STATIC AND DYNAMIC APPLICATIONS OF A FIVE NODED HORIZONTALLY CURVED BEAM ELEMENT WITH SHEAR DEFORMATION

A Þve-noded thirteen DOF horizontally curved beam element with or without an elastic base is presented. One set of fourth-degree Lagrangian polynomials in natural co-ordinates is used for interpolation of beam geometry and vertical displacement while the angles of transverse rotation and twist are interpolated by another set of third-degree polynomials. For elastic subgrade, the reactive forces o¤ered at any point are assumed to be proportional to the corresponding displacements at that point. The e¤ect of shear deformation is accounted for in the sti¤ness matrix. In mass matrix evaluation, for dynamic problems, translational as well as rotary intertias have been considered and studied separately. For numerical integration of the sti¤ness matrix, a four-point Gaussian scheme has been found to be adequate. Numerical results for a number of sample problems and their comparison with analytical solutions have been presented for circular as well as for non-circular curved beams. Displacements, bending moment and torque for static loading with or without elastic foundation, as well as natural frequencies and mode shapes are computed for di¤erent cases. Examples include the problem of a cantilever beam of spiral geometry with di¤erent parametric values of the spiral and the agreement with the analytical results establishes the e¦cacy of the element. The performance of the element has been found be be excellent in both static and dynamic conditions. Su¦cient details are presented so that the formulation may be readily used. It is hoped that the large number of numerical illustrations will elucidate the validity and the range of applicability of the element and will also serve as benchmark for future researchers