A new method for determining the buckling resistance in the nonlinear range of strains for a column supported by rotational stiffeners

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An innovational method for solving the Euler–Bernoulli problem of an overall buckling of the uniform column supported by rotational springs of stiffnesses γ1, γ2, N ∙ m free from traditional simplifications (invariable flexural stiffness and length) is given. It is based on a natural and comprehensive constraint on the restored axis length. A system of algebraic equations relating the critical stress σcr to the nonlinear compression diagram ε(σ) of the material, the slenderness of the column λ and the values γ1, γ2 has been obtained, solved and verified in important special cases. It is shown that columns of the same material with the same so-called the reduced spring stiffnesses have identical dependencies σcr(λ). It is shown that columns with λ ≤ λmin(γ1,γ2) cannot be buckled by any axial load F for various types of ε(σ) (Ramberg–Osgood, rational fraction, polynomial, etc.).

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Sobre autores

V. Chistyakov

Physical and Technical Institute of RAS named after A.F.Ioffe

Autor responsável pela correspondência
Email: v.chistyakov@mail.ioffe.ru
Rússia, Saint-Peterbourg

S. Soloviev

Physical and Technical Institute of RAS named after A.F.Ioffe

Email: serge.soloviev@mail.ioffe.ru
Rússia, Saint-Peterbourg

Bibliografia

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2. Fig. 1. a) Loading diagram of a curved column; b) bending profile in the ratio scale: vertical black dash-dotted line – boundaries of convexity regions I, II and III y(z), red dots – slope of pf 1 axis at point A, red dotted line – slope q1 at inflection point (IP) 1, blue dots – slope of pf 2 axis at point B, blue dotted line – slope q2 at IP 2, green long dotted line – line of inflection points, horizontal black dotted line – tangent at the point of maximum deviation y.

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3. Fig. 2. a) Dependence σcr(λ), Pa (6.1) for an I-beam S = 51.3 cm2, ix = 3.6 cm, Al 6061 T6 alloy [10] for the same spring stiffnesses in the range γ = 0–50 MN m; b) 3D graph σcr(λ,γ) for bamboo columns 100×100×20 [11], green – physical sheet, red – non-physical.

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4. Fig. 3. a) Compression diagram of the Al + 15% Al2O3 composite: circles are experimental points [12], black dotted line is σ(ε), Pa, fourth-order polynomial, gray solid line is Hooke's law, red is the ε(σ) dependence of the fifth-order polynomial, green (right) is the tangent modulus of elasticity Et, Pa; b) minimum flexibility as a function of the stiffness of identical springs γ, N ∙ m for a 20K1 I-beam made of the composite.

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5. Fig. 4. a) Dependences σ(λ), Pa (7.4) for column support on an ideal hinge and rigid fixing for linear (gray) and polynomial n = 5 (black) compression diagrams, I-beam 20K1, Al + 15% Al2O3; b) dependence of σ, Pa, on λ, γ1, N ∙ m (8.3) for the same profile.

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6. Fig. 5. a) Curves σ(γ1 = γ2, N ∙ m), Pa and σ(γ1,γ2 = 0), Pa for I-beam 20K1 made of composite Al + 15 wt.% Al2O3 at λ = 50 (~2.5 m); b) projection of surface “ridge” σ, Pa, from γ1,γ2, N ∙ m with values ​​χ i according to (9.1) onto coordinate plane (Oσγ1) (border of blue with upper blue-gray) and line of its intersection with (Oσγ1) (border of blue with lower gray).

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