## On the optimal design of columns against buckling (1992)

Venue: | SIAM J. Math. Anal |

Citations: | 11 - 2 self |

### BibTeX

@ARTICLE{Cox92onthe,

author = {Steven J. Cox and Michael L. Overton},

title = {On the optimal design of columns against buckling},

journal = {SIAM J. Math. Anal},

year = {1992},

pages = {287--325}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract – We establish existence, derive necessary conditions, and construct and test an algorithm for the maximization of a column’s Euler buckling load under a variety of boundary conditions over a general class of admissible designs. We prove that symmetric clamped–clamped columns possess a positive first eigenfunction and introduce a symmetric rearrangement that does not decrease the column’s buckling load. Our necessary conditions, expressed in the language of Clarke’s generalized gradient [10], subsume those proposed by Olhoff and Rasmussen [25], Masur [22], and Seiranian [34]. The work of [25], [22], and [34] sought to correct the necessary conditions of Tadjbakhsh and Keller [37] who had not foreseen the presence of a multiple least eigenvalue. This remedy has been hampered by Tadjbakhsh and Keller’s miscalculation of the buckling loads of their clamped-clamped and clamped–hinged columns. We resolve this issue in the appendix. In our numerical treatment of the associated finite dimensional optimization problem we build on the work of Overton [26] in devising an efficient means of extracting an ascent direction from the column’s least eigenvalue. Owing to its possible multiplicity this is indeed a nonsmooth problem and again the ideas of Clarke [10] are exploited. 1.

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Citation Context ...se eigenvalues and eigenvectors are computed by subspace iteration with a block size of two, with the necessary linear systems solved directly using the Cholesky factors of B(σ) (see Bathe and Wilson =-=[5]-=- for details). Define the approximate multiplicity t of λ1 by t = 2 if λ2(σ) − λ1(σ) ≤ τλ1(σ) and t = 1 otherwise. Here τ is a positive tolerance which may be adjusted during the optimization process.... |

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Citation Context ...< 0). Taking ℓ2 = ν2/ν1 we arrive at ∫ 1 (ˆσ − σ)( ˆ ξ + ℓ 2 ) dx ≤ 0, ∀ σ ∈ C. 0 The subsequent reduction to pointwise optimality conditions follows a well known course, see e.g., Cea and Malanowski =-=[8]-=-. In particular, ˆσ(x) = α ⇒ − ˆ ξ(x) ≤ ℓ 2 α < ˆσ(x) < β ⇒ − ˆ ξ(x) = ℓ 2 ˆσ(x) = β ⇒ − ˆ ξ(x) ≥ ℓ 2 (4.5) (4.6) (4.7) for almost every x ∈ (0, 1). To appreciate this result we must recall that ˆ ξ ∈... |

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Citation Context ... to maximize a functional of Auchmuty, and hence to consider λ −1 1 . A different proof of the unconstrained version of this theorem was given (when K = I) by Overton [26], following work of Fletcher =-=[14]-=-. The n × n matrix Û is known as a “dual matrix” by analogy with “dual variables” (Lagrange multipliers) familiar from mathematical programming. The t × t matrix U may be called a “reduced dual matrix... |

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Citation Context ...ive first eigenfunction and introduce a symmetric rearrangement that does not decrease the column’s buckling load. Our necessary conditions, expressed in the language of Clarke’s generalized gradient =-=[10]-=-, subsume those proposed by Olhoff and Rasmussen [25], Masur [22], and Seiranian [34]. The work of [25], [22], and [34] sought to correct the necessary conditions of Tadjbakhsh and Keller [37] who had... |

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Citation Context ...cal results attest. 287. Computational Results The algorithm outlined in the previous section has been implemented in Fortran 77 and tested extensively. Subroutines from the Linpack [13] and Eispack =-=[17]-=- libraries were used to (a) perform the QR factorizations required during the partial LP solution process (for matrices with at most four columns), (b) obtain the Cholesky factorizations of B(σ) neede... |

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Citation Context ...ntroduces computational difficulties. Although they did go on to suggest how 16π 2 /3 was incorrectly obtained, a number of workers have remained unconvinced, e.g., Myers and Spillers [24] and Barnes =-=[4]-=-. Upon fleshing out the relevant remarks of Olhoff and Rasmussen we shall see, in work relegated to an appendix, that the buckling load for the column proposed by Tadjbakhsh and Keller does not exceed... |

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Citation Context ...e necessary for a multiple eigenvalue to attain its extremum? The greatest advances on this question have come in finite dimensions and lie in the apparently little known work of Bratus and Seiranian =-=[6]-=-. These conditions, later discovered independently in a more general form by Overton [26], will be discussed in detail in §5. For now, we note that Bratus and Seiranian, upon applying their finite dim... |

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