Computational methods in structural engineering

Abstract

The present research is focused on computational methods in structural engineering. The main work includes three parts: 1) the development of cubic B-spline finite elements and the applications in vibration analysis of beams and coupled vibration analysis of axially loaded beam-columns; 2) the development of analytical trapezoidal Fourier p-elements and the applications in vibration analysis of beams, membranes and plates; and 3) the development of an analytical method based on symplectic series expansion and the applications in three-dimensional static problems of transversely isotropic piezoelectric media. The cubic B-spline functions are used to construct the displacement field. The finite element matrices are formed. The assembly of elements and the introduction of boundary conditions follow the standard finite element procedure. The results under various boundary conditions are compared with those obtained by the exact method and the finite difference method. They show that the results obtained by the developed cubic B-spline finite element method are in excellent agreement with the analytical results and much more accurate than the results obtained by the finite difference method, especially for higher order modes. The developed cubic B-spline finite element method is also extended to investigate the coupled vibration problem of axially loaded beam-columns under various boundary conditions. Instead of equal share assumption, it is proposed that the end torque should be shared in x and y axes as initial stresses in the rigidity ratio. Clamped-free, clamped-pinned and clamped-clamped boundary conditions are taken into consideration. Other boundary conditions can be readily investigated by the developed codes. Extensive interactive diagrams are constructed for analysis purpose. The numerical results are in excellent agreement with the data available in literature, indicating that the developed method is efficient and accurate. Analytical trapezoidal Fourier p-elements, using trigonometric functions as shape functions instead of polynomials to avoid ill-conditioning problems, are developed for the longitudinal vibration analysis of beam element, the transverse vibration analysis of membranes, and the in-plane and out-of-plane vibration analysis of elastic and viscoelastic plates. The element matrices are analytically integrated in closed form. With the additional DOFs in Fourier series, the accuracy of computed natural frequencies is greatly increased in a stable manner. Rectangular elements are good for the vibration analysis with regular shapes such as square, L- and H- shapes, while for irregular polygonal shapes, triangular elements are useful. But the existing triangular Fourier p-element cannot be integrated analytically and error from the numerical integration will be introduced. A triangle can be fortunately divided into three trapezoids by drawing three lines parallel to the edges from any point inside the triangle. Thus, the range of application of the developed element is much wider than the rectangular Fourier p-element. Any plane problem with polygonal shape can be analysed by a combination of rectangular and trapezoidal elements. Numerical examples show that the present element possesses a very fast convergence rate, and produces more accurate natural frequencies than the conventional FEM with the same number of DOF. The results are in excellent agreement with the analytical results and benchmark numerical results available in literature. In practical implementation of the present method, the symbolic operation involved in analytical integration is very time-consuming with the increasing of the number of trigonometric terms. Fortunately, the present method with only a few trigonometric terms can produce adequately accurate results. For out-of-plane vibration problems of thin plates, the present method fails to predict the natural frequencies because of the loss of accuracy. But it behaves well for thick plates. In the framework of Hamiltonian system, an analytical method based on symplectic series expansion is developed to investigate three-dimensional static problems of piezoelectric media with general boundary conditions. By introducing the new concept of sub-symplecticity, the method of separable variables is generalized to totally separate the three special coordinates so that the tangling problem of the unknowns and their high-order partial differentiations with respect to the three spatial coordinates in the governing equations is eliminated. The governing equations with separable variables are first derived in Hamiltonian form and symplectic eigenvectors are directly obtained through analytical method. All solutions of the problem are reduced to determine eigenvalues and eigenvectors. The homogenous solutions consist of the solutions of derogatory zero eigenvalues and those of well-behaved non-zero eigenvalues. The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with averaged global behaviors such as rigid-body translation, rigid-body rotation or bending. The non-zero-eigenvalue solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle. The significant zero- and non-zeroeigenvalue solutions for the present problem are presented. Linear combinations of zero- and non-zero-eigenvalue solutions completely cover all kinds of solutions with any boundary conditions along the edges. The method shows an analytical and rational process which is different from the traditional semi-inverse methods. The analytical solutions can serve as bench mark examples. To verify advantages of the present method, some numerical examples are presented. Diagrams of distributions of generalized displacements and stresses at the end of the circular cylinder for axisymmetric and non-axisymmetric problems are presented for the first time.