Keywords: complex structures, Rayleigh-Ritz method, penalty function method, inertial penalty parameters, negative penalty parameters, vibration.

The very well known Rayleigh-Ritz method is often used to solve eigenproblems of simple continuous systems, usually built by only one element. This is the main disadvantage of the Rayleigh-Ritz method in comparison with the more popular finite element method. This work addresses this disadvantage defining beam, plate and shell elements in free conditions using the Rayleigh-Ritz method together with the penalty function method to define all constraints of the system as well as to join beam, plate and shell elements as needed to build a complex structure. The main contributions of this work is that in all cases the same non-dimensional set of admissible functions are used in the Rayleigh-Ritz method to model the deflection of beam, plate and shell elements and that four different types of penalty parameters are used to define boundary conditions and to join structural elements. This set of functions consists of a combination of a second order polynomial and a Fourier cosine series. These functions may be the simplest set of admissible functions that can be used to model beam, plate and shell elements in a completely free condition. The main characteristics of this set of functions are that it satisfies the boundary conditions only as a sum, avoids orthogonalization and allows the use of a large number of terms without causing round-off errors or ill-conditioning. This is rarely found in other publications. Thus, this methodology simplifies the stiffness, mass and penalty matrices of the complex structure, because of the use of the same simple set of non-dimensional functions regarding the type of element and boundary conditions and/or connection to other elements. The penalty function method was chosen over the Lagrangian multiplier method, because it does not change the size of the matrices nor adds zeros in the main diagonal of the stiffness matrix of the structure. In addition to the traditional penalty parameters representing springs (positive penalty parameters of stiffness type), the use of inertial penalty parameters representing masses and moments of inertia will be investigated as well as the use of negative penalty parameters. Recent publications have proved that the use of penalty parameters of the same type and opposite sign can be used to predict the error of the solution due to constraint violation. This is due to the monotonic convergence from above or from below with respect to the absolute penalty parameter value (depending on the type and sign of the penalty parameters) in the absence of round-off errors and ill-conditioning. Results of vibration problems of structures built with plate and shell elements obtained using the methodology presented in this work are compared with results obtained using the finite element program ABAQUS.

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