AXIAL FORCE INFLUENCE ON RESPONSE TO THE DISTRIBUTED MOVING LOAD OF TIMOSHENKO BEAMS WITH GENERAL BOUNDARY CONDITIONS

Abstract

In this thesis, the problem of the transverse motions of Timoshenko beam-type resting on Vlasov foundation and under the actions of partially distributed masses is studied. In particular, Uniform and Non-uniform Timoshenko beams are considered. Enormous amount of work have beam done on dynamical problems involving Bernoulli-Euler and other beam types under moving masses, lumped or distributed whereas, works involving Timoshenko beams under moving masses are scanty in literature. This class of problems is governed by simultaneous second order partial differential equations with variable and singular coefficients. The main objective of this study is to obtain closed form solutions to this class of dynamical problems. In order to achieve the above purpose, the Heaviside function which describes the partially distributed load was first expressed in series form so as to simplify the transformation of the governing simultaneous partial differential equations. An approach based on the Generalised Galerkin’s method was first used to reduce the simultaneous second order partial differential equations governing the dynamical problem of uniform Timoshenko beam resting on Vlasov foundation and under the action of moving distributed loads to a sequence of coupled second order ordinary differential equations with variable coefficients. Since the resulting simultaneous second order ordinary differential equations do not yield readily to classical methods, a modified asymptotic method of Struble was used to simplify the simultaneous ordinary differential equations while the Laplace transformation in conjunction with the initial conditions and convolution theory are used to obtain analytical solutions to the dynamical problems. Similar approach was used to obtain the closed from solution to the problem of Non-uniform prestressed Timoshenko beam resting on Vlasov foundation and under the action of moving distributed loads. An important feature of this technique is that it is capable of tackling this class of dynamical problems for all variants of commonly encountered classical boundary conditions. From the closed form solutions, it is observed that for the same natural frequency, the critical speed for the prestressed uniform Timoshenko beam traversed by moving distributed force is greater than that under the influence of a moving distributed mass. Hence, resonance is reached earlier in the moving distributed mass problems. The same analyses are obtained for the dynamical system of prestressed non-uniform Timoshenko beam traversed by moving distributed load. Furthermore, the transverse displacements for the moving distributed force and the moving distributed mass models were calculated for various times t and presented in plotted curves and from all illustrative examples, it is found that the moving distributed force solution is not an upper bound for the accurate solution of the moving distributed mass problems. Analyses further show that an increase in the values of the axial force N, reduces the response amplitude of the dynamical system for all the illustrative examples in the dynamical problems of uniform prestressed Timoshenko beam under distributed load. Similarly, in this same dynamical system, as one increase shear modulus G and foundation stiffness K, in all the illustrative examples of classical boundary conditions, the transverse displacement reduces whether for moving distributed force or moving distributed mass models. It is however, noted that higher values of the structural parameter are required for a more noticeable effect on the structural response in the case of other boundary conditions than the case of simply supported boundary conditions for both moving force and moving mass problems. The same results and analysis characterize the non-uniform Timoshenko beam resting on Vlasov foundation and under the actions of moving loads. Finally, it is concluded that relying on the moving force solutions as a good approximation to a moving mass problem could not only be misleading but tragic.