18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS DEVELOPING ANISOTROPIC BEAM ELEMENT FOR DESIGN OF COMPOSITE WIND TURBINE BLADES T. Kim 1 *, K. Branner 1 & A. M. Hansen 1 1 Risø National Laboratory for Sustainable Energy, Technical University of Denmark, Roskilde, Denmark * Corresponding author (tkim@risoe.dtu.dk) Keywords : Anisotropic beam element, HAWC2, Wind turbine 1 Introduction For wind turbines blades, composite materials are widely used because they can reduce the total weight while retaining the structural properties and because they have good tailoring and fatigue life characteristics. The tailoring capability of the composite blade could be used to passively control Fig.1: A sketch of the coordinate system the wind turbine response and results in a decrease 2.1 New beam element of fatigue loads and the risk of flutter. However, the aeroelastic codes in the wind energy fields such as In order to compute the element stiffness and HAWC2 still use the classical beam models. It mass matrix, the elastic energy and the kinetic cannot be used to investigate the coupling effects of energy are considered. anisotropic materials. It is shown in [1] that a typical 2.1.1 Stiffness matrix wind turbine blade has very small couplings, but that The elastic energy of the beam is defined as these can be introduced easily by adding angled follows unidirectional layers. In this paper a new beam element which is able L T 2 U S dz (1) to consider anisotropic characteristics of a beam is 0 developed and implemented into the structural part where S is the cross-sectional stiffness matrix of the HAWC2. defined by diagonal matrix into current HAWC2. 2 Methods For the classical Timoshenko beam S is addressed by the diagonal matrix as follows The classical Timoshenko beam element is used in HAWC2 by considering Finite Element GJ S diag k GA { , k GA , EA , EI , EI , } (2) Analysis (FEA). In order to compute the shape x y x y functions, static equilibrium equations are solved where k x and k y are shear factors related to forces in with the geometric boundary conditions. The x and y direction, respectively. principle of virtual displacements is used to derive In Eq. (1), ε the generalized strains of the the element stiffness with obtained shape functions. Timoshenko beam is expressed as More detailed equations are represented in [2]. T However, the beam model in the current HAWC2 { , , , , , } x y z x y z (3) can generally not be extended to an anisotropic { u , u , u , , , } beam model because the shape functions do not x y y x z x y z necessarily capture the coupled motions. Therefore, In FEA the displacement and rotation can be new beam element with new shape functions should expressed by an interpolating polynomial in terms of be introduced to observe coupled behaviors. Figure generalized degrees of freedom as follows 1 shows a sketch of the coordinate system in HAWC2.
T 1 1 q x ( ) { u , u , u , , , } N x ( ) d N N N x y z x y z (4) 1 1 1 2 6 1 N 6 6 N i i 0 I 2 2 where N is the polynomial matrix in which, 0 I 2 n 1 where n =1 for linear 12 (6 N 12) N x x x 12 12 i 1 2 0 I polynomial, α is the generalized degrees of freedom, (11) 6 N 1 i 12 1 (6 N 12) 1 i (6 N 12) 12 (6 N 12) (6 N 12) i i i and N i is the highest power in the polynomial + 1. A A 1 2 From the Eqs. (3) and (4) the generalized strain A A can be expanded in terms of strain-displacement 1 1 2 2 6 N 12 12 1 6 N (6 N 12) (6 N 12) 1 matrix and generalized degrees of freedom as i i i i 1 1 A N d A N N A follows 1 1 1 1 2 2 2 2 To compute α vector, the total energy B x ( ) (5) 6 N 1 dU 6 6 N i i minimization approach in terms of 2 , , is 0 d where B is the strain-displacement matrix which 2 considered. From Eqs. (7) and (11), the total elastic includes polynomial matrix and its derivative terms energy of the beam is obtained as follows as follows 1 B x ( ) B N x ( ) B N x ( ) T T T T T T T T T U d N A N N A A D 0 1 (6) 1 1 2 2 1 1 2 2 2 6 6 N T (12) i 1 1 A N d A N N A By substituting Eq. (5) into Eq. (1), the elastic 1 1 1 1 2 2 2 2 energy of the beam can be illustrated as follows Resulting from the total energy minimization, L T T 2 U B SB dz 2 vector is obtained as follows (7) 0 D dU 0 In order to find α in Eq. (7) the boundary d 2 conditions at the beam ends are satisfied and the T T T 1 T 1 N N A DA N A DA N d beam sections are in equilibrium which can be 2 1 1 1 1 2 1 1 obtained when the total elastic energy is minimized. P T T T 1 T 1 By applying boundary conditions the nodal degrees (13) N N A DA N N A DA N N 2 1 1 1 1 2 2 1 1 2 of freedom are obtained as follows 2 T T T T N N A DA A DA 2 1 1 2 2 2 Q 1 d N N N (8) d 1 2 1 Q P d 12 1 6 N 1 2 2 12 6 N i i 12 1 6 N 12 12 6 N 12 6 N 12 i i i Here N 1 is a 12 by 12 matrix which is assumed By substituting Eq. (13) into Eq. (11), α vector to be invertible. Therefore, N 1 and N 2 become as a function of the nodal degrees of freedom is represented as follows I 0 0 0 0 L I N , N (9) 1 2 2 3 N I LI L I L I i 1 1 1 1 A N d A N N Q Pd A Q Pd 12 12 12 6 N 12 1 1 1 1 2 2 i 1 1 1 1 A N A N N Q P A Q P d where L is the length of the beam element. 1 1 1 1 2 2 (14) From Eq. (8) α 1 can be rewritten as N d 6 N 12 1 N d N (10) i 1 1 2 2 Therefore α can be expressed as follows
DEVELOPING ANISOTROPIC BEAM ELEMENT FOR DESIGN OF COMPOSITE WIND TURINE BLADES Finally, the elastic energy of the beam is Table 2. More detailed information about the obtained in terms of nodal degrees of freedom by material properties and geometries are addressed in substituting Eq. (14) into Eq. (7) as follows [3, 4]. 1 L T T T U d N B SB dz N d 2 0 (15) 1 T d Kd 2 L T T where K matrix, K N B SB dz N , is the (a) Case1: [0°] T layup with arbitrary isotropic 0 material element stiffness matrix. 2.1.2 Mass matrix The method to compute the element mass matrix is similar to the definition of the stiffness matrix. The element mass matrix is obtained from the kinetic energy as follows 1 T T r rdV 2 V (16) 1 L (b) Case2: [30°] T layup with Graphite/Epoxy T r Erdz 2 0 where , , r V and E are the mass density, velocity of , the body, volume of body and the cross-sectional mass matrix respectively. By applying the same shape function as the stiffness matrix, Eq. (16) can be extended as follows 1 L T T T T d N N x ( ) EN x dz N d ( ) 2 0 (17) 1 T d Md 2 L T T where M matrix, M N N EN dz N , is the 0 (c) Case3: [45°/0°] 3s layups with Graphite/Epoxy element mass matrix. Fig.2: A sketch of considered cases 3 Results After implementing this new beam element Table1: Structural properties of Case 1 into HAWC2, three different cases are investigated Material Arbitrary material in order to validate the new beam model. The effect E 11 , E 22 , E 33 100 psi of using anisotropic material is studied as well. G 12 , G 13 , G 23 41.6667 psi Three different cases are considered for this study. ν 12 , ν 13 , ν 23 0.2 Figure 2 (a), (b), and (c) show a sketch of the ρ 1 lb-sec 2 /in 4 considered cases. Table 1 shows the detailed Width 0.1 in structural properties and cross-sectional stiffness Height 0.1 in matrix for the first example. For Case 2 and Case 3, Length 7.5 in only sectional stiffness information is displayed in 3
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