Design and experimental analysis of hat-stiffened panels for thermoplastic wind turbine blades. A. Blanken

Transcription

1 Design and experimental analysis of hat-stiffened panels for thermoplastic wind turbine blades A. Blanken

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3 Design and experimental analysis of hat-stiffened panels for thermoplastic wind turbine blades Master Thesis April, 2010 Graduation committee Prof.dr.ir. G.J.W van Bussel (TU Delft-LR-WE) ir. M. Zaayer (TU Delft-LR-WE) ir. F.J.J.M.M. Geuskens (TU Delft-LR-DPCS) ir. A.W. Hulskamp (TU Delft-LR-DPCS) Delft University of technology Faculty of Aerospace Engineering Section Wind Energy Chair of Design and Production of Composite Structures Msc program Sustainable Energy Technology Eindhoven University of Technology i

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5 Abstract Developments in large wind turbine blades point towards a redesign of conventional blade structures into a rib-spar-skin design. Additionally research on thermoplastic glass fibre composites shows promising results, concerning material properties as well as manufacturing and assembly processes, for the application of thermoplastic composites in a rib-spar-skin design of a turbine blade. Thermoplastics are not compatible with traditional core materials of sandwich constructions. A monolithic design on the other hand mixes very well with capabilities of thermoplastic technology, such as rubber forming and welding, and has a potential in cost and weight reduction. This thesis is focused on the design of a hat-stiffened panel as an alternative for a conventional sandwich structure, which is used as a skin in modern wind turbine blades. Although thermoset glass fibre composites are used in this thesis the research is aimed at development of thermoplastic wind turbine blades. A sandwich panel was isolated from a turbine blade design in order to be used as a reference for the design of a hat-stiffened panel. The sandwich panel has been analyzed on failure modes due to longitudinal compression and a reference compression load was established. Several methods were used to develop an optimization methodology for the design of a mass-efficient hat-stiffened panel. In order to validate the theory, a scaled version of the designed hat-stiffened panel was constructed in order to be used in experiments. Furthermore a test rig was designed to provide simply supported boundary conditions during the experiments. Several elastic buckling experiments have been carried out as well as a post-buckling failure experiment. iii

6 Contents Abstract... iii Nomenclature... vi 1 Introduction Modern wind energy Blade structure Thermoplastic composites Objective and thesis structure Theory on failure modes of compression panels Compression failure Buckling of flat plates in pure compression Buckling of stiffened panels in pure compression Orthotropic plate theory Finite strip method Wrinkling of sandwich plates Sandwich laminate reconstruction and failure analysis Provided material properties Estimation of the ply properties Laminate design Laminate stiffness matrix Specially orthotropic stacking sequence Experimental analysis Tensile tests Compression tests Test results Failure analysis of the reference sandwich panel Development and validation of analytical models Development of analytical models Global buckling Local buckling Compression failure Design tools Model validation Simulations using a hat-stiffened panel Simulations using I-stringers and hat-stringers Simulations using small I-stringer stiffened panels Local buckling Design study for hat-stiffened panels Parameter study using FSM Panel optimization methodology Reference hat-stiffened panel design Test panel construction and test setup Design of the hat-stiffened test panel Construction of the test panel Support structure Measurement setup iv

7 7 Buckling experiments Elastic buckling: simply supported boundary conditions Pre-buckling behaviour Buckling behaviour Discussion Elastic buckling: simply supported boundary conditions at the loaded ends in combination with free sides Pre-buckling behaviour Buckling behaviour Discussion Post-buckling failure Pre-buckling behaviour Post-buckling behaviour Conclusions and recommendations References Appendix A Equations for the laminate stiffness matrix Appendix B Specimen measurement results Appendix C Plate transverse bending stiffness Appendix D Panels used in simulations Appendix E Parameters used in parameter study Appendix F Panel designs Appendix G Results of the buckling experiments v

8 Nomenclature A cross-sectional area m 2 A ij elements of the extensional stiffness matrix Nm -1 B ij elements of the coupling stiffness matrix N C coefficient dependent on boundary conditions - D flexural rigidity Nm D ij elements of the flexural stiffness matrix Nm E Young s modulus Nm -2 F compression load N G shear modulus Nm -2 H 11,H 22 transverse (through-the-thickness) shear stiffnesses Nm with respect to yz- and yx- planes, respectively K 0 elastic buckling coefficient - L fraction - M x, M y, M xy moments per unit width Nmm -1 N x, N y, N xy direct and shear loadings Nm -1 V f fibre volume fraction - Z Nq modal coefficient - a panel length m a ij elements of the inverse extensional stiffness matrix mn -1 a mq coefficient - b panel width d thickness m k numerical factor - m mass g n number of stringers - p number of longitudinal half waves - q number of transverse half waves - s number of laminate plies - x x-coordinate m y y-coordinate m z z-coordinate m w out of plane deflection m Greek letters δ auxiliary quantity - ε, ε x, ε y, γ xy strains - θ abbreviation - κ x, κ y, κ xy curvatures m -1 ν Poisson s ratio - ρ density kgm -3 σ stress or strength Nm -2 φ ply orientation rad vi

9 Super- and subscripts UCS ultimate compression strength UTS ultimate tensile strength b buckling bf bottom flange c core cr critical cf compression failure f foam or fibre n neutral plane p plate r resin s stringer subpl subplate wr wrinkling neutral plane of whole panel z 0 vii

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11 1 Introduction 1.1 Modern wind energy Wind energy is becoming as affordable as conventional energy [1] since the efficiency of wind turbines increases and the prices of fossil fuels rise. Moreover there is a growing concern about steady energy supply for the growing population, the limited energy resources that are left and pollution of the earth [2]. Therefore wind energy is starting to form a significant part of the mainstream energy and it is believed that it will deliver 12% of the global consumed electricity by 2020 and even 23% by 2040 [3]. In order to achieve this tremendous growth the wind energy sector is now turning its interest from onshore wind turbines to large scale offshore wind farms. The offshore conditions are very different from onshore conditions. The harsh remote environments make operation and maintenance more complicated and expensive and consequently require special offshore wind power systems like health monitoring and active load damping of the wind turbine blades [3, 4]. Additionally in order to lower the cost of wind energy the length of the wind turbine blades has been increased in the past years, as the power of a wind turbine is (theoretically) proportional to the rotor swept area [1]. In order to maintain a reduction in cost and weight now reconsideration of the structural design and materials of wind turbine blades is needed. 1.2 Blade structure Common modern wind turbine blades are made from thermoset glass fibre composites and consist of very few parts: a top skin, a spar box and a bottom skin, which are joined using a structural adhesive (see figure 1.1). The blades have been designed to withstand extreme conditions caused by aerodynamic and gravitational forces and are equipped with thick spar caps (flanges) and sandwich structures to endure the high bending moments. Figure 1.1: Wind turbine blade structure [5]. In [1] a topology optimization was used to find optimum preliminary designs for megawatt size wind turbine blades using the structural integrity as the main design driver. Like most wind turbine blade designers the author used (among others) the extreme gust condition in order to analyze the flapwise loads. This situation occurs when the wind turbine is in parked mode and the blades are hit by the 50-year extreme gust wind. Normally, while in parked 1

12 Chapter 1 - Introduction mode, the blades are pitched to feather to minimize the aerodynamic loads, but in order to consider the worst case scenario the loads are often calculated with the blades positioned with the chord parallel to the rotor plane. The optimum solution as a result of combined flapwise loads showed a rib-spar configuration (see figure 1.2) which is very common in aircraft structures as can be seen in figure 1.3. Figure 1.2: Topology optimization result of a blade section for a multiple flapwise load case [1]. Figure 1.3: Two-spar aircraft wing construction. 1.3 Thermoplastic composites A rib-spar-skin configuration requires a lot of bonding since it is build up from many parts, as opposed to the current designs of wind turbine blades, where the number of joints is minimized since bonding of thermoset composites is labour intensive and time consuming [4]. Therefore thermoset composites are not very suitable for a rib-spar-skin design. The use of a thermoplastic matrix on the other hand seems very promising for this design as small thermoplastic composite parts, like ribs, can be melt-processed using rubber forming (see figure 1.5) and assembled by means of welding (although large bond surfaces are not feasible), as shown schematically in figure 1.4. Figure 1.4: Process schematics for resistance welding of thermoplastic composites [4]. In the past years there has already been interest in the use of thermoplastics in wind turbine blades due to several structural advantages, like good impact properties, abrasion and chemical resistance and high toughness, even at low temperatures. Moreover with some thermoplastics it is possible to recover the monomers from the polymers again, providing a recyclable wind turbine blade at the end of its lifetime and recyclability of manufacturing waste. But due to shortcomings of the technology thermoplastics have not been used to 2

13 1.3 Thermoplastic composites manufacture wind turbine blades up till now. Melt processing for instance is performed at high temperatures requiring expensive temperature resistant tooling while feasible part sizes and thicknesses are limited due to the use of heavy presses. Furthermore processing of continuous fibre reinforced thermoplastic requires the use of costly intermediate materials and manufacturing wind turbine blades would require new processing methods and expensive equipment. Furthermore a structural downside of thermoplastic composites is the low fatigue performance due to poor fibre-to-matrix bonding [4]. Figure 1.5: Rubber forming of thermoplastic composite parts. A recent development is to combine the material properties of thermoplastics with the process technology of thermosets. Thermoset turbine blades are mostly manufactured using vacuum infusion. In this process dry fibres are placed into a rigid mould and covered with a vacuum bag. Then a vacuum is applied to the mould cavity on one side and on the other side a resin is allowed to flow from one end to the other forming a matrix between the fibres. As thermoplastic polymers are highly viscous and therefore not suitable for vacuum infusion, low viscous thermoplastic monomers are used for the infusion followed by in situ polymerization. A problem that is still under development is the weak fibre-to-matrix bonding which results in low fatigue and compression performance of the material. In [4] anionic polyamide-6 (APA-6) has been selected to be most suitable in order to manufacture thermoplastic wind turbine blades. The additional advantages of APA-6 infusion technology are the short curing time (minutes instead of hours compared to thermoset infusion), low resin price, availability and the fact that it allows for the use of a conventional production process. APA-6 already has a strong fibre-to-matrix bond (mechanical combined with chemical bonding) however it still requires further improvement to become competitive with thermoset materials regarding fatigue performance. The main challenge that remains is the performance in wet conditions as the material properties strongly decay due to moisture absorption. 3

14 Chapter 1 - Introduction 1.4 Objective and thesis structure Objective Having an optimum solution for the preliminary design and a material that is promising to fit the requirements, actual redesigning can start. In current wind turbine blades sufficient buckling capacity of the skins outside the spar box is achieved by constructing them as sandwich panels in order to withstand part of the flapwise bending load that occurs during the extreme gust condition. Such a sandwich panel consists of two composite face sheets on both sides of a foam core (see figure 1.6). This construction provides a very stiff solution, however the production of these sandwich panels is a very costly operation as the foam has to be milled to fit the double curved shapes of the turbine blade. For thermoplastics it is very hard to find a sandwich core material that has the desirable mechanical properties and that is both chemically and thermally resistant to the production process of thermoplastics. Monolithic construction of the panels could form a solution for a redesign of these skins. A monolithic design mixes very well with capabilities of thermoplastic technology such as rubber forming of modular parts and welding. In addition it has a potential in cost and weight reduction. Figure 1.6: Section of a sandwich panel. Figure 1.7: Section of a hat-stiffened panel [6]. The main objective of this thesis is to design a hat-stiffened panel (see figure 1.7) made out of glass fibre-reinforced plastic as an alternative for the sandwich construction of the skin in modern wind turbine blades. A virtual turbine blade, designed in the Upwind project in order to be used to estimate representative stresses and strains for subcomponents and to identify hotspots in the blade, has been used as a reference for the design of the hatstiffened panel. This reference wind turbine blade has been analyzed and a reference section of the blade, which is most critical, has been isolated at the suction side where the chord is at maximum. At this location, the panel of interest is almost flat and of constant width (2.3m), see figure 1.8. The critical load in this panel is a compression load in longitudinal direction, due to flapwise bending of the turbine blade. Figure 1.8: Cross section of the reference turbine blade (arrow indicates the reference section). 4

15 1.4 Objective and thesis structure Thesis structure The theory on failure modes of compression panels, as found during a literature study, has been described in chapter two. In order to analyze the possible failure modes of the sandwich panel, theory on wrinkling and plate buckling was researched, while theory on buckling of stiffened panels was researched to be able to design a hat-stiffened panel. Chapter three describes the design, construction and experimental analysis of a laminate that was designed to represent the face sheets of the reference sandwich panel. Thereafter the determined material properties have been used to analyze the critical compression load of the reference sandwich panel. In chapter four several analytical models were developed using the theory on failure modes of stiffened panels. The reliability of these models has been researched using comparisons between different panel designs. In chapter five a parameter study was conducted in order to gain knowledge about the effect of certain parameters on the mass-efficiency of the hat-stiffened panel. Furthermore an optimization methodology was established and a design of the hat-stiffened panel as an alternative for the reference sandwich panel has been presented. Chapter six describes the design and construction of a hat-stiffened test panel. Furthermore the design of the support structure, which provided simply supported boundary conditions during the experiments, is described as well as the used measurement setup. In chapter seven the results of the experiments are shown. Several elastic buckling experiments have been carried out as well as a post-buckling failure experiment. 5

16 2 Theory on failure modes of compression panels Monolithic and sandwich structures have several ways of failing due to longitudinal compression. In order to make a proper analysis of the sandwich panel thatt is used as a reference and to design a hat-stiffened panel, knowledge of the possible failure modes is required. For that reason several failure modes of compression panels are explained in this chapter. 2.1 Compression failure When a laminate fails due to compression failure, the material is simply crushed due to the high stresses in the material. Compression failure of composites can occur in different modes, like fibre failure, elastic micro buckling, matrix failure and plastic micro buckling [7]. It is out of the scope of this project to separately analyze all these failure modes, therefore the ultimate compression strengths ( ) of the materials as given by Upwind and found during the experiments are used as the critical stress for compression failure. 2.2 Buckling of flat plates in pure compression Buckling is a failure mode that is characterized by sudden failure due to a highh compressive load, which is lower than the ultimate compressive load the material can withstand. Buckling can also be referred to as failure due to elastic instability. The critical buckling load is the critical value of compression force at which the flat form of equilibrium of the plate becomes unstable and the plate starts to buckle. As imperfections are inevitable, the plate is assumed to have some initial curvature or some lateral loading, therefore the critical buckling load can also be defined as the load at which the deflections have a tendency to grow indefinitely [8, 9]. In [10] it is concluded thatt linear buckling analysis may not be sufficient in determining buckling loads and stresses, as the inevitable presence of imperfections is neglected. Imperfections such as minor production errors can decrease the buckling resistance of the material. Therefore the result of the linear buckling analysis has to be interpreted with care and should possibly be lowered with a certain factor if used in a real design. Analytical formulations describing the buckling load are often derived using the energy method. In this method the plate is assumed to experience little lateral bending as a result of compression forces. The work done by the forces needed to bend the plate is compared to the strain energy of bending of every possible shape of lateral deflection. Elastic instability occurs when the work done by the forces becomes larger than the strain energy of bending. Figure 2.1: Schematic view of clamped (left) and simply supported (right) boundary conditions. The shape of the deflection of the plate depends on the boundary conditions at the edges of the plate. There are three kinds of boundary conditions considered in plate buckling: 6

17 2.2 Buckling of flat plates in pure compression clamped conditions, simply supported conditions (see figure 2.1) and free edges. With clamped boundary conditions, rotation of the normals to the plate is restricted as well as out of plane deflection at the line of support. Simply supported boundary conditions also restrict out of plane deflection at the line of support but rotation of the normals to the plate is allowed. Therefore the bending moment acting about the line of support is always equal to zero. With free edges there is no restriction at the edges causing the plate to behave consistent with Euler or column buckling. Figure 2.2: Simply supported plate buckling in three half waves. A plate that is simply supported at all edges and compressed in longitudinal direction buckles into a double curved shape, as can be seen in figure 2.2. The deflection surface w can be represented by the double trigonometric series sin sin (2.1) where and are the numbers of half waves in respectively longitudinal and transverse direction, and represent the plate length and width and represents a coefficient. In [11] an expression is derived for the critical buckling load of a simply supported isotropic plate using the energy method:, (2.2) where is the flexural rigidity of the plate. The load at which the plate buckles depends on the number of half waves formed in both longitudinal and transverse direction. From equation 2.2 it can be seen that the minimum critical buckling load will contain only one half wave in transverse direction. In figure 2.3 buckling curves for a simply supported plate with one half wave in transverse direction are shown for several numbers of half waves in longitudinal direction as a function of the plate aspect ratio. The number of half waves in which the plate will buckle is determined by the curve which provides the lowest buckling load at the aspect ratio of the plate. 7

18 Chapter 2 - Theory on failure modes of compression panels Figure 2.3: Buckling curves for a simply supported plate with one transverse half wave. The equation for the buckling mode with one half wave in longitudinal and one in transverse direction can be written as: where, (2.3) (2.4) When a plate A with simply supported boundary conditions buckles in two longitudinal half waves then both halve waves are in exactly the same conditions as a plate B, with a length half that of plate A and all else equal, that buckles in one longitudinal half wave. Therefore equation 2.3 can be used to describe the buckling load for a mode with two half waves in longitudinal direction by replacing with /2. This can be in done in a similar way for higher numbers of longitudinal half waves. In [12] an equation is provided to calculate the critical buckling load of thin flat rectangular specially orthotropic plates with several loading and edge support conditions (see paragraph for more information on special orthotropy):, 2 (2.5) where is an element of the flexural bending stiffness matrix (see paragraph 3.3.1). The buckling coefficient and constant can be found in the provided curves in [12] and are respectively 19.7, for long plates, and 2.0, when using simply supported boundary conditions. In equation 2.5 the through-the-thickness shear stiffnesses H 11 and H 22 are assumed to have negligible effect and are therefore assumed to be infinite. This assumption is only valid if the following equation holds:, 0.1 (2.6) 8

19 2.3 Buckling of stiffened panels in pure compression In case this inequality is not met a computer program [13] can be used. The program uses the same theory as used in the analysis described above, only now the through-thethickness shear stiffnesses are taken into account. These shear stiffnesses can be calculated using another computer program [14]. In figure 2.4 the buckling load is plotted as a function of the shear stiffness. The figure shows that the buckling load is estimated highest for infinite through-the-thickness shear stiffnesses and therefore overestimated in case shear stiffnesses are significantly low and equation 2.5 is used. Figure 2.4: Buckling load as a function of the through-the-thickness shear stiffness. 2.3 Buckling of stiffened panels in pure compression Orthotropic plate theory The design of stiffened panels has been a subject of interest for many constructions in the 20 th century. For instance bridge decks and aircrafts needed the benefits of saving weight. Most commonly used are longitudinal stringers placed parallel to the in-plane load to increase the bending stiffness in that direction, transverse stringers to subdivide the panel in smaller panels or a combination of both. There are two different modes of buckling that can occur in stiffened panels depending on the structure: local and global buckling (see figure 2.5), respectively half waves between the stringers and half waves bending both plate and stringer simultaneously [15]. According to Mittelstedt [16] a mass-optimal configuration can be found when both local and global buckling can occur at the same critical load, while in [17] it is stated that when the critical local and global buckling loads lie within the same range, the two modes interact reducing the critical buckling load with a significant amount. Figure 2.5: Schematic view of a compressed stiffened plate in a local (left) and a global (right) buckling mode. Orthotropic plate theory, valid for anisotropic plates of uniform thickness, has been used repeatedly to describe global buckling of stiffened plates. In this theory the stringers are 9

20 Chapter 2 - Theory on failure modes of compression panels smeared over the plate. For most stiffened plates the neutral planes in the two orthogonal directions do not coincide and the way this problem is solved is not generally agreed [18]. In order to solve this problem Timoshenko [11] suggested calculating the moment of inertia of the stringer about the top of the plate where the plate and stringer coincide and using this in the (isotropic) solutions valid for stringers with their neutral planes located in the plate middle surface. Furthermore in [19] Seide suggested using an expression to calculate the effective moment of inertia of the stringers around a plane between the neutral plane of the stringer and the neutral plane of the plate. In [16] Mittelstedt derived equation 2.7 for the global buckling load of orthotropic plates in compression in longitudinal direction stiffened by I-stringers. The moment of inertia of the I- stringers is calculated around the neutral plane of the stringers, a simplification that leads to fairly rough results., (2.7) with and sin 1 (2.8) (2.9) where is the area moment of inertia of the stringer cross section, the number of stringers, and are the Young s moduli of respectively the stringers and the plate, and and the surface areas of respectively one stringer and the plate. 10

21 2.3 Buckling of stiffened panels in pure compression Finite strip method In order to generate accurate and reliable results as a reference for the orthotropic plate theory, the finite strip method (FSM) has been used. This method can be considered as a simplified version of the finite element method [20] and can be used for structures that have a constant cross section along its length. The cross section of the structure is divided into elements by means of nodes, see figure 2.6. The elements represent longitudinal plate strips, each with an individual thickness and degrees of freedom, which are interconnected by means of the nodal lines running along the full length of the structure. Figure 2.6: Finite strip model of a stiffened panel. For the in-plane displacements of the elements, linear shape functions are used in transverse direction and sinusoidal shape functions in longitudinal direction. For the out of plane displacement a cubic polynomial is used [21]. It is out of the scope of the project to fully research this method. Therefore a computer program [22] has been used that is based on the finite strip method. According to [23] a previous study has shown that results generated with this program agree with the finite element method very well. The program calculates the critical buckling load as a function of the half wave length for one longitudinal half wave. As already seen in paragraph 2.2, a panel buckling in two or more half waves in longitudinal direction experiences identical boundary conditions as a simply supported panel, with a length equal to the half wave length, that buckles in only one half wave. Therefore the output of the program, a buckling curve for one longitudinal half wave, can be used to create the buckling curves for longer panels buckling in multiple half waves in longitudinal direction. For each plate aspect ratio the critical buckling load is determined by the curve with the lowest buckling load, as illustrated in figure 2.7. The first part of this graph (plate aspect ratio < 2.1) is dominated by local buckling while the second part is dominated by global buckling. Critical load [kn] m = 1 m = 2 m = 3 m = 4 m = Plate aspect ratio Figure 2.7: Buckling curves for several numbers of half waves (left) and the resulting critical buckling load (right). 11

22 Chapter 2 - Theory on failure modes of compression panels 2.4 Wrinkling of sandwich plates Wrinkling is a specific instability phenomenon that occurs in the faceplates of sandwich plates under compression. Wavelengths are in the order of magnitude of the core thickness. Wrinkling in sandwich plates as a result of compression is found to occur in three different modes [8, 9]: wrinkling of the compression face, antisymmetrical wrinkling and symmetrical wrinkling. The first mode occurs due to differences in stiffness of the faceplates or differences in loads and is therefore not considered. Antisymmetrical and symmetrical wrinkling, as shown in figure 2.8, occur at about equal load and are therefore treated as one and the same wrinkling mode. Figure 2.8: Symmetrical and antisymmetrical wrinkling of sandwich face sheets. Fagerberg [24] investigated the transition from face sheet wrinkling failure to face sheet compression failure. He used the following equation, derived by Plantema [9], to calculate the critical wrinkling load per face sheet:, (2.10) where is the flexural rigidity of the faceplates and and are respectively the Young s and the shear modulus of the core. Fagerberg found that for lower core densities wrinkling is the predominant failure mode rather than compression failure. An increase in core density results in a steady increase in the wrinkling load. From a certain core density compression failure will become the predominant failure mode. An interesting conclusion coupled to this transition hypothesis is that at this point the optimum combination of face sheet properties and core density can be found. The transition point can be found by equalling the ultimate compressive strength (σ UCS ) to the critical wrinkling stress of the sandwich face sheet. 12

23 3 Sandwich laminate reconstruction and failure analysis In this report theoretical failure analyses are made of three different panels: the reference sandwich panel, isolated from the Upwind turbine blade, a hat-stiffened panel, designed to form an alternative for the sandwich panel and a hat-stiffened test panel, used in experiments to validate the theoretical models. The hat-stiffened test panel has been constructed using a fabric (UT-E300 UD fibres) that was available at the Delft Aerospace Structures and Materials Laboratory. In order to create comparable analyses of the different panels the ply properties of this fabric are used for the design of the laminates of the panels. Since the sandwich panel is used as a reference, the laminate of the faceplates has been reconstructed and tested in tension and compression in order to determine the ply properties of the fabric and to validate the ultimate compressive strength. 3.1 Provided material properties The faceplates of the reference sandwich panel are triaxial laminates made out of glass fibre reinforced plastic and the core is made out of foam. The material properties of the sandwich panel were provided by Upwind as shown in table 3.1. Table 3.1: Provided material properties. E 11 [MPa] E 22 [MPa] G 12 [MPa] v 12 [-] ρ [kg/m 3 ] σ UTS [MPa] σ UCS [MPa] d [mm] UD laminate Triaxial laminate Foam core Estimation of the ply properties To be able to design the stacking sequence of the laminate that represents one face plate of the sandwich panel an estimation of the ply properties of the provided triaxial laminate is needed. This estimation has been calculated using a concentric cylinder analytical model (CCA) and several assumptions like the fibre volume fraction (50%), epoxy and fibre materials (epoxy and E-glass) and the ratio of the ply orientations (50% of 0 plies, 25% of +45 plies and 25% of -45 plies). In table 3.2 the estimation of the ply properties is shown. Table 3.2: Estimation of the ply properties of the provided laminate. E 11 [MPa] E 22 [MPa] G 12 [MPa] v 12 [-] Estimation of ply properties

24 Chapter 3 - Sandwich laminate reconstruction and failure analysis 3.3 Laminate design Three triaxial laminates have been designed using the estimated ply properties. For these designs classical laminate theory was used in order to calculate the material properties as described in paragraph Furthermore a specially orthotropic stacking sequence was used as described in paragraph Laminate stiffness matrix The laminate stiffness matrix (see equation 3.1), also referred to as ABD matrix, is a combined matrix of the extensional stiffness matrix (A matrix), the coupling stiffness matrix (B matrix) and the bending stiffness matrix (D matrix). The elements in the A matrix relate inplane stress resultants to the in-plane midplane strains, very similar to the moduli of elasticity, although the A 12 and A 16 terms couple shear strains to normal stresses and normal strains to shear stresses. The B matrix relates bending and extension which means that midplane curvature occurs when normal and shear forces are applied to the midplane or midplane strain occurs when bending and twisting moments are applied. The terms in the D matrix relate the midplane curvatures to the bending moments. The equations that are used to form the ABD matrix can be found in appendix A. κ κ κ (3.1) where the 6x6 matrix consisting of the elements, and represents the ABD matrix,, and and, and are respectively the force and moment resultants,, and the strains and, and the curvatures. Using equations the properties of the designed laminate can be calculated: (3.2) (3.3) (3.4) (3.5) where are elements of the inverse ABD matrix, and are respectively the longitudinal and transverse Young s Moduli, is the shear modulus, the Poisson s ratio and the laminate thickness. 14

25 3.4 Experimental analysis Specially orthotropic stacking sequence For the design of the stacking sequence (order of orientation of the plies) of the laminate the elastic modulus in longitudinal direction and the ratio between plies oriented at +/-45 and 0 were used as design drivers. Furthermore a stacking sequence for specially orthotropic laminates was used since this is required for several calculations used throughout the report. A laminate is specially orthotropic if the following conditions are satisfied [25]: the extensional and bending stiffness matrices A and D are orthotropic, meaning A 16, A 26, D 16 and D 26 are zero and bending and extension are uncoupled, hence B = 0. In this case the laminate stiffness matrix is filled as shown in figure Figure 3.1: Laminate stiffness matrix of a specially orthotropic laminate. In [25] examples of stacking sequences for specially orthotropic laminates are given for up to 21 plies. All of these sequences may be enlarged by adding a pair of orthotropic plies, aligned with either x- or y-axis, to the outsides of the sequence, while some sequences can also be enlarged within the sequence. A specially orthotropic stacking sequence was chosen, that allowed the addition of plies in the middle of the stacking sequence to be able to change the sequence with one ply at a time. Because the bending stiffness of the reference laminate has not been provided, the thickness is used as a reference for this material property, since the bending stiffness of a laminate increases with the thickness to the power third. The number of plies, needed to create the required thickness, was established using an expected ply thickness. As a result three triaxial laminates of 14, 15 and 16 plies were designed together with a unidirectional laminate of 16 plies. The stacking sequences for these laminates can be found in table 3.3. Table 3.3: Stacking sequences of the designed laminates. Laminates Stacking sequence Visualization of stacking sequence U - 16 plies [0 16 ] **************** S - 16 plies [(0 2,+45,-45) s,(0 2,-45,+45) s ] ** **** ** T - 15 plies [0 2,(+45,-45) s,0 3,(-45,+45) s,0 2 ] ** *** ** L - 14 plies [0 2,(+45,-45) s,0 2,(-45,+45) s,0 2 ] ** ** ** 3.4 Experimental analysis The designed laminates have been produced using vacuum infusion and were cut into tensile and compression specimens (see figure 3.2). Both tensile and compression specimens have been tested on the Zwick 250kN test facility using respectively the standards ASTM and ASTM D (e1), which provided guidelines and requirements for the dimensions of the specimens as well as for the execution of the tests. 15

26 Chapter 3 - Sandwich laminate reconstruction and failure analysis (a) (b) (c) (d) Figure 3.2: Production of the test specimens showing: (a) dry fibers, (b) vacuum infusion, (c) laminate, (d) tensile specimen. For each laminate the fibre volume fraction has been determined. Small specimens have been cut from the laminates and were heated in an oven up to 570 degrees Celsius for several hours, evaporating all of the resin. The difference in mass before and after heating the specimens accounts for the mass of the resin. The fibre volume fraction was its turn calculated using: (3.6) where and are the masses and and are the densities of respectively the fibres and the resin. Furthermore the width and thickness of all the specimens have been measured to be able to calculate the stress from the measured force Tensile tests From each laminate two series of tensile specimens were cut: specimens oriented at 0 and specimens oriented at 90 with respect to the fibre orientation. The Young s modulus and Poisson ratio in longitudinal direction as well as the ultimate tensional strength were determined using the specimens oriented at 0, while the Young s modulus and Poisson ratio in transverse direction were determined using the specimens oriented at 90. The tensile specimens have been provided with glued aluminium tabs for a gradual introduction of the tensional forces. The length of the tensile specimens was made up out of the lengths of the tabs and the gage section, respectively 50mm (2x) and 100mm. The required width has been determined using the tensile specimen geometry recommendations of the test standard and is close to 25mm. For each of the specimen series of 15 and 16 plies one specimen was provided with one strain gage oriented in longitudinal direction and one strain gage in transverse direction, while the other specimens of these series were only provided with a strain gage in transverse direction. Initially an extensometer was used in order to measure the strain in longitudinal direction while the specimens with a strain gage oriented in longitudinal were used as a reference for the extensometer. A significant difference, though, was found between the measurement data of the longitudinal strain gages and the extensometer. Therefore only the measurement data of the specimens with both longitudinal and transverse strain gages were used to determine the material properties, while the measurement data of the extensometer was used to monitor the deviation of the 16

27 3.4 Experimental analysis measurements. The specimens consisting of 14 plies were all provided with strain gages in longitudinal as well as transverse direction. For each tensile test the stress has been plotted as a function of the strain in order to determine the young s modulus which was found from the slope of a linear fit of about 200 measurement points of each curve around 0.15% strain, since in most cases the presence of noise obstructed the use of more measurements points. The Poisson ratio was found from the slope of a linear fit of the lateral strain as a function of longitudinal strain for the same measurement points as used for the Young s modulus. The highest measured stress was used for the ultimate tensional strength Compression tests The compression tests have been carried out in order to determine the ultimate compression strength of the laminate as well as the Young s modulus in longitudinal direction, which should be similar to the modulus, determined using the tensile tests. From the tensile tests it had become clear that the laminate consisting of 14 plies has material properties that are closest to the theoretical material properties of the faceplates of the reference sandwich panel. Therefore compression tests were only carried out using this laminate. Eight compression specimens were produced. As provided by the test standard the specimen compression ends had to be parallel within 0.03 mm and also perpendicular to the longitudinal axis of the specimen within 0.03 mm. Special care was taken when cutting the specimens in order to meet these tolerances. Each specimen was provided with two strain gages in longitudinal direction, one at each side, to be able to compensate for differences in strain due to bending when determining the Young s modulus. The highest measured stress was used for the ultimate compressive strength Test results The average results of both tensile and compression measurements are found in table 3.4, while the complete list with results is found in tables B.1 - B.3 in Appendix B. The consistency of the measurements is good as can be seen from the relative standard deviations calculated for the measured material properties (see table B.4 in Appendix B). It is concluded that the laminate consisting of 14 plies has sufficiently similar material properties to those of the reference triaxial laminate. Although the Young s moduli are slightly higher, the thickness of the laminate as well as the ultimate tensile and compression strengths correspond very well with the provided data. Table 3.4: Measured material properties Specimens E 11 [MPa] E 22 [MPa] V 12 [-] V 21 [-] σ UTS [MPa] σ UCS [MPa] d [mm] V f [-] ρ [kg/m 3 ] L T S U The earlier estimation of the ply properties, as shown in table 3.2, was based on a 50% fibre volume fraction. From the measurements of the laminate consisting of 14 plies a fibre volume fraction of 58% was found. Therefore another calculation was performed by means 17

28 Chapter 3 - Sandwich laminate reconstruction and failure analysis of CCA in order to obtain the final ply properties of the used fabric. These properties are shown in table 3.5 together with the theoretical properties of the laminate consisting of 14 plies, which are calculated using the final ply properties and classical laminate theory as described in paragraph When compared to the data in table 3.3 it can be seen that the theoretical laminate properties are very close to the measured laminate properties. Table 3.5: Final ply properties of the used fabric and calculated laminate properties of the laminate consisting of 14 plies. E 11 [MPa] E 22 [MPa] G 12 [MPa] V 12 [-] V 21 [-] d [mm] Ρ [kg/m 3 ] Final ply properties Theoretical laminate Failure analysis of the reference sandwich panel To be able to design the hat-stiffened panel the critical compression load of the reference sandwich panel is needed. In chapter two it is seen that three failure modes need to be considered: buckling, wrinkling and compression failure. In order to calculate the critical buckling load the analytical approach was used, as described in paragraph 2.2. The bending stiffness matrix has been calculated for the sandwich panel using the laminate consisting of 14 plies for each faceplate and a foam core with properties as given in table 3.1. As already mentioned in chapter two this approach does not take the through-the-thickness shear stiffnesses into account. Therefore the computer programs, as described in paragraph 2.2, were used to calculate the critical buckling loads for infinite- as well as the calculated through-the-thickness shear stiffnesses in order to examine the difference. It was found that for infinite through-the-thickness shear stiffnesses the buckling load is similar to the one calculated with the analytical approach, while based on the calculated through-the-thickness shear stiffnesses the buckling load is lower. In order to research the worst case scenario the values using infinite through-the-thickness shear stiffnesses are used for the critical buckling load. Furthermore the critical wrinkling load has been calculated as described in paragraph 2.4, while the load at compression failure was calculated using the ultimate compression strength of the laminate in combination with the total cross-sectional area of the faceplates. As can be seen from table 3.6, failure due to buckling occurs at the lowest load and therefore buckling is considered the dominating failure mode. Therefore the reference critical compression load to be used for the design of the hat-stiffened panel is 1 MN while the mass should not be higher than 44.1 kg per meter length of the panel. Table 3.6: Calculated mass and critical loads of the reference sandwich panel. Mass sandwich panel per meter length 44.1 kg/m Buckling load analytical 1.06 MN Buckling load calculated through-the-thickness shear stiffnesses 0.88 MN Buckling load infinite through-the-thickness shear stiffnesses 1.08 MN Wrinkling load 6.34 MN Load at compression failure 4.58 MN 18

29 4 Development and validation of analytical models In chapter two it was found that there are three failure modes concerning stiffened panels in compression on which the design should be focused: global buckling, local buckling and compression failure. Different analytical models have been studied in order to establish a combined model that predicts the critical global buckling load of a hat-stiffened panel. Several definitions were used in the explanatory texts for the components of a hat-stiffened panel. These definitions have been illustrated in figure : Illustration of the components of a hat-stiffened panel. 4.1 Development of analytical models Global buckling Since composite laminates are orthotropic, equation 2.7 is used as a basis to calculate the global buckling load of the stiffened panel. For the sake of simplicity Mittelstedt used this equation to calculate the buckling load of plates stiffened by I-stringers on one side of the plate, while in fact the equation describes the buckling load of a plate with stringers positioned with their moments of inertia around the neutral plane of the plate. In other words equation 2.7 describes a plate with stringers on both sides of the plate. This simplification results in a lower buckling load since the longitudinal stiffness of the panel is underestimated. The methods to correct for this underestimation suggested by Timoshenko and Seide as described in paragraph are researched on their performance in paragraph 4.3 and explained further below. Timoshenko Timoshenko suggested correcting the moment of inertia of the stringers for the distance of their neutral plane to the top of the plate. These planes are illustrated in figure 4.2a. The moment of inertia of the stringers is in this case calculated using:, (4.1) Seide Seide on the other hand argued that the effective moment of inertia of a stringer can vary between the moment of inertia taken about neutral plane of the stringer and the moment of inertia taken about neutral plane of the plate, as illustrated in figure 4.2b. The following expression was used to correct the moment of inertia of the stringers according to Seide s theory: 19

30 Chapter 4 - Development and validation of analytical models, 1 1 (4.2) where represents the distance between the neutral plane of the plate and neutral plane of the stringer and is the modal coefficient which can be obtained from [19]. (a) (b) (c) 4.2: Illustration of the planes which are used in the different methods: (a) Timoshenko, (b) Seide and (c) Centroid. Centroid In addition a method was developed to adjust the longitudinal bending stiffness (D 11 ) of the plate as well as the moment of inertia of the stringers with respect to the neutral plane of the cross-sectional area of the total panel (also referred to as centroid). As can be seen from figure 4.2c the neutral plane of the total structure is found in between the neutral plane of the plate and the neutral plane of the stringers. The position of this plane can be calculated using the following equation: (4.3) where and are respectively the cross sectional areas and Young s moduli of all parts of the panel while z i represents the distances of their neutral plane to a chosen z-coordinate. The moment of inertia of the stringers is calculated around the centroid of the total panel in a similar manner as in which the correction of Timoshenko is applied. Since the hat-stringers are composed of a top flange, webs and bottom flanges, the equation for the effective moment of inertia of the stringers is given as a summation of the effective moments of inertia of these elements:,, (4.4) where represents the area moments of inertia of each part of the stringer and, the z- coordinates of their neutral planes. In order to adjust the longitudinal bending stiffness of the plate classical laminate theory is used as explained in appendix A. Only now the neutral plane of the plate is placed at an offset from the xy-plane, which is in this case positioned at the centroid of the total panel, as shown in figure

31 4.1 Development of analytical models Figure 4.3: Schematic illustration showing a cross section of the plate. Transverse bending stiffness When the panel buckles it bends in both transverse and longitudinal direction. The stringers mainly influence bending in longitudinal direction, but also contribute to the bending stiffness in transverse direction. In equation 2.7 this contribution is not taken into account. In order to compensate for this the plate is assumed to have infinite transverse bending stiffness (D 22 ) underneath the stringers excluding the stringer bottom flanges. The adjusted transverse bending stiffness can be calculated using equation 4.5, which has been derived using beam bending theory. The complete derivation can be found in appendix C. (4.5) where and are respectively the fraction of the subplates and the fraction of plates underneath the stringer (excluding the stringer bottom flanges) of the total width of the panel Local buckling In [16] Mittelstedt used an equation, similar to equation 2.5, for orthotropic plate buckling for the calculation of the local buckling load of a stiffened plate. In this approach he uses the assumption that during local buckling the subplates between the stringers behave like regular simply supported orthotropic plates. In addition in [26] he provided a method to calculate the boundary conditions formed by hat-stringers due to their torsional rigidity, being elastic constraints that vary between simply supported and clamped boundary conditions. Since the total panel has simply supported boundary conditions at all sides, the subplates at the sides are restricted with stringer dependent boundary conditions on one side and simply supported boundary conditions on the other side. Therefore this method is expected to overestimate the critical local buckling load. Furthermore the stringer bottom flanges should be taken in account since they can be of significant dimensions compared to the subplates. In order to compensate for this the bottom flanges are suggested to be smeared over the subplate using equation 4.6. Using this equation the material of the bottom flanges is added to the plate by increasing the plate ply thickness., 2 (4.6) where, is the smeared ply thickness, the width of a bottom flange, the thickness of the bottom flange, which is taken half the stringer thickness due to its triangular 21

32 Chapter 4 - Development and validation of analytical models shape,, the width of the subplate, the plate laminate thickness and the number of plies in the plate laminate Compression failure In order to exclude failure due to compression, the compression strain can be used as an indicator. The designed laminates of the stiffened panels have a similar ratio between plies oriented at +/-45 and 0 as used in the test specimens consisting of 14 plies (see chapter 3). It is assumed that during the compression tests of these specimens the plies oriented at +/- 45 failed before the plies oriented at 0 failed. Furthermore it is assumed that in a laminate with a similar ratio between plies oriented at +/-45 and 0 the plies oriented at +/-45 will fail at about the same strain as occurred during the compression tests. The stress at which this strain occurs depends on the compression stiffness of the laminate and therefore on the stacking sequence. Therefore if the stacking sequence is altered also the ultimate compression strength has changed. The load at which compression failure is expected to occur can be calculated using: (4.7) where represents the strain at which the test specimens of 14 plies failed, and and are respectively the Young s modulus and cross-sectional area of the laminate that is being analyzed. 4.2 Design tools The developed analytical models, referred to as Timoshenko, Seide and Centroid, have been used to create two design tools, a case simulator and a panel optimizer, based on matlab and excel. Both tools can be used with a configuration interface in excel where the parameters of the panel can be adjusted. Matlab in its turn is used to carry out the calculations and the results are transferred to another excel file. The case simulator is used to calculate the mass and critical loads of a designed panel as a function of the plate aspect ratio. Each of the developed analytical models can be used, while the finite strip method can be used simultaneously as a validation. With the panel optimizer it was tried to create an automatic panel design tool based on the analytical models. The panel optimizer calculates the most mass-efficient panel from a series of panels as a function of the plate aspect ratio and a defined minimum critical load. For two parameters, that is the stringer thickness and the number of stringers, a range can be used in the configuration file, while all other parameters are filled in as a constant. For each combination of these parameters the plate thickness is calculated that is needed for the panel to be able to withstand the minimum critical load. The most mass-efficient panel for each plate aspect ratio is used for the output. It was found however that this design tool does not provide the most mass-efficient panels. This is caused by inaccurate predictions of the critical buckling load by the analytical models as described in the following paragraphs. Despite the inaccurate predictions the panel optimizer is still a helpful tool to get an idea of required plate and stringer thicknesses. 22

33 4.3 Model validation 4.3 Model validation In this paragraph the performance of the analytical models has been researched, using the finite strip method as a reference for simulations of several stiffened panel designs. In paragraph the models have been compared using a hat-stiffened panel that can withstand the reference load. From this comparison several differences were found between the finite strip method and the analytical models. The source of these differences has been analyzed in paragraphs and The analytical models are based on a derivation for I-stringer stiffened panels. Therefore an I-stringer stiffened panel was simulated in paragraph This panel has been designed to withstand the reference load as well. In addition a hat-stiffened panel was designed with the same number of stringers as used on the I-stringer stiffened panel. The hat-stringers of this panel have been designed in such a way that the centroids of both panels are positioned at equal distance from the top of the plate. This comparison has been carried out since, except for the correction of the plate transverse bending stiffness, the analytical models only use the moment of inertia of the stringers, while the actual shape of the stringers is not taken into account. Therefore it is expected that, when the correction of the transverse bending stiffness of the plate has not been applied, the analytical models give exactly the same critical loads for both panels. The finite strip method on the other hand does take the shape of the stringers in account and is therefore used in the comparison. Despite the use of I-stringers, a difference was still found between the analytical models and the finite strip method. In order to understand more about the cause of these differences, a comparison was carried out in paragraph using three I-stringer stiffened panels of a smaller size. All panels in this paragraph were designed using simply supported boundary conditions and for the purpose of comparison only. The dimensions of the used panels can be found in Appendix D Simulations using a hat-stiffened panel For this comparison a hat-stiffened panel with five stringers was used, which was designed (using the analytical models) to withstand the reference load at a plate aspect ratio of 1.6. This panel is also used as a base case for the parameter study in paragraph 5.1. When the analytical models (Timoshenko, Seide and Centroid) are compared to each other similar buckling curves are found. The finite strip method on the other hand gives a significantly higher value for the horizontal asymptote of the curve, as can be seen in figure 4.4. Furthermore, the transition points where the number of half waves changes do not occur at a similar plate aspect ratio when comparing the analytical models with the finite strip method. 23

34 Chapter 4 - Development and validation of analytical models Critical load [MN] FSM Seide Timoshenko centroid Critical load [MN] FSM Seide Timoshenko centroid Plate aspect ratio Plate aspect ratio Figure 4.4: Critical buckling curves for a hat-stiffened panel with five stringers. The figure at the right is a zoomed view of the figure at the left Simulations using I-stringers and hat-stringers The analytical models were originally derived to be used for panels with I-stringers. Therefore a panel with I-stringers was designed for the reference load at a plate aspect ratio of 1.6. As I-stringers are very narrow compared to the hat-stringers, 13 I-stringers were needed in the design to avoid local buckling of the subplates. Furthermore a hat-stiffened panel was designed with 13 stringers to use as a comparison for the I-stringer stiffened panel. The dimensions of the plate are the same for both panels and the stringers are dimensioned in such a way that for both panels the centroids of the panels are positioned at equal distance from the plate. Furthermore the cross sectional areas of both panels are identical. The buckling loads have been calculated as a function of the plate aspect ratio using both the finite strip method and analytical models. Since the analytical models (Timoshenko, Seide and Centroid) provided very similar results, only the results for the analytical model referred to as Centroid are shown in figure 4.5 in order to maintain a clear view in the graph. Critical load [MN] FSM hat Centroid hat FSM I Centroid I Plate aspect ratio Figure 4.5: Critical buckling curves for a hat-stiffened panel and a I-stringer stiffened panel, both with 13 stringers. 24

35 4.3 Model validation Stringer shape and torsional rigidity When comparing the results for the hat-stiffened panel of the analytical model and the finite strip method (see figure 4.5, curves Centroid hat and FSM hat ), a significant difference is again found between the horizontal asymptotes of these curves. Although in a smaller amount, this difference is also found when comparing the results for the I-stringer stiffened panel of the analytical model and the finite strip method (see curves Centroid I and FSM I ). Since the centroid of both panels is positioned at equal distance from the plate and the panels have equal cross-sectional areas, the difference between the curves FSM hat and FSM I is entirely caused by the different shape of the stringers. Due to the closed shape of the hat-stringers their torsional rigidity is significantly higher than the torsional rigidity of the I-stringers causing the critical load to be much higher. The torsional rigidity of stringers in a stiffened panel is of great influence on the shape of the buckling mode and consequently on the critical compression load. Transverse bending stiffness of the plate The analytical models were used in two ways for the hat-stiffened panel: with and without compensation for the influence of the hat-stringers on the transverse bending stiffness of the plate (see equation 4.5). The simulation using the original transverse bending stiffness of the plate resulted in exactly the same critical loads as resulted from the simulation of the I- stringer stiffened panel ( Centroid I in figure 4.5) and is therefore not shown in the figure. The curve Centroid hat shows the results where the compensation has been taken into account. The effect of the compensation of the transverse bending stiffness is consequently clearly found when comparing the curves Centroid hat and Centroid I. The adjustment results in higher critical loads and is therefore closer to the curve for the hat-stiffened panel using the finite strip method. The transition points for the number of longitudinal half waves however have been shifted towards a lower plate aspect ratio, while these points are found at equal plate aspect ratios for the other simulations. It is actually surprising to see that the shape of the stringers, and consequently the torsional rigidity of the stringers, does not influence the transition points for the number of half waves Simulations using small I-stringer stiffened panels In order to validate the performance of the theory closer to its origin several smaller panels with relatively small I-stringers were designed for arbitrary buckling loads. Three cases are used to illustrate mutual differences between the analytical models as well as differences with the finite strip method: a panel with three normal I-stringers, a panel with eight normal I-stringers and a panel with eight relatively thick I-stringers. The dimensions of the plate are equal for all three panels. In figure 4.6 it can be seen that, for the first two panels, the analytical models Seide and Centroid perform very well with respect to the finite strip method, while the model using Timoshenko s adjustment differs from the other models. 25

36 Chapter 4 - Development and validation of analytical models Critical load [kn] FSM Seide Plate aspect ratio Timoshenko centroid Critical load [kn] FSM Seide Plate aspect ratio Timoshenko Centroid (a) (b) Figure 4.6: Critical buckling curves for panels with three (a) and eight (b) normal I-stringers. In figure 4.7 a similar offset is seen between the curves for the finite strip method and the curves for the analytical models Seide and Centroid as found in the previous paragraphs. This confirms the theory that the torsional rigidity is of great influence on the critical compression load. The I-stringers are only three times as thick compared to the stringers used in the panels of figure 4.6, but already a significant effect can be noticed. Furthermore it can be seen that the analytical models Seide and Centroid give very similar results for the panel with relatively thick stringers. The model using Timoshenko s adjustment however was found to be more sensitive to changes in number and size of stringers and therefore the other two models are thought to be more reliable. Critical load [kn] FSM Seide Timoshenko Centroid Plate aspect ratio Figure 4.7: Critical buckling curves for a panel with eight thick I-stringers 26

37 4.3 Model validation Local buckling Several methods, as described in paragraph 4.1.2, were used to calculate the critical local buckling load for several panels. It was found however that in most cases the local buckling load was overestimated significantly. Only the method which simulates the subplate as a simply supported plate without the addition of smeared stringer bottom flanges resulted, in specific cases, in similar loads when compared to the finite strip method. It was found that, when a thick plate is used relative to the stringers, the local buckling load is overestimated significantly using this method. This is expected to be caused by the fact that local buckling occurs in the stringers in these cases. Furthermore when many small stringers are used in the design this method also overestimates the local buckling load. This is believed to be caused by the fact that in these cases local buckling occurs with transverse half wave lengths larger than the width of the subplate. Eventually local buckling is believed to be a phenomenon that is too complicated to catch in such a model. 27

38 5 Design study for hat-stiffened panels In chapter four the differences between the analytical models and the finite strip method have been researched. The finite strip method is thought to give more reliable results since it is considered being a simplified version of the finite element method, while the analytical models are far more simplified models. It was found that the design tool panel optimizer based on the analytical models does not produce reliable results. Since an optimization tool has not been designed for the finite strip, all simulations needed to be carried out one by one. In order to gain knowledge about the effect of certain parameters a parameter study has been carried out. Such a parameter study gives insight in an efficient design process and is therefore a helpful tool when designing the hat-stiffened panel. 5.1 Parameter study using FSM For the parameter study a hat-stiffened panel has been designed which is used as a base case for the simulations. This panel is used as a starting point and for each series of simulations only one parameter of the panel is varied. The base case panel has been designed for the same conditions as the reference panel, although it has a slightly higher critical buckling load at a plate aspect ratio of 1.6. It was designed in such a way that local buckling occurs only at extreme cases to be able to gather insight on the relation between several parameters and global buckling. The simulations are compared based on their mass per load ratio at three different plate aspect ratios, while the mass is expressed in weight per meter length. The parameters for the base case as well as for the other simulations can be found in appendix E. Number and size of stringers First it is researched whether a panel is preferred to contain many small stringers or a few large stringers. Therefore several configurations have been simulated using different numbers and sizes of stringers. It was chosen to keep the shape of the stringers constant by keeping the stringer angle invariable and setting the height of the stringer to half the stringer width. Furthermore it was chosen to use stringers that are as large as possible depending on the number of stringers. This means that for a certain number of stringers, the stringer width is increased to such a large value that the width of the subplates is very small and local buckling is less likely to occur. It has been taken into account that the bottom flanges of the stringers may not overlap. The results of the simulations are plotted in figure 5.1 as a function of the number of stringers and in table 5.1 accompanying data of the simulations can be found. Table 5.1: Panel configurations and accompanying data for the simulations of figure 5.1. Number of stringers [-] Stringer width [mm] Panel mass [kg/m] Buckling force for a/b = 1.6 [kn] Mass/load ratio for a/b = 1.6 [g/n]

39 5.1 Parameter study using FSM Mass per load ratio [g/nm] a/b = 2.5 a/b = 1.6 a/b = 1 local buckling Number of stringers [-] Figure 5.1: Mass per load efficiency curve varying size and number of stringers. From these simulations it can be concluded that a panel with a few large stringers is preferred since the mass per load ratio decreases with the number of stringers, as can be seen in figure 5.1. If the number of stringers becomes too low though, local buckling becomes the dominating failure mode. This is caused by the fact that the stringer top flanges, -webs, -bottom or the subplates become too wide compared to the laminate thickness. This is for instance the case for the simulations at a plate aspect ratio of 1 with 4 and 5 stringers. In these two cases the mass per force ratio is still efficient, but when only three even larger stringers are used the critical local buckling load becomes very low increasing the mass per load ratio to inefficient values. Stringer height Another set of simulations was carried out varying only the stringer height. It was found that high stringers are most efficient concerning mass and force as the mass per force ratio decreases with increasing stringer height. Also for this parameter local buckling forms a limit, as can be seen in figure 5.2 where the plate aspect ratio is equal to one and the height of the stringers is 120 to 160 mm. The webs of the stringers are now so long compared to the laminate thickness that local buckling is expected to occur here. Mass per load ratio [g/nm] Stringer height [mm] Figure 5.2: Mass per load efficiency curve varying the height of the stringers. 29 a/b = 2.5 a/b = 1.6 a/b = 1 local buckling

40 Chapter 5 - Design study for hat-stiffened panels Stringer and plate thickness Two more parameters that have been researched are the plate and stringer laminate thicknesses, see figure 5.3 and 5.4. These thicknesses are built up from the ply thickness, the stacking sequence and a ply thickness factor. The ply thickness is used as determined from the specimens described in chapter three and the stacking sequence determines the number of plies used in the laminate. In order to change the thickness but maintain the stacking sequence and therefore comparable laminate properties, the ply thickness factor is used to increase or decrease the thickness of the plies and therefore the thickness of the laminate. For both parameters it was found that increasing the ply thickness factor decreases the mass per load ratio only very little from a certain value. For the design this means that at a certain point these parameters can be used in order to increase or decrease the critical buckling load without seriously affecting the mass to force ratio. Furthermore in both cases local buckling forms a limit, as the plate or stringer will buckle locally when thicknesses become too small. Mass per load ratio [g/nm] a/b = 2.5 a/b = 1.6 a/b = 1 local buckling Mass per load ratio [g/nm] a/b = 2.5 a/b = 1.6 a/b = 1 local buckling Ply thickness factor [-] Ply thickness factor [-] Figure 5.3: Mass per load efficiency curve varying the ply thickness of the stringers. Figure 5.4: Mass per load efficiency curve varying the ply thickness of the plate. 5.2 Panel optimization methodology In this paragraph a method is described that can be used as a guideline when designing a hat-stiffened panel for simply supported boundary conditions. At the start of the design a few basics are needed like material properties of the used fabric, the width of the panel, the shape of the stringers and the load the panel is required to withstand. Furthermore a specially orthotropic stacking sequence needs to be chosen which allows for expanding or decreasing, although it can still be changed during the design process. In order to gain insight in required laminate thicknesses and the number of stringers that is needed the optimization tool, based on the analytical models, can be used. As described in paragraph 4.2 the optimization model will not provide the most mass efficient panel but it can still be used as a start-up since it takes relatively little time compared to using the case simulator based on the finite strip method. 30

41 5.3 Reference hat-stiffened panel design After this the knowledge gained from the parameter study, described in the previous paragraph, can be used in the case simulator. As it was concluded that a panel with a few large stringers is most mass efficient it should be researched if the number of stringers can be decreased. In order to avoid local buckling laminate thicknesses can be increased but also the width of the stringer or bottom flanges can be increased. Furthermore parameters like stringer height and angle and the stacking sequence can be changed in order to design a more efficient panel. A very important aspect that remains is the plate aspect ratio, which is dependent on the application on one hand and the mass per load ratio compared to the mass of the needed ribs on the other hand. The characteristic buckling curve can be divided into two sections of interest depending on the plate aspect ratio. For high plate aspect ratios the panel will buckle in multiple half waves. For these cases the buckling curve gives a horizontal asymptote. Therefore the plate aspect ratio can be chosen as high as possible in order to avoid additional weight of ribs. For low plate aspect ratios the panel will buckle in only one half wave. As the curve is very steep at this point, decreasing the plate aspect ratio results in a large increase in the critical buckling load. The highest value of the global buckling load is, depending on the design, limited by the occurrence of local buckling as the critical global buckling load becomes higher than the critical local buckling load at very low plate aspect ratios. An increase in buckling load allows for mass reducing parameter changes in the design. When designing in this section of the buckling curve an optimum can be found between the additional weight of the needed number of ribs and the panel mass both depending on the plate aspect ratio. 5.3 Reference hat-stiffened panel design In order to design the reference hat-stiffened panel the methodology of the previous paragraph was used. From the optimizer the order of laminate thicknesses was found, where after the number of stringers was decreased and the size of the stringers was increased. It was found that configurations with six, five or four stringers were desirable. For these configurations several parameters were varied and simulations were carried out in order to find an optimal design. It was found that the addition of plies, oriented at 90 degrees, to the plate is desirable. The length of the panel was chosen to be 3.68m, resulting from a chosen plate aspect ratio of 1.6 and the reference width of 2.3m. At this point the buckling mode is dominated by one half wave in both longitudinal and transverse direction or by local buckling, since both critical compression loads are similar. In this way the benefits of the steep curve are used while the number of ribs remains acceptable. The critical local buckling load for this design is 1.00MN, while the critical global buckling load is 1.08MN according to the finite strip method, as can be seen in figure 5.6. Figure 5.5: Cross section of the optimized hat-stiffened panel. 31

42 Chapter 5 - Design study for hat-stiffened panels The optimized design (see figure 5.5) has five stringers and a weight of 37.5kg per meter length. Assuming one rib for each panel this would mean that in order to have equal mass to the sandwich panel (44.1kg/m) the rib may weigh (44.1kg/m kg/m) *3.68m= 24.3kg. This is slightly more than the mass of three stringers as used in this design over the full width of the panel. It is not expected that a rib of this weight will be needed, although this has not been researched since it is out of the scope of this project. Critical load [MN] Plate aspect ratio Figure 5.6: Critical buckling curve for the optimized hat-stiffened panel. 32

43 6 Test panel construction and test setup To validate the theories a hat-stiffened test panel was constructed to be tested on elastic buckling on the 80 tons compression testing machine using uniform longitudinal compression. A test rig was designed to simulate simply supported boundary conditions at all four sides of the panel. 6.1 Design of the hat-stiffened test panel The test panel is a scaled version of the designed reference hat-stiffened panel and therefore has five stringers, see figure 6.1. The capacities of the test facility concerning load and specimen size were used as the limiting factors for the design of the panel. The width of the plate is limited to 600mm, while the plate length is determined by the chosen plate aspect ratio of 1.6 resulting in 960mm. Due to the design of the test rig however, as described in paragraph 6.3, the panel was constructed with a length of 914mm. The panel has been designed, using FSM, to have a critical global buckling load of about 109kN. The critical local buckling load is designed to be much higher (+/- 500kN) in order to avoid local buckling or interaction between local and global buckling modes. The ultimate compression strength was monitored using the method described in paragraph and is expected to be about 850 kn. Figure 6.1: Cross section of the hat-stiffened test panel. The stacking sequence of the laminates has been designed using the guidelines as described in paragraph 3.3. For simplicity it was chosen to use an equal stacking sequence for both the plate and stringers. The laminate consists of 11 plies and the ply orientations can be described as [ ]. The stringers are equally spaced and the subplates at the edges are designed to have the same width as the subplates in between the stringers. An additional +/-10mm is added to these sides to allow movement in the side supports (see paragraph 6.3). 6.2 Construction of the test panel Construction of the dry product The test panel has been constructed with the same fabric as used for the test specimens in chapter 3. Since this fabric is only 500mm wide the laminate plies had to be cut out in parts as can be seen in figure 6.3a. In order to create the shape of the stringers a light foam was sawn into long angled strips to form the inside shape of the stringers (see figure 6.3b). The foam strips have been provided with two rows of 1mm drilled holes from the top to the bottom spaced at about 20 mm. This was done in order to transport the resin from the flow mesh through the stringer laminate and through the drilled holes to the plate laminate beneath the stringers during the infusion process. The foam strips have been attached to the plate fibres using fixation spray and the stringer laminate was glued onto the foam strips layer by layer to ensure the plies to stay in place during infusion. Extra care was taken not to use fixation spray at the bottom flanges of the stringers in order to allow movement of the 33

44 Chapter 6 - Test panel construction and test setup plies towards the stringer foam when applying the vacuum and therefore avoiding runners during infusion. The plies of the stringer laminate have been dimensioned in such a way that, at the attachment of the stringer bottom flanges to the plate, each ply overlaps the ply underneath. The first ply has about 5mm of contact surface with the plate at each bottom flange and each ply on top overlaps the one underneath with about 1.5mm per stringer bottom flange, creating a 20mm long bottom flange. In this way the ply drop-offs are enclosed inside the laminate (see figure 6.2), which is beneficial in relation to e.g. delamination, contamination and avoiding runners during vacuum infusion. 6.2: Close up of a stringer bottom flange, showing the enclosed ply drop-offs. Vacuum infusion process During the infusion of the panel difficulties were encountered and three infusion attempts were needed to produce a test panel of sufficient quality. Since the panel is relatively large the supply of resin is a very important aspect in the infusion process. Therefore several measures were taken to ensure a sufficient supply of resin as discussed below. The panel has been infused in longitudinal direction. It was chosen to use two resin inlets in combination with two resin reservoirs to ensure an even distribution of the resin supply over the panel end at the inlet. Furthermore a very open spiral, which allows the resin to flow from the inlets to the product, was used in order to minimize pressure drops over the inlets. This results in a higher pressure drop over the product, which is beneficial for the resin supply. A fabric peel ply was used, since the use of a plastic peel ply resulted in a large void content in the product just below the resin supply holes of the foil. Both flow mesh and peel ply were cut into long strips, one strip for each stringer and one strip for each subplate. This was done to allow movement of the flow mesh and peel ply towards the inside corners of the stringers when applying vacuum and therefore to avoid runners during infusion. Flow mesh and peel ply was used over the whole surface of the panel except for the last +/-5cm at the side of the outlet, as can be seen in figure 6.3c. There only peel ply was used in order to slow down the resin flow towards the outlet and therefore to allow the laminate to become completely filled with resin. Moreover the connection between the product and the outlet was established using a double layer of peel ply for the same argument. During infusion the resin front moved faster at the upper part of the product compared to the side of the mould. This resulted in enclosures of dry spots at the end of the stringers. An attempt was made to resolve this, using small channels at the bottom end of the stringer foam to increase the flow rate at the bottom of the panel. The dry spots still occurred and it is therefore recommended to produce significantly longer panels than needed when using a similar method. 34

45 6.2 Construction of the test panel In the end two panels were manufactured: one with a high visible void content due to the use of plastic peel ply as described above and one panel without a visible void content which was found to be of sufficient quality. After the infusion the panels have been post cured in an oven for four hours at 40 degrees Celsius. (a) (b) (c) Figure 6.3: Production of the test panel showing: (a) plate fabric, (b) stringers, (c) vacuum infusion End blocks In order to create a gradual introduction of the compression load both panels have been provided with end blocks, made of a casting resin, of about 20mm thick around the ends of the panel, as can be seen in figure 6.4a. The loaded panel ends have been machined afterwards to make the end surfaces (top and bottom) parallel and perpendicular to the longitudinal axis of the panel. There after the panel ends have been cut in different sections (see figure 6.4b), using cuts of about 40mm high, to allow variation in rotation of the sections. This is explained further in the next paragraph. (a) (b) Figure 6.4: Test panel with resin blocks (a) for gradual introduction of compression loads, cut in different sections (b) to allow for discrete rotation. 35

46 Chapter 6 - Test panel construction and test setup 6.3 Support structure Side supports The design of the side supports for the unloaded edges, as shown in figure 6.5, is based on several examples of rounded knife edge supports, as described in [27]. Rounded knife edge supports allow for movement in the in-plane direction as well as rotation of the plate, but obstruct movement in the out of plane direction, forming a simply supported boundary condition. (a) (b) Figure 6.5: Schematic representation (a) as well as a picture (b) of the side supports. The square and L-shaped beams, see figure 6.5, provide high stiffness in the out of plane direction and are used to position the knife edge supports by means of bolts. One of the two L-shaped beams is bolted to a steel plate, which is clamped to the upper compression plate of the compression test machine. At the lower compression plate the same L-shaped beam is positioned by means of a sliding slot allowing movement in vertical direction. A small gap of about 10mm is left between the panel end blocks and the knife edge supports to allow vertical strain in the plate to occur without compressing the knife edge support structures. Loaded end supports For the loaded end supports at the upper and lower end of the test panel a similar thrust bearing arrangement was used as described in [28]. The thrust bearing consists of a bearing rail, five bearing pads and a cylinder and is mounted to the steel plate, which is clamped to the compression test machine (see figure 6.6). The bearing rail and pads are provided with a v-groove to fit the cylinder in between. This configuration allows the bearing pads to rotate around the axis of the cylinder and therefore to follow the flexural buckling deformations of the test panel. Figure 6.6: Schematic representation of the loaded end supports. 36

47 6.4 Measurement setup The bearing pads only allow for a discrete rotation of the loaded ends of the panel. This is a simplification of a real simply supported condition which would allow for a continuous rotation. The discrete rotation results in local plate deformations at the panel ends during the global buckling mode since neighbouring bearing pads have different rotation angles. In order to reduce these deformations the test panels have been provided with cuts (see figure 6.4b), dividing the ends of the panels in seven sections. At the outer two sections of the panel end, a combination of side supports and bearing pads would result in a clamped boundary condition, since the side supports do not allow for out of plane deflections and would consequently block the rotation of the outer bearing pads. Therefore bearing pads were not used for the outer two sections of the panel ends causing these sections to remain unloaded during compression. Since these sections are relatively small compared to the other sections, it was assumed that this would not cause unexpected behaviour. The test panel can be positioned with the neutral plane of its cross-sectional area aligned with the axis of the cylinder to ensure load transfer through this plane. Since the test panel rotates around the axis of this cylinder, the length of the panel that is used in the buckling calculations is represented by the distance between the axes of the upper and lower cylinder. Due to the simply supported boundary conditions, the internal moment in the panel ends is negligible. Therefore the influence of the rigid panel ends on the buckling behaviour is minimal, while for the calculation of strains the actual panel length can be used. 6.4 Measurement setup During the measurements the following measurement techniques have been used: - strain gages - linear variable differential transformers (LVDT) - a setup of two cameras for Digital Image Correlation (DIC) The applied load has been measured in the compression test machine itself and is recorded to a computer together with the other measurement data. Strain gages Eight strain gages have been used on the panel. Since the setup is symmetrical and the buckling mode is also expected to be symmetrical, the strain gages are positioned along the transversal axis of symmetry (indicated as a dotted line in figure 6.7). The strain gages are placed in four pairs on the subplates in between the stringers. The pairs are knowingly not placed on the stringers as deformation of the stringer cross section could affect the measurements. The plate on the contrary has little room for deformation other than bending or compression strain. Each strain gage pair has one strain gage positioned at the front side and one strain gage positioned at the back side of the plate, respectively strain gages 1,3,5,7 and 2,4,6,8 as shown in figure. During the tests the combination of strain due to compression and strain due to bending is measured by each strain gage. From the strain difference in each pair the direction of bending can be derived at the four positions, resulting in a rough determination of the buckling mode. 37

48 Chapter 6 - Test panel construction and test setup Figure 6.7: Test panel in the support rig, with strain gages indicated on the transversal axis of symmetry. Strain gages 1,3,5 and 7 are positioned on the front side, while strain gages 2,4,6 and 8 are positioned on the back side of the plate. Linear variable differential transformers During compression the lower compression plate of the test facility moves upwards, while the upper plate of the test facility remains in position. The vertical displacement of the lower plate is measured using an LVDT, which is positioned in between the two compression plates close to the middle of the test setup in order to avoid deviation between measured and actual displacement. The lateral deflection of the panel has been measured in the middle of the transversal axis of symmetry at the top of the middle stringers. A thin thread was glued to the point of interest on the stringer and laid over a small pulley. At the other end the thread is connected to a vertically positioned LVDT which has been provided with a small weight in order to be pulled lightly downwards in case of lateral deflection towards the pulley (see figure 6.8). Camera setup for Digital Image Correlation The flat side of the panel has been painted white and is provided with a speckle pattern using a spray can. At this side the optical method Digital Image Correlation (DIC) is used to measure the deformation of the plate. This method uses speckle images, captured by two cameras during the measurements, in order to track the gray value pattern in small areas of the images and calculates changes like strain and displacements by comparing the images of each measurement step. A stereo-system consisting of two cameras is used to be able to measure the lateral deflection, as can be seen in figure 6.9. The measurement system is connected to the load output of the test facility and records load data synchronously with the images. The surface area that can be recorded is limited by the resolution of the cameras in combination with the desired scale of the measurements. When a large surface area is recorded, relatively large dots need to be used in the speckle pattern. This results in measurements of a larger scale when compared to measurements on a small surface area with small dots in the speckle pattern. Therefore a balance between the measurement scale and the size of the recorded surface area needs to be found. For the performed tests a surface area of 400mm by 400mm was recorded at the upper half op the plate, since the 38

49 6.4 Measurement setup measurement scale using this configuration was sufficient to observe local effects and a large part of the panel could be monitored. Figure 6.8: Setup to measure the lateral deflection of the middle stringer using an LVDT. Figure 6.9: Setup for the optical method Digital Image Correlation using stereo-system consisting of two cameras. 39

50 7 Buckling experiments Three types of experiments have been carried out. The first two types are referred to as elastic buckling, as the panels were loaded until elastic instability occurred (referred to as loading trajectory), where after the panels were unloaded again (referred to as unloading trajectory). The first experiment type was carried out using simply supported boundary conditions, while the second experiment type was carried out using simply supported boundary conditions only at the loaded ends in combination with free sides. The third experiment type was carried out using simply supported boundary conditions at all sides and is referred to as post-buckling failure, since the panel was loaded even beyond the event of buckling until final failure of the panel occurred. For the first two experiment types two hat-stiffened test panels were used, one with a high visible void content (panel 1) and one with a low visible void content (panel 2) (see paragraph 6.2). Panel 1 was at first used to test the setup, however at the reached stresses and strains negative effects of the voids were not explicitly noticed and the test results were very comparable to those of panel 2. For the post-buckling failure experiment only panel 2 was used. 7.1 Elastic buckling: simply supported boundary conditions Three elastic buckling experiments have been carried out with simply supported boundary conditions: one using panel 1 and two using panel Pre-buckling behaviour Vertical displacement The panels have been compressed in load-based steps until elastic buckling occurred. The vertical displacement of the compression plate of the test machine was measured using an LVDT. This displacement has been plotted as a function of the compression load for the loading trajectory of the three experiments, as shown in figure 7.1. It can be seen that during the first few loading steps the curves show non-linear behaviour. This behaviour is caused by the tolerances in the test setup as several gaps are closed during these loading steps. The rest of the loading trajectory is characterized by linear compression behaviour. It is remarkable to see the accurate correlation between the curves of both panels, although the maximum compression load was significantly higher for panel 1 compared to panel 2. Compression load [kn] Panel 1 Panel 2 -test 1 Panel 2 -test Vertical displacement [mm] Figure 7.1: Vertical displacement of the compression plate during the loading trajectory

51 7.1 Elastic buckling: simply supported boundary conditions Out of plane displacement During the loading trajectory the out of plane displacements, at the middle of the transversal axis of symmetry of the panels, have been measured using an LVDT as well as using the camera setup (referred to as DIC in the figures). These measured displacements have been plotted for both panels as a function of the applied compression load, as shown in figures 7.2 and 7.3. It can be seen that the direction in which the panels deflected is opposite: panel 1 deflected in the direction of the plate while panel 2 deflected in the direction of the stringers. In the beginning of the loading trajectory the deflection behaviour is quite linear. At higher compression loads, though, an increase in load constitutes an ever larger increase in out of plane deflection, causing instability of the panel to become inevitable at the point of elastic instability. Compression load [kn] DIC LVDT Compression load [kn] Out of plane deflection [mm] Out of plane deflection [mm] Figure 7.2: Out of plane deflection of panel 1 Figure 7.3: Out of plane deflection of panel 2 during the loading trajectory. during the loading trajectory of test 2. DIC LVDT Young s modulus Since during the loading trajectory the panel deflections are very small relative to the dimensions of the test panel, the influence of these deflections on the average compression strain in the panel is assumed to be negligible. Consequently the measured vertical displacement can be considered as a representation of the total longitudinal strain over the length of the panel. Using this knowledge the average absolute longitudinal strain can be calculated as a function of the compression load by dividing the measured vertical displacement with the panel length. Furthermore the compression loads can be converted to stresses using the calculated cross sections of the panels. The Young s modulus in its turn can be found from the slope of the resulting stress-strain curve. The determined Young s moduli for both panels are very similar (see table 7.1). Table 7.1: Test results for simply supported boundary conditions. Load before buckling [kn] Load after buckling [kn] Critical buckling load [kn] Young s modulus [GPa] Panel Panel 2 test Panel 2 test

52 Chapter 7 - Buckling experiments Buckling behaviour Buckling mode At the moment of buckling the out of plane deflection of the panels suddenly increased to significantly higher values. Since the compression test machine is displacement-controlled the measured vertical displacement remained the same during this event. As a result the compression load decreased. Both panels buckled globally forming a double curved buckling mode as can be seen in figures 7.4 and 7.5. These figures only show part of the panel since they are created using the data from the camera setup, as described in paragraph 6.4. The unsmooth shape of the transverse half wave in the middle of panel 2 reveals the position of the stringers. Three stringers can be distinguished, one in the middle where x = 0 and two others at x = 108 and x = These areas are relatively flat and it can be concluded that the curvature is mainly established in the subplates. Panel 1 on the other hand shows a less curved transverse half-wave, indicating that the three middle stringers deflected in a similar manner. Figure 7.4: Out of plane deflection of panel 1 before (left) and after (right) elastic buckling. Figure 7.5: Out of plane deflection of panel 2 before (left) and after (right) elastic buckling. 42

53 7.1 Elastic buckling: simply supported boundary conditions Critical buckling load Elastic instability is caused by asymmetric flaws in the panel, an external force or a bending moment. Since the test panels are relatively thick and have a high longitudinal bending stiffness, these instability sources must be relatively large in order to cause the panel to buckle at the critical buckling load. The significant differences in load before and after buckling, as shown in table 7.1, confirm that the test panels remained stable far beyond the critical buckling load. After buckling occurred the test panels have been unloaded in steps in order to measure the unloading behaviour. In figure 7.6 and 7.7 the out of plane deflections of both panels are plotted as a function of the compression load for the unloading trajectory. It can be seen that for the first few unloading steps the panel shows distinctive behaviour compared to the rest of the curve. During these steps the load decreased significantly while the deflection only decreased little compared to the following steps. This could imply that part of the compression strain is released during these steps. Compression load [kn] Out of plane deflection [mm] DIC Compression load [kn] Figure 7.6: Out of plane deflection of panel 1 Figure 7.7: Out of plane deflection of panel 2 during the unloading trajectory. during the unloading trajectory of test 2. LVDT DIC Out of plane deflection [mm] 0 The rest of the curves first show a large decrease of the deflection with a decrease in compression load, transitioning smoothly to a lesser decrease of the deflection with the decrease in compression load. The critical buckling load described by theory can be found using these curves. Since the panels were unloaded from a buckling mode position, the unloading curves do not show a sharp critical buckling point. But when the asymptotes of the curves are drawn the critical buckling loads can be found at the crossings of these lines, see figures 7.6 and 7.7. Despite the large difference in compression load at the moment of instability and the opposite buckling direction the determined critical buckling loads are very similar for panel 1 and panel 2, as can be seen in table

54 Chapter 7 - Buckling experiments Discussion Compression and bending strain During the loading and unloading trajectories the strain at both sides of the plate has been measured using strain gages, as described in paragraph 6.4. The measured strain forms a combination of bending and compression strains. For both panels the measured strain has been analyzed and the results are explained below using strain gage pair 2, consisting of strain gages 3 and 4, as an example in figures 7.8 and Compression load [kn] SG 3 SG Compression load [kn] SG 3 SG Absolute strain [-] Figure 7.8: Absolute strain for panel 1 measured by strain gages 3 and 4 during the loading (black lines) and unloading (grey lines) trajectories Absolute strain [-] Figure 7.9: Absolute strain for panel 2 measured by strain gages 3 and 4 during the loading (black lines) and unloading (grey lines) trajectories. During the loading trajectory both panels showed little out of plane deflection, resulting in relatively low bending strains and relatively high compression strains. Consequently for both panels negative strains were measured during this trajectory. After buckling however the out of plane deflections were relatively large and therefore the bending strains were also significantly larger. Panel 1 buckled in the direction of the plate. As both sides of the plate are in this case positioned at the outer side of the curvature at a certain distance from the neutral plane of the total panel the resulting bending strains in both strain gages were positive. Strain gage 4 measured a larger value in strain compared to strain gage 3, since it is positioned at the back of the plate being further from the neutral plane of the panel. As the measured strain for panel 1 is positive in the first part of the unloading trajectory (see figure 7.8) it can be concluded that at that moment the bending strain was higher than the compression strain. As the panel was unloaded further the bending strains decreased and the measured strain turned back to negative values again since the values of the compression strain were higher than those of the bending strains from that point. Panel 2 buckled in the direction of the stringers. As both sides of the plate are now positioned at the inner side of the curvature at a distance from the neutral plane, the resulting bending strains in the plate were negative. Therefore the measured strains in the plate were negative during both the loading and unloading trajectories as can be seen in figure 7.9. The difference in measured strain between the two strain gages again indicates their position on the plate (back or front side). 44

55 7.1 Elastic buckling: simply supported boundary conditions Compression load at elastic instability The compression load at the point of elastic instability was found to be significantly higher for panel 1 compared to panel 2. This is presumably caused by the asymmetric bending mode of panel 1 during the loading trajectory, which can be shown using the measurement data of the strain gages as well as the data from the camera setup. It is assumed that the strain due to compression is equal for two strain gages in a pair, e.g. for strain gage 1 and 2, while the remaining difference in measured strain is caused by bending. Figures 7.10 and 7.11 show the differences in strain, and therefore the bending strain, for each strain gage pair of both tested panels. As can be seen in figure 7.10, panel 1 showed an asymmetric bending mode during the loading trajectory. This is concluded from the fact that the difference in strain for pair 1 has an opposite sign compared to the other strain gage pairs. This means that this part of the plate bent in opposite direction compared to the parts of the plate were the other strain gages are positioned. In figure 7.12 the asymmetric mode is shown using the data from the camera setup. Strain gage pair 1 is positioned where x = +/ During the unloading trajectory on the contrary panel 1 showed a (more) symmetrical buckling mode, as can be seen from the figures for the strain gage data as well as the figures for the data of the camera setup. Panel 2 showed a (more) symmetrical buckling mode during both loading and unloading trajectories, as can be seen in figure 7.11 where all pairs show the same sign in strain difference. Compression load [kn] Pair 1 Pair 2 Pair 3 Pair Strain difference [-] Figure 7.10: Strain difference in each strain gage pair for panel 1 during the loading (black lines) and unloading (grey lines) trajectories. Compression load [kn] Pair 1 Pair 2 Pair 3 Pair E-19 Strain difference [-] Figure 7.11: Strain difference in each strain gage pair for panel 2 during the loading (black lines) and unloading (grey lines) trajectories

56 Chapter 7 - Buckling experiments Figure 7.12: Out of plane deflection of panel 1 before (left) and after (right) elastic buckling. Material damage Panel 2 has been tested twice under the same conditions resulting in compression loads at the moment of instability of respectively 176kN and 130kN. From this difference it can be concluded that the material has been affected during the tests. 7.2 Elastic buckling: simply supported boundary conditions at the loaded ends in combination with free sides During these experiments simply supported boundary conditions were only used at the loaded ends, while at the sides free boundary conditions were used. As a result the panel buckled in a single curved mode, which means that one half-wave occurred only in longitudinal direction. Also for these boundary conditions three experiments were carried out: one using panel 1 and two using panel Pre-buckling behaviour Similar to the tests using simply supported conditions at all sides, these experiments revealed a linear behaviour of the panels regarding the vertical displacement during the loading trajectory (see figure 7.13). The correlation between the curves of both panels is again very exact and also the determined Young s moduli, which can be found in table 7.2, are similar. Furthermore it can be seen in figure 7.13 that the compression load for panel 1 at the moment of instability is significantly lower compared to the test using simply supported boundary conditions at all four sides. 46

57 7.2 Elastic buckling: S.S. B.C. at the loaded ends in combination with free sides Compression load [kn] Panel 1 Panel 2 -test 1 Panel 2 -test 2 Figure 7.13: Vertical displacement of the compression plate of the compression test machine during the loading trajectory. The deflection curves for the loading trajectory (see Appendix G) of the tests using simply supported boundary conditions at the loaded ends in combination with free sides, show similar behaviour as well compared to the simply supported tests: the panels buckled when the increase in out of plane displacement with an increase in compression load became very large. Table 7.2: Test results for simply supported boundary conditions at the loaded ends in combination with free sides. Load before buckling [kn] Load after buckling [kn] Critical buckling load [kn] Young s modulus [GPa] Panel Panel 2 test Panel 2 test Vertical displacement [mm]

58 Chapter 7 - Buckling experiments Buckling behaviour The buckling mode of panel 2 right after elastic instability is shown in figure The curve shows a smooth symmetrical global buckling mode with one half wave only in longitudinal direction. Figure 7.14: Out of plane deflection of panel 2 after elastic buckling. The critical buckling loads, see table 7.2, have again been determined using the asymptotes of the curves for the out of plane displacement as a function of the compression load, see figures 7.15 and The critical buckling loads were found to be relatively high compared to the critical loads determined for the tests using simply supported boundary conditions at all sides. The finite strip method predicted a difference in critical buckling loads of 38kN between the tests using simply supported boundary conditions at the loaded ends in combination with free sides and the tests with simply supported boundary conditions at all sides, while the found critical buckling loads (see table 7.1 and 7.2) only differ 10-15kN Compression load [kn] DIC LVDT Out of plane deflection [mm] Out of plane deflection [mm] Figure 7.15: Out of plane deflection of panel 1 Figure 7.16: Out of plane deflection of panel 2 during the unloading trajectory. during the unloading trajectory. Compression load [kn] DIC LVDT

59 7.2 Elastic buckling: S.S. B.C. at the loaded ends in combination with free sides Discussion Local buckling During the loading trajectory local buckling of the sides of the plate was observed as can be seen in figure This occurred due to the free boundary conditions at the unloaded edges of the panel. After elastic instability occurred a global buckling mode was shown without the observed local half waves. Figure 7.17: Local buckling of the sides of the plate during the loading trajectory. Instability The compression load at which panel 1 became unstable during the test was found to be much lower than for the test with simply supported boundary conditions. It is concluded that the asymmetrical buckling mode of panel 1, as described in paragraph 7.1.3, could only remain stable due to the side supports. 49

60 Chapter 7 - Buckling experiments 7.3 Post-buckling failure In order to research failure modes after buckling, panel 2 was compressed, using simply supported conditions at all four sides, until final failure of the panel occurred Pre-buckling behaviour The pre-buckling behaviour during this test was very similar to the behaviour observed at the first tests with simply supported boundary conditions. A young s modulus of 25.9GPa has been determined and the maximum pre-buckling load was 143 kn Post-buckling behaviour After the event of elastic buckling the vertical displacement of the compression plate of the compression test machine has been increased until final failure of the panel occurred. Several failure events were observed in between elastic buckling and final failure. The different events have been illustrated in the curve for the vertical displacement as a function of the compression load in figure The vertical displacement of the compression plate has been displayed in positive values instead of negative values for illustration purposes. 150 Compression load [kn] A Vertical displacement [mm] B Measurement points Failure events Failure events C D E F Figure 7.18: Illustration of the events that occurred during the test using the measured vertical displacement. Event A in figure 7.18 represents elastic instability where the panel bent in a double curved mode similar to previous tests. The load has decreased while the out of plane deflection of the middle stringer has increased to about 12mm. Thereafter the panel has been compressed further, resulting in an out of plane displacement of about 44mm in the middle of the panel at the moment of event B. At this event a small crack arose in an upper corner of the panel, starting at one of the two most outside cuts in the panel end, progressing a few centimetres towards the middle of the panel under a 45 50

61 7.3 Post-buckling failure angle (see figure 7.19). Since the outer blocks of the panel ends are not loaded the compression load has to be distributed to the sides of the plate. Therefore peak loads arise at the ends of the cuts that separate the outer blocks of its neighbouring block. These peak loads caused the laminate the fail at this point. Figure 7.19: Event B, a small crack arose at a cut in the panel end due to peak loads. At event C the plate buckled locally at one side of the panel. One local half wave arose in opposite direction compared to the global buckling mode as can be seen in figure The stringer at that position however remained in place resulting in local separation of the stringer and plate contact surface. Since the global buckling mode is pointed towards the side of the stringers the plate experiences extra compression as a result of bending due to its position with respect to the neutral plane of the panel. The contact surface with the stringer prevents the plate from buckling locally up to the point that the bond between the two breaks. Figure 7.20: Event C, a local half wave in the plate as a result of high compression load due to bending. 51

62 Chapter 7 - Buckling experiments At event D a small crack arose in the local half wave which grew to a large crack over the width of the plate during event E as can be seen in figure The out of plane deflection of the local half wave increased significantly during event E resulting in further separation of the stringer and additionally separating a small part of the neighbouring stringer. During event F the crack grew over the complete width of the panel (see figure 7.22a). The outer bottom flange of the other outer stringer also separated from the plate during this event, as can be seen in figure 7.22b. (a) (b) Figure 7.21: Event D (a), a small crack in the local half wave. Event E (b), the crack expands over the width of the panel. (a) (b) 7.22: Event F, a crack over the full width of the panel (a) and separation of the outer stringers (b) 52