TABLE OF CONTENTS A. M. Lyapunov thought that having put the physical initial problem it should be solved mathematically and it would be best if the precise solution was reached I. V. Andrianov and J. Awrejcewicz (2000).
The increasing popularity if the finite element method conceals in it a danger of extreme infatuation with it in damage to analytic methods. Consideration of discrete methods and analytical ones as opposite techniques, that appears in the literature, is not justified. Both methods have characteristic advantages and drawbacks; they mutually complement each other I. F. Obraztsov (1975).
Exact analytical solutions are available but only for simple structures subjected to static loads M. Shinozuka (1989).
Structural finite element mechanics provides good arguments for choosing the bending compliance of a beam as an input rather than the bending stiffness A. M. Hasofer, O. Ditlevsen, and N. J. Tarp Johansen (1998).
Although the above statement by Professor Shinozuka was made in 1989, the situation regarding exact solutions, let alone closed-form solutions, has not changed drastically. However, here we offer considerably wider class of exact solutons than the one that was available in 1989. In this chapter we derive several exact solutions for stochastic beams, in order to use them as benchmark solutions for comparing with them the results furnished by the finite element method. We start with the shear beams and proceed to the Bernoulli-Euler beams with stochastic stiffness. We consider beams under either deterministic or stochastic excitation.
The random vibration...
