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Author:Niemi, Antti Henry Matias
Title:Matalan kuoren pistekuormaongelma
The Point Load Problem in Shallow Shell Deformations
Publication type:Master's thesis
Publication year:2004
Pages:57      Language:   fin
Department/School:Teknillisen fysiikan ja matematiikan osasto
Main subject:Matematiikka   (Mat-1)
Supervisor:Pitkäranta, Juhani
Instructor:
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark TF80     | Archive
Keywords:shallow shells
point load
asymptotics
layers
Fourier series
finite element method
matalat kuoret
pistekuorma
asymptotiikka
reunahäiriöt
Fourier-sarjat
elementtimenetelmä
Abstract (eng): The objective of this thesis is to study a set of model problems which represents extensively the different geometries (parabolic, elliptic, hyberbolic) of a shell.
The external load is taken to be a periodic point load acting on the shell midsurface.
The starting point of the study is a simplified shallow shell model.
The geometry of the shell can be described with three constant parameters in the model.
The fourth active parameter of the model is the shell thickness.

The problem is first formulated and solved exactly in general geometry.
The solution is then analysed more carefully in five different model cases.
The shell is hyperbolic in three of these.
Parabolic and elliptic geometries are represented by the remaining two cases.
The solutions are dominated by layers due to the load irregularity.
Moreover, it turns out that in almost all cases the shell falls into so-called intermediate state, where the total strain energy of the shell is divided into membrane and bending components.
The one exceptional case is bending-dominated.

The problem is approached also from the point of view of numerical analysis.
Indeed, standard finite elements of both low and high order are tested in the model cases by comparing displacements with the reference values obtained from the series solution.
As predicted by the theory, locking is strongly present in the lowest-order formulations but higher orders (>=4) give good results -at least when approximating displacements.
Abstract (fin): Työssä tutkitaan malliongelmasarjaa, jossa kuoren erilaiset geometriat (parabolinen, hyberbolinen, elliptinen) ovat kattavasti edustettuina, Malliongelmassa kuoren keskipintaan kohdistetaan periodinen pistekuormitus.
Tarkastelujen lähtökohdaksi otetaan yksinkertaistettu matalan kuoren malli, jossa kuoren keskipinnan geometria voidaan esittää kolmen vakioparametrin avulla.
Mallin neljäs aktiivinen parametri on kuorenpaksuus.

Ongelma formuloidaan aluksi yleisessä geometriassa ja sille määritetään analyyttinen sarjakehitelmäratkaisu.
Saatua ratkaisua analysoidaan tarkemmin viidessä erilaisessa esimerkkitapauksessa.
Kolmessa näistä kuori on hyberbolinen.
Kaksi muuta tapausta edustavat parabolista ja elliptistä geometriaa.
Ulkoisen kuorman epäsäännöllisyyden takia ratkaisuissa esiintyy voimakkaita reunahäiriöiden luontoisia ilmiöitä.
Lisäksi osoittautuu, että kuori asettuu miltei kaikissa tapauksissa nk. välitilaan, jossa kuoren venymäenergia jakautuu sekä kalvo- että taipumakomponentteihin.
Poikkeuksen muodostaa yksi tapaus, jossa kuori asettuu taipumadominoituun tilaan.

Ongelmaa lähestytään myös numeerisen analyysin näkökulmasta testaamalla tavanomaista elementtimenetelmää esimerkkitapauksissa.
Approksimaatioiden laskennassa käytetään sekä matala- että korkeampiasteisia elementtejä ja siirtymiä verrataan sarjaratkaisun mukaisiin tarkkoihin referenssiarvoihin.
Ennusteiden mukaisesti matala-asteiset elementit lukkiutuvat kaikissa tapauksissa melko pahasti, mutta korkeammilla asteluvuilla (>=4) saavutetaan - ainakin siirtymien osalta - hyviä tuloksia.
ED:2004-11-26
INSSI record number: 26530
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