Convex cross-section beam with shell finite elements

the steady-state solution with the elongation of the blade resulting from the centrifugal forces will be examined. The formulation for the analytical approach is provided by Eq. (9.8). The required linear and constant portions of the line force n x) in the lengthwise direction depicting the centrifugal effects are 

The contained constant m represents mass per length of the beam. For the two different cross-sections under consideration, it takes the following form. 

Rectangular cross-section Convex cross-section  

Since the essential parts of the right-hand side of the differential equation system, given by Eqs. (9.27), have demonstrated their operability, the homogeneous solution will be examined in detail to complete the inspection of the left-hand side. As there is no anal dic approach available to capture the dynamic behavior, the subsequent comparison comprises the formulations with the developed beam finite elements and with the commercial shell finite elements. The resulting natural frequencies w for all modes up to the third torsional mode are given in Table 10.9 for the non-rotating system as well as in Table 10.10 for the rotating system. 

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Fig. 10.11. Discretization of the convex cross-section beam with shell finite elements.

As outlined in Section 4.2.5, the implementation of anisotropic thermal effects in commercial finite element codes may be utilized to simulate the implications of the piezoelectric effect. To capture the behavior of thin-walled beams with cross-sections as defined above, spatial shell elements may be employed. With this methodology, however, it is not possible to examine problems with dynamic actuation. The beams with rectangular and convex cross-sections have been discretized with 2200, respectively 2300, SHELL99 elements of ANSYS as exemplarily shown for the latter case in Figure 10.11. 

Validation of the beam finite elements by comparison of solution components to those attainable with commercial shell finite elements for example configurations with a rectangular single-cell and a convex double-cell cross-section approximating the properties of an actual helicopter rotor blade. 

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