One-dimensional models for the spacial behaviour of tapered thin-walled bars with open cross sections : static, dynamic and buckling analyses

摘要

Tapered thin-walled bars are extensively used in the fields of civil, mechanical and aeronauticalengineering. The competitiveness of tapered structural members is hindered by the factthat their spatial behaviour is still poorly understood and by the lack of rational andefficient methods for their analysis and design. The present thesis aims at providing acontribution to overcome these drawbacks, by (i) developing one-dimensional models (i.e.,models having a single independent spatial variable) to perform linear static, dynamic andlateral-torsional buckling analyses of tapered thin-walled bars with open cross-sections,(ii) supplying physical interpretations for the key behavioural features implied by thesemodels and (iii) offering a detailed examination of several illustrative examples that will beuseful for benchmarking purposes.The first part of the thesis is devoted to bars whose shape allows them to resist biaxialbending by the membrane action of their walls (I-section or C-section bars, for instance). Itstarts with the development, based on the induced-constraint approach, of a linear onedimensionalmodel for the stretching, bending and twisting of tapered thin-walled bars witharbitrary open cross-sections under general static loading conditions. A two-dimensionallinearly elastic membrane shell model is adopted as parent theory. The Vlasov assumptions,extended to the tapered case in such a way as to retain an intrinsic geometrical meaning, aretreated consistently as internal constraints, that is, a priori restrictions, of a constitutivenature, on the possible deformations of the middle surface of the bar. Accordingly, (i) themembrane forces are decomposed additively into active and reactive parts, and (ii) theconstitutive dependence of the active membrane forces on the membrane strains reflectsthe maximal symmetry compatible with the assumed internal constraints.For a large class of tapered thin-walled bars with open cross-sections, the membrane strainand force fields implied by the internal constraints do not have the same form as inVlasov’s prismatic bar theory – they feature an extra term, involving the rate of twist.Consequently, the torsional behaviour (be it uncoupled or coupled with other modes ofdeformation) predicted by our tapered model is generally at odds with that obtained using a piecewise prismatic (stepped) approach. The discrepancies may be significant, as illustratedthrough examples.The developed linear model is then extended into the dynamic range. The contributions ofrotatory inertia and torsion-warping inertia are fully taken into account. The inclusion of aviscous-type dissipative mechanism is briefly addressed.Subsequently, we derive a model for the elastic lateral-torsional buckling of singly symmetrictapered thin-walled beams with arbitrary open cross-sections, loaded in the plane ofgreatest bending stiffness. The adopted kinematical description rules out any local ordistortional phenomena. Moreover, the effect of the pre-buckling deflections is ignored.Since isolated beams with idealised support conditions are seldom found in actual designpractice, an archetypal problem is used to show how the presence of out-of-plane restraintscan be accommodated in the one-dimensional buckling model. The restraints may (i) havea translational, torsional, minor axis bending and/or warping character, and (ii) be eitherlinearly elastic or perfectly rigid.The second part of the thesis is concerned with strip members (i.e., members with a narrowrectangular cross-section) exhibiting constant thickness and varying depth. It deals withthree problems of increasing complexity:(i) the elastic lateral-torsional buckling of cantilevered beams with linearly varying depth,acted at the free-end section by a conservative point load;(ii) the elastic lateral-torsional buckling of cantilevers (ii1) whose depth varies according to anon-increasing polygonal function of the distance to the support and (ii2) which aresubjected to an arbitrary number of independent conservative point loads;(iii) the elastic flexural-torsional buckling of linearly tapered cantilever beam-columns,acted by axial and transverse point loads applied at the free-end section.These three problems are tackled analytically – we obtain exact closed-form solutions tothe governing differential equations and, thereby, establish exact closed-form characteristicequations for the buckling loads. However, in the third problem, the analytical approach issuccessful only for certain values of the ratio between the minimal and maximal depth of thestrip beam-column – in the remaining cases, it is necessary to resort to a numerical procedure.