Abstract
The plastic collapse of thin shells of revolution under axisymmetric loads is considered. A rigid perfectly-plastic material is assumed. Methods of limit analysis are applied to find collapse loads. The Tresca-Mohr yield criterion, as closely approximated by Nakamura [1, 2] is used.
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Abbreviations
- ε s , εθ :
-
Plastic strain-rates of the middle surface of a shell
- f i , g i , h i :
-
Yield functions
- H :
-
Thickness of a sheJI
- H i :
-
A constituent regular face of a piecewise regular yield hypersurface for which h i = 1
- H 1∩H 2 :
-
An intersection of two regular faces
- H 1→H 2 :
-
A transition between two regular faces corresponding to either a weak or a strong discontinuity
- k :
-
Geometrical parameter:\(\frac{{{M_0}}}{{L{N_0}}} = \frac{H}{{4L}}\)
- K s , K θ :
-
Plastic rates of curvature of the middle surface x s =K s M 0/N 0, x θ = K θ M 0/N 0
- L :
-
A typical shell length
- λ, λ i :
-
Deformation parameters in Mises flow law
- M 0 :
-
Yield bending momentjlength, \(\frac{{{\sigma _0}{H^2}}}{4}\)
- M s , M θ :
-
Bending moments/length, m s = M s /M 0, m θ = M θ/M 0
- N 0 :
-
Yield normal foree/Iength, σ0 H
- N s , N θ :
-
Normal forces/length, n s = N s /N 0, n θ = N θ/N 0
- Q s :
-
Transverse shear force/Iength, q s = Q s /N 0
- Q :
-
Generalized stress vector
- q :
-
Generalized strain-rate vector
- R :
-
Radius of a parallel eirele, r = R/L
- R 1, R 2 :
-
Prineipal radii of curvature of a meridian and parallel cirele, respeetively, r l = R 1/L, r 2 = R 2/L
- S :
-
Length measured along a meridian, s = S/L
- σ 0 :
-
Yield stress in simple tension
- θ :
-
Coordinate defining the position of a meridian
- V, W :
-
Veloeities in the S-direetion and the direetion normal to the shell middle surfaee, respeetively, v = V/L, w = W/L
- ()′:
-
\(\frac{{d()}}{{ds}}\)
References
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Flügge, W., Gerdeen, J.C. (1969). Axisymmetric plastic collapse of shells of revolution according to the Nakamura yield criterion. In: Hetényi, M., Vincenti, W.G. (eds) Applied Mechanics. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85640-2_15
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DOI: https://doi.org/10.1007/978-3-642-85640-2_15
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