Keywords: reinforced concrete, frame, non-linear analysis, cross-sectional analysis, hybrid approach.

Reinforced concrete frames under spatial action exhibit distinct non-linear behaviour. The material non-linearity can be captured at cross-sectional level. A cross-sectional analysis determines the strain and stress state corresponding to given sectional forces. Existing cross-sectional models can be classified into resultant models, truss models, uniaxial models, wall models, as well as models based on finite element analyses. Resultant models describe the cross-sectional behaviour directly in terms of internal forces and generalised strains. Most of the concrete codes use truss models in the design for transverse forces and torque. However, truss models only apply to the ultimate limit state. Uniaxial models are the most common non-linear analysis type for frame elements. They are generally restricted to biaxial bending and normal force. Wall models discretise the cross section into coupled membrane elements. A locally plane strain and stress state is assumed, and thus transverse forces and Saint-Venant torsion can be covered. However, no generally accepted model for arbitrary cross-sections under spatial sectional forces exists so far.

Here, the authors propose a new hybrid approach. The approach combines the extended opportunities of a wall model with the advantages of a simple uniaxial model. The model considers all six internal forces of a spatial frame element with an arbitrary cross section. The shear distribution produced by transverse forces conforms with equilibrium, compatibility and the constitutive laws. The cross section is divided into two components. The stirrups, the adjacent concrete and longitudinal rebars form a notional, thin-walled cross section assembled from membrane elements. The second, uniaxially stressed component comprises the remaining concrete and longitudinal reinforcement. While the uniaxially stressed areas only contribute to biaxial bending and normal force, the membrane elements contribute to all six sectional forces including torsion and transverse forces. The strain state of the cross section is described by generalised strain measures, their longitudinal derivatives, and additionally by the transversal and shear strains of each membrane element. The strain state follows from the equations of equilibrium, continuity, and the constitutive laws. The non-linear problem is solved by a damped Newton-Raphson iteration in four levels.

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