Optimization of a composite cylinder under bending by tailoring stiffness properties in circumferential direction

Composites: Part B 41 (2010) 157–165

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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

Optimization of a composite cylinder under bending by tailoring stiffness properties in circumferential direction Adriana W. Blom a,b,*, Patrick B. Stickler c, Zafer Gürdal a a

Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands Stork Fokker AESP, Industrieweg 4, 3351LB Papendrecht, The Netherlands c The Boeing Company, 3003 W. Casino Road, Everett, WA 98204, USA b

a r t i c l e

i n f o

Article history: Received 24 October 2008 Received in revised form 20 August 2009 Accepted 7 September 2009 Available online 20 October 2009 Keywords: B. Variable-stiffness E. Advanced fiber placement C. Cylinder C. Bending B. Buckling C. Optimization

a b s t r a c t A fiber-reinforced cylindrical shell with given geometry and material properties is optimized for maximum load-carrying capability under bending. The shell is assumed to be built using an advanced fiber-placement machine, which allows in-plane steering and overlapping of fibers, resulting in a so-called variable-stiffness shell. The design methodology for strength and stiffness variation in circumferential direction by means of fiber placement is explained and restrictions on the manufacturability are specified. Implementation in the commercially available finite element package ABAQUSÒ for structural analysis is described. Subsequently, the cylinder is optimized to carry a maximum buckling load under bending, while applying a strength constraint. Constraints on the global stiffness are imposed by means of comparison with a baseline quasi-isotropic shell, while a matrix dominated lay up is avoided at all locations in the laminate in order to ensure that the laminate is strong enough in all directions in case a hole is present. Optimization is done using a surrogate model in order to minimize the amount of finite element analyses. Improvements of up to 17% are obtained by changing the load path. The tension side is made stiffer and the compression side softer in longitudinal direction by changing the fiber orientation from near zero at the upper (tension) side to higher fiber angles at the lower (compression) side, such that load is relieved from the compression side. This results in a higher load-carrying capability of the cylinder. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Many large aerospace structures, such as airplane fuselages and rocket stages, consist of cylindrical sections. These structures are subject to numerous combined load cases, which complicate structural optimization. In this paper only one primary load case for a fuselage is applied to an unreinforced cylindrical shell in order to show the methodology and to demonstrate its potential for realistic problems. Due to the demand for lighter structures in the aerospace industry, increased use is made of fiber-reinforced composites. The directionality of these materials is used to increase structural performance or reduce weight. Furthermore, structures can be more integrated, which reduces part count and eliminates complete assembly steps, thereby saving time and cost. Automation of the production process improves the quality and reproducibility of fiber-reinforced composites, while reducing manufacturing costs when compared to manual production.

* Corresponding author. Address: Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands. E-mail address: [email protected] (A.W. Blom). 1359-8368/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2009.10.004

Among the automated production processes, placing fibers using an advanced fiber-placement machine is becoming increasingly popular, because of the vast capabilities of the machine. A typical fiber-placement machine [1] has seven computer controlled primary axes that allow for very precise control of the fiber orientation, even on complex shapes. Current generation of machines has a fiber delivery system with a capacity of up to 32 individually directed 1/8 in. tows, which can all be cut and restarted independently. This control of individual tows provides the possibility of varying the thickness locally. In addition, the machine head is capable of steering the fibers within the planes of the laminate. Consequently the stiffness of the laminate can be varied locally, creating so-called variable-stiffness laminates. This results in an increased design space when compared to the conventional straight-fiber, constant-stiffness laminates. Variable-stiffness laminates using advanced fiber placement technology have been used to optimize the buckling load of flat plates by Tatting, Gürdal and Jegley [1–4], showing considerable improvements in load-carrying capabilities. Similar methods applied by Alhajahmad et al. [5,6] also proved to be beneficial for the pressure pillowing problem of fuselage panels with and without cutouts. The concept of continuously varying fiber angles has been extended to conical and cylindrical shells by Blom et al. [7]

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who varied the fiber angle as function of the longitudinal coordinate. They showed improvements in fundamental frequency for conical shells with different cone angles and sizes [8]. In this paper a cylindrical shell will be optimized for buckling under constant bending by varying the fiber angles as a function of the circumferential coordinate. Earlier work on circumferential stiffness variation has been carried out by Tatting [9] who used a linear membrane solution to find the optimum fiber angle variation for cylinders with multiple (combined) load conditions. He noted that a circumferential stiffness variation was most beneficial for load conditions that also contained variation in the circumferential direction. A linear angle variation was applied within segments of the cylinder circumference, where the design variables could vary in 15° increments from one segment to another. Optimization was done with genetic algorithms. Sun and Hyer [10] tailored the circumferential stiffness to improve the buckling capacity of elliptical composite cylinders under axial load and showed that the entire cylinder could be made to participate in the buckling deformations thereby increasing the buckling load. The current paper will optimize the buckling load of a cylinder in pure bending by using a multiple-segment constant curvature fiber angle variation in circumferential direction, while taking into account manufacturing constraints, an equivalent 10% rule for composites, and assuring that buckling occurs before material failure. Structural analysis is being carried out by means of a finite element model in ABAQUSÒ [11] and optimization is done with a surrogate-model-based method in Design Explorer [12,13] in order to minimize the number of finite element analyses. In the following, first an introduction to advanced fiber placement technology will be given, as well as an explanation of the path definitions that will be used to create the fiber-steered laminates. Then the implementation of the variable-stiffness properties in the finite element model will be described. Furthermore the design problem will be defined and the solution procedure will be given. Finally the optimization results will be presented and recommendations for future work will be offered.

2. Advanced fiber placement technology and path definitions for fiber steering Advanced fiber-placement machines place up to 32 tows of 1/ 8 in. pre-preg slit tape on the surface, which all together are called a course. The course centerline follows a specified path, normally straight, but in the case of variable-stiffness composites curved paths are implemented. The path coordinates are defined by the desired fiber orientation angle u, which is linked to the path tangent as follows:

dy ¼ tan u dx

ð1Þ

The amount of angle variation that is possible is limited by the manufacturing process. If fibers are steered too much, they might start wrinkling at the inside of the turn, thereby reducing the quality of the product. Since this is undesirable, a minimum turning radius, qmin, for the fiber path is defined. If the turning radius of a fiber path is larger than this value, no wrinkling of fibers will occur. This constraint is known as the curvature constraint (where curvature is the inverse of the turning radius), because the limit can also be expressed in terms of the curvature, j = 1/q. A complete ply is constructed by placing multiple courses that are generated by shifting a course perpendicular to the direction of variation. Since the courses will not be parallel (unless they are straight), either gaps or overlaps will occur. In Fig. 1 two courses with an angle variation as function of the horizontal coordinate are depicted. Hence they are shifted in vertical direction. If

the course width w is kept constant the effective width in the direction of shift we varies with the fiber angle, as can be seen in Fig. 1b. Assuming gaps are undesirable (Fig. 1a), the amount of shift is chosen such that gaps are avoided and only overlaps exist (Fig. 1b). In many situations however, also overlaps are undesired. This can be solved by using the tow cut/restart capability of the machine such that a constant thickness ply is created (Fig. 1c). On a cylinder the fiber orientation angle u is defined with respect to the longitudinal axis, as shown in Fig. 2. In this work the fiber angle is varied as a function of the circumferential coordinate h and varies over multiple circumferential segments of the cylinder. In order to easily evaluate the curvature constraint for manufacturability the path definition is chosen to have a constant inplane curvature within a segment. The angle variation then becomes:

cos uðhÞ ¼ cos T i þ ðcos T iþ1  cos T i Þ

h  hi hiþi  hi

ð2Þ

In this equation Ti is the fiber angle at the hi location around the circumference. At this point there is a transition between two segments, where the fiber angle itself is continuous, but the in-plane curvature changes. For a cylinder with radius R the in-plane curvature j within one segment is:

j¼

cos T i  cos T iþ1 Rðhiþ1  hi Þ

ð3Þ

The short length of the cylinder L in Fig. 2 prevents the path from wrapping around, but if the cylinder would have been longer, eventually a path that starts at h = 0° would reach h = 2p°. This is illustrated in Fig. 3 in which an expanded cylinder is shown with one complete path on it (shown in blue1) which is then shifted horizontally to create a complete ply. The fiber angle is varied as a function of the azimuth angle h (on the vertical axis) and therefore the basic path is shifted in longitudinal direction. The amount of shift depends on the minimum effective width we,min with all 32 tows being placed and the distance L2p at which the fiber path would wrap around the cylinder. Ideally the shift between two courses is exactly we,min in order to minimize the amount of overlap, but a regular pattern requires a discrete number of courses N within the distance L2p, i.e.:

N ¼ ceil



jL2p j we;min



ð4Þ

where ceil rounds the expression in the brackets to the nearest larger integer. Then the shift Dx becomes:

Dx ¼

jL2p j N

ð5Þ

The resulting fiber paths thus form a regular pattern. Even though the paths on the cylinder with length L = 32 in., shown in black in Fig. 3, seem to be completely different paths, they all originate from the basic path shown in blue. The basic path itself is defined by the design variables Ti which are spaced around the circumference at 45° increments, as shown on the right of the figure. In order to maintain periodicity the fiber angle at h = 2p, T8, is defined to be the same as the fiber angle at h = 0, T0. The effective course width we depends on the fiber orientation and is approximated by:

we 

w sin u

ð6Þ

The exact calculation of the effective course width can be found in Ref. [7]. If a constant thickness ply is required the course width w has to be adjusted such that the effective course width is equal 1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.

159

e

Direction of shift

w

Ov er l ap

Direction of shift

Direction of shift

A.W. Blom et al. / Composites: Part B 41 (2010) 157–165

w

p

Ga

we,min

Direction of variation

Direction of variation

Direction of variation

Fig. 1. Laminate construction by shifting courses.

T

0

angle approaches zero the course width needs to be reduced to almost zero as well. In practice this is not possible because a full course consists of 32 discrete tows that are 1/8 in. wide and a near zero course width would result in placing no fibers at all. Therefore the minimum fiber orientation in this paper is set to be 10°, such that the minimum course width is at least 5 tows.

θ0 Stage 1

θ1

ϕ(θ)

Direction of variation

Stage 2 T1

θ

3. Implementation of variable-stiffness properties in a finite element model

θ2 Direction of shift

Once the laminate geometry has been defined the stiffness properties have to be included in a finite element model to perform structural analysis. For this purpose the finite element package ABAQUSÒ [11] has been selected. Use is made of an S4 shell element in combination with the UGENS user subroutine. The S4 shell element is a fully integrated, general-purpose, finite-membranestrain shell element, with four nodes and four integration points. The UGENS subroutine is a user-written FORTRAN subroutine that passes the shell stiffness for each integration point to ABAQUSÒ. The local stacking sequence is calculated based on the position of the element, the material properties and the laminate definition. Then the ABD matrix is calculated with Classical Lamination Theory and the ABD matrix is provided to ABAQUSÒ. Since ABAQUSÒ does not have any information about the stacking sequence post

Stage 3 θ3

T2

Fig. 2. Fiber angle and segment definition.

to the amount of shift Dx. This is realized by cutting and restarting tows along the path. From Eq. (6) it can be seen that as the fiber

paths within real cylinder paths outside real cylinder

Azimuth angle, θ [deg]

L

baseline path

L 2π

360

T8 = T0

315

T7

270

T

225

T5

6

180

T4

shift 135

T

90

T2

45

T1

3

0 0

20

40

60

80

100

120

140

Length along the shell, x [in]

Fig. 3. Fiber paths on an expanded cylinder.

160

180

200

T0

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processing of the results to obtain stresses or to evaluate strength constraints is done using a Python script. An overview of the analysis sequence is given in Fig. 4. A performance comparison was done between the approach described above and a method that used direct input of the stacking sequence through a composite shell element, but the latter one needed more than 20 times the amount of computation time needed for the USER subroutine method. Hence the latter is the preferred method for the implementation of variable-stiffness properties. The local stacking sequence is calculated using the layup that results from the manufacturing process, which deviates a little from the ideal circumferential variation. This is shown in Fig. 5. The fiber angles are plotted in a surface contour plot. Ideally, the fiber angle is only a function of the circumferential coordinate, as shown in Fig. 5a. However, due to the finite width of the courses there is some deviation from the ideal fiber angle distribution (Fig. 5b). The latter one is modeled by discrete elements in ABAQUSÒ (Fig. 5c). Once the stiffness is defined for all the elements, any regular analysis can be done within ABAQUSÒ. A study on the influence of course width and resulting deviations from the ideal fiber angle distribution was performed by Blom et al. [14] and showed that sensitivity of the calculated buckling load to deviations in fiber orientation due to course width is very small.

4. Design and optimization formulation Only one load case is considered in this paper, which is pure bending around the horizontal axis of the cylinder, such that the upper side is loaded in tension and the lower side is loaded in compression. This load case especially lends itself for a demonstration of circumferential stiffness tailoring as the load distribution varies around the circumference as well. 4.1. Cylinder and laminate configuration The cylinder has a diameter of 24 in. and is 32 in. long. It consists of 24 plies, made of AS4/8773 material [15] for which the mechanical properties are given in Table 1. For simplicity the material strength properties for the steered configuration are assumed to be the same as for the traditional configuration. Since for the steered plies the thickness is kept constant by cutting and restarting tows these locations might weaken the laminate locally, thereby invalidating the assumption of equal strength properties. Future research is needed to address this issue. At the two ends the cylinder is assumed to be fixed to end plates so that the cross section remains circular and in one plane, while rigid body motion

of the end plates as a whole is allowed. The bending moment is applied at the center of the end plates. The finite element model consists of 71 elements in longitudinal direction and 170 elements in circumferential direction, which is sufficient to capture the varying stiffness and results in a run time of approximately 20 min per analysis including pre and post processing on a quad core AMD Athlon with 1 GHz processors and 4 GB RAM per CPU. The 24 ply laminate is defined to be symmetric and balanced and therefore contains only six different ply definitions. In order to have some impact resistance and a smooth outer surface, the outer plies are set to ±45°, because of the impact resistance of ±45 plies and because an outer ply without tow drops is desirable. This leaves five plies that can be varied. These are picked to be either 0j90 or ±u(h). The variable-stiffness plies are defined to vary in eight segments around the circumference with symmetry about the vertical axis resulting in five design variables per ply: T0, T1, T2, T3 and T4, as indicated in Fig. 6. Multiple laminates consisting of combinations of 0j90° and steered plies were optimized. The three laminate definitions that came out best are given in Table 2. In this table only half of the symmetric layup is given. 4.2. Optimization using a surrogate model The optimization is done using the surrogate model optimizer in Design Explorer [12,13] in order to minimize the number of finite element (FE) analyses. A general overview of the optimization process is shown in Fig. 7. First a design of experiments is generated to systematically sample the design space. Secondly, surrogate models are constructed in order to approximate the responses given by the expensive finite element analysis. These models are being used to analyze the influence of the design variables on the responses, as well as to serve as a basis for the optimization. After the design of experiments is performed, the optimizer itself selects a set of points that serve as a starting point for local optimizations. At these specific points the FE analysis is carried out and the outcomes are being analyzed by the optimizer again. Also additional sets of random points will be evaluated using the FE analysis in order to reduce the chances of ending up in a local optimum. Once the new results are added the models are updated and a new iteration starts, until one of the termination criteria is met. For laminate 1 the maximum fineness of the poll grid which was set to 64 was reached, while for laminates 2 and 3 the maximum number of finite element analyses was reached, which was set to 250. The surrogate model optimizer is suitable for the current design problem because derivative information that is needed for the optimization can be taken from the surrogate models, removing

ABAQUS CAE ANALYSIS

Fortran program Data input: geometry, materials laminate definition

Fortran UGENS subroutine

ABAQUS input

Coordinates

ABD matrix

ABAQUS finite element analysis

ABAQUS output

Python postprocessing

Analysis output

Fig. 4. Analysis scheme.

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Upper side, tension T0 T1

T1

θ

T2

T2

M

T3

T3

T4

Lower side, compression Fig. 6. Definition of the design variables per ply.

the need to determine the derivatives by means of finite differences. Furthermore, the surrogate models introduce smoothness and filter out noise that could be present in the objective and constraint functions. Since the structure is being optimized for the lowest buckling load, the objective function might not be completely smooth due to changes of the first buckling mode between different designs. Also, strength and effective stiffness (described in the next section) are determined for each element, after which the most critical value over the entire domain is used to evaluate the constraints. This could also introduce irregularities in the constraint functions. Hence, the use of surrogate models increases the chance that the optimization converges. 4.3. Optimization formulation

Fig. 5. Fiber angle distributions within one ply.

Table 1 AS4/8773 Mechanical properties [15]. E1 E2 G12

m12 tply

q Xt Xc Yt Yc S

18.83  106 psi 1.317  106 psi 7.672  105 psi 0.32 0.0072 in. 0.0574 lbs/in3 2.999  105 psi 1.680  105 psi 1.925  104 psi 2.898  104 psi 1.688  104 psi

The objective of the optimization is to maximize the buckling load. Since one of the cylinders resulting from the optimization presented here will be built and tested, it is determined that it should be buckling critical and not strength critical. For this purpose a first ply failure criterion based on a strain equivalent of the Tsai-Wu criterion is used. A full description of this method can be found in Ref. [16]. Furthermore the laminate should not be matrix dominated in any of the four major directions (0, 90, ±45), so that no problems occur if a fastener is put into the structure. This constraint is enforced by requiring the effective stiffness in each of the four directions E0,90,±45 to be larger than a certain threshold that is derived from the 10% rule for traditional laminates (E* = 4.207  106 psi). Besides the strength requirements the global stiffness of the variable-stiffness cylinders should be comparable to the best traditional laminate, which is accomplished by limiting the amount of deflection at a one in-kips bending moment to the deflection of the baseline cylinder. The deflection of the baseline cylinder at a one in-kips bending moment as calculated by ABAQUS was 6.22  105 in. Finally the manufacturability constraints are imposed by restricting the amount of in-plane curvature j to be less than 1/20 in1 (i.e. a minimum turning radius of 20 in.). Mathematically the optimization problem is defined as:

Maximize

M cr

Subject to M f P M cr

in all elements

d 6 d Ed P E jji j 6 j

for d ¼ 0; 90; þ45; 45; in all elements for i ¼ 1; 2; . . . ; N ð7Þ

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Table 2 Variable-stiffness laminate definitions. Ply number (half of the symmetric layup)

Laminate number

1 2 3

1

2

3

4

5

6

7

8

9

10

11

12

+45 +45 +45

45 45 45

+u1 +u1 0

u1 u1 90

+u2 0 +u2

u2 90 u2

+u3 +u3 0

u3 u3 90

+u4 0 +u4

u4 90 u4

+u5 +u5 0

u5 u5 90

OPTIMIZATION Define domain: T per ply i

Define experiment

Run experiment ANALYSIS

Build surrogate models

Define optimization

Run optimization ANALYSIS

Fig. 7. Optimization scheme.

In this formulation Mcr and Mf, respectively, represent the buckling moment and the moment at which first ply failure occurs, d is the deflection at a one in-kips bending moment and d* is the maximum allowed deflection at this load level, E* is the threshold for the equivalent laminate stiffness, and j represents the in-plane curvature for a steered fiber path within a given ply and within a given segment, where N is the total number of segments over which it is being steered (i.e. N is equal to four times the number of steered ply definitions). As mentioned before, the design variables are T0, T1, T2, T3, and T4 for each unique steered ply definition present in the laminate. These angles are allowed to vary between 10° and 89°. 4.4. Traditional baseline laminate Before optimizing a laminate containing variable-stiffness plies, a traditional laminate with 0°, 90°, and ±45° plies is optimized with the same constraints (except the deflection constraint) as described in the previous section. The optimum configuration will serve as the baseline laminate against which the variable-stiffness laminates are compared. As with the steered laminate, the outer plies are constrained to be ±45°. Some basic stacking rules are applied to these traditional laminates, i.e.:  Ten percent rule: each fiber direction (0°, 90°, and ±45) should constitute at least 10% of the total number of plies within the laminate.  No more than three plies with the same fiber orientation can be grouped together.  Every +45° ply is accompanied by a 45° ply. The optimum laminate determined is: [±45, 02, ±45, 02, 90, ±45, 90]S, which has a buckling moment of 5293 in-kips and a material failure moment of 5851 in-kips. The downward deflection at a bending moment of one in-kips is 6.22  105 in. 5. Optimum variable-stiffness configurations The optimization results for the variable-stiffness laminates listed in the previous section are summarized in Table 3. In this

table the buckling moment and material failure moment are given. The safety factor for material failure is calculated at all elements and the lowest value for the bending moment is reported as being the material failure moment. The same procedure is followed for the equivalent stiffness values. Since all analyses are linear, deflections are compared at a bending moment of one in-kips. For brevity the in-plane curvature values are not displayed here; all values are within the limit of 0.05 in1, some of them are active constraints. For all steered cylinders the strength constraint was active, while the deflection constraint was not. The fact that in Table 3 there is still a small margin between the strength values and the buckling values is due to scaling of the constraint values. Also, optimizations without any constraints applied showed that the buckling load could be increased if no constraint on strength was present. The equivalent stiffness constraint was active only in laminate 1; due to the number of 0° and 90° plies in laminates 2 and 3 the equivalent stiffness constraint was already close to being satisfied and therefore not active in the optimization. From Table 3 it can be seen that laminate 2 shows the most improvement, even though it only has 15 design variables compared to 25 design variables of laminate 1. This can be explained by the fact that the design space for laminate 2 is not a subset of the design space for laminate 1: A ±u(h) ply combination can never result in a 0j90 ply combination. Furthermore, the requirement of a minimum fiber angle of 10° in a steered ply significantly reduces the longitudinal stiffness of the ply combination, unless the fiber angle of both plies is close to 10°. Since the equivalent stiffness requirement rules out this possibility the ±u(h) ply combination will always be less effective than the 0j90 combination. In the following the importance of the laminate longitudinal stiffness will become evident. Since laminate 2 shows the most improvement, this laminate definition is examined more closely. The other variable-stiffness laminates show similar behavior and therefore they are not discussed here. Laminate 2 contains three steered ply definitions, for which the design variables are listed in Table 4 and for which the fiber angle variations are shown in the form of course centerlines in Fig. 8. The T0 and T1 variables are located at the upper side of the cylinder, which is in tension (see Fig. 6), while T2 is close to the neutral axis and T3 and T4 are located at the compression side

Table 3 Summary of results.

Laminate number

Baseline 1 2 3

Mcr (in-kips)

M f (in-kips)

E0 (msi)

E90 (msi)

E45 (msi)

E45 (msi)

Displacement (in.)

Improvement in buckling load (%)

5293 6077 6188 5938

5851 6095 6198 5949

8.65 7.49 7.45 6.70

5.99 4.26 4.94 6.40

7.25 4.99 4.82 4.83

7.25 4.99 4.82 4.83

6.22  105 5.15  105 5.10  105 5.44  105

– 15 17 12

A.W. Blom et al. / Composites: Part B 41 (2010) 157–165 Table 4 Optimum ply variables for laminate 2: [±45, ±u1(h), 0, 90, ±u3(h), 0, 90, ±u5(h)]S

u1(h) u3(h) u5(h)

T0

T1

T2

T3

T4

10.0 10.0 10.0

10.0 10.0 10.0

10.0 10.0 22.0

11.7 18.2 37.5

25.1 40.1 71.1

of the cylinder. In all three steered plies the fiber angle is constant on the upper side (i.e. T0 = T1 = 10°) and changes to larger orientation angles toward the lower side. Obviously small fiber angles are preferred in tension as the fibers are most effective in carrying tension loads. The increasing orientation in the lower side causes the laminate to soften, thereby relieving the amount of load that is being carried by this region. This is shown in Fig. 9 in which both the longitudinal strain component, one of the fiber angle variations, and the normalized section forces are presented as function of the azimuth angle h. Since the cylinder ends have to remain planar the strain variation with the azimuth angle is sinusoidal (left plot in Fig. 9). When the laminate is made of constant stiffness plies the section forces also have a sinusoidal variation. The stiffness variation in the steered laminate however causes the load distribution not to be sinusoidal anymore. The fiber angle variation of the middle steered ply that is shown in Fig. 9 is representative for the other steered plies in the laminate. The small fiber angles on the tension side of the cylinder cause a higher stiffness, and since the section force is a multiplication of the strain with the local stiffness, the amount of load carried by this part of the laminate is increased. The larger fiber orientation angles on the compression side of the cylinder reduce the stiffness locally and thus the amount of load being carried there does not increase, even though the strains are increasing toward the bottom of the cylinder. Hence, the stiffness variation in the steered laminate causes the upper half to pick up more loads and relieves the compression side, thereby increasing the total load-carrying capability of the cylinder. Since the neutral axis is shifted upward, the T2 design variables start playing a role as well. By keeping these angles small the neutral line is prevented from moving up too far and the region around h = 90° is carrying part of the compression loads as well. The three steered ply definitions shown in Fig. 8 all show the same trend of hardening the laminate on the tension side of the

163

cylinder and softening the laminate on the compression side, but the amount of angle variation differs per ply. It can be observed that fiber angles on the compression side of the cylinder are smaller in the plies closer to the laminate surface (e.g. u1) than the plies close to the middle of the laminate (e.g. u5). This sequence results in the highest laminate bending stiffness and is therefore best for resisting the formation of buckles. The principle of increasing structural performance by changing load paths through fiber steering will work for all fiber-reinforced materials that can be manufactured using AFP, although the exact laminate architecture and the amount of improvement depend on the material properties. An example can be found in Ref. [17] where the same optimization described in this paper was performed for a different material system, resulting in a 18% improvement in buckling load. The first buckling mode is shown in Fig. 10 and displays a buckling pattern that has two half waves in circumferential direction on the compression side of the cylinder. This might be explained by the fact that these two regions carry the highest compressive loads as can be seen from the distribution of the section forces in Fig. 9. Since more energy is needed to form two buckles instead of one for a constant-stiffness laminate, the buckling load is increased. The buckling modes presented here are based on a linear analysis in which geometrical imperfections not included. Since these can have a large effect on the non-linear response of the cylinder, more research is needed to investigate the differences in response between the baseline cylinder and the tailored cylinder. Also postcure deformations and stresses are different for both cylinders, having an effect on the structural behavior. 6. Conclusions and recommendations A composite cylinder was optimized for maximum buckling under pure bending by varying the stiffness in circumferential direction. This was achieved by steering fibers within different plies using advanced fiber placement. Manufacturability was ensured by imposing constraints on the optimization. Design criteria were implemented by requiring the global stiffness to be comparable to a baseline laminate that consisted of 0°, 90°, and ±45° plies, and by requiring the equivalent stiffness in the 0,90 and ±45° directions to be larger than a threshold value. The material failure

Fig. 8. Ply angle variations in: [±45, ±u1(h), 0, 90, ±u3(h), 0, 90, ±u5(h)]S.

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TENSION

TENSION

315 Neutral axis

Neutral axis

Azimuth angle, θ [deg]

270

225

180

COMPRESSION

COMPRESSION

135

90 Neutral axis

Neutral axis

45

0

−1

−0.5

TENSION 0 0.5

TENSION −0.2 −0.1 0 0.1

1

Longitudinal strain ε at 1,000 in−lbs, [−] x

+/− ϕ3(θ)

0.2

0.3

Section forces in longitudinal direction, 2 normalized by M/R , [−]

Fig. 9. Longitudinal strain and normalized section forces for laminate 2.

Fig. 10. First buckling mode for laminate 2.

moment was determined by using a strain-based Tsai-Wu criterion, which had to be higher than the buckling moment in order to ensure the cylinder to be buckling critical. The latter constraint was the dominating one, while in some designs the effective stiffness and curvature constraints were active. In this paper the strength properties of the fiber-steered laminate were assumed to be the same as for a conventional laminate, although the presence of tow drop locations in the steered laminate are likely to reduce the strength properties. Further research including mechanical testing is needed to quantify the effect of tow drop locations on the strength properties of a laminate. It can be concluded that varying the laminate stiffness of a composite cylinder circumferentially is effective for increasing the buckling load of a cylinder subjected to pure bending. Improvements of up to 17% with respect to a baseline laminate are obtained while the structural mass remains the same. The resulting laminates have a stiff laminate at the tension side of the cylinder and a soft laminate at the compression side of the cylinder, thereby relieving load from the buckling critical compression area and changing the buckling mode.

Manufacturability limits the design by requiring a minimum fiber orientation angle and by imposing a constraint on the amount of steering that can be applied. Relaxing the latter constraint is expected to result in small improvements when compared to the current design. In addition, a formulation of a variable-stiffness laminate different than the shifted course method might be used to allow 0° fibers to be placed on the tension side of the cylinder, thereby further improving the load-carrying capability. In this paper only laminates with a constant thickness were optimized for a maximum buckling load under pure bending. In the future also variable-stiffness laminates including overlaps will be optimized, as these overlaps are believed to be even more beneficial for the structural performance. A baseline and fibersteered shell have been built and are tested in bending. A nonlinear analysis will be done for the selected laminate, including the influence of curing and geometric imperfections. The main purpose of the test will be to validate the presented analysis methods. Finally, optimization for combined load cases, as well as for stiffened shell structures will be needed to investigate the efficiency of steered laminates in real structures. References [1] Tatting BF, Gürdal Z. Design and manufacture of elastically tailored tow placed plates. Technical report. NASA/CR-2002-211919; 2002. [2] Tatting BF, Gürdal Z. Automated finite element analysis of elastically-tailored plates. Technical report. NASA/CR-2003-212679; 2003. [3] Jegley DC, Tatting BF, Gürdal Z. Optimization of elastically tailored tow-placed plates with holes. In: Proceedings of the 44th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials (SDM) conference. Norfolk (VA, USA); 2003. [4] Gürdal Z, Tatting BF, Wu KC. Variable stiffness composite panels; effects of stiffness variation on the in-plane buckling response. Compos Part B: Eng 2008;39(5):911–22. [5] Alhajahmad A, Abdalla MM, Gürdal Z. Design tailoring for pressure pillowing using tow-placed steered fibers. AIAA J Aircraft 2008;45(2):630–40. [6] Alhajahmad A, Abdalla MM, Gürdal Z. Optimal design of a pressurized fuselage panel with a cutout using tow-placed steered fibers. In: Proceedings of the international conference on engineering optimization. Rio de Janeiro, Brazil; 2008. [7] Blom AW, Tatting BF, Hol JMAM, Gürdal Z. Path definitions for elastically tailored conical shells. Compos Part B: Eng 2008. [8] Blom AW, Setoodeh S, Hol JMAM, Gürdal Z. Design of variable-stiffness conical shells for maximum fundamental eigenfrequency. Comput Struct 2008;86(9).

A.W. Blom et al. / Composites: Part B 41 (2010) 157–165 [9] Tatting BF. Analysis and design of variable stiffness composite cylinders. Ph.D. Thesis. Virginia Polytechnic Institute and State University; 1998. [10] Sun M, Hyer M. Use of material tailoring to improve buckling capacity of elliptical composite shells. AIAA J 2008;46(3):770–82. [11] ABAQUS, Inc. ABAQUS Version 6.7 User’s Manual. Pawtucket, RI, USA; 2005. [12] Booker AJ, Dennis Jr JE, Frank PD, Serafini DB, Torczon V, et al. A rigorous framework for optimization of expensive functions by surrogates. Struct Optimization 1999;17(1):1–13. [13] Audet C, Dennis Jr JE, Moore DW, Booker A, Frank PD. A surrogate-model-based model for constrained optimization. In: Proceedings of the 8th AIAA/USAF/ NASA/ISSMO symposium on multidisciplinary analysis and optimization. Long Beach (CA, USA); 2000.

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[14] Blom AW, Stickler PB, Gürdal Z. Design and manufacture of a variable-stiffness cylindrical shell. In: Proceedings of the SAMPE Europe 2009 conference. Paris (France); 2009. [15] Lopes CS, Camanho PP, Gürdal Z, Tatting BF. Progressive failure analysis of tow-placed, variable-stiffness composite panels. Int J Solids Struct 2007;44(25–26):8493–516. doi:10.1016/j.ijsolstr.2007.06.029. [16] IJsselmuiden ST, Abdalla MM, Gürdal Z. Implementation of strength based failure criteria in the lamination parameter design space. In: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials (SDM) conference. Honolulu (HI, USA); 2007. [17] Blom AW, Rassaian M, Stickler PB, Gürdal Z. Modal testing of a composite cylinder with circumferentially varying stiffness. In: Proceedings of the 50th AIAA/ASME/AHS/ASC structures, structural dynamics and materials (SDM) conference. Palm Springs (CA, USA); 2009.