Variational Principles And Its Application With Mixed Variables Of Straight Beams With Large Deflection

With the continuous innovation of the world’s science and technology,various kinds of infrastructure construction have also been developed rapidly.In practical engineering application,it is necessary to guarantee the strength and rigidity of some bending members,so as to ensure the safe use of structural facilities and to meet people’s visual effects.Straight beam is the most simple component that is commonly used in practical engineering,straight beam with large deflection is applied extensively in engineering more,such as the application in the aerospace field and large space grid structure,people also pay more attention to the large deflection problem.The correctness of the variational principle of mixed variables has been verified in the field of stability,vibration and bending of rectangular plates.The number of equivalent equations of getting the extreme value for a functional of mixed variable is more than that of Lagrange and E.Reissner algorithm.The number of the weakly admissible mixed variables of using weak allowable deflection is more than the weakly admissible displacement and the weakly admissible deflection.All these are easy to derive the variational principle of mixed variables for large deflection straight beams.Then we deduce the variational stress-strain relation of the large deflection straight beam line elastic mechanics variational principle of mixed variables,and then apply the variational principle of mixed variables derived,that is principle of minimum potential energy with mixed variables of straight beams with large deflection,principle of stationary complementary energy with mixed variables,principle of minimum potential action with mixed variables,principle of stationary complementary action with mixed variables,to solve the problem of deformation stability of straight beams with large deflections,a new algorithm was given.The calculations were performed through MATLAB,and the calculation results were compared with the ANSYS finite element simulation software to verify the correctness of two types of variational principle with mixed variables of straight beams with large deflection.On this basis we use the reciprocal theorem of large deflection work to deduce the zero variation relation of straight beam in different cases,that is three types of variational principle with mixed variables.Beam is a special case of plate.In the analysis of the nonlinear process of the beam,applying the variational principle with mixed variables to solve the forced vibration of a cantilever beam with large deflection under uniformly distributed load,the algorithm is more flexible,accurate and reliable.It can solve the problems of balance and vibration of large deflection straight beam under different boundary conditions and loads.At the same time,the derivation of four kinds of mixed variational principle has certain theoretical significance,and provides a new train of thought to solve the problem of large deflection straight beam in engineering practice.