Nonlinear Stability Analysis In Plane Of Shallow Arch Based On Catastrophe Theory

Arch structure is widely applied because of its loaded state is more reasonable, shallow arch is also applied in practical engineering as one special structure, but its stable characteristics is different from ordinary arch. On the function of arranging ralated literatures and summarizing predecessor’s research results, the advantege of catastrophe theory is taked to solve nonlinear problems in this paper, the cusp catastrophic model of nonlinear transformation and vibration equations of shallow arch are builded, and the nonlinear stability characteristics in plane of shallow arch is analyzed combined finite element simulation. The major job content as follows:Catastrophe theory and its fundamentals are shortly introduced, fold catastrophe, cusp catastrophe, swallowtail catastrophe and catastrophe indications are described underline. The basic characteristics of catastrophe models that including multimodal, inaccessible properity, sudden jump, divergency, hysteresis and multiple paths are indicated. The applications of catastrophe theory in natural science, biological science and social science are reviewed.The nonlinear stability characteristics of shallow arch under vertical uniformly distributed load are analyzed. The stability conditions of shallow arch are obtained by established nonlinear equations and catastrophe equations. The impacts of axis’ s section, span, rise and load to shallow arch are compared, the mechanical properties of shallow arch strengthen with the increase of axis’ s section, span and with the decrease of rise. The size of the released elastic energy when shallow arch jumping can be expressed as the area that arounded with cruves including the balance surface of cusp catastrophic modal and state variables, control variables.The nonlinear stability characteristics of shallow arch under hamonic load are analyzed. Nonlinear vibration modal is established and is equivalent to stanard form of cusp catastrophic modal. The characteristics of resonance condition distortion, amplitude mutation, amplitude change hysteresis and so on are obtained. Stable areas decrease along with the increase of span, hamonic load and increase along with the increase of rise, damping ratio, otherwise, the nonlinear response of shallow arch increase along with the increase of span, hamonic load and decrease along with the increase of rise, damping ratio. Srtain energy discriminant is obtained by energy principle to judge the instability situation of shallow arch, the two methods have each advantege.The finite element software is used to numerical simulation of shallow arch. The nonlinear stability problems under vertical uniformly distributed load are analyzed by arc length method, combined with modal analysis, steady and transient state dynamic analysis for shallow arch under hamonic load, the difference is small compared with the theoretical analysis results. The bearing conditions, arch axis’ s shape and material nonlinear that can not be considered in theoretical analysis are considered, the conclusion that the stability performance of clamped shallow arch preferably, the impacts of arch axis’ s shape to nonlinear response and so on are received.