Elastic plastic analysis and optimization of axisymmetric plates
Date
2016-09-28
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Abstract
Insenerimehaanikast on teada, et konstruktsioonielementide (talade, plaatide, koorikute) projekteerimisel on materjali kokkuhoiuks mõistlik arvestada lisaks elastsetele deformatsioonidele ka plastseid. Elastse deformatsiooni korral taastub keha esialgne kuju pärast koormuse eemaldamist, plastse deformatsiooni korral mitte.
Antud töös vaadeldakse nn sandwich-tüüpi ümar- ja rõngasplaate, millele mõjub telgsümmeetriline ristkoormus. Ümarplaat on ringsilindriline keha, mille kõrgus on teiste mõõtmetega võrreldes väike. Sandwich-tüüpi plaadiks nimetatakse ideaalset kahekihilist plaati, mille kandva kihi paksus h on kihtidevahelise kaugusega H võrreldes väike.
Kogu plaat on elastne, kui talle rakendatakse väikseid koormusi. Koormuse suurendamisel tekib plaadis üks või mitu plastset piirkonda.
Doktoritöös uuritakse erinevate kinnitusviisidega tükati konstantse paksusega elastseid-plastseid ümar- ja rõngasplaate tükati lineaarsete voolavustingimuste korral.
Elastse-plastse konstantse paksusega välisservast vabalt toetatud rõngasplaadi paindeülesaande lahendamiseks leitakse erinevate koormuste korral analüütiliselt ja numbriliselt plaadi läbipainded ning radiaal- ja tangentsiaalsuunalised paindemomendid.
Selgub, et tükati konstantse paksusega elastse-plastse seest jäigalt kinnitatud ja välisservast täiesti vaba astmelise rõngasplaadi pingeseisundi saab jagada kolme erinevasse staadiumisse. Leitakse läbipainde ja paindemomentide avaldised vastavalt elastse, elastse-plastse ja täiesti plastse plaadi pingeseisundi korral.
Numbriliselt lahendatakse ühe astmega elastsete homogeensest ja anisotroopsest materjalist ümarplaatide optimeerimisülesanded, kus etteantud plaadi ruumala korral arvutatakse optimaalsed kandvate kihtide paksused ning astme asukoht nii, et plaadi keskpunkti läbipaine oleks minimaalne. Samuti leitakse ringikujuliste lisatugede optimaalsed asukohad vabalt toetatud elastse ümarplaadi puhul nii, et plaadi läbipaine oleks minimaalne.
When modelling the structural behaviour of the structural elements (e.g. beams, plates, shells), it is necessary to account for both the plastic and elastic deformations. In the case of elastic deformation the body recovers its initial shape after removing the loading, in the case of plastic deformation it does not recover. In this work, the so-called sandwich-type circular and annular plates under axisymmetric transverse loading are studied. A circular plate is a cylinder with a much smaller height compared to its radius. A sandwich-type plate is an ideal two-layered plate where the height of the carrying layers h is much smaller than the thickness of the core material H. Under small loads, the entire plate is elastic. Increasing of the load may cause the appearance of one or several plastic areas in the plate. In this thesis, the circular and annular plates with the piecewise constant thickness and different support types are investigated using piecewise linear yield conditions. The bending problem for the elastic plastic annular plate, simply supported at the outer edge, is solved by finding the deflections, the radial and circumferential bending moments for different loadings. It appears that the stress strain state of the elastic plastic annular plate with the piecewise constant thickness, clamped at the inner edge and absolutely free at the outer edge, can be divided into three stages. The expressions of the deflections and the bending moments are found in the cases of elastic, elastic plastic and entirely plastic stress strain states, respectively. The optimization problems regarding to the stepped circular plates made of homogeneous and anisotropic materials are solved numerically. For the fixed plate volume, the optimal values for the heights of the carrying layers and the location for the step are calculated while requiring minimal deflection at the centre of the plate. Also, the optimal locations for additional circular supports corresponding to the minimum of the mean deflection are found in the case of the elastic simply supported plate.
When modelling the structural behaviour of the structural elements (e.g. beams, plates, shells), it is necessary to account for both the plastic and elastic deformations. In the case of elastic deformation the body recovers its initial shape after removing the loading, in the case of plastic deformation it does not recover. In this work, the so-called sandwich-type circular and annular plates under axisymmetric transverse loading are studied. A circular plate is a cylinder with a much smaller height compared to its radius. A sandwich-type plate is an ideal two-layered plate where the height of the carrying layers h is much smaller than the thickness of the core material H. Under small loads, the entire plate is elastic. Increasing of the load may cause the appearance of one or several plastic areas in the plate. In this thesis, the circular and annular plates with the piecewise constant thickness and different support types are investigated using piecewise linear yield conditions. The bending problem for the elastic plastic annular plate, simply supported at the outer edge, is solved by finding the deflections, the radial and circumferential bending moments for different loadings. It appears that the stress strain state of the elastic plastic annular plate with the piecewise constant thickness, clamped at the inner edge and absolutely free at the outer edge, can be divided into three stages. The expressions of the deflections and the bending moments are found in the cases of elastic, elastic plastic and entirely plastic stress strain states, respectively. The optimization problems regarding to the stepped circular plates made of homogeneous and anisotropic materials are solved numerically. For the fixed plate volume, the optimal values for the heights of the carrying layers and the location for the step are calculated while requiring minimal deflection at the centre of the plate. Also, the optimal locations for additional circular supports corresponding to the minimum of the mean deflection are found in the case of the elastic simply supported plate.
Description
Väitekirja elektrooniline versioon ei sisalda publikatsioone.
Keywords
ehitusmehaanika, tööstusmatemaatika, plaadid, elastsus- ja plastsusteooria, structural mechanics, engineering mathematics, plates, theory of elasticity and ductility