Berechnung von elastisch orthogonal bewehrten maxwell-Schalen mit kleinem bewehrungsanteil (z.B. Stahlbeton-Schalen) — teil I

Abstract

The constitutive equations for shallow shells of viscoelastic (Maxwell) material with elastic reinforcement are derived. With respect to reinforced concrete shells shrinkage is considered. (Constitutive equations here mean relations between stress resultants, as membrane forces, bending and twisting moments, and strains or distortions of the shell middle surface, and further, formulae for the determination of the stress resultants related to the elastic and to the viscoelastic part of the section, respectively.) Concerning strain geometry, the usual assumptions of the theory of elastic shells are made. The reinforcement shall be orthogonal, but may be unsymmetrical to the middle surface and locally variable. Influences of transverse contraction are not considered. The derivations are based on the tri-dimensional constitutive equations for viscoelastic media of the type of reinforced concrete as established by D. Hilliges, who proved, that such a medium behaves in a different manner under shearing stresses than under normal stresses. This characteristic behaviour is also expressed in the constitutive shell equations given in this paper. The elastic and inelastic strains of a compound shell uder bending moments and normal forces are much smaller than those of a corresponding shell without reinforcement, and there is a decrease of the stress resultants of the viscoelastic part of the section (relaxation) and a corresponding increase of the stress resultants of the elastic part, whereas under shearing and twisting loads the (elastic and inelastic) rigidity of a compound shell element is only slightly higher than that of an element without reinforcement, and there is no remarkable shift of the stresses from the viscoelastic to the elastic part of the section. Using the derived constitutive equations there are given the displacement equations of the orthotropic Maxwell shell. Exact solutions can only be found for some simple cases (i.e. for a simply supported rectangular plate). Therefore these equations are transformed in a way that perturbation calculation can be applied. Thus the solution is reduced to the solution of the corresponding elastic problem and to the finding of particular integrals. This method of analysis of orthotropic Maxwell shells is applied to the following examples: circular cylindrical shell, rectangular plate, and circular disk with locally variable reinforcement. It is found that partly the stress resultants are considerably different (more than 20%) from those of an elastic isotropic shell and after creeping even different from those of the corresponding orthotropic elastic shell. It may even occur, that the stress resultants in the viscoelastic component of the material act against the reinforcement stress resultants. The present work represents an extension and generalization of derivations by R. Trostel for the case of compound plates.