Abstract:
A Unified Beam finite element for vibration of Timoshenko beam on elastic foundation was developed in this work. The proposed unified beam finite element was used to formulate the stability and free vibration analysis of beams with shear effect deformation with and without elastic foundation. For this purpose, the Timoshenko beam was divided into Euler-Bernoulli beam and shear layer beam elements. To apply the finite element method, the displacement functions were assumed to be cubic polynomials for the Euler Bernoulli beam and quadratic for the shear layer beam and were used to develop the element stiffness matrix, geometric stiffness matrix, mass matrix, rotary inertia matrix and foundation matrix of the proposed beams. The analytical relationship between bending and proposed shear rotations of the Euler Bernoulli beam and shear layer beam was established through the use of a Bending-Shear rotation interdependent factor . Desirable results were obtained when evaluating the efficiency and accuracy of the element for free vibration and stability analysis of beam. The element was free from shear locking affecting other Timoshenko beam finite elements and was seen to give excellent results up to 95% using one element in a mesh. On the stability of Timoshenko Beam element, it was observed that the critical buckling load and the effective length factor of the Timoshenko beam element were proportional to the interdependent interpolation factor for all the support conditions. Also it was discovered that the critical buckling load Pcr of Timoshenko beam did not only depend on the elasticity, second moment of area, the boundary conditions and the length of the beam as in the case of classical beam, but also the shear deformation parameter Ф. On the natural frequency of beam, the effect of shear deformation was evident when rotary inertia was considered. The frequency ratio reduces as span to depth ratio (L/d) decreases from 100 to 4, while reverse was the case for L/d less than 4, when the rotary inertia was neglected the frequency ratio increases with increase in L/d. Furthermore, it was observed that at L/d of 5 about 10 - 30% of the natural frequency of vibration was lost to rotary inertia effect. On the vibration of Timoshenko beam on elastic foundation, it was observed that the natural frequency of the beam increases with the increase in the value of modulus of the subgrade. While the introduction of the axial load causes a reduction in the natural frequency of the beam. For the purpose of design, for beam under any static load, shear deformation may be ignored for L/d greater than 5.While for beam under dynamic load, shear deformation may be ignored for L/d greater than 5. When L/d ratio is less than 5, shear deformation contributes significantly and should be accounted for in such exceptional cases.
