Stability of elastic stepped beams with cracks

Abstract

Käesolevas väitekirjas vaadeldakse elastsete astmeliste talade stabiilsust. Töö aluseks on autori kuus teaduslikku publikatsiooni, millest kolm on avaldatud viimase kolme aasta jooksul. Väitekiri koosneb neljast osast: kokkuvõtvast osast ehk kokkuvõtteartiklist, publikatsioonide koopiatest, kirjanduse ülevaatest ja autori elulookirjeldusest. Antud töös uuritakse elastseid talasid, millele mõjub teljesuunaline koormus. Talad on astmelised ning astme kohtades asuvad defektid ehk praod, mis antud uurimuses on stabiilsed. Pragude sügavus ja asukoht mõjutab talade stabiilsust ning stabiilsuse tundlikkust antud parameetrite suhtes on analüüsitud kombineerides elastsusteooria ja lineaarse purunemismehaanika meetodeid. Esimeses peatükis tuuakse ajalooline ülevaade kirjandusest. Teises peatükis esitatakse uurimuse põhialused prao mõju analüüsiks. Praoga tala uurimiseks kasutatakse nn. kaalutu väändevedru mudelit. Selle mudeli kohaselt asendatakse praoga tala konstruktsiooniga, mis koosneb kahest tala tükist (elemendist). Need elemendid on omavahel ühendatud kaalutu väändevedruga, mille jäikus on võrdeline pinge intensiivsuse koefitsendiga prao tipu juures. Järgnevas neljas peatükis uuritakse kriitilise koormuse sõltuvust prao parameetritest erinevate talade ja kinnitustingimuste korral. Esimesel juhul on vaatluse all konsooltala, teisel juhul on vabale otsale lisatud elastne kinnitus. Kolmandaks uuritakse konsooltala, mis asub elastsel alusel ning lõpetuseks tala, mis on seest õõnes (nelinurkne toru).

In the present thesis critical buckling loads of stepped beams are studied and the sensitivity of the critical load on the parameters of stable cracks as location and depth is analysed. Combining the methods of the elastic beam theory and of the linear elastic fracture mechanics an approximate method for the stability analysis of beams and columns subjected to the axial pressure is developed. Introducing the additional compliance matrix the flexibility of the beam in the vicinity of a crack is prescribed by means of the compliance of the structure. This, in turn, is coupled with the stress intensity factor which can be calculated by methods of the linear elastic fracture mechanics. Critical buckling loads of stepped columns subjected to the axial pressure and weakened with cracks emanating from re-entrant corners of steps are established. Numerical results are presented for uniform and hollow beams with single step of the cross section, also for two-stepped beams. The beams under consideration are simply supported or clamped at the ends, also cantilevers, elastically fixed. The case of beams resting on elastic foundation is studied separately. The dissertation is based on the six papers of the author (two of these are published during the last two years). The dissertation consists of the review of the obtained results, the copies of the papers, the list of literature and CV of the author. The dissertation is organized as follows. Section 1 contains historic background of the stability analysis, the aim and the structure of the dissertation. In section 2 the concept of local flexibility is described in detail. In sections 3, 4, 5 and 6 the method is applied to partcular cases of beams. The first case concerns elastic beams that are clamped at one end and free at another end. Secondly elastically fixed beams are studied in greater detail. In section 5 beams resting on elastic foundation are considered. Finally, in section 6 beams with hollow cross sections are studied. The influence of crack length and step location on the stability of the beams has been analyzed.