Discrete models such as the lumped parameter model and the finite element model are widely used in the solution of soil amplification of earthquakes. However, neither of the models will accurately estimate the natural frequencies of soil deposit, nor simulate a damping of frequency independence. This research develops a new discrete model for onedimensional viscoelastic response analysis of layered soil deposit based on the mode equivalence method. The new discrete model is a one-dimensional equivalent multi-degree-of-freedom（MDOF） system characterized by a series of concentrated masses, springs and dashpots with a special configuration. The dynamic response of the equivalent MDOF system is analytically derived and the physical parameters are formulated in terms of modal properties. The equivalent MDOF system is verified through a comparison of amplification functions with the available theoretical solutions. The appropriate number of degrees of freedom（DOFs） in the equivalent MDOF system is estimated. A comparative study of the equivalent MDOF system with the existing discrete models is performed. It is shown that the proposed equivalent MDOF system can exactly present the natural frequencies and the hysteretic damping of soil deposits and provide more accurate results with fewer DOFs.
A unified stress function for bi-modulus beams is proposed based on its mechanic sense on the boundary of beams. Elasticity solutions of stress and displacement for bi-modulus beams under combined loads are derived. The example analysis shows that the maximum tensile stress using the same elastic modulus theory is underestimated if the tensile elastic modulus is larger than the compressive elastic modulus. Otherwise, the maximum compressive stress is underestimated. The maximum tensile stress using the material mechanics solution is underestimated when the tensile elastic modulus is larger than the compressive elastic modulus to a certain extent. The error of stress using the material mechanics theory decreases as the span-to-height ratio of beams increases, which is apparent when L/h ≤ 5. The error also varies with the distributed load patterns.
