Minimum stiffness of optimally located supports for maximum value of column buckling loads

Abstract

A b s t r a c t The problem of maximizing the buckling load of Bernoulli-Euler columns, with their continuously distributed selfweight taken into account, is considered in cases where the stiffnesses and positions of discrete lateral spring supports are used as design variables. Necessary conditions for optimum support stiffness and position are derived, and they are discussed in the light of some classical results. Several numerical examples are presented. I n t r o d u c t i o nIt is well-known (Courant and Hilbert 1953) that optimum locations of a given number of rigid interior lateral supports which maximize the fundamental natural frequency of transverse vibrations of a string or beam, or the bifurcation load of an axially compressed column, are nodal points of an appropriate higher-order vibration of buckling mode of the original structure.In a recent paper on transversely vibrating beams, /~ke-sson and Olhoff (1988) demonstrated how the stiffnesses of supports placed in the described way may be reduced to finite minimum values without a decrease of the fundamental natural frequency. Thus optimum characteristics can be maintained at a reduced cost since a decrease of support stiffnesses normally implies a cheaper structure. However, if the support stiffnesses are decreased below the aforementioned minimum values, which are problem-dependent, then the fundamental frequency decreases from its maximum value. Also the optimum positions of the supports will no longer be the nodal points mentioned, but will be governed by different conditions.In the present paper we show that the problem of the optimal support of columns exhibits features similar to those of vibrating beams and we determine the optimum positions and stiffnesses of a set of discrete lateral supports in several problems where the selfweight of a vertical column is taken into account.In Section 2, we formulate the buckling eigenvalue problem, establish the pertinent Rayleigh quotient, and derive necessary conditions for the optimum stiffness and location of flexible lateral supports by variational analysis. In Section 3, these conditions are discussed relative to some known conditions for optimally placed rigid lateral supports by way of the detailed example of a cantilever column under pure selfweight loading, and we identify the aforementioned features.

Sensitivity computation and shape optimization for a non‐linear arch model with limit‐points instabilities

“…Three linear cases where analytic solutions for the optimization problem are known. [39][40][41][42] 2. Thickness optimization of a clamped-hinged circular arch subjected to point load.…”

Singular divergence instability thresholds of kinematically constrained circulatory systems

“…Our study confirms this observation, because the maximal buckling load of the Ziegler pendulum with respect to the kinematic constraints corresponds to the higherorder (second) buckling mode of the non-constrained system while the minimal load is attained at the lower-order (first) one. This effect holds also in case of the non-conservative systems, which were not considered, however, in [35][36][37]. The possible destabilizing effect of additional constraints was not anticipated in these works, neither the fact that the optimal constraints in non-conservative systems are chosen by the second-order work criterion.…”

“…4(a). It is worth noting that Liu et al [35], following the results of Rozvany and Mroz [37] and Olhoff and Akesson [36] had found that the optimal locations of internal supports for maximizing the buckling load of a column are at the nodal points of an appropriate higher-order buckling mode. Our study confirms this observation, because the maximal buckling load of the Ziegler pendulum with respect to the kinematic constraints corresponds to the higherorder (second) buckling mode of the non-constrained system while the minimal load is attained at the lower-order (first) one.…”

Sensitivity of frames to variations of hinges in dynamic and stability problems

“…It has been shown that the generalized formulation of optimization problems taking into account not only dimensional or configuration variables, but also variable supporting conditions, can lead to better results since the structural response to external loads and to distortions is sensitive to support variations. Extensions of the formulation to problems of minimum stiffness of optimally located supports for dynamic and stability response has been presented by/~kesson and Olhoff (1988) and Olhoff and .~kesson (1991).Considering a support as a joint between a structure and a foundation, the above problems can be generalized for joints between substructures. Surprisingly, the latter problem has not been discussed in literature from the point of view of structural sylithesis.…”