In-Plane Nonlinear Theory And Elastic-Plastic Stability Of Pined Circular Arches

Starting from Bernoulli's assumption for straight beams and introducing the Green strain definition in the nonlinear analysis of solid mechanics, in-plane nonlinear equilibrium equations for circular arches are derived using the principle of virtual work. Besides the second order effect of longitudinal normal stresses, second-order effects of transverse stresses and shear stresses are also incorporated in the derivation. The proposed theory is most perfect because no nonlinear terms are neglected. Differential equations for buckling of circular arches are derived from the nonlinear equilibrium equations in this thesis.In recent years, solutions different from the classical one by Timoshenko and Vlasov for buckling of circular arches under uniform radial pressure appeared. This thesis presented also a new analysis to this classical problem and revealed that the main cause leading to the discrepancy between recent solutions and the classical solution is that the recent investigation has not considered the effect of transverse stresses. Again the theory of this thesis verified once more the important conclusion that the second-order effects of all stresses, which are necessary to keep the structure in equilibrium (including transverse stress), should be considered in stability theory, otherwise some terms that cancel each other will lurk in the equations, which lead to wrong results.By using the linear equilibrium equations of infinitesimal element of arch, the nonlinear equilibrium equations are linearized. Linearized buckling equations are obtained. The critical loads of rings and pined circular arches are calculated using the linearized buckling equations. The effects of three kinds of uniform radial load (load remains parallel to its original direction, load is directed toward the initial center of arch, load remains normal to the deflected axis of arch) are discussed.The widely used assumption of inextensibility of the axial line of arch during buckling is discussed. It is found that this assumption can only be used for uniform radial load that remains normal to the deflected axis of arch. It cannot be used for the other two radial uniform loads.The linear exact solutions of circular arches under several kinds of loads are investigated. The figures of inner forces and displacements are given. It is found that the linear solutions can be used to separate deep arches and shallow arches. The critical loads of anti-symmetric buckling of pined circular arches under different loads are obtained.The geometrical nonlinear buckling and post-buckling behaviors of pined arches are investigated. It is found there are five types of buckling behaviors of elastic arches under symmetric loads. A finer criterion for the classification of buckling behavior is given. Equations that can be used for symmetric buckling of deep arches and anti-symmetric buckling of shallow arches are provided.This thesis investigated also the amplification factor of moments and displacements of pined arches considering the second order effects. Approximate formulae for the amplification factors of moments and displacements under symmetric and anti-symmetric loads are presented. Nonlinear moments and displacements under non-symmetric loads are studied. It is found that nonlinear moments and displacements under non-symmetric loads can be obtained approximately by superposing the amplified linear moments and displacements of the symmetric and the anti-symmetric parts of non-symmetric loads.The thesis studied further the ultimate strengths of axially loaded pinned arches, considering the effects of slenderness ratio, central angle, initial crookedness, residual stress and sectional shapes. Stability factors for axially loaded pinned arches are provided. It is found that stability factors of compressive pined arches can be calculated by stability factors of compressive straight member in GB50017-2003.Ultimate strengths of pinned arches with both axial force and moment are studied, also considering the effects of slenderness ratio, radius angle, initial crookedness and residual stress. Design formulas of pinned arches under symmetric and anti-symmetric and non-symmetric loads are given. The formulas have the same form as the in-plane stability design formula of straight beam-columns.