Study On Coupling Dynamics Of Foundation For Long-Span Arch Bridge

Long-span arch bridge has rapidly developed and been built in mountainous areas,but theoretical research is relatively under-developed.What’s worse,when foundation/support is built in weak soils/rocks,overall stability and safety of arch bridge is confronted with even more challenges.Due to the huge horizontal force of arch bridge,the stiffness capacity of foundation is required to be high,while complex geological conditions in mountainous areas make it difficult to satisfy perfectly this stiffness condition.For the ideal case,arch is involved with fixed boundary conditions.By contrast,if foundation/support being built in weak soils/rocks,arch is connected with non-ideal boundary,which possibly leads to coupling dynamics between arch and foundation.Therefore,investigations into coupling dynamics between arch and foundation have great significance for both science and engineering.The study not only enriches coupled dynamic theory of arch bridges,but also help to solve various technical problems in site selection,design and construction.In this thesis,aiming at a nonlinear arch-foundation coupled structure system,the foundation is modeled,step by step,as fixed end,an elastic oscillator,a torsional-constrained rigid body,and an elastic foundation beam,and correspondingly,various coupling model with arch structure are established,which are comprehensively explored for understanding nonlinear coupling dynamics between arch and foundation.The main contents of this thesis are summarized as follows(1)A refined asymptotic analysis of softening-hardening transition in nonlinear shallow arch:arch is a typical geometrically nonlinear structure,i.e.,a nonlinear system with both quadratic and cubic nonlinear effects for large amplitude vibration.Initial curvature(or rise)of shallow arch will affect the competition mechanism between quadratic nonlinearity and cubic nonlinearity,and then affect consequently modify nonlinear softening/hardening effects.In particular,it is a key difficulty to correctly predict nonlinear dynamic behaviors close to softening-hardening transition.Seemingly linear symmetric responses of standard perturbation analysis(up to 3rdorder)fail to capture the nonlinear characteristics near softening-hardening transition.Essentially,high-order nonlinear modulation will dominate the near-transition dynamics,which should be well captured through high-order refined perturbation procedures,being similar to correction of failure of linear stability analysis via highorder nonlinear analysis for dynamical systems with vanishing eigenvalues.(2)Arch-elastic oscillator nonlinear coupled dynamics:based upon a key weak foundation motion assumption,an arch-foundation(elastic oscillator)coupling model is proposed,aiming to investigate nonlinear coupling dynamics between arch and elastic foundation.After being non-dimensionalized,an asymptotically reduced model of this coupling system is derived by direct multiple scale mothed,which is regarded as the slow dynamics of the system.It turns out that the foundation/arch mass ratio characterizes coupling intensity and plays a key role in understanding arch-foundation coupled dynamics.Based upon the asymptotic model,arch-foundation’s coupled responses are fully studied,with stability and bifurcation characteristics being determined for both equilibrium solutions(coupled system’s periodic behaviors)and periodic solutions(coupled system’s quasi-periodic behaviors).Of particular interest is that,both Hopf and PD bifurcations,and thus their induced complicated periodic/quasi-periodic behaviors,occur only in a limited parameter range,which has been explicitly determined through a two-parameter bifurcation analysis with respect to mass ratio M and detuning parameter σ1.(3)Arch-torsional rigid body nonlinear coupled dynamics:the elastic foundation/support is modeled as a rigid body constrained by a torsional spring,and a relatively refined arch-rigid body coupling model is then proposed,which provides a basis for revealing dynamic effect induced by finite depth of elastic foundation.It turns out that the moment of inertia of rigid body characterizes the coupling intensity and plays a key role.Nonlinear modulation equations of the coupling system are obtained via asymptotic expansion,i.e.,differential equations characterize the slowly modulated amplitude and phase of coupling system.The nonlinear dynamic characteristics of arch-rigid body primarily resonant responses are fully investigated by constructing frequency response diagrams.Furthermore,a detailed comparison study between the refined model(arch-rigid body coupled model)and the preliminary model(archoscillator coupled model),unveils a potential limit of the latter:the coupling intensity is underestimated by the arch-oscillator coupled model.An interpretation of this model difference is also given by resorting it to the so-called mass reduced factor(i.e.,1/3)naturally associated with the rigid body(foundation),i.e.,only partial mass of the elastic support is involved with the dynamic coupling due to finite depth,rather than the full mass as assumed in the unrefined arch-oscillator model.(4)Arch-elastic beam on Winkler foundation nonlinear coupled dynamics:the weakly moving foundation of arch is modeled as a Winkler elastic foundation beam,and a refined arch-elastic beam coupling model is proposed,which is further refinement of arch-oscillator coupled and arch-rigid body coupling models.Archelastic beam coupling model well captures,in addition to finite stiffness and inertia of foundation,finite depth and distributed elastic effect.Both foundation/arch mass ratio and height of beam characterizes coupling intensity and plays a key role in arch-elastic foundation coupling system.Nonlinear dynamic responses and bifurcations of this coupling model are fully investigated through direct perturbation method.Furthermore,the arch-oscillator coupled model and arch-rigid body coupled model are both compared with arch-elastic beam coupled model,revealing their underlying differences,leading to a relatively comprehensive theoretical framework for research of archelastic foundation coupling dynamics(refined modelling of elastic foundation is realized,step by step,by following a route consisting of oscillator-rigid bodycontinuous beam).