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Behaviour of the extensible elastica solution

magnusson, Anders; Ristinmaa, Matti LU and Ljung, Christer (2001) In International Journal of Solids and Structures 38(46-47). p.8441-8457
Abstract
The general form of the virtual work expression for the large strain Euler–Bernoulli beam theory is derived using the nominal strain (Biot's) tensor. From the equilibrium equations, derived from the virtual work expression, it turns out that a linear relation between Biot's stress tensor and the (Biot) nominal strain tensor forms the differential equation used to derive the elastica solution. Moreover, in the differential equation one additional term enters which is related to the extensibility of the beam axis. As a special application, the well-known problem of an axially loaded beam is analysed. Due to the extensibility of the beam axis, it is shown that the buckling load of the extensible elastica solution depends on the slenderness,... (More)
The general form of the virtual work expression for the large strain Euler–Bernoulli beam theory is derived using the nominal strain (Biot's) tensor. From the equilibrium equations, derived from the virtual work expression, it turns out that a linear relation between Biot's stress tensor and the (Biot) nominal strain tensor forms the differential equation used to derive the elastica solution. Moreover, in the differential equation one additional term enters which is related to the extensibility of the beam axis. As a special application, the well-known problem of an axially loaded beam is analysed. Due to the extensibility of the beam axis, it is shown that the buckling load of the extensible elastica solution depends on the slenderness, and it is of interest that for small slenderness the bifurcation point becomes unstable. This means the bifurcation point changes from being supercritical, which always hold for the inextensible case, i.e. the classical elastica solution, to being a subcritical point. In addition, higher order singularities are found as well as nonbifurcating (isolated) branches. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Stability, Elastica, Bifurcation
in
International Journal of Solids and Structures
volume
38
issue
46-47
pages
8441 - 8457
publisher
Elsevier
external identifiers
  • scopus:0035834460
ISSN
0020-7683
DOI
10.1016/S0020-7683(01)00089-0
language
English
LU publication?
yes
id
44c46e41-6c39-4f06-8ab2-0819d91d8bad (old id 2223709)
date added to LUP
2011-12-06 13:19:53
date last changed
2017-01-01 08:13:41
@article{44c46e41-6c39-4f06-8ab2-0819d91d8bad,
  abstract     = {The general form of the virtual work expression for the large strain Euler–Bernoulli beam theory is derived using the nominal strain (Biot's) tensor. From the equilibrium equations, derived from the virtual work expression, it turns out that a linear relation between Biot's stress tensor and the (Biot) nominal strain tensor forms the differential equation used to derive the elastica solution. Moreover, in the differential equation one additional term enters which is related to the extensibility of the beam axis. As a special application, the well-known problem of an axially loaded beam is analysed. Due to the extensibility of the beam axis, it is shown that the buckling load of the extensible elastica solution depends on the slenderness, and it is of interest that for small slenderness the bifurcation point becomes unstable. This means the bifurcation point changes from being supercritical, which always hold for the inextensible case, i.e. the classical elastica solution, to being a subcritical point. In addition, higher order singularities are found as well as nonbifurcating (isolated) branches.},
  author       = {magnusson, Anders and Ristinmaa, Matti and Ljung, Christer},
  issn         = {0020-7683},
  keyword      = {Stability,Elastica,Bifurcation},
  language     = {eng},
  number       = {46-47},
  pages        = {8441--8457},
  publisher    = {Elsevier},
  series       = {International Journal of Solids and Structures},
  title        = {Behaviour of the extensible elastica solution},
  url          = {http://dx.doi.org/10.1016/S0020-7683(01)00089-0},
  volume       = {38},
  year         = {2001},
}