- Email: [email protected]
Tailoring for strength of composite steered-fibre panels with cutouts
Composites: Part A 41 (2010) 1760–1767
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Tailoring for strength of composite steered-fibre panels with cutouts C.S. Lopes a,⇑, Z. Gürdal b, P.P. Camanho c a
INEGI – Instituto de Engenharia Mecânica e Gestão Industrial, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal Faculteit Luchtvaart- en Ruimtevaarttechniek, Technische Universiteit Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands c DEMec, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal b
a r t i c l e
i n f o
Article history: Received 10 May 2009 Received in revised form 9 August 2010 Accepted 20 August 2010
Keywords: A. Laminates B. Buckling B. Strength C. Finite element analysis (FEA) Steered-fibre panels
a b s t r a c t A large number of composite parts include cutouts to accommodate windows, doors, and bolted joints. These regions are hot-spots in terms of design because they concentrate stresses, hence becoming critical in terms of the structural integrity of the part. A traditional approach to the problem of stress concentrations around cutouts is to locally increase the laminate thickness in order to improve the strength margins. Often this practice attracts more loads to the cutout besides increasing part weight. A more effective solution is to tailor the panel in-plane stiffness by means of fibre-steered laminates, and avoid the stress concentrations altogether. The present research demonstrates that it is possible to design and manufacture composite panels whose buckling and first-ply failure responses are insensitive to the existence of a central hole. Moreover, it is shown that the structural performance of these designs more than doubles that of straight-fibre configurations. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Tailoring of the in-plane stiffness of composite laminates, in the attempt to improve their structural performance, as been pursued by several researchers. For example, Biggers et al. [1–4] proposed a piecewise uniform stiffness tailoring concept wherein the loadaligned ply material is redistributed across the panel width and through the panel thickness towards the load-aligned edges. By using this optimisation technique Biggers et al. managed to more than double the initial buckling loads of a given baseline configuration with uniform thickness, and increase their postbuckling stiffness [1,2]. Moreover, the postbuckling first-ply failure and ultimate failure loads of tailored panels with and without cutouts where significantly increased [3,4]. The tailoring method described above does not result in a change in panel mass. However, the panel width becomes non-uniform, i.e. the supported panel edges become thicker with the added load-aligned ply material, while the panel central sections become thinner. As a result, the stiffer panel edges attract more loads which are diverted from the buckling and failure sensitive central sections. The increased panel thickness might not be desirable in many applications. Moreover, the zones of thickness drop are prone to stress concentrations which might result in premature delaminations and other localised damage modes. A better concept ⇑ Corresponding author. Tel.: +351 939884128. E-mail addresses: [email protected] (C.S. Lopes), [email protected] (Z. Gürdal), [email protected] (P.P. Camanho). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.08.011
would be if the panel in-plane stiffness could be varied continuously. During the past couple of decades the aerospace industry has witnessed the development of automated processes for the manufacturing of composite parts. Such automations improve part quality and repeatability, and lead to reduction of labour costs. One of these processes, the Automated Fibre Placement (AFP), offers the capability of steering individual fibre-tows over the planform of a laminate panel. The design flexibility allowed by the AFP has been explored by some researchers in order to tailor the in-plane stiffness of laminates. However, the increased freedom of design also increases the complexity and requires novel approaches to many common problems in composite laminates engineering. Several different approaches have been attempted to adequately model and predict the response of curvilinear fibre laminates (e.g. [5–7]). A simple method of modeling complete tow paths was developed in the research started by Gürdal and Olmedo [8–10]. Due to the variation of stiffness properties associated with the change in fibre orientation throughout the laminate plane, these structures were termed variable-stiffness panels. Previous analytical investigations by Gürdal et al. [8–11] predicted significant advantages offered by variable-stiffness panels in terms stiffness and buckling loads. Experimental research [12– 15] demonstrated the superiority of these designs concerning stiffness and buckling responses as well as strength characteristics. The improvements were attributed to the redistribution of the applied in-plane loads to the stiff support regions in order to avoid buckling in the critical central panel sections. Furthermore, it was found
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
that the residual thermal stresses due to the laminate curing cycle can have a significant positive effect on the buckling response of variable-stiffness panels [13]. Numerical investigations in which thermal stresses were taken into account [13,16,17] achieved good correlation between simulated and experimental results. The knowledge about the fibre-steered configurations that lead to the highest first-ply failure loads is still rather limited. Furthermore, the full details about the mechanisms that lead to failure of variable-stiffness panels are not known yet, particularly the influence of local effects resulting from the manufacturing of these composites. The objective of this paper is to characterise the elastic response envelope of fibre-steered panels under compression. This means evaluating the boundaries of the behaviour of variablestiffness panels under edge shortening, in the elastic regime up to the onset of damage, which eventually occurs in the postbuckling regime. This task involves the evaluation of the response characteristics of variable-stiffness panels, for all possible combinations of the fibre orientations, concerning buckling response and first-ply failure in postbuckling. Furthermore, the influence of a central circular hole on these responses is predicted. To achieve these purposes, numerical analyses are carried out using the commercial finite element (FE) package ABAQUS [18]. The onset of damage in the laminates is evaluated using stress-interactive, physically-based first-ply failure criteria which identify the failure mechanisms. As a first approach, only ‘ideal’ variable-stiffness configurations are analysed which ignore some of the manufacturing aspects. A more refined approach consists of introducing inevitable tow overlaps or cuts, which result in small fibre-free areas, and evaluating their influence on the panel failure response. 2. Response characteristics Gürdal et al. [11,17] investigated the stiffness and buckling response characteristics of variable-stiffness panel configurations with linearly varying fibre orientations [8–10], defined by
jxj d
hðxÞ ¼ T 0 þ ðT 1 þ T 0 Þ
ð1Þ
It was assumed that the angle of the tow-course reference path, h(x), varied linearly from the value T0 at the centre of the panel to T1 at a specified distance d, as illustrated in Fig. 1. This distance was taken as half of the panel width along the x direction. The orientation of a single curvilinear fibre path was denoted by hT0jT1i. A ply was considered to be constructed of fibres oriented similarly to the reference fibre path, hence the description of the reference fibre path also served to describe the ply. By carrying parametric studies over the whole range of possible angles T0 and T1, the present investigation follows the same proceθ=T1
x' y θ=T0
φ
y0
x0
d
x
course width Fig. 1. Linear fibre angle variation reference path.
1761
dure used by Gürdal et al. [11,17]. However, the goal here is to determine the first-ply failure response characteristics of panels with holes in the postbuckling regime. Here, residual thermal stress, eigenvalue extraction and nonlinear postbuckling, and failure calculations are carried out using the commercial FE package ABAQUS [18]. Within the framework of the parametric studies presented in this section, the finite width of the tow-course, i.e. the course of the AFP machine head pass, is not considered. Instead, in a simplified approach, each course is considered to have an infinitesimal width, or the width of a single fibre. This means that there is a continuous shifting of the reference path in the direction perpendicular to the fibre orientation variation instead of a discrete shifting as imposed by a finite width tow-course. In this ‘ideal’ design, the fibre paths do not have a constant distance between them, and appear to converge and diverge from one another. Such a fibre path variation can only be possible if the fibre volume fraction of the laminate changes. However, since the AFP machines that are used to build panels of the type studied here have finite width heads with a fixed number of fibre tows, the fibre paths are parallel within a given course. These aspects will be addressed in Section 3 of this paper, wherein the response of ‘ideal’ and ‘manufacturable’ configurations is compared. A set of physically-based criteria developed at the NASA – Langley Research Centre (LaRC) is used in the analysis of first-ply failure behaviour of constant and variable-stiffness composite panels in the postbuckling regime, a method previously employed by the authors [19]. The LaRC failure criteria [20,21] are implemented in a special-purpose ‘‘FORTRAN” subroutine ‘‘UVARM” to obtain User defined VARiables at FE Material points, and to predict the main failure modes of fibre-reinforced plastics: longitudinal tension (fibre breakage), longitudinal compression (fibre kinking), transverse tension (matrix cracking perpendicular to the plane of the ply), and transverse compression (matrix cracking at an angle from the perpendicular to the plane of the ply). The LaRC failure criteria take into account the in situ effect characterised by higher transverse tensile and shear strengths of a ply when it is constrained by plies with different fibre orientations in a laminate, as compared with the strength of the same ply in a unidirectional laminate [22]. The ‘‘UVARM” subroutine is run at every load increment along the prebuckling and postbuckling regimes for every integration point of the FE model, postprocessing the stress field components and calculating the values of the LaRC failure indices. A refined step-wise load incrementation ensures an accurate capture of the load level at which the first failure criterion is met. This point corresponds to the onset of damage in the laminate. Further load incrementation would cause a degraded stiffness response, eventually leading to final collapse. This nonlinear progressive damage behaviour is not analysed in this paper. The analyses are performed on simply supported 24-ply composite panels under uniform edge shortening, v(x, b/2) = v0, as depicted in Fig. 2. Two panel geometries are considered: (i) a flat square panel and (ii) a square panel with a central hole. Both cases are similar except for the central circular hole with a diameter equal to one-third of the panel width. The panel is considered to lay in the x–y plane. The transverse edges are free to expand in the x direction. The fibre orientation, h, is a function of the x-coordinate only (h = h(x), / = 0°). The laminate adopts the general form [±hT0jT1i]6s. This set-up is different from the one adopted in previous works by Gürdal et al. [11,17], where the loading was applied along the x-axis and the fibre orientation varied with the y-axis. In [11,17] the whole panel was rotated by 90°, and the configuration was of the form [90 ± hT0jT1i]6s. The results presented here are in non-dimensional form, following [11,17], in order to simplify the comparison between straightfibre and variable-stiffness laminates. Furthermore, the analyses
1762
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
v=v 0 , w=0
v=v 0 , w=0
φ=a/3
a
a
v= 0, w=0
v= 0, w=0
w=0
x
b
w=0
x
w=0
y
b
w=0
y
Fig. 2. Variable-stiffness panel geometry and boundary conditions. a/b = 1. The transverse edges are free to deform in-plane. The stiffness variation occurs in the x direction (perpendicular to the loading direction).
are based on the lamina properties of AS4/9773 carbon-epoxy material provided in [14] and presented in Table 1. The nominal layer thickness, t, is 0.2 mm and the nominal laminate thickness, tl, is 4.8 mm. FE calculations are performed discretely at each 5° for each T0 and T1 in the 0–90° range. The FE models make use of quadrilateral shell elements of reduced integration (‘‘S4R”) with a typical edge length of 1/30 of the panel characteristic dimension. For fibresteered laminates, each element of the FE model is associated with a particular lay-up, as opposed to straight-fibre formats where the same lay-up applies to the entire FE domain. The local stacking sequence is calculated based on the centroid of each element, i.e. in practical terms the fibre orientation variation is discrete. Between each two neighbouring elements, the maximum fibre angle difference is 6°. The residual thermal stresses are taken into account by defining a thermal step with DT = 137.5° prior to the application of the mechanical loading. In the loading case represented in Fig. 2, the transverse stress resultant, Nx, and the in-plane shear resultant, Nxy, are zero over the entire panel, and it expands uniformly in the x direction leaving the transverse edges straight. However, unlike a traditional straight-fibre laminate, the variation of the u displacement is not linear along the x direction but it depends on the variation of the in-plane stiffness terms along this coordinate [11]. One of the most important consequences of the present configuration (v(x,b/2) = v0 and h = h(x)) is a favourable stress resultant along the load direction, as will be shown. 2.1. Buckling response Although uniform edge displacements are used for loading the panels, the following results of the buckling analyses are compared Table 1 Physical properties of the AS4/9773 carbon-epoxy material system [14]. Elastic properties Thermal expansion (°C1) Ply strengths (MPa)
E1 = 129.8 GPa; E2 = 9.2 GPa; G12 = 5.1 GPa; m12 = 0.36 a11 = 34.2 108; a22 = 34.4 106 XT = 2070; XC = 1160; YT = 29.0; YC = 157.9; SL = 91.0
to the straight-fibre format panels by using the axial stress resultant Ny corresponding to the lowest eigenvalue, Ncr. As Ny = Ny(x), in order to describe the critical loading in terms of the axial stress resultant, an average critical load is defined as:
Nacrv ¼
1 a
Z
a=2
Ny;cr ðx; b=2Þ dx
ð2Þ
a=2
The overall panel stiffness modulus is also function of the x-coordinate, Ey = Ey(x). Similarly to Ny, for a laminate with small angle T0 and a larger angle T1, Ey is larger at x = ±a/2 than at x = 0. In order to compare the panel stiffness modulus to the constant-stiffness value of straight fibre panels, the overall stiffness, Eeq y , is defined by:
Eeq y ¼
b
R a=2 a=2
Ny ðx; b=2Þ dx tl av 0
ð3Þ
The calculation of the panel critical buckling loads and buckling modes in ABAQUS [18] follows a linear eigenvalue extraction analysis. The critical axial stress resultants of panels normalised by 2 b =ðE1 t 3l Þ ðNcr Þ, as a function of the respective overall axial stiffness normalised by the longitudinal ply stiffness ðEy ¼ Eeq y =E1 Þ, are plotted in Fig. 3, for the flat and the cutout panels. The thick lines in the graphs correspond to straight-fibre configurations with a ±h lay-up, and show the variation of both the panel stiffness and buckling load as the fibre angle is changed from 0° to 90° (measured from the x-axis), respectively at the left and the right ends of the lines. For flat constant-stiffness panels, the maximum stiffness is achieved for an all 90° laminate (fibres parallel to the loading direction). The maximum normalised critical buckling load is 1.63 and corresponds to a [±45°]6s laminate with overall normalised stiffness value Eeq y =E1 ¼ 0:14. For variable-stiffness panels, there is a family of curves corresponding to various values of T0, from 0° to 90° with increments of 10°. Each curve is generated by varying the value of T1 between 0° and 90° for a given value of T0. Points to the left of the intersection of these curves with the curve corresponding to straight-fibre panels, identify configurations where T1 > T0 and vice versa. For flat variable-stiffness panels, the maximum normalised value of the critical buckling load is 3.53, obtained for the configuration [±h0j75i]6s. This corresponds to
1763
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
4
4 StraightFibres
3.5
StraightFibres
3.5
T0=0
3
T0=0
3
T0=10
T0=10
T0=20
2.5
T0=30
2
Ncr*
Ncr*
2.5
T0=40 T0=50
T0=30
2
T0=40
T0=60
1.5
T0=20
1.5
T0=50 T0=60
1
1
T0=90 T0=80
T0=80
0.5 0
T0=90
T0=70
T0=70
0.5
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
Fig. 3. Stiffness and buckling performances of variable-stiffness panels under uniform edge shortening,
a 117% improvement in critical buckling load when comparing variable and constant-stiffness designs. Furthermore, the overall axial stiffness of the best curvilinear-fibre configuration in terms of buckling load, Eeq y =E1 ¼ 0:21, is about 54% higher than that of the best straight-fibre format panel. Most variable-stiffness panels with T0 = 0° have higher buckling loads than those of straight-fibre designs. Typically, there are many variable-stiffness configurations that have a critical load greater than the straight-fibre configuration with the same Eeq y . Additionally, for a given value of Ncr, there are only one or two configurations with straight fibres but many with steered fibres. This provides an added advantage in the design process by allowing more freedom to tailor the stiffness and critical load of the structure simultaneously [11]. The reason for the increase in the critical load is due to a favourable distribution of the applied load across the panel height as a function of the x-coordinate, as shown in Fig. 4. The configurations for which the value of Ncr is high are characterised by small values of the fibre orientation angle at the panel centreline (x = 0) and large values at x =± a/2, just as the configuration exemplified in Fig. 2. The local axial stiffness of the regions of the panel is high
SF, SF2 (Avg: 75%) −10.1 −371.6 −733.0 −1094.5 −1455.9 −1817.4 −2178.8
0.8
1
Ey*
Ey*
v0
av 2
N cr ¼ NEcr tb3 ; Ey ¼ 1 l
eq
Ey E1
.
at locations where the fibres are closely aligned with the loading direction. As a result, most of the applied load introduced via uniform edge shortening is carried by the high stiffness regions of the panel that are located near the simply-supported edges of the panel where out-of-plane deflections are suppressed, i.e. the central section of the panel carries only a small fraction of the applied load, which makes it highly resistant to buckling. For the panels with a central hole, the curves corresponding to the normalised critical stress resultants as a function of the overall normalised axial stiffness, shown in Fig. 3b, have similar shapes to the ones for flat panels. However, due to the softening effect of the hole, the maximum normalised stiffness has a value lower than 1 (0.72 only). Though, this significantly affects only the configurations with high T0 (typically, T0 > 50°) and low T1. For these laminates, the central sections of the panels are more loaded than the edges. Therefore, the axial stiffness (and stress resultant) are greatly reduced by the presence of the cutout. On the opposite side, on configurations with low T0 and high T1, Ey and Ny are much less affected by the hole. These considerations are also reflected on the values of the critical buckling loads. This means that, concerning
SF, SF2 (Avg: 75%) 22.8 −343.8 −710.4 −1076.9 −1443.5 −1810.1 −2176.7
Fig. 4. Ny stress resultant on flat and cutout variable-stiffness panels [±h0j75i]6s at buckling load.
1764
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
the buckling behaviour, variable-stiffness designs with low T0 and high T1 are nearly insensitive to the presence of the central hole. The constant and variable-stiffness optimum configurations for buckling resistance are the same for panels with and without a central hole. However, while for the [±h0j75i]6s configuration, Eeq y and Ncr are unaffected by the presence of the hole, for the [±45]6s design they are reduced by 18% and 14%, respectively. This means that the ratio of buckling performance between both designs is 2.62. 2.2. Postbuckling first-ply failure response The LaRC failure criteria [20,21] is used here to study the firstply failure characteristics of the panels illustrated in Fig. 2, eventually occurring in the nonlinear postbuckling regime. In order to generate the nonlinear solutions along the loading path, a linear eigenvalue extraction analysis is initially performed, just as in the analyses previously described, and the first two buckling modes are calculated. Then, the FE model is re-defined with the buckling modes introduced as initial imperfections with small amplitudes as compared to the panel thickness (typically 5%). Finally, geometrically nonlinear solutions are determined for loads up to first-ply failure, as predicted by the LaRC failure criteria. The ‘Riks path following method’ is used to trace the equilibrium paths in the postbuckling collapsing behaviour. Similarly to the buckling analyses, the following results of the failure analyses are compared to the straight-fibre format panels by using the axial stress resultant Ny corresponding to the firstply failure load, Nfpf, defined by:
Nafpfv ¼
1 a
Z
a=2
Ny;fpf ðx; b=2Þ dx
ð4Þ
a=2
The Nfpf of panels is, in the same way as before, normalised by 2 b ¼ E1 t 3l in order to compare critical buckling and first-ply failure loads. The normalised failure stress resultants as function of the overall normalised panel axial stiffness, Eeq y =E1 , for flat and cutout panels, are plotted in Fig. 5. The thick lines in the graphs correspond to straight-fibre designs. The families of curves now have substantially different shapes than those seen earlier, indicating that the mechanisms that trigger buckling and failure are different. Under compressive loads, not only the laminate longitudinal stiffness
but also the transverse stiffness terms play an important role in the resistance to buckling. Therefore, unidirectional laminates, and those that are close to such configuration, are poor designs in terms of buckling. On the other hand, the better the alignment of fibres with the loading, the stronger is the laminate because of the great disparity between longitudinal and transverse, or shear strengths. However, Fig. 5 shows that for the variable-stiffness format, the most promising designs in resisting failure roughly coincide with the best configurations in resisting buckling. The maximum normalised value of the failure load is 3.51, both for flat and cutout panels, obtained for the configuration [±h0j80i]6s. A very close value (3.38) is obtained for the optimum design for buckling resistance, corresponding to the [±h0j75i]6s laminate. The differences between these two configurations are 5° in fibre angle T1, and overall axial stiffness of 0.24 versus 0.20. On the other hand, for constant-stiffness panels, the best configuration for failure resistance is the [±80]6s ðEeq y =E1 ¼ 0:66Þ laminate, with a normalised first-ply failure load of 2.92, which is quite different from the best straight-fibre panel for buckling resistance, the [±45]6s laminate ðEeq y =E1 ¼ 0:11Þ, with a normalised first-ply failure load of 1.60. In terms of first-ply failure loads, this corresponds only to a 20% improvement when comparing constant and variable-stiffness configurations, a value that contrasts with the 162% improvement in the critical buckling load of panels with a central hole. Furthermore, the overall axial stiffness of the best fibre-steered panel in terms of failure load is about 64% lower than the best straight-fibre panel. However, for most values of Eeq y , there are many variable-stiffness configurations that have a failure load greater than the corresponding straight-fibre configuration. Additionally, for a given value of Nfpf, there are only one, two or at most three configurations with straight fibres but many with steered fibres. This provides an added advantage in the design process by allowing more freedom to tailor the stiffness, buckling and failure loads of the structure simultaneously. The normalised failure load for the variable-stiffness panels with a hole as a function of the fibre orientation angles T0 and T1 is plotted in Fig. 6. Typically, for the best results, variable-stiffness panels need a low T0 and a high T1, similarly to the buckling case. On the opposite side of the scale are the configurations with high T0 and a low T1. The variation in results is as high as 600%. In the specific cases of constant-stiffness panels which lie along the diagonal line shown in the figure, the best results are obtained for fibre
4
4 T0=0 T0=10
3.5
T0=20
T0=30 T0=40
3
T0=30
3
T0=40
2.5 T0=90
Nfpf*
Nfpf*
2.5 2 T0=80
1.5
T0=70
0.2
T0=70
0.4
0.6
Ey*
0.8
T0=60
0.5
StraightFibres
0
T0=80
1
T0=50
0.5
T0=90
2 1.5
T0=60
1
0
T0=0 T0=10
3.5
T0=20
1
0
T0=50
0
0.2
StraightFibres
0.4
0.6
0.8
Ey*
Fig. 5. Stiffness and first-ply failure performances of variable-stiffness panels under uniform edge shortening,
v0
N pfp ¼
v b2 N afpf E1 t 3l
; Ey ¼
eq
Ey E1
.
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
90
Nfpf* 3 2.5 2 1.5 1 0.5
75
T1 [deg]
60
45
30
15
0
0
15
30
45
60
75
90
T0 [deg] N av b2 Fig. 6. Normalised first-ply failure load N pfp ¼ Efpft3 field contours in the linear 1 l
fibre angle variation domain. The straight line identifies straight fibre panel solutions.
orientation angles above 60°. For fibre orientation angles between 35° and 55°, the failure loads of constant-stiffness panels is reduced. This is also evident from the depression in the curve corresponding to these configurations in Fig. 5b. A closer look at the FEM results reveal that this dip in the Nfpf field corresponds to panels that fail (prematurely) due to stress concentrations around the hole edge. The failure response curve for the straight-fibre flat panel version, shown Fig. 5a, lacks such a depression. The most promising variable-stiffness panels in terms of buckling and failure under pure compressive loads do not fail at the hole edge. On the other hand, the constant-stiffness panels that perform better in terms of buckling loads end up by failing by the hole edge. Actually, with exception from a small range of fibre orientation angles around 30°, all straight-fibre panels typically with h < 60° fail at such locations. As for the buckling response, the reason for the superior performance of variable-stiffness panels in terms of failure loads, as compared to the straight-fibre counterparts, is a favourable distribution of the applied load across the panel height as a function of the x-coordinate. The configurations for which the value of Nfpf is high are characterised by small values of the fibre orientation angle at the panel centreline (x = 0) and large values at the edges x = ±a/2. The local axial stiffness of the regions of the panel is high at locations where the fibre orientation is closer to 90°. As a result, most of the applied load introduced via uniform edge shortening is carried by the highly stiff regions of the panel located near their simply-supported edges. The introduction of the central hole does not greatly disturb the stress field and does not create stress concentrations higher than the ones forming near the panel transverse edges. Due to the favourable fibre alignment, at these locations the load is mostly supported along the local longitudinal directions. This architecture leads to strong laminates since the longitudinal strength of plies is much higher than their transverse strength, and also the strain-to-failure is higher in the transverse direction than in the longitudinal direction. There is a second reason for the superior failure performance of the tow-steered format: their superior postbuckling response. First-ply failure in the [±h0j80i]6s panel, as an example, occurs where the added influences of the directly applied longitudinal compressive stresses and the bending induced ones (due to buck-
1765
ling) is the highest, i.e. fibre kinking is triggered by bending-induced compressive stresses on the concave side of the buckles, in regions close the transverse edges where the direct compressive loads are higher. Furthermore, failure initiates at face plies because the bending induced stresses are higher there and their in situ strength is lower than for their embedded (constrained) counterparts [22]. These considerations lead to the conclusion that, for failure in the postbuckling regime in general, high buckling loads promote high failure loads. Hence, the improved failure performance of variable-stiffness panels is also a consequence of their improved postbuckling response. The [±80]6s laminate, for instance, has a favourable fibre distribution in terms of strength but its critical buckling load is rather low. As a consequence, face ply fibres at the concave side of the buckle start to kink prematurely due to added effects of direct and bending induced compressive loads.
3. Response of manufacturable panels The analyses reported in the previous section concern fibresteered designs with ‘ideal’ fibre courses, i.e. where there is a continuous shifting of the reference path in the direction perpendicular to the fibre orientation variation. Real manufacturing conditions impose some constraints to the designs idealized by Gürdal et al. [8–10]. The AFP heads typically have the capability of accommodating up to 36 fibre tows within a 76.2 mm (3 in.) course width. This means that within each head pass or course, up to 36 tows are placed parallel to each other and to the reference path. The shifting is, therefore, made discretely at multiples of the course width. The boundaries of neighbouring courses do not match for all locations along the x-axis. This leads to irregular regions between tow courses that can be accounted for by prescribing the shift distance so that no gaps occur. However, for curved reference paths, this practice produces overlap regions. To prevent the overlap regions the tow-placement machine can be instructed to cut tows individually so that no thickness builds up. This practice results in laminates that contain small wedge-like areas free of fibres due to the dropping of the individual, finite thickness tows. These small fibre-free areas are likely to create resin-rich regions during the laminate curing process. The difference between ‘ideal’ and ‘manufacturable’ fibre-steered plies is exemplified in Fig. 7. In order to avoid the collocation of course edges and of resinrich areas, that would occur at the same places through-the-thickness of a laminate in clustered plies with the same fibre angle distribution, Gürdal et al. [23] proposed to stagger the layers, i.e. to shift the fibre placement courses of one layer, in one of the planar directions, in relation to the adjacent layer with the same fibre orientation distribution. Experimental testing [14,15] revealed no influence of ply staggering on panel buckling response. In this section, the most promising designs, in terms of first-ply failure performance, are analysed in detail. That is, the [±h0j80i]6s variable-stiffness panel is compared with the [±80]6s straight-fibre laminate. Additionally, the [±45]6s straight-fibre design is analysed as well, since this configuration may be preferable to the [±80]6s laminate due to its superior buckling performance. Some of the issues that arise due to the practical manufacturing of these panels are taken into account. A nominal course width approximately equal to 1/4th of the panel height is adopted. This means that, on average, each ply can be laid completely with four AFP machine head passes. The effects of ply staggering on panel response are also analysed. The results for the cases studied in this section are reported in Table 2. The predicted overall first-ply failure load, N afpfv , for the [±h0j80i]6s ‘manufacturable’ panel is 469.5 N/mm, only 1.2% higher
1766
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
Fig. 7. Fibre directions on ‘ideal’ and ‘manufacturable’ tow-steered plies with a central hole (T0 = 0°, T1 = 80°). The course width on the manufacturable ply is approximately equal to 1/4th of the panel height.
Table 2 Buckling and strength results for the [±h0j80i]6s ‘ideal’ and ‘manufacturable’ designs. The strength performance is compared with that for the [±80]6s and [±45]6s laminates. VSP – variable-stiffness panel. Design
N acrv (N/mm)
N afpfv (N/mm)
vs. [±80]6s (%)
vs. [±45]6s (%)
Straight fibres ([±45]6s) Straight fibres ([±80]6s)
232.7 148.8
244.7 464.1
47.2 –
– 89.7
Ideal VSP Manufacturable VSP Manufacturable VSP (staggered plies)
555.8 548.6 543.9
559.6 469.5 523.4
20.0 1.2 12.8
120.5 91.9 113.9
than the 464.1 N/mm calculated for the [±80]6s laminate, and much lower than what was expected, having in mind the results achieved by the ‘ideal’ design. In terms of predicted critical buckling loads, the difference between ‘ideal’ ðN acrv ¼ 555:8 N=mmÞ and ‘manufacturable’ ðN acrv ¼ 548:6 N=mmÞ designs is only about 1%. The longitudinal compressive failure index field for the critical surface ply of the [±h0j80i]6s panel with tow-drops, corresponding to N afpfv ¼ 469:5 N=mm, is depicted in Fig. 8a. The figure shows values of the failure index approaching 1.0 near the course edges, representing the occurrence of first-ply failure. This means that stress
UVARM7 fraction = − 0.98 0.81 0.65 0.49 0.32 0.16 −0.00
concentrations develop in those areas, because of the fibre angle mismatch between adjacent courses. These discontinuities trigger premature first-ply failure. The repeatability of plies in the laminate determines that the layerwise stress concentrations develop at the same (x, y)-coordinates through-the-thickness of the panel, therefore there are no alternative paths for stress relief. In an attempt to lower stress concentrations, by smearing them over wider areas of the panel and promoting alternative load paths, the technique of ply staggering is applied to the [±h0j80i]6s laminate. The predicted first-ply failure for this design is
UVARM7 fraction = 0.99 0.83 0.66 0.50 0.33 0.16 −0.00
Fig. 8. Longitudinal compressive failure index field, at first-ply failure load, for the critical surface ply of the [±h0j80i]6s panel with tow-drops.
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
N afpfv ¼ 523:4 N=mm, 12.8% higher than for the [±80]6s laminate while the critical buckling load is kept nearly the same as that of the non-staggered panel (less than 1% difference). This failure load is still lower than the value predicted for the ‘ideal’ case, but higher than for the configuration without ply staggering. The fibre compression failure index for this design, corresponding to N afpfv ¼ 523:4 N=mm, is shown in Fig. 8b. The highly stressed regions are now dispersed over wider areas resulting in a more uniform loading of the panel. This lowers the peak stresses and postpones failure to higher applied loads.
1767
stiffening panel edges, tow-steering might also be a good approach in increasing the effectiveness of bonded or bolted joints. Acknowledgement Partial funding of this research through the Scholarship SFRH/ BD/16238/2004 from the Portuguese Foundation for Science and Technology (FCT) is gratefully acknowledged. Additionally, the financial support of the FCT under the project PTDC-EME-PME64819-2006 is acknowledged by the third author. References
4. Conclusions The buckling and postbuckling first-ply failure response characteristics of variable-stiffness panels fabricated by AFP technology were analysed. Both in terms of buckling and first-ply failure, important advantages in comparison with constant-stiffness formats were demonstrated. Fibre-steered panels offer significant performance improvements in terms of critical buckling loads. This is due to a favourable redistribution of the applied loads towards the supported panel edges resulting in an unloading of its central section. This is the effect already reported in the framework of stiffness tailoring efforts made by other researchers such as Biggers et al. [1–4]. Moreover, the gains in structural performance are of the same order of magnitude. The advantage of the current tailoring concept is that the in-plane stiffness properties vary continuously, avoiding critical areas as abrupt changes in panel thickness, and ply drop-offs. The effect of load transfer towards the panel edges can be so pronounced that the optimum variable-stiffness designs with a central hole have nearly the same initial buckling loads as panels with the same distribution of fibre orientations but no hole. The same lack of sensitivity to the presence of a cutout was demonstrated in terms of panel failure. This is one of the major advantages of tow-steered designs in comparison with traditional designs. In principle, these results should be relatively little affected by the shape of the cutout, e.g. having a central rectangular window with rounded corners instead of a circular hole of similar size. This is because of the effect of the stiffness variation on the global behaviour of a panel. However, the structural performance is expected to decrease if the cutout should not be centrally located. There are clear benefits in using tow-steered designs in terms of nominal panel strength, although not as pronounced as in terms of critical buckling loads. For panel failure in the postbuckling regime, these improvements have two different causes. One is the load redistribution towards the panel supported edges and a favourable fibre alignment at these locations such that the load is mostly supported along the local longitudinal direction. The other reason is related to the higher buckling loads. Panel buckling adds bending induced stresses to the directly applied loads, therefore, the high critical loads of some variable-stiffness panels also improves their failure performance. The manufacturability of fibre-steered laminates imposes some design constraints that limit their advantages in terms of first-ply failure performance, specifically the fibre angle mismatches at course edges due to the finite width of the AFP machine head passes that create stress concentrations. These stress concentrations can be mitigated by using narrower tow courses or by staggering the laminate plies. The results reported herein are encouraging of applying the technique of tow-steering to several other structural cases, such as tensile loaded panels and composite panels containing notches or multiple neighbouring cutouts, in the expectation of achieving similar gains in performance. Moreover, due to the possibility of
[1] Biggers SB, Srinivasan S. Compression buckling response of tailored rectangular composite plates. AIAA J 1993;31(3):590–6. [2] Biggers SB, Srinivasan S. Postbuckling response of piece-wise uniform tailored composite plates in compression. J Reinf Plast Compos 1994;13:803–21. [3] Baranski AT, Biggers SB. Postbuckling analysis of tailored composite plates with progressive damage. Compos Struct 1999;46:245–55. [4] Xie D, Biggers SB. Postbuckling analysis with progressive damage modeling of tailored laminated plates and shells with a cutout. Compos Struct 2003;59:199–216. [5] Hyer MW, Charette RF. Use of curvilinear fiber format in composite structure design 1991;29(6):1011–5. [6] Hyer MW, Lee HH. The use of curvilinear fiber format to improve buckling resistance of composite plates with central circular holes. Compos Struct 1991;18:239–61. [7] Nagendra S, Kodiyalam S, Davis JE, Parthasarathy VN. Optimization of tow fiber paths for composite design. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 36th structures, structural dynamics and materials conference, AIAA 95-1275, New Orleans, LA; 1995. p. 1031–41. [8] Gürdal Z, Olmedo RA. Composite laminates with spatially varying fiber orientations: variable stiffness panel concept. In: Proceedings of the AIAA/ ASME/ASCE/AHS/ASC 33rd structures, structural dynamics and materials conference, AIAA 92-2472, vol. 2, Dallas, TX; 1992. p. 798–808. [9] Gürdal Z, Olmedo R. In-plane response of laminates with spatially varying fiber orientations: variable stiffness concept. AIAA J 1993;31(4):751–8. [10] Olmedo RA, Gürdal Z. Buckling response of laminates with spatially varying fiber orientations. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 34th structures, structural dynamics and materials conference, AIAA 93-1567, La Jola, CA; 1993. p. 2261–9. [11] Gürdal Z, Tatting BF, Wu KC. Variable stiffness panels: effects of stiffness variation on the in-plane and buckling responses. Composites Part A: Appl Sci Manuf 2008;39:911–22. [12] Wu KC, Gürdal Z. Thermal testing of tow-placed, variable stiffness panels. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 42nd structures, structural dynamics and materials conference, AIAA 2001-1190, Seattle, WA; 2001. [13] Wu KC, Gürdal Z, Starnes Jr JH. Structural response of compression-loaded, tow-placed, variable stiffness panels. In: Proceedings of the AIAA/ASME/ASCE/ AHS/ASC 43rd structures, structural dynamics and materials conference, AIAA 2002-1512, Denver, CO; 2002. [14] Jegley DC, Tatting BF, Gürdal Z. Optimization of elastically tailored tow placed plates with holes. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 44th structures, structural dynamics and materials conference, AIAA 2003-1420, Norfolk, VA; 2003. [15] Jegley DC, Tatting BF, Gürdal Z. Tow-steered panels with holes subjected to compression or shear loading. In: Proceedings of the AIAA/ASME/ASCE/AHS/ ASC 46th structures, structural dynamics and materials (SDM) conference, AIAA 2005-2017, Austin, TX; 2005. [16] Lopes CS, Camanho PP, Gürdal Z, Tatting BF. Progressive failure analysis of tow-placed, variable-stiffness composite panels. Int J Solids Struct 2007;44:8493–516. [17] Abdalla MM, Gürdal Z, Abdelal GF. Thermo-mechanical response of variablestiffness composite panels. J Therm Stress 2009;32:187–208. [18] Abaqus, Inc., Abaqus Version 6.8 User’s Manual, Pawtucket, RI, USA; 2008. [19] Lopes CS, Gürdal Z, Camanho PP. Variable-stiffness composite panels: buckling and first-ply failure improvements over straight-fibre laminates. Comput Struct 2008;86:897–907. [20] Dávila CG, Camanho PP, Rose CA. Failure criteria for FRP laminates in plane stress. J Compos Mater 2005;39:323–45. [21] Pinho ST, Dávila CG, Camanho PP, Iannucci L, Robinson P. Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity, Technical report, NASA, Langley Research Center, Hampton, VA, NASA/TM-2005-213530; February 2005. [22] Camanho PP, Dávila CG, Pinho S, Iannucci L, Robinson P. Prediction of in situ strengths and matrix cracking in composites under transverse tension and inplane shear. Composites Part A: Appl Sci Manuf 2006;37:165–76. [23] Gürdal Z, Tatting BF, Wu KC. Tow-placement technology and fabrication issues for laminated composite structures. In: Proceedings of the AIAA/ASME/ASCE/ AHS/ASC 46th structures, structural dynamics and materials conference, AIAA 2005-2017, Austin, TX; 2005.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Tailoring for strength of composite steered-fibre panels with cutouts C.S. Lopes a,⇑, Z. Gürdal b, P.P. Camanho c a
INEGI – Instituto de Engenharia Mecânica e Gestão Industrial, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal Faculteit Luchtvaart- en Ruimtevaarttechniek, Technische Universiteit Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands c DEMec, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal b
a r t i c l e
i n f o
Article history: Received 10 May 2009 Received in revised form 9 August 2010 Accepted 20 August 2010
Keywords: A. Laminates B. Buckling B. Strength C. Finite element analysis (FEA) Steered-fibre panels
a b s t r a c t A large number of composite parts include cutouts to accommodate windows, doors, and bolted joints. These regions are hot-spots in terms of design because they concentrate stresses, hence becoming critical in terms of the structural integrity of the part. A traditional approach to the problem of stress concentrations around cutouts is to locally increase the laminate thickness in order to improve the strength margins. Often this practice attracts more loads to the cutout besides increasing part weight. A more effective solution is to tailor the panel in-plane stiffness by means of fibre-steered laminates, and avoid the stress concentrations altogether. The present research demonstrates that it is possible to design and manufacture composite panels whose buckling and first-ply failure responses are insensitive to the existence of a central hole. Moreover, it is shown that the structural performance of these designs more than doubles that of straight-fibre configurations. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Tailoring of the in-plane stiffness of composite laminates, in the attempt to improve their structural performance, as been pursued by several researchers. For example, Biggers et al. [1–4] proposed a piecewise uniform stiffness tailoring concept wherein the loadaligned ply material is redistributed across the panel width and through the panel thickness towards the load-aligned edges. By using this optimisation technique Biggers et al. managed to more than double the initial buckling loads of a given baseline configuration with uniform thickness, and increase their postbuckling stiffness [1,2]. Moreover, the postbuckling first-ply failure and ultimate failure loads of tailored panels with and without cutouts where significantly increased [3,4]. The tailoring method described above does not result in a change in panel mass. However, the panel width becomes non-uniform, i.e. the supported panel edges become thicker with the added load-aligned ply material, while the panel central sections become thinner. As a result, the stiffer panel edges attract more loads which are diverted from the buckling and failure sensitive central sections. The increased panel thickness might not be desirable in many applications. Moreover, the zones of thickness drop are prone to stress concentrations which might result in premature delaminations and other localised damage modes. A better concept ⇑ Corresponding author. Tel.: +351 939884128. E-mail addresses: [email protected] (C.S. Lopes), [email protected] (Z. Gürdal), [email protected] (P.P. Camanho). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.08.011
would be if the panel in-plane stiffness could be varied continuously. During the past couple of decades the aerospace industry has witnessed the development of automated processes for the manufacturing of composite parts. Such automations improve part quality and repeatability, and lead to reduction of labour costs. One of these processes, the Automated Fibre Placement (AFP), offers the capability of steering individual fibre-tows over the planform of a laminate panel. The design flexibility allowed by the AFP has been explored by some researchers in order to tailor the in-plane stiffness of laminates. However, the increased freedom of design also increases the complexity and requires novel approaches to many common problems in composite laminates engineering. Several different approaches have been attempted to adequately model and predict the response of curvilinear fibre laminates (e.g. [5–7]). A simple method of modeling complete tow paths was developed in the research started by Gürdal and Olmedo [8–10]. Due to the variation of stiffness properties associated with the change in fibre orientation throughout the laminate plane, these structures were termed variable-stiffness panels. Previous analytical investigations by Gürdal et al. [8–11] predicted significant advantages offered by variable-stiffness panels in terms stiffness and buckling loads. Experimental research [12– 15] demonstrated the superiority of these designs concerning stiffness and buckling responses as well as strength characteristics. The improvements were attributed to the redistribution of the applied in-plane loads to the stiff support regions in order to avoid buckling in the critical central panel sections. Furthermore, it was found
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
that the residual thermal stresses due to the laminate curing cycle can have a significant positive effect on the buckling response of variable-stiffness panels [13]. Numerical investigations in which thermal stresses were taken into account [13,16,17] achieved good correlation between simulated and experimental results. The knowledge about the fibre-steered configurations that lead to the highest first-ply failure loads is still rather limited. Furthermore, the full details about the mechanisms that lead to failure of variable-stiffness panels are not known yet, particularly the influence of local effects resulting from the manufacturing of these composites. The objective of this paper is to characterise the elastic response envelope of fibre-steered panels under compression. This means evaluating the boundaries of the behaviour of variablestiffness panels under edge shortening, in the elastic regime up to the onset of damage, which eventually occurs in the postbuckling regime. This task involves the evaluation of the response characteristics of variable-stiffness panels, for all possible combinations of the fibre orientations, concerning buckling response and first-ply failure in postbuckling. Furthermore, the influence of a central circular hole on these responses is predicted. To achieve these purposes, numerical analyses are carried out using the commercial finite element (FE) package ABAQUS [18]. The onset of damage in the laminates is evaluated using stress-interactive, physically-based first-ply failure criteria which identify the failure mechanisms. As a first approach, only ‘ideal’ variable-stiffness configurations are analysed which ignore some of the manufacturing aspects. A more refined approach consists of introducing inevitable tow overlaps or cuts, which result in small fibre-free areas, and evaluating their influence on the panel failure response. 2. Response characteristics Gürdal et al. [11,17] investigated the stiffness and buckling response characteristics of variable-stiffness panel configurations with linearly varying fibre orientations [8–10], defined by
jxj d
hðxÞ ¼ T 0 þ ðT 1 þ T 0 Þ
ð1Þ
It was assumed that the angle of the tow-course reference path, h(x), varied linearly from the value T0 at the centre of the panel to T1 at a specified distance d, as illustrated in Fig. 1. This distance was taken as half of the panel width along the x direction. The orientation of a single curvilinear fibre path was denoted by hT0jT1i. A ply was considered to be constructed of fibres oriented similarly to the reference fibre path, hence the description of the reference fibre path also served to describe the ply. By carrying parametric studies over the whole range of possible angles T0 and T1, the present investigation follows the same proceθ=T1
x' y θ=T0
φ
y0
x0
d
x
course width Fig. 1. Linear fibre angle variation reference path.
1761
dure used by Gürdal et al. [11,17]. However, the goal here is to determine the first-ply failure response characteristics of panels with holes in the postbuckling regime. Here, residual thermal stress, eigenvalue extraction and nonlinear postbuckling, and failure calculations are carried out using the commercial FE package ABAQUS [18]. Within the framework of the parametric studies presented in this section, the finite width of the tow-course, i.e. the course of the AFP machine head pass, is not considered. Instead, in a simplified approach, each course is considered to have an infinitesimal width, or the width of a single fibre. This means that there is a continuous shifting of the reference path in the direction perpendicular to the fibre orientation variation instead of a discrete shifting as imposed by a finite width tow-course. In this ‘ideal’ design, the fibre paths do not have a constant distance between them, and appear to converge and diverge from one another. Such a fibre path variation can only be possible if the fibre volume fraction of the laminate changes. However, since the AFP machines that are used to build panels of the type studied here have finite width heads with a fixed number of fibre tows, the fibre paths are parallel within a given course. These aspects will be addressed in Section 3 of this paper, wherein the response of ‘ideal’ and ‘manufacturable’ configurations is compared. A set of physically-based criteria developed at the NASA – Langley Research Centre (LaRC) is used in the analysis of first-ply failure behaviour of constant and variable-stiffness composite panels in the postbuckling regime, a method previously employed by the authors [19]. The LaRC failure criteria [20,21] are implemented in a special-purpose ‘‘FORTRAN” subroutine ‘‘UVARM” to obtain User defined VARiables at FE Material points, and to predict the main failure modes of fibre-reinforced plastics: longitudinal tension (fibre breakage), longitudinal compression (fibre kinking), transverse tension (matrix cracking perpendicular to the plane of the ply), and transverse compression (matrix cracking at an angle from the perpendicular to the plane of the ply). The LaRC failure criteria take into account the in situ effect characterised by higher transverse tensile and shear strengths of a ply when it is constrained by plies with different fibre orientations in a laminate, as compared with the strength of the same ply in a unidirectional laminate [22]. The ‘‘UVARM” subroutine is run at every load increment along the prebuckling and postbuckling regimes for every integration point of the FE model, postprocessing the stress field components and calculating the values of the LaRC failure indices. A refined step-wise load incrementation ensures an accurate capture of the load level at which the first failure criterion is met. This point corresponds to the onset of damage in the laminate. Further load incrementation would cause a degraded stiffness response, eventually leading to final collapse. This nonlinear progressive damage behaviour is not analysed in this paper. The analyses are performed on simply supported 24-ply composite panels under uniform edge shortening, v(x, b/2) = v0, as depicted in Fig. 2. Two panel geometries are considered: (i) a flat square panel and (ii) a square panel with a central hole. Both cases are similar except for the central circular hole with a diameter equal to one-third of the panel width. The panel is considered to lay in the x–y plane. The transverse edges are free to expand in the x direction. The fibre orientation, h, is a function of the x-coordinate only (h = h(x), / = 0°). The laminate adopts the general form [±hT0jT1i]6s. This set-up is different from the one adopted in previous works by Gürdal et al. [11,17], where the loading was applied along the x-axis and the fibre orientation varied with the y-axis. In [11,17] the whole panel was rotated by 90°, and the configuration was of the form [90 ± hT0jT1i]6s. The results presented here are in non-dimensional form, following [11,17], in order to simplify the comparison between straightfibre and variable-stiffness laminates. Furthermore, the analyses
1762
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
v=v 0 , w=0
v=v 0 , w=0
φ=a/3
a
a
v= 0, w=0
v= 0, w=0
w=0
x
b
w=0
x
w=0
y
b
w=0
y
Fig. 2. Variable-stiffness panel geometry and boundary conditions. a/b = 1. The transverse edges are free to deform in-plane. The stiffness variation occurs in the x direction (perpendicular to the loading direction).
are based on the lamina properties of AS4/9773 carbon-epoxy material provided in [14] and presented in Table 1. The nominal layer thickness, t, is 0.2 mm and the nominal laminate thickness, tl, is 4.8 mm. FE calculations are performed discretely at each 5° for each T0 and T1 in the 0–90° range. The FE models make use of quadrilateral shell elements of reduced integration (‘‘S4R”) with a typical edge length of 1/30 of the panel characteristic dimension. For fibresteered laminates, each element of the FE model is associated with a particular lay-up, as opposed to straight-fibre formats where the same lay-up applies to the entire FE domain. The local stacking sequence is calculated based on the centroid of each element, i.e. in practical terms the fibre orientation variation is discrete. Between each two neighbouring elements, the maximum fibre angle difference is 6°. The residual thermal stresses are taken into account by defining a thermal step with DT = 137.5° prior to the application of the mechanical loading. In the loading case represented in Fig. 2, the transverse stress resultant, Nx, and the in-plane shear resultant, Nxy, are zero over the entire panel, and it expands uniformly in the x direction leaving the transverse edges straight. However, unlike a traditional straight-fibre laminate, the variation of the u displacement is not linear along the x direction but it depends on the variation of the in-plane stiffness terms along this coordinate [11]. One of the most important consequences of the present configuration (v(x,b/2) = v0 and h = h(x)) is a favourable stress resultant along the load direction, as will be shown. 2.1. Buckling response Although uniform edge displacements are used for loading the panels, the following results of the buckling analyses are compared Table 1 Physical properties of the AS4/9773 carbon-epoxy material system [14]. Elastic properties Thermal expansion (°C1) Ply strengths (MPa)
E1 = 129.8 GPa; E2 = 9.2 GPa; G12 = 5.1 GPa; m12 = 0.36 a11 = 34.2 108; a22 = 34.4 106 XT = 2070; XC = 1160; YT = 29.0; YC = 157.9; SL = 91.0
to the straight-fibre format panels by using the axial stress resultant Ny corresponding to the lowest eigenvalue, Ncr. As Ny = Ny(x), in order to describe the critical loading in terms of the axial stress resultant, an average critical load is defined as:
Nacrv ¼
1 a
Z
a=2
Ny;cr ðx; b=2Þ dx
ð2Þ
a=2
The overall panel stiffness modulus is also function of the x-coordinate, Ey = Ey(x). Similarly to Ny, for a laminate with small angle T0 and a larger angle T1, Ey is larger at x = ±a/2 than at x = 0. In order to compare the panel stiffness modulus to the constant-stiffness value of straight fibre panels, the overall stiffness, Eeq y , is defined by:
Eeq y ¼
b
R a=2 a=2
Ny ðx; b=2Þ dx tl av 0
ð3Þ
The calculation of the panel critical buckling loads and buckling modes in ABAQUS [18] follows a linear eigenvalue extraction analysis. The critical axial stress resultants of panels normalised by 2 b =ðE1 t 3l Þ ðNcr Þ, as a function of the respective overall axial stiffness normalised by the longitudinal ply stiffness ðEy ¼ Eeq y =E1 Þ, are plotted in Fig. 3, for the flat and the cutout panels. The thick lines in the graphs correspond to straight-fibre configurations with a ±h lay-up, and show the variation of both the panel stiffness and buckling load as the fibre angle is changed from 0° to 90° (measured from the x-axis), respectively at the left and the right ends of the lines. For flat constant-stiffness panels, the maximum stiffness is achieved for an all 90° laminate (fibres parallel to the loading direction). The maximum normalised critical buckling load is 1.63 and corresponds to a [±45°]6s laminate with overall normalised stiffness value Eeq y =E1 ¼ 0:14. For variable-stiffness panels, there is a family of curves corresponding to various values of T0, from 0° to 90° with increments of 10°. Each curve is generated by varying the value of T1 between 0° and 90° for a given value of T0. Points to the left of the intersection of these curves with the curve corresponding to straight-fibre panels, identify configurations where T1 > T0 and vice versa. For flat variable-stiffness panels, the maximum normalised value of the critical buckling load is 3.53, obtained for the configuration [±h0j75i]6s. This corresponds to
1763
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
4
4 StraightFibres
3.5
StraightFibres
3.5
T0=0
3
T0=0
3
T0=10
T0=10
T0=20
2.5
T0=30
2
Ncr*
Ncr*
2.5
T0=40 T0=50
T0=30
2
T0=40
T0=60
1.5
T0=20
1.5
T0=50 T0=60
1
1
T0=90 T0=80
T0=80
0.5 0
T0=90
T0=70
T0=70
0.5
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
Fig. 3. Stiffness and buckling performances of variable-stiffness panels under uniform edge shortening,
a 117% improvement in critical buckling load when comparing variable and constant-stiffness designs. Furthermore, the overall axial stiffness of the best curvilinear-fibre configuration in terms of buckling load, Eeq y =E1 ¼ 0:21, is about 54% higher than that of the best straight-fibre format panel. Most variable-stiffness panels with T0 = 0° have higher buckling loads than those of straight-fibre designs. Typically, there are many variable-stiffness configurations that have a critical load greater than the straight-fibre configuration with the same Eeq y . Additionally, for a given value of Ncr, there are only one or two configurations with straight fibres but many with steered fibres. This provides an added advantage in the design process by allowing more freedom to tailor the stiffness and critical load of the structure simultaneously [11]. The reason for the increase in the critical load is due to a favourable distribution of the applied load across the panel height as a function of the x-coordinate, as shown in Fig. 4. The configurations for which the value of Ncr is high are characterised by small values of the fibre orientation angle at the panel centreline (x = 0) and large values at x =± a/2, just as the configuration exemplified in Fig. 2. The local axial stiffness of the regions of the panel is high
SF, SF2 (Avg: 75%) −10.1 −371.6 −733.0 −1094.5 −1455.9 −1817.4 −2178.8
0.8
1
Ey*
Ey*
v0
av 2
N cr ¼ NEcr tb3 ; Ey ¼ 1 l
eq
Ey E1
.
at locations where the fibres are closely aligned with the loading direction. As a result, most of the applied load introduced via uniform edge shortening is carried by the high stiffness regions of the panel that are located near the simply-supported edges of the panel where out-of-plane deflections are suppressed, i.e. the central section of the panel carries only a small fraction of the applied load, which makes it highly resistant to buckling. For the panels with a central hole, the curves corresponding to the normalised critical stress resultants as a function of the overall normalised axial stiffness, shown in Fig. 3b, have similar shapes to the ones for flat panels. However, due to the softening effect of the hole, the maximum normalised stiffness has a value lower than 1 (0.72 only). Though, this significantly affects only the configurations with high T0 (typically, T0 > 50°) and low T1. For these laminates, the central sections of the panels are more loaded than the edges. Therefore, the axial stiffness (and stress resultant) are greatly reduced by the presence of the cutout. On the opposite side, on configurations with low T0 and high T1, Ey and Ny are much less affected by the hole. These considerations are also reflected on the values of the critical buckling loads. This means that, concerning
SF, SF2 (Avg: 75%) 22.8 −343.8 −710.4 −1076.9 −1443.5 −1810.1 −2176.7
Fig. 4. Ny stress resultant on flat and cutout variable-stiffness panels [±h0j75i]6s at buckling load.
1764
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
the buckling behaviour, variable-stiffness designs with low T0 and high T1 are nearly insensitive to the presence of the central hole. The constant and variable-stiffness optimum configurations for buckling resistance are the same for panels with and without a central hole. However, while for the [±h0j75i]6s configuration, Eeq y and Ncr are unaffected by the presence of the hole, for the [±45]6s design they are reduced by 18% and 14%, respectively. This means that the ratio of buckling performance between both designs is 2.62. 2.2. Postbuckling first-ply failure response The LaRC failure criteria [20,21] is used here to study the firstply failure characteristics of the panels illustrated in Fig. 2, eventually occurring in the nonlinear postbuckling regime. In order to generate the nonlinear solutions along the loading path, a linear eigenvalue extraction analysis is initially performed, just as in the analyses previously described, and the first two buckling modes are calculated. Then, the FE model is re-defined with the buckling modes introduced as initial imperfections with small amplitudes as compared to the panel thickness (typically 5%). Finally, geometrically nonlinear solutions are determined for loads up to first-ply failure, as predicted by the LaRC failure criteria. The ‘Riks path following method’ is used to trace the equilibrium paths in the postbuckling collapsing behaviour. Similarly to the buckling analyses, the following results of the failure analyses are compared to the straight-fibre format panels by using the axial stress resultant Ny corresponding to the firstply failure load, Nfpf, defined by:
Nafpfv ¼
1 a
Z
a=2
Ny;fpf ðx; b=2Þ dx
ð4Þ
a=2
The Nfpf of panels is, in the same way as before, normalised by 2 b ¼ E1 t 3l in order to compare critical buckling and first-ply failure loads. The normalised failure stress resultants as function of the overall normalised panel axial stiffness, Eeq y =E1 , for flat and cutout panels, are plotted in Fig. 5. The thick lines in the graphs correspond to straight-fibre designs. The families of curves now have substantially different shapes than those seen earlier, indicating that the mechanisms that trigger buckling and failure are different. Under compressive loads, not only the laminate longitudinal stiffness
but also the transverse stiffness terms play an important role in the resistance to buckling. Therefore, unidirectional laminates, and those that are close to such configuration, are poor designs in terms of buckling. On the other hand, the better the alignment of fibres with the loading, the stronger is the laminate because of the great disparity between longitudinal and transverse, or shear strengths. However, Fig. 5 shows that for the variable-stiffness format, the most promising designs in resisting failure roughly coincide with the best configurations in resisting buckling. The maximum normalised value of the failure load is 3.51, both for flat and cutout panels, obtained for the configuration [±h0j80i]6s. A very close value (3.38) is obtained for the optimum design for buckling resistance, corresponding to the [±h0j75i]6s laminate. The differences between these two configurations are 5° in fibre angle T1, and overall axial stiffness of 0.24 versus 0.20. On the other hand, for constant-stiffness panels, the best configuration for failure resistance is the [±80]6s ðEeq y =E1 ¼ 0:66Þ laminate, with a normalised first-ply failure load of 2.92, which is quite different from the best straight-fibre panel for buckling resistance, the [±45]6s laminate ðEeq y =E1 ¼ 0:11Þ, with a normalised first-ply failure load of 1.60. In terms of first-ply failure loads, this corresponds only to a 20% improvement when comparing constant and variable-stiffness configurations, a value that contrasts with the 162% improvement in the critical buckling load of panels with a central hole. Furthermore, the overall axial stiffness of the best fibre-steered panel in terms of failure load is about 64% lower than the best straight-fibre panel. However, for most values of Eeq y , there are many variable-stiffness configurations that have a failure load greater than the corresponding straight-fibre configuration. Additionally, for a given value of Nfpf, there are only one, two or at most three configurations with straight fibres but many with steered fibres. This provides an added advantage in the design process by allowing more freedom to tailor the stiffness, buckling and failure loads of the structure simultaneously. The normalised failure load for the variable-stiffness panels with a hole as a function of the fibre orientation angles T0 and T1 is plotted in Fig. 6. Typically, for the best results, variable-stiffness panels need a low T0 and a high T1, similarly to the buckling case. On the opposite side of the scale are the configurations with high T0 and a low T1. The variation in results is as high as 600%. In the specific cases of constant-stiffness panels which lie along the diagonal line shown in the figure, the best results are obtained for fibre
4
4 T0=0 T0=10
3.5
T0=20
T0=30 T0=40
3
T0=30
3
T0=40
2.5 T0=90
Nfpf*
Nfpf*
2.5 2 T0=80
1.5
T0=70
0.2
T0=70
0.4
0.6
Ey*
0.8
T0=60
0.5
StraightFibres
0
T0=80
1
T0=50
0.5
T0=90
2 1.5
T0=60
1
0
T0=0 T0=10
3.5
T0=20
1
0
T0=50
0
0.2
StraightFibres
0.4
0.6
0.8
Ey*
Fig. 5. Stiffness and first-ply failure performances of variable-stiffness panels under uniform edge shortening,
v0
N pfp ¼
v b2 N afpf E1 t 3l
; Ey ¼
eq
Ey E1
.
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
90
Nfpf* 3 2.5 2 1.5 1 0.5
75
T1 [deg]
60
45
30
15
0
0
15
30
45
60
75
90
T0 [deg] N av b2 Fig. 6. Normalised first-ply failure load N pfp ¼ Efpft3 field contours in the linear 1 l
fibre angle variation domain. The straight line identifies straight fibre panel solutions.
orientation angles above 60°. For fibre orientation angles between 35° and 55°, the failure loads of constant-stiffness panels is reduced. This is also evident from the depression in the curve corresponding to these configurations in Fig. 5b. A closer look at the FEM results reveal that this dip in the Nfpf field corresponds to panels that fail (prematurely) due to stress concentrations around the hole edge. The failure response curve for the straight-fibre flat panel version, shown Fig. 5a, lacks such a depression. The most promising variable-stiffness panels in terms of buckling and failure under pure compressive loads do not fail at the hole edge. On the other hand, the constant-stiffness panels that perform better in terms of buckling loads end up by failing by the hole edge. Actually, with exception from a small range of fibre orientation angles around 30°, all straight-fibre panels typically with h < 60° fail at such locations. As for the buckling response, the reason for the superior performance of variable-stiffness panels in terms of failure loads, as compared to the straight-fibre counterparts, is a favourable distribution of the applied load across the panel height as a function of the x-coordinate. The configurations for which the value of Nfpf is high are characterised by small values of the fibre orientation angle at the panel centreline (x = 0) and large values at the edges x = ±a/2. The local axial stiffness of the regions of the panel is high at locations where the fibre orientation is closer to 90°. As a result, most of the applied load introduced via uniform edge shortening is carried by the highly stiff regions of the panel located near their simply-supported edges. The introduction of the central hole does not greatly disturb the stress field and does not create stress concentrations higher than the ones forming near the panel transverse edges. Due to the favourable fibre alignment, at these locations the load is mostly supported along the local longitudinal directions. This architecture leads to strong laminates since the longitudinal strength of plies is much higher than their transverse strength, and also the strain-to-failure is higher in the transverse direction than in the longitudinal direction. There is a second reason for the superior failure performance of the tow-steered format: their superior postbuckling response. First-ply failure in the [±h0j80i]6s panel, as an example, occurs where the added influences of the directly applied longitudinal compressive stresses and the bending induced ones (due to buck-
1765
ling) is the highest, i.e. fibre kinking is triggered by bending-induced compressive stresses on the concave side of the buckles, in regions close the transverse edges where the direct compressive loads are higher. Furthermore, failure initiates at face plies because the bending induced stresses are higher there and their in situ strength is lower than for their embedded (constrained) counterparts [22]. These considerations lead to the conclusion that, for failure in the postbuckling regime in general, high buckling loads promote high failure loads. Hence, the improved failure performance of variable-stiffness panels is also a consequence of their improved postbuckling response. The [±80]6s laminate, for instance, has a favourable fibre distribution in terms of strength but its critical buckling load is rather low. As a consequence, face ply fibres at the concave side of the buckle start to kink prematurely due to added effects of direct and bending induced compressive loads.
3. Response of manufacturable panels The analyses reported in the previous section concern fibresteered designs with ‘ideal’ fibre courses, i.e. where there is a continuous shifting of the reference path in the direction perpendicular to the fibre orientation variation. Real manufacturing conditions impose some constraints to the designs idealized by Gürdal et al. [8–10]. The AFP heads typically have the capability of accommodating up to 36 fibre tows within a 76.2 mm (3 in.) course width. This means that within each head pass or course, up to 36 tows are placed parallel to each other and to the reference path. The shifting is, therefore, made discretely at multiples of the course width. The boundaries of neighbouring courses do not match for all locations along the x-axis. This leads to irregular regions between tow courses that can be accounted for by prescribing the shift distance so that no gaps occur. However, for curved reference paths, this practice produces overlap regions. To prevent the overlap regions the tow-placement machine can be instructed to cut tows individually so that no thickness builds up. This practice results in laminates that contain small wedge-like areas free of fibres due to the dropping of the individual, finite thickness tows. These small fibre-free areas are likely to create resin-rich regions during the laminate curing process. The difference between ‘ideal’ and ‘manufacturable’ fibre-steered plies is exemplified in Fig. 7. In order to avoid the collocation of course edges and of resinrich areas, that would occur at the same places through-the-thickness of a laminate in clustered plies with the same fibre angle distribution, Gürdal et al. [23] proposed to stagger the layers, i.e. to shift the fibre placement courses of one layer, in one of the planar directions, in relation to the adjacent layer with the same fibre orientation distribution. Experimental testing [14,15] revealed no influence of ply staggering on panel buckling response. In this section, the most promising designs, in terms of first-ply failure performance, are analysed in detail. That is, the [±h0j80i]6s variable-stiffness panel is compared with the [±80]6s straight-fibre laminate. Additionally, the [±45]6s straight-fibre design is analysed as well, since this configuration may be preferable to the [±80]6s laminate due to its superior buckling performance. Some of the issues that arise due to the practical manufacturing of these panels are taken into account. A nominal course width approximately equal to 1/4th of the panel height is adopted. This means that, on average, each ply can be laid completely with four AFP machine head passes. The effects of ply staggering on panel response are also analysed. The results for the cases studied in this section are reported in Table 2. The predicted overall first-ply failure load, N afpfv , for the [±h0j80i]6s ‘manufacturable’ panel is 469.5 N/mm, only 1.2% higher
1766
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
Fig. 7. Fibre directions on ‘ideal’ and ‘manufacturable’ tow-steered plies with a central hole (T0 = 0°, T1 = 80°). The course width on the manufacturable ply is approximately equal to 1/4th of the panel height.
Table 2 Buckling and strength results for the [±h0j80i]6s ‘ideal’ and ‘manufacturable’ designs. The strength performance is compared with that for the [±80]6s and [±45]6s laminates. VSP – variable-stiffness panel. Design
N acrv (N/mm)
N afpfv (N/mm)
vs. [±80]6s (%)
vs. [±45]6s (%)
Straight fibres ([±45]6s) Straight fibres ([±80]6s)
232.7 148.8
244.7 464.1
47.2 –
– 89.7
Ideal VSP Manufacturable VSP Manufacturable VSP (staggered plies)
555.8 548.6 543.9
559.6 469.5 523.4
20.0 1.2 12.8
120.5 91.9 113.9
than the 464.1 N/mm calculated for the [±80]6s laminate, and much lower than what was expected, having in mind the results achieved by the ‘ideal’ design. In terms of predicted critical buckling loads, the difference between ‘ideal’ ðN acrv ¼ 555:8 N=mmÞ and ‘manufacturable’ ðN acrv ¼ 548:6 N=mmÞ designs is only about 1%. The longitudinal compressive failure index field for the critical surface ply of the [±h0j80i]6s panel with tow-drops, corresponding to N afpfv ¼ 469:5 N=mm, is depicted in Fig. 8a. The figure shows values of the failure index approaching 1.0 near the course edges, representing the occurrence of first-ply failure. This means that stress
UVARM7 fraction = − 0.98 0.81 0.65 0.49 0.32 0.16 −0.00
concentrations develop in those areas, because of the fibre angle mismatch between adjacent courses. These discontinuities trigger premature first-ply failure. The repeatability of plies in the laminate determines that the layerwise stress concentrations develop at the same (x, y)-coordinates through-the-thickness of the panel, therefore there are no alternative paths for stress relief. In an attempt to lower stress concentrations, by smearing them over wider areas of the panel and promoting alternative load paths, the technique of ply staggering is applied to the [±h0j80i]6s laminate. The predicted first-ply failure for this design is
UVARM7 fraction = 0.99 0.83 0.66 0.50 0.33 0.16 −0.00
Fig. 8. Longitudinal compressive failure index field, at first-ply failure load, for the critical surface ply of the [±h0j80i]6s panel with tow-drops.
C.S. Lopes et al. / Composites: Part A 41 (2010) 1760–1767
N afpfv ¼ 523:4 N=mm, 12.8% higher than for the [±80]6s laminate while the critical buckling load is kept nearly the same as that of the non-staggered panel (less than 1% difference). This failure load is still lower than the value predicted for the ‘ideal’ case, but higher than for the configuration without ply staggering. The fibre compression failure index for this design, corresponding to N afpfv ¼ 523:4 N=mm, is shown in Fig. 8b. The highly stressed regions are now dispersed over wider areas resulting in a more uniform loading of the panel. This lowers the peak stresses and postpones failure to higher applied loads.
1767
stiffening panel edges, tow-steering might also be a good approach in increasing the effectiveness of bonded or bolted joints. Acknowledgement Partial funding of this research through the Scholarship SFRH/ BD/16238/2004 from the Portuguese Foundation for Science and Technology (FCT) is gratefully acknowledged. Additionally, the financial support of the FCT under the project PTDC-EME-PME64819-2006 is acknowledged by the third author. References
4. Conclusions The buckling and postbuckling first-ply failure response characteristics of variable-stiffness panels fabricated by AFP technology were analysed. Both in terms of buckling and first-ply failure, important advantages in comparison with constant-stiffness formats were demonstrated. Fibre-steered panels offer significant performance improvements in terms of critical buckling loads. This is due to a favourable redistribution of the applied loads towards the supported panel edges resulting in an unloading of its central section. This is the effect already reported in the framework of stiffness tailoring efforts made by other researchers such as Biggers et al. [1–4]. Moreover, the gains in structural performance are of the same order of magnitude. The advantage of the current tailoring concept is that the in-plane stiffness properties vary continuously, avoiding critical areas as abrupt changes in panel thickness, and ply drop-offs. The effect of load transfer towards the panel edges can be so pronounced that the optimum variable-stiffness designs with a central hole have nearly the same initial buckling loads as panels with the same distribution of fibre orientations but no hole. The same lack of sensitivity to the presence of a cutout was demonstrated in terms of panel failure. This is one of the major advantages of tow-steered designs in comparison with traditional designs. In principle, these results should be relatively little affected by the shape of the cutout, e.g. having a central rectangular window with rounded corners instead of a circular hole of similar size. This is because of the effect of the stiffness variation on the global behaviour of a panel. However, the structural performance is expected to decrease if the cutout should not be centrally located. There are clear benefits in using tow-steered designs in terms of nominal panel strength, although not as pronounced as in terms of critical buckling loads. For panel failure in the postbuckling regime, these improvements have two different causes. One is the load redistribution towards the panel supported edges and a favourable fibre alignment at these locations such that the load is mostly supported along the local longitudinal direction. The other reason is related to the higher buckling loads. Panel buckling adds bending induced stresses to the directly applied loads, therefore, the high critical loads of some variable-stiffness panels also improves their failure performance. The manufacturability of fibre-steered laminates imposes some design constraints that limit their advantages in terms of first-ply failure performance, specifically the fibre angle mismatches at course edges due to the finite width of the AFP machine head passes that create stress concentrations. These stress concentrations can be mitigated by using narrower tow courses or by staggering the laminate plies. The results reported herein are encouraging of applying the technique of tow-steering to several other structural cases, such as tensile loaded panels and composite panels containing notches or multiple neighbouring cutouts, in the expectation of achieving similar gains in performance. Moreover, due to the possibility of
[1] Biggers SB, Srinivasan S. Compression buckling response of tailored rectangular composite plates. AIAA J 1993;31(3):590–6. [2] Biggers SB, Srinivasan S. Postbuckling response of piece-wise uniform tailored composite plates in compression. J Reinf Plast Compos 1994;13:803–21. [3] Baranski AT, Biggers SB. Postbuckling analysis of tailored composite plates with progressive damage. Compos Struct 1999;46:245–55. [4] Xie D, Biggers SB. Postbuckling analysis with progressive damage modeling of tailored laminated plates and shells with a cutout. Compos Struct 2003;59:199–216. [5] Hyer MW, Charette RF. Use of curvilinear fiber format in composite structure design 1991;29(6):1011–5. [6] Hyer MW, Lee HH. The use of curvilinear fiber format to improve buckling resistance of composite plates with central circular holes. Compos Struct 1991;18:239–61. [7] Nagendra S, Kodiyalam S, Davis JE, Parthasarathy VN. Optimization of tow fiber paths for composite design. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 36th structures, structural dynamics and materials conference, AIAA 95-1275, New Orleans, LA; 1995. p. 1031–41. [8] Gürdal Z, Olmedo RA. Composite laminates with spatially varying fiber orientations: variable stiffness panel concept. In: Proceedings of the AIAA/ ASME/ASCE/AHS/ASC 33rd structures, structural dynamics and materials conference, AIAA 92-2472, vol. 2, Dallas, TX; 1992. p. 798–808. [9] Gürdal Z, Olmedo R. In-plane response of laminates with spatially varying fiber orientations: variable stiffness concept. AIAA J 1993;31(4):751–8. [10] Olmedo RA, Gürdal Z. Buckling response of laminates with spatially varying fiber orientations. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 34th structures, structural dynamics and materials conference, AIAA 93-1567, La Jola, CA; 1993. p. 2261–9. [11] Gürdal Z, Tatting BF, Wu KC. Variable stiffness panels: effects of stiffness variation on the in-plane and buckling responses. Composites Part A: Appl Sci Manuf 2008;39:911–22. [12] Wu KC, Gürdal Z. Thermal testing of tow-placed, variable stiffness panels. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 42nd structures, structural dynamics and materials conference, AIAA 2001-1190, Seattle, WA; 2001. [13] Wu KC, Gürdal Z, Starnes Jr JH. Structural response of compression-loaded, tow-placed, variable stiffness panels. In: Proceedings of the AIAA/ASME/ASCE/ AHS/ASC 43rd structures, structural dynamics and materials conference, AIAA 2002-1512, Denver, CO; 2002. [14] Jegley DC, Tatting BF, Gürdal Z. Optimization of elastically tailored tow placed plates with holes. In: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 44th structures, structural dynamics and materials conference, AIAA 2003-1420, Norfolk, VA; 2003. [15] Jegley DC, Tatting BF, Gürdal Z. Tow-steered panels with holes subjected to compression or shear loading. In: Proceedings of the AIAA/ASME/ASCE/AHS/ ASC 46th structures, structural dynamics and materials (SDM) conference, AIAA 2005-2017, Austin, TX; 2005. [16] Lopes CS, Camanho PP, Gürdal Z, Tatting BF. Progressive failure analysis of tow-placed, variable-stiffness composite panels. Int J Solids Struct 2007;44:8493–516. [17] Abdalla MM, Gürdal Z, Abdelal GF. Thermo-mechanical response of variablestiffness composite panels. J Therm Stress 2009;32:187–208. [18] Abaqus, Inc., Abaqus Version 6.8 User’s Manual, Pawtucket, RI, USA; 2008. [19] Lopes CS, Gürdal Z, Camanho PP. Variable-stiffness composite panels: buckling and first-ply failure improvements over straight-fibre laminates. Comput Struct 2008;86:897–907. [20] Dávila CG, Camanho PP, Rose CA. Failure criteria for FRP laminates in plane stress. J Compos Mater 2005;39:323–45. [21] Pinho ST, Dávila CG, Camanho PP, Iannucci L, Robinson P. Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity, Technical report, NASA, Langley Research Center, Hampton, VA, NASA/TM-2005-213530; February 2005. [22] Camanho PP, Dávila CG, Pinho S, Iannucci L, Robinson P. Prediction of in situ strengths and matrix cracking in composites under transverse tension and inplane shear. Composites Part A: Appl Sci Manuf 2006;37:165–76. [23] Gürdal Z, Tatting BF, Wu KC. Tow-placement technology and fabrication issues for laminated composite structures. In: Proceedings of the AIAA/ASME/ASCE/ AHS/ASC 46th structures, structural dynamics and materials conference, AIAA 2005-2017, Austin, TX; 2005.