UBC Theses and Dissertations
Finite deflection dynamic response of axially restrained beams Schubak, Robert Brian
The deformation response of symmetrically supported, axially restrained beams subjected to uniformly distributed pulse loads is studied herein, leading to the development of an analytical procedure to predict the character and magnitude of such response. The procedure is valid for beams of any singly symmetric or doubly symmetric cross-section, and is based upon the assumption that the beam material can be approximated as behaving in a rigid-perfectly plastic manner. The governing equations of motion are derived from variational statements consisting of the principle of virtual work and d'Alembert's principle, and include the effects of finite geometry changes. From the static analysis of axially restrained beams it is found that the yield curve of a beam section may be replaced by a linear approximation thereof to obtain a good estimate of the beam's load capacity. Incorporating the linear yield curve approximation in a dynamic analysis of an axially restrained beam results in the uncoupling of the response into two distinct phases — an initial small deflection phase in which the beam retains bending resistance and deforms as a mechanism formed by plastic hinges, and a subsequent large deflection phase in which the beam has no bending resistance and deforms as a plastic string. The results of such an analysis for a rectangular beam subjected to a rectangular load pulse compare well with the results of a previous solution which used the true quadratic yield curve. The linear yield curve approximation further results in linear differential equations of motion, and the response to load pulses of general load-time history may be solved in closed form. Blast-type pulses of varying shape are found to induce significantly different permanent deflections in a beam than a rectangular pulse. On the other hand, the effect of finite rise time of the pulse's load intensity is found to be small if the the rise time is less than about twenty to thirty percent of the pulse duration. A procedure developed by CK. Youngdahl is used to obtain rough estimates of the permanent deformation response by converting a pulse of triangular shape to an "effective" rectangular pulse. These estimates compare well with results obtained by the complete analysis of a triangular pulse developed herein. The use of Youngdahl's procedure combined with the analysis of a rectangular pulse developed herein can provide a quick, simple solution to the permanent deformation of a dynamically loaded beam which is amenable to hand computation.
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