An analytical model is proposed which consists of two Euler–Bernoulli beams joined by a torsional spring with linear and cubic stiffness. The method of harmonic balance is used to find an approximate solution supported and clamped end conditions. Specifically, a one term harmonic balance solution is developed for two beams with unsymmetric properties and symmetric distributed loading. Solutions are given for three cases: no non-linearity, and two cases with different levels of non-linearity. For all cases, frequency–amplitude relations are given, and the effects of the cubic non-linearity are observed and discussed.

A new symmetric beam bending vibration theory which explicitly includes bending warping is presented. The governing differential equations are of sixth order and the boundary conditions allow enforcement of total fixity of the end of a beam. The motivation for the study was the desire to determine if consideration of bending warping restraint in bending theory is as important as is torsion warping restraint in torsion theory. Natural frequency predictions are given and compared with other theories for constant cross-section, cantilevered “beams” of two types: short, deep rectangular beams; and, short, thin, shallow, circular cylindrical shells. The accuracy of the new theory is comparable with Murty's, Levinson's, Bickford's and Rehfield's theories for beams with rectangular cross-sections. It is also comparable with Cowper's theory for general symmetric cross-sections.

The ‘reduced bending stiffness’ (RBS) method has been used on occasions in the past as a means of simplifying the analysis of the flexural behavior of unsymmetrically laminated composite plates. However, the validity of the method has never been established. This paper makes direct comparisons between relatively simple, exact solutions for the static deflections, buckling loads and vibration frequencies of simply-supported plates and those arising from the RBS method. Extensive calculations are made for wide ranges of the physical parameters involved (aspect ratio, moduli ratio, lamination orientation angle, numbers of plies). The RBS method is found to yield sufficient accuracy for cross-ply plates, but errors up to 29% are obtained for angle-ply plates constructed with materials currently under development.

A great deal of published literature exists which analyzes the free vibrations of turbomachinery blades by means of one-dimensional beam theories. Recently, a more accurate, two-dimensional analysis method has been developed based upon shallow shell theory. The present paper summarizes the two types of theories and makes quantitative comparisons of frequencies obtained by them. Numerical results are presented for cambered and/or twisted blades of uniform thickness. Significant differences between the theories are found to occur, especially for low aspect ratio blades. The causes of these differences are discussed.