Keywords: nonlinear random response, thermal loading, high-cycle-fatigue, reduced order methods.

Direct numerical simulation of nonlinear random response in physical degrees-of-freedom (DoFs) is computationally intensive for even the simplest structures. Its use for design of high-cycle-fatigue tolerant aerospace vehicle structures is considered impractical. Accordingly, much effort has been spent in recent years to develop accurate reduced order analyses, which could be suitable for use in design environments.

Finite-element-based nonlinear modal numerical simulation methods have been the focus of much of the research effort as they offer the ability to investigate practical structures. Such methods may be viewed as being in one of two categories; those in which the nonlinear modal stiffness is directly evaluated from the nonlinear finite element stiffness matrix (so-called direct methods), and those in which the nonlinear modal stiffness is indirectly evaluated. Direct methods are typically implemented in special purpose finite element codes in which the nonlinear stiffness is known, see for example references [1,2]. Indirect stiffness evaluation methods are typically implemented for use with commercial finite element codes in which the nonlinear stiffness is unavailable, see for example reference [3]. For both direct and indirect stiffness evaluation approaches, the crux of the problem lies in the selection of the proper basis, through which the nonlinear modal stiffness may be determined. Through comparison with numerical simulation in physical DoFs, the authors recently demonstrated [4] the ability of a reduced order method to accurately predict geometrically nonlinear random displacement and stress response, provided that a suitable modal basis is utilized. The focus of this study is to extend that work to determine the effect of modal basis selection on high-cycle-fatigue life estimation.

The problems of interest are those that exhibit nonlinear bending-membrane coupling. This coupling dramatically changes the stress response characteristics with increasing load. For low-level excitation, the total surface stress response has a zero mean, a Gaussian probability density function (PDF) distribution, and is typically dominated by the bending component. As the excitation level increases, the contribution of membrane stress to the total becomes more significant, resulting in a non-zero mean, which skews the total stress PDF from Gaussian. Additionally for the spring hardening system, the stress power spectral density (PSD) exhibits peak broadening and shifting to higher frequencies, and indicates significant peaks due solely to the membrane component. From a mechanics point of view, the reduced order analysis must be capable of accurately predicting all of these behaviors. However, from the fatigue point of view, the stress range PDF and stress ratio govern the fatigue life estimate for a given material system. Thus, for fatigue analysis, the reduced order method should be sufficient if it can accurately predict these quantities.

In an effort to determine what selection of basis functions most accurately predicts the stress range PDF and stress ratio, several bases are considered including (i) bending modes only; (ii) coupled bending and companion modes; (iii) three variations of uncoupled bending and companion [5] modes; and (iv) bending and membrane modes. Results are compared with those obtained from numerical simulation in physical DoFs. A planar aluminum beam in a pre-buckled regime is considered to keep the computational cost of the numerical simulation in physical DoFs reasonable. Two locations along the span of the beam are investigated - one at the clamped end where the bending stress component is dominant and one at a location close to the quarter span, where the effect of membrane stress is more significant.

1

Mei, C., Dhainaut, J.M., Duan, B., Spottswood, S.M., Wolfe, H.F., "Nonlinear random response of composite panels in an elevated thermal environment," Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, AFRL-VA-WP-TR-2000-3049, October 2000.

2

Przekop, A., "Nonlinear response and fatigue estimation of aerospace curved surface panels to combined acoustic and thermal loads," Ph.D. Dissertation, Old Dominion University, August 2003.

3

Muravyov, A.A., Rizzi, S.A., "Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures," Computers and Structures, Vol. 81, No. 15, pp. 1513-1523, 2003. doi:10.1016/S0045-7949(03)00145-7

4

Rizzi, S.A., Przekop, A., "The effect of basis selection on thermal-acoustic random response prediction using nonlinear modal simulation," Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA-2004-1554, Palm Springs, CA, 2004.

5

Mignolet, M.P., Radu, A.G., Gao, X., "Validation of reduced order modeling for the prediction of the response and fatigue life of panels subjected to thermo-acoustic effects," Structural Dynamics: Recent Advances, Proceedings of the 8th International Conference, Southampton, UK, 2003.

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