Dual numbers and dual vectors are widely used in spatial kinematics [3,5-15,18]. Plücker line coordinates of a straight line can be represented by a dual unit vector located at the dual unit sphere (DUS). By this way, the trajectory of the screw axis of a rigid body in 3 R (the real three space) corresponds to a dual curve on the DUS. This correspondence is done through Study Mapping [8,9]. Conversely a dual curve on DUS obtained from the rotations of the DUS represents a rigid body motion in 3 R [8]. The dual Euler parameters are used in defining the screw transformation in 3 R [8], but originally in this paper these parameters are constructed from the Rodrigues and the dual Rodrigues parameters [15].
The finite-difference method is a numerical technique for obtaining approximate solutions to differential equations. The main objective of the present study is to give a new aspect to the finite-difference method by using a variational derivative. By applying this formulation, accurate values of the buckling loads of beams and frames with various end supports are obtained. The performance of this formulation is verified by comparison with numerical examples in the literature Keywords: variational derivative, finite difference, buckling load, stability, beam, frame 1. Introduction. Structures are designed so that their maximum stresses and deflections remain within tolerable limits. Common experience indicates that it is necessary to analyze structures by adding stability considerations. The stability of structures can be analyzed and the buckling loads predicted by formulating the governing differential equation and obtaining the exact solution. However, finding exact solutions is often difficult. In such instances, approximate methods, such as the finite-element, finite-difference, Ritz, and energy methods, should be employed [2,9,17]. Many studies are based on these methods [3,4,7,8,10,13]. The finite-element method may appear to be a better choice than the finite-difference method owing to the easier definition of boundary conditions. This is because in the finite-difference method, some of the points, termed as fictitious points, fall outside the domain of solution and, therefore, cannot easily be defined for all kinds of supports. Many researchers overcome this difficulty by using the finite-difference approach with the principle of virtual work or minimum potential energy, referred to as the finite-difference energy (FDE) method and applied to static, vibration, and instability problems for plates [1,15,16].The variational method is a powerful method that can be used for both formulation and approximate solution of problems [11]. In the present study, we use this method to overcome the above-mentioned difficulty. In fact, it will be shown that the variational derivative and the finite-difference methods are equivalent at inside points of the domain. They differ at the boundaries of the beam. Therefore, the main aim of the present study is to give a new aspect to the finite-difference method by using the variational derivative. In this formulation, the functional, obtained in [7] for an axially loaded beam, is used. This functional, which has only first-order derivative terms, is developed using the Gâteaux differential [12,14] and equations transformable to the classical energy. The moment and deflection in this functional are chosen as independent variables. The element matrix is derived based on the variational derivative. Then, the system matrix is obtained and the boundary conditions are included using a coding technique. The problem of determining the buckling loads of a structural system is reduced to a standard eigenvalue problem. The accuracy and validity of this element matrix is investigated for a...
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