By Natalia I. Obodan, Olexandr G. Lebedeyev, Visit Amazon's Vasilii A. Gromov Page, search results, Learn about Author Central, Vasilii A. Gromov,
This ebook specializes in the nonlinear behaviour of thin-wall shells (single- and multilayered with delamination parts) below numerous uniform and non-uniform loadings.
The dependence of severe (buckling) load upon load variability is published to be hugely non-monotonous, displaying minima while load variability is as regards to the eigenmode variabilities of resolution branching issues of the respective nonlinear boundary problem.
A novel numerical technique is hired to investigate branching issues and to construct fundamental, secondary, and tertiary bifurcation paths of the nonlinear boundary challenge for the case of uniform loading. the weight degrees of singular issues belonging to the trails are thought of to be serious load estimates for the case of non-uniform loadings.
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Extra info for Nonlinear Behaviour and Stability of Thin-Walled Shells
Example text
6 System behaviour at bifurcation point (ascending postcritical branch) a2 ( ) a3( ) cr a1( ) a cr Fig. 7 System behaviour at bifurcation point (descending postcritical branch) a a2( ) cr a1( ) a3( ) acr aj ( ) a Fig. 8 System behaviour at limit point cr,u cr, l a cr,u a cr,l a The type of behaviour depends upon the layout and stability of the solution branches, upon its full energy levels, upon the difference between these levels (energetic barriers), and, last but not least, upon the value and type of perturbations.
Inasmuch as the solution Y ð xÞ is completely determined by the given initial vector Y 0 ; so the functions S of boundary conditions may be considered as functions of Y 0 : À Á À Á S ¼ S YðY 0 Þ; k ¼ S Y 0 ; k ð4:10Þ Due to the nonlinearity of operators Y ðY 0 Þ; the solution Y ð xÞ; and S as well, are nonlinear functions of Y 0 : The form of S ¼ SðY 0 ; kÞ remains unknown but the algorithm of calculation of S values by given Y 0 vector and parameter k value is known. Such an algorithm represents the numerical integration of the Cauchy problem from the point x0 to the points x ¼ a and x ¼ b where the boundary conditions are formulated.
10) and, respectively, its limit point is close to the secondary bifurcation path (branch) bifurcation point. As shown in Fig. ). 25) in vector form LðU; kÞ ¼ 0; ð3:1Þ SðU; kÞjC ¼ 0; ð3:2Þ 34 3 Branching of Nonlinear Boundary Problem Solutions where L is the differential operator of the Eqs. 12); S is the operator corresponding to boundary conditions [chosen from Eqs. 25)] defined on contour C; U ¼ U ð X Þ: Singular points (kcr, acr) are associated with indifferent (neutral) equilibrium that implies that an elastic body is able to deviate from the initial equilibrium U0 ~ while the load parameter is constant k = kcr.