Behavior of spinning pretwisted composite plates using a nonlinear finite element approach

The behavior of spinning pretwisted plates is significantly different from that of spinning flat plates. The coupling between bending, twisting, and extension behavior, introduced because of the geometry of these plates, has a significant effect on their static and dynamic behavior. This, in conjunction with coupling due to material asymmetry, can result in fairly nonintuitive, though physically understandable, behavior. Since existing plate/shell finite elements have numerical deficiencies when modeling twisted plates, a six-node triangular element is developed for the analysis of spinning laminated composite structures. The element is based on a first-order shear deformable laminated plate theory. Complete cubic polynomials are used to represent the transverse deflections and complete quadratic polynomials are used to model the rotations and the in-plane displacements. Linear transformation relations are developed to move four nodes associated with transverse deflections to the element interior, and these nodes are then eliminated by using static and dynamic condensation to obtain a six-node element. The element is shown to be free of shear locking and superior to existing commercially available triangular elements. Numerical results are presented for the analysis of spinning laminated composite plates. NTEREST in the vibrational behavior of turbomachinery blades has led to a number of studies in the analysis of pretwisted blades. However, most of these studies have modeled the blades as cantilever pretwisted beams. This modeling is inaccurate for low-aspect-ratio blades that need to be modeled as pretwisted plates. Also, being essentially one-dimensional models, the beam models are unable to incorporate higher modes like those involving chordwise or edgewise bending. As these modes couple with bending modes for pretwisted blades, the beam model can lead to inaccurate results. A typical pretwisted plate is shown in Fig. 1. It has been shown that pretwist causes an increase in torsional stiffness and decrease in bending stiffness. This was explained for beams by Zickel,1 who pointed out that twist causes an inclination of lines parallel to the central line, reducing their radial distance from the initial twist axis. This causes smaller strain for a given curvature and also causes the stress in the direction of the lines to be concentrated more towards the beam axis. This reduces the resisting moment and, consequently, the bending stiffness. In the same way, resistance to torsion is increased with increasing pretwist. MacBain2 modeled spinning twisted isotropic cantilever plates using NASTRAN showing that pretwist resulted in a large increase in torsional frequencies, whereas it had little effect on bending frequencies and the bending frequencies were more strongly affected by the angular velocity than the torsional frequencies. Sreenivasamurthy and Ramamurti3'4 studied the vibration of rotating pretwisted low-aspect-ratio tapered cantilever plates. They observed an increase in frequency with rotational speeds for all vibration modes except edgewise bending. The decrease in edgewise bending frequency with speed is to be expected since the centrifugal stiffness term causes a decrease in the in-plane stiffness of the plate with increase in rotation speed. Lapid et al.5 presented results of experiments conducted at NASA Lewis for strains, deflection, and vibration behavior in nonspinning and spinning pretwisted laminated composite plates. For nonspinning pretwisted plates it was observed that pretwist had little effect