Based on principles of virtual work, a general method of deriving STIFFNESS MATRIX of non-prismatic curved Eluer- Bemoulli beam element is presented. Exact .satisfaction of equilibrium equations in any interior point of the element is the main characteristic of this method. Presented relations could be transformed based on the super-parametric formulation via piecewise mapping of element axis. This coordinate transformation helps to generate elements of complicated geometry and common to use. Various numerical examples proves that the results obtained by this method are compatible/comparable in both efficiency and economy with those obtained by other proposed formulations, if any.

The present paper attempts to develop a new analytical integration to evaluate the element STIFFNESS MATRIX for the finite elements with irregular shapes. Most of the finite elements developed by Sabir are based on the strain rather than displacement approach. They are characterized by a regular form and appropriate coordinates with the form of the element. Together, they tend to decrease the elements utilization domain. Hence, for reasons of importance and particularity of these elements (higher order shape functions expressed in terms of independent strains); it is necessary to introduce irregular forms, which require a special integration technique, and a specific classification in programming level for different geometric forms. To overcome this geometrical inconvenience; the paper presents a new integration solution routine. This will help to know how the elements will behave when they have irregular form, and to extend their applications domain for the curved structures no matter what the geometrical shape of the element might be.

In this paper, the exact STIFFNESS MATRIX of curved beams with non-uniform cross section is derived using direct method. The considered element has two nodes and twelve degrees of freedom with three forces and three moments applied on each node. The effect where the shear center .and center of area do not coincide is considered in this element. The cross sections across the beam are deformed by bending, torsion, axial and shear loads. The curve representing center of area can have any curvature in the space and the cross section properties may change arbitrarily along it. The STIFFNESS of three kinds of beams have been determined by this method. In one type the results have been compared with previous works and theoretical analyses. Finally it has been .shown that the determined STIFFNESS MATRIX is exact and all kinds of beams can be analyzed by this method.

In this paper, a new canonical form is introduced for efficient analysis of structures with special geometric properties. Using the properties of this MATRIX, the number of operations needed for the MATRIX inversion is considerably reduced employing the decomposition of the block STIFFNESS matrices. The condition for applicability of the presented method is also discussed. For the previously developed canonical forms, the Kronecker products and the corresponding theorems could be used for certain class of repeated structures. Here this class is extended to the STIFFNESS matrices having more general block tri-diagonal form where the diagonal blocks are not necessarily identical, requiring a different treatment. Two examples of finite element models are analyzed to illustrate the efficiency of the presented method.

A Newtonian (vectorial) approach is used to develop the governing differential equations of motion for a three layer sandwich beam in which the uniform distribution of mass and STIFFNESS is dealt with exactly. The model allows for each layer of material to be of unequal thickness and the effects of coupled bending and longitudinal motion are accounted for. This results in an eighth order ordinary differential equation whose closed form solution is developed into an exact dynamic member STIFFNESS MATRIX (exact finite element) for the beam. Such beams can then be assembled to model a variety of structures in the usual manner. However, such a formulation necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm, whose implementation is discussed in detail. The algorithm enables any desired natural frequency to be converged upon to any required accuracy with the certain knowledge that none have been missed. The accuracy of the method is then confirmed by comparison with five sets of published results together with a further example that indicates its range of application. A number of further issues are considered that arise from the difference between sandwich beams and uniform single material beams, including the accuracy of the characteristic equation, co-ordinate transformations, modal coupling and the application of boundary conditions.

Curved beams are widely used in combination with the linear elements of various civil engineering structures. Many researchers attempted to analyze beam curved in plan, beam curved in elevation, and spatial curved beam using different methods and different approaches and presented analytical exact solution and approximate numerical solution. The analytical exact integration of the governing differential equations is the major difficulty for the analysis of the geometrically non-linear curved beams. To overcome this difficulty, a finite displacement transfer method is proposed to eliminate analytical differentiation and integration, completely. This paper deals with the STIFFNESS MATRIX of 3D curved beam with varying curvature and varying cross-sectional area. A novel finite displacement transfer method is used to determine displacements of the freely supported node of the cantilever 3D curved beam. The flexibility MATRIX is derived using the finite displacement transfer method. The STIFFNESS MATRIX is derived by employing equilibrium and transformation MATRIX. The finite difference method is used for the numerical solution of the differential equations. Results of the calculation method are compared with the results of other methods in the literature and the FEM based analysis software. For the circular helix with uniformly varying cross-sectional area and 3600 elements, the maximum and minimum percentage difference in the STIFFNESS coefficient is 2.89% and −0.65% respectively. For the elliptic helix with the uniform cross-sectional area and 720 elements, the maximum and minimum percentage difference in the STIFFNESS coefficient is 2.69% and −2.65% respectively. The novel of this study lies in the generation of the STIFFNESS MATRIX of the 3D curved beams without tedious analytical differentiation and integration of governing equations. The STIFFNESS MATRIX of the spatial curved beam is applicable to the planer curved beam also.