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Linearized Theory of Buckling
The linearized theory of buckling is illustrated here, aimed to determine the critical load of prestressed elastic structures. Under the fundamental hypothesis of negligibility of the precritical strains, a quadratic expression is determined for the total potential energy. It differs from that of the linear theory for the presence of an additional peculiar term, representing the second-order work of the prestress. The variational procedure leads to a linear eigenvalue problem in the unknown critical load. Reference is made both to discrete and continuous systems. The eigenvalue problem for the Cauchy continuum is derived as an example. In view of studying nonconservative systems, an alternative formulation is described, grounded on the virtual work principle. Finally, a direct derivation of the equations is also illustrated, useful to interpret the equilibrium of forces in the adjacent configuration. In the last part of the chapter, imperfect systems, with a single or several degrees of freedom, are addressed, whose linearized response to incremental loads is determined. The solution suggests the definition of an amplification factor of the linear response, which accounts for the geometric effects not included in the linear theory. An example of imperfect two degrees of freedom system is worked out.
Linearized Theory of Buckling
The linearized theory of buckling is illustrated here, aimed to determine the critical load of prestressed elastic structures. Under the fundamental hypothesis of negligibility of the precritical strains, a quadratic expression is determined for the total potential energy. It differs from that of the linear theory for the presence of an additional peculiar term, representing the second-order work of the prestress. The variational procedure leads to a linear eigenvalue problem in the unknown critical load. Reference is made both to discrete and continuous systems. The eigenvalue problem for the Cauchy continuum is derived as an example. In view of studying nonconservative systems, an alternative formulation is described, grounded on the virtual work principle. Finally, a direct derivation of the equations is also illustrated, useful to interpret the equilibrium of forces in the adjacent configuration. In the last part of the chapter, imperfect systems, with a single or several degrees of freedom, are addressed, whose linearized response to incremental loads is determined. The solution suggests the definition of an amplification factor of the linear response, which accounts for the geometric effects not included in the linear theory. An example of imperfect two degrees of freedom system is worked out.
Linearized Theory of Buckling
Luongo, Angelo (author) / Ferretti, Manuel (author) / Di Nino, Simona (author)
Stability and Bifurcation of Structures ; Chapter: 6 ; 161-183
2023-02-17
23 pages
Article/Chapter (Book)
Electronic Resource
English
Linearized theory , Prestressed systems , Precritical strains negligible , Prestress elastic energy , Incremental equilibrium equations , Virtual work principle for prestressed bodies , Direct equilibrium in the adjacent configuration , Linearized imperfect paths , Modal decomposition of the response , Amplification factor due to imperfections Engineering , Mechanical Statics and Structures , Solid Mechanics , Mechanical Engineering , Structural Materials , Solid Construction , Building Construction and Design
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