Composite Structures 92 (2010) 1614–1619

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Process-induced deformation of composite T-stiffener structures

Chensong Dong *

Department of Mechanical Engineering, Curtin University of Technology, GPO Box 1987, Perth, WA 6845, Australia

a r t i c l e i n f o

Article history:Available online 24 November 2009

Keywords:CompositeT-stiffenerDeformation

0263-8223/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.11.026

* Tel.: +61 8 92669204; fax: +61 8 92662681.E-mail address: c.dong@curtin.edu.au

a b s t r a c t

Composite T-stiffener structures are widely used structural elements in weight sensitive structural, aero-space and marine applications for the purpose of weight reduction. A study on the process-induced defor-mation of composite T-stiffener structures is presented in this paper. The deformation was calculatednumerically by Finite Element Analysis (FEA), and the resulting displacements show that the deformationcan be represented by a single angular displacement: the spring-in of the skin. A parametric study wasconducted by Design of Experiments (DOE) and FEA to investigate the influence of design parameterson the spring-in of the skin. It is shown that the spring-in of the skin increases with the radius andthe bonding length, and decreases with the fiber volume fraction.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Composites offer the advantages of low density, high strength,high stiffness to weight ratio, excellent durability, and design flex-ibility. Composite T-stiffener structures are widely used structuralelements in weight sensitive structural, aerospace and marineapplications for the purpose of weight reduction. They are oftencalled integrated structures because the parts connected by thefasteners are replaced by molded parts. These integrated structuresare often made of flat or curved panels with co-cured or adhesivelybonded frames and stiffeners.

T-stiffener composite structures can be made by the prepreg/autoclave or the resin transfer molding (RTM) process. The pre-preg/autoclave process can be further categorized into three differ-ent processing methods: co-curing, co-bonding and secondarybonding [1]. When the RTM process is used, the whole part ismolded as one piece or the spar and skin are molded separatelyand bonded together with an adhesive [2].

Anisotropy, the main characteristic of composite materials, is anadvantage from the perspective of structural design, but is also amain cause of process-induced deformation. Past research on theprocess-induced deformation of composites was mainly focusedon the spring-in of angled parts, i.e. the reduction in the enclosedangle. The spring-in of composites was either studied by elasticor viscoelastic models. When the elastic models were used, bothanalytical and numerical methods were developed. Hahn andPagano [3] performed an elastic analysis of the residual stressesin a thermoset matrix composite. Radford and Diefendorf [4]developed a simple mathematical formula to predict the spring-

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in of curved shaped parts, which was used by Huang and Yang[5] in their experimental studies. Kollar [6] presented an approxi-mate analysis of spring-in. Jain and Mai [7,8] developed a mechan-ics-based model using modified shell theory. For a more complexshape, e.g. T-stiffener, numerical methods are often needed. Wanget al. [9] conducted a finite element analysis of spring-in usingABAQUS. Ding et al. [10] developed a 3-D finite element analysisprocedure to predict spring-in resulting from anisotropy for boththin and thick angled composite shell structures.

The mechanical behavior of composite materials is betterrepresented by a viscoelastic model. Some studies calculated theresidual stress developed during the cooling down stage by ther-mo-viscoelastic models [11–13]. Others [14–20] also addressedthe residual stresses developed before cooldown, i.e. during thecuring process. White and Hahn [14,15] studied the residual stressdevelopment during the curing of thin laminates by a 2-D finitedifference thermo-chemical model and a generalized plane-strainfinite element model. Li et al. [16] used a plane-strain, linearly elas-tic finite element model with temperature-dependent matrixproperties to analyze the evolution of residual stresses in graph-ite-PEEK composites during curing. Wiersma et al. [17] developeda thermo-elastic model and extended it into a thermo-viscoelasticmodel. A plane-strain finite element process model COMPRO wasdeveloped to simulate the spring-in and warpage in the autoclaveprocess [18,19]. Zhu et al. [20] developed a fully 3-D coupled ther-mo–chemo-viscoelastic finite element model to simulate the heattransfer, curing, and residual stress development during the man-ufacturing cycle of thermoset composite parts.

From the literature, it can be seen that although the process-induced deformation of composites such as spring-in was studiedeither analytically or numerically, the deformation of T-stiffenerstructures has not received much attention. Both elastic and

C. Dong / Composite Structures 92 (2010) 1614–1619 1615

viscoelastic models were developed for calculating the process-in-duced deformation of composites. Compared with elastic models,viscoelastic models are more complicated to use. During the curingprocess, approximately 60% of the total shrinkage occurs prior tothe gel point [21]. In addition, after demolding, the composite isusually post-cured at an elevated temperature above the glasstransition temperature Tg. During this process, the stresses inducedby curing can be significantly relaxed [22–24]. Thus, only a smallfraction of curing shrinkage contributes to the residual stress andwarpage development. For this reason, an elastic approach wasemployed in this study. The deformation was computed by FEA,and the resulting displacements show that the deformation canbe represented by a single angular displacement: the spring-in ofthe skin. A parametric study was conducted by Design of Experi-ments and FEA to investigate the influence of design parameterson the spring-in of the skin. The paper is organized as follows: Sec-tion 2 presents the modeling of deformation; the deformation ischaracterized in Section 3; the parametric study is presented inSection 4 and finally the conclusions are given in Section 5.

2. Deformation modeling

2.1. Governing equations

In order to model the deformation, the material properties ofcomposites need to be derived. A composite part normally consistsof a number of laminas stacked in a certain sequence. The orienta-tion of an individual lamina is represented by h, as shown in Fig. 1.

In the global coordinate system, the stress–strain relationship isgiven by the constitutive law for a monotropic material [25], i.e.

�C11�C12

�C13 0 0 �C16

�C12�C22

�C23 0 0 �C26

�C13�C23

�C33 0 0 �C36

0 0 0 �C44�C45 0

0 0 0 �C45�C55 0

�C16�C26

�C36 0 0 �C66

2666666664

3777777775

exx � axxDTeyy � ayyDT

ezz � azzDT

cyz

cxz

cxy � 2axyDT

2666666664

3777777775¼

rxx

ryy

rzz

syz

sxz

sxy

2666666664

3777777775

ð1Þ

where ½�Cij� is the stiffness matrix; exx, eyy, ezz, cyz, cxz, and cxy are thestrain; axx, ayy, azz, and axy are the CTE; DT is the temperature differ-ence; and rxx, ryy, rzz, syz, sxz, and sxy are the stress. In order to cal-

Fig. 1. Definition of principal material axes and loading axes for a lamina.

culate the deformation, the stiffness matrix and CTE need bederived from the lamina properties.

2.2. Lamina properties

The effective lamina properties of composites are dependent onthe constituent properties and fiber volume fraction Vf. Generally,the properties of fibers are independent of temperature. As for res-ins, the CTE and Poisson’s ratio is constant below Tg while the elas-tic modulus is temperature-dependent. The mechanical propertiesof an epoxy resin from Dynamic Mechanical Analysis (DMA) are asshown in Fig. 2, from which it is shown that the modulus decreasesapproximately linearly from the room temperature to Tg.

In this study, the materials are assumed to be AS4 graphite fi-bers and epoxy resin. Their properties and CTE are shown in Table1.

Based on these constituent properties, the lamina properties,including the longitudinal modulus E11, the transverse modulusE22 or E33, and the shear moduli G12, G13, and G23, are derived byHashin’s model [25] for two temperatures 20 �C and 150 �C,respectively. These lamina properties are used to derive the stiff-ness matrix �C. At any temperature T between 20 �C and 150 �C,�Cij is given by:

�Cij;T ¼�Cij;20ð150� TÞ þ �Cij;150ðT � 20Þ

130ð2Þ

2.3. Coefficient of thermal expansion

The longitudinal CTE of the lamina is given by:

a11 ¼afLEfLVf þ amEmð1� Vf Þ

EfLVf þ Emð1� Vf Þð3Þ

where afL is the longitudinal CTE of fiber; am is the CTE of resin; EfL isthe longitudinal modulus of fiber; and Em is the modulus of resin.

The transverse CTE of the lamina is calculated by Hashin’s con-centric cylinder model [27].

a22 ¼ a22 þ ð�S12 � S12Þ½ðafL � amÞP11 þ 2ðafT � amÞP12�þ ð�S22 � S22Þ½ðafL � amÞP12 þ ðafT � amÞðP22 þ P23Þ�þ ð�S23 � S23Þ½ðafL � amÞP12 þ ðafT � amÞðP22 þ P23Þ� ð4Þ

where �Sij and Sij are the effective and average composite compli-ance; afT is the transverse CTE of fiber; and Pij is given by:

Fig. 2. Modulus of epoxy vs. temperature.

Table 1Material properties of AS4 fiber and epoxy resin [26].

EfL (GPa) EfT (GPa) Gf (GPa) GfTT (GPa) mf mfTT afL (10�6 �C�1) afT (10�6 �C�1)

AS4 fibres235 14 6.917 5 0.2 0.4 �0.4 18

Em at 20 �C (GPa) Em at 150 �C (GPa) mm am (10�6 �C�1)

Epoxy2.581 1.620 0.265 64

Fig. 4. Stacking sequences.

1616 C. Dong / Composite Structures 92 (2010) 1614–1619

P11 ¼ ðA222 � A2

23Þ=det AP12 ¼ ðA12A23 � A22A12Þ=det A

P22 ¼ ðA11A22 � A212Þ=det A

P23 ¼ ðA212 � A11A23Þ=det A

ð5Þ

where A ¼ SðfÞ � SðmÞ; det A ¼ A11ðA222 � A2

23Þ þ 2A12ðA12A23� A22A12Þ;and S(f) and S(m) are the compliance matrix of fiber and resin,respectively.

2.4. Curing shrinkage

During the curing process, resin will undergo substantialshrinkage (up to 5% for epoxy resin). However, because in thebeginning of a typical liquid composite molding process, resin isfully uncured and behaves as viscous fluid. During the curing pro-cess, resin is heated up to a temperature usually above the glasstransition temperature. A significant increase in modulus and areduction in specific volume begin to occur. However, only a smallfraction of curing shrinkage contributes to the residual stress andwarpage development. Thus, in this study, the effective curingshrinkage of epoxy is assumed to be 0.6%.

2.5. Finite element analysis

In this study, the deformation of composite T-stiffener struc-tures was calculated by FEA. The validation of the FEA model is pre-sented in an earlier study [28]. A composite T-stiffener part usuallyconsists of a number of repeating stiffener sub-units, as shown inFig. 3. Each sub-unit consists of the skin and two L-frames. In thisstudy, a four-stiffener structure is studied as an example. The rel-evant dimensions are: L = 400 mm; Lb = 200 mm; H = 150 mm;and h = 2 mm. The thickness of the skin is 2h.

As shown in Fig. 4, the skin consists of 16 laminas and the stack-ing sequence is [45/0/�45/90/45/0/�45/0]s; the left and right ribsconsist of eight laminas and the stacking sequences are [�45/0/45/90/�45/0/45/0] and [45/0/�45/90/45/0/�45/0], respectively. The

Fig. 3. A composite part consisting of repeating stiffener sub-units.

fiber volume fraction is 50%. The noodle area is usually filled withunidirectional fibers and it is assumed that the noodle fiber volumefraction is 30%.

A commercial FEA package MSC.Marc Mentat was employed inthis study. Eight-noded, isoparametric, three-dimensional brickelements are used. Because of the symmetry, only half of the struc-ture is modeled and the part is restrained at the left-hand edge. Themesh is shown in Fig. 5.

In order to simulate the cooling process, the initial and finaltemperatures are defined to be 150 �C and the room temperature.After computation, the resulting deformation is shown in Fig. 6.The contour shows the total displacement in mm. The deformationis like ‘‘spring-in”, i.e. reduction in the angle between the skin andthe stiffeners. This is caused by the in-plane and through-thicknessCTE mismatch of the L-frames, and constraints due to the skin andthe noodle areas.

3. Deformation characterization

In order to characterize the deformation, the y-displacement ofthe skin is plotted versus corresponding x coordinate in Fig. 7 (left),from which it is shown that the dy–x curve can be divided intothree regions. dy is approximately zero in the fist zone and linearly

Fig. 5. Mesh for FEA.

Fig. 6. Deformation of the stiffener structure (mm).

Fig. 8. Angular displacements.

C. Dong / Composite Structures 92 (2010) 1614–1619 1617

increases with x in the second and third zones. Likewise, the x-dis-placements of the stiffeners are plotted versus corresponding ycoordinate in Fig. 7 (right), from which it is shown that for bothstiffeners, dx linearly decreases with y.

These displacements as shown in Fig. 7 are given by the follow-ing equations, where the unit is mm.

Skin:

dy ¼

0 0 6 x 6 200�0:9121þ 4:64� 10�3x 200 < x 6 600�3:876þ 9:55� 10�3x 600 < x 6 800

8>>><>>>:

ð6Þ

Fig. 7. Left: dy of the skin vs. x; ri

ener 1:

dx ¼ �5:418� 10�2 � 2:31� 10�3y ð7Þ

Stiffener 2:

dx ¼ �0:1614� 6:95� 10�3y ð8Þ

It can be seen Eqs. (6)–(8) that the displacements are linearfunctions of x or y. Thus, the deformation of the stiffener structureis characterized by the angular displacements hr1, hr2, hs1, and hs2, asshown in Fig. 8.

From Eqs. (6)–(8) the following relationships can be derived:

hs1 ¼ 2hr1 ð9Þ

hr2 ¼ hs1 þ hr1 ¼ 3hr1 ð10Þ

hs2 ¼ 2hs1 ð11Þ

For this AS4 graphite/epoxy composite T-stiffener structure, itwas calculated that hs1 = 0.2763�. These linear relationships sug-gest that hs1 is independent of L and H. This is further confirmedby another FEA case study when L = 200 mm, which showshs1 = 0.2724�. Thus, for a T-stiffener structure, the deformationcan be characterized by the spring-in of the skin hs.

4. Parametric study

The influence of the design parameters on the spring-in of theskin hs was investigated by design of experiments. For a givenmaterial system, the potential variables affecting the process-in-duced deformation include L, Lb, H, R, h, and Vf. The aforementionedanalysis suggests that the spring-in hs is independent of L and H.Thus, the dimensionless bond length Lb/L is introduced. It is alsoshown from FEA that the spring-in is constant when the model isscaled up or down, i.e. the radius/thickness ratio is constant. Thus,a dimensionless variable R/h is introduced. With the introductionof dimensionless variables, the design factors to be investigatedare reduced to Lb/L, R/h, and Vf. For all the cases, it is assumed that

ght: dx of the stiffeners vs. y.

Fig. 10. Main effects plot.

Table 2Levels of the 23 factorial design.

Parameter Low High

Lb/L 0.125 0.875R/h 1.5 5Vf 40% 60%

1618 C. Dong / Composite Structures 92 (2010) 1614–1619

Vfn = 30%. A 23 factorial design with center point is chosen and thelevels are shown in Table 2. Factorial designs are most efficient forexperiments involving the study of the effects of two or more fac-tors. In a factorial design, all possible combinations of the levels ofthe factors are investigated. Thus, it allows studying the effect ofeach factor on the response variable, as well as the effects of inter-actions between factors on the response variable [29].

For each parameter combination, the deformation was calcu-lated by FEA. The complete data are shown in Table 3 and theyare analyzed by MINITAB

�[30].

The Pareto chart for the effects is shown in Fig. 9 and it can beused to compare the relative magnitude and the statistical signifi-cance of both main and interaction effects.

In Fig. 9, A, B, and C represent the main effect of Lb/L, R/h, and Vf,respectively, which is defined as the change in response producedby a change in the level of that factor averaged over the levels of allthe other factors. AB, BC, and AC are the two-factor interactions.For example, AB represents the interaction between Lb/L and R/h,which is defined as the average difference between the effect ofR/h at the high level of Lb/L and the effect of R/h at the low levelof Lb/L. Similarly, ABC is the three-factor interaction, which is de-fined as the average difference between the AB interactions forthe two different levels of C.

These effects are plotted in decreasing order of the absolute va-lue of the standardized effects and a reference line is drawn on thechart. Any effect that extends past this reference line is significant.In this case, the a-level was set to be 0.10. It can be seen that all

Table 3Complete data from the 23 factorial design with centre point.

Order Lb/L R/h Vf hs (�)

1 0.125 1.50 0.4 0.29302 0.875 1.50 0.4 0.35443 0.125 5.00 0.4 0.39114 0.875 5.00 0.4 0.46085 0.125 1.50 0.6 0.22626 0.875 1.50 0.6 0.26817 0.125 5.00 0.6 0.31128 0.875 5.00 0.6 0.35879 0.500 3.25 0.5 0.3196

Fig. 9. Pareto chart for the effects.

three factors Lb/L, R/h, and Vf are significant. In addition, the mostsignificant factor is R/h because it extends the farthest.

The main effects plot is shown in Fig. 10 for determining whichfactors influence the response and comparing the relative strengthof the effects. In Fig. 10, the response means for each factor levelare plotted and the points for each factor are connected. A refer-ence line is drawn at the overall (grand) mean. It is seen that hs in-creases with Lb/L and R/h, and decreases Vf. It is also shown fromFig. 10 that the center point deviates from the straight line, whichindicates the existence of non-linear relationships. This needs fur-ther investigations.

5. Conclusions

A study on the process-induced deformation of compositeT-stiffener structures is presented in this paper. The deformationwas calculated numerically by Finite Element Analysis (FEA), andthe resulting displacements show that the deformation can be rep-resented by a single angular displacement: the spring-in of theskin. A parametric study was conducted by Design of Experiments(DOE) and FEA to investigate the influence of design parameters onthe spring-in of the skin. It is shown that the spring-in of the skinincreases with the radius and the bonding length, and decreaseswith the fiber volume fraction.

Acknowledgement

The author acknowledges the support from the Curtin ResearchFellowship.

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