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Abstract. The buckling load of a structure may usually be computed with an eigenvalue problem: it is the eigenvalue of smallest absolute value. In optimizing structures with a constraint on the buckling load, repeated eigenvalues are likely to occur. We prove continuity and differentiability results of eigenelements with respect to design variables using the variational characterization of eigenvalues. We illustrate these results with a classical problem: buckling of a beam. Application to arch buckling is presented in another article.

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We study some spring mass models for a structure having some unilateral springs of small rigidity ε. We obtain and justify mathematically an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: T ε ∼ 1/ε as usual; or, for a new critical case, we can only expect: T ε ∼ 1/ √ ε. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes" (NNM); this algorithm provides results close to the asymptotic expansions but enables us to compute NNM even when ε becomes larger.

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The effective dose of hypobaric ropivacaine combined with sufentanil 5 µg providing 95% success in spinal anesthesia for traumatic femoral neck surgery in the elderly is ED95 = 9 mg (95% confidence interval, 8-14). Using doses exceeding the ED95 may increase the incidence of hypotension. If doses less than the ED95 are chosen, the use of additional analgesia may be necessary.

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