To study the effect of the transverse stress and strain distribution and steel ties (stirrups) upon the ultimate bending moment and bending ductility, a three-dimensional finite element analysis of a cross section slice is carried out. The slice consists of a layer of eight-node isoparametric elements, whose axial displacements are constrained so that the cross sections remain planar but not orthogonal. This allows interpreting the results in terms of curvature, bending moment, axial force and shear force. Each element within the layer is allowed to independently undergo cracking when its tensile strength limit is exceeded, and the incremental inelastic stiffness matrix of the cracked material is derived. The inelastic behavior of uncracked concrete or concrete between the cracks is modeled by the previously published endochronic theory, which allows representing the inelastic dilatancy due to shear, the hydrostatic pressure sensitivity, and the strain-softening (decrease of stress at increasing strain). The use of a constitutive relation that is capable of describing these effects is essential, since the dilatancy of concrete is opposed by ties which thus produce hydrostatic pressure in concrete thereby increasing its ductility. Transverse reinforcement is modeled either as reinforcement smeared throughout an element or as a steel bar connecting the nodes. Special measures are taken to eliminate spurious shear effects in the finite element model. A computer program to calculate the moment-curvature diagram of a given beam has been written using the incremental loading procedure. The calculated results compare satisfactorily with the available published test data on the effect of tie spacing upon the moment-curvature diagrams and flexural ductility.
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