composite Panels for HypersonicVehicles", AIAA 4th International Aerospace Planes

Conference, Orlando, FL, December 1-4, 1992, Paper No.AIAA-92-5015

4 Percy,W.C. and Fields,R.A., "Buckling of Hot Struc-tures", 8th NASP Symposium, Monterey, CA, March 26-30, 1990

5 Jones,R.M., Mechanics of Composite Materials, Hemi-sphere Publishing Corporation, Washington, DC, 1975

6 Halpin,J.C., Primer on Composite Materials: Analysis,Technomic Publishing Company, Inc., Lancaster, PA,1984

7 Tsai,S.W., Mechanics of Composite Materials, Part II-Theoretical Aspects, Technical Report AFML-TR-66-149, Part II, Nov. 1966

8 Stroud,J.W. and Arganoff,N., Minimum-Mass Designof Filamentary Composite Panels Under CombinedLoads: Design Procedure Based on Simplified BucklingEquations, NASA TN D-8257, Oct. 1976

9 Hahn,H.T. and Pagano,N.J., "Curing Stresses inComposite Laminates", J. Composite Materials, Vol.9,1975 ,p.91

10 Dennis,S.T., Unsymmetric Laminated Gr/EpComposite Plate and Beam Analysis for DeterminingCoefficients of Thermal Expansion, M.S. Thesis MIT,Feb. 1983.

11 Ishikawa,T. and Chou,T.W., "In-plane Thermal Expan-sion and Thermal Bending Coefficients of FabricComposites," Proceedings of the ASME Winter AnnualMeeting, Nov. 14-19, 1982, p.115-120

12 The MacNeal-Schwendler Corporation, MSC/-NASTRAN User's Manual, Vol 1, Version 67, TheMacNeal-Schwendler Corporation, Los Angeles, CA,Aug. 1991

13 Advanced Composites Design Handbook, The BoeingCompany, Seattle, WA, Sept., 1985

14 Haftka,R.T., Gurdal,Z., and Kamat,M.P., Elements ofStructural Optimization, Kluwer AcademicPublishers, The Netherlands, 1990

TRADEMARKS

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