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A beam with nonlinearly‐elastic lateral restraints attached at discrete points along its span is investigated via analytical and numerical methods. Previous results for the critical moment and the deflected shape based on an eigenvalue analysis of a similar beam with linearly‐elastic restraints are discussed, along with a validation of these results against an equivalent finite element model and results from numerical continuation. A beam with nonlinearly‐elastic restraints is then analysed with treatments for both quadratic and cubic restraint force–displacement relationships being provided. After formulation of the potential energy functionals, the governing differential equations of the system are derived via the calculus of variations and appropriate boundary conditions are applied. The equations are then solved using the numerical continuation software AUTO‐07p for a standard I‐section beam. The variation in elastic critical buckling moment with the linear component of the restraint stiffness is tracked via a two‐parameter numerical continuation, allowing determination of the stiffness values at which the critical buckling modes changes qualitatively. Using these stiffness values, subsequent analyses are conducted to examine the influence of the nonlinear component of the restraint stiffness, from which post‐buckling equilibrium paths and deformation modes are extracted. The results of these analyses are then compared with an equivalent Rayleigh–Ritz formulation whereby the displacement components are represented by Fourier series. Equilibrium equations are derived by minimizing the potential energy functional with respect to the amplitudes of the constituent harmonics of the Fourier series. The amplitudes are solved for in the post‐buckling range by AUTO‐O7p and equilibrium paths are produced and compared to the equivalent solutions of the differential equations, with good agreement observed.
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There are several novel pharmacological approaches to asthma therapy which have resulted from a further understanding of airway smooth muscle function and inflammatory mechanisms involved in asthma. The most effective bronchodilators currently available are beta 2-adrenoceptor agonists and drugs with a prolonged duration of action after inhalation will be useful. Other possible bronchodilator approaches include vasoactive intestinal peptide, prostaglandins, novel xanthines, selective muscarinic antagonists, drugs interfering with intracellular calcium release and K+-channel activators. The most effective anti-inflammatory treatment is corticosteroids and efforts are being made to improve the topical potency of these drugs. Many other ways of reducing inflammatory effects in asthma are being explored, such as mediator antagonists (especially thromboxane and PAF antagonists), inhibitors of inflammatory cell activation (especially of eosinophils) and inhibition of neurogenic inflammation. Use of anti-inflammatory therapies earlier in the disease should become more widespread.