Article

# Moment curvature relationship of reinforced concrete sections under combined bending and normal force

(Impact Factor: 1.71). 08/1988; 21(5):341-351. DOI: 10.1007/BF02472160

ABSTRACT

The modelling of tension stiffening effects is important for the verification of serviceability limit states of reinforced
or prestressed concrete structures or even for the computation of the stability limit state of particular classes of slender
concrete structures (bridge piles, towers, masts, etc.). The case of pure bending has received extensive analytical and experimental
consideration, but little attention has been paid, up to now, to the case of combined bending and normal force. A series of
tests on rectangular reinforced concrete beams submitted to bending and constant compressive normal force is reported in this
paper. The moment-curvature relationships are evaluated and compared with the prediction of two theoretical models. The first
model is the CEB model proposed by Favre and Koprna. It is a simplified model which refers to the uncracked and fully cracked
stiffnesses in pure bending only. The second model is a proposition made by the authors which takes into account the tension
stiffening effects, the variation in position of the neutral axis as a function of the eccentricity of the normal force, and
the non-linear behaviour of concrete in compression.

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• "This method may be found in classical literature of concrete structures, such as Nilson et al. (2010) and Calavera (2008), among others, or in classical papers of moment-curvature relationship of reinforced concrete sections, such as Carreira and Chu (1986) and Espion and Halleux (1988). The standard method states a rectangular concrete compressive stress block, with a compressive stress 0.85f c ' until a straight line located parallel to the neutral axis at a distance a = β 1 c from the fiber of maximum compressive strain. "
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ABSTRACT: The problem of a concrete cross section under flexural and axial loading is indeterminate due to the existence of more unknowns than equations. Among the infinite solutions, it is possible to find the optimum, which is that of minimum reinforcement that satisfies certain design constraints (section ductility, minimum reinforcement area, etc.). This article proposes the automation of the optimum reinforcement calculation under any combination of flexural and axial loading. The procedure has been implemented in a program code that is attached in the Appendix. Conventional-strength or high-strength concrete may be chosen, minimum reinforcement area may be considered (it being possible to choose between the standards ACI 318 or Eurocode 2), and the neutral axis depth may be constrained in order to guarantee a certain sectional ductility. Some numerical examples are presented, drawing comparisons between the results obtained by ACI 318, EC 2 and the conventional method.
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