In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods.
The effects of mismatch parameters on isochromatic fringe patterns were studied using the technique of photoelasticity. First, the mathematical equations of isochromatic fringes were derived for singular stress field near a bi-material notch. These equations were used to study the effects of mismatch parameters on the shape of isochromatic fringes theoretically. Analytical results indicated that the mismatch parameters have a significant effect on the shape of the isochromatic fringe patterns around the bi-material notch tip. In order to assess the accuracy of the analytical results, a photoelastic test program was conducted on the V-notched bi-material Brazilian disc specimens. A very good agreement was shown to exist between the experimental results and the analytical reconstructions.
The aim of the present work was to develop a guideline for approving the railway axles made of C35 steel and containing surface and/or in-body defects after manufacturing. First, several through and part-through circular cracks were modeled on the surface and in the body of the axle at its critical cross-section. Then, the permissible size of such cracks was determined by using the fracture mechanics. To verify the validity of the guideline, the theoretical result for the semi-circular surface crack was compared with the allowable size prescribed by the international railway standard. A very good agreement was found to exist between the predicted and the standard values.