Predicting the dynamic response of elastic-plastic beams by dimensional analysis

Elastic-plastic beams subjected to dynamic loading are widely found in engineering. However, due to the complex coupling of geometric and material nonlinearity, there is no complete analytical solution available. In this paper, based on energy conservation and dimensional analysis, a dominant dimensionless number, the loading intensity ξ, is proposed. Including both the effects of geometry, material and load, ξ could be used to predict the significant responses of an elastic-plastic beam under uniformly pressure loading, such as energy ratio, deflection ratio and deformation mechanism. With combination of loading intensity ξ and dimensionless stiffness β, the dimensionless maximum and final deflections of elastic-plastic beams under pressure loading can be predicted directly. Dimensionless numbers of beams with different cross-sections, or under different loading forms are also analyzed and confirmed.

Then, elastic effect over dynamic response of elastic-plastic beams is analyzed. For final deflection, it is found $\xi >5$ is the applicability of theoretical solution based on rigid, perfectly plastic (R-PP) material simplification. However, R-PP results can't be used in initial plastic hinge location and subsequent plastic zone evolution, even for very large ξ.

Using these dimensionless numbers, experimental or simulation data could be expressed in a normalized and comparable form, and the physical mechanisms will be more intuitive and clearer, which is valuable for scaled dynamic tests.