212 M.E. Asl et al. .GAx/com V00 C 0x D ARV (21.1a) .EIx/com 00x .GAx/com V0 C x D Ix R x (21.1b) where x denotes the rotation of the cross section with respect toxaxis shown in Fig. 21.2, V the displacement inydirection, q the distributed load, the prime (0 ) is used to indicate differentiation with respect to z and (.) is time differentiation. Density and area of cross section are expressed by andArespectively andIx is moment of inertia with respect tox-axes. (GAx)comand (EIx)com are shear and flexural rigidity of thin walled composite with respect tox, respectively which could be expressed as: .EIx/com D A˛ 11 y .2/ ˛ 2B˛ 11 y˛ CD˛ 11 b˛ C b.3/ 3 12 A3 11 (21.1c) .GAx/com DA˛ 55b˛ CA3 66b3 (21.1d) where A11 , A66 , A55 , B11 andD11 are elements of extensional, coupling and bending stiffness matrices for a composite layup [27]. The superscript in the parenthesis ( ) denotes the power of the exponent, and the repeated index denotes summation. Index˛varies from 1 to 3 where the indices 1 and 2 represent the top and bottom flanges, and 3 is for the web, respectively as shown in Fig. 21.2 and b˛ denotes width of the flanges and web. The closed-form solution for flexural vibration frequencies in the y-direction may be directly calculated for the free-free boundary condition as [28]: !yn Ds A .EIx/com L4 .nC0:5/4 4 C A .GAx/com L2 .nC1/2 2 1 (21.2) Natural frequencies for vibration of a shear deformable composite I-beam in free-free boundary conditions are described by Eq. (21.2). To derive the scaling laws, it is assumed that all the variables of the governing equations for the prototype (xp) can be connected to their corresponding variables in a scaled model (xm) by a one to one mapping. Then, the scale factor for each variable can be defined as x Dxp/xm which is ratio of each variable of the prototype to that of the scaled model. Rewriting Eq. (21.2) for the model and prototype and applying similarity transformation, the scaling laws can be extracted as the following based on the standard similitude procedure [12]: l 2 D EI GA (21.3) ! Ds n 4 EI A l 4 (21.4a) ! Ds n 2 GA A l 2 (21.4b) Eq. (21.3) which is referred to as the design scaling law is a prerequisite for deriving constitutive response scaling laws Eqs. (21.4a) and (21.4b). Design scaling law Eq. (21.3) denotes that for having complete similarity between two shear deformable beams, ratio of the flexural to shear stiffness must be equal to the square of the length for the two scales. This condition is shown to be satisfied if the prototype and model have a unidirectional layup in their flanges and also the dimensions of the model are proportional to those of the prototype [25]. Having Eq. (21.3) satisfied, the ratio of natural frequencies between two scales can be obtained using constitutive scaling laws Eqs. (21.4a) and (21.4b). 21.3 Experimental Results Based on the similitude analysis in the previous section, three I-beams having three different scales (i.e. small, medium and large) were manufactured from the prototype. The prototype geometry and lay-up scheme considered in this study emulate the spar-cap flanges of the Sandia 100 m wind turbine blade [24] near its max chord. The prototype is assumed
RkJQdWJsaXNoZXIy MTMzNzEzMQ==