Free vibrations of stepped cylindrical shells containing cracks
Date
2010-11-19
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Käesolev väitekiri koosneb sissejuhatusest, neljast peatükist, kirjanduse loetelust ja kokkuvõttest.
Väitekiri on pühendatud tükiti konstantse paksusega silindriliste koorikute võnkumiste uurimisele juhul, kui koorikus on praod. Praod eeldatakse olevat stabiilsed konstantse sügavusega ringikujulised praod, mis asuvad nendes ristlõigetes, kus kooriku paksus muutub hüppeliselt.
Sissejuhatuses on esitatud ajalooline ülevaade plaatide ja koorikute uurimisele pühendatud töödest. Põhitähelepanu on siin pööratud töödele, kus uuritakse konstruktsiooni elementide võnkumisi ja stabiilsust juhul, kui konstruktsioonis on praod.
Töö esimeses peatükis käsitletakse otstest vabalt toetatud ringsilindriliste koorikute vabavõnkumisi. Eeldatakse, et kooriku paksus on tükiti konstantne ja et paksuse hüppekohtades asuvad praod. Esitatakse probleemi uurimiseks vajalikud põhivõrrandid. Need koosnevad tasakaaluvõrranditest, geomeetrilistest seostest ja Hooke’i seadusest. Prao mõju kooriku käitumisele arvestatakse lokaalse järeleandlikkuse koefitsiendi abil, mis seotakse purunemismehaanikast tuntud pinge intensiivsuse koefitsiendiga.
Põhivõrrandite süsteem lahendatakse Fourier meetodil. Rahuldades seejärel rajatingimused ja vastavad pidevuse ning katkevuse tingimused ristlõigetes, kus asuvad praod, jõutakse algebraliste võrrandite süsteemini. Leides vastava maatriksi omaväärtused saadakse kooriku omavõnkesagedused.
Väitekirja teises peatükis uuritakse astmeliste koorikute võnkumisi mitmessuguste kinnitustingimuste korral arvestades paksuse astme kohtades asuvate pragudega. Detailsemalt analüüsitakse juhte, kui kooriku mõlemad otsad on jäigalt kinnitatud ning kui üks ots on jäigalt kinnitatud, aga teine vaba. Tulemuste võrdlemisel kirjandusest teada olevate lahenditega on näidatud, et konstantse paksusega kooriku korral on tulemused heas kooskõlas teiste autorite töödega.
Töö kolmandas peatükis vaadeldakse komposiitmaterjalist valmistatud ning kihiliste koorikute võnkumisi tükiti konstantse paksuse korral. On lahendatud konkreetne näide klaasplastikust silindrilise kooriku võnkumise kohta.
Väitekirja neljandas peatükis käsitletakse ringsilindriliste koorikute mitte-telgsümmeetrilisi võnkumisi. On töötatud välja ligikaudne meetod omavõnkesageduste määramiseks kasutades Donnelli poolt esitatud põhivõrrandite süsteemi approksimatsiooni. Ka siin arvestatakse prao mõjuga koorikute dünaamilisele käitumisele pinge intensiivsuse koefitsiendi abil.
The current study consists of the introduction and four chapters. In the first chapter free vibrations of simply supported circular cylindrical shells are considered. First of all, the governing equations for analysis of axisymmetric deformations of cylindrical shells are presented. Resorting to the concept of separation of variables a solution technique is suggested. According to this concept the solution of partial differential equations is transformed into a problem with ordinary equations. In the next section the model of the crack is discussed. The idea of an equivalent spring and local compliance is used herein in order to quantify the relationship between loading and stress concentration around the crack tip. As regards the stress concentration it is measured by stress intensity coefficients known from the linear elastic fracture mechanics. Accounting for the local compliance one can solve the problem up to the end. The results of calculations are compared with those obtained by other methods. In the second chapter vibrations of circular cylindrical shells with various boundary conditions are studied. The cases of shells clamped at both ends, also cantilever shells, e.g. shells clamped at the left end and absolutely free at the right hand end are treated in greater detail. It was assumed that the stress strain state of the tube remains axisymmetric during deformation. The shells of piece wise constant thickness are treated under the condition that at re-entrant corners of steps circular cracks of constant length are located. Cracks are considered to be stationary surface cracks, problems related to re-distribution of stresses and strains due to the extension of crack are not treated herein. The influence of cracks on the behaviour of the shell is prescribed via local compliances of the shell coupled with stress intensity factors. Calculations showed that the cracks have essential influence on vibration characteristics. It was established that if the crack location is fixed then the maximal value of the characteristic number k is achieved if the crack length is equal to zero. On the other hand, for fixed crack length and variable ratio of thicknesses maximum of the number k is achieved when h0=h1 thus for the shell of constant thickness. When ratio of thicknesses is fixed whereas h1<h0 then the minimum of k is achieved when the step and crack location tends to the free end. The third chapter is devoted to layered and composite cylindrical shells. It is assumed here that the stress-strain state of a circular cylindrical shell remains axisymmetric during deformations. The natural frequency of vibrations is determined for various non-homogeneous materials. In the fourth chapter non-axisymmetric vibrations of circular cylindrical shells are studied. The simplified system of equilibrium equations presented by Donnell is used. Numerical results are obtained for cylindrical shells of stepped thickness containing cracks at re-entrant corners of steps.
The current study consists of the introduction and four chapters. In the first chapter free vibrations of simply supported circular cylindrical shells are considered. First of all, the governing equations for analysis of axisymmetric deformations of cylindrical shells are presented. Resorting to the concept of separation of variables a solution technique is suggested. According to this concept the solution of partial differential equations is transformed into a problem with ordinary equations. In the next section the model of the crack is discussed. The idea of an equivalent spring and local compliance is used herein in order to quantify the relationship between loading and stress concentration around the crack tip. As regards the stress concentration it is measured by stress intensity coefficients known from the linear elastic fracture mechanics. Accounting for the local compliance one can solve the problem up to the end. The results of calculations are compared with those obtained by other methods. In the second chapter vibrations of circular cylindrical shells with various boundary conditions are studied. The cases of shells clamped at both ends, also cantilever shells, e.g. shells clamped at the left end and absolutely free at the right hand end are treated in greater detail. It was assumed that the stress strain state of the tube remains axisymmetric during deformation. The shells of piece wise constant thickness are treated under the condition that at re-entrant corners of steps circular cracks of constant length are located. Cracks are considered to be stationary surface cracks, problems related to re-distribution of stresses and strains due to the extension of crack are not treated herein. The influence of cracks on the behaviour of the shell is prescribed via local compliances of the shell coupled with stress intensity factors. Calculations showed that the cracks have essential influence on vibration characteristics. It was established that if the crack location is fixed then the maximal value of the characteristic number k is achieved if the crack length is equal to zero. On the other hand, for fixed crack length and variable ratio of thicknesses maximum of the number k is achieved when h0=h1 thus for the shell of constant thickness. When ratio of thicknesses is fixed whereas h1<h0 then the minimum of k is achieved when the step and crack location tends to the free end. The third chapter is devoted to layered and composite cylindrical shells. It is assumed here that the stress-strain state of a circular cylindrical shell remains axisymmetric during deformations. The natural frequency of vibrations is determined for various non-homogeneous materials. In the fourth chapter non-axisymmetric vibrations of circular cylindrical shells are studied. The simplified system of equilibrium equations presented by Donnell is used. Numerical results are obtained for cylindrical shells of stepped thickness containing cracks at re-entrant corners of steps.
Description
Keywords
dissertatsioonid, matemaatika, võnkumine, silindriline koor, pragu, axisymmetric vibration, non-axisymmetric vibration, stepped cylindrical shells, flaws