A P-Finite element method of a three dimensional non-uniform asymmetric beam structure of arbitrary polynomial functions

Abstract

Tapered beams are commonly used in civil, aerospace or mechanical engineering structures as they can reduce its structural weight without sacrificing the strength and flexibility. Tapered beams are also used to satisfy aesthetic or architectural requirement. Most of numerical methods to analyze tapered beam structures are using a Galerkin’s finite element approach where the beam is divided into a number of elements to obtain accurate result. The beam stiffness matrix is usually obtained through integration of each element by assuming a shape function for the beam transversal deformations. Since the number of elements are big, therefore such an approach may affect computational time. In the present research, a different approach is conducted where an analytical formulation of a finite element stiffness matrix for a tapered, asymmetric beam element is developed by using a flexibility approach. The beam stiffness matrix is first divided into bending, axial and torsional matrices. For the bending stiffness matrix, to simplify the formulation and therefore to accelerate the numerical calculation, it is necessary to divide further the bending stiffness matrix into four sub-matrices. Each of the sub-matrices is a 4-by-4 matrix representing the bending stiffness matrix in three dimensional coordinate system. The key to the present approach lays on the formulation of the first sub-matrix, whereas the other three sub-matrices can be obtained from the first sub matrix by using direct, simple matrix operations. The first sub-matrix is constructed based on the flexibility approach where a two-steps analytical integration of second order, partial differential equations is performed. The partial differential equations are derived based on the Euler-Bernoulli governing equations for the three-dimensional bending deformations, where the transversal deformations of the beam are coupled due to the properties of the asymmetric cross section. After rearranging the transversal deformations in matrix forms, the resulting explicit forms of the differential equations contain rational functions with multi-polynomial functions on both numerator and denominator of the rational function. It is found that, in order to ensure the robustness of the integrations, the denominator functions should be expressed as the multiplication factor of their roots. By properly considering the boundary conditions of the beam under various load conditions, the results of the analytical integration are a 4-by-4 flexibility matrix. The final form of the first sub-matrix is the stiffness matrix which can be obtained by matrix inversion of the flexibility matrix. For the axial and torsional stiffness matrices, a similar approach is conducted but it is much simpler since it involves only first order differential equations. It is found that the present stiffness matrix contains logarithmic terms which are not occurred if one use direct Galerkin’s finite element approach. The present finite element method can be considered as an analytical stiffness matrix formulation since no assumed shape functions used for the whole process of the formulation. Therefore, if the tapered functions of the beam geometry is given, only one element is sufficient to accurately simulate the beam deformation. To validate the present finite element method, a number of structural tapered beam having symmetric and asymmetric cross section are used and the results are compared with available analytical result or other software’s such as Nastran. The results show that the present method gives the accuracy of more than 7 significant digits compared with the analytical solution. In all cases, the present method by using one element gives the result similar to Nastran convergent result where, in order to achieve the convergence, a number of elements in Nastran are needed. It is expected that the finding of the present method can contribute further the development of finite element numerical simulation.