Abstract
An accurate beam finite element is used to solve nonlinear vibration of arched beams and framed structures. The nonlinear governing equations of a skeletal structure are integrated numerically using symplectic integration schemes so that the Poincaré integral invariant of a Hamiltonian flow are preserved during the evolution. The element stiffness matrices are not required to be assembled into global form, because the integration is completed on an element level so that many elements can be handled in core by a small computer. Testing examples include arched beams and frames with and without damping in free and forced vibration. The dynamic symmetry breaking phenomena are noted at the dynamic buckling point.
Highlights
The geo.metric no.nlinear analyses o.f framed structure fo.r bo.th static and dynamic Io.ading are o.f co.nsiderable interest and practical impo.rtance. Chajes and Churchill (1987) and Meek and Tan (1984) co.nsidered the influence o.f higher o.rder no.nlinear terms and gave the stiffness matrix expressio.ns accurately
In o.ur previo.us paper (Leung and Mao., 1995), the seco.nd- and thirdo.rder no.nlinear stiffness matrices were introduced by the order o.f no.dal displacements
The following terms are new: the quadratic nodal displacement the second-order stiffness matrix where p is the mass density of material, t is the time variable, and F,k(, t) and FAx, t) are the axial and transverse loads, respectively
Summary
The geo.metric no.nlinear analyses o.f framed structure fo.r bo.th static and dynamic Io.ading are o.f co.nsiderable interest and practical impo.rtance. Chajes and Churchill (1987) and Meek and Tan (1984) co.nsidered the influence o.f higher o.rder no.nlinear terms and gave the stiffness matrix expressio.ns accurately. The geo.metric no.nlinear analyses o.f framed structure fo.r bo.th static and dynamic Io.ading are o.f co.nsiderable interest and practical impo.rtance. In o.ur previo.us paper (Leung and Mao., 1995), the seco.nd- and thirdo.rder no.nlinear stiffness matrices were introduced by the order o.f no.dal displacements. The equilibrium equatio.ns were accurately discretized and an explicit beam finite element was established. The dynamic problems o.f beams with large displacement and small strain and rotatio.n have been so.lved. A previo.usly develo.ped beam finite elements fo.rmulatio.n (Leung and Mao., 1995) is extended to. .lve the equatio.ns element by element witho.ut fo.rming the glo.bal matrices. .lve no.nlinear vibratio.ns o.f large framed structures using a small computer.
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