Potential of cross section varying Ω stringer made of carbon fibre reinforced plastics

Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Potential of cross section varying Ω stringer made of carbon fibre reinforced plastics E. Petersen n, C. Hühne German Aerospace Center, Institute for Composite Structures and Adaptive Systems, Lilienthalplatz 7, D-38108 Braunschweig, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 25 June 2015 Received in revised form 27 October 2015 Accepted 28 December 2015

Modern lightweight structures commonly use stiffeners of constant cross section to increase the stability of thin shells. This paper introduces a particular Ω stringer with variable cross section, allowing an increase of local skin buckling load, through an effective partitioning of the skin fields. The present work contains a methodology to develop a feasible design for stiffened shells. Parametric studies including criteria like stability, strength and damage tolerance, are performed. Additionally, a static compression experiment is conducted. Considering actual sizing criteria it can be shown that the introduced type of stringer leads to improved load carrying capacity through weight quotients. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Omega stringer Stiffened panel Stability Varying cross section Non-rectangular skin fields

1. Introduction In lightweight engineering composite materials are often the first choice, as high in-plane loads are occurring besides lower outof-plane loads. Therefore, these materials are considered appropriately for thin walled structures like stiffened shells. Modern airplane structures are usually built according to the semimonocoque concept, where a pressurized skin is stiffened transversely by frames and longitudinally by stringers. Due to the slenderness, loss of stability is one of the most influential sizing criterion. According to the above-mentioned field of application, stiffeners are used to increase overall stiffness and buckling load, while preserving a low mass. Many different types of stiffeners, such as T-, Z- or I-profiles are used. These exhibit open cross sections. Fig. 1 shows stiffeners with closed cross sections. These exhibit high bending and torsional stiffening attributes due to the parallel-axis theorem and to their closed cross section when connected to the skin. Methods for the investigation of the mechanical behaviour of closed cross section stiffeners have already been developed and conducted. The buckling behaviour of omega shaped stringer stiffened panels was point of research by Perret et al. [11,12] using numerical simulation and static testing. Mittelstedt and Schröder in [10] took a closer look at the omega postbuckling behaviour under transverse compression. They developed a closed form n

Corresponding author. E-mail address: [email protected] (E. Petersen).

solution for this loading type. Bertolini et al. [1] and Bisagni and Davilla [2] investigated the debonding behaviour of single omega stringers with bending tests and buckling experiments, respectively. For the manufacturing process with carbon fibre reinforced plastics (CFRP), the round Ω shape is more feasible than the hat or omega shape, as larger radii facilitate draping. It has not been in the focus of research yet. Anisogrid or Lattice structures are another promising stiffening concept for shells, see e.g. Vasiliev et al. [17]. The grid consists of axially loaded stiffeners with different orientations. An advantage over the semimonocoque design is the absence of Mouseholes at the crossing of transversal and longitudinal stiffening elements because of integral manufacturing and intersection of all rib elements. The buckling coefficient of triangular and hexagonal skin fields is increased compared to rectangular skin fields. To determine the buckling load of triangular skin fields, solutions are provided by Tan et al. [14] and Wang and Liew [18], who investigated isotropic triangular fields. More recent investigations were performed by Totaro [15,16], who presents analytical solutions for triangular and hexagonal grids that are not bonded to a skin. Semi-analytical solutions for skin buckling of grid-stiffened shells are given by Weber and Middendorf [19]. In this paper a concept is presented, which is based on the semimonocoque concept with frames and stringers, but uses some principles of anisogrid structures. An Ω stiffener with a varying cross-section along its length is introduced. Considered is an application region in air planes as stringer. The stringers' width and height can be increased in trigonometric relation to the stringer

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located at the stringer centre. Width and height are maximum at the centre position between two frames and decrease back to their initial dimensions at the next intersection point. The cross section variation is not characterized in a linear relation. Instead, cosine functions are used to describe the surface. Eqs. (1) and (2) describe the stringer geometry with the coordinate origin in stringer centre (φ ¼ 0). Fig. 1. Different closed cross sections for stiffeners.

centre, which leads to non-rectangular skin fields and a varying height distribution. Here, only flat stringer stiffened panels are considered as they are a relatively simple subcase of generally stiffened shells. In a first step, methods to describe the mechanical behaviour of structures with varying cross sections are discussed. One approach is to simplify the geometry to a solid tapered beam. Stability analysis can be found in the literature, as in Dube and Dumir [6], who investigated tapered beams on elastic foundation. In the Aeronautical Engineering Handbook (HSB 41113-01 [7]) the Euler buckling load of a solid column with tapered cross section is given. Unfortunately, solutions for solid cross sections are not applicable because of the thin walled shape of the Ω stringer. Another approach is the determination of the buckling load of non-rectangular skin fields as part of the stiffened shell. For trapezoidal skin fields under compression a solution can be taken from HSB 4522001 [8]. To obtain an exact solution for a whole stiffened panel, a finite element model containing skin and stringers is a suitable method. For a new type of structure, a design process has to be conducted to find optimal attributes. Sizing of a stiffened panel with the objective of a high buckling load at minimal weight leads to a stiffness distribution problem. A very theoretical method to find the ideal stiffness distribution of a plate is presented by Setoodeh et al. [13]. Weißgraeber et al. [20] presented a different approach to determine adequate properties of stiffeners in order to constrain global buckling. As the methods are simplifying the buckling problem supposing constant stiffener attributes, not all required aspects are included in the design process. Deniz et al. [5] have already developed a method which combines the impact of manufacturing and static finite element analysis in an evolutionary optimization and applied it to the geometry that is considered in this paper. Results are the maximum possible stringer length and highest fibre angle deviation due to the curvature. To date, no complete solution or feasible design method is available in the literature. Therefore, it is necessary to develop methods and criteria for the type of stringer and stiffened shell investigated here. The objective of the work performed in this paper is to show the mechanical principle of the particular Ω stringer with varying height and width, and to unfold the possible methods to find an ideal design. Part of this work is a parameric study to find a geometry with the best mechanical properties. For specific configurations, a comparison between an omega-, a constant cross-section Ω stringer and the variable Ω stringer is performed considering actual aerospace design philosophies for composite structures. Finally, in situ experience about the structural behaviour is exploited through a static experiment with a stiffened panel.

2. Idea and physical principal The idea of the cross section varying Ω stringer was developed by Kolesnikov et al. [9]. Fig. 2 illustrates the basic idea. The introduced type of stringer is small in height and width at the location where the frames are crossing. Subscripts 0 and ±l/2 denote the position along the stringer length with respect to the origin

h(φ) = (ϕI − 1)·cos(φ)·h±l /2 + h±l /2 ,

(1)

b(φ) = (ϕII − 1)·cos(φ)·b±l /2 + b±l /2 .

(2)

With −90° ≤ φ ≤ 90° and ϕI = h0 /h±l /2, ϕII = b0 /b±l /2 being the expansion factors in height (ϕI), respectively width (ϕII). This leads to a double curved surface (DCS) for the whole stringer geometry. One difficult aspect, even of a regular Ω stringer stiffened panel, is the computation of the moment of inertia. As opposed to the hat or omega stringer, no closed analytical solution is available for the DCS. Therefore, a computational solution is sought, such as shown in Fig. 3. The stringer is partitioned into small sub-parts, for which the moment of inertia can be calculated. The moment of inertia is then found by adding the moment of inertia of the subparts: n

Iy =

∑ Iyi. i=1

(3)

For the DCS this has to be repeated at different positions in x-direction because of the changing width and height. The DCS stiffened panel exhibits some attributes of grid structures, which are marked in Fig. 4 with dotted lines. Similar to a grid structure, the load paths adjust to the investigated stringer geometry leading to mainly axial loads. Contrary to grid structures however, there is no load transfer between the stringers. Instead the geometry is used to produce a load redistribution. Therefore, the main idea of the DCS is inspired by the positive aspects of grid structures, but is avoiding the manufacturing challenges through the intersection points. An increased stiffener height leads to a greater moment of inertia at the centre of the panel and thus to an increased panel stiffness. Since the load follows the curvature of the variable cross section stiffener, a load component perpendicular to the plane of the plate results. This components acts as a perturbation load with respect to stability as depicted in Fig. 5. Hence, the varying height leads to a higher eccentricity than for the regular shape. Consequently, the column buckling load of a single DCS may be lower than the buckling load of a comparable omega or Ω stringer. As it dominates the stiffened panels behaviour, the lower stringer buckling load decreases the global buckling load. Therefore, the relation between height and global buckling has to be investigated. The stiffened shell is partitioned into several skin fields by the stringers as shown in Fig. 4. These fields are sensitive to local skin buckling. A design with DCS yields to different sized fields due to the varying area between and underneath the stringers. In conventional stringer design, the fields under the stringer are smaller than the fields between two stringers. Due to the expansion in width, DCS can have a large area under the stringers. Hence, the field between the stringers is getting smaller. A maximum skin buckling load is theoretically achieved, when the size of the fields under the stringers equals the size of the fields between the stringers, without buckling of the stringer itself. With this additional possibility to divide the stiffened shell into differently sized sub-parts, the width expansion factor provides an additional design parameter for the skin fields shape.

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Fig. 2. Shape definition: (a) front view and (b) isometric view.

Fig. 3. Partitioning of the cross section to calculate the moment of inertia Iy with respect to the centre of gravity.

Fig. 6. Geometry and material parameters. Fig. 4. Possible stiffened panel with different sized Mouseholes for transverse frames, bottom: grid, middle: comparison of grid and DCS, top: regular Ω-stringer compared to DCS.

3. Analysis of different DCS configurations To find an optimal configuration and to determine the sensitivity to different parameters, parametric studies with finite element models are performed using a generic modelling tool. Criteria like stability, strength and damage tolerance are considered. The tree in Fig. 6 shows the different variables used to find a

mechanically acceptable and mass efficient design. All parameters related to geometry can be found on the left side, whereas all parameters describing the composite material are summarized on the right side. An optimization of the stacking sequence and, hence, the variables related to material properties, can be found in Bold et al. [3].

Fig. 5. Load paths in DCS.

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3.1. Finite element model Programs written in the “PATRAN Command language” (PCL) are used to automatically generate DCS finite element models in the preprocessor PATRAN. These programs compute the DCS stiffened panel geometry data based on the parameters shown in Fig. 6 and using Eqs. (1) and (2). Material draping along the stringers' curved surfaces is simulated with a numerical tool included in PATRAN (Laminate Modeller). It estimates the correct layer orientation and thickness direction. Skin and stringers are modelled with layered linear shell elements based on the geometry data. These are connected by a thin sheet (0.2 mm) of linear solid elements with the properties of an epoxy based adhesive. Material properties of skin and stringers are defined by the chosen carbon fibre reinforced plastics (CFRP) layups, which are kept constant for the geometry study. Load and boundary conditions are applied and a linear buckling analysis is started. With the tool many different geometry configurations can be generated and compared with manageable effort. 3.2. Loadcases For the compression load case the panel is fixed and clamped at x = − l/2, clamped at x = l/2 and simply supported at the edges allowing a translation in x-direction. The load is applied through a rigid body at x = l/2, which couples the x-translation of the included nodes. In Fig. 7 the load and boundary conditions for compression are marked with the letter c. The shear load case is similar to the compression load case. Loading is applied in a transverse manner using a rigid body at x = l/2. The y-translation is coupled, so that all other DOF are restraint through the boundary conditions. At x = − l/2 all DOF are restraint and the edges are simply supported free to move in ydirection, see Fig. 7. Load and boundary conditions related to shear are marked with the letter s.

Table 1 Damage tolerance restraints. Load direction Strain in (με ) Tension Compression

4.500  4.000

⎡⎛ ⎞2 ⎛ τ ⎞2⎤ ⎢⎜ σ1 ⎟ ⎥ RFF ·⎢⎜ t, c ⎟ + ⎜⎜ 12 ⎟⎟ ⎥ = 1 R ⎝ R∥⊥ ⎠ ⎦ ⎣⎝ ∥ ⎠

(4)

⎡ ⎤ ⎡⎛ ⎞2 ⎛ τ ⎞2⎤ σ2 1 ⎞⎥ ⎥ ⎢ ⎛1 ⎢ σ RFM2·⎢⎜⎜ 1t, c ⎟⎟ + t 2 c + ⎜⎜ 12 ⎟⎟ ⎥ + RFM·⎢σ2⎜ t + c ⎟⎥ = 1 R⊥ ⎠ 2R R⊥·R⊥ ⎝ R∥⊥ ⎠ R ⎦ ⎣ ⎝ ⊥ ⎦ ⎣⎝ ∥ ⎠

(5)

2

with the material strengths R∥ parallel to the fiber, R⊥ perpendicular to the fiber, R∥⊥ in shear and the indices t , c marking tension or compression. The damage tolerance criterion restricts the strains to the limits in Table 1. This is a conservative industrial approach and can be used like a strength criterion in the design process. 3.4. Linear buckling analysis

A linear eigenvalue analysis is used to obtain the buckling loads for the sizing process. More detailed information about the buckling behaviour, especially the postbuckling regime are obtained by the incremental iterative Newton–Raphson method for specific configurations. The criteria in Eqs. (4) and (5) are evaluated to describe material strengthening with respect to reserve factors for matrix failure (RFM) and fibre failure (RFF)

All geometry parameters depicted in Fig. 6 are varied within realistic ranges. This leads to 7 different variables for the geometry related side of the tree. To allow a direct comparison, the efficiency is introduced, which relates the buckling load or Eigenvalue λ to the structural weight (λ/weight). An example of the parametric study is shown in Fig. 8. The buckling load is depicted as a function of different parameters varied in a specific range. Here, the compression buckling efficiency for different stringer heights over varying stringer width is depicted, while all other values are kept constant. The compression load case shows a maximum λ /weight = 1.35 and is nearly insensitive to the height of the stringer. For the shear loadcase (Fig. 9) a dependency of stringer height and width on the efficiency can be seen. The maximum improves for a larger stringer height to a smaller width. Such variable dependencies are the reason for indefinite results. This makes comprehensive combinations of the variables necessary and time consuming. To expose the influence of the expansion factors, these are varied in a range between 1.0 and 2.0 and all other parameters are kept constant. As can be seen in Fig. 10, the values for compression reach a maximum at ϕI = ϕII = 1.6. Global stringer buckling occurs at ϕI = ϕII = 2.0. The values for shear are showing a steady increase

Fig. 7. Load and boundary conditions for the finite element model, c: compression, s: shear.

Fig. 8. Compression buckling efficiency for varying stringer height and width.

3.3. Methods and criteria

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Fig. 9. Shear buckling efficiency for varying stringer height and width.

5

Fig. 12. Compression buckling efficiency for varying stringer pitch/width, nonlinear analysis.

3.5. Nonlinear analysis

Fig. 10. Compression and shear efficiency for varying expansion factors, solid dots mark stringer loss of stability.

with higher stringer expansion factors. Loss of stringer stability under shear occurs above ϕI = ϕII = 1.8. Fig. 11 points out the efficiency of the expanding stringer under shear loading for different stringer pitch ratios p/pmax . At all, efficiency is increasing with a smaller stringer pitch. Interesting is to mention the increasing shear buckling load for higher width, expanding independently from the expansion of the height. Hence, the single increase of the width leads to an increase of the shear buckling load.

The postbuckling behaviour is investigated using an iterative Newton–Raphson method with increasing displacement in addition to the buckling analysis obtained by linear eigenvalue analysis. Objective is to investigate the effect of the height increase, which impacts the stringer stability and hence, the global buckling behaviour. Global buckling is defined as the first significant drop in the overall loadbearing capability observed in the force displacement diagram. Within this study ϕII has been defined as 1.8 and two different configurations, ϕI = 1.0 and ϕI = 1.8, have been evaluated. The results given in Fig. 12 verify the observed insensitivity to ϕI, of the local skin buckling coefficient for different values of the ratio p/b0 . For the global stability behaviour, a clear load increase due to ϕI can be observed.

4. Comparison of different stringers with closed cross sections under aspects of practical application Below, stringers with a number of closed cross sections are compared with respect to design rules, meaning the allowed load or the load carrying capacity. In this section the use of DCS under typical design constraints of aerospace applications is presented. 4.1. Sizing criteria The state of the art sizing philosophy for modern CFRP pressurized stiffened shell structures constrains the following sizing criteria:

 No local buckling below Limit Load (LL)  No global buckling below Ultimate Load (UL) 4.2. Comparison of omega stringer and DCS

Fig. 11. Shear buckling efficiency for different combinations of ϕI and ϕII and varying stringer pitch.

A typical configuration of a CFRP fuselage panel design with classical omega stringer is compared with an appropriate and complementary DCS configuration. The volume and hence, the weight are approximately coincident (2.7% deviation). Table 2 compares the main geometrical properties of the investigated cases. Fig. 13 shows the force displacement graphs of a nonlinear analysis of a DCS (1.0/1.8) and a typical omega stringer from aerospace applications related to Table 2. For the global buckling case, the efficiency of the omega stringer is around 50% higher

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Table 2 Comparison of DCS properties with omega stringer properties.

Table 3 Panel dimension for numerical analysis.

Property

DCS vs. omega (%)

Part

b (mm)

l/h [mm]

t (mm)

p (mm)

ϕI

ϕII

Stringer cross section at x = ± l/2 Panel cross section at x = ± l/2 Overall panel volume Inertia of Stringer at x = ± l/2 Inertia of overall panel at x = ± l/2 Inertia of stringer at x ¼0 Inertia of overall panel at x¼ 0

 2.9

Panel Stringer

600 30

600 20

2 1.75

150 (4 str.)

1/2

1/2

 0.6 þ 2.7  25.8  30.3 þ 4.6 þ 2.5

maximum performance benefit is observed for a ratio of 2.07 independently of the investigated wall thickness. Also a tendency of increased efficiency is observed for thicker structures. For small p/b0 ratios, no beneficial configuration could be found. 4.3. Comparison between DCS and

Fig. 13. Load–displacement curves of DCS and omega stringer, normalized to global buckling of omega stringer.

than that of the DCS. The local buckling load for the investigated configuration shows the opposite effect from the local buckling load of the DCS, as it is around 50% higher than for the omega stringer. Following the sizing criteria “No local buckling up to limit load” and applying the safety factor for aerospace primary structures of 1.5, it can be proven that local buckling is the critical load case for both stringer configurations. Therefore, the increased global buckling resistance of the omega stringer cannot be utilized. Instead, the higher local buckling load of the DCS can be exploited, when designing a stiffened panel. In Fig. 14, the results of the nonlinear simulation discussed in Section 3.5 are set in relation to the critical local buckling load of the typical omega stringer. The simulation has been performed for different stringer wall thicknesses to investigate the general influence of thickness and to define a potential application region. Fig. 14 shows a clear tendency of increased performance for the DCS with a higher stringer pitch to stringer width ratio (p/b0). The

Ω stringer

Two specific configurations are compared. The panels have the dimensions listed in Table 3. Modelling and simulation are performed analogous to Section 3.1. The same load and boundary conditions as shown in Fig. 7 are used. Linear buckling analyses lead to the results in Fig. 15. Both first eigenvalue modes show nine skin halfwaves. The reference stringer shows the highest out of plane displacement in the skin field in panel centre. As the centre skin fields of the DCS panel are smaller in the panel centre, the highest out of plane displacement is occurring in the fields near the top and the bottom. The ratio buckling load/weight is increased by almost 17% for the DCS. Information about the post-buckling regime can be extracted from a nonlinear analysis. Fig. 16 pictures the load–displacement curves of the Ω stringer compared to the DCS. The identified skin buckling loads are higher than in the eigenvalue analysis and the DCS buckling mode is superimposed with an overall bending. The regular Ω stringer shows a slightly stiffer behaviour in the linear portion, but a 13% lower skin buckling efficiency. In the postbuckling regime, the DCS shows the initial loss of global stability, which here is induced by stringer buckling. Fig. 17 shows the stringer reserve factor for both criteria of Eqs. (4) and (5) for the related buckling loads. Material failure is not the sizing factor for the investigated configurations. Lowest reserve factors occur in the load introduction region, where the cross sectional area is smaller than in DCS centre. For the DCS strains and stresses in thickness direction in the bondline are increased in comparison to the Ω stringer.

5. Static experiment To gain experimental experience about the DCS behaviour and to compare it with numerical results, a static test is performed at the DLR buckling test facility in Brunswick. The difficulty of manufacturing the DCS leads to a different layup in comparison to the parametric studies and following, through the limitations of the buckling test facility even the geometry deviates from earlier studies. 5.1. Specimen manufacturing

Fig. 14. Comparison of omega stringer and DCS ϕI = 1.8, ϕII = 1.0 for different thickness and pitch/width values, critical load from nonlinear analysis.

For this purpose a plane stiffened panel with three stringers is manufactured. A 0.4 mm thick layer of the pasty epoxy adhesive Henkel–Hysol is used to bond the precured stringers onto the precured skin. Ultrasonic inspection is used for quality control and reveals a homogeneous bondline. A HT-fibre is combined with the epoxy resin SR1710 to produce the panel and stringers. Skin and stringers are manufactured manually via the resin transfer moulding method (RTM). 745°layer is required to support the DCS on the inner and the outer

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Fig. 15. Results of linear eigenvalue analysis related to structural weight. Table 4 Panel dimension for static experiment.

Fig. 16. Load–displacement curves related to structural weight, reference stringer in comparison to DCS.

sides to avoid slipping. The inner 0° layers are oriented along the panel length. As a consequence, the stringers exhibit a thickness t¼1.97 mm and hence, a higher stiffness than used in the parametric studies. The skin layup is quasi-isotropic with a 2 mm thickness. Due to the higher thickness, the expected test specimen buckling load exceeded the limitations of the test facility. Therefore, the design was scaled. The length was increased so as to decrease the expected buckling load. Consequently, although the design is based on the afore described investigations, the properties differ from the initially defined configurations. The experimental panel dimensions are listed in Table 4. The top and bottom edges of the panel are potted in polyurethane reinforced with a siliceous filling material so as to generate a clamped boundary condition. Other sides are not constrained. To determine the quality of the manufactured part, including the imperfection pattern, both sides of the panel are measured with the 3D sensor system ATOS resulting in a point cloud which

Part

Width (mm)

Length/height (mm)

ϕI

ϕII

Panel Stringers

450 30

800 20

1.8

1.8

represents the geometry. This point cloud and the CAD DCS stringer geometry are overlapped so as to have three coincident points at x = ± l/2 and x¼ 0 for each stringer. So doing, the geometric deviation between the manufactured part and the CAD part can be measured over the whole panel. It is concluded that the stringer discrepancies are acceptable as the maximum surface deviation does not exceed 1.28 mm at the outer surface. However, it also discovered that the overall panel is curved about its x-axis with an amplitude of about 1 mm at y = ± b/2. This is probably induced during the bonding process. Axial shortening u is measured by two transducers which are used to apply the displacement driven load via a hydraulic cylinder. The test procedure used at the DLR panel test facility in Brunswick is described in the literature [21,4] and was validated through several tests in the past. 5.2. Measuring equipment During the experiment, measurements are taken with digital image correlation (DIC) using an ARAMIS system and strain gauges at several positions on the stringers and the panel-front and back. The displacements in z-direction and axial strains on the skin and stringers are recorded with DIC. Since the ARAMIS system cannot evaluate the whole panel areas, two representative areas on the front- and backside (painted region in Fig. 19, right picture) are measured. The strain gauges are positioned on the skin field and mainly under and on the middle stringer. On the middle stringer comb 3 linear strain gauges, which record the axial strains, are positioned. Additionally, two strain gauge rosettes are positioned on each side of the stringer.

Fig. 17. Static reserve factors for applied buckling loads: (a) Reference Ω stringer RF¼ 2.3 and (b) DCS stringer RF ¼ 1.07.

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Fig. 18. Load–displacement curves of static experiment and simulation.

5.3. Test results Fig. 18 shows the load–displacement curve of the experiment and compared it to the results of a nonlinear simulation using the finite element model from Section 3.1 with adapted material properties as the fibre parallel stiffness was decreased to E∥c = 110, 000 N/mm2. The experimental curve shows a similar behaviour as the finite element simulation before debonding occurs, which is not implemented in the model. Failure takes place at a displacement of u ¼2.56 mm as can be identified at the strong load drop in the force–displacement diagram. The FE curve shows a small stiffness decrease beginning at about u¼ 1.1 mm, whereas the experimental curve shows a stiffness decrease at about u ¼1.3 mm. The displacement in z-direction for different applied axial displacements is depicted in Fig. 19. For the experimental curve, only the area measured with DIC is shown. As the buckling onset is progressive, an exact buckling load cannot be determined, even for skin buckling. Specific displacements, where a significant buckling pattern displacement occurs, are determined. In the simulation, skin buckling is occurring at a load step of

about u¼ 1.1 mm. The experiment shows a slightly later occurrence of skin buckling coinciding with the later stiffness decrease. For u ¼2.0 mm, the buckling pattern in simulation and experiment is fully developed and only an increase of the amplitude can be observed at u ¼2.5 mm. There are minor differences in the shape of buckling patterns between experiment and FE simulation. The experiment displays an even number of waves in its deformation pattern, whereas the simulation shows an odd number. In Fig. 20, the recorded strains on the panel front side are depicted. Fig. 20(a) shows the strains on the top of the stringer along the crest line of the centre stringer at three positions. The gauge DMS0 near the stringers' clamped edge shows compression strain in all load steps. Gauges DMS1 and especially DMS2 at the centre position start with compressive strain, which turns into tensile strain at about u ¼1.5 mm shortening. Fig. 20(b) shows the strains in the skin fields recorded by triaxial rosettes. Around the load stage u ¼1.5 mm, the tensile strains in x-direction show a higher increase than in the portion before (R1a, R0a). In transverse panel direction (R1c, R0c), the compression strain turns into a tensile strain at about u ¼1.5 mm. At all, the strain gauges show a change of the behaviour around u ¼ 1.5, where a stiffness decrease in the load–displacement curve could be observed and the DIC recording shows the development of a buckling pattern. The finite element simulation tend to show a slightly stiffer behaviour with earlier appearance of stiffness decrease and buckling pattern.

6. Discussion The studies in Section 3 indicate the interaction of many design parameters. A clear optimum under consideration of all parameters and constraints could not be achieved. However, a design space could be delimited in which regular Ω stringers and DCS show an adequate structural behaviour. For the material properties used, the best geometric attributes for the stringers could be found at small width/height ratios. The length and pitch of the stringers have a great influence on the buckling load. Both attributes are strongly depending on each other, as they characterize the skin fields size. Therefore, this parameters should be adjusted to the chosen field of application.

Fig. 19. Buckling pattern in experiment from ARAMIS (top) and simulation (bottom), right failure in experiment.

Please cite this article as: E. Petersen, C. Hühne, Potential of cross section varying Thin-Walled Structures (2016), http://dx.doi.org/10.1016/j.tws.2015.12.026i

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probably provoked through the stringers eccentricity leading to a behaviour similar to that of a plane panel with strong surface or load imperfections. Also the unsupported sides play a significant role for the global behaviour and are not representing realistic boundary conditions. Loss of global stability can occur as a load drop, if the overall panel shows buckling, or through buckling on the stringers themselves. In the experiment, debonding occurred and marked the end of the load carrying capacity.

7. Conclusion

Fig. 20. Strains on panel top.

Both expansion factors are varied combined and individually. Their corresponding expansion directions have a specific influence on the buckling behaviour. An optimum for the local skin buckling load can be reached by varying the stringer pitch and width shape factor ϕII to a value where the buckling modes nearly appear simultaneously between and under the stringers. The nonlinear analysis reveals that the height expansion factor ϕI influences the global buckling load in correlation with the increase of the moment of inertia. With the present layup, expansion factors should not exceed 1.8, so as to prevent instabilities of the stringer itself as was shown in Fig. 10. Smaller basic stringer geometries and material properties might lead to different results and even allow a larger expansion. A direct numerical comparison of DCS with omega or Ω stringer stiffened panel leads to a higher skin buckling load for the DCS panel. Conversely, the global buckling load of a DCS stiffened panel can be lower, when the height expansion factors are of the same magnitude as the width expansion factors. Buckling on the DCS was found to be sensitive to high expansion factors, as higher surface areas decrease the buckling load. Sufficient wall thickness can prevent this failure type. The static experiment demonstrated the static behaviour of a DCS stiffened plate, which nearly coincides with the corresponding numerical simulation. Both, experiment and nonlinear simulation reveal a progressively occurring skin buckling pattern with a slightly decreasing stiffness, rather than a sudden occurrence with significant load drop. Also a global bending could be observed, which leads to a highly nonlinear load–displacement curve. This is

The main advantage of the concept with a variable cross section lies in the increased skin buckling load compared to conventional omega or hat stringers. A higher load carrying capacity related to weight is achieved, when assuming skin buckling to be the sizing criterion as required by today's design philosophy for CFRP structures. This benefit can be related to the evenly distributed partitioning of the panel by the expanding stringer width. Higher global buckling loads can also be achieved through an expansion in height, such that the influence of the higher moment of inertia dominates the higher eccentricity. Additionally, a possible increase of the load carrying capacity under shear is exposed in the parametric studies. This positive aspect could be observed for panels with width expanding stringers and seems to be independent of the expansion in height. Further investigations on this topic should be performed to reveal the potential for shear loads. The design studies illustrated the need for well defined boundary conditions and load applications. A defined application for a specific structure with distinct constraints could lead to a clear design solution. It has been determined that the application for the DCS is short panels with large width, where the higher skin buckling load increases the allowed load carrying capacity. The static behaviour in the performed buckling experiment reveals additional crucial topics to be investigated. Manufacturing, especially the bonding process has to be improved to minimize the influence of imperfections. Additional experimental investigations have to be performed with a reference Ω stringer stiffened panel for a direct comparison of the in situ structural behaviour. As the considered design was influenced by manufacturing and test constraints, these aspects should be considered early in the specimen design process and would result in a different DCS panel configuration for testing. Varying the cross section leads to a higher manufacturing effort, especially if high production rates have to be realized. However, the structural weight saving and following the associated lower costs of operation must prevail the higher manufacturing costs. Further issues concerning design and manufacturing have to be investigated in greater detail. For example, a reduced stringer height at the intersection points could lead to secondary weight savings. Today's sizing philosophy, which limits the load carrying capacity of CFRP structures to the level of local skin buckling, allows a weight saving potential for the DCS in comparison to omega stringers. This can be achieved with the single expansion in width.

Acknowledgements The present work was funded by the German Ministry of Defence under the research project FFS5. The authors want to thank Dr. Boris Kolesnikov for the initiation and source of ideas for the “colani stringers”. Further acknowledgements are given to Jens

Please cite this article as: E. Petersen, C. Hühne, Potential of cross section varying Thin-Walled Structures (2016), http://dx.doi.org/10.1016/j.tws.2015.12.026i

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E. Petersen, C. Hühne / Thin-Walled Structures ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Bold for considering the geometry generation tool and Michael Fabian for manufacturing. Thomas Kruse deserves acknowledgement for the discussion of the industrial possibilities and the continuous support of the topic.

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Please cite this article as: E. Petersen, C. Hühne, Potential of cross section varying Thin-Walled Structures (2016), http://dx.doi.org/10.1016/j.tws.2015.12.026i

Ω stringer made of carbon fibre reinforced plastics,