Analysis of Elastic Wave Propagation Through Thin Shell Modelled Flexible Pipes
Author
Morsbøl, Jonas
Term
4. term
Education
Publication year
2011
Submitted on
2011-05-31
Pages
59
Abstract
Fleksible rør bruges i olie- og gasindustrien til at transportere fossile væsker. Et typisk eksempel er et stigerør, der fører råolie eller naturgas fra havbunden til en platform eller et tankskib. Når væsken strømmer, kan den skabe vibrationer, som forplanter sig som elastiske bølger langs røret. Disse bølger kan nå rørenderne og med tiden give træthedsbrud i de komponenter, der er fastgjort der. For at kunne dæmpe disse vibrationer ved enderne er det nødvendigt at forstå røret som bølgeleder – dvs. hvordan elastiske bølger bevæger sig gennem strukturen. Som første skridt undersøger afhandlingen bølgelednings-egenskaberne for en infinit toroidal skal, en matematisk model af en donut-formet, tyndvægget rørflade. En lille sektion af en sådan torus kan bruges som idealisering af en bøjning i et tyndvægget rør, og de generelle resultater kan senere overføres til en begrænset sektion via en metode med randintegralligninger. Røret modelleres som en skal for at medtage fleksibiliteten af tværsnittet. Modellen for den toroidale skal er benchmarket mod klassisk Bernoulli–Euler bjælke-teori og teori for krumme bjælker. Sammenligningen med Bernoulli–Euler er især relevant, når torus-radien er meget stor, fordi strukturen da nærmer sig en lige rørsektion. Af samme grund er der også opstillet en skalmodel af en tyndvægget cylinder og foretaget sammenligning med den toroidale skal. Benchmarkingen sker ved at sammenligne dispersionskurver for nogle af de enkleste vibrationsmodi, som teorierne har til fælles. Dispersionskurverne beregnes ved at anvende prøve-løsninger på de styrende differentialligninger og derefter bruge Galerkins metode til at få et ligningssystem med samme antal ukendte og ligninger. For den toroidale skal er der brugt afkortede komplekse Fourier-rækker som prøve-løsninger, der beskriver henholdsvis bøjning i planet og uden for planet, og konvergensen af de tilhørende dispersionskurver for enkle vibrationsmodi er undersøgt. Studiet viser, at de afledte styrende differentialligninger for den toroidale skal er gyldige.
Flexible pipes are used in the oil and gas industry to transport fossil fluids. A common case is a riser that carries crude oil or natural gas from the seabed to a platform or tank ship. Flowing fluid can generate vibrations that travel as elastic waves along the pipe. These waves can reach the pipe ends and, over time, cause fatigue failure in attached components. To reduce vibrations at the ends, we need to understand the pipe as a waveguide—that is, how elastic waves move through the structure. As a first step, this thesis studies the waveguide properties of an infinite toroidal shell, a mathematical model of a donut-shaped, thin-walled pipe surface. A small section of such a torus can idealize a bend in a thin-walled pipe, and general results can later be transferred to a finite section using a boundary integral equations method. The pipe is modeled as a shell to account for flexibility of the cross-section. The toroidal shell model is benchmarked against classical Bernoulli–Euler beam theory and curved beam theory. The comparison to Bernoulli–Euler is particularly relevant when the torus radius is very large, because the structure then approaches a straight pipe section. For the same reason, a shell model of a thin-walled cylinder is also established and compared with the toroidal shell. Benchmarking is done by comparing dispersion curves for some of the simplest vibration modes covered by both theories. Dispersion curves are obtained by applying trial solutions to the governing differential equations and then using Galerkin’s method to form a system with equal numbers of equations and unknowns. For the toroidal shell, truncated complex Fourier series are used as trial solutions to represent in-plane and out-of-plane bending, and the convergence of the resulting dispersion curves for simple vibration modes is examined. From these studies, it is concluded that the derived governing differential equations for the toroidal shell are valid.
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