The Analysis of the Non-linear Deflection of Non-straight Ludwick type Beams Using Lie Symmetry Groups
Proceedings of the 3rd International Conference of Control, Dynamic Systems, and Robotics (CDSR'16)
Nonlinear deflection of beams under the various forces and boundary conditions has been widely studied. The prediction of deflection of beams has been of great interest to generations of researchers. This seems to be a mundane problem as it is subject of textbooks on elementary mechanics of materials. Although both analytical and numerical solution have been found for specific type of loads, the general problem for beams that are not geometrically perfectly straight has not been approached so
... een approached so far of a systematic fashion. The present work presented a general method based on Lie symmetry groups that yields an exact solution to the general problem involving any arbitrary loading non-straight Ludwick type micro-beams. Lie symmetry method is used to reduce the order of the ODE describing the large deflection of the beam. The solution is validated against the particular cases of loading for which the large deflection problem has been solved and presented in the open literature. In this work an approach in solving the general problem based on Lie symmetry groups and a general analytical solution of the problem is presented below. The objective of the present work is to investigate a versatile mathematical method into the deflection of geometrically non-straight cantilever beams subjected point loads and moments applied at the free end while experiencing non-linear deflection. Lie symmetry method presented below can be used to any geometry of the bended beams under to condition that there is no residual stress in the unloaded beam. The deflection equation was calculated based on Ludwick experimental strain-stress curve. The integral equation was solved numerically and the end beam deflection and rotation were calculated. The same problem of large deflection cantilever beams made from materials behaving of nonlinear fashion under the tip point force was solved by finite difference methods.